elliptic diophantine equations

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De Gruyter Series in Discrete Mathematics

and Applications 2

Editor Colva M. Roney-Dougal, St Andrews, United Kingdom



Nikos Tzanakis

Elliptic Diophantine Equations

A Concrete Approach via the Elliptic Logarithm

De Gruyter



Mathematics Subject Classification 2010: 11D25, 11D41, 11D88, 11G05, 11G07, 11G50,

11H06, 11J86, 11Y16, 11Y50, 11-04, 14E05, 14H52, 14Q05, 33E05, 52C07, 68W30

Author Nikos TzanakisUniversity of Crete

Department of Mathematics

Voutes Campus

70013 Heraklion, Crete

Greece

[email protected]

ISBN 978-3-11-028091-3

e-ISBN 978-3-11-028114-9

Set-ISBN 978-3-11-028115-6

ISSN 2195-5557

Library of Congress Cataloging-in-Publication DataA CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

© 2013 Walter de Gruyter GmbH, Berlin/Boston

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Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen

Printed on acid-free paper

Printed in Germany

www.degruyter.com



To my beloved wife Maro,and to our family’s younger mathematicians,

Eleni & Alexis, and Giorgos,with whom I share in life mathematics and so much more.



Preface

This book 17 0 is about elliptic Diophantine equations, the most standard instance

of such an equation being the equation y2 D x3 C Ax C B in integers x , y, where

A, B 2 Q and the right-hand side has no multiple roots. Many more Diophantine

equations are elliptic Diophantine equations; in Chapter 1 the term is explained in its

generality.

More specifically, the main theme of this book is the explicit resolution of such

equations and the resolution method that is exposed in detail is called elliptic log-arithm method or, briefly, Ellog, in accordance with the terminology and notation

introduced in [55]. The method has two main characteristics which are, first, the ex-

ploitation of the group structure with which the points of a non-singular cubic curve

are endowed and, second, the use of linear forms in elliptic logarithms. The method

owes its name to its second characteristic, the most modern. Immediately below we

give more explanations.

The first main characteristic (or ingredient) of the Ellog, which makes possible the

transition from the elliptic Diophantine equation to linear forms in elliptic logarithms

– the second main characteristic – is the fact that, on the one hand, an elliptic Dio-

phantine equation can be transformed, by an appropriate “change of variables”, into anon-singular cubic equation of a special shape1 and, on the other hand, that the set of

points2 of the curve defined by this cubic equation, is endowed with the structure of

a finitely generated abelian group. The operation of “addition” in this group is a con-

sequence of the simple observation that the third point of intersection with the curve

of a line joining two points of the curve with coordinates in a number field also has

coordinates in this same number field; if the two points coincide, as “the line joining

them” we understand the tangent at the point. Thus, if we know two distinct ratio-

nal, say, solutions (points), we can obtain a third one, also with rational coordinates,

which is the third intersection with the curve of the line joining the two known points;and if we know only one solution (point), as a “line joining the two known points”

we consider the tangent at the known point. This is the chord and tangent method

for generating new solutions from known ones. At this point I refer the reader to the

beautiful booklet of I. G. Bashmakova [3]. Bashmakova seems to believe that, in the

two problems below, found in Arithmetica, Diophantus applied the chord and tangent

method consciously. I am very cautious about this view; nevertheless, one might ex-

1 The Weierstrass equation.

2 With coordinates rational or, more generally, in a number field.



viii Preface

plain Diophantus’s solution to these problems as an application of this method, which

is what I do below, following Bashmakova’s exposition [3, Chapter 6].

Problem Δ-24 of Diophantus’s Arithmetica [61, pp. 242–244].3

To divide a given number into two numbers such that their product is acube minus its side.Original text: Δοθέντα ἀριθμὸν διελεῖν εἰς δύο ἀριθμούς, καὶ ποιεῖν τὸνὑπ’ αὐτῶν κύβον παρὰ πλευράν.

Sketch, in modern language, of Diophantus’s solution. As a “given number” he takes

6 and the two numbers in which 6 is “divided” are 6 and 6 x. Let y be the “side of

the cube”, so that, by the problem, x.6 x/ D y3 y. Diophantus puts y D ax 1

with a temporarily not specified.4 His first attempt, setting a D 2, is not successful

because the coefficients of x are not cancelled out. Therefore he takes a D 3 andthen his equation becomes 27x3 26x2 D 0, which gives the non-zero solution x D26=27, y D 136=27.

A deeper explanation of the above solution. Consider x .6 x/ D y3 y as a curve

possessing the obvious point .x , y/ D .0, 1/. The tangent to the curve at this point

meets the curve at three points, two of them being .0, 1/ counted twice and the third

will be the new sought-for point. The equation of the tangent is y D 3x 1 and we

are led to Diophantus’s choice a D 3.

In this solution, only one point, namely .0, 1/, was a priori known, so that the tangent

was used. An analogous use of tangent explains the duplication formula of Bachet(1621), by which he was able to find rational solutions .x, y/ with xy ¤ 0 to the

equation y2 D x3 Cc for any integer c ¤ 1, 432, once he knew one rational solution

.x1, y1/ with x1y1 ¤ 0; see [46, Introduction].

Problem Δ-26 of Diophantus’s Arithmetica [61, pp. 248–250].5

To find two numbers such that their product augmented by either gives acube.Original text: Νὰ εὑρεθῶσι δύο ἀριθμοί, ὅπως τὸ γινόμενον αὐτῶν σὺνἑκάτερον σχηματίζει κύβον.

Sketch, in modern language, of Diophantus’s solution. As the first number he takes a

multiple of a cube;6 specifically, 8x. He takes the second number equal to x2 1. The

conditions of the problem require that both 8x.x2 1/C8x and 8x.x2 1/Cx2 1 be

cubes. The first condition is satisfied by every x, while the second gives 8x3Cx28x3 Also [51, p. 199].4 “I form a cube by arbitrary times minus its side” is my rough translation from Greek of Diophantus’s

statement.5 Also [51, p. 203].

6 “I form the first by an arbitrary cube” is my rough translation from Greek of Diophantus’s statement.



Preface ix

1 D y 3. Again, Diophantus sets y D 2x 1, obtaining thus the solution x D 14=13;

hence the sought-for numbers are 8 14=13 D 112=13 and .14=13/2 1 D 27=169.

A deeper explanation of the above solution. Why the substitution y D ax 1 with

a D

2? What is special about this value for a? In projective coordinates .X : Y : Z /

the above cubic equation becomes7 8X 3CX 2Z8XZ 2Z3 D Y 3 and the equation of

the above line through the point .0, 1/ – which is the projective point .0 : 1 : 1/ – is

Y D aX Z. On this line, the “point at infinity” is .1 : a : 0/, and the requirement that

this be also a point on the cubic curve forces a D 2. Thus, the solution of Diophantus

is the third point of intersection of the (projective) cubic curve with the (projective)

line joining the points .0 : 1 : 1/ and .1 : 2 : 0/.

The necessary theory and tools related to the above are discussed mainly in Chapter 1 and, also, in Chapter 2.

The second characteristic (or ingredient) relevant to the method from which this

book takes its name, consists in the fact that to each elliptic Diophantine equation oneor more linear forms in elliptic logarithms are attached, and the computation of upper

and lower bounds for them is a major part of the method. The necessary theory and

tools for this are developed in Chapter 3.

A first complete image of Ellog, in general, which results from the combination

of the two ingredients, is given in Chapter 4. Specialising the application of Ellog

to various classes of elliptic Diophantine problems results in Chapters 5, 6, 7 , and

8. Each of these chapters leads up to a theorem furnishing an upper bound for the

absolute value of the linear form L, involved in the Diophantine problem, in terms of

a critical parameter M > 0.

In Chapter 9, a major step is achieved due to a Theorem of S. David [12], namely, a

lower bound for jLj (the same L as above), again in terms of M (the same M as above)

is obtained. All quantities in both the upper and the lower bound of jLj, except for M ,

are explicit ; moreover, as it will turn out, the lower bound runs faster to infinity withM than the upper bound and this fact clearly implies an explicit upper bound for M .

Why is this important? At this point, it is not possible to explain in a few sentences, the

meaning of M . For the present, the reader should consider that an explicit bound for

M would reduce the resolution of the Diophantine problem to that of checking which

lattice points in a hyper-cube of side 2M satisfy a certain condition. This is what I call

in this book an effective resolution to the Diophantine problem. It is important to stressthe fact that this checking can be performed in practice only if M is “very small”; this

issue is discussed later in this preface.

Four specific examples corresponding, respectively, to Chapters 5, 6, 7, and 8 are

discussed, resulting in explicit very large upper bounds for M ; in all cases this is larger

than 1040.

Unfortunately, this is a general fact: In all specific Diophantine problems, the effec-

tive upper bound for M is so large (something of the size of 1030, say, would be very

7 On setting x D X=Z and y D Y =Z.



x Preface

“friendly”) that, in practice, the checking of lattice points in the hyper-cube, mentioned

a few lines above, is impossible.

Is it possible to reduce the upper bound of M to a manageable size (say of a few

decades)? This would lead to an explicit resolution of the Diophantine problem. The

answer is, in principle, positive due to a reduction method developed by B. M. M. de

Weger [72], which is based on the LLL-basis reduction algorithm of Lenstra–Lenstra–

Lovász [27]. Chapter 10 presents everything related to the reduction of the upper

bound of M . The reduction method is then applied to the examples of Chapter 9,

leading to upper bounds for M at most 17. This small upper bound leads then, very

easily, to the complete explicit solution.

The theoretical idea of the method8 goes back to Lang [25], who also explains it in

[26]; its brief explanation is found in [45, Chapter IX.5, “Linear forms in elliptic log-

arithm”]. The discussion so far and the contents of this book confirm that from theory

to practice a long way had to be covered; the method became practical only in 1994,after the work of R. J. Stroeker and the author [54], and, independently, the work of

J. Gebel, A. Peth˝ o and H. G. Zimmer [15]. These two 1994 papers would not have

appeared if the work of N. Hirata-Kohno [17] and S. David [12], on lower bounds of

linear forms in elliptic logarithms, had not been previously published. In a correspon-

dence9 during 1991–1992, I asked S. David if he could make explicit the constants

involved in the results of [17]; I am extremely grateful to him for his accepting my

far from non-trivial “challenge”, which turned out to be a really heavy work [12] of

more than 130 pages! For a clear and relatively short description of the practical “1994

method” I refer to [50, Chapter XIII].

Chapter 11 is special. In it, the resolution of the Weierstrass elliptic equation in S -

integers is discussed. What do we mean by S -integers? If S is a finite set of primes,

an S -integer is, by definition, a rational number, with the property that the prime de-

composition of its denominator allows only primes belonging to S ; in particular, every

usual (rational) integer is an S -integer.

One has to develop a theory of p-adic elliptic logarithms (with p a prime) for the

points of an elliptic curve, in analogy with the theory of elliptic logarithms developed

in Chapter 3; this is done in Section 11.1. In Section 11.2 linear forms in p-adic ellipticlogarithms are introduced and the p-adic version of Ellog, is developed. This section is

inspired by the papers of N. Smart [47] and Peth˝ o–Zimmer–Gebel–Herrmann [36]; thesecond paper completes the project set up in the first paper. When [36] was published,

no explicit lower bound for linear forms in p-adic elliptic logarithms – the analogue in

the p-adic case of S. David’s theorem – existed, except for that in [39] which, however,

is applicable only to elliptic curves of rank at most two.10 Therefore the authors of [36]

8 Without its “details”: Lower bounds for linear forms in elliptic logarithms, reduction process and,

in general, all computational aspects.9 Handwritten letters of the good old days!

10 This result is improved in [18], but still treats the case of two p-adic elliptic logarithms.



Preface xi

turned to the recent, for those days, paper [16]. Very recently,11 N. H. Kohno released

a valuable paper [19], in which “the p-adic analogue of S. David’s theorem” is proved.

This is what is used in Chapter 11 instead of [16]. I am grateful to Noriko, who, meeting

my desire, worked hard in order to provide me with her theorem before the date that I

had to send my manuscript to the editors.

The chapter includes a specific example with the primes p D 2, 3, 5, 7 involved.

About the style of the book. To what extent should my exposition take for granted

standard (more or less) material found in the literature? I had this speculation mainly

concerning Section 1.2 of Chapter 1, Chapters 2, 3 and 9, Section 10.1 of Chapter 10,

and Section 11.1 of Chapter 11. My decisions are detailed below.

The basic theory of elliptic curves is so beautifully written in various text-books –

few of them (only) are included in my bibliography –, that my hypothetical contribu-

tion could be described by C 1

! Therefore, Section 1.2 of Chapter 1 includes only

the very basic facts that will be needed and gives references.

For the theory of heights, in Chapter 2, only a moderate use of p-adic theory is

required. I found that standard texts either include so much material that reference to

them would disorientate (with respect to this book’s aim) my reader, or they adopt

a point of view not very appropriate for the present book, as they build the relevant

theory by considering extensions of Qp .12

Concerning Weierstrass equations over C and R and the Weierstrass -function,

treated in Chapter 3, I do the following. Since the basic general theory is so neatly

written, for example, in [1, Chapter 1], I decided that I need not provide any expla-

nation. However, specialisation of the general theory to Weierstrass equations withreal coefficients is absolutely necessary in order to build a practical theory of ellip-

tic logarithms. I decided to discuss this issue, rather in detail, guided by my personal

taste. I adopted a very classical point of view, mainly based on the old (still in print)

nice book [73] and the personal notes of N. Kritikos from A. Hurwitz’s 1916–1917

E. T. H. lectures on elliptic functions.13

The hard core of Chapter 9 is a special – very important though – case of Sinnou

David’s Theorem. As I already mentioned, this is the result of his memoir [12] of

more than 130 pages. The theorem, in the form appropriate for the needs of this book

(Theorem 9.1.2), is stated only and aspects of its application in practice are discussed.

For the applications of Chapter 10 the main tool is the reduction technique of

B. M. M. de Weger [72], which is based on the LLL-algorithm [27]. The style of [72]

is very appropriate for this book,14 but discusses many more applications than those

11 Actually, when I was ready to send my manuscript to the editors!12 For this book’s purposes, working with non-archimedean absolute values on (finite) extensions of Q

is much more appropriate.13 These notes in Greek [23], prepared by the late Dr. I. Ferentinou-Nikolakopoulou, circulated around

1980 in the Department of Mathematics of the University of Crete.

14 Is it accidental that B. M. M. de Weger and I had a congenial collaboration for years?



xii Preface

needed here. Therefore, in Chapter 10, among other issues, I expose the reduction

process focusing on the particular applications of the book.

In Section 11.1 of Chapter 11, p-adic elliptic logarithms and their linear forms play

the fundamental role. The theory on which the construction of such logarithms is based

is described in Chapter IV of J. Silverman’s valuable book [45], though from a point

of view somewhat more general than necessary for this book. What I decided to do

was state only the absolutely necessary facts from Silverman’s exposition “translated”

into a language appropriate for practical applications.

As in the case with S. David’s Theorem in Chapter 11, the very recent and extremely

important theorem of N. Hirata-Kohno, mentioned before, is only stated in the form

appropriate for the needs of this book (Theorem 11.2.5).

Although a main characteristic of the book is its use of computational methods, it is not a book on Computational Number Theory; issues such as – to mention only a

few examples – the actual computation of Mordell–Weil bases, the search for rational

points on elliptic curves up to a certain bound, the computation of canonical heights,

various aspects of the implementation of the LLL-algorithm, and/or improvements of

existing methods and algorithms, are beyond the scope of the book. To this “rule” I

allowed three exceptions: In Chapter 3, first, I did not refrain from discussing in detail

the actual computation of a fundamental pair of periods for a Weierstrass equation

with real coefficients, an issue that fits very well in the framework of the chapter.15

Second, again in Chapter 3, I did not resist the temptation to describe the very clever

algorithm of D. Zagier [74] for the computation of elliptic logarithms. Third, in Chap-

ter 8, I present an algorithm of J. Coates related to the computation of the coefficientsof Puiseux series.

Suggestions for reading this book. The Diophantine problems treated in this book

are classified to five classes: Weierstrass, quartic elliptic, simultaneous Pell, general

elliptic, and Weierstrass in S -integers; let us use for these problems the symbols P i ,

where i D 5, 6, 7, 8, 11, respectively, with this numbering justified by the chapter

where the corresponding problem is mainly (but not exclusively) discussed. Since

Ellog is applied in the most direct manner to P 4, I would suggest that the reader starts

by understanding the resolution of this problem. In general, in order to understand the

complete and explicit resolution of problem P i , I suggest the following scheme: Read carefully Chapter 1.

Make a first superficial reading of Chapter 2 to become acquainted with heights,

so that you can read Section 2.6; if you already know about heights, go directly to

Section 2.6.

Pay attention to the content of Section 3.5. An understanding of the previous sections

of Chapter 3 is necessary, with the exception of Section 3.4 which you will need only

if you are interested in the actual computation of periods.

15 Besides, a detailed treatment of this issue is not easily found in the literature.



Preface xiii

Comprehend the content of the short Chapter 4.

Read carefully Chapter i. If i = 11 do not proceed to Theorem 11.2.6; instead, pro-

ceed to the following step.

From Chapter 9 read carefully Sections 9.1 and 9.2.If i = 11 go back to Theorem 11.2.6 and complete your study of Chapter 11. END!

If i is not 11, read that Section among 9.3, 9.4, 9.5 and 9.6 which corresponds to the

chosen P i .

If i is different from 11, proceed to Chapter 10, read Section 10.1 and chose among

the subsections of Section 10.2 the one that corresponds to the chosen P i . END!

Software packages. My frequent reference to the software packages PARI (free),

MAGMA and MAPLE is because I happen to have been acquainted with them for years.

Alternatively, for the applications of this book, one could turn to SAGE (free). As this

was developed very recently – comparatively to the previously mentioned packages –,I had not the time to gain experience with it; this is the only reason why SAGE is not

mentioned in my applications.

Final acknowledgments. The materials of Chapters 4, 5, 8, 9 and 10 are mostly

based on joint-papers with Roel Stroeker published between 1994 and 2003. It was a

real pleasure to cooperate with Roel, noble friend and brave co-traveller in the long

and adventurous but beautiful trip in the field of elliptic Diophantine equations.

Around that same period, other people worked independently on various aspects of

elliptic Diophantine equations, from a similar point of view; I have in mind mainly (in

alphabetic order) J. Gebel, E. Herrmann, A. Peth˝ o, N. P. Smart and H. G. Zimmer. Wealways had fruitful, and friendly communication; also their work was an inspiration

source in writing Chapter 11.

All serious computations in the examples of this book, besides their obvious debt

to the software packages mentioned above, owe much, though indirectly, to people

on whose work the routines that I have used are, more or less, based; let me mention

(alphabetically) J. Cremona, M. van Hoeij, A. K. Lenstra, H. W. Lenstra, L. Lovász,

J. Silverman, M. Stoll, B. M. M. de Weger, D. Zagier and many anonymous (to me, at

least) heroes who are behind the algorithms’ implementation in various packages.

Generally speaking, this book owes something to every author whose name appears

in the bibliography; to some of them it owes much more, as becomes clear from the

frequent references to their work. I also thank Y. Thomaidis who, shared with me his

professional views about some issues of Diophantus’s Arithmetica.

Warm thanks to De Gruyter for its continued collaboration and to D. Poulakis for

inciting me to write this book and his warm encouragement.

I am grateful to P. Voutier for his careful reading of Chapters 2 and 3. Of course, I

am absolutely responsible for anything wrong that possibly escaped his attention.



xiv Preface

I am indebted to my wife Maro for her lifelong support, and for her warm encour-

agement and patience when I was writing this book; this has been a main factor for its

completion!

Heraklion, Crete, May 12, 2013 Nikos Tzanakis



Contents

Preface vii

1 Elliptic curves and equations 1

1.1 A general overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Elliptic curves and the Mordell–Weil Theorem . . . . . . . . . . . . . . . . . . 5

2 Heights 9

2.1 Notations and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Absolute values in a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Heights: Absolute and logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 A formula for the absolute logarithmic height . . . . . . . . . . . . . . . . . . . 18

2.5 Heights of points on an elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 The canonical height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Weierstrass equations over C and R 29

3.1 The Weierstrass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 The Weierstrass equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 : E.C/ 7! C=ƒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Weierstrass equations with real coefficients . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.3 Explicit expressions for the periods . . . . . . . . . . . . . . . . . . . . . 41

3.4.4 Computing !1 and !2 in practice . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 : E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1 . . . . . . . . . . . . . . . . . . 47

4 The elliptic logarithm method 54

5 Linear form for the Weierstrass equation 57

6 Linear form for the quartic equation 60

7 Linear form for simultaneous Pell equations 69



xvi Contents

8 Linear form for the general elliptic equation 78

8.1 A short Weierstrass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.2 Puiseux series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.3 Large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8.4 The elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.5 Computing in practice B1 of Proposition 8.3.2 . . . . . . . . . . . . . . . . . . . 89

8.6 Computing in practice B2 and c9 of Proposition 8.4.2 . . . . . . . . . . . . . 91

8.7 The linear form L.P / and its upper bound . . . . . . . . . . . . . . . . . . . . . . 94

9 Bound for the coefficients of the linear form 98

9.1 Lower bound for linear forms in elliptic logarithms . . . . . . . . . . . . . . . 98

9.2 Computational remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9.3 Weierstrass equation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.4 Quartic equation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

9.5 Simultaneous Pell equations example . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.6 General elliptic equation: A quintic example . . . . . . . . . . . . . . . . . . . . 118

10 Reducing the bound obtained in Chapter 9 121

10.1 Reduction using the LLL-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.2.1 Weierstrass equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

10.2.2 Quartic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.2.3 System of simultaneous Pell equations . . . . . . . . . . . . . . . . . . . 131

10.2.4 General elliptic equation: A quintic example . . . . . . . . . . . . . . 134

11 S-integer solutions of Weierstrass equations 137

11.1 The formal group of C and p-adic elliptic logarithms . . . . . . . . . . . . . 137

11.2 Points with coordinates in ZS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11.3 The p-adic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

11.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

List of symbols 165

Bibliography 173

Index 177



Chapter 1

Elliptic curves and equations

1.1 A general overview

In this section we make an overview of general facts, terminology and conventions

that will be used in this book.

Let g.X , Y / be a non-zero polynomial with coefficients in a subfield K of C (in

most cases, K D Q), irreducible over C, and let R be a subring of K (usually, but

not always, R D Z) which will be fixed throughout this chapter. We are interested insolving the Diophantine equation

g.u, v/ D 0, .u, v/ 2 R R. (1.1)

The characterisation of the above equation as “Diophantine” comes from the require-

ment that the unknowns u, v belong to the prescribed ring R and not to the whole C or

R. Solving the Diophantine equation is far different from solving the algebraic equa-

tion g.u, v/ D 0, in which the unknowns belong to C. The solutions of the algebraicequation define a curve or, more precisely, a model C of a curve; we state this by

writing

C : g D 0; g.X , Y / D a specific polynomial in X , Y

and we say that C or, more precisely, the model C is defined by the polynomial

g.X , Y /, or by the equation g D 0. Thus, we view C as the set C .C/ D ¹.u, v/ 2C C : g.u, v/ D 0º and the elements of C.C/ are called points of (the model)C . Sometimes we wish to focus our interest to the “real part” of C , which is the set

C.R/ D ¹.u, v/ 2 R R : g.u, v/ D 0º of real points of (the model) C . In gen-

eral, if A is a subring of C, we set C.A/ D ¹.u, v/ 2 A A : g.u, v/ D 0º and if

.u, v/ 2 C.A/, we say that .u, v/ is an A-point of (on) C . The fact that the model C

is defined by means of the polynomial g, whose coefficients belong to R, is expressed

by saying that C is defined over R.Sometimes (actually very rarely) we will need to refer to the projective equation or,

equivalently, to the projective model corresponding to equation (1.1). This results from

the so-called homogenisation of the variables u and v, which consists in considering

the equation

g.U : V : W / D 0, (1.2)

g.U : V : W / def D W ng.U=W , V = W /, n D max¹degug, degvgº.

Note that g.U , V , W / is homogeneous in U , V , W of degree n and g.u : v : 1/ Dg.u, v/.



2 Chapter 1 Elliptic curves and equations

If g.U : V : W / D 0, then and only then g.kU : k V : k W / D 0 for every k 2 C,

therefore it is more appropriate to view the solutions of the equation g.U : V : W / D0 projectively, i.e. as points .U : V : W / 2 P 2.C/ rather than as solutions or affine

points .U , V , W / 2 C3. If g.U : V : W /

D 0 for some .U : V : W /

2 P 2.C/ and

there exists a k 2 C such that kU , kV , kW 2 R, then .U : V : W / is a projective

solution (point) over R.

The dehomogenisation process from g D 0 to g D 0 consists in dividing

g.U , V , W / D 0 through by W n and putting .U=W , V = W / D .u, v/.

In the homogenisation process, from every solution .u, v/ of g D 0 we obtain a

projective solution .u : v : 1/ of g D 0. In the dehomogenisation process, a solution

.U : V : W / of g D 0 furnishes a solution .u, v/ of g D 0 if, and only if, W ¤ 0, the

solution in this case being .u, v/ D .U=W , V = W /; but projective solutions of the form

.U : V : 0/ cannot be “dehomogenised” to solutions .u, v/ of (1.1). Such solutions

.U : V : 0/ are characterised as solutions (points) at infinity of the equation (1.1).

Now we proceed to discussing the important fact that different equations may de-

fine the same curve. In order to make this more precise, we need first the following

definition:

Definition 1.1.1. For i D 1, 2, let gi .X , Y / be non-zero polynomials in CŒX , Y ,

irreducible over C and consider the models C i : gi D 0.

We say that a birational transformation exists between C 1 and C 2 or, equivalently,

that the models C 1 and C 2 are birationally equivalent if, for .i , j /

D .1, 2/, .2, 1/,

rational functions U ij ,V ij 2 C.X , Y / exist such that: for .i , j / as above, if .ui , vi / 2C i .C/ and we define .uj , vj / D . U ij .ui , vi /,V ij .ui , vi //, then .uj , vj / 2 C j .C/ and

. U j i .uj , vj /,V j i .uj , vj // D .ui , vi /.

Actually, the above definition of birational equivalence, though satisfactory for the

needs of this book, is not very precise: What about points .ui , vi / 2 C i .C/ for which

U ij .ui , vi / or V ij .ui , vi / is not defined (i.e. .ui , vi / is a zero of the denominator)? We

overcome these difficulties if, for any model C : g D 0, we consider its function field C.C / (see a few lines below) and we define the notion of birational equivalence of two

models C 1 and C 2 by means of their function fields C.C 1/ and C.C 2/. The function

field of the model C : g D 0 is, by definition, the fieldC. , /, where is transcenden-

tal over C and is algebraic over C./, satisfying g. , / D 0. Equivalently, C.C /

can be defined as the quotient field of the integral domain CŒX , Y =I , where I is the

ideal g.X , Y /CŒX , Y of CŒX , Y . If two models C 1 and C 2 are birationally equiv-

alent, then there exists an isomorphism of their function fields which fixes C. For a

treatment of these issues, very appropriate for the theoretical background of this book,

we refer the reader to §§3,4 of the classical book [66]. For an alternative, or comple-

mentary, exposition the interested reader can refer to [45, Chapters I, II.1, II.2].



Section 1.1 A general overview 3

Let us come back to Definition 1.1.1 and impose the further conditions that gi 2RŒX , Y for i D 1, 2 and the four rational functions U ij , V ij have coefficients in

K . Then we say that C 1 and C 2 are birationally equivalent over K . If K D Q.R/,

the quotient field of R, then, obviously, any R-point .ui

, vi/ of C

i is mapped by the

birational transformation of Definition 1.1.1 to a Q.R/-point .uj , vj / of C j .

Clearly, birational equivalence of models of curves is an equivalence relation. The

equivalence class of a model C will be called a curve, denoted by C . Thus, for this

book, a curve is an equivalence class of a model and will be denoted by a capital calli-

graphic letter. If C 1, C 2 are models of the same curve C , then C 1 and C 2 share a number

of important properties, a very important one being that they have the same genus and

we thus speak about the genus of the curve C rather than the genus of this or that model.

The notion of genus is not easy to define but, fortunately, an in-depth knowledge of

this notion is not absolutely necessary for the main purpose of this book, which is the

practical resolution of elliptic Diophantine equations. Anyway, the interested reader

can refer, for example, to [45, Chapter II.5], especially Theorem 5.4. A more classical

approach is found in [66, Chapter VI, §5.3]. Another classical analytic point of view

for genus, very much in the spirit of the present book, is found in [4, Chapter III, §21];

see especially Corollary 2. A “picturesque” geometrical notion of genus, easy for ev-

erybody, is presented in [38, Appendix to Chapter 1]. Although this approach to genus

is not practical for the purposes of this book, we recommend the reader unaccustomed

with this notion to have a look at it.

In this book, we will deal with Elliptic Diophantine Equations. These are equa-

tions (1.1) defining a model C : g D 0 of an elliptic curve over K , which means

that the curve has genus one and the corresponding projective equation (equivalently

stated, the corresponding projective model) (1.2) has a solution (has a point) inP 2.K/.

In our applications, K D Q but, as a tool, we will sometimes need to consider

elliptic curves defined over a number field K as well. Also, in our applications R D Z,

except for Chapter 11, in which R D ZS , the ring of S -integers, where S is a finite

set of primes; see the beginning of that chapter.

When we solve an elliptic Diophantine equation, defined by an equation (1.1), we

employ properties of the elliptic curve C , a model of which is C : g

D 0. Let us put

g D g1 and C D C 1; the method of solution that we will apply requires working withone or two further models C i : gi D 0 (i D 2, 3) of C , also defined over R, with any

two of C 1, C 2, C 3 birationally equivalent by means of transformations U ij ,V ij (cf. Def-

inition 1.1.1) having coefficients in R (in other words, the said models are birationally

equivalent over Q.R/). These birational transformations will be fixed during the pro-

cess of the resolution of the Diophantine equation g1 D 0. We will exclude all the

finitely many (and easily computed) exceptional points of C 1.C/ at which at least one

of the rational functions U 1j ,V 1j is not defined, focusing on non-exceptional points of

C 1.C/, i.e. on points which map to points on C j for j ¤ 1 by means of the birational




map

.u1, v1/ ! .uj , vj / D . U 1j .u1, v1/,V 1j .u1, v1//

.u1, v1/

D. U j 1.u1, v1/,V j 1.u1, v1//

.uj , vj /

For the convenience of our notation we will avoid the use of subscripts in curve no-

tation and denote the various (two or three) birationally equivalent models that will

be used during the resolution of an elliptic Diophantine equation by, for example,

C : g.u, v/ D 0, E : f .x, y/ D 0, D : f 1.x1, y1/ D 0 instead of C i : gi D 0

(i D 1, 2, 3); note the use of different letters for the variables of different models. In

accordance with the chosen letters, the birational transformation e.g. from C to E will

then be denoted by . U ,V / and its inverse from E to C by .X ,Y /, so that

.u, v/ ! .x, y/ D .X .u, v/,Y .u, v// (1.3)

.u, v/ D . U .x, y/,V .x, y// .x, y/. (1.4)

If we denote by C the curve, models of which are C and E , then, in accordance with

what was previously said, a point P of (on) C is “visualised” by both a pair .u0, y0/

satisfying g.u0, y0/ D 0 and a pair .x0, y0/ satisfying f .x0, y0/ D 0 (and analogously

if a third model of C is used), where .u0, y0/ and .x0, y0/ are related by (1.3) and (1.4),

with .u0, v0/ in place of .u, v/ and .x0, y0/ in place of .x, y/. It will be convenient to

express this by writing P C D .u.P /, v.P// D .u0, v0/ and P E D .x.P /, y.P// D.x0, y0/.

To sum up: In general, if we have to solve a certain elliptic Diophantine equationg.u, v/ D 0, we consider the model C : g.u, v/ D 0 and the elliptic curve C cor-

responding to the model C , along with one or two further models E : f .x, y/ D 0,

D : f 1.x1, y1/ D 0 which will remain fixed in the process of the resolution of the Dio-

phantine equation. We will consider fixed birational transformations between any pair

of models C , E, D as in (1.3) and (1.4). We will say that “P is a point of (on) C ” and

will state this by writing P 2 C , if there exist points P C D .u.P /, v.P// 2 C.C/,

P E D .x.P/, y.P// 2 E.C/ such that the rational functions U .u, v/ and V .u, v/

are defined at .u.P /, v.P// (in other words, the point .u.P /, v.P// of the model C is

non-exceptional) and the pairs .u.P/, v.P// and .x.P /, y.P// are related by means

of (1.3) and (1.4), with .u.P /, v.P// in place of .u, v/ and .x.P /, y.P// in place of

.x, y/.1

Consequently, in accordance with these conventions/notations, we have

P 2 E , P E D .x.P/, y.P// 2 E.C/, P C D .u.P /, v.P// 2 C.C/

P C D . U .P E /,V .P E // D . U .x.P/, y.P//,V .x.P/, y.P//(1.5)

P E D .X .P C /,Y .P C // D .X .u.P /, v.P//,Y .u.P /, v.P//

1 And analogously if it is necessary to make use of a third model D of C .



Section 1.2 Elliptic curves and the Mordell–Weil Theorem 5

Finally, if, as in Chapter 3, we use only one model, say E, then we will omit the

superscript E . Moreover, and in accordance with this “rule”, the term “point P ” will

mean “point of the model E” and we will simply write P instead of P E .

We emphasise, once again, that a deep comprehension of the notion of genus is not

absolutely necessary for the purposes of this book. From a practical point of view, the

following fact is very important.

Fact 1.1.2. If C : g.u, v/ D 0 is a model of genus one, where g has rational co-efficients, then C is birationally equivalent over Q to a “short Weierstrass model”E : y2 D x3 C Ax C B with A, B 2 Q by means of a birational transforma-tion (1.5) , the coefficients of which are real algebraic numbers of degree at most min¹degug, degvgº. These coefficients, as well as A and B , can be explicitly computed.

A practical general algorithm of M. van Hoeij for computing the short Weierstrass

model and the birational transformation, mentioned in Fact 1.1.2, is described in [22];

an implementation of this algorithm is included in the package algcurvesof the com-

puter algebra system MAPLE. Given a non-singular point .u0, v0/ 2 C.Q/ and the co-

efficients of g as input, this algorithm returns the coefficients A, B of the equation

of the short Weierstrass model and the coefficients of the birational transformation

(1.3)–(1.4), which belong to Q.u0, v0/. Although this is not very explicit in [22], it

can be verified by careful scrutiny of [20, §§ 1 and 2.1] and [21, §§ 1–3.1] on which

the algorithm of [22] is based.

The inclusion A, B 2 Q.u0, v0/ can be further improved to A, B 2 Q: by an argu-ment found on pages 93–95 of [6], a simple transformation .x, y/ 7! .t 2x, t 3y/, for a

conveniently chosen t 2 Q, maps the equation y 2 D x3 C Ax C B to an equivalent

short Weierstrass equation with coefficients in Q.

The above-mentioned algorithm, can be applied to quite “exotic” examples of mod-

els C : g.u, v/ D 0, like those appearing in [56]. If g has a “more ordinary” form,

one can see more directly the necessary birational transformation between C and

the short Weierstrass model, or even do the necessary computations “by hand”; see

[10, Section 7.2].

1.2 Elliptic curves and the Mordell–Weil Theorem

In this section we focus exclusively on elliptic curves. We confine ourselves to stating

the main facts of the basic (only) theory, with which we assume that the reader is

accustomed. For these facts and the relevant theory one may refer to standard books

on elliptic curves, such as [6, 24, 45, 46],2 or even from sections in some number

2 Indicated titles; only a few with which the author happens to be more acquainted.




theory textbooks devoted to elliptic curves, e.g. [7, Section 5.9], [32, Section 5.5]3. In

this chapter we will mostly refer to [45, Chapter III].

Firstly, any elliptic curve E over K has a model

E : y 2 C a1xy C a3y D x3 C a2x2 C a4x C a6, (1.6)

where a1, : : : , a6 2 K . We call this equation a Weierstrass equation and the model

E is a Weierstrass model. Weierstrass equations with a1 D a2 D a3 D 0 are called

short Weierstrass equations and the corresponding models short Weierstrass models(cf. Fact 1.1.2). The first condition characterising E as an elliptic curve is that its

genus be one (see page 3); equivalently, this is expressed by the condition E ¤ 0,

where E is the discriminant of E .4 Also, we denote by E.K/ the projective model

corresponding to C and observe that O D .0 : 1 : 0/ 2 E.K/ so that the second

requirement for characterising E as an elliptic curve is also fulfilled. The point O

is called the zero point of the curve for reasons that will become clear below; since

O is not visualised on the affine plane, its traditional name, coming from Projective

Geometry, is the point at infinity.

The crucial fact about elliptic curves is that in E.K/ we can define by a very natural

geometrical way, the so-called chord-tangent method , an operation C, called addition.

Then, .E.K/, C/ is an abelian group with O its neutral element, which explains why

we call it zero point; see [45, Section III.2], [46, Section I.2], [24, Section III.3], [7,

Sections 5.9, 5.10], [6, Chapter 7], [32, Section 5.5]. The result of the addition of two

points is their sum; the coordinates of the sum of two points are expressed as rational

functions with coefficients in K of the coordinates of the two points; see [45, GroupLaw Algorithm 2.3], [24, Section III.4], [7, Section 5.10], [32, Definition 5.159]. The

extremely important fact is that, when K is a number field, the group E.K/ is finitelygenerated , as L. J. Mordell [33] and A. Weil [71] proved (Mordell in the case K D Q

and Weil for a general number field K ); this is the famous Mordell–Weil Theorem:

Theorem 1.2.1 (Mordell–Weil Theorem). Let K be a number field and consider theelliptic curve model E given by (1.6). Consider also the abelian group E.K/ and denote by Etors.K/ the torsion subgroup of E.K/. There exists a non-negative integer

r , the rank of E over K , such that the following group isomorphismE.K/ Š Etors.K/ Zr ,

is valid.

Proof. For the original proofs we refer to [33] and [71]. A proof of a very classical

flavour in the case K D Q is found in [34, Chapter 16]. For a modern proof when

K D Q we refer to [45, Chapter VIII], especially Section VIII.4.

3 Comment analogous to that of footnote 2.

4 [45, pages 42–43].



Section 1.2 Elliptic curves and the Mordell–Weil Theorem 7

Practically, the Mordell–Weil Theorem means the following. If r 1, there exist

points P E1 , : : : , P Er in E.K/, such that:

(1) P Ei

2E.K/ and P Ei is of infinite order (i

D1, : : : , r).

(2) For every point P E 2 E.K/, there exist integers m1, : : : , mr and T E 2 Etors.K/,

such that

P E D m1P E1 C C mr P Er C T E (1.7)

and .m1, : : : , mr , T E / is unique.

(3) If r D 0, then E.K/ D Etors.K/.

By another famous theorem due to B. Mazur (see [29, 30], or [45, Theorem 7.5]),

Etors.Q/ is isomorphic as a group either to Z=N Z, with 1 N 12, N ¤ 11, or toZ=2Z Z=2N Z with 1 N 4. Moreover, for every specific elliptic curve (1.6)

defined over Q, Etors.Q/ is explicitly calculable by the Lutz–Nagell Theorem; see, for

example, [24, Theorem 5.1] or [45, Corollary 7.2].

If r 1, then the union of the set of points ¹P E1 , : : : , P Er º and a set of generators

for Etors.K/ form a Mordell–Weil basis over K of E. The proof of Mordell–Weil

Theorem is not constructive (even in the case K D Q), hence there is no guarantee

that one can compute effectively a Mordell–Weil basis for any given E. However,

very powerful techniques have been developed for elliptic curves over Q which are

implemented in routines of standard software packages. The most standard reference

about these techniques is John Cremona’s book [11], a true treasury of elliptic curves!

Thus, although the explicit computation of a Mordell–Weil basis is not theoretically

guaranteed, nevertheless, for “any reasonable” elliptic curve over Q, it is very likely

that we can obtain such a basis by applying the relevant routines of software packages,

like MAGMA, PARI, SAGE and others; see the item “software packages” on J. Cremona’s

home page. Therefore, in this book we will not discuss further this issue and will

take for granted that, when we solve an elliptic Diophantine equation with rational

coefficients, a Mordell–Weil basis for the related elliptic curve is at our disposal.

Addition of points in general elliptic models. We revisit Fact 1.1.2, keeping the

same notation etc. Thus, we work now with two models C , defined by equation (1.1)

with rational coefficients, and E , as in (1.6), again with rational coefficients, or as in

Fact 1.1.2. We denote by E the elliptic curve, two models of which are C and E . The

fact that we work simultaneously with two distinct models forces us to use superscripts

on points (see Section 1.1). The transition from a point P C of C to the point P E of

E and vice-versa is ruled by (1.5).

Suppose that Q1, Q2 are points on E . In view of the discussion in Chapter 1.1, this

implies that, for i D 1, 2, there exist points P Ei on E and points P C i on C , related by

a birational transformation (1.4)–(1.3) and (1.5). Implicitly, it is understood that U , V

are defined on P E

and X , Y are defined on P C

.




By the previous discussion we can form the sum QE1 C QE

2 , which is a point, say

P E , of the model E. If the birational transformation (1.4) is also defined on P E , then,

in accordance with the conventions adopted in Section 1.1, P E is the representative

of a point P of E . Then, this will allow us to write Q1 C

Q2 D

P , i.e. by definition,

Q1 C Q2 D P def ” QE

1 C QE2 D P E .

If, moreover, the birational transformation (1.5) is defined on QE1 , QE

2 and P E , then

the following makes sense:

QC 1 C QC

2 D P C def ” QE1 C QE

2 D P E .

Therefore, in view of these definitions along with (1.5), we see that

.Q1 C Q2/C D P C D . U .P E /,V .P E // D . U .QE1 C QE

2 /,V .QE1 C QE

2 //.

The above definitions and conclusions are obviously generalised to an arbitrary(finite) number of points. More particularly, let Q1, : : : , Qs be points on E and let

m1, : : : , ms be arbitrary integers. Then,

m1Q1 C C ms Qs D P def ” m1QE

1 C C msQEs D P E . (1.8)

and

.m1Q1 C C ms Qs /C D . U ..m1QE1 C Cms QE

s /,V ..m1QE1 C CmsQE

s //.

(1.9)

Finally, we give a simple useful lemma, that we will apply in subsequent chapters.

Lemma 1.2.2. If the coefficients a1, a2, a3, a4, a6 in (1.6) are inZ and .x, y/ 2 E.Q/ ,then .x, y/ D .x1=z2, y1=z3/ , where x1, y1, z 2 Z , z > 0 and gcd.x1, z/ D 1 Dgcd.y1, z/.

Proof. Let us write .x, y/ D .x1=w1, y1=w2/, where x1, w1, y1, w2 are integers with

w1, w2 positive and gcd.x1, w1/ D 1 D gcd.y1, w2/. Replacing for x and y in (1.6)

we get

w21 .y2

1 w1

Ca1x1y1w2

Ca3y1w1w2/

Dw2

2.x31

Ca2x2

1 w1

Ca4x1w2

1

Ca6w3

1 /.

We note that w21 is relatively prime to the number in the parenthesis in the right-hand

side, therefore w1jw2 and we write w2 D zw1, where z is a positive integer relatively

prime to x1y1. Replacing for w2 in the above equation we obtain

w1.y21 C a1x1y1z C a3y1w1z/ D z2.x3

1 C a2x21 w1 C a4x1w2

1 C a6w31/.

Now we observe that w1 is relatively prime to the number in the parenthesis in the

right-hand side, hence w1jz2, and, on the other hand, z2 is relatively prime to the

number in the parenthesis in the left-hand side, hence z2jw1. It follows that w1 D z 2

and w2

Dzw1

Dz3, as claimed.



Chapter 2

Heights

In all Diophantine applications of the theory of elliptic curves we need a “measure” for

the points of the elliptic curves involved in our investigations. This is accomplished

with the notion of height of a point.1 In this chapter we develop the relevant theory

from a point of view appropriate for the purposes of this book.

2.1 Notations and facts

We assume familiarity with standard algebraic number theory, as exposed, for exam-

ple, in [35, Chapters 4 §3, 5, 6] and (partially) in [40, Chapter V]. For the convenience

of the reader, we collect the basic facts that we will need without giving special refer-

ence to each of them.

We will need to work with various relative finite extensions L=K=Q, where K and

L may vary. The ring of integers of K will be denoted by ZK and analogously for L.

For the elements of the number fields we will use small Greek letters.

For the ideals of ZK we will use small fracture letters (like p, for example) and for

the ideals of ZL we will use capital fracture letters (like P, for example). The symbolj between ideals means a divisibility relation, and is equivalent to the relation .

If a is a non-zero ideal of ZK , then ZK =a is finite and we write NK=Q.a/ :Dcard.ZK =a/; this is the absolute norm of the ideal a. If ˛ 2 K , then NK=Q.h˛i/ DjNK=Q.˛/j, where NK=Q.˛/ is the usual element norm of ˛.

By default, the term ideal refers to integral ideal, i.e. ideal of the ring of integers.

But we will also use fractional ideals, for which we will use fracture letters as well; the

fact that an ideal is not necessarily integral will be emphasised by using the adjective

“fractional”. If a is an ideal (integral or fractional) in K and we want to view it as an

ideal in L, we will write aZL. If ˛ 2

K , we will writeh˛i

for the principal ideal ˛ZK .

If we view ˛ as an element of L rather than as an element of K , then we will write

˛ZL for the principal ideal that ˛ generates in L.

We will write P for the set of rational primes and P K ,P L for the set of prime ideals

of ZK and the set of prime ideals of ZL, respectively. If a is a non-zero fractional

ideal in K and p 2 P K , we define the exponent of p in a and denoted by p.a/ as the

exponent of p in the factorisation of a into prime ideals. If 2 K , we will write p.˛/

instead of p.h˛i/. Analogous notations will be used for the non-zero ideals of L.

1 Actually, two notions of height of a point will be discussed.



10 Chapter 2 Heights

For P 2 P L there exists precisely one p 2 P K , such that PjpZK (the ideal P is

over p or, in equivalent wording, p is below P). This p, in turn, is over a prime p 2 P ,i.e. p is the unique rational prime divisible by p.

With p, p and P related as just above, let us view p2P

K as an ideal (not necessarily

prime) of ZL, i.e. let us consider the ideal pZL of ZL and decompose it into prime

ideals, as follows:

pZL DYPjp

PeL=K .P/; (2.1)

where eL=K .P/ is the ramification index of P relatively to L=K (hence eL=K .P/ DP.pZL). The following relation is valid if P 2 P L is over p 2 P K :

eL=Q.P/ D eL=K .P/ eK=Q.p/. (2.2)

Further, the finite field ZL=P is a finite extension of the finite field ZK =p. The degree

of this extension, which is called the degree of P relative to the extension L=K , is

denoted by f L=K .P/. Similarly to (2.2) we have the relation

f L=Q.P/ D f L=K .P/ f K=Q.p/. (2.3)

Then the ideal norm of P relative to the extension L=K is, by definition,

NL=K .P/ D pf L=K .P/; (2.4)

note that, if in place of L and K we respectively have K and Q, this definition agrees

with the one given above for the absolute norm. The definition of the ideal norm is

extended by multiplicativity to all non-zero fractional ideals of L. Namely, if A is anon-zero fractional ideal in L and

A DY

P2P L

PP.A/

is its prime decomposition, then, by definition, the ideal norm of A relative to the

extension L=K is

NL=K .A/ def D

Yp2P K

YPjp

pP.A/f L=K .P/ (2.5)

and the norm of the zero ideal is defined to be the zero ideal. Two important propertiesof the norm are the following: The function NL=K from the group of fractional ideals

of L to the group of fractional ideals of K is a homomorphism; moreover, for any

fractional ideal A of L we have

NL=Q.A/ D NK=Q.NL=K .A//. (2.6)

If we have the decomposition (2.1), then, for every P dividing p we have the im-

portant relation

XPjp d L=K .P/

DŒL : K , d L=K .P/

def

DeL=K .P/

f L=K .P/; (2.7)



Section 2.2 Absolute values in a number field 11

d L=K .P/ is called local degree of L=K at P. Using (2.2), (2.3) and (2.7) we obtain

another useful relation:

XPjp d L=Q.P/

D XPjp eL=Q.P/f L=Q.P/

D eK=Q.p/eL=K .P/ f K=Q.p/f L=K .P/

D eK=Q.p/f K=Q.p/XPjp

eL=K .P/f L=K .P/

D d K=Q.p/XPjp

d L=K .P/ D ŒL : K d K=Q.p/. (2.8)

2.2 Absolute values in a number field

Consider a number field K (K may coincide with Q). An absolute value on K is a

function j j : K ! R satisfying: (1) jxj 0 for every x 2 K , and jxj D 0 if and

only if x D 0, (2) jxyj D jxjjyj, and (3) jx C yj jxj C jyj, for all x and y in K .

Two absolute values j j1 and j j2 are equivalent if there is a positive constant c 2 Rsuch that jxj2 D jxjc

1 for all x 2 K . This is equivalent to the fact that the metrics

ıi .x, y/ D jx yji on K induce equivalent topologies on K . An equivalence class

of an absolute value on K is a place on K . All (different) places in K are obtained as

follows:

If p 2 P K and p is over p 2 P , then the place corresponding to p is represented bythe absolute value j jp which is defined by

j˛jp D´

pp.˛/=eK=Q.p/ D NK=Q.p/p.˛/=d K=Q.p/ if ˛ ¤ 0

0 if ˛ D 0. (2.9)

This is a non-archimedean absolute value, which means (by definition) that it satisfies

j˛ C ˇjp max¹j˛jp, jˇjpº

and, if j˛jp ¤ jˇjp, then strict inequality holds. In this way all non-archimedean placeson K are obtained.2

The archimedean places on K are obtained as follows. Let ŒK : Q D d . There exist

precisely d embeddings i : K ,! C (i D 1, : : : , d ). An embedding i : K ,! C is

called real if .K/ R, otherwise it is called complex. If s embeddings are real and

2 Easily then, every absolute value j j equivalent to j jp has the same property that characterises non-

archimedean absolute values: j˛ C ˇj max¹j˛j, jˇjº and, if j˛j ¤ jˇj, then strict inequality holds.

Therefore it makes sense to speak about a non-archimedean place.




2t are complex, then d D s C 2t and we enumerate the embeddings as follows:

1, : : : , s

„ ƒ‚ …real

sC1, : : : , sCt , sCtC1 D sC1, : : : , sC2t D sCt

„ ƒ‚ …complex

,

where means complex-conjugate, so that .x/ :D .x/.

Let E K be the set of all embeddings K ,! C and let E oK be the subset of E K consist-

ing of all pairwise non-conjugate embeddings; in particular card.E K / D s C 2t and

card.E oK / D s C t .

For each 2 E o we define the absolute value

j˛j D j.˛/jd

where in the right-hand side j j denote the usual absolute value of C.

These sC

t absolute values are the archimedean absolute values; they are represen-

tatives of all possible archimedean places of K ; an equivalent terminology is to say

that these absolute values (places) are defined by the infinite primes of K .

The product formula is the relationYp2P K

j˛jd K=Q.p/p

Y 2E oK

j˛jd D 1, .˛ ¤ 0/. (2.10)

where

d D ´1 if is real

2 if is complex.

The proof of the product formula follows immediately fromYp2P K

j˛jd K=Q.p/p D

Yp2P K

NK=Q.p/p.˛/ D NK=Q

Yp2P K

pp.˛/

D NK=Q.h˛i1/ D jNK=Q.˛/j1

and Y 2E oK

j˛jd D

Y 2E K

j.˛/j D jNK=Q.˛/j. (2.11)

The following lemma is very useful, as it relates the absolute values in relative exten-sions.

Lemma 2.2.1. Let L=K be an extension of number fields and p 2 P K . Then,(1) for any ˛ 2 K and any P dividing p ,

j˛jP D j˛jp,

(2) for any ˛ 2 K ,

j˛jŒL:Kd K=Q.p/p D

YPjpj˛jd L=Q.P/

P .



Section 2.3 Heights: Absolute and logarithmic 13

Proof. Obviously, it suffices to consider a non-zero ˛ 2 K . With the aid of (2.2), (2.3)

and the definitions of d L=Q.P/ and d K=Q.p/ we observe that

d L=Q.P/

DeL=Q.P/f L=Q.P/

DeK=Q.p/eL=K .P/

f K=Q.p/f L=K .P/

D eK=Q.p/f K=Q.p/ eL=K .P/f L=K .P/ D d K=Q.p/ eL=K .P/f L=K .P/

which shows that

eL=K .P/f L=K .P/ D d L=Q.P/

d K=Q.p/.

We also note that, since the exponent of p in h˛i is p.˛/ and the exponent of P

in p is eL=K .P/, it follows that the exponent of P in ˛ZL, i.e. P.˛/, is equal to

p.˛/eL=K .P/.

Using (2.4), the multiplicativity of the ideal norm and (2.6) we now calculate

j˛jd L=Q.P/

P D NL=Q.P/P.˛/ D ¹NK=Q.NL=K .P//ºP.˛/

D ¹NK=Q.pf L=K .P//ºP.˛/ D NK=Q.p/f L=K .P/P.˛/

D NK=Q.p/f L=K .P/eL=K .P/p.˛/ D NK=Q.p/d L=Q.P/p.˛/=d K=Q.p/

D j˛jd L=Q.P/p .

It follows then that

YPjp j˛jd L=Q.P/

P D j˛jPPjp d L=Q.P/

p D j˛jŒL:Kd K=Q.p/

p ,

where the right-most equality holds because of (2.8). The proof is now complete.

2.3 Heights: Absolute and logarithmic

The K -height of a projective point .x0 : x1 : : xn/ 2 P n.K/, where K is a number

field, is defined as follows. First the finite-prime factor of the K -height is defined by

H K ,fin.x0 : x1 : : xn/ D Yp2P K

max¹jx0jp, jx1jpj, : : : jxnjpºd K=Q.p/,

and the infinite-prime factor of the K -height is defined by

H K ,1.x0 : x1 : : xn/ DY

2E oK

maxi

¹jxi jd º.

Then, the K -height of the point is, by definition,

H K .x0 : x1 : : xn/ D H K ,fin.x0 : x1 : : xn/ H K ,1.x0 : x1 : : xn/. (2.12)




Due to the product formula above, it is straightforward to check that, if ˛ 2 K and

.x0 : x1 : : xn/ 2 P n.K/, then

H K .x0 : x1 :

: xn/

DH K .˛x0 : ˛x1 :

: ˛xn/,

which shows that the K -height of a projective point is independent from the choice

of its projective coordinates. Moreover, due to the proposition below, for any number

field K containing x0, x1, : : : , xn, the value of H K .x0 : x1 : : xn/1=ŒK :Q is the

same.

Proposition 2.3.1. H K .x0 : x1 : : xn/1=ŒK :Q is independent from the number field K containing the coordinates x0, x1, : : : , xn.

Proof. We will show that, if L is a finite extension of K , then

H L.x0 : x1 : : xn/1=ŒL:Q D H K .x0 : x1 : : xn/1=ŒK :Q.

Actually, this is true for both the “finite-prime” and the “infinite-prime” factor sepa-

rately.

For the “finite-prime” factor it suffices to prove that

H L,fin.x0 : x1 : : xn/ D H K ,fin.x0 : x1 : : xn/ŒL:K . (2.13)

We calculate the left-hand side; to simplify notation, we write d P (respectively d p)instead of d L=Q.P/ (respectively d K=Q.p/):

H L,fin.x0 : x1 : : xn/ DY

P2P L

maxi

¹jxi jPºd P DY

p2P K

YPjp

maxi

¹jxi jPºd P .

By (1) of Lemma 2.2.1, if Pjp, then jxi jP D jxi jp, therefore, in the right-most side of

the above displayed equation, maxi ¹jxi jPº D maxi ¹jxi jpº D (say) jxi0jp. Then

H L,fin.x0 : x1 :

: xn/

D Yp2P K YPjp jxi0

jp

ºd P

D jxi0

jPPjp d P

p

D jxi0

jŒL:Kd pp

(by (2.8))

D Y

p2P K

max¹jxi jpºd p

ŒL:K

D H K ,fin.x0 : x1 : : xn/ŒL:K

as claimed.

Next we prove that

H L,1.x0 : x1 : : xn/ D H K ,1.x0 : x1 : : xn/ŒL:K

. (2.14)




First we note that every embedding : K ,! C has exactly ŒL : K distinct extensions

L ,! C. If is an extension of we write j . Now we have

H L,1.x0 : x1 :

: xn/

D Y 2E oL

maxi ¹j

xi

j

ºd

D Y 2E L

maxi ¹

.xi /º

DY

2E K

Y j

max¹j .xi /jº

D Y

2E K

max¹j .xi /jºŒL:K

D Y 2E oK

max¹j .xi /jºd

ŒL:K

D H K ,1.x0 : x1 : : xn/ŒL:K .

By (2.13), (2.14) and (2.12) we see that H L.x0 : : xn/ D H K .x0 : : xn/ŒL:K ,

as required.

Definition 2.3.1. Let .x0 : x1 : : xn/ 2 P n.K/, where K is a number field. Then,

H.x0 : x1 : : xn/ def D H K .x0 : x1 : : xn/1=ŒK :Q

is called the absolute height of the projective point .x0 : x1 : : xn/. According toProposition 2.3.1, this definition is independent from K .

The absolute logarithmic height of .x0 : x1 : : xn/ 2 P n is, by definition,

h.x0 : x1 : : xn/ def D log H.x0 : x1 : : xn/.

The absolute logarithmic height of an algebraic number ˛ is, by definition,

h.˛/ def D h.1 : ˛/ D log H.1 : ˛/.

The very basic properties of the absolute logarithmic height are stated in the follow-ing proposition:

Proposition 2.3.2. If ˛, ˇ are algebraic numbers, then

h.˛ˇ/ h.˛/ C h.ˇ/, (2.15)

h.˛ C ˇ/ log2 C h.˛/ C h.ˇ/ (2.16)

and, for every n 2 Z ,

h.˛n

/ D jnjh.˛/. (2.17)




Proof. Let K be any number field containing ˛ and ˇ. Denote by P the set of prime

ideals of ZK and by E K the set of all embeddings K ,! C; note that card.E K / D ŒK :

Q. For every p 2 P , let d p D d K=Q.p/.

We have

H.˛ˇ/ŒK :Q D H K .˛ˇ/ DYp2P

max¹1, j˛ˇjpºd p Y

2E K

max¹1, j.˛ˇ/jº.

Using the inequality max¹1, xyº max¹1, xº max¹1, yº, valid for every pair

of non-negative real numbers x, y, we obtain from the above displayed inequality

H.˛ˇ/ŒK :Q H.˛/ŒK :QH.ˇ/ŒK :Q, from which the inequality (2.15) follows im-

mediately on taking logarithms.

In order to bound H.˛ C ˇ/ŒK :Q we first observe the following. If p 2 P , then

max¹1, j˛ C ˇjp max¹1, max¹j˛jp, jˇjpºº max¹1, j˛jp max¹1, jˇjp.

For 2 E K ,

max¹1, j .˛ C ˇ/jº max¹1, j.˛/j C j.ˇ/jº 2 max¹1, j.˛/jº max¹1, j.ˇ/jº.

Note that, if in the last inequality we let run through E K and we multiply the resulting

relations, then the factor 2ŒK :Q will appear in the right-hand side.

Using the above two displayed inequalities in H K ,fin.˛ C

ˇ/ and H K ,1.˛ C

ˇ/,

respectively, we get

H.˛ C ˇ/ŒK :Q H K ,fin.˛/ H K ,fin.ˇ/ 2ŒK :QH K ,1.˛/ H K ,1.ˇ/

D 2ŒK :QH K .˛/H K .ˇ/ D .2H.˛/H.ˇ//ŒK :Q,

which completes the proof of (2.16).

Concerning (2.17), this is clear if n 0, so it suffices to prove that h.˛1/ D h.˛/.

We use the identity max¹1, x1º D x1 max¹1, xº, valid for every real x > 0. We

have

H.˛1/ŒK :Q D H K .˛1/ D Yp2P

max¹1, j˛1jpºd p Y 2E K

max¹1, j .˛1/jº

DYp2P

j˛1jp max¹1, j˛jpºd p Y

2E K

j .˛1/j max¹1, j.˛/jº

D Y

p2P

j˛1jpY

2E K

j˛1j

H K .˛/ D H.˛/ŒK :Q,

where for the right-most equality we used the product formula (2.10). Thus H.˛/ DH.˛

1/, as required.




Heights over Q. Let us specialise our previous discussion to the important case of

rational numbers.

First, let us note that, for a point .x0 : x1 : : xn/ 2 P n.Q/, its absolute height is

H.x0 : x1 : : xn/ D max¹jx0j, jx1j, : : : , jxnjº Yp2P

max¹jx0jp, jx1jp, : : : , jxnjpº,

where P is the set of (rational) primes. For ˛ 2 Q, j˛jp D pp.˛/ , p.˛/ is the

exponent of p in the prime decomposition of ˛, and j0jp D 0.

Now, let a0, : : : , an 2 Z, not all zero, with gcd.a0, : : : , an/ D 1. Then, for every

prime p, jai jp 1 for all i D 0, : : : , n and for at least one index i0, ai0 is not divisible

by p, so that jai0jp D 1. Therefore, for every prime p, max¹ja0jp, ja1jp , : : : , janjpº D

1. Consequently, H.a0 : a1 : : an/ D max¹ja0j, ja1j, : : : , janjº.

In general, for a0, a1, : : : , an

2Z, not all zero,

H.a0 : a1 : : an/ D max¹ja0j, ja1j, : : : , janjºgcd.a0, a1, : : : , an/

. (2.18)

For points of the form .1 : x1 : : xn/, a useful expression is the following.

If xi D ai =bi , where ai , bi are relatively prime integers (i D 1, : : : , n), and b Dlcm.b1, : : : , bn/ 1, then

H.1 : x1 : : xn/ D max¹b,bja1j

b1

, : : : ,bjanj

bnº D bmax¹1, jx1j, : : : , jxnjº. (2.19)

Indeed, .1 : x1 :

: xn/ D

.b : bja1

j=b1 : : : : : b

jan

j=bn/. Observe that

gcd.b, bja1j=b1, : : : , bjanj=bn/ D 1 and apply (2.18). For the last claim about the

gcd it suffices to show that, for every prime p dividing b, some bai =bi is not divisible

by p . For, if pk is the highest power of p dividing b (k 1), then pk is the highest

power of p dividing bi for at least one i . But, since gcd.ai , bi / D 1, p does not divide

ai ; hence bai =bi is an integer not divisible by p, as claimed.

The absolute logarithmic height of a rational number q takes the form:

h.q/ D log H.1 : q/ D log max¹1, jqjº CX

p

log max¹1, jqjpº. (2.20)

If q D a=b, where a, b 2 Z, we have, in view of (2.18), a second expression for h.q/

h.q/ D H.1 : a=b/ D H.b : a/ D logmax¹jaj, jbjº

gcd.a, b/ . (2.21)

A third expression for h.q/ results as follows. Let q D a=b, where a, b 2 Z and write

q as a fraction a1=b1 in lowest terms. Then, specialising to n D 1 the right-most side

of (2.19), we have H .1 : q/ D jb1j max¹1, jqjº and, since b1 D b= gcd.a, b/, we

conclude that

h.q/ D log max¹1, jqjº C logjbj

gcd.a, b/. (2.22)




2.4 A formula for the absolute logarithmic height

This section aims at proving a very handy formula for the absolute logarithmic height.

Our treatment is essentially that of D. Masser’s lectures [28], with slight modifications.We will need the notion of the height of a polynomial.

Height of a polynomial. Let K be a number field and, as before, letP K be the set of

prime ideals of the ring of integers ZK . The ring KŒX of polynomials over K can be

viewed also as the subset of K 1 (infinite-tuples) .ai /i0, “almost all” (all but finitely

many) of whose coordinates are equal to zero. Of course, the operations are defined

by

.ai / C .bi / D .ai C bi /

and

.ai / .bi / D .ci /, ci D Xj CkDi

aj bk .i 0/. (2.23)

Let P D .ai / be a polynomial over K . The finite-prime height of P over K is, by

definition,

H K ,fin.P / DY

p2P K

max¹jai jd pp º,

where d p :D d K=Q.p/. We note that, for every ˛ 2 K ,

H K ,fin.X ˛/ D H K ,fin.1 : ˛/.

Also, if we view ˛ 2 K as the constant polynomial, then

H K ,fin.˛/ DY

p2P K

j˛jp D jNK=Q.˛/j1, (2.24)

where the right-most equality holds due to the product formula (2.10) and (2.11).

The following proposition is, actually, a generalised and more sophisticated version

of the so-called Gauss Lemma for Polynomials.

Proposition 2.4.1. If P , Q

2KŒX , then H K , fin.PQ/

DH K , fin.P/H K , fin.Q/.

Proof. Obviously it suffices to prove our claim when both P and Q are non-zero

polynomials. Let P D .ai /, Q D .bi / and P Q D .ci / (cf. (2.23)). Fix p 2 P K . Let

jaj 0jp D maxj ¹jaj jpº, jbk0jp D maxk¹jbkjpº and set a1

j 0P D .a0j /, b1

k0Q D .b0

k/.

Obviously, ja0j jp 1 for every j and jb0kjp 1 for every k, with equality holding for

at least one subscript in both cases.

Now a1j 0

b1k0

PQ D .c 0i /, where c 0i DP

j CkDi a0j b0k

for every i , so that jc0i jp 1.

Let j1 and k1 be respectively the least j for which ja0j jp D 1 and the least k for

which jb0kjp D 1. We have c0

j 1Ck1DPj CkDj 1Ck1

a0j b0k

. Consider .j , k/ such that

j C k D j1 C k1. If j < j1, then ja0j jp < 1 and, since jb

0kjp 1, this implies



Section 2.4 A formula for the absolute logarithmic height 19

ja0j b0kjp < 1. If j > j1, then k < k1, so that jb0

kjp < 1 and ja0j b

0kjp < 1. It follows

that, in c 0i DP

j CkDi a0i b0j , all summands but a0j 1b0

k1have p-absolute value strictly

less than 1, while ja0j 1b0

k1jp D 1, leading to the conclusion that jc0

j 1Ck1jp D 1. But,

since jc0i jp 1 for every i , we conclude that maxi¹jc

0i jpº D 1 so that,

maxi

¹jci jd pp º D jaj 0jd p

p jbk0jd pp D max

i¹jai jd p

p º maxi

¹jbi jd pp º.

Since the above relation holds for every p 2 P K , the proof is now complete.

Proposition 2.4.2. Let ˛ be an algebraic number and f .X/ D a0X d C C ad

its minimal polynomial over Z. By this we mean that f has integer coefficients, witha0 > 0 and gcd.a0, : : : , ad / D 1; f is irreducible over Q , and f .˛/ D 0. Denoteby ˛.i / , i

D1, : : : , d the roots of f in C. Then,

h.˛/ D 1

d log

a0

d YiD1

max¹1, j˛.i /jº

.

Proof. We view ˛ as a complex number. Let K D Q.˛/, so that ŒK : Q D d , and

let let L be the splitting field of f over K , so that f .X/ D a0

Qd iD1.X ˛i /, where

˛ D ˛.1/, say.

Viewing a0 as a polynomial over L and applying (2.24) we see that H L,fin.a0/ D

ja0

jŒL:Q.

Since the coefficients of f are relatively prime in Z, they are relatively primealso in L, therefore maxi¹jai jPº D 1 for every prime ideal P of ZL; consequently,

H L,fin.f / D 1. Also, since all fields Q.˛.i // are isomorphic to Q.˛/, it follows that

H L,fin.X ˛i / D H L,fin.X ˛/.

Now we apply Proposition 2.4.1:

1 D H L,fin.f / D H L,fin.a0/ Y

i

H L,fin.X ˛i / D ja0jŒL:QH L,fin.X ˛/d

which shows that H L,fin.1 : ˛/d

DH L,fin.X

˛/d

D ja0

jŒL:Q.

By Proposition 2.3.1, H L,fin.1 : ˛/ D H K ,fin.1 : ˛/ŒL:K , hence

ja0jŒL:Q D H L,fin.1 : ˛/d D H K ,fin.1 : ˛/ŒL:Kd

D H K ,fin.1 : ˛/ŒL:Q,

hence H K ,fin.1 : ˛/ D ja0j. By definition,

h.˛/ D log H.1 : ˛/ D 1

d log H K .1 : ˛/ D 1

d log.H K ,fin.˛/ H K ,1.˛//

D 1

d

log.a0

H K ,1.˛//. (2.25)




It remains to compute H K ,1.˛/. There are exactly d embeddings i : K ,! C

(i D 1, : : : , d ) and they are characterised by their image of ˛, namely, .˛/ D ˛.i /.

Therefore, H K ,1.˛/ D

Qd iD1 max¹1, j˛.i /jº. By inserting this into (2.25) we com-

plete the proof.

2.5 Heights of points on an elliptic curve

As always in this chapter, H . : : / and h./ denote absolute height and absolute

logarithmic height, respectively.

Consider an elliptic curve E and its short Weierstrass model

E : y2 D x3 C Ax C B, A, B 2 Q. (2.26)

Use will be made also of the following slightly different model of E

D : y02 D 4x03 g2x0 g3, g2 D 4A, g3 D 4B , .x0, y0/ D .x, 2y/.

For points P of E , such that P E 2 E.Q/, the naive or Weil height of P is, by

definition,

h.P/ def D h.x.P //.

In the special important case that P E 2 E.Q/, we put P E D .x, y/, so that P D D.x0, y0/ D .x, 2y/, with x D a=b, a, b 2 Z, gcd.a, b/ D 1. Then,

h.P/

def

D h.x/ D log max¹1, jxjº CXp log max¹1, jxjpº D log max¹jaj, jbjº. (2.27)

For S. David, whose main result in [12] is extremely important for the application of

Ellog, it is more convenient to use a somewhat different height of P , namely,

hD.P / def D h.1 : x 0 : y 0/ D h.1 : x : 2y/.

In the special case that EP D .x, y/ 2 E.Q/,

hD.P / D h.1 : x 0 : y 0/ D h.1 : x : 2y/

Dlog max

¹1,

jxj,j2y

jº CXp

log max¹

1,jxjp

,j2y

jpº. (2.28)

From the point of view of Diophantine equations, it is more convenient to work with

the model E than with the model D, this last model being more appropriate for [12].

Therefore we need to see how the two heights are related. This we do in the following

lemma.

Lemma 2.5.1. Let E as in (2.26) and let P E D .x, y/ 2 E.K/ , where K is a number field. Then

hD.P /

5

2

h.P/

C2log2

C 1

2

.h.A/

Ch.B//.



Section 2.5 Heights of points on an elliptic curve 21

Proof. We will make extensive use of Proposition 2.3.2, without special reference

each time we use any of the relations (2.15), (2.16) or (2.17). Moreover, we will make

use of the fact that, if x1, x2 are algebraic numbers, then

h.1 : x1 : x2/ h.x1/ C h.x2/ (2.29)

Indeed, if K is a number field containing both x1 and x2, then it suffices to prove

that H .1 : x1 : x2/ H.1 : x1/H.1 : x2/ which, in turn, amounts in proving that

H K .1 : x1 : x2/ H K .1 : x1/H K .1 : x2/. The proof is left to the reader, as it is very

similar to the proof of (2.15).

Now, we have

2h.y/ D h.y2/ D h.x3 C Ax C B/ log2 C h.x3 C Ax/ C h.B/

D log2 C .h.x/ C h.x

2

C A// C h.B/ D h.x

2

C A/ C h.x/ C h.B/ C log 2D .log2 C 2h.x/ C h.A// C h.x/ C h.B/ C log 2

D 3h.x/ C h.A/ C h.B/ C 2log2.

Using this and (2.29) we have,

hD.P / D h.1 : x : 2y/ h.x/ C h.2y/ h.x/ C h.2/ C h.y/

D h.x/ C h.y/ C log2 5

2h.x/ C 1

2.h.A/ C h.B// C 2log2.

Since in our future study we will make use almost exclusively of points with rationalcoordinates, it is worthwhile to give a more accurate relation between hD.P / and

h.P/, when P E 2 E.Q/. This will be used in the proof of Proposition 2.6.1.

Lemma 2.5.2. Let E be the short Weierstrass model (2.26). There exist constantsc1, c2 , depending only on E , such that, for any point P 2 E with P E 2 E.Q/ , wehave

c1 C 3

2h.P/ hD.P / c2 C 3

2h.P/.

Proof. Let P E D .x, y/ 2 E.Q/, so that P D D .x0, y0/ D .x, 2y/ 2 D.Q/. In

analogy with (2.28) we define (only for the needs of this proof)

hE .P / def D h.1 : x : y / D log max¹1, jxj, jyjº C

Xp

log max¹1, jxjp, jyjpº.

We observe that, for every prime p,

1

2max¹1, jxjp, jyjpº max¹1, jxjp , j2yjpº

max¹1, jxjp , jyjpº < 2 max¹1, jxjp, jyjpº




and

1

2max¹1, jxj, jyjº < max¹1, jxj, jyjº max¹1, jxj, j2yjº 2max¹1, jxj, jyjº,

from which it follows that

log2 C hE .P / hD.P / log 2 C hE .P /. (2.30)

Now we will compare hE .P / with h.P / (cf. (2.27)).

Firstly, we will prove that there exist positive constants C 1, C 2, depending only on

E and not on P , such that

C 1 max¹1, jxjº3=2 max¹1, jxj, jyjº C 2 max¹1, jxjº3=2. (2.31)

Here is the proof for the right-most inequality (2.31). We have y 2 D x3 C Ax C B.If jxj 1, then y 2 1 C jAj C jBj; and if jxj > 1, then y 2 < .1 C jAj C jBj/jxj3,

therefore, in both cases, max¹1, jyjº p

1 C jAj C jBj max¹1, jxjº3=2. Obviously,

max¹1, jxjº p

1 C jAj C jBj max¹1, jxjº3=2, therefore,

max¹1, jxj, jyjº D max¹max¹1, jxjº, max¹1, jyjºº C 2 max¹1, jxjº3=2,

where C 2 Dp

1 C jAj C jBj.Now the proof of the left-most inequality (2.31). There exists a sufficiently large M >

1 such that, if

jt

j M then

jt 3

CAt

CB

j > 1

4 jt

j3. If

jx

j 1 then max

¹1,

jy

jº 1 D max¹1, jxjº D max¹1, jxjº3=2. If 1 < jxj < M , then M 3=2 max¹1, jyjº M 3=2

> jxj3=2 D max¹1, jxjº3=2, hence max¹1, jyjº > M 3=2 max¹1, jxjº3=2. If jxj M ,

then jyj 1

2jxj3=2, hence max¹1, jyj º jyj 1

2jxj3=2 D 1

2 max¹1, jxjº3=2. Therefore,

max¹1, jxj, jyjº max¹1, jyjº C 1 max¹1, jxjº3=2,

where C 1 D min¹1

2, M 3=2º.

Next, we will compute a lower and an upper bound for hE .P /, both of which are of

the form “constant

C

3

2h.x/”. These bounds will be combined with (2.30) to complete

the proof of the lemma. For this purpose, let d be the least positive integer such thatboth d 4A and d 6B are integers. Obviously, d is a divisor of the denominators of A

and B , which implies that d is independent from the point P D .x, y/. Now, y 2 Dx3CAxCB implies that .d 3y/2 D .d 2x/3C.d 4A/.d 2x/C.d 6B/, hence .d 2x, d 3y/

satisfies a short Weierstrass equation with integer coefficients. By Lemma 1.2.2, it

follows that d 2x D r=t 2 and d 3y D s= t 3, where r , s, t are integers, t > 0 and

gcd.r, t / D gcd.s, t / D 1. We have

hE .P / D h.x, y/ D log H.1 : x : y / D log H.1 : r=.dt/2 : s=.dt/3/

D log H..dt/

3

: r.dt/ : s/. (2.32)



Section 2.6 The canonical height 23

By (2.18),

H..dt/3 : r.dt/ : s/ D max¹.dt/3, dt jrj, jsjºı

D .dt/3

ı max¹1, jxj, jyjº, (2.33)

where

ı :D gcd..dt/3, dt r , s/ gcd.d 3t 3, d 3t r , d 3s/ d 3 gcd.t 3, t r , s/ D d 3

and we have used the fact that gcd.r, t / D 1 D gcd.s, t /.

By (2.33) and (2.31),

C 1.dt/3

ı max¹1, jxjº3=2 H..dt/3 : r.dt/ : s/ C 2

.dt/3

ı max¹1, jxjº3=2. (2.34)

Applying (2.22) with q D

x D

r=.dt/2, we get

h.x/ D log max¹1, jxjº C logd 2t 2

gcd.r , d 2t 2/ D log max¹1, jxjº C log

d 2t 2

gcd.r, d 2/,

hence

max¹1, jxjº3=2 D gcd.r, d 2/3=2

.dt/3 exp. 3

2h.x//.

Then, (2.34) becomes

C 1

gcd.r, d 2/3=2

ı e

3

2h.x/

H..dt/3 : r.dt/ : s/

C 2

gcd.r, d 2/3=2

ı e

3

2h.x/

where, obviously,1

d 3 gcd.r, d 2/3=2

ı d 3.

Combining the last two displayed relations with (2.32) we get

log C 1 3log d C 3

2h.x/ hE .P / log C 2 C 3log d C 3

2h.x/.

Finally, combining the above relation with (2.30) and remembering that, by definition,

h.x/

D h.P/ (cf. (2.27)), we obtain the relation in the announcement of the lemma,

with c1 D log 2 C log C 1 3log d and c2 D log2 C log C 2 C 3log d .

2.6 The canonical height

Let now E be an elliptic curve over a number field K and P 2 E . By Fact 1.1.2 of

Section 1.1 there exists a Weierstrass model

C : y2 C a1xy C a3y D x3 C a2x2 C a4x C a6

with coefficients in K . Working with the model C of E and its function field K.C/, we




have the degree-two function x 2 K.C/, defined by C.K/ 3 P C D .x.P/, y.P// 7!x.P/ 2 K and this function is even (x.P / D x.P /). Therefore, if P 2 E is such

that P C 2 C.K/, then according to [45, pages 247–248], the canonical height of P C

can be defined by the following limit:

1

2lim

N !1

h.x.2N P C //

4N , (2.35)

where h./ denotes the absolute logarithmic height. This limit is independent from the

Weierstrass model. Indeed, let C 0 : W .x0, y0/ D 0 be another Weierstrass model over

K of E . By [45, Proposition 3.1(b), Chapter III.3], x0.P / D 2x.P/C for convenient

, 2 K . Working now with the model C 0 of E and its function field K.C 0/, we have

the degree-two function x 0 2 K.C 0/, defined by C.K/ 3 P C 0 D .x0.P /, y0.P// 7!x0.P / 2 K and this function is even (x0.P / D x0.P /). Therefore, as before, the

canonical height of P C

0

can be defined by

1

2lim

N !1

h.x0.2N P C 0//

4N .

But x 0 is an even, degree-two function of K.C / also, in view of the simple relation

between x0.P / and x.P /, therefore, by [45, Proposition 9.1, Chapter VIII.9], the two

limits displayed above are equal and their common value (2.35) is defined as the canon-ical or Néron–Tate height of the point P 2 E , denoted by Oh.P/, without any model

indication on P . Thus, taking also into account (2.27), we have

Oh.P/ D 12

limN !1

h.2N

P C

/4N

. (2.36)

It is worthwhile to note, although we will not need it in this book, that for the definition

of the canonical height it is not necessary to use the function x 2 E.K/, but any even

function in E.Q/; see [45, VIII.9].

As already noted, for S. David it is more convenient to work in [12] with models D

like the one on page 20 and adopt hD as the absolute logarithmic height of a point, as

defined in (2.28). As a consequence, in the notation just before the relation (2.28), the

canonical height used by S. David is defined by

OhD.P / D limN !C1

hD.2N P /

4N . (2.37)

As an immediate consequence of the above relation and Lemma 2.5.2 we have proved

the following proposition:

Proposition 2.6.1. Let E : y2 D x3 C Ax C B

be a model of an elliptic curve E , where A, B 2 Q , and let

D : y02

D 4x03

g2x0

g3, g2 D 4A, g3 D 4B ,




so that D is also a model of E . Then, for any point P 2 E , the canonical heights Oh.P/

and OhD.P / , defined by (2.36) and (2.37) , respectively, are related by

OhD.P /

D3

Oh.P/.

A very important property of the canonical height is that, by means of it, a positive-

definite quadratic form is defined as follows: First, one defines the so-called Néron–Tate (or Weil) pairing by

hP , Qi D Oh.P C Q/ Oh.P/ Oh.Q/.

The following important properties for the canonical height and the Néron–Tate pair-

ing hold (see [45, Theorem 9.3]):

The Néron–Tate pairing is bilinear. For any P 2 E with P E 2 E.Q/ and any m 2 Z, Oh.mP / D m2 Oh.P/; in particular,

Oh.P / D Oh.P/.

Oh.P/ 0 and Oh.P/ D 0 if and only if P E is a torsion point.

Using these properties, it is not difficult to see that, if P E is expressed as in (1.7), then

Oh.P/ D 1

2

X1i ,j r

hP i , P j imi mj . (2.38)

The function Oh can be extended to the r-dimensional real vector space E .Q/ ˝ R,where E.Q/=Etors.Q/ sits as an r -dimensional lattice, and defines a positive definite

quadratic form on this vector space; see [45, items 9.4-9.7]. Then, the height pairingmatrix of .P 1, : : : , P r / is, by definition, the positive definite matrix

H D H.P 1, : : : , P r / D . 1

2hP i , P j i/rr (2.39)

(cf. (2.38)).

The following result is crucial for the applications of this book:

Proposition 2.6.2. Let P

E

be expressed as in (1.7). ThenOh.P/ max1ir

m2i ,

where is the least eigenvalue of the height pairing matrix H defined in (2.39).

Proof. According to (2.38) we have

Oh.P/ D mTHm,

where m is the column vector with components m1, : : : , mr . As H is symmetric, a

diagonal matrix ƒ of eigenvalues 0 < def

D 1 < 2 < < r of H and an




orthogonal matrix Q exist such that H D QTƒQ. Writing n D Qm and observing

that QTQ D I r (identity matrix), we deduce

Oh.P/ D m

T

QT

ƒQm D n

T

ƒn Dr

XiD1

i n2

i

rXiD1

n2i D nT

n D mTQ

TQm D mT

m

D

rXiD1

m2i c1 max

1irm2

i ,

as claimed.

Proposition 2.6.3. Let E : y2

D x3

C Ax C B with A, B 2 Q be an elliptic curvemodel. Let

D : y21 C a1x1y1 C a3y1 D x3

1 C a2x21 C a4x1 C a6, (2.40)

be any Weierstrass model of the same elliptic curve, with a1, a2, a3, a4, a6 rationalintegers , such that the equation of D is related to the equation of E by a change of variables

x1 D 2x C , y1 D 3y C 2x C ,

for appropriate rational numbers , , , .3 Let

D 1

12log jj C 1

12logC jj j C 1

2logC jb2=12j C 1

2log2

C 1

2.log2 C h.// C h./ C log jj C 1.07, (2.41)

where and j are, respectively, the discriminant and j -invariant of the model E ,b2 D a2

1 C 4a2 , 2 D 1 or 2 according to whether b2 vanishes or not, respectively,and logC is defined for any real ˛ > 0 by logC˛ D log max¹1, ˛º.Then, for every P E D .x.P/, y.P// 2 E.Q/ , we have

Oh.P/ 12

h.x.P // . (2.42)

Proof. Note that the two models D and E have equal j -invariants, while their corre-

sponding discriminants 1 and are related by

1 D 12. (2.43)

Now we apply Silverman’s Theorem 1.1 in [44] to the model (2.40). That theorem

requires that the ai ’s be algebraic integers. In our proposition, these coefficients are

3 One can easily see that there always exists such a model D.




rational integers, which permits to replace h./ and h1.j /, h1.b2=12/, which ap-

pear in Silverman’s theorem, by log jj and logC jj j, logC jb2=12j, respectively. We

thus obtain the following inequality

Oh.P/ 12

h.x1.P// 0 (2.44)

with

0 D 1

12 log j1j C 1

12 logC jj1j C 1

2 logC jb2=12j C 1

2 log2 C 1.07,

where 1 and j1 are, respectively, the discriminant and the j -invariant of the model

D. As already noted, (2.43) holds and j1 D j , the j -invariant of E . Therefore,

0

D 1

12 log

j

j Clog

j

j C 1

12 logC

jj

j C 1

2 logC

jb2=12

j C 1

2 log 2

C1.07. (2.45)

Now,

h.x1.P// D h.2 x.P/ C /

log2 C 2h./ C h./ C h.x.P // (by Proposition 2.3.2),

therefore,

Oh.P/ 1

2h.x.P // Oh.P/ 1

2h.x1.P// C 1

2.log2 C h.// C h./

0 C 1

2 .log2 C h.// C h./ (by (2.44)).

On combining the last inequality with (2.45) we obtain the inequality (2.42).

Remark. According to the announcement of Proposition 2.6.3, the Weierstrass model

D must fulfil certain conditions, but otherwise it is arbitrary. What we do in practice

when we have to solve a specific Diophantine equation, is to choose (2.40) in such

a way that the value of is as small as possible. Usually we choose for (2.40) the

(global) minimal Weierstrass model (see [45, Section VIII.8]), which can be computed

by various packages, for example, PARI, MAGMA, SAGE; not rarely, it happens that theminimal Weierstrass model is (4.1).

The following proposition is useful in order to compute an upper bound for the

canonical height for the point of an elliptic curve, if the coordinates of the point belong

to a number field. Computing the canonical height of such a point is more difficult

compared with the analogous task for a rational point (see [43]); anyway, standard

packages such as PARI and MAGMA refuse to work with points over a number field.

Fortunately, for the needs of this book it suffices that we know a “reasonably good”

upper bound for the canonical height.




Proposition 2.6.4. Let D : y21 C a1x1y1 C a3y1 D x3

1 C a2x21 C a4x1 C a6 be a

model of an elliptic curve, with a1, a2, a3, a4, a6 rational integers. Let P D 2 D.K/ ,where K is a number field. Then

Oh.P/ h.x1.P// C 112

log jj C 112

logC jj j C 12

logC jb2=12j C 12

log2,

where, 1 and j1 are, respectively, the discriminant and j -invariant of the model D ,b2 D a2

1 C 4a2 , 2 D 1 or 2 according to whether b2 vanishes or not, respectively,and logC is defined for any real ˛ > 0 by logC˛ D log max¹1, ˛º.

Proof. Immediate application of Silverman’s theorem [44, Theorem 1.1]. Note that,

in our proposition, the coefficients of the equation defining D are rational inte-

gers.4 Therefore, 1 2 Z and j1, b2 2 Q, so that we can replace h.1/ and

h1.j1/, h1.b2=12/, appearing in Silverman’s theorem, by log

j1

j and logC

jj1

j,

logC jb2=12j, respectively.

4 Silverman’s theorem demands only that these coefficients be algebraic integers in K .



Chapter 3

Weierstrass equations over C and R

3.1 The Weierstrass function

For the basic background of this section we refer to [1]. Let !1, !2 be two complex

numbers, such that !2=!1 62 R, which we will call periods and let ƒ D Z!1 CZ!2 C be the lattice generated by these numbers, which we call a period lattice. The paral-

lelogram … D ¹x1!1 C x2!2 : 0 x1, x2 < 1º and all its translations by vectors!0!

with ! 2 ƒ are called period parallelograms of ƒ.1

The lattice ƒ has infinitely manyperiod parallelograms, since there are infinitely many bases .!1, !2/ of ƒ; actually,

any pair .! 01, !0

2/ obtained from .!1, !2/ by a unimodular linear transformation with

integer coefficients is another basis for ƒ.

Definition 3.1.1. The Weierstrass -function corresponding to the lattice ƒ is defined

by the series

.z/ D z2 CX

!2ƒX¹0º

..z !/2 !2/. (3.1)

The Weierstrass -function defined above is an even, doubly periodic function, withset of periods ƒ. It has a double pole at each ! 2 ƒ and is analytic at every z 2 CXƒ;

see [1, Theorem 1.10 ]. Since is a meromorphic and doubly periodic function, it is,

by definition, an elliptic function. Any basis .!1, !2/ of ƒ is also called a fundamen-tal pair of periods of . If we translate the closure of a period parallelogram of ƒ by

a vector!0z, (the end point z need not be a lattice point) and from each pair of paral-

lel sides of the resulting (closed) parallelogram we remove exactly one side (and its

vertices), we obtain a fundamental parallelogram of . Thus, from the four vertices

which bound a fundamental parallelogram, exactly one is contained in it. Obviously,

any period parallelogram of ƒ is a fundamental parallelogram of the corresponding

-function but, clearly, the converse is not true.

As noted just before Definition 3.1.1, for the function , there are infinitely many

fundamental pairs of periods, hence there are infinitely many period parallelograms

and fundamental parallelograms. In view of periodicity it is clear that may be viewed

as an 1-1 function on any of its fundamental parallelograms rather than as a function

on C.

1 Note that, in a period parallelogram, only the “left-lower” vertex is included; equally well, however,

instead of this vertex, one could include a different one.



30 Chapter 3 Weierstrass equations overC and R

Let r D min¹j!j : ! 2 ƒ, ! ¤ 0º. Then, for 0 < jzj < r an alternative expression

for .z/ is

.z/

D 1

z2

C

1

XnD1

.2n

C1/G2nC2z2n, (3.2)

where, for n 3,

Gn DX

!2ƒX¹0º

1

!n; (3.3)

see [1, Theorem 1.11]. The function satisfies the differential equation

0.z/2 D 4.z/3 g2 .z/ g3, (3.4)

where, by definition,

g2 D 60G4, g3 D 140G6 ; (3.5)

see [1, Theorem 1.12].

Remark. .z/, Gn, g2 and g3 are defined by means of a given lattice ƒ (of rank 2),

therefore it would be more precise to respectively write .z; ƒ/, Gn.ƒ/, g2.ƒ/ and

g3.ƒ/. Actually, sometimes we will use this more precise notation if we want to em-

phasise the role of ƒ; otherwise, for the sake of simplicity in our notation, we will

omit the indication of ƒ.

Note that, by (3.1),.z/0 D 2

X!2ƒ

1

.z !/3,

which is an odd function. Further, the roots of the polynomial 4X 3 g2X g3, which

we will denote by e1, e2, e3, are expressed in terms of the three non-zero half-periods,

as follows:

e1 D .!1

2/, e2 D .

!1 C !2

2/ e3 D .

!2

2/ (3.6)

and later on we will make use of the points

Qidef D .ei , 0/ 2 E.C/, .i D 1,2,3/ (3.7)

The roots (3.6) are distinct, hence the discriminant of the cubic polynomial 4X 3 g2X g3 is non-zero: D g3

2 27g23 ¤ 0; see [1, §1.10]. By [1, Theorem 1.13],

the coefficients G2nC2 in (3.2) are expressible as polynomials in g2, g3 with positive

rational coefficients, therefore the Weierstrass function is uniquely defined by the

pair .g2, g3/ satisfying the condition g32 27g2

3 ¤ 0. The converse is also true. More

precisely, if two given complex numbers a2, a3 given satisfy a32 27a2

3 ¤ 0, then

there exists a lattice ƒ D Z!1 CZ!2, with !2=!1 62 R, such that g2.ƒ/ D a2 and

g3.ƒ/ D a3; see [1, Theorem 2.9].



Section 3.2 The Weierstrass equation 31

We summarise:

Fact 3.1.2. Given a lattice ƒ D Z!1 C Z!2 , with !1=!2 62 R , we define the Weier-strass function .z ; ƒ/

D .z/ by means of (3.1) , and the parameters g2.ƒ/

D g2

and g3.ƒ/ D g3 by means of (3.5) and (3.3). Then, the discriminant .ƒ/ D def Dg3

2 27g23 is non-zero and the differential equation (3.4) is satisfied.

Conversely, given two complex numbers a2, a3 such that a32 27a2

3 ¤ 0 , there existsa lattice ƒ D Z!1 C Z!2 , with !1=!2 62 R , such that g2.ƒ/ D a2 and g3.ƒ/ D a3.The Weierstrass function .z ; ƒ/ D .z/ defined by (3.1) will be called Weierstrass function with parameters g2 D a2 and g3 D a3.

It is sometimes convenient to replace the lattice ƒ by its homothetic lattice ƒ DZCZ , where D !2=!1, so that ƒ D !1ƒ . The simple equations below relate the

values of g2, g3 and that correspond to the lattices ƒ and ƒ . Note, however, that,

instead of writing g2.ƒ /, we simply write g2./ and similarly for g3 and . We have

g2 D g2.ƒ/ D !41 g2. /

g3 D g3.ƒ/ D !61 g3. /

D .ƒ/ D !121 ./ (3.8)

(see, for example, [1, §1.11]). Moreover, if is in the upper half-plane, then there is

a Fourier expansion for ./

./ D .2/12

1XnD1

t.n/e2in ,

where t .n/ 2 Z for all n and a product expansion

./ D .2/1224./, (3.9)

where

./ D ei=12

1YnD1

.1 e2in / (3.10)

(see [1, Theorem 3.3]).

3.2 The Weierstrass equation

Consider now an equation

y2 D x3 C Ax C B , A, B 2 C, 4A3 C 27B 2 ¤ 0 (3.11)

in complex numbers x, y. Since .4A/3 27.4B/2 ¤ 0, Fact 3.1.2 implies that

there exists a lattice ƒ D Z!1 CZ!2 with g2.ƒ/ D 4A and g3.ƒ/ D 4B . The

corresponding Weierstrass function has parameters g2 D 4A and g3 D 4B.




Then, for every z 2 C X ƒ we have .0.z/=2/2 D .z/3 C A.z/ C B , in view

of (3.4). It is a very important fact that the converse is also true: If .x, y/ satisfies

y2 D x3 C Ax C B (hence .2y/2 D 4x3 g2x g3), then there exists a unique

z mod ƒ such that .x, 2y/D

..z/, 0.z//.

The proof is a consequence of a general property common to all elliptic functions,

namely, that in a fundamental parallelogram with no poles or zeros on its boundary,the number of zeros is equal to the number of poles, with multiplicities of poles and zeros taken into account ; see [1, Theorem 1.18]).

Having this in mind we fix a pair .x, y/ as above and consider a fundamental par-

allelogram F. We will prove that there exists exactly one z 2 F such that .x, 2y/ D..z/, 0.z//. Obviously, it suffices to prove this property for any particular funda-

mental parallelogram. Therefore, for the needs of our proof we may assume that F is

such that the points 0, !1=2, !2=2 and .!1

C!2/=2 are contained in its interior.

We consider first the case y D 0 which implies that x D ei for some i 2 ¹1,2,3º(cf. (3.6)). Depending on whether i D 1, 2 or 3, we see that, if w 2 F with w !1

2 ,

!1C!2

2 ,

!2

2 .mod ƒ/, respectively, then the relation .ei , 0/ D ..z/, 0.z// is sat-

isfied by z D w. Moreover, for fixed i 2 ¹1,2,3º, no other point z 2 F satisfies this

relation. Indeed, since is an elliptic function with 0 as its only pole in F, and this

pole is double, it is not difficult to see that 0 is an elliptic function with fundamental

parallelogram F, its only pole in F is 0, and the order of this pole is 3. Hence, 0 has

exactly three zeros in F. But we already know that !1

2 ,

!1C!2

2 ,

!2

2 are three distinct

zeros of 0, hence these are the only ones; in particular, for the chosen i , there is no

z 2 F other than w satisfying .ei , 0/ D ..z/,

0

.z//.Next, let y ¤ 0. Then we consider the function z 7! .z/ x which, obviously,

is elliptic and F, as above, is a fundamental parallelogram for this function. Since

0 2 F is a double pole for this function and no other poles exist in F, it follows that

the equation .z/ x D 0 has exactly two solutions z1, z2 2 F (note that z1, z2 62¹0,

!1

2 ,

!1C!2

2 ,

!2

2 º). But since z1 is a solution, the same is true for the w 2 F satisfying

w z1 .mod ƒ/, since is an even function. Thus x D .z1/ D .w/ and then,

if in (3.4) we successively put z D z1 and z D w, we obtain 2y D ˙0.z1/ and 2y D˙0.w/ with appropriate choice of signs. It follows that 0.w/ D ˙0.z1/, hence

2y is equal to either 0.z/ or to 0.w/. Note that w

¤ z1.2 Therefore, necessarily,

w D z2, z2 z1 .mod ƒ/ and our discussion above showed that there exists exactly

one z 2 F satisfying .x, 2y/ D ..z/, 0.z// and this is either z D z1 or z D z2.

Our discussion is summarised as follows:

Fact 3.2.1. Let A, B 2 C such that 4A3 C 27B 2 ¤ 0 and consider the elliptic curvemodel E : y 2 D x3CAxCB .Let be the Weierstrass function with parameters g2 D4A and g3 D 4B , let F be any fundamental parallelogram for (in particular, a

2 Otherwise, 2z1 2 ƒ which easily implies that z1 !1

2,

!1C!2

2,

!2

2.mod ƒ/ and, consequently,

2y D 0.z1/ D 0, a contradiction.



Section 3.3 : E.C/ 7! C=ƒ 33

period parallelogram) and let ! be the only pole of in F. Then the map

F 3 z 7!

´..z/,

1

20.z// if z ¤ !

O if z

D!

2 E.C/ (3.12)

is one-to-one and onto the group E.C/.3

3.3 : E.C/ 7! C=ƒ

In this section E will be the elliptic curve model considered in Fact 3.2.1. According

to Section 1.1, E is just one out of many models of an elliptic curve E , therefore, any

point on E , being a representative of a certain abstract point P (say) of E , should be

denoted by P E . However, since in the present chapter we will not make use of any

other model of E except for E , and for the sake of simplifying the notation, we willomit the superscript E on P , simply writing P instead of P E .

Our main purpose is to define a group isomorphism : E.C/ 7! C=ƒ, which is

the inverse of that we defined in Fact 3.2.1.

Let … be the period parallelogram with vertices 0, !1, !2, !1 C !2 and consider the

map (3.12) with F D …. Obviously, we can view … as the abelian group C=ƒ. Then,

Fact 3.2.1 is part of a far stronger result ([24, Theorem 6.17]) which says that the map is a group isomorphism; even more, it is an isomorphism of Riemann surfaces (see

[24, Theorem 6.14], or [45, Proposition 3.6]). We will not need the full strength of this

result, but only that is a group isomorphism. In view of Fact 3.2.1, for the proof of this very last claim it remains to prove that is a homomorphism. It is more convenient

to prove that the inverse map of , which we will denote by , is a homomorphism.

First, let us make more explicit . As always, “zero point” is the pointO. According

to Fact 3.2.1, for any non-zero point .x , y/ 2 E.C/ there exists exactly one r 2 …,

such that .x, y/ D ..r/, 1

20.r//. Therefore we have the well-defined bijection :

E.C/ ! C=ƒ mapping O to ƒ and any non-zero point .x , y/ D ..r/, 1

20.r// to

r C ƒ.

We have to show that is a group homomorphism. We will make use of the so-

called “addition formula” and “duplication formula” for Weierstrass functions, re-ferring, on the one hand to [73, §20.3] and, on the other hand, to the formulas for

the addition and duplication of points on a Weierstrass model of an elliptic curve, as

found, for example, in the “Group Law Algorithm 2.3” of [45].

Let P 1, P 2 be two points on E.C/. Then, for i D 1, 2, there exists zi 2 … such that

.x.P i /, y.P i // D ..zi /, 1

20.zi //. We have to prove that .P 1 C P 2/ D .P 1/ C

.P 2/. This is obviously true if at least one P i is the zero point, therefore we assume

that both P 1, P 2 are non-zero points. We distinguish the cases .z1/ ¤ .z2/ and

.z1/ D .z2/.

3 Actually, is a group isomorphism; see the beginning of next section.




(a) .z1/ ¤ .z2/. Then,

x.P 1 C P 2/ D

y.P 2/ y.P 1/

x.P 2/

x.P 1/

2

x.P 1/ x.P 2/

D 1

4

0.r/ 0.z2/

.z1/ .z2/

2

.z1/ .z2/

D .z1 C z2/,

the last equality being true because of the “addition-theorem” [73, §20.31]. On the

other hand, by the first determinant formula in [73, §20.3], we haveˇˇ

.z1/ 0.z1/ 1

.z2/ 0.z2/ 1

.z1

Cz2/

0.z1

Cz2/ 1

ˇˇD 0

and after some standard calculations we find that

1

20.z1 C z2/ D 1

2

0.z2/ 0.z1/

.z2/ .z1/ .z1 C z2/ 1

2

0.z1/.z2/ 0.z2/.z1/

.z2/ .z1/

D y.P 2/ y.P 1/

x.P 2/ x.P 1/x.P 1 C P 2/ y.P 1/x.P 2/ y.P 2/x.P 1/

x.P 2/ x.P 1/

D y.P 1 C P 2/.

This proves that .x.P 1 C P 2/, y.P 1 C P 2// D ..z1 C z2/, 1

20.z1 C z2//, hence

.P 1 C P 2/ D .z1 C z2/ C ƒ D .P 1/ C .P 2/.

(b) .z1/ D .z2/. Then, 0.z2/ D ˙0.z1/. If the minus sign holds, then

..z2/, 1

20.z2// D ..z1/, 1

20.z1//, hence P 2 D P 1 and .P 1CP 2/ D .O/ D

ƒ. On the other hand, .P 1/ C .P 2/ D .z1 C z2/ C ƒ D ƒ and this last equality

holds because ..z2/, 1

20.z2// D ..z2/, 1

20.z2// D ..z1/,

1

20.z1// which

implies that z1 D z2 .mod ƒ/.

If the plus sign holds, then z1 C ƒ D z2 C ƒ, hence, P 1 D P 2 D (say) P and

we put .x.P/, y.P// D ..r/, 1

20.r// with r 2 P . We have then to show that

.2P / D

2.P/. If P D

.ei , 0/ for some i 2 ¹

1,2,3

º (cf. (3.6)), then, on the

one hand, z D z0 2 ¹!1

2 , !1C!2

2 , !2

2 º and, on the other hand, 2P D O. Consequently,

.2P / D ƒ D 2.z0Cƒ/ D 2.P/. If for every i D 1,2,3wehave P ¤ .ei , 0/, then

y.P/ ¤ 0 and 0.r/ ¤ 0. Then, differentiation of 0.r/2 D 4.r/3 g2.r/ g3 D4.r/3 C 4A.r/ C B gives 00.r/ D 2.3.r/2 C A/. By the “duplication formula”

[73, §20.311],

.2r/ D 1

4

00.r/

0.r/

/2 2.r/ D

3.r/2 C A

0.r/

2

2.r/

D

.3x.P/2 C A/2 8xy2

4y2

Dx.2P /.



Section 3.3 : E.C/ 7! C=ƒ 35

We differentiate the relation .2r/ D . 3.r/2CA

0.r/ /2 2.r/ above and take into

account the relations 00.r/ D 2.3.r/2 C A/, .r/ D x.P/ and 0.r/ D2y.P/. After some elementary calculations we find that 0.2r/ D 2y.2P /. Thus,

.x.2P /, y.2P // D ..2r/,

1

2 0

.2r//, which shows that .2P / D 2r Cƒ D 2.P/,as required.

Combining our discussion above with Fact 3.2.1 we obtain the following theorem:

Theorem 3.3.1. We refer to Fact 3.2.1 , in which we put F D … , where … is the period parallelogram at the beginning of this section. We identify F D … with the groupC=ƒ. Then : C=ƒ ! E.C/ is a group isomorphism, whose inverse isomorphism : E.C/ ! C=ƒ is as follows: .O/ D ƒ and, for a non-zero point P D .x, y/ 2E.C/ , we define .P/ D r C ƒ , where r is the unique non-zero r 2 … with the property ..r/,

1

20.r//

D.x, y/.

Definition 3.3.2. Fix any fundamental parallelogram F of . Theorem 3.3.1 implies

that, for any P D .x, y/ 2 E.C/ there exists a unique r 2 F such that P D..r/,

1

20.r//. This r is called elliptic logarithm of the point P (belonging to F).

In analogy with complex logarithms, any point has infinitely many elliptic loga-

rithms, depending on the fundamental parallelogram which we choose. Any two of

them, however, differ by an element of ƒ. This situation is similar to that of complex

logarithms, in which any non-zero complex number has infinitely many logarithms

and any two of them differ by an integral multiple of 2i .

Now a natural problem arises:

Problem 3.3.3. Given a point P D .x.P/, y.P// 2 E.C/ and a fundamental par-allelogram F , compute the elliptic logarithm of P belonging to F. In other words,compute r 2 F such that ..r/,

1

20.r// D .x.P/, y.P//.

Once again we turn to the classical book [73]. First, we cut the complex plane along

a line segment joining any two of e1, e2, e3 and along any half-line with origin the

remaining ei which (half-line) has no common points with the (closed) line segment.

What remains then is a cut plane which we denote by C; we understand that e1, e2, e3

are not points of C. If we fix some z0 2 C, such that 4z30 g2z0 g3 ¤ 0, along withone of the two square roots of 4z3

0 g2z0 g3, let us denote it by w0, then there exists

an analytic function h on C such that h.z/2 D 4z3 g2z g3 for every z 2 C and

h.z0/ D w0. This is a very classical result. For obvious reasons, this analytic function

h is denoted byp

4z3 g2z g3, despite the ambiguity of the notation coming from

the fact that, for the chosen z0 there are two choices for w0. However, in using the

square root notation, we understand that a certain pair .z0, w0/, as above, has been

fixed.

Suppose now that w 2 C. Let z.w/ D

R 1

wdtp

4t 3g2tg3

, where the integration

path, except for 1 and, possibly, its end point w, is contained in C and does not pass




through any ei , i.e. only its end point w might coincide with a root ei . Then, according

to [73, §20.221], .z/ D w . This is the so-called integral formula for .z/, which is

expressed by the following implication:

z D Z 1w

dtp 4t 3 g2t g3

) .z/ D w. (3.13)

Using (3.13) we can partially solve Problem 3.3.3. Indeed, setting in (3.13) w D x.P/

we have .z/ D x.P/. Consequently, 0.z/2 D 4x.P/3 g2x.P/g3 D .2y.P//2,

implying "0.z/ D 2y.P/, where " 2 ¹1, 1º. Then, since 0 is an odd func-

tion, 0."z/ D 2y.P/. If we choose r 2 F such that r D "z .mod ƒ/, then

.x.P/, y.P// D ..r/, 1

20.r//, as required.

This answer is partial in the sense that, given the point .x, y/ we do not know a

priori how to choose "

2 ¹1, 1

º. However, in the case that A and B are real numbers

and P 2 E.R/ we will give a very satisfactory answer in Section 3.5.

3.4 Weierstrass equations with real coefficients

In view of Fact 1.1.2, we intend to study Diophantine equations

E : y2 D x3 C Ax C B , A, B 2 Q, 4A3 C 27B 2 ¤ 0. (3.14)

In terms of the Weierstrass function, we have to find all z 2 C, such that ..z/, 1

20.z//

is an integral (or rational) point, where is the Weierstrass function with parameters

g2 D 4A, g3 D 4B. (3.15)

In such a case, .x, y/ D ..z/, 1

20.z// is a sought for point on E. It is natural,

therefore, that we focus our study on Weierstrass -functions whose period lattice

ƒ D Z!1 C Z!2 is such that g2 D g2.ƒ/ and g3 D g3.ƒ/ are real numbers, forget-

ting for the moment that, in the context of our Diophantine study, these are actually

rational numbers.

In this section we will study the elliptic curve model E defined by the equation (3.11)

when A, B

2 R. We will adopt the notation at the beginning of Section 3.2 and will

refer to Fact 3.2.1, according to which every point .x, y/ 2 E.C/ is of the shape

.x, y/ D ..z/, 1

20.z// for a unique z belonging to a fundamental parallelogram.

The problem that will concern us in this section is the following: When A, B 2 R, to

specify exactly those z in a conveniently chosen fundamental parallelogram for which

the point ..z/, 1

20.z// has real coordinates, that is, it belongs to E.R/.

In our study which we start immediately below, we will distinguish between the

case of positive discriminant (Subsection 3.4.1) and the case of negative discriminant

(Subsection 3.4.2). Finally, in Subsection 3.4.3 we will find explicit expressions for a

fundamental pair of periods. To a large extent our exposition follows closely that of

A. Hurwitz [23].



Section 3.4 Weierstrass equations with real coefficients 37

We have ƒ D Z!1 C Z!2, where !1=!1 62 R and both g2 D g2.ƒ/ D 4A and

g3 D g3.ƒ/ D 4B are real. By definition, G4 D g2=60, G6 D g3=140 (cf. (3.5)).

Moreover, by [1, Theorem 1.13], G2nC2 2 QŒg2, g3, hence, all coefficients of the

series in the right-hand side of (3.2) are real, which implies that .z/ D

.z/. We

claim now that ƒ contains both real and purely imaginary periods. Surely, ƒ contains

an ! such that <.!/ ¤ 0 and =.!/ ¤ 0. For such an ! and every z 2 C we have

.z C !/ D .z C !/ D .z C !/ D .z/ D .z/, which shows that ! is a period

for , i.e. ! 2 ƒ. But then, ˙.! C !/ 2 R and ˙.! !/ 2 iR are periods for .

Changing our notation, we adopt the following notation:

Throughout Section 3.4 and in the sequel of the chapter, we will denote by !1 the

least real positive period, and by !2 the totally complex period with least positive

imaginary part.

Before proceeding, let us make a comment about our temporary terminology. Imme-diately below, by “parallelogram with vertices 0, z1, z1Cz2, z2” (for some z1, z2 2 C),

we mean the set ¹x1z1 C x2z2 : 0 x1, x2 < 1º.

Now, let P be the parallelogram with vertices 0, !1, !1 C !2, !2. It is easy to see

that on the contour of P there are no periods other than the vertices. We distinguish

two cases.

Case 1. No period is in the interior of P. Then, P is a period parallelogram of the

lattice ƒ and, consequently, .!1, !2/ is a fundamental pair of periods for . We put

D !2=!1

2 iR so that e i

2 R. Then, by (3.9), ./ is a positive real number

and, consequently, by (3.8), > 0.

Case 2. Periods are included in the interior of the parallelogram P. Let ! D r C si 2ƒ belong to the interior of P. Then, 0 < r < !1 and 0 < s < =.!2/. According

to our discussion above, ! 2 ƒ, hence 2r D ! C ! and 2s D ! ! are periods

with 0 < 2r < 2!1 and 0 < 2s < 2=!2. These conditions imply, respectively,

2r D !1 and 2si D !2; hence ! is the centre of P: ! D !1C!2

2 . It is easy to see then

that the parallelogram with vertices 0, !1, !1 C !, ! contains no lattice points in its

interior, hence it is a period parallelogram of ƒ and !1, ! D !1C!2

2 is a fundamental

pair of periods with D

!=!1

D 1

2

.1 C

i t /, where i t D

!2=!1

2 iRC. Then,

e2 i D e t 2 R and, consequently, by (3.9) and (3.10), ./ is a negative real

number.

We summarise our conclusions:

.!1, !2/ is a fundamental pair of periods if and only if > 0 and .1, 2/ D

.!1, !1C!2

2 / is a fundamental pair of periods if and only if < 0.

We examine the two cases ( > 0 and < 0) separately.




3.4.1 > 0

In Figure 3.1 we translate the closure of the period parallelogram marked out by ver-

tices

, by the vector with initial point

1

2.!1

C!2/ and end point 0, and then we remove

the lower and the left side (with their vertices) of the resulting parallelogram. We thusobtain the fundamental parallelogram, the vertices of which are marked by ı in Fig-

ure 3.1; by its definition, this fundamental parallelogram contains only the upper-right

vertex. We examine for which z 2 P we have .z/ 2 R:

.z/ 2 R , .z/ D .z/ , .z/ D .z/ ) 0.z/ D ˙0.z/.

Therefore, if z 2 P and .z/ 2 R, then the point ..z/, 0.z// is equal to ei-

ther ..z/, 0.z// or ..z/, 0.z//, hence, either z D z .mod ƒ/ or z D z

.mod ƒ/; equivalently,1

2 .z ˙ z/ 2 1

2 ƒ. (3.16)

Remember now that P includes only the right vertical and the upper horizontal side.

If z is on one of these sides of P, then (3.16) is true with the plus or the minus sign,

respectively. Next, consider points z in the interior of P. Since the only point of 1

2ƒ

lying in the interior of P is 0, it is easy to see that (3.16) is true with the plus or minus

sign if and only if z is, respectively, on the line segment joining !2

2 with

!2

2 or the

line segment joining !1

2 with

!1

2 .

Now we consider the “small grey” closed parallelogram P0 with vertices 0, !1

2 ,

!1C!2

2

, !2

2

. Our discussion above, along with the fact that the function is even, imply

−

12(ω1 + ω2)

12(ω1 − ω2)

12(ω2 − ω1)

12(ω2 + ω1)

−

12ω1

12ω1

ω1

ω2 ω1 + ω2

0

period

parallelogram

P fundamental

parallelogram

P0

Figure 3.1. The fundamental parallelogram and the period parallelogram: Case > 0.




that, as z runs along the contour of P0, all possible real values of .z/ are obtained.

Therefore, we study the behaviour of the function on every side of P0.

The function .0, !1

2 3 z 7! .z/ 2 R is 1-1. Indeed, suppose that z1 ¤ z2 are on

the domain of this function and .z1

/ D

.z2

/. Then 0.z2

/ D ˙

0.z1

/. If the plus

sign holds, then z1 D z2 .mod ƒ/, clearly impossible. If the minus sign holds, then

..z2/, 0.z2// D ..z1/, 0.z1// and, consequently, z2 Cz1 2 ƒ, again impossi-

ble, since 0 < z1Cz2 < !1. Observe now that, in view of (3.2), limz!0C .z/ D C1and, in view of (3.6), . !1

2 / D e1. Therefore, the continuous function above is 1-1 and

takes on all values in the interval .C1, e1. It follows that e1 is the largest among the

roots (3.6) of the polynomial 4X 3 g2X g3. For, otherwise, some z 2 the interval

.0, !1

2 / would exist such that .z/ is one of the numbers (3.6), other than e1. This

would imply that either z !2

2 or z !1C!2

2 belongs to ƒ, which is impossible.

Next consider the function restricted to the side with end points !1

2 and

!1C!2

2 .

With similar arguments we conclude the following: Along this side, the function is

1-1; at the vertex !1

2 it takes the value e1 which, as already seen, is the least among

the roots (3.6); at the other vertex it takes the value . !1C!2

2 / (D e2, by (3.6)) which,

therefore, must be the second larger root; consequently, e3 D . !2

2 / is the least root

of this polynomial.

Similarly, as z runs along the side with end points !1C!2

2 and !2

2 the values of .z/

vary in 1-1 way from the middle root e2 to the smallest root e3.

Finally, as z runs along the side with end points !2

2 and 0, the values of .z/ vary

in a 1-1 way from the smallest root e3 to1

.

Now we study the behaviour of the function 0.

We have seen that .z/ is strictly decreasing as z runs along the side with end points

0 and !1

2 , hence, for these z’s we have .z/ e1. Then, 0.z/2 is a non-negative real

number, hence 0.z/ is a real number; more precisely, since .z/ is strictly decreasing,

we have 0.z/ 0 with equality holding if z D !1

2 .

Next, we see that the values of .z/ as z runs along the points of the upper horizontal

side of P0 following the direction from !1C!2

2 to

!2

2 , coincide with the values of the

real function Œ0, !1

2 3 x 7! .x C !1C!2

2 / 2 Œe3, e2, which is continuous, mapping

0 to e2 and !1

2

to e3. Therefore, this is a strictly decreasing function, hence its derivative

takes negative values in the open interval .0, !1

2 / and becomes zero at 0 and !1

2 . But

the derivative values of the above function coincide with those of 0.z/ as z runs along

the upper horizontal side of P0, hence 0.z/ < 0 for z in the interior of this side and

0.z/ D 0 if z is a vertex.

When z runs along the vertical sides of the parallelogram P0, the values of .z/

belong to either the interval Œe2, e1 (if z is on the right vertical side) or to the interval

.1, e3/ (if z is on the left vertical side), hence 0.z/2 0 and, except if z is an end

point of a side, ..z/, 0.z// 62 E.R/.

Summing up: The only points z in P0 for which ..z/, 1

20.z//

2 E.R/ are those

of the two horizontal sides. For those z’s, we have 0.z/ 0, with equality only on




Table 3.1. Case > 0. All points ..z/, 12

0.z// 2 E.R/.

z runs along half-open line segments

. !2!1

2. . . . . . . . . . . . . . . . . . .

!2C!1

2 .

!1

2. . . . . . . . . . . . . . . . . . . . . . .

!1

2

z !2!1

2Ý

!2

2Ý

!2C!1

2 !1

2Ý 0 Ý

!1

2

.z/ e2 & e3 % e2 e1 % C1 & e1

C10.z/ 0 > 0 0 < 0 0 0 % infinite # jump 0

1 %

the vertices. Also, .z/ is decreasing from C1 to e1 or from e2 to e3, according to

whether z runs along the lower side or the upper side, respectively.Now, by ..z/, 0.z// D ..z/, 0.z//, we see that, if z moves from !1

2 to

0 along the real axis, then .z/ increases from e1 to C1 and 0.z/ increases from 0

to C1; and since ..z/, 0.z// D ..z C !2/, 0.z C !2//, we conclude that,

if z moves from !2!1

2 to

!2

2 along the horizontal line joining these points, then .z/

decreases from e2 to e3 and 0.z/ is non-negative.

In Table 3.1 we schematically summarise our results.

3.4.2 < 0

In this case, .1, 2/ D .!1, !1C!2

2 / is a fundamental pair of periods (see page 37);

note that 2 1

21 2 iRC. In Figure 3.2, a fundamental parallelogramP is obtained as

follows: the closure of the period parallelogram marked out by vertices is translated

by the vector 1

2

!01 and then, from the resulting parallelogram, the sides left of the

vertical diagonal are removed, together with their vertices. The parallelogram P is

1

2Ω1−

1

2Ω1

−Ω2 + 1

2Ω1

Ω2 −1

2Ω1 Ω2

Ω2

Ω10

period

parallelogram

P fundamental

parallelogram

Figure 3.2. The fundamental parallelogram and the period parallelogram: Case < 0.




Table 3.2. Case < 0. All points ..z/, 12

0.z// 2 E.R/.

z runs along half-open line segment

.

1

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

z 1

2Ý 0 Ý

1

2

.z/ e1 % C1 & e1

C10.z/ 0 % infinite # jump 0

1 %

marked out by the vertices

ı, but only the vertex

1

2

is included. We examine for

which z 2 P (Figure 3.2) we have .z/ 2 R. As in the case of > 0, we see that

.z/ 2 R is equivalent to 1

2.z ˙ z/ 2 P \ 1

2ƒ. It can be easily checked that the last

relation is true with the plus or the minus sign if and only if z is on the vertical or the

horizontal diagonal of P, respectively. By (3.6), . 1

2 / D . !1

2 / D e1, hence this is

the only real root of 4X 3 g2X g3.

Now, assume that z runs along the horizontal diagonal of P. Then .z/ 2 R and, by

(3.2), limz!0 .z/ D C1. Moreover, is 1-1 on the intervals .1

2 , 0/ and .0,

1

2 ,

respectively. Therefore, as z moves from 1

2 to 0 along the real axis, .z/ increases

from e1 to

C1. In particular, this implies that 0.z/ > 0. On the other hand, as z

runs along the interval .0, 1

2 /, .z/ decreases from C1 to e1 which, in particular,

implies that 0.z/ < 0.

Next, assume that z runs along the vertical diagonal of P. If z tends to 2 1

21,

then .z/ tends to . 1

21/ D e1. For the interior points z of the upper half of this

diagonal we have, by (3.2), .z/ < 0 and limz!0i .z/ D 1, hence the values

of .z/ belong to the interval .1, e1/. It follows that 4.z/3 g2.z/ g3 < 0.

Consequently, 0.z/ 2 iR and no real point on E arises when z is on the open upper

half of the vertical diagonal of P. Similarly the same is true if z is in the open lower

half of this diagonal. In Table 3.2 we schematically summarise our results.

3.4.3 Explicit expressions for the periods

We refer to our discussion, notation etc. following Problem 3.3.3 up to the end of

Section 3.3.

Computation of !1. We cut C in the appropriate way, so that the real half-line from

e1 to C1 is contained in C and considerp

4x3 g2x g3 as the analytic continu-

ation on C of the positive real-valued function .e1, C1/ 3 x 7! p 4x3 g2x g3.

With w D e1 in (3.13) and integration path the half-line from e1 to C1 we have




.z/ D e1 D . 1

2!1/, hence

!1

2

Z 1

e1

dt

p 4t 3

g2t

g3

.mod ƒ/

(observe that !1

2 !1

2 .mod ƒ/). Denote, temporarily, by I the integral in the

right-hand side of the above relation. Since I 2 RC, this relation can be written as

I D 1

2!1 C n!1 for some non-negative integer n. We claim that n D 0. Suppose that

n > 0. Consider the real function

t 7! I.x/ DZ C1

x

dtp 4t 3 g2t g3

defined on the interval .e1, C1/, which is continuous and strictly decreasing. Since

limx!e1C I.x/ D I , this function takes values in the interval .0, 1

2!1 C n!1/, there-

fore, there exists x0 > e1 such that I.x0/ D 12 !1. This says that the relation (3.13)

holds with w D x0 and z D 1

2!1, which implies that x0 D .!1=2/ D e1, a contra-

diction. Therefore, n D 0 and

!1

2DZ C1

e1

dtp 4t 3 g2t g3

, (3.17)

hence, in terms of the coefficients of our Diophantine equation (3.14), we obtain, in

view of (3.15),

!1 DZ C1

e1

dt

p t 3

CAt

CB

. (3.18)

Computation of !2. We distinguish the two cases > 0 and < 0. We put

q.t/ D 4t 3 g2t g3.

Let > 0. In this case we choose C to be the complex plane from which we

have removed the open line segment with end points e2, e3 and the open half-line on

the real axis from e1 to C1. What remains in C from the real axis are the intervals

.1, e3 and Œe2, e1 on which the real non-negative functionp

q.x/ is defined.

We consider therefore the analytic function C 3 z 7! i

p q.z/ 2 C which, for

z 2 .1, e3 [ Œe2, e1, takes values in iRC

. Clearly, this is an analytic branch inC of the square root of q.x/. Therefore, if in (3.13) we take i

p q.t/ in place of p 4t 3 g2t g3, e3 in place of w and, as an integration path, the half-line on the real

axis from e3 to 1, then we have

z DZ 1

e3

dt

ip

q.t/, .z/ D e3 D .!2=2/.

It follows that

i

Z 1e3

dt

p q.t/ !2

2.mod ƒ/.




Since !2 2 iR, the last relation is equivalent to

Z e3

1

dt

p q.t/D s

2C ns, where !2 D is, s 2 RC and n 2 Z.

The left-hand side is positive, therefore, n 0. We will show that n D 0. Indeed,

suppose n > 0 and consider the real continuous function I.x/ D R x1

dtp q.t/

, de-

fined on the interval .1, e3/. It is an increasing function with values in the inter-

val .0, . 1

2 C n/s/. Therefore, there exists x0 < e3 such that I.x0/ D s=2, hence

iI.x0/ D !2=2. This means thatR 1

x0

dt

ip q.t/

D !2=2 and, consequently, x0 D.!2=2/ D e3, a contradiction. Therefore, n D 0 and we conclude that:

Case > 0

!2 D 2i Z e3

1

dtp q.t/

D i Z e3

1

dtp .t 3 C At C B/

. (3.19)

Let < 0. Now we cut the complex plane along the half-line on the real axis

from e1 to C1 and along the segment joining e2 with e3 D e2. In analogy to the

previous case we observe that q.x/ > 0 for x < e1, the real function .1, e1 3x 7!

p q.x/ 2 Œ0, C1/ is well defined and is extended analytically on C . As

before, if z D R 1e1

dt

ip q.t/

, then .z/ D e1 D .!1=2/. Now, a fundamental pair of

periods is .1, 2/

D.!1, .!1

C!1/=2/, therefore i R

e1

1dtp q.t/

D R 1

e1

dt

ip q.t/

D!1

2 C m!1 C n !1C!2

2 D . 1

2 C m C n

2/!1 C n

2!1, for some m, n 2 Z. The left-hand

side, as well as !2 belong to iRC, therefore, the coefficient of !1 must be zero, n is

odd andR e1

1dtp q.t/

D n

2s, where !2 D i s. In the last but one equality, the integral

is positive, hence n 1. If n 3, then, arguing as in the previous cases, we can find

x0 < e1 such thatR x0

1dtp q.t/

D s

2, hence

R 1x0

dt

ip q.t/

D s

2 and, consequently,

x0 D . !2

2 / D . !1C!2

2 !1

2 / D .!1

2 / D e1, a contradiction. Therefore n D 1

and we conclude that

Case < 0

!2 D 2i Z e1

1

dtp q.t/

D i Z e1

1

dtp .t 3 C At C B/

. (3.20)

Summing up we have the following theorem.

Theorem 3.4.1. Let q.X / D 4X 3 g2X g3 D 4.X 3 C AX CB/ D 4f.X/ , whereA, B 2 R and D 16.4A3 C 27B2/ ¤ 0. Denote by e1, e2, e3 the roots of q.X/ –which are also the roots of f .X/ – where e3 < e2 < e1 if > 0 , and e1 2 R , e3 D e2

if < 0. Let be the Weierstrass function with parameters g2 D 4A , g3 D 4B.Finally, let !1 be the least positive real period and let !2 be the totally complex period

with least positive imaginary part. Then




!1 is given by (3.17) or, equivalently, by (3.18); !2 is given by (3.19) if > 0 and by (3.20) if < 0.

A fundamental pair of periods for is .!1, !2/ if > 0 and .!1, 1

2.!1

C!2// if

< 0.

3.4.4 Computing !1 and !2 in practice

We keep the notation of Theorem 3.4.1. Our purpose in this section is to indicate a

fast method for computing !1 and !2. This will be accomplished by means of the

arithmetic-geometric mean (AGM), about which we immediately state the basic facts.

Let a, b be two positive real numbers. We define the sequences .an/ and .bn/ as

follows:

a0 D a, b0 D b,

anC1 D an C bn

2, bnC1 D

p anbn .n 0/.

In view of bn bnC1 anC1 an for n 1 we conclude that both sequences are

convergent and then, taking limits in 2anC1 D an C bn we see that lim an D lim bn.

This common limit is denoted by M.a, b/ and is called the arithmetic-geometric mean

(AGM) of a, b. The computation of M.a, b/ is very fast in view of the relation anC1 bnC1 D 1

8b1.an bn/2. The following formula is due to Lagrange and Gauss:

Z 2

0

dsp a2 cos2 s C b2 sin2 s

D 2

M.a, b/. (3.21)

For an elementary proof we refer to [5].

We consider first the case > 0. Our problem is the practical computation of the

integrals in (3.17) and (3.19). In this case we follow [5, Section 2.1]. First we compute

the integral in (3.17). The change of variable

t 0

D e2t e1e2 C e1e3 e2e3

t e2

gives Z C1e1

dtp q.t/

DZ e3

e2

dtp q.t/

and then, the change of variable t D e3 C .e2 e3/ sin2 s transforms the last integral

into1

4

Z 2

0

ds

p .e1 e3/ cos2 s C .e1 e2/ sin2 s

,

which, by (3.21) is equal to 2=M.p e1 e3, p e1 e2/.




In a similar way we see that the integral in (3.19) is equal toR e1

e2

dtp q.t/

which, in

turn, is equal to 2=M.p

e1 e3,p

e2 e3/.

Therefore, by (3.17) and (3.19) we conclude:

Case > 0. A fundamental pair of periods is .!1, !2/, where

!1 D

M.p

e1 e3,p

e1 e2/ (3.22)

!2 D i

M.p

e1 e3,p

e2 e3/ . (3.23)

Now, let < 0. Our model

E1 : y2

D 4x3

g2x g3

is 2-isogenous with the model

E2 : Y 2 D 4X 3 4.15e21 g2/X 2.7e1g2 C 11g3/, (3.24)

via the isogeny : E1 ! E2 defined by

.x, y/ 7! .X , Y / D

4x3 8e1x2 C .16e21 g2/x 3g3 2e1g2

4x2 8e1x C 4e21

,

8e

2

1 y g2y 4yx

2

C 8e1yx4x2 8e1x C 4e2

1

with dual isogeny O : E2 ! E1 defined by

.X , Y / 7! .x, y/ D

2X 3 C 8e1X 2 C 2.e21 C g2/X 3g3 C e1g2

8X 2 C 32e1X C 32e21

,

g2Y 7e21 Y YX 2 4e1YX

8X 2 C 32e1X C 32e21

.

The discriminant of E2 is equal to 2 D 18e21 g

22 4g

32 C27g

23 27e1g2g3 D 4.12e

21

g2/.3e21 g2/2. The discriminant of E1 is 1 D D .g3

2 27g22 /=16, hence 27g2

3 g2

2 > 0. Since g3 D 4e31g2e1, the last inequality becomes .3e2

1g2/.12e21g2/2 > 0.

Consequently, 3e21 g2 > 0 and a fortiori 12e2

1 g2 > 0. Thus 2 > 0. Moreover,

we see that the three real roots of the polynomial in the right-hand side of (3.24) are

e1 p

12e21 g2 < 2e1 < e1 C

p 12e2

1 g2 . (3.25)

Since 2 > 0, we conclude that E2 has a fundamental pair of periods .w1, w2/ with

w1 2 RC and w2 2 iRC. Therefore we calculate w1, w2 in a very efficient way

using the AGM algorithm in analogy with (3.22) and (3.23), but now with the roots




(3.25) in place of e3 < e2 < e1, so that, in place of e1 e3 we have 2p

12e21 g2 D

4p

3e21 C A, in place of e1 e2 we have 3e1 C

p 12e2

1 g2 D 3e1 C2p

3e21 C A, and

in place of e2 e3 we have 3e1 Cp

12e21 g2 D 3e1 C 2

p 3e2

1 C A. Therefore,

w1 D

2M

4

q 3e2

1 C A , 1

2

q 3e1 C 2

p 3e2

1 C A

(3.26)

w2 D i

2M

4

q 3e2

1 C A , 1

2

r 3e1 C 2

q 3e2

1 C A

! .

Now, how can we calculate a fundamental pair of periods for E1?

Let ƒ1, ƒ2 be the period lattices for E1 and E2 respectively. The fact that : E1

!E2 is a 2-isogeny implies that ƒ1 is a sublattice of ƒ2 D Zw1 C Zw2 of index 2. By

Section 3.4.2 we know that ƒ1 D Z1 CZ2 with 1 2 RC and 1 1

21 2 iRC.

Therefore 1

2

D V

w1

w2

,

where V is a 2 2 matrix with integer entries and determinant 2. We easily check that

every such matrix V is of the form V D UA, where U 2 SL2.Z/ and A is one of the

following matrices:

2 00 1 , 1 0

0 2 , 2 01 1 .

Therefore, without loss of generality, we may assume that .1, 2/ has one of the

following values:

.w2, 2w1/, .2w2, w1/, .2w1, w1 C w2/.

The first two cases are easily excluded in view of the conditions 1 2 RC and 1 1

21 2 iRC, so that the third case remains. Instead of .1, 2/ D .2w1, w1 C w2/

we prefer to take as a fundamental pair of periods .!

0

1, !

0

2/ D .2, 1 2/ D.w1 C w2, w1 w2/.

Now we relate !1, !2 with w1, w2. Since the pair .! 01, !0

2/ generates all periods, we

have !1 D m!01 C n! 0

2 D .m C n/w1 C .m n/w2 for some integers m, n. Since

!1 is a real period, we must have m D n and !1 D m.2w1/. But 2w1 D !01 C !0

2 is

a positive real period, while !1 is the least real period. Therefore, m D 1, !1 D 2w1

and, consequently, from (3.26) we obtain

!1 D

M 4q 3e21

CA ,

1

2r 3e1

C2q 3e2

1

CA!

. (3.27)



Section 3.5 : E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1 47

With completely analogous arguments we conclude that

!2 D i

M 4p 3e2

1 C A ,

1

2q 3e1 C 2p 3e2

1 C A. (3.28)

Since .!01, !0

2/ D .w1 C w2, w1 w2/ D ..!1 C !2/=2, .!1 !2/=2, we can take the

last pair as a fundamental pair of periods. Note that .!01 C!0

2, !01/ D .!1, .!1 C!2/=2

is also a fundamental pair of periods.

Case < 0. As a fundamental pair of periods we can take any of

.!1 C !2

2,

!1 !2

2/, .!1,

!1 C !2

2/,

where !1, !2 are given by (3.27) and (3.28), respectively.

3.5 : E.R/ 7! C=ƒ and l : E.R/ ! R=Z!1

We keep the notation of Theorem 3.4.1 and go back to the end of Section 3.3. We

will revisit Theorem 3.3.1 now focusing our interest exclusively on points of E.R/.

Our first task will be, given a point P 2 E.R/, to compute explicitly .P/; see

Theorem 3.5.1 below. Next, in Theorem 3.5.2 we will focus on the set E0.R/ of real

points .x , y/, such that x e1, showing that, actually, the restriction of to E0.R/establishes a group isomorphism between E0.R/ and R=Z!1. Finally, we will show

that, in the case of a positive discriminant, this last isomorphism can be extended to a

“two-to-one” group epimorphism E.R/ ! R=Z!1.

Note first that E0.R/ is indeed a subgroup of E.R/. For, if < 0, then E0.R/ DE.R/. If > 0, then E.R/ D E0.R/ [ E1.R/, where E1.R/ is the bounded set of

real points .x , y/, such that e3 x e2, i.e. the set of points on the “egg”, in pop

terminology. It is geometrically obvious that any line of the plane having a common

point with the “egg” must have “two” common points with it, where in “two” we also

include double points, in which case the line is tangent to the egg. Consequently, if

a line joins “two” points belonging to E0.R/ (with the meaning of “two” as above),

then the third point of intersection must also lie on E0.R/ and not on the “egg”; for

otherwise, the line would have four common points – multiplicities taken into account

– with the cubic model, which is absurd. From this we conclude that E0.R/ is closed

under addition of points and, consequently, E0.R/ is indeed a subgroup of E.R/,

hence a subgroup of E.C/ as well. Moreover, the sum of two points on the “egg” is a

point on E0.R/ and the sum of a point on the “egg” and a point on E0.R/ is a point

on the “egg”; in other words the group E.R/=E0.R/ is isomorphic to Z2.

First, let P D .x.P/, y.P// 2 E0.R/. From Tables 3.1 and 3.2 and the conclu-

sions of the relevant subsections we see that there exists a unique real number r in




the interval . 1

2!1,

1

2!1 such that .x.P /, y.P// D ..r/,

1

20.r//. We find now an

explicit formula for r . Consider the real integral

z D Z

C1

x.P/

dtp 4t 3 g2t g3 D 1

2 Z C1

x.P/

dtp t 3 C At C B.

According to the discussion following the integral formula (3.13) for .z/, with a

convenient " 2 ¹1, 1º, we have .x.P/, y.P// D .."z/, 1

20."z//. On the other

hand, since P 2 E0.R/, we have x.P / e1, therefore 0 < z 1

2!1, in view of

(3.17). From Tables 3.1 and 3.2 we see that 0.z/ 0, concluding thus that " D 1

or 1 according to whether y.P / 0 or y.P / 0, respectively. Therefore, in this

case,

r D 1

2"P Z

C1

x.P/

dt

p t 3

CAt

CB

, (3.29)

where "P D 1 or 1, according to whether y.P / > 0 or y.P / 0, respectively.

Next, let P D .x.P/, y.P// 2 E1.R/. Note that this can occur only in the case

of positive discriminant. In (3.7) we defined the points Qi D .ei , 0/ (i D 1, 2, 3).

Theorem 3.3.1 and (3.6) imply that .Q1/ D !1

2 , .Q2/ D !1C!2

2 and .Q3/ D !2

2 ,

therefore we may assume that P ¤ Q2, Q3. We refer to Section 3.4.1. From Table

3.1 and the conclusions of that section we see that there exists an r on the line segment

with end points !2!1

2 (excluded) and

!2C!1

2 (included), such that .x.P /, y.P// D

..r/, 1

20.r//. We will find this r .

Observe first that P C

Q2

2 E0.R/ and y.P

CQ2/ have the same sign as y.P /.

We have, in view also of our conclusion about points of E0.R/,

.P/ D ..P C Q2/ C Q2/ D .P C Q2/ C .Q2/

D 1

2"P

Z C1x.P CQ2/

dtp t 3 C At C B

C !1 C !2

2C ƒ,

where "P D 1 or 1, according to whether y.P / > 0 or y.P/ 0, respectively.

Therefore we are looking for integers m, n such that

r

D..P

CQ2/

CQ2/

D.P

CQ2/

C.Q2/

D 1

2"P

Z C1x.P CQ2/

dtp t 3 C At C B

C !1 C !2

2C m!1 C n!2

belongs to the line segment with end points !2!1

2 and

!2C!1

2 as mentioned above.

This condition is equivalent to !1

2 <.r/ !1

2 and i=.r/ D !2

2 . The second condition

immediately implies n D 0 and the first condition implies that m D 0 if "P D 1 and

m D 1 if "P D 1. Therefore,

r D

1

2"P Z

C1

x.P CQ2/

dt

p t 3 C At C B C !2 C "P !1

2. (3.30)




Note that, if we adopt the convention x .O/ D C1, then formula (3.30) with P DQ2, gives r D !2!1

2 D !2C!1

2 .mod ƒ/. Also, with P D Q3 we similarly obtain

x.Q3 C Q2/ D x.Q1/ D e1 and then formula (3.30), in combination with (3.18),

gives r D

!2

2 . Thus, formula (3.30) is valid for all points on P

2E

1.R/.

Our previous discussion leads to the following theorem:

Theorem 3.5.1. Let E be the short Weierstrass model

E : y2 D x3 C Ax C B, A, B 2 R, D 16.4A3 C 27B 2/ ¤ 0,

let be the Weierstrass function with parameters g2 D 4A and g3 D 4B

(cf. Fact 3.1.2) and let !1, !2 be as in Theorem 3.4.1 , so that the period lattice of is ƒ D Z!1 C Z!2 or ƒ D Z!1 C Z

!1C!2

2 according to whether the discrimi-

nant of E is positive or negative, respectively. In the case that > 0 we also set

Q2 D .e2, 0/. Then the isomorphism of Theorem 3.3.1 , restricted to the subgroupE.R/ of E.C/ , is the monomorphism : E.R/ ! C=ƒ , given by

E.R/ 3 P D .x.P/, y.P// 7! r C ƒ 2 C=ƒ,

where r is as follows: if P 2 E0.R/ then r is given by (3.29) and if P 2 E1.R/ (whichcan happen only if > 0) then r is given by (3.30) , where in both cases, "P D 1 if y.P / > 0 and "P D 1 if y.P / 0.

Finally, by Theorem 3.3.1 , .x.P /, y.P// D ..r/, 1

20.r//.

The isomorphism E.R/

$ C=ƒ that is guaranteed by Theorem 3.5.1 is certainlyimportant. However, from this book’s practical point of view, it has the following dis-

advantage: If P 2 E1.R/, then .P/ is not a real number, as !2 makes its appearance.

This would imply one more unknown integer involved in the linear form (4.4) that we

will meet in Chapter 4.4 That linear form is the basic tool for Ellog and, as we will

see in Chapter 9, where we will obtain an upper bound for M , this bound depends

exponentially on the number of the integer unknowns involved in the linear form. A

“slight modification” of can rid us of this disadvantage at the cost of losing the “one-

to-one” property in the case > 0 (only). More specifically, in this section we will

define a “two-to-one” epimorphism E.R/ l

$R=Z!1 with which we will work in the

following chapters; as it will turn out, actually, the linear form mentioned above is a

linear form in l-values of points of E .

Theorem 3.5.2. We keep the notation and assumptions of Theorem 3.5.1. In partic-ular, for any P 2 E.R/ we consider the complex number r , such that P and r arerelated as described in the above theorem.

We define the map l0 : E0.R/ ! R=Z!1 by setting

l0.P / D r CZ!1,

4 More precisely, instead of m0!1 (see (4.4)) we would have m0!1 C m00!2.




and the map l : E.R/ ! R=Z!1 as follows:

l.P /

D ´l0.P / if P 2 E0.R/

l0.P C Q2/ if P 2 E1.R/.

(3.31)

(a) The map l0 is a group isomorphism.

(b) The map l is a group epimorphism; moreover, the relation l.P 1/ D l.P 2/ impliesthat either P 2 D P 1 or P 2 D P 1 C Q2.

Next, we identify R=Z!1 with the interval I D .!1

2 ,

!1

2 , so that the sum of two

numbers in I is the usual sum of real numbers plus an appropriate integral multiple of !1 in order that the result falls in I. Then, for every P 2 E.R/ we will view l.P / as anumber of I.

(c) If K R is a number field, P 1, : : : , P k 2 E.K/ are Z-linearly independent points of infinite order and T 2 E.K/ is torsion point, then, a relation n1l.P 1/C C nk l.P k/ C l.T / C n0!1 D 0 , where n0, n1, : : : , nk 2 Z is possible only if n1 D : : : D nk D 0.

Proof. Two preliminary observations: First, if P 2 E.R/, then l0.P / C ƒ D .P /.

Indeed, let .P / D r C ƒ. By the definition of l0, we have l0.P / D r C Z!1,

therefore, l0.P / C ƒ D r CZ!1 C ƒ D r C ƒ D .P/.

Second, if P , P 0 2 E.R/ and l0.P / D l0.P 0/, then P D P 0. Indeed, in view of

the previous observation, we have .P / D l0.P / D l0.P

0

/ D .P

0

/ and, since isone-to-one, it follows that P D P 0.

Proof of (a). Using the first preliminary observation and the fact that is a homo-

morphism, we show that l0 is a group homomorphism, as follows. Let P 1, P 2 2 E0.R/.

Then,

l0.P 1 C P 2/ C ƒ D .P 1 C P 2/ C ƒ D .P 1/ C .P 2/ C ƒ

D ..P 1/ C ƒ/ C ..P 2/ C ƒ/

D .l0.P 1/ C ƒ/ C .l0.P 2/ C ƒ/.

Let .P i / D ri C ƒ (i D 1, 2) and .P 1 C P 2/ D r C ƒ, where r1, r2 and r are

defined according to Theorem 3.5.1; by that theorem, r1, r2, r 2 R. On the other hand,

l0.P i / D ri C Z!1 (i D 1, 2) and l0.P 1 C P 2/ D r C Z!1, therefore the above

displayed relation implies that r r1 r2 2 ƒ D Z!1 C Z!02, where ! 0

2 D !2 if

> 0 and ! 02 D .!1 C !2/=2 if < 0. Let us write r r1 r2 D m!1 C n! 0

2,

where m, n 2 Z. The fact that the left-hand side is real and !2 2 iR forces n D 0 and

r r1 r2 D m!1 2 Z!1, which is equivalent to l0.P 1 C P 2/ D l0.P 1/ C l0.P 2/.

We show that l0 is one-to-one. If P 2 E0.R/ and l0.P / D 0 C !1Z, then, by the

first preliminary observation, .P / D 0 C ƒ. Since is one-to-one, it follows that

P D O.




Finally, is also onto, for, if r is a real number and we set P D ..r/, 1

20.r// (if

r 2 ƒ we mean, of course, that P D O) then, by Theorem 3.5.1, .P / D r C ƒ and,

consequently, by the definition of l0, we have l0.P / D r CZ!1.

Proof of (b). The map l is a group epimorphism because, on the one hand, its re-

striction to E0.R/ is, in view of (a), a group isomorphism and, on the other hand,

the group E.R/=E0.R/ is isomorphic to Z2. For example, if P 1, P 2 2 E1.R/ then

P 1 C P 2 2 E0.R/ and, therefore, l.P 1 C P 2/ D l0.P 1 C P 2/. On the other hand,

l.P 1/Cl.P 2/ D l0.P 1CQ2/Cl0.P 2CQ2/ D l0.P 1CP 2C2Q2/ D l0.P 1CP 2CO/ Dl.P 1 C P 2/, as required.

Finally, let P 1, P 2 2 E.R/ with l.P 1/ D l.P 2/. If P 1, P 2 2 E0.R/, then l0.P 1/ Dl.P 1/ D l.P 2/ D l0.P 2/, therefore, by the second preliminary observation, P 1 D P 2.

Next, suppose that > 0. If P 1, P 2 2 E1.R/, then, P i CQ2 2 E0.R/ for i D 1, 2, so

that l0.P 1

CQ2/

Dl.P 1/

Dl.P 2/

Dl0.P 2

CQ2/ and, as before, P 1

CQ2

DP 2

CQ2,

from which P 1 D P 2. If exactly one among P 1, P 2, say the second, belongs to E1.R/,then l0.P 1/ D l.P 1/ D l.P 2/ D l0.P 2 C Q2/, hence P 1 D P 2 C Q2, which is

equivalent to P 2 D P 1 C Q2.

Proof of (c). Let n1l.P 1/ C C nkl.P k/ C l.T / C n0!1 D 0. Since l is a homo-

morphism, we have

l.n1P 1 C C nkP k C T / D n1l.P 1/ C C nkl.P k/ C l.T / C `!1,

for an appropriate integer ` which makes the left-hand side falling in the interval I D.1

2!1 ,

1

2!1. In view of the hypothesis, the right-hand side of the last equality is

equal to .` n0/!1; since the left-hand side belongs to I , this forces ` D n0 andl.n1P 1 C C nk P k C T / D 0. Then, in view of (b), n1P 1 C C nkP k C T is

equal either to O or to Q2. The second alternative is impossible if Q2 62 E.K/; and if

Q2 2 E.K/, both alternatives imply that n1P 1C Cnk P k is a torsion point of E.K/,

which contradicts the hypothesis that P 1, : : : , P k are Z-linearly independent.

Conclusions and remarks

(1) In all subsequent applications of this book we will identify R=Z!1 with the in-

terval I D .!1

2 , !1

2 , as explained in the announcement of Theorem 3.5.2. Notethat, with this convention, l.P / is identified with the elliptic logarithm of P or of

P C Q2, depending on whether P belongs to E0.R/ or to E1.R/, respectively.

Moreover, the discussion of this section which led us to (3.31) along with Theorem

3.5.1 immediately imply the following expression of l.P / as an integral:

l.P / D "P

2

Z C1xP

dtp f.t/

, xP D´

x.P/ if P 2 E0.R/

x.P C Q2/ if P 2 E1.R/(3.32)

where, as before, "P D 1 if y.P / > 0 and "P D 1 if y.P / 0 (note that y.P /

and y.P C Q2/ have equal signs).




(2) Let f .X/ have rational coefficients. Since Q2 D .e2, 0/ and e2 is a root of f .X/,

we see that Q2 has coordinates in an extension of Q of degree at most three. There-

fore, in view also of (1), above, we conclude that: If P 2 E0.Q/ , then l.P / isequal to the elliptic logarithm of a point (this is P ) with rational coordinates. If P 2 E1.Q/ , then l.P / is equal to the elliptic logarithm of a point (this is P CQ2)with coordinates belonging to a number field of degree at most three.

(3) We defined the elliptic logarithm by means of a short Weierstrass model. In the

chapters that follow, we often work with two different models of the same elliptic

curve E , namely, a short Weierstrass model E and another model, say C . There-

fore, in accordance with Section 1.1, any abstract point P of E has a representative

P E on E and a representative P C on C , the coordinates of which satisfy the equa-

tions of E and C , respectively. Having fixed the short Weierstrass model E , we

will always understand as elliptic logarithm of a certain point P of E the ellip-

tic logarithm of P E . Therefore we will always omit the superscript E from thearguments of l and will write l.P /, instead of l.P E /.

Practical computation of l.P/. Obviously, it suffices to compute l0.P / for any point

P 2 E0.R/. We present here an algorithm due to D. Zagier [74].

Let l.P / D r.P/ C Z!1, where r.P / D r is given by (3.29) if P ¤ O and

r.P/ D 0 if P D O. In view of (3.18) and (3.29), r.P / 2 .!1

2 ,

!1

2 . We set now

s.P/ D 8<:0 if P D O

r.P/

!1if P

¤0 and y.P /

0

r.P/

!1C 1 if P ¤ 0 and y.P / > 0

.

Clearly, s.P / 2 .0, 1

2/ if y.P / < 0; s.P / D 1

2 if y.P / D 0 and s.P / 2 . 1

2, 1/ if

y.P / > 0. Further, s.P / D l0.P /!1

.mod Z/. Since l0 is a homomorphism, if P 1, P 2 2E0.R/, then s.P 1CP 2/ D s.P 1/Cs.P 2/Ck, where k 2 ¹1,0,1º and, consequently,

for P 2 E0.R/ and n 2 Z, s.n P / D n s.P / C an with an 2 Z. In order to

compute l.P / it suffices to compute s.P/. If P ¤ Owe write s.P/ as a binary number

s.P/ D d 12 C d 2

22 C d 323 C , where d i 2 ¹0, 1º for all i ’s. The digits d 1, d 2, d 3, : : : are

computed successively as follows.

If y.P / D 0 then d 1 D 1, d i D 0 for every i > 1 and the calculation of s.P / is

complete. If y.P / < 0 then 0 < s.P/ < 1

2, forcing d 1 D 0; and if y.P / > 0 then

s.P / > 1

2, hence d 1 D 1.

Suppose that, for some i 2 we have already computed d 1, : : : , d i1. We have

s.2i1 P / D 2i1s.P/ C k with k 2 Z, hence s.2i1 P / D ai1 C d i2 C d iC1

22 C , where ai1 2 Z. If 2i1 P D O, then s.2i1 P / D 0 and this can happen

only if ai1 D 1 and d j D 1 for all j i . Now suppose that 2i1 P ¤ O.

If y.2i1 P / D 0 then s.2i1 P / D 1

2, implying ai1 D 0, d i D 1, d j D 0

for every j > i and the calculation of s.P/ is finished. If y.2

i1

P / < 0 then




0 < ai1 C d i2 C d iC1

22 C < 1

2, forcing ai1 D 0 and d i D 0. Analogously, if

y.2i1 P / > 0 then 1

2 < ai1 C d i

2 C d iC1

22 C < 1 which can occur only if

ai1 D 0 and d i D 1.

The above discussion suggests the following algorithm:

13 7 Computing the elliptic logarithm

DESCRIPTION: Compute the elliptic logarithm l.P / D l0.P / of P 2 E0.R/.

INPUT: P 2 E0.R/, N D number of desired binary digits for l.P /.

OUTPUT: l.P /.

INITIAL VALUES: u 0, s 0, Q P , i 1.

while i N do

if Q DO then goto ()

else

if y.Q/ D 0 then s s C 1

2i : goto ()

else

if y.Q/ > 0 then s s C 1

2i end if

end if

end if

if i < N then Q D 2 Q endif

i i C 1

end while

() u D !1s.

if P

¤O and y.P/ > 0 then u

u

!1 end if

l.P / D u.END



Chapter 4

The elliptic logarithm method

In this chapter we make a general description of the elliptic logarithm method , briefly

Ellog, which will be used in later chapters for solving various types of elliptic equa-tions in integers. In this chapter we will refer to the resolution of any such equation as

the Diophantine problem.

To each Diophantine problem, we will attach a convenient short Weierstrass model

E : y2

D x3

C Ax C B , A, B 2 Q; (4.1)

this is done separately for each problem in Chapters 5, 6, 7, 8 and 11, respectively.

We denote by r the rank of the elliptic curve E .Q/. We also denote by r0 the least

common multiple of the orders of the non-zero points of Etors.Q/; by Mazur’s theorem

(see [29, 30], or [45, Theorem 7.5]), 1 r0 12, r0 ¤ 11.

We refer to Theorem 1.2.1, which we apply for K D Q. According to that theorem,

if r > 0, then there exist points of infinite order P E1 , : : : , P Er in E.Q/, with the

property that every point P E

D .x.P/, y.P//

2 E.Q/ is written uniquely in the

formP E D m1P E1 C C mr P Er C T E (4.2)

for some integers m1, : : : , mr and T E D .x.T /, y.T // 2 Etors.Q/.

Thus, (4.2) establishes a map P E 11! .m, T E /, where m D .m1, : : : , mr / is a

lattice point in Rr and T E 2 Etors.Q/.

A first important feature of Ellog is that the solutions of the Diophantine problems

in Chapters 5, 6, 7, 8 and 11 are associated in some way, particular to each Diophantine

problem, to points P E $ .m, T E / with m belonging to a (finite) cube of Rr ; that is,

if P E is the point that “covers” (comes from) a solution of the Diophantine problem

and P E is expressed as in (4.2), then M def D max1ir jmi j is bounded. Our task will

be to compute an explicit upper bound for M ; a huge one first, and then with the use

of it, a bound of “manageable” size.

A second important feature of Ellog is that, given any point P E 2 E.Q/, one can

easily (in principle) check whether it “covers” an actual solution of the Diophantine

problem. Therefore, if an explicit upper bound, say K , of M is known, then, for each

choice K mi K , i D 1, : : : , r and T E 2 Etors.Q/, a point P E 2 E.Q/ is

calculated by means of (4.2) and then, this P E is checked whether it comes from a

solution of the Diophantine problem. In particular, one can explicitly find the set T

consisting of all points T of E , such that T E

2 Etors.Q/ and T covers a solution of



Chapter 4 The elliptic logarithm method 55

the Diophantine problem. As the calculation of the set T is a rather trivial task, we

will henceforth assume that, once we are given a certain Diophantine problem, T has

been computed already from the beginning and we are left with the difficult task of

explicitly computing a finite set P of points of E with the following properties: Each

solution of the Diophantine problem not covered by points of T , is covered by some

point of P , and P E 62 Etors.Q/ for every P 2 P . The last condition is, of course,

equivalent to

r 1 & .m1, : : : , mr / ¤ .0, : : : , 0/. (4.3)

In this way we will effectively solve the Diophantine problem under consideration.

Here, “effectively” means “after finitely many steps”, no matter how many they are.

Take a very modest typical example, in which r D 2, the torsion points are two and,

by some means we calculate K D 1030, which is rather “very small”. Then the num-

ber of checks will be, approximately, 23

1060. Of course, there is no hope to solve

explicitly our Diophantine problem this way, but, at least, we have an effective or, if

you prefer, a constructive proof of the fact that there exist finitely many solutions.

Certainly, this is far more satisfactory for our mind than an existential (in other words,

non-constructive) confirmation which would merely assure the existence of finitely

many integer solutions. At the same time, an effective proof of the finiteness of so-

lutions suggests, at least in theory, a method of resolution of the problem. We will

show that one can exploit this effective method to finally obtain an explicit method of

resolution of the Diophantine problem, i.e. a method whose output will be a finite set

S consisting precisely of all solutions of the Diophantine problem.

Below we make a very general description of how an upper bound for M is ob-tained. We recommend the reader to revisit very quickly Chapter 1 in order to recall

the terminology, conventions and facts that will be used. In accordance with them:

We will view E as a model of an elliptic curve. Referring to (4.1), we put g2 D4A, g3 D 4B and we adopt the notations and assumptions of Theorem 3.4.1. In

particular, q.X/ D 4X 3 g2X g3 and e1, e2, e3 denote the roots of q.X/, where

e1 > e2 > e3 if all three roots are real; otherwise e1 is the only real root. Further, we

put f .X/ D q.X/=4 D X 3 C AX C B .

In order that the group relation (4.2) provide us with a numerical relation, we applythe group epimorphism l : E.R/ ! R=Z!1 of Theorem 3.5.2. We remind here that,

according to this theorem, we view the group R=Z!1 as the interval .!1

2 , !1

2 in

which the group operation is the usual addition “adjusted” by an integral multiple of

!1, so that the final sum falls in the above interval.

The group relation (4.2) implies the following numerical relation1

0 ¤ l.P / D m1l.P 1/ C C mr l.P r / C l.T / C m0 !1, (4.4)

1 Below we omit the indication E from the points when they are an argument of the elliptic logarithm

function l; see Conclusions and remarks 3, page 52.



56 Chapter 4 The elliptic logarithm method

where m0 !1 is the “adjusting summand” mentioned a few lines above. This is a linear form in elliptic logarithms of points with algebraic coordinates of degree at most three;

see (2) in Conclusions and remarks, page 52. The non-vanishing of l.P / is justified

by (4.3) and Theorem 3.5.2 (c).

If t 2 is the (group) order of a torsion point T E ¤ O, then t l.T / D l.t T / Dl.O/ D 0 2 R=Z!1, which shows that l.T / D s

t!1 with 1

2 < s

t 1

2. Thus, in

general,

l.T / D s

t , t jr0, t > 0, jsj r0

2, 1

2<

s

t 1

2. (4.5)

Therefore, we can rewrite (4.4) as

0 ¤ l.P / D

m0 C s

t

!1 C m1l.P 1/ C C mr l.P r /. (4.6)

Since every l-value is absolutely less than !1

2

, it follows from (4.6) and (4.5) that

jm0j 1 C .jm1j C C jmr j/=2. Therefore, if we set

M D max1ir

jmi j, M 0 D max0ir

jmi j (4.7)

then

M 0 max®

M , 1

2rM C 1

¯. (4.8)

Closely related to l.P / is a linear form L.P /, which we will define for each partic-

ular case in Chapters 5, 6, 7, 8 and 11. In each case we will compute an explicit upper

bound for

jL.P/

jin terms only of M ; denote it (temporarily) by Ub.M /. On the other

hand, the application of Theorem 9.1.2 to L.P / will furnish either an explicit upperbound for M itself, and we are done, or an explicit lower bound for jL.P/j in terms

only of M , which we will temporarily denote by Lb.M /. In the second case we will

thus arrive to

Lb.M / jL.P/j Ub.M /.

Both functions Lb.x/ and Ub.x/ are decreasing for x positive and sufficiently large.

What is very important is that there exists an explicit positive constant K such that

Lb.x/ > Ub.x/ for every x > K . This, combined with the last displayed relation,

clearly implies that M

K , and we obtain the required upper bound for M .

Remark on terminology. In the chapters that follow, the term “constant” always

means “number depending on the particular Diophantine problem”. Once the Dio-

phantine problem is expressed in concrete numerical terms, the “constant” has a spe-

cific numerical value.

In the constants of the form ci , which are introduced in this book, we use subscripts

i 7 in order to avoid confusion with the standard symbols c4 and c6, that are used

in the general theory of elliptic curves (see [45, p. 42]).



Chapter 5

Linear form for the Weierstrass equation

In this chapter we study the Weierstrass model of an elliptic curve

C : g.u, v/ D 0 ; g.X , Y / D Y 2Ca1XY Ca3Y .X 3Ca2X 2Ca4X Ca6/, (5.1)

where a1, a2, a3, a4, a6 are rational integers. According to our discussion in Chapter

4, to the generic point P C D .u.P /, v.P// with integer coordinates we will attach

a linear form in elliptic logarithms L.P / (cf. page 56 immediately after (4.8)). Our

purpose is to prove Theorem 5.2, which gives an explicit upper bound for jL.P/j.This will be the first main step towards the explicit determination of all points P C as

above. The presentation in this chapter is based on joint work of R. Stroeker and the

author [54].

Throughout the chapter we keep the notations, assumptions and results of Chapter 4.

For the basic background of this chapter we refer, explicitly or not, to [45, Chapter III].

As mentioned in Section 1.2, since C is a model of an elliptic curve, its discriminant

C is non-zero. In the special important cases a1 D a2 D a3 D 0 and a4 D A, a6 DB , the discriminant is C D 16.4A3 C 27B2/ ¤ 0 and we have seen in Section 3.1

how crucial the non-vanishing of this discriminant is.A linear substitution

u D 2x C , v D 3y C 2x C , (5.2)

with conveniently chosen rationals ¤ 0, , , , makes the model (5.1) birationally

equivalent to a short Weierstrass model E defined by (4.1). The two discriminants,

of the model C , and E D 16.4A3 C 27B2/ of the model E are related by D1612.4A3 C 27B 2/. For example, a possible choice for , , , is . , , , / D.1, 1

12a2

1 1

3a2, 1

2a1,

1

24a3

1 C 1

6a1a2 1

2a3/. Thus, C and E are models of the

same elliptic curve, say E .A basic fact is that, given a point P 2 E with u.P / 2 Z, sufficiently large, then its

canonical height Oh.P/ is “close” to the half of the logarithm of x.P/. This is expressed

by the following proposition:

Proposition 5.1. Let , be as in (5.2) and let ı be the least common multiple of thedenominators of 2 and = 2. Then, for any P 2 E with u.P / an integer 2 C ,

Oh.P/ 1

2 log x.P/ C 1

2 log ı,

where is defined by (2.41).



58 Chapter 5 Linear form for the Weierstrass equation

Remark. The proposition requires that the integer u.P / is at least 2 C . This is not

a serious restriction in practice, because, as is easily seen, the points P with u.P / an

integer < 2 C are finitely many and can be explicitly computed very easily.

Proof. We have x.P/ D .u.P //=2 1, hence, by the definition of ı, ı x.P/ D.ı=2/ u.P / ı.=2/ is an integer. Therefore, if we write x.P / D a=b, where

a, b are relatively prime positive integers, then bjı and, in particular, b ı. Now,

applying (2.21), we have h.x.P // D log max¹a, bº D log a, because x.P / 1. But

0 < a ı .a=b/ D ı x.P/, therefore,

h.x.P // log ı C log x.P/.

Now we apply Proposition 2.6.3 with C in place of D and .u, v/ in place of .x1, y1/.

By (2.42) and the above relation we have

Oh.P/ 1

2 h.x.P // Oh.P/ 1

2 log x.P/ 1

2 log ı,

as claimed.

The main result of this section is the following.

Theorem 5.2. Let E be the elliptic curve represented by the model C in (5.1). Let E : y2 D f.x/ be the short Weierstrass model (4.1) which is obtained from C bymeans of the transformation (5.2) , so that everything in Chapter 4 , especially relations(4.4) through (4.8) , refers to this particular model E .

Assume that P

2E is such that P C

D.u.P /, v.P// has integer coordinates with

u.P / > 22 max¹1

2, je1j, je2j, je3jº C ,

where, as usual, e1, e2, e3 are the roots of f .X/.Write P E as in (4.2). With l.P / as in (4.6) , set L.P / D l.P /. Then, with M , ,

and ı defined, respectively, by the first relation (4.7) and by Propositions 2.6.2 , 2.6.3

and 5.1 , we have

jL.P/j 2p

2 exp. C 1

2 log ı M 2/. (5.3)

Proof. First we observe that all x 2 max¹je1j, je2j, je3jº satisfy the inequality

0 < Z 1x

dtp f.t/

4p 2jxj1=2. (5.4)

Indeed, for such an x and t x we have 0 < f.t/ D jt e1jjt e2jjt e3j. As t is

larger than the absolutely largest root of the polynomial f.X/, it follows that jt e1j t je1j t =2, and likewise for e2 and e3. Consequently, 1=

p f.t/ 23=2t3=2 and

hence, for all N > x,

0 <

Z N

x

dt

p f.t/

Z N

x23=2t3=2dt D 4

p 2.x1=2 N 1=2/.

Letting N tend to infinity we obtain (5.4).



Chapter 5 Linear form for the Weierstrass equation 59

Now, in view of (5.2) and the hypothesis about the lower bound of u.P / we have

x.P / > 2 max¹je1j, je2j, je3jº, therefore, by (5.4),

0 < Z 1

x.P/

dtp f.t/ 4p 2 x.P/

1=2

.

But, x.P / e1, therefore P E 2 E0.R/ and consequently, by the definition of l (see

(3.32)), the left-hand side in the above displayed relation is equal to 2jl.P /j, so that

jl.P /j 2p

2 x.P/1=2. By Propositions 5.1 and 2.6.2,

log¹x.P/

1

2 º C 1

2 log ı Oh.P/ C 1

2 log ı M 2,

which completes the proof.

To continue with the resolution of the Weierstrass equation, one can safely avoid Chapters 6, 7 and 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.3.



Chapter 6

Linear form for the quartic equation

In this chapter we will deal with the integer solutions of Diophantine equations

F. U/ D V 2, where F denotes a quartic polynomial with rational coefficients and

the equation F.x/ D y2 defines an elliptic curve over Q. This implies, first, that the

genus of the corresponding curve is one, which is equivalent to the non-singularity

of the curve, hence to the non-vanishing of the discriminant of the polynomial F.X/,

and second, that we know a rational solution .U , V / D .u0, v0/. In practice, the search

for such a solution can be done by some powerful computational tool like ratpoints[53] due to M. Stoll [52], or the routine Points of MAGMA; of course, although it is

highly probable that the search will be successful, there is no guarantee for this. Any-

way, assuming this, we then reduce our equation to solving in integers the equation

Q.u/ D v 2, where Q.u/ D F .u C u0/, .u, v/ D .U u0, V / and the constant term

of the polynomial Q is a square of an integer. Therefore, we will consider the elliptic

curve E , a model of which is on the quartic elliptic model

C : g.u, v/

D0, g.u, v/

Dv2

Q.u/, (6.1)

Q.u/ D au4 C bu3 C cu2 C du C e2, a, b, c, d , e 2 Q , a, e > 0 , discr.Q/ ¤ 0.

(6.2)

Before proceeding we note that we may assume that a is not a perfect square, for, in

this case, the equation Q.u/ D v2 in integers can be solved very efficiently by totally

elementary means; see [37].

According to our discussion in Chapter 4, to P C D .u.P /, v.P//, the generic point

on C with integer coordinates, we will attach a linear form in elliptic logarithms L.P/;

see page 56 immediately after (4.8). We intend to prove Theorem 6.8, which gives an

explicit upper bound for jL.P/j, as a first main step towards the explicit determination

of all points P C as above. Our presentation is based on the author’s paper [62].

Throughout the chapter we keep the notations, assumptions and results of Chapter

4. Wherever the symbolp

is used, always, care is taken that the radicand be non-

negative and thenp

denotes the non-negative square root of the radicand.

The short Weierstrass model (4.1) with

A D 1

3 c2

C bd 4ae2

, B D 2

27 c3

1

3 bcd 8

3 ace2

C b2

e2

C ad 2

(6.3)



Chapter 6 Linear form for the quartic equation 61

is birationally equivalent to (6.2) and the birational transformations (1.3) and (1.4)

between the two models (cf. Fact 1.1.2) are the following:

x

DX .u, v/

D 6ev C cu2 C 3du C 6e2

3u2

,

(6.4)

y DY .u, v/ D ebu3 C 2ecu2 C 3ed u C duv C 4e2v C 4e3

u3

u D U .x, y/ D .12e2x C 8ce2 3d 2/=.6y e 3dx C cd 6be2/ (6.5)

v D V .x, y/ D1

3.32c3e3 108b2e5 C 108bcde3 C 27 d 3y C 216 be4y

108 de2cy C 216e3x2c 81ex2d 2 C 9ec 2d 2 C 216e3x3

27bed 3 108e3y2/=.6y e 3dx C cd 6be2/2. (6.6)

In the above relations, D ˙1; for both D C1 and D 1 a valid birational

transformation is obtained. However, for the needs of this chapter it will be useful to

chose the value of according to the following lemma.

Lemma 6.1. At least one of the two numbers

d p

a C eb, 8e3p

a C 4e2c d 2

is non-zero, hence we define

D ´sgn.d p a C eb/ if d p a C eb ¤ 0

sgn.8e3p

a C 4e2c d 2/ if d p

a C eb D 0.

For u such that Q.u/ > 0 , let

R.u/ D beu3 C 2ceu2 C 3deu C 4e3 C .4e2 C du/p

Q.u/. (6.7)

Then, for sufficiently large u > 0 we have Q.u/ > 0 and sgn.R.u// D .

Proof. If d p

a C eb D 0 and 8e3p

a C 4e2c d 2 D 0, then b D d p

a=e and

c

D 2e

p a

Cd 2=.4e2/. Consequently, Q.u/

D .

p au2

.d=2e/u

e/2, which

contradicts the hypothesis discr.Q/ ¤ 0. Therefore, at least one of d p a C eb D 0and 8e3

p a C 4e2c d 2 D 0 is non-zero.

Suppose first that d p

a C be ¤ 0. Then limu!1R.u/=u3 D d p

a C be , which

proves our claim in this case. Next, let d p

a C be D 0. Then

R.u/

u2 D .be C d

r a C b

u C /u C .2ce C 4e2

r a C b

u C / C 3de

u C 4e3

u2 .

As u tends to infinity the right-hand side tends to bd=.2p

a/C.2ce C4e2p

a/C0C0,

which, in view of d p

a C be D 0, is equal to .8e3p

a C 4e2c d 2/=.2e/, hence of

the same sign as .



62 Chapter 6 Linear form for the quartic equation

Now, let

x0 D 2ep

a C c

3. (6.8)

Following the notation in Chapters 3 and 4, we set f .X/

D X 3

C AX

C B and

denote by e1, e2, e3 the roots of f.X/, enumerating them as explained on page 55. Thefollowing proposition describes the position of x0 relatively to e1, e2, e3.

Lemma 6.2. If d p

a C eb ¤ 0 , then either x0 > e1 or e3 < x0 < e2. If d p

a Ceb D 0 , then x0 D e1 and D C1.

Proof. It is easy to check that f .x0/ D x30 C Ax0 C B D .d

p a C be/2.

If d p

a C be ¤ 0, then f .x0/ > 0 and, consequently, either x0 > e1, or (in the

case that e2, e3 2 R) e3 < x0 < e2.

If d p

a

Ceb

D 0, then x0 is a root of f .X/ and we set f .X/

D .X

x0/h.X/,

where h.X/ D X 2 C x0X C A C x20 . In particular, e2, e3 are the roots of h.X/. But

h.x0/ D 3x20 C A D

p a .8e3

p a C 4ce 2 d 2/=e; for the last equality we used

b D d p

a=e. By Lemma 6.1, 8e3p

a C 4ce 2 d 2 ¤ 0, hence x0 ¤ e2, e3 and,

consequently, x0 D e1. Moreover, if e2, e3 62 R, then h.x0/ > 0; and if e2, e3 2 R,

then x0 > e2, so that again h.x0/ > 0. Hence, D sgn.8e2a C 4cep

a C bd / Dsgn.h.x0// D C1.

Lemma 6.3. Let u0 1 be such that for u u0 the conclusion of Lemma 6.1 holds.For u

u0 define

x.u/ D cu2 C 3du C 6e2 C 6ep

Q.u/

3u2 . (6.9)

Then, in the interval .u0, C1/ , x is strictly increasing or strictly decreasing, accord-ing to whether D 1 or C1 , respectively. Moreover, limu!C1 x.u/ D x0.

Proof. For the first claim, just observe that dx.u/

du D R.u/

u3p

Q.u/and combine this with

Lemma 6.1. The claim about the limit is obvious.

Now we are in a position to prove a proposition which subsequently will help us to

express the integral R C1udwp Q.w/

as a linear forms in elliptic logarithms.

Proposition 6.4. Let u0 be as in Lemma 6.3. Then, a constant u u0 exists suchthat, for u u we haveZ C1

u

dwp Q.w/

D

Z x.u/

x0

dtp f.t/

. (6.10)

Proof. It is an elementary exercise to see with the aid of Lemmas 6.3 and 6.2 that, for

u u0, the following cases are possible:




d p

a C b e ¤ 0 and x0 > e1. Then, x is strictly increasing when D 1 and

strictly decreasing when D C1, with limiting value (in both cases) x0, as u !C1. Hence, for an appropriate u u0, it is indeed true that u u ) x.u/ 2.x

0,C1

/

.e1

,C1

/.

d p

a C be D 0 and x0 D e1. Then, necessarily, D C1, hence x is strictly

decreasing with limiting value x0, as u ! C1. Same conclusion as in the previous

“bullet”.

d p

a C be ¤ 0 and x0 2 .e3, e2/. Then x is strictly decreasing when D C1 and

strictly decreasing when D 1 with limiting value (in both cases) x0 2 .e3, e2/, as

u ! C1. Therefore, for an appropriate u u0, it is true that u u ) x.u/ 2.x0, e2/ .e3, e2/ if D C1 and x.u/ 2 .e3, x0/ .e3, e2/ if D 1; hence, in

both cases, x.u/ 2 .e3, e2/.

Let u be as in the above “bullets”. Then, for u u0, the interval .u, C1/ ismapped by x onto the open interval with end points x.u/ and x0, which is a subset

of .e1, C1/ in the case of the first two “bullets” and a subset of .e3, e2/ in the case

of the last one. Therefore, in the right-hand side of (6.10) we can make the change of

variable t D x.w/ D .6ez C 3t d C ct 26e2/=.3t 2/, z Dp

Q.w/,

so that

dt D .6e dz C 3d dw C 2cw dw/w 2.6ez C 3dw C cw 2 C 6e2/ dw

3w3 .

By z2

DQ.w/ we have dz

D.4aw3

C3bw2

C2cw

Cd/=z

dw and, on replacing

for dz in the last displayed equation, we obtain after some elementary calculations,

dt D bew3 C 2cew2 C 3dew C dwz C 4e3 C 4e2z

w3 dw

z D R.w/

w3 dw

z ,

where R is defined in (6.7). A symbolic computation shows that .R.w/=w3/2 Df .x.w// D f.t/ and, since w u0, we know that sgnR.w/ D . Therefore,

R.w/=w3 D p

f.t/ and the last displayed equation becomes

dt

p f.t/D dw

z D dw

p Q.w/,

which implies (6.10).

Proposition 6.5. Let u be as in Proposition 6.4. Let P be a point of E , such that P C has real coordinates .u.P /, v.P// with u.P / u and v.P / 0. Then, for thecoordinates .x.P /, y.P// of P E we have x.P / D x.u.P // and y.P / 0.

Proof. Let us write for simplicity, u, v, x, y instead of u.P /, v.P/, x.P/, y.p/. Cer-

tainly, v Dp

Q.u/ and, by (6.4), we have

x D 6ev

Ccu2

C3du

C6e2

3u2 D 6ep Q.u/

Ccu2

C3du

C6e2

3u2 D x.u/




and y D R.u/=u3, whereR.u/ is given in (6.7). In view of Lemma 6.1, R.u/ > 0,

as claimed.

From now on, P will denote a point on E such that v.P / > 0 and u.P / is an integer

u, where u is as in Proposition 6.4.Also, we define P 0 2 E by

P E0 D .x.P 0/, y.P 0// D .x0, .be C d p

a// (6.11)

(cf. beginning of proof of Lemma 6.2).

We will use, implicitly, Proposition 6.5 as well as the conclusions of the “bullets”

in the proof of Proposition 6.4. We will also use the following lemma.

Lemma 6.6. If e2, e3 2 R and e3 x e2 , then

Z x

e2

dtp f.t/

D Z C1x0

dtp f.t/

,

where

x0 D e2 C .e1 e2/.e2 e3/

e2 x 2 Œe1, C1/

In particular, in the case that x D x.P / 2 Œe3, e2 , we have x 0 D x .P C Q2/ , whereQ2 is the point of E with QE

2 D .e2, 0/ (see (3.7)).

Proof. For t 2 Œx, e2/ we make the change of variable

t D e2 C .e1

e2/.e2

e3/

e2 t 0 so that t0

D e2 C .e1

e2/.e2

e3/

e2 t .

The function t 7! t 0 is strictly increasing in the interval Œx, e2/, with values in

Œx0, C1/. A simple calculation now proves the equality of the integrals, as stated

in the announcement of the lemma.

Concerning the identity x 0 D x.P C Q2/, its proof is a matter of an easy (even by

hand) symbolic calculation.

Now we proceed further, supposing first that eb C d p

a ¤ 0. According to

Lemma 6.2, either x0 > e1, or e2, e3 2 R and x0 2 .e3, e2/.

If x0 > e1, then (first “bullet” in the proof of Proposition 6.4), x.P / D x.u.P // >e1, so that we can writeZ x.P/

x0

dtp f.t/

DZ C1

x0

dtp f.t/

Z C1

x.P/

dtp f.t/

DZ C1

x.P 0/

dtp f.t/

Z C1

x.P/

dtp f.t/

.

(6.12)

If e2, e3 2 R and e3 < x0 < e2, then, for t 2 Œe3, e2/ we write

t 0 D e2 C .e1 e2/.e2 e3/

e2

t




(cf. proof of Lemma 6.6). By the third “bullet” in the proof of Proposition 6.4, x.P/ Dx.u.P // 2 .e3, e2/ and, using Lemma 6.6, we compute

Z x.P/

x0

dtp f.t/ D Z e2

x0

dtp f.t/ Z e2

x.P/

dtp f.t/

DZ C1

x00

dtp f.t/

Z C1

x.P/0

dtp f.t/

DZ C1

x.P 0CQ2/

dtp f.t/

Z C1

x.P CQ2/

dtp f.t/

. (6.13)

Next, suppose that be C d p

a D 0. Then, by Lemma 6.2, x0 D e1, D C1 and

the second “bullet” in the proof of Proposition 6.4 x.P / D x.u.P // > e1. Therefore

we can write Z x.P/

x0

dtp f.t/

DZ C1

e1

dtp f.t/

Z C1

x.P/

dtp f.t/

. (6.14)

All three integrals in the left-hand sides of relations (6.12)–(6.14) are, by Proposi-

tion 6.4, equal to

Z C1u.P/

dwp Q.w/

and, moreover, in (6.14) we know that D C1.

On the other hand, we look at the right-hand sides of relations (6.12)–(6.14). Forboth relations (6.12) and (6.13), the right-hand side is equal to 1

2.l.P 0/"P 0 Cl.P/"P /,

in view of the relation (3.32). Further, in this case, by the definition of P 0, we have

y.P 0/ > 0, hence "P 0 D 1. Also, since u.P/ > u0 and v.P / > 0, Proposition 6.5

asserts that y.P / > 0, hence "P D 1 and, consequently, the right-hand side of both

(6.12) and (6.13) is equal to 1

2l.P 0/ C 1

2l.P /. In the case of the relation (6.14), the

first integral is equal to !1, in view of (3.18). Then, using (3.32) as before, we conclude

that the right-hand side of (6.14) is !1 C 1

2l.P /.

A straightforward combination of our conclusions above proves the following

proposition.

Proposition 6.7. Let P 2 E be such that u.P / u and v.P / > 0.

If either x0 > e1 , or e2, e3 2 R and e3 < x0 < e2 , thenZ C1u.P/

dwp Q.w/

D

2.l.P / l.P 0//.

If x0 D e1 (so that, necessarily, D C1), then

Z C1

u.P/

dw

p Q.w/ D!1

C

1

2l.P /.




Now we are ready for the main result of this chapter.

Theorem 6.8. Let E be the elliptic curve represented by the model C in (6.1). Let E : y2

D x3

CAx

CB

D f.x/ be the short Weierstrass model with A, B defined

in (6.3) and consider that, in Chapter 4 , equation (4.1) as well as everything else, in particular relations (4.4)–(4.8) , refers to this particular model E .

Choose a number u following the guidelines in the proof of Proposition 6.4 and then choose u u and c7 1 such that,

jx.u/j c7 for all u u.

Assume that P 2 E is such that P C D .u.P /, v.P// has integer coordinates withv.P / > 0 and u.P / u and write P E as in (4.2). With l.P / as in (4.6) , defineL.P/ by

L.P/ D´l.P / l.P 0/ if either x0 > e1 , or e2, e3 2 R and e3 < x0 < e2

l.P / C 2!1 if x0 D e1,

where e1, e2, e3 are the roots of f .X/ , as described in page 55 , !1 is the least positivereal period of the Weierstrass function associated with the short Weierstrass modelE , and P 0 is the point of E with P E0 as in (6.11). Then,

jL.P/j 4p a

exp. 1

2 log.3c7/ C M 2/, (6.15)

where and are defined in Propositions 2.6.3 and 2.6.2 , respectively.

Proof. First we show that there exist pairs .u, c7/ as in the announcement of the

theorem. Indeed, let any u u. Then u u0 (see Lemma 6.3) and, consequently,

Q.u/ > 0. Then, by (6.9),

jx.u/j D jcu2 C 3du C 6e2 C 6ep

Q.u/j3u2

jcju2 C 3jd ju C 6e2 C 6ep

Q.u/

3u2 .

Obviously, there exists u

u such that, if u

u, then 0 < Q.u/

2au4 and,

consequently, by the above displayed relation, jx.u/j c7, where c7 D max¹1, .jcjC3jd j C 6e2 C 6e

p 2a/=3º; we ask the reader to have a look at the remark after the

proof.

Now, consider a point P as in the announcement of the theorem. By the definition

of L.P / and Proposition 6.7 we have

jL.P/j D 2

Z C1u.P/

dwp Q.w/

4p a

u.P/1, (6.16)

where one can prove the inequality on the right exactly as we did for the upper bound

of the integral in Proposition 5.4.




Since v.P / > 0, we have v.P / Dp

Q.u.P // and then, in view of (6.4) and (6.9),

x.P/ D x.u.P //. Let x.P / D m=n, where m, n are relatively prime integers. But

also, x.P / D .x.u.P // 3u.P/2/=.3u.P/2/ where the numerator and denominator

are integers; actually, the numerator is equal to cu.P/2

C3d

u.P/

C6e2

C6e

v.P/.

Therefore, jmj 3jx.u.P //j u.P/2 3c7 u.P/2, jnj 3u.P/2 3c7 u.P/2

and, consequently, for the logarithmic height of the rational number x.P / we have

(see (2.21))

h.x.P // D log max¹jmj, jnjº log.3c7 u.P/2/ D log.3c7/ C 2log u.P/. (6.17)

Then,

log.u.P //1 1

2log.3c7/ 1

2h.x.P // (by (6.17)/

1

2

log.3c7/

C

Oh.P/ (by Proposition 2.6.3)

1

2log.3c7/ C M 2 (by Proposition 2.6.2).

Now, a straightforward combination of the last inequality with (6.16) proves (6.15).

Remark. The value of c7 given in the proof was obtained under the assumption that u

is so large that Q.u/ 2au4. Obviously, this choice is arbitrary; we could have chosen

any constant > 1 and then assumed that u is so large that Q.u/ constant u4.

Solutions .u, v/ with u < 0. On the first glance, Theorem 6.8 seems to treat integersolutions .u, v/ to (6.1) with “large” positive u, only. However, as we will show below,

with a very little extra effort we can treat all integer solutions .u, v/ with “large” jujand obtain an exactly analogous upper bound in which the assumption concerning

the “large” unknown point P C D .u.P /, v.P// is that “ju.P/j is sufficiently large”

rather than “u.P/ is sufficiently large”.

It is clear what one should do; it suffices to replace the coefficients a, b, c, d , e of

the given equation by a, b, c, d , e, respectively, and consider the equation v2 Dau4 bu3 C cu2 du C e2 in integer .u, v/ with sufficiently large u > 0. How does

the replacement of .a, b, c, d , e/ by .a,

b, c,

d , e/ affect the crucial “quantities” (=

parameters and auxiliary functions) involved in the discussion so far? This is shown in

Table 6.1. The symbols of the “quantities” involved in our discussion are on the left,

while on the right the analogous symbols are listed with a bar on them, denoting

the analogous “quantities” which result when the coefficients b, d are replaced by

b, d , respectively. We avoid the bar indication if the replacement does not affect

the “quantity”.




Table 6.1. Parameters and auxiliary functions for the solution of the quartic elliptic equation.

0 10 Q.u/ D au4 C bu3 C cu2 C du C e2 Q.u/ D au4 bu3 C cu2 du C e2

(see Lemma 6.1) D ´ if d p a C be ¤ 0 if d

p a C be D 0

A, B (see (6.3)) A, B

x0 (see (6.8)) x0

R.u/ D beu3 C 2ce u2 C 3deu C 4e3 R.u/ D beu3 C 2ce u2 3deu C 4e3

C.4e2 C du/ Q.u/1=2 C.4e2 du/ .Q.u//1=2

x.u/ D cu2 C 3du C 6e2 C 6e .Q.u//1=2

3u2 x.u/ D cu2 3du C 6e2 C 6e .Q.u//1=2

3u2

Choose u0

1 so that Choose u0

1 so that

u u0 ) Q.u/ > 0 & sgn.R.u// D u u0 ) Q.u/ > 0 & sgn.R.u// D

Choose u u0 so that Choose u u0 so that

In the case that x0 e1: In the case that x0 e1:

u > u ) x.u/ > e1 u > u ) x.u/ > e1

In the case that e3 < x0 < e2: In the case that e3 < x0 < e2:

u > u ) e3 < x.u/ < e2 u > u ) e3 < x.u/ < e2

Choose u u and c7 > 0 so that Choose u u and c7 > 0 so that

jx.u/j c7 for all u u

jx.u/j c7 for all u u

P E0 D .x0, .be C d p

a// P E

0 D´

P E0 if d p

a C be ¤ 0

P E0 if d p

a C be D 0

l.P 0/ l.P 0/ D´

l.P 0/ if d p

a C be ¤ 0

l.P 0/ if d p

a C be D 0

L.P / L.P /

if either x0 > e1 if either x0 > e1

or e2, e3 2 R and x0 2 .e3, e2/ or e2, e3 2 R and x0 2 .e3, e2/

then L.P/ D l.P / l.P 0/ then

L.P/ D l.P / l.P 0/ L.P / D´l.P / C l.P 0/ if d

p a C be ¤ 0

l.P / l.P 0/ if d p

a C be D 0

if x0 D e1 then if x0 D e1 then

L.P/ D l.P / C 2!1 L.P/ D l.P / C 2!1

To continue with the resolution of the quartic elliptic equation, one can safely avoid Chapters 7 and 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.4.



Chapter 7

Linear form for simultaneous Pell equations

Let A1, A2 be given non-zero integers and let D1, D2 be given positive non-square

integers. In this chapter we will deal with the system

U 2 D1V 2 D A1, W 2 D2V 2 D A2 (7.1)

of simultaneous Pell equations, where .U , V , W / are positive integers, under the as-

sumption that we know a rational solution .U 0, V 0, W 0/ with V 0 > 0 and U 0, W 0

0,

not both zero. This assumption is not really restrictive in practice. All known exam-ples possess at least one small solution, easily found by a direct search; see e.g. the

table in [60, Appendix]. An alternative method to discover a rational, or even integral,

solution .U 0, V 0, W 0/ is to consider the equation .D1V 2 C A1/.D2V 2 C A2/ D Z2

(Z D U W ) and search for rational solutions. For example, if we take .D1, A1/ D.7, 2/ and .D2, A2/ D .32, 23/ (an example from Z. Y. Chen, mentioned in [60, Ap-

pendix]), then the MAGMA routine Points applied to the corresponding curve1 imme-

diately returns the points .V , Z/ D .1, 9/, .271, 1099161/, which furnish the solutions

.U , V , W / D .3,1,3/, .717, 271, 1533/.

In [60] an alternative method for the explicit resolution of (7.1) is developed, which

requires the solution of a number of quartic Thue equations; moreover, that paper

contains a rich bibliography on simultaneous Pell equations, to which we refer the

interested reader. Anyway, choosing between Ellog and Thue equations for the reso-

lution of a certain Diophantine problem does not admit a general answer; this issue is

discussed in the beautiful paper [58].

Now, let us come back to (7.1). To each solution .U , V , W / we will associate a

point P on a certain short Weierstrass model, along with a linear form L.P / and then

we will prove Theorem 7.1, which will provide us with an explicit upper bound for

jL.P/

j(cf. page 56 immediately after (4.8)). This chapter’s exposition is based on the

author’s paper [63].Throughout the chapter we keep the notations, assumptions and results of Chapter 4.

Clearly, for a solution .U 0, V 0, W 0/ as above,

sgn.U 0 V 0p

D1/ D sgn.A1/, sgn.W 0 V 0p

D2/ D sgn.A2/. (7.2)

Moreover, we will assume that A1D2 A2D1 ¤ 0, otherwise the system is rather

trivially solved. Indeed, if A1D2 A2D1 D 0, then (7.1) implies D2U 2 D D1V 2,

so that A2=A1 D D2=D1 D q2 for some q 2 Q and, consequently, the solutions

1 Viewed as an “hyperelliptic curve” in MAGMA’s language.



70 Chapter 7 Linear form for simultaneous Pell equations

.U , V , W / of (7.1) are given by .U , V , qU /, where .U , V / is a solution of the first

equation, satisfying qU 2 Z.

In view of symmetry of the pairs .A1, D1/ and .A2, D2/, we may assume, without

loss of generality, that A1

D2

A2

D1

> 0. If in this relation we replace A1

by U 20 D1V 2

0 and A2 by W 20 D1V 20 we see that

U 0p

D2 W 0p

D1 > 0. (7.3)

Now let .U , V , W / be any solution (not necessarily integral) to (7.1), such that .V :

W / ¤ ˙.V 0 : W 0/. Combining the two equations (7.1) we obtain

A2U 2 A1W 2 C .A1D2 A2D1/V 2 D 0. (7.4)

We put

u D U 0V V 0U W 0V C V 0W

, (7.5)

so that

V D V 0.U C uW /

U 0 uW 0. (7.6)

Then we substitute for V in (7.4) to obtain a quadratic equation in U=W . One solution

of that equation is, obviously, U =W D W 0=U 0, hence the other solution is

U

W D

A2U 0u2 2A1W 0u C A1U 0

A2W 0u2

2A2U 0u C A1W 0

. (7.7)

Then, using (7.6) we obtain

V

W D .A2u2 A1/V 0

A2W 0u2 C 2A2U 0u A1W 0. (7.8)

We have 1 D2. V

70W /2 D A2

80W 2 , hence, (7.8) and the relations

U 20 D1V 20 D A1, W 2

0 D2V 20 D A2 , (7.9)

imply that

v2 D au4 C bu3 C cu2 C du C e2 def D Q.u/, (7.10)

where

v D A2W 0u2 2A2U 0u C A1W 0

W , (7.11)

D sgn.A1/ D sgn.U 0 V 0p

D1/, (7.12)

.a, b, c, d / D .A22, 4A2U 0W 0, 4A1W 20 2A1A2 C 4A2U 20 , 4A1U 0W 0, jA1j/.

(7.13)

Equation (7.10) has been studied in Chapter 6. There we focused our interest on integer

solutions .u, v/ with arbitrarily large u > 0, whilst now the u and v in (7.5) and (7.11),



Chapter 7 Linear form for simultaneous Pell equations 71

respectively, are certainly rationals but, in general, not integral and, moreover, as we

will see, u belongs to a finite interval instead of approaching infinity. Nevertheless, we

will exploit part of the results of our study in Chapter 6 in order to solve our present

problem.

In (7.11) we replace u by its expression in (7.5) in order to express v as a rational

function of U , V , W (we also take into account (7.9)):

v D 2V 20

A2U 0U C .D2A1 D1A2/V 0V C A1W 0W

.V 0W C W 0V /2 . (7.14)

Conversely, we need also express U , V , W as rational functions of u, v. By the defi-

nition of v,

W D A2W 0u2 2A2U 0u C A1W 0

v (7.15)

and this, combined with (7.8) and (7.7) gives

V D .A2u2 C A1/V 0

v , U D

A2U 0u2 2A1W 0u C A1U 0

v . (7.16)

Now, using (6.4), (6.5) and (6.6), we establish a birational transformation between

(7.10) and

E : y2 D x3 C Ax C B def D f.x/, (7.17)

with A and B expressed by the formulas (6.3) as functions of a, b, c, d , e taken from

(7.13).

As in Chapter 6, we will express an elliptic integral R dt

p f.t/

, with appropriate up-

per and lower limits, in terms of another integral in which the variable V is directly

involved. This will be accomplished with the aid of a function x.V /, defined on an in-

terval ŒV , C1/, for a sufficiently large positive V . Below we proceed to construct

this function.

Let .U , V , W / ¤ .U 0, V 0, W 0/ be any positive real solution to (7.1), so that U Dp A1 C D1V 2 and W D

p A2 C D2V 2. Then (7.5) and (7.11) suggest that we define

the functions

V 7! u.V / D U 0V V 0p

A1 C D1V 2

W 0V

CV 0p A2

CD2V 2

(7.18)

and

V 7! v.V / D A2W 0 u.V /2 2A2U 0 u.V / C A1W 0p

A2 C D2V 2, (7.19)

which satisfy the relation v.V /2 D Q.u.V //. The substitutions U p

A1 C D1V 2

and W p

A2 C D2V 2 in (7.5) makes u D u.V /, v D v.V / (by (7.15)), and

consequently, in view of (7.14),

v.V / D 2V 20

A2U 0p

A1 C D1V 2 C .A1D2 A2D1/V 0V C A1W 0p

A2 C D2V 2

.W 0V

CV 0p A2

CD2V 2/2

.

(7.20)




After some standard computations, in which use is made of the definition of (see

(7.12)), as well as of the relations (7.2) and (7.3), we verify the following (dashes

denote derivatives):

(a) u1def D lim

V !C1u.V / D U 0 V 0p D1

W 0 C V 0p

D2

.

(b) limV !C1

V 3 u0.V / D .p

D2U 0 Cp

D1W 0/.U 0 V 0p

D1/V 0p D1D2.W 0 C V 0

p D2/

.

(c) limV !C1

v.V / D 0.

(d) lim

V !C1

V 2

v0.V /

D

2V 20 jU 0 V 0

p D1j.U 0

p D2 C W 0

p D1/

W 0 C V 0p D2

.

(e) Q.u1/ D 0.

(f) Q0.u1/ D 8V 30

p D1D2.U 0 V 0

p D1/.U 0

p D2 C W 0

p D1/

W 0 C V 0p

D2

.

We note first that, by (e), the polynomial Q has four real roots which, in view of

(a), are actually the algebraic conjugates of u1 in the quartic field Q.p

D1,p

D2/;

temporarily, let us denote these roots by r4 < r3 < r2 < r1.

If

D 1, then, by (f), Q is strictly increasing in an interval with u1 as its centre,

therefore, u1 D r1 or r3 and Q.u/ > 0 in an interval I with u1 as left end point

and right end point appropriately close to u1 (or even C1 if u1 D r1). If D C1,

then an analogous argument shows that u1 D r2 or r4 and Q.u/ > 0 in an interval I

with u1 as right end point and left end point appropriately close to u1 (or even 1if u1 D r4).

Then, with the interval I as above, it is clear that

I 3 u 7! x.u/ def D 6e

p Q.u/ C cu2 C 3du C 6e2

3u2 , (7.21)

I 3 u 7! y.u/ def D ebu3 C 2ecu2 C 3edu C 4e3 C .du C 4e2/p Q.u/

u3 (7.22)

are meaningful as real-valued functions; note that the formula for x.u/ is the same as

that in (6.9), but the domains of definition differ: There, the domain is a neighbourhood

of C1, here the domain is the interval I . We also remark that, if in the right-hand

sides of (6.4) we replace v byp

Q.u/, then we obtain the right-hand sides of (7.21)

and (7.22), respectively, except that is defined differently. Note, however, that a

different choice of in (6.4), (6.5) and (6.6) does not affect the fact that these relations

establish a birational transformation between v 2 D Q.u/ and y 2 D x3 C Ax C B;

cf. the comment just before the announcement of Lemma 6.1.




Next we note that, in view of (c) and (d) on page 72, if V is sufficiently large, then

v.V / > 0.

Having in mind the above remarks, let .u, v/ D .u.V /, v.V //. In view of (a) and

(b) on page 72, if V is sufficiently large, then u D

u.V / 2

I and v D

v.V / > 0.

Since v 2 D v.V /2 D Q.u.V // D Q.u/, we conclude then that v D p Q.u/ and,

consequently, from (7.21) and (6.4) we see that x.u/ D X .u, v/ D X .u,p

Q.u//.

Similarly, (7.22) and (6.4) imply that y.u/ D Y .u, v/ D Y .u,p

Q.u//. Therefore,

for sufficiently large V , the functions

V 7! x.V / def D .x ı u/.V /, V 7! y.V /

def D .y ı u/.V / (7.23)

are meaningful as real-valued functions and

.x.V /, y.V // D .X .u.V /, v.V //,Y .u.V /, v.V ///, v.V / D

p Q.u.V // ;

in particular, .x.V /, y.V // 2 E.R/; just remember that, if u, v 2 R andv2 D Q.u/, then .x, y/ D .X .u, v/,Y .u, v// is a point of E.R/.

Summing up: Any integer solution .U , V , W / to (7.1) with V > 0 and U , W 0

corresponds to a point .x, y/ D .x.V /, y.V // 2 E.Q/. Conversely, from .x, y/ we

can recover the solution .U , V , W /, first, by obtaining .u, v/ using (6.5) and (6.6), and

then by using (7.16) and (7.15).

Next note that we can also define R.u/ by (6.7) and we check that

R.u1/ D 8V 30 .U 0

CV 0p D1/

jU 0

V 0p D1

j3.U 0p D2

CW 0p D1/p D1D2

.W 0 C V 0p D2/2 ,

whence R.u/ > 0. But then, similary to the proof of Lemma 6.3, we see that the

function x is strictly decreasing in the interval I . On the other hand, by (b) on page

72, for sufficiently large V , u is strictly increasing or decreasing, according to whether

D C1 or D 1, respectively. Therefore, we finally conclude that, for sufficiently

large V , the function V 7! x. V/ is strictly increasing or decreasing according to

whether D 1 or D C1, respectively.

Further, we compute

x0def D lim

V !C1x.V / D X .u1,p Q.u1// (7.24)

D 4V 20

3

D2U 20 C D1D2V 20 C D1W 20 C 3

p D1D2U 0V 0

C 3D1

p D2V 0W 0 C 3

p D1D2U 0W 0

> 0

and

y0def D lim

V !C1y.V / D Y .u1,

p Q.u1// (7.25)

D 8V

3

0 p D1D2.W 0p D1 C U 0p D2/.U 0 C V 0p D1/.W 0 C V 0p D2/ > 0.




Note that .x0, y0/ is a point of E.R/; more precisely, a point of E.Q.p

D1D2//.

Indeed, look at (7.24) and (7.25) and, once again, remember that any .x, y/ D.X .u, v/,Y .u, v// with real u, v and v2 D Q.u/ is a point of E.R/. The point

P E0def

D .x0, y0/ 2 E.Q.p D1D2// (7.26)

plays a role in our method; see Theorem 7.1 below. In the applications we will find

important the property that 2P E0 2 E.Q/; more specifically,

2P E0 D . 4

3.D1D2V 2

0 C D1W 20 C D2U 20 /V 20 , 8D1D2U 0W 0V 4

0 /. (7.27)

Coming back to x.V / and y.V /, we conclude that, according to our above dis-

cussion, if V 2 .V , C1/ for a sufficiently large V , then x.V / 2 I.x0/, where

I.x0/ is an interval .x1, x0/ or .x0, x1/, according to whether D C1 or D 1,

respectively, for an appropriate x1. Moreover, for V

2.V ,

C1/ we have y.V / > 0,

whence y.V / D p f .x.V //.

Now we focus our interest on the integralR x0

xdtp f.t/

when x belongs to the interval

I.x0/ as above. According to our previous discussion, the interval with end points x

(included) and x0 (excluded) is the image under the function x D x ı u of an interval

ŒV , C1/, where x D x.V /. Therefore, for t 2 Œx, x0/ we can make the change of

variable t D x.T / D x.u.T //, where T 2 ŒV , C1/. Then y.T / Dp

f.x.T// Dp f.t/ and

dt

dT D dx

du du

dT .

From (7.21) we calculate

dx

duD eb u3 C 2ec u2 C 3ed u C 4e3 C .d u C 4e2/

p Q.u/

u3p

Q.u/

D (by (7.22)) y.u/p

Q.u/D

y

vD

p f.t/

v.

From (7.18) we compute

du

dT D

V 0.A2U 0

p A1 C D1T 2 C .A1D2 A2D1/V 0T C A1W 0

p A2 C D2T 2

.V 0p A2 C D2T 2 C W 0T /2p A1 C D1T 2p A2 C D2T 2

D (by (7.20))

2V 0 vp

.A1 C D1T 2/.A2 C D2T 2/.

On combining the above relations we obtain

dt

dT D 1

2V 0

p f.t/p

.A1 C D1T 2/.A2 C D2T 2/

and, consequently,

Z x0

x.V/

dt

p f.t/ D 1

2V 0 Z C1

V

dT

p .A1 C D1T 2/.A2 C D2T 2/. (7.28)




Now we examine more closely the short Weierstrass model E defined by (7.17).

First we note that the polynomial f .X/ has the following three rational roots:

1

D

4

3

.D2X 20

2D1D2Y 20

CD1Z2

0 /Y 20

D

4

3

.A1D2

CA2D1/Y 20

2 D 4

3.2D2X 20 C D1D2Y 20 C D1Z2

0 /Y 20 D 4

3.A2D1 2A1D2/Y 20

3 D 4

3.D2X 20 C D1D2Y 20 2D1Z2

0 /Y 20 D 4

3.A1D2 2A2D1/Y 20 .

Note that, with our ordinary notation, the roots of f.X/ are denoted by e1, e2, e3, where

e1 > e2 > e3; therefore, .e1 e2 e3/ is a permutation of .1 2 3/. Now we consider

the point P 0 with P E0 D .x0, y0/ 2 E.R/ defined in (7.26). We easily check that

x0 i > 0 for i D 1, 2, 3, therefore, P E0 2 E0.R/ and, consequently, according to

our comment immediately after (7.25), we may assume that, if V is sufficiently large,then the point P V , which is defined by P E D .x.V /, y.V //, belongs to E0.R/ and

y.V / > 0. Then, we can write the left-hand side of (7.28) as follows:Z x0

x.V /

dtp f.t/

DZ C1

x.V/

dtp f.t/

Z C1

x0

dtp f.t/

D 2.l.P / l.P 0// (by (3.32)).

Easily, the right-hand side of (7.28) is absolutely less than a constant times V , provided

that V is not “very small”. For example, if V maxiD1,2¹p

2jAi j=Di º, then the right-

hand side of (7.28) is absolutely less than V 1=.V 0

p D1D2/ and, consequently,

jl.P / l.P 0/j < V 1

2V 0p

D1D2

. (7.29)

Now we are ready to prove the main result of this chapter.

Theorem 7.1. Let .U 0, V 0, W 0/ , with V 0 > 0 and U 0, W 0 0, U 0 C W 0 > 0 , be aknown rational solution of the system (7.1).

Let .U , V , W / be an integer solution of the system (7.1) with U , V , W positive, sothat U D

p A1 C D1V 2 , W D

p A2 C D2V 2 and, consequently, any function of

.U , V , W / can be viewed as a function of V only. Assume that V is sufficiently large,so that every condition below is fulfilled:

For i D 1, 2 it is true that p

Ai C Di V 2 is bounded by a constant times V .2

u.V / (cf. (7.18)) and v.V / (cf. (7.19)) satisfy Q.u.V // > 0 and v.V / > 0 , respec-tively.

V belongs to an infinite interval .V , C1/ in which the functions x and y , defined in (7.23) , have the following properties: The function x is strictly monotonous, x.V /

is larger than the maximum of the i ’s displayed at page 75 and y.V / > 0.

2 For example, this constant can be chosen to be 2p Di if Ai > 0 and p Di if Ai < 0.




Let E be the Weierstrass model defined in (7.17) and consider that, everything inChapter 4 , in particular the relations (4.1) and (4.4)–(4.8) , refers to this particular model E . Consider also the points P E D .x.V /, y.V // 2 E.Q/ and P E0 defined in(7.26). Express P E as in (4.2) and l.P / by (4.6) , and set L.P/

Dl.P /

l.P

0/. Then,

jL.P/j 1

2V 0p

D1D2

exp. 1

2c8 C M 2/, (7.30)

where 12 0 and are as in Propositions 2.6.2 and 2.6.3 , respectively, and c8 is a pos-itive constant which can be explicitly calculated as explained below, in the paragraphimmediately before relation (7.32).

Proof. Firstly, by what we have already seen on pages 73 and 74, .x.V /, y.V // and

.x0, y0/ are, indeed, points of E.R/. Further, we saw on page 75 that .x0, y0/ 2 E0.R/

and, if V is sufficiently large, then also .x.V /, y.V // belongs to E0.R/. More pre-cisely, .x0, y0/ 2 E0.Q.

p D1D2// and, since U , V , W are integers, .x.V /, y.V // 2

E0.Q/.

Next, a quick look at our previous discussion after page 71 and before the relation

(7.29) easily reminds that, indeed, for sufficiently large V all conditions marked by

“bullets” are fulfilled, so we proceed to the main part of the proof, noting the following:

If u and v are given by (7.5) and (7.14), respectively, then

x. V/ D 6ev C 6u2 C 3du C 6e2

3u2 . (7.31)

The right-hand side can be written as an element of Q.U , V , W /, the numerator

and denominator of which are both of total degree 2. On multiplying numerator and

denominator by `2, where ` is the least common multiple of the denominators of

U 0, V 0, W 0, we obtain a relation of the form x.V / D P 1.U , V , W /=P 2.U , V , W /,

where P 1, P 2 are polynomials in U , V , W with explicit integer coefficients belong-

ing to ZŒD1, D2, .`U 0/, .`V 0/, .`W 0/. Then, since U Dp

A1 C D1V 2 and W Dp A2 C D2V 2, it is clear that, for i D 1, 2, we have jP i .U , V , W /j consti V 2,

where consti is an explicitly computable positive real number.3 Let now x.V / D m=n,

where m, n are relatively prime integers, so that h.x.V //

D log max

¹jm

j,

jn

jº. We

have m=n D x.V / D P 1.U , V , W /=P 2.U , V , W / and the right-most quotient hasinteger numerator and denominator. Therefore,

max¹jmj, jnjº max¹P 1.U , V , W /, P 2.U , V , W /º max¹const1, const2º V 2,

from which it becomes clear that

h.x.V // c8 C 2log V hence log V 1 1

2c8 1

2h.x.V // (7.32)

3 Here we need the requirement set by the first “bullet” in the announcement of the theorem.




for some explicitly computable positive constant c8; in the notation of the previous

lines,

c8 D log max¹const1, const2º.

By Proposition 2.6.3, Oh.P/ 1

2 h.x.V // ; by Proposition 2.6.2, Oh.P/ M 2 andcombining these with (7.32) gives

log V 1 1

2c8 1

2h.x.V // 1

2c8 C Oh.P/ 1

2c8 C M 2.

By the definition of L.P/ and (7.29) we have L.P / < V 1=.2V 0p

D1D2/ and, com-

bining this with the last displayed inequality, gives the inequality (7.30).

To continue with the resolution of the simultaneous Pell equations, one can safelyavoid Chapter 8 and go directly to Chapter 9, Sections 9.1, 9.2 and 9.5.



Chapter 8

Linear form for the general elliptic equation

In this chapter we treat general elliptic equations,

C : g.u, v/ D 0, (8.1)

where g is a polynomial with coefficients in Z, irreducible over Q and the curve rep-

resented by the model (8.1) is of genus one and possesses a non-singular point with

rational coordinates.

Throughout the chapter we keep the notations, assumptions and results of Chapter 4.This chapter is based mainly on the joint work [56] of R. Stroeker and the author.

It is well known that the equation (8.1) has finitely many solutions (see, for ex-

ample, [2] or [41]), but here, in analogy to Chapters 5, 6 and 7, we intend to prove

Theorem 8.7.2, which gives an explicit upper bound for jL.P/j, where L.P / is an

appropriate linear form in elliptic logarithms; cf. page 56 immediately after (4.8).

Without loss of generality we assume that degug degvg D n. We will make the

further assumption that C .R/ possesses infinite (= unbounded) branches along both

the u-direction and the v-direction. This is very natural to assume, for, otherwise, the

problem of solving g D 0 in integers is, obviously, trivial.Before proceeding to the study of (8.1), we collect all the assumptions that we set

so far, which will be implicitly understood throughout the whole chapter.

g is a polynomial with coefficients in Z, irreducible over Q.

The curve represented by the model (8.1) is of genus one and a non-singular point

with rational coordinates is known.

degug degvg D n.

C.R/ possesses infinite branches along both the u-direction and the v- direction.

8.1 A short Weierstrass model

As in previous chapters, a basic tool of the Ellog is the use of a short Weierstrass

model birationally equivalent to the model C with which we started.

Proposition 8.1.1. The model C is birationally equivalent over Q to a model

E : y2 D f.x/ D x3 C Ax C B , A, B 2 Q. (8.2)

The birational transformation, as well as A and B , can be explicitly constructed.



Section 8.1 A short Weierstrass model 79

Proof. We refer to van Hoeij’s [22], where an algorithm is presented for calculating

the birational transformation between C and a short Weierstrass model E , provided

that a non-singular point .u0, v0/ of C with algebraic coordinates is known. According

to Section 3.2 of that paper, the coefficients of the transformation which is constructed,

as well as the coefficients of the Weierstrass model E , belong to the field Q.u0, v0/;

this is even more clear if one looks at van Hoeij’s papers [20] and [21] on which the

algorithm of [22] is based. Since we have assumed that a non-singular point of C with

rational coordinates is known, it follows that, if we insert this particular point in van

Hoeij’s algorithm [22], we will get a birational transformation with rational coeffi-

cients between C and a short Weierstrass model E which will be defined over Q.

The birational transformation of Proposition 8.1.1 will be denoted by

.X ,Y /

!C 3 .u, v/ .x, y/ 2 E. U ,V /

, (8.3)

meaning that

u.P/ D U .x.P/, y.P//,

x.P/ D X .u.P /, v.P//,

v.P/ D V .x.P/, y.P//

y.P/ D Y .u.P /, v.P//,

in accordance with Section 1.1.

ExampleStep 1. In this chapter, parallel to the general exposition, we will discuss the

methodical solution in integers of the equation g.u, v/ D 0, where

g.u, v/ D 3v5 C 3uv3 271uv 3u2. (8.4)

With the aid of MAPLE we find out that the genus of the curve C : g D 0 is one. Obvi-

ously, u D .0, 0/ is a point, so that C is a model of an elliptic curve overQ. The MAPLE

implementation of van Hoeij’s algorithm [22] gives the birational transformation (8.3)

between C and the model

E : y2

Df.x/

Dx3

17846163x

C

120408061761

4

. (8.5)

The functions X and Y are the following:

X .u, v/ D 2710u 9225uv3 43821uv 5904u2 C 21867uv2 C 726v4 C 2169u3

C 8883u2v2 48429vu2 C 2349v4u/=u.u 1/2

Y .u, v/ D 79947u4 C 14069376vu3 829440u3v2 C 1802709u3 C 137565u2

6952716v3u2 C 140667858vu2 39726990u2v2 1224558v4u2

22481070uv2 4683906v4u C 11958534uv3 C 34375266uv

C 81029u 395286v4

/=.u.u 1/3, (8.6)



80 Chapter 8 Linear form for the general elliptic equation

and the inverse functions1 are

U .x, y/ D

2x5 C 776970x4 C 6389652843x3 C 2050yx 3 11183395769802x2

C126072288yx 2

123607085812402995x

199252067022xy

C 268148092563378425394

930481914765852y/=2.x C 4365/2.x 2169/3

V .u, v/ D 3.x2 5121x 3270699 C 242y/=.x C 4365/.x 2169/. (8.7)

Example continued on page 82.

8.2 Puiseux series

Our intuition suggests that it should be possible to “solve for v” the equation g.u, v/ D0 over the real numbers. Actually this is guaranteed by the implicit function theorem.

Since, in our case, the real function .u, v/ 7! g.u, v/ is very special, namely, a poly-

nomial function, we can say much more about the solutions v D v.u/, which can be

expressed in a very concrete way by means of the Puiseux series. In general, let K be

a subfield of the complex numbers and a 2 K . Roughly speaking, we obtain a Puiseux

series around a if in a usual power seriesPC1

kDm qk.X a/k (m 2 Z may be nega-

tive) we replace X a by .X a/1= for some positive integer . For a systematic

presentation of the relevant theory we refer to [66, Chapter IV] and/or [4, Chapter II];2

useful references are also [67, §3], [68], [69, §2], [70, §3]. We collect below all theinformation that is necessary for the purposes of the present chapter.

Fact 8.2.1.

(a) There is a finite Galois extension K=Q , which we view as a subfield of C , and ndistinct formal power series (Puiseux expansions at infinity)

vi .u/ D1X

kDi

˛k,i uk=i , with ˛i ,i ¤ 0 .i D 1, : : : , n/, (8.8)

where for each i , the following hold: i , i 2 Z , i 1 , all ˛k,i ’s belong toK , the formal identity g.u, vi .u// D 0 holds, and i is minimal subject to therestriction that no proper divisor of i divides all k i with ˛k,i ¤ 0.

(b) Any formal power series v.u/ satisfying the formal identity g.u, v.u// D 0 and having properties analogous to those of the series (8.8) , even without the require-ment that the coefficients of v.u/ be algebraic, necessarily coincides with one of the above n series.

1 We will need them only at the final stage of the resolution, in Section 10.2.4.

2 We note that the terminology “Puiseux series” is absent from these two books.



Section 8.2 Puiseux series 81

(c) The formal identity

g.u, v/ D p0.u/

n

YiD1

.v vi .u//

holds, where p0.u/ is the coefficient of vn in g.u, v/.

(d) Each series (8.8) converges for juj > B0 , where B0 is the maximum modulus of the roots of the polynomial Resv.g,

@g

@v/ 2 ZŒu.

(e) For each i the function t 7! vi .ti / D P1kDi

˛k,i t k is analytic and one-to-

one into the punctured disk with centre at the origin and radius B1=i0 .

Proof. We do not present here a proof in the strict sense of the word. The proofs of

statements (a) through (d) are scattered in the bibliography.

Statements (a), (b), and (c) are classical results about Puiseux series. They can be

found in classical books such as [4, Chapter II] and [66, Chapter IV], though in slightly

different form. In these books the notion of parameterisation is used in order to express

the solutions .u, v/. More specifically, instead of (8.8), the parameterisation

u D ti

v D vi .ti / D1X

kDi

˛k,i t k

is used. Here we prefer to follow [67, §3].

Statement (d) seems widely known. However, we could not find an easily accessiblereference where this is explicitly stated and proved. Implicitly it can be derived, for

example, from [4, Chapter II] (especially §13).

Statement (e) is found, for example, in [4, Theorem 13.1] and what precedes this

theorem. For the injectivity proof of t 7! vi .ti / we need the somewhat technical

requirement in (a) on the minimality of i .

Remarks.

(1) The Puiseux expansions (8.8) can be computed algorithmically by means of

Newton polygons; see for instance [66, Chapter IV, §3]. An interesting refine-

ment of this process is found in [69] with an added discussion on complexitymatters; see also [68, §2] and [70, §3].

(2) The routinepuiseuxof MAPLE computes the Puiseux expansions of an algebraic

function or, rather, the conjugacy classes of these expansions; by this we mean

the following.

Fix a primitive i -th root of unity . For every m 2 ¹0, 1, : : : , i 1º and every

2 Gal.K=Q/ define the formal series

vi .u, m, /

D

1

XkDi

.˛k,i / mk uk=i ;




in particular,

vi .u,0,id/ D vi .u/ .i D 1, : : : , n/.

The series vi .u, m, / also satisfies g.u, vi .u, m, // D 0, therefore, by Fact 8.2.1(b),

it must coincide with another series (8.8), say with vj .u/ D vj .u,0,id/. We then saythat vj .u/ and vi .u/ belong to the same conjugacy class. The n series (8.8) are thus

partitioned into disjoint conjugacy classes (see after relation (3.5) in [67]). Therefore,

the computation of a smaller set of series (8.8) composed of representatives of each

conjugacy class is sufficient for the computation of all Puiseux series (8.8).

ExampleStep 2. (continued from the end of Step 1, page 80)With the aid of MAPLE we compute two conjugacy classes of Puiseux series solving

g.u, v/ D 0. The first class consists of the series

v1.u/ D i u1=2 C 12

110524

i u1=2 138u1 C 65291211152

i u3=2 C 3902669

u2 C ,

where i Dp

1, and

v2.u/ D i u1=2 C 1

2C 1105

24i u1=2 138u1 6529121

1152i u3=2 390266

9u2 C ,

The second class consists of the series10 5

v3.u/ D u1=3 1

3C 275

9u1=3 3317

81u2=3 C 92u1 1271288

729u4=3

9364963

6561 u5=3

C ,

v4.u/ D !2u1=3 1

3C 275

9!u1=3 3317

81!2u2=3 C 92u1 1271288

729!u4=3

9364963

6561!2u5=3 C ,

v5.u/ D !u1=3 1

3C 275

9!2u1=3 3317

81!u2=3 C 92u1 1271288

729!2u4=3

9364963

6561 !u

5=3

C ,where 10 0 ! is a primitive cubic root of unity.

In the notation of Fact 8.2.1(a), K D Q.i , !/. Also, mi D 1, i D 2 for i D 1, 2

and mi D 1, i D 3 for i D 3, 4, 5. The constant B0, defined in Fact 8.2.1(d) is

approximately equal to 483.945; we round to B0 D 484.


Computing the number field K . The extension K=Q mentioned in Fact 8.2.1(a),

can be computed by an algorithm of J. Coates which is implicitly contained in the

proof of [8, Lemma 3].



Section 8.2 Puiseux series 83

First, define the polynomials with integer coefficients pj , j D 1, : : : , n by writing

g.u, v/ Dn

Xj D0

pj .u/vnj .

For any i D 1, : : : , n it suffices to show that there exists an index k0,i 0 such that,

˛kC 2 Q.˛, : : : , ˛Ck0,i/ for all k k0,i . Therefore, let us fix i 2 ¹1, : : : , nº and

subsequently, in order to simplify notation, omit the subscript i from i , ˛k,i , i and

vi .u/ in (8.8). Compute the integers D i and D i by the first few steps in the

construction of the Newton polygon.

Choose a non-negative integer N such that

degpj C .n j / C N 0 .j D 0, : : : , n/

with equality for at least one subscript j , and put

P j .X/ D pj .X / X .nj /CN .j D 0, : : : , n/

and

G.X , Y / DnX

j D0

P j .X/ Y nj .

Then G 2 ZŒX , Y , and it is straightforward to check that

G.X , y.X// D 0 identically in X , where y.X/ D1X

kD0

˛kCX k . (8.9)

Before giving Coates’ algorithm, let us make a notation remark: For any polynomial hover any field, in one or more variables X , : : :, when we write r D ordX .h/, we mean

that X r appears in some monomial of h with non-zero coefficient, and r is minimal

with respect to this property.

10 6 Computing the finite extension Q.˛, ˛C1, : : : /

DESCRIPTION: Compute k0 0 such that ˛Ck 2 Q.˛, : : : , ˛Ck0/ 8k k0.

INPUT: G.X , Y / as above, m D degX G, D .2n 2/m C 1.

OUTPUT: k0.

INITIAL VALUES: G0.x, y/ G.x, y/, k 0.() H k.Y / Gk .0, Y /.

if degH k D 1 then go to ()

else

Define uk by H k.uk / D 0

r ordX .Gk.X , uk C XY//

Gk .X , Y / X r Gk.X , uk C XY /

k k C 1

goto ()

endif

(

) k0

k

END




By the proof of [8, Lemma 3], the polynomials H k .k D 0,1,2, : : : / are not identi-

cally zero, their degrees are in non-increasing order and degH k0 D 1 for some k0 ,

so that for all k k0 the polynomial H k is of degree 1. Now for each k 0, the

algebraic number ˛kC

in (8.9) is a root of H k

, and therefore assumes one of the

possible values of uk . By the linear character of H k for all k k0, it follows that

for k > k0, the coefficient ˛kC is uniquely determined by the previous ˛’s and

˛kC 2 Q.˛, ˛C1, : : : , ˛Ck0/.

8.3 Large solutions

We focus our interest to “large” integer solutions .u, v/ of (8.1), since solutions with

bounded juj are easily located. A first condition that we impose is that juj > B0, where

B0 is defined in Fact 8.2.1(d). Since the solutions .u, v/ of g D 0 with u < 0 are thesolutions of g D 0 with u > 0, where g.u, v/ D g.u, v/, in the description of the

method we can assume that u > B0. We need first a lemma concerning real solutions

.u, v/ with u > B0.

Lemma 8.3.1. Let g.u0, v0/ D 0 with u0, v0 2 R and u0 > B0. Among the n series(8.8) there is exactly one vs .u/ with all its coefficients ˛k,s real algebraic numberssuch that v0 D vs.u0/ , where u1=s is the real s -th root of 1=u.

Proof. By Fact 8.2.1(c) we have 0 D g.u0, v0/ D p0.u0/

QniD1.v0 vi .u0//. We

assumed that u0 > B0, therefore u0 is not a root of the polynomial

Resv.g,@g

@v/ D .1/n.n1/=2p0.u/ discv.g.u, v//; (8.10)

in particular, p0.u0/ ¤ 0. Consequently, v0 D vs.u0/ for some s 2 ¹1, : : : , nº. By

Fact 8.2.1(a), the coefficients ˛k,s (k D s , sC1, : : :) in (8.8) are algebraic num-

bers and now it suffices to show that all of them are real. We see this as follows. Let

0 2 Gal.K=Q/ be the automorphism which is obtained by restricting the complex

conjugation automorphism of C to K . Then, by Remark 2 on page 81, vs .u, 0, 0/ Dvj .u/ for some j . This means that the coefficients of the series vj .u/ are complex-

conjugates of the corresponding coefficients of vs.u/. Suppose now that some co-efficients of this last series are not real. Then the series vj .u/ and vs .u/ are dis-

tinct. Consequently, j ¤ s and vj .u0/ D vs .u0/ D v0 D v0 D vs.u0/, so that

g.u0, v/ D p0.u0/Qn

iD1.v vi .u0// 2 CŒv is a non-zero polynomial divisible by

.v vs .u0//.v vj .u0// D .v v0/2, hence it has a root of multiplicity greater than

1. But, as already mentioned, u0 is not a root of the right-hand side of (8.10), therefore

0 ¤ discv.g.u, v// juDu0 D discv.g.u0, v/,

which means that g.u0, v/ has no multiple root. This contradiction proves that all

coefficients of the series vs.u/ are real and completes the proof of the lemma.



Section 8.3 Large solutions 85

Notations and assumptions. From now on and until the end of the chapter, E will

denote the elliptic curve, a representative of which is the model C defined in (8.1),

another one being the model E referred to in Proposition 8.1.1. We will denote by P

the generic point of E such that the coordinates of P C

D .u.P /, v.P// are integers

and u.P/ > B0; later we will require that u.P / be larger than an explicit constant

larger than B0. Then, by Lemma 8.3.1, v.P / D vs .u.P // for some s 2 ¹1, : : : , nº,

where vs is as in Fact 8.2.1(a), with all its coefficients real algebraic numbers.

For a positive real number u, when we write uk=s we mean the unique s-th real

root of uk .

Referring to the birational transformation (8.3), we have the following proposition:

Proposition 8.3.2. The limit limu!C1X .u, vs.u// exists in R[¹˙1º. If this limit is

¤ ˙1 , then it is a real algebraic number that can be explicitly computed.

Next, definex.u/ D X .u, vs.u//, y.u/ D Y .u, vs.u//.

Then, there exists a B1 B0 such that, in the interval .B1, C1/ the functions x and y are continuous, x is strictly monotonous and y does not change sign.

Proof. First note that g.u, v/ cannot be a factor of either the numerator or the denomi-

nator of the rational functionX .u, v/. For, otherwise, the model C could be injectively

mapped into a straight line, which is impossible for a curve of genus 1.

Next, put u

D ts with t

2 R and 0 < t < B

1=s0 . Then, by Lemma 8.3.1 and

Fact 8.2.1(a),

x.u/ D X .ts ,

1XkDs

˛k,suk=s / D ˇt C ˇ0t 0 C ˇ00t 00 C : : :

t C 0t 0 C 00t 00 C : : : D x.t /, (8.11)

where ˇ , ˇ0, ˇ00, : : : , , 0, 00, : : : are non-zero real algebraic numbers and < 0 <

00 < : : : , < 0 < 00 < : : : are rational integers. This shows that

limu!1

x.u/D 8<:

ˇ= if D ,

0 if > ,

sgn.ˇ=/1 if < .

By Proposition 8.1.1, the rational function X is explicitly computable and the same is

true for the ˛k,s due to the algorithm on page 83. It follows that ˇ , and x0s can be

explicitly computed.

Next, since the polynomial g does not divide neither denominator of X or Y , there

exists a positive constant M 1 B0 such that, if g.u, v/ D 0 and u > M 1, then

.u, v/ is not a zero of neither denominator of X or Y . This implies that in the inter-

val .M 1 , C1/, the functions x and y are continuous, because X and Y are rational

functions and vs is continuous (cf. Fact 8.2.1(e)).




Concerning the strict monotonicity of the function x near C1, it suffices to show

the strict monotonicity of the right-hand side of (8.11) near 0C. But this function in t is

of the form t C 0t 0 C 00t 00 C , where , 0, 00, : : : are non-zero real numbers

and

D

< 0 < 00 < : : :. As t !

0C the derivative of this function is

non-zero with constant sign, and this proves our claim.

In complete analogy, we can express y as a series t C 0t 0 C 00t 00 C , where

, 0, 00, : : : are non-zero real numbers and < 0 < 00, : : :. Sufficiently close to 0C,

such a series, clearly, does not change sign.

Remark. In specific numerical examples the constant B1 can be explicitly calculated,

as described in Section 8.5.

Now we define a point P 0 that will be involved in the linear form L.P/ of the main

result of this chapter, namely, Theorem 8.7.2.

Definition 8.3.3. Let x0sdef D limu!1X .u, vs.u// and denote by P 0 D P 0s the point

of E with

P E0 D´

.x0s , y0s / if x0s ¤ ˙1O if x0s D ˙1 ,

where y 20s D f .x0s/ and y0s 0.

ExampleStep 3. (continued from the end of Step 2, page 82)The only Puiseux series with all its coefficients real is v3.u/. Therefore, according

to Lemma 8.3.1, for every integer solution .u, v/ of (8.4) with juj 484 we havev D v3.u/. Thus, in the notation of Proposition 8.3.2, x.u/ D X .u, v3.u// and, putting

u D t3 D t3 we write x.u/ as a series in t (cf. relation (8.11))

x.t/ D 2169 C 8883t 52002t 2 C 546057t 3 C O.t 4/.

Then, according to Definition 8.3.3, P 0 D P 03 D .2169, 79947=2/.


8.4 The elliptic integrals

In this section we find an explicit relation between two elliptic integrals. The proposi-

tion below is analogous to Proposition 6.4 in the case of the quartic elliptic equation.

We keep the notation of previous sections; in particular that in (8.2) and (8.3).

Proposition 8.4.1. Let P 2 E with u.P / > B0 and v.P / D vs .u.P //.3 Let x, y and B1 be as in Proposition 8.3.2. For u > B1 we define

G.u, v/ D 2Y u.u, v/ gv.u, v/ Y v.u, v/ gu.u, v/

3X 2.u, v/ C A ,

3 Cf. “Notations and assumptions” on page 85.



Section 8.4 The elliptic integrals 87

where subscripts u or v in functions indicate partial derivatives with respect to u or v , respectively, and

g.u/ D G.u, vs.u//.

Let " D sgn.y.u// , so that we have

y.u/ D "

q f .x.u//.

Then, Z C1u.P/

g.u/ du

gv.u, vs.u// D "

Z x0s

x.P/

dxp f.x/

. (8.12)

Proof. In this proof, if R 3 u 7! h.u/ 2 R is a differentiable function, then h0.u/

will denote the derivative of h.

Differentiating y.u/ D Y .u, vs .u// and y.u/ D "p f .x.u// we respectively get

y0.u/ D Y u.u, vs.u// C Y v.u, vs.u// v0s.u/. (8.13)

and

y0.u/ D " 3 x.u/2 C A

2p

f .x.u// x0.u/. (8.14)

The relation g.u, vs.u// D 0 implies gu.u, vs .u// C gv.u, vs.u//v0s .u/ D 0. Solving

for v0s.u/ and substituting in the relation which results on equating the right-hand sides

of (8.13) and (8.14), gives

x0.u/ D " g.u/ p

f .x.u//

gv.u, vs.u//. (8.15)

In view of the strict monotonicity of the function x we can make the change of variable

in the integral belowZ x0s

x.P/

dx

p f.x/

xDx.u/DZ C1

u.P/

x0.u/ du

p f .x.u//

.8.15/D "

Z C1u.P/

g.u/ du

gv.u, vs .u//,

as claimed.

For Ellog it is important that the integrand in the left-hand side of (8.12) tends to

zero as u ! C1. This is made precise in the proposition below.

Proposition 8.4.2. There exists a sufficiently large constant B2 B1 such that, if u B2 , then ˇ

ˇg.u/

gv.u, vs.u//

ˇˇ c9juj1 , (8.16)

where 1

s and c9 is a positive constant independent from u.




Proof. We write (8.15) as an equation of differentials

dx

yD g.u/

gv.u, vs.u// du.

The equation y2 D f .x/ is parameterised by

x D x.t/ D t2, y D y.t / D t3 C 1

2At C 1

2Bt 3 1

8A2t 5 1

4ABt 7 C O.t 9/,

hence, Z dx

yDZ

x0.t /

y.t / dt D 2 C At 4 C Bt 6 C O.t8/

and we see that there are no singularities in the expansion of

R dx=y. This, by defini-

tion, means that the elliptic integral is of the first kind (see [4, §24]). Therefore, the

same is true for the integral R .g.u/=gv.u, v// du, implying that, for any parameteri-sation .u, v/ D .u.t/, v.t// of g.u, v/ D 0, the t -expansion of

g.u/

gv.u.t/, v.t// du

dt

has no negative powers of t . Applying this conclusion to the parameterisation

u.t/ D ts , v.t/ D vs .ts / D1X

kDs

˛k,st k,

and taking into account that du=dt D sts1, we are led to the inequality

ordtg.ts /

gv.ts , vs .ts // s C 1.4

This means that

g.ts /

gv.ts , vs.ts// D t 1CsC .ˇ0 C ˇ1t C ˇ2t 2 C /, 0, ˇ0 ¤ 0.

Putting t

s

D u in the relation above we conclude thatg.u/

gv.u, vs .u// D u1 .ˇ0 Cˇ1u1=s Cˇ2u2=s C /, D .1C/=s 1=s .

For sufficiently large u, the absolute value of the right-hand side is bounded above by

c9juj1 , as claimed.

Remark. In specific numerical examples the constants B2, c9 and of Proposition 8.4.2

can be made explicit according to Section 8.6.

4 The function ord has been defined just before the algorithm on page 83.



Section 8.5 Computing in practice B1 of Proposition 8.3.2 89

8.5 Computing in practice B1 of Proposition 8.3.2

In Proposition 8.3.2 we introduced the constant B1, the existence of which is guaran-

teed by a non-constructive proof. In any specific equation, however, we must explicitlycompute such a B1, and in this section we show how we can do this. First we state a

simple lemma.

Lemma 8.5.1. Let F be a polynomial in two variables with real coefficients and let V : R ! R be a continuous function, such that F .u, V.u// D 0 for juj > U 0 ,where U 0 is a positive constant. Let R be the set of all real roots of the polynomialF .0, Y / , and S D ¹u : juj > U 0 & V.u/ 2 Rº. Put .U min, U max/ D .U 0, U 0/

or .U min, U max/ D .minS ,maxS / according to whether S is or is not empty. Thenthe function V keeps a constant sign in the interval .U max ,

C1/ and does so in the

interval .1 , U min/.

Proof. Suppose that u2 > u1 > U max and V .u2/V.u1/ < 0. Then, by the continuity

of V , there exists a u0 2 .u1, u2/, such that V .u0/ D 0. This means that u0 2 S , hence

u0 U max. But u0 > u1 > U max, a contradiction.

Similarly we arrive at a contradiction if we assume that u2 < u1 < U min and

V .u2/V.u1/ < 0.

We will use this lemma in order to show how to compute M 1,min and M 1,max abso-

lutely larger than B0, such that y keeps a constant sign in the interval .M 1,max ,

C1/

and does so in the interval .1 , M 1,min/.

Let D.u, v/ be the square-free part of the product of the denominators of X and Y .

Since g is an absolutely irreducible polynomial and does not divide neither denomi-

nator of X or Y , the resultant Resv.D, g/ is a non-zero polynomial in u; we denote

by R the maximum absolute value of its real roots. Let any u with juj > max¹B0, Rº.

After clearing out the denominator in y.u/ D Y .u, vs.u//, we obtain the polynomial

relation H 1.u, vs.u/, y.u// D 0. On the other hand, we also have g.u, vs.u// D 0 and

we can eliminate vs.u/ from the last two relations, using resultants. We thus obtain

a polynomial relation R1.u, y.u//

D 0. Now, applying Lemma 8.5.1 with F

D R1

and V D y we compute values U min, U max (in the notation of the lemma) and we takeM 1,max D dmax¹U max, B0, Rºe and M 1,min D bmin¹U min, B0, Rºc.

Next we show how to compute M 2,mi n and M 2,max such that in both intervals,

.M 2,max , C1/ and .1 , M 2,mi n/, x is strictly monotonous.

Again, let any u with juj > B0. Clearing out the denominator in x.u/ DX .u, vs .u// gives a polynomial relation H 2.u, vs.u/, y.u// D 0. From this relation

and g.u, vs .u// D 0 we eliminate vs .u/, using resultants. Thus we obtain a polyno-

mial relation R2.u, x.u// D 0. Differentiating with respect to u gives

@R2

@u .R2.u, x.u// C @R2

@x .R2.u, x.u// x0

.u/ D 0, (8.17)




where x0 means derivative with respect to u. This is a polynomial relation

H 3.u, x.u/, x0.u// D 0. Eliminating x.u/ from the last relation and R2.u, x.u// D 0,

we get a polynomial relation R3.u, x0.u// D 0 and we apply Lemma 8.5.1 with

F D

R3

and V D

x0 to obtain the values U min

and U max

(in the notation of

the lemma). Then, it suffices to take M 2,max D dmax¹U max, B0, Rºe and M 2,min Dbmin¹U min, B0, Rºc.

Finally,

B1 D max1i2

¹jM i ,minj , jM i ,maxjº.

ExampleStep 4. (continued from the end of Step 3, page 86)The constant B0 appearing in Fact 8.2.1(d) is already computed: B0 D 484. Therefore

we will search for integer solutions .u, v/ with

ju

j 484. By (8.6), R

D1, so that the

requirement u > R is already fulfilled.An easy search shows that the only integer solutions with juj 483 are .u, v/ D

.0, 0/, .1, 3/, .243, 3/.

We first calculate M 1,max and M 1,min. In the notation of our previous discussion,

we calculate H 1.u, vs .u/, y.u//, a straightforward task. It turns out that H 1 is the

sum of 19 monomials and deguH 1 D 4, degvsH 1 D 4, degyH 1 D 1. Then, us-

ing MAPLE, the computation of R1.u, y.u// presents no difficulty; the result is the

product of 81u5.u 1/12 with a polynomial in u and y, let us denote it by R10,

which is the sum of 24 monomials with some coefficients having 36 decimal digits,

degu.R10/ D 3 and degy D 5. Since we have already assumed that juj > 484, we musthave R10 D 0 and we apply Lemma 8.5.1 with F D R10, U 0 D B0 D 484, V D y.

We have F .0, y/ D .2y C 81029/.2y 35626473/4, therefore R D ¹1, 2º, where

1 D 81029=2 and 2 D and 2 D 35626473=2. The solutions of the equation

y.u/ D 1 are approximately 2339.17722, 346.44760 and 0. Hence, in the nota-

tion of Lemma 8.5.1, S D ¹2339.17722º, U 1,min D U 1,max D 2339.17722 and,

consequently, M 1,max D 484 and M 1,min D 2340, which implies that y does not

change sign in the interval .1 , 2340/, and the same is true for y in the interval

.484, C1/.

Next we compute M 2,max and M 2,min. The polynomial H 2.u, vs.u/, x.u// has 13

monomial terms and deguH 2 D 3, degvsH 2 D 4, degxH 2 D 1. The resultant

R2.u, x.u// is the product of 81u5.u 1/8 with a polynomial R20.u, x/ which is

the sum of 18 monomials of degree 2 with respect to u and degree 5 with respect to x;

its coefficients have at most 23 decimal digits; then, necessarily, R20 D 0.

According to our discussion before the example, the left-hand side of (8.17) is a

polynomial H 3 in u, x and x0, linear in x0. Eliminating x from R20 D 0 and H 30 D 0,

we obtain R3.u, x0.u// D 0, where R3.u, x0.u// D Resx.R20, H 3/. It turns out (with

the aid of MAPLE) that R3 is the product of a 22 decimal-digit integer with the square of

a cubic polynomial in u (only), the roots of which belong to the interval .540 , 311/,

and a polynomial R30 in u and x0

which is the sum of 35 monomials, having coeffi-



Section 8.6 Computing in practice B2 and c9 of Proposition 8.4.2 91

cients with at most 33 decimal digits; also, degu.R30/ D 10 and degx0.R30/ D 5.

If we assume that juj 540, then, necessarily, R30 D 0 and we apply Lemma

8.5.1 with F D R30, V D x0, U 0 D B0 D 484. Now F .0, x0/ D .271x0 299/

times a large constant, therefore R D ¹

299=271º

. The approximate solutions of

R30.x, 299=271/ D 0 are 932.85436, 738.24160, 173.11938 and 0, therefore,

S D ¹932.85436 : : : , 738.24160 : : :º. Consequently, M 2,mi n D 933, M 2,max D484 and Lemma 8.5.1 implies that x0 is non-zero in the interval .1 , 933/ and has a

constant sign, hence x is strictly monotonous. Similarly, x is strictly monotonous in the

interval .484, C1/. Finally, by the displayed formula for B1 we obtain B1 D 2340.

Therefore, from now on, we will assume that juj > B1 D 2340.


8.6 Computing in practice B2 and c9 of Proposition 8.4.2At first we work as in Section 8.5. We denote by I D I .u/ the left-hand side of (8.16)

and clear out the denominator to obtain a polynomial relation H 4.u, vs .u/, I / D 0.

Using resultants we eliminate vs .u/ from the last relation and g.u, vs.u// D 0. Thus

we obtain a polynomial relation

I m C q1.u/ I m1 C C qm1.u/ I C qm.u/ D 0, (8.18)

where qi .u/ 2 Q.u/ for i D 1, : : : , m. We distinguish the cases of positive and

negative I . Let I be positive. For j

uj

B2

, where B2

B1

is sufficiently large

(how large can be made explicit in each specific numerical example), each qi .u/ has

constant sign. Let qj .u/, qj 0.u/, : : : with 1 j < j 0 < : : : m be the strictly

negative coefficients in (8.18) and let k be their number. By the so-called Cauchyrule,5 we have

I max¹.kjqj .u/j/1=j , .kjqj 0.u/j/1=j 0 , : : :º,

from which we find an inequality of the form I constant u1 , where the “con-

stant” and > 0 is explicit. Next, we replace I by I and repeat the argument to

obtain a similar upper bound for I . Combining the two upper bounds we get an

upper bound forj I

jof the form c

9 u1 , where c

9 is an explicit positive constant.

ExampleStep 5. (continued from the end of Step 4, page 91)Following the instructions above we compute

H 4.u, vs.u/, I / D u2.u 1/4H 40.u, vs.u/, I /,

5 By this we mean the following: Let f .X/ D X n C a1X n1 C C an 2 RŒX and let al , am, : : :

be the negative coefficients, their total number being k , where l > m > : : :. Then every real root of f .X/ satisfies max¹.kjal j/1= l , .kjamj/1=m , : : :º; the result and its naming is found in [31,

Chapter V.4]; probably, to other people’s minds, “Cauchy’s rule” (for the roots of a polynomial) means

a result of this type, but not quite the same.




where H 40.u, vs.u/, I / is the sum of 84 monomials, degu.H 40/ D 7, degvs .H 40/ D 12

and, of course, H 40 is linear in I . Then,

Resvs.H 40.u, vs.u/, I /, g.u, vs .u//

Dconstant u12.u 1/16.81000u C 19902511/.784u 17846163/h.u/R4.u, I /,

where h.u/ is a second degree polynomial. In the right-hand side, the roots of the

non-constant factors, except for R4.u, I / belong to the interval .696 , 22763/. Also,

R4.u, I / is the sum of 12 monomials, degu.R4/ D 7 and deg I .R4/ D 5. If we assume

that juj 22763, then R4.u, I / D 0 and we write the last relation as follows:

I 5 C q2.u/ I 3 C q3.u/ I 2 C q4.u/ I C q5.u/ D 0, (8.19)

(q1.u/ D 0) with

q2.u/ D 49

2025u2

C 1491042u C 318440176uq.u/

q3.u/ D 32

9 3u C 1355

uq.u/

q4.u/ D 1175056

243 u3q.u/

q5.u/ D 32

243 u3q.u/,

whereq.u/ D26244u4 C 20515275u3 C 1099852416u2 2071134904704u

374185039449856.

Now we work as follows. Consider q2.u/. Its numerator has no real roots and those

of the denominator belong to the interval .484 , 318/. But we have already assumed

that juj 22763 and we compute that q2.22763/ < 0 and q2.22763/ > 0. Hence, if

u 22763, then q2.u/ < 0 and if u 22763, then q2.u/ > 0. In an analogous way

we check that, if juj 22763, then q3.u/ > 0, and if u 22763, then both q4.u/

and q5

.u/ are negative, and if u

22763, then both q4

.u/ and q5

.u/ are positive.

In view of the above discussion, if u 22763, then qi .u/ > 0 for i D 2,3, 4, 5 and,

consequently, in (8.19), I < 0. Setting I D J < 0 we obtain the equation

J 5 C q2.u/J 3 q3.u/J 2 C q4.u/J q5.u/ D 0, (8.20)

where now the strictly negative coefficients are q3.u/ and q5.u/. By Cauchy’s rule,

0 < J < max¹.2q3.u//1=3 , .2q5.u//1=5º.

We have

q3.u/ D 32

9 3u4

C1355u3

q.u/ u

4

< .2=27/

3

u

4



Section 8.6 Computing in practice B2 and c9 of Proposition 8.4.2 93

and

q5.u/ D 32

243 u4

q.u/ u7 < .23=313/ u7.

Hence, by Cauchy’s rule,

0 < I D J < max¹.24=3=27/ u4=3 , .24=5=313=5/ u7=5º < 0.1001 u4=3

Consider now (8.19) when u < 22763. The strictly negative coefficients in (8.19)

are q2.u/, q4.u/ and q5.u/.

Let first I 0. Then, by Cauchy’s rule,

0 < I < max¹.3jq2.u/j/1=2 , .3jq4.u/j/1=4 , .3jq5.u/j/1=5º.

We have

jq2.u/j D 4

9 j2025u4 C 1491042u3 C 318440176u2j

jq.u/j juj3

jq4.u/j D 1175056

243 u4

jq.u/j juj7

jq5.u/j D 32u4

243 jq.u/j juj7.

From these relations we easily obtain the following upper bound for I :

0 < I < 0.11 juj7=5.

Next, let I < 0. We put I D J with J > 0, so that the equation (8.20) holds. Since

u 22763, the strictly negative coefficients in (8.20) are now qi .u/ with i D 2,3,4.

By Cauchy’s rule,

0 < J < max¹.3jq2.u/j/1=2 , .3jq3.u/j/1=3 , .3jq4.u/j/1=4º.

Working as we did a few lines above, we obtain 0 < I D J < 0.11juj4=3

.

Combining all our partial results above we come to the following conclusion:

If juj 22763, then the left-hand side of (8.16) is bounded above by 0.11 juj4=3,

hence, in the notation of Proposition 8.4.2, B2 D 22763, D 1=3 D 1=s and

c9 D 0.11.

Remark. An easy search for integer solutions of g.u, v/ D 0 in the range juj < 22763

reveals no further solutions than those already found, namely, .u, v/ D .0, 0/, .1, 3/

and .243, 3/.





8.7 The linear form L.P/ and its upper bound

The goal of this section is the definition of a linear form L.P / in elliptic logarithms

and the proof of Theorem 8.7.2 which provides us with an explicit upper bound forjL.P/j.We keep the “Notations and assumptions” on page 85, except that now the condition

u. P / > B0 is replaced by the stronger condition u.P/ > B2 (see the conclusion

in page 93). Note that the case u.P/ < B2 is treated in a completely analogous

manner, as the solutions of g.u, v/ D 0 with negative u coincide with the solutions of

g.u, v/ D 0 with positive u, where g.u, v/ D g.u, v/.

We also keep the notations, assumptions and results of Chapter 4. In particular, e1

denotes the largest real root of the polynomial f .X/ in (8.2) if this polynomial has

three real roots and e1 is the only real root of f .X/ otherwise. We distinguish two

cases regarding the relative position of e1 and x0s (cf. Definition 8.3.3 and Proposition8.3.2).

We have P C D .u.P /, vs .u.P /// and P E D .x.P/, y.P// D .x.u.P //,

y.u.P ///; see Proposition 8.3.2. By that proposition, the continuous function u 7!x.u/ is strictly monotonous in the interval .B1 , C1/ with limu!C1 x.u/ D x0s. A

moment’s thought then shows that, if x0s e1 then x.P / D x.u.P // e1, and if (in

the case that also e2 and e3 are real) e3 x0s e2, then e3 x.P/ D x.u.P // e2.

In other words, P E is a point of E0.R/ or E1.R/ (cf. beginning of Section 3.5), ac-

cording to whether P E0 is a point of E0.R/ or E1.R/, respectively.

By Proposition 8.3.2 we also know that y.u/ has a constant sign when u 2.B1 , C1/, which we will denote by ".

By (8.12) we have

"

Z C1u.P/

g.u/ du

gv .u, vs.u// DZ x0s

x.P/

dxp f.x/

(8.21)

and we distinguish two cases.

(1) P 0 2 E0.R/. Then P E 2 E0.R/ and using (3.32),6 we write the right-hand side

of (8.21) as follows:

Z x0s

x.P/

dxp f.x/ D Z C1x.P/

dxp f.x/ Z C1x0s

dxp f.x/ D 2"P l.P / 2 l.P 0/.

(2) P 0 2 E1.R/.7 Then, P E 2 E1.R/. Making use of the point Q2, that point of

E with QE2 D .e2, 0/ (cf. Lemma 6.6), we write the right-hand side of (8.21) as

follows:Z x0s

x.P/

dxp f.x/

DZ e2

x.P/

dxp f.x/

Z e2

x0s

dxp f.x/

6 The fact that in the definition of P 0 we assumed y.P 0/ 0 is also used.

7 Case possible only if all three roots of f .X/ are real.



Section 8.7 The linear form L.P/ and its upper bound 95

DZ C1

x.P CQ2/

dxp f.x/

Z C1

x.Q2CP 0/

dxp f.x/

(by Lemma 6.6)

D 2"P l.P /

2 l.P 0/. (by (3.32))

Thus, in both cases (1) and (2) we have, in view of (8.21),

jl.P / C "P l.P 0/j D 1

2

ˇZ C1u.P/

g.u/ du

gv.u, vs .u//

ˇ. (8.22)

Now we will prove a proposition which will permit us to compute an upper bound for

the right-hand side of (8.22) in terms of u.P /.

Proposition 8.7.1. Let us write the relation g.u, v/ D 0 in the form

vn

C a1.u/vn1

C C an1.u/v C an.u/ D 0 (8.23)

where the ai ’s are polynomials in u. Let B3 be a constant larger than every root of every non-zero polynomial ai .

If P C 2 C.Z/ and u.P/ > max¹B2, B3º , then

h.x.P // c10 C c11 log ju.P/j, (8.24)

where, h./ denotes absolute logarithmic height and c10, c11 are explicitly computable positive constants, independent from P .

Proof. We have v.P / D vs.u.P //. We write X D F 1=F 2 for some relatively primepolynomials with rational integer coefficients. For simplicity in the notation of this

proof let us put u.P / D u and v.P / D v, so that

h.x.P // D h.X .u.P /, v.P// log max¹jF 1.u, v/j, jF 2.u, v/jº D log jF j .u, v/jfor the proper choice of j D 1,2.

Let 1 i n be such that ai is a non-zero polynomial. Then ai .u/ does not

change sign as u runs through the values > B2. Therefore, for u > B2, it makes sense

to distinguish all the (strictly) negative coefficients, say am1 .u/, am2.u/, : : : , amk.u/,

in the left-hand side of (8.23). If v 0, Cauchy’s rule

8

implies that

0 v max1ik

.kjamij/1=mi

and, clearly, the right-hand side is bounded by ˛0uˇ 0

, where ˛0 and ˇ0 are explicit

positive constants (ˇ0 2 Q). If v < 0, we put v D w with w > 0 and we rewrite

(8.23) as w n C b1.u/wn1 C C bn1.u/w C bn.u/ D 0, where bi D ˙ai for

i D 1, : : : , n. As before, Cauchy’s rule implies a relation of the form 0 < w ˛00uˇ 00

and, combining the two upper bounds we obtain an explicit bound jvj ˛uˇ .

8 See Footnote 5.




Next, we write F j .u, v/ D P.k,l/ ak,l ukvl and let c11 D max.k,l/¹k C lˇº, where

the maximum runs over all pairs .k, l/ for which ak,l ¤ 0. Then, jF j .u, v/j c10uc11 ,

where C 10 D

P.k,l/ jak,l j˛l and (8.24) holds with c10 D log C 10.

ExampleStep 6. (continued from the end of Step 5, page 93)We have a1 D 0, a2 D u, a3 D 0, a4 D 271

3 u and a5 D u2, hence we can take

B3 D 1.

We put .u.P /, v.P// D .u, v/ and, according to Proposition 8.7.1, we assume that

juj max¹B2, B3º D B2 D 22730.

Assume u > 22730.

If v 0, then Cauchy’s rule implies 0 v max¹.2ja4j/1=4 , .2ja5j/1=5º Dmax¹. 542

3 u/1=4 , 5

p 2 u2=5º D 5

p 2 u2=5.

If v < 0, we put v D w with w > 0 so that g.u, v/ D 0 is written as w5 C uw3 271

3 uw u2 D 0. In the notation of our previous discussion, b1 D 0, b2 D u, b3 D

0, b4 D 271

3 u, b5 D u2, hence 0 < v D w . 271

3 u/1=4, by Cauchy’s rule.

Combining the two upper bounds we conclude that jvj 5p

2 u2=5.

Next, assume u 22730. Then, we consider g.u, v/ D g.u, v/ instead of

g.u, v/. Working as above we obtain the bound jvj p

2 juj1=2.

Thus, in general, for juj 22730 we have jvj p

2 juj1=2 and, consequently, the

absolute value of the numerator of X .u, v/ is, easily, bounded by

2710juj C 66879juj5=2p 2 C 43821juj3=2p 2 C 52542juj2 C 29331juj3 < 30000juj3

and, clearly, 30000juj3 is an upper bound for the absolute value of the denominator

u.u 1/2 of X .u, v/. Thus,

h.x.P // D log jx.P/jD log max¹numer.jX .u.P /, v.P//j/ , denom.jX .u.P /, v.P//j/º log.30000juj3/ D log.30000/ C 3log ju.P/j

and, consequently, c10 D log.30000/ and c11 D 3.Example continued on page 118.

Theorem 8.7.2. Let E be the elliptic curve represented by the model C in (8.1) , subject to the conditions at the beginning of this chapter.

Let E : y2 D x3 C Ax C B def D f.x/ be the short Weierstrass model with A, B

defined in Proposition 8.1.1 and consider that everything in Chapter 4 , in particular,equation (4.1) and relations (4.4)–(4.8) , refer to this particular model E .

Let X be the rational function defined in Section 8.1 and let P 0 be the point of E in

Definition 8.3.3.



Section 8.7 The linear form L.P/ and its upper bound 97

Assume that P 2 E is such that P C D .u.P /, v.P// has integer coordinateswith u.P / max¹B2, B3º 9 and v.P / D vs .u.P // for some Puiseux series vs (see

Fact 8.2.1 and (8.8)) all the coefficients of which are real (see “Notations and assump-

tions” on page 85).Write P E as in (4.2) , consider l.P / as in (4.6) and define

L.P/ D l.P / C "P l.P 0/,

where "P D 1 or 1 , according to whether y.P / > 0 or y.P / 0 , respectively.Then

jL.P/j c9

1 C exp

.2 C c10/

c11

2

c11

M 2

, (8.25)

where c9 and are as in Proposition 8.4.2 ,10 c10 and c11 are as in Proposition 8.7.1

and and are defined in Propositions 2.6.3 and 2.6.2 , respectively.Proof. By the definition of L.P/ and relations (8.22) and (8.16) we see that jL.P/j

c9

1Cju.P/j . By (8.24), ju.P/j1 exp¹.c10 h.x.P ///=c11º and, therefore,

jL.P/j c9

1 C exp

.c10 h.P //

c11

. (8.26)

By Proposition 2.6.3, h.x.P // 2 2 Oh.P/ 2 2M 2, where the right-most

inequality holds because of Proposition 2.6.2 and (4.7). Replacing for h.P/ in (8.26)

completes the proof.

9 For B2 see Proposition 8.4.2 and Section 8.6; for B3 see Proposition 8.7.1.

10 See also Section 8.6.



Chapter 9

Bound for the coefficients of the linear form

In each of Chapters 5, 6, 7 and 8 we defined a convenient linear form L.P /, where

P denotes the point which “covers”1 the generic (“sufficiently large”) solution of the

Diophantine problem considered in the chapter. Theorems 5.2, 6.8, 7.1 and 8.7.2 of the

corresponding chapters provide us with an upper bound for jL.P/j as a function of M ,

the maximum absolute value of the coefficients of L.P/. In the present chapter we will

give a lower bound for jL.P/j, again as a function of M . As it will turn out, comparing

the upper and lower bound will result to an upper bound for M and, consequently, tothe effective solution of the Diophantine problem under consideration; see Chapter 4

immediately after (4.8).

9.1 Lower bound for linear forms in elliptic logarithms

In each of the Theorems 5.2, 6.8, 7.1 and 8.7.2, the linear form L.P / to which the

corresponding theorem refers, has a direct simple relation with the linear form

l.P / D .m0 C s

t /!1 C m1l.P 1/ C C mr l.P r /,

defined in (4.6). For the above displayed linear form the relations (4.3)–(4.8) are valid;

in particular M D max1ir jmi j and M 0 D max0ir jmi j. We put `i D l.P i /

for i D 1, : : : , r . Possibly in Theorem 6.8 and, anyway, in Theorems 7.1 and 8.7.2,

another point P 0 is involved. The coordinates of P E0 belong to a number field which

is of degree at most two in the case of Theorem 6.8 and four in the case of Theorem

7.1; for the case of Theorem 8.7.2 there is no a priori bound for the degree of the

coordinates of P E0 .

We will also put `0 D l.P 0/. Finally, we will put m00 D m0, except in that sub-case

of Theorem 6.8 when x0 D e1, in which we will put m00 D m0 C 2.

Having agreed to the above, we can give a first uniform shape to L.P /, namely

L.P/ D

m00 C s

t

!1 C m1`1 C mr `r C "`0, (9.1)

1 For this terminology, see a few lines below (4.2).



Section 9.1 Lower bound for linear forms in elliptic logarithms 99

where

" D

8ˆ<:

0 if L.P / comes from Theorem 5.2

0 if L.P / comes from Theorem 6.8 when x0 D e1

˙1 if L.P / comes from Theorem 6.8 when x0 ¤ e1

1 if L.P / comes from Theorem 7.1

˙1 if L.P / comes from Theorem 8.7.2

and

m00 D

´m0 C 2 if L.P / comes from Theorem 6.8 when x0 D e1

m0 otherwise.

Below we will write (9.1) in the more convenient form

L.P/ D n0

dr0!1 C

n1

d `1 C C nk

d `k (9.2)

where the meaning of the various quantities involved in it will be made explicit by

distinguishing three cases.

Let " D 0. Remember the definition of r0 at the beginning of Chapter 4, a few lines

after (4.1), and the relation (4.5), according to which, t jr0. Therefore, we can write

m00 C s

t D n0

r0

, n0 2 Z.

On putting k D r , d D 1, ni D mi .i D 1, : : : , k/,

we transform (9.1) into (9.2).

Next, we will find an elementary upper bound for the maximum of the jmi j’s in

terms of M . By (4.7) we have max1ik jni j M and, by (4.5) and (4.8),

jn0j t jm00j C jsj N 0

def D r0 max¹M , 1

2rM C 1º C 1

2r0 (9.3)

where

D ´5 if L.P / comes from Theorem 6.8 when x0 D e1

1 otherwise.

Therefore, by (9.3),

N def D max¹jni j, i D 0,1, : : : , kº N 0.

Let " D ˙1 and the points P E1 , : : : , P Er , P E0 areZ-linearly independent. Clearly, a

necessary condition for this case is that P E0 62 E.Q/ but, certainly, this condition is

by no means sufficient. In this case, m00 D m0 and, in analogy with the case " D 0,

we put

m0 C s

t D n0

r0 , n0 2 Z



100 Chapter 9 Bound for the coefficients of the linear form

and

k D r C 1, d D 1, ni D mi .i D 1, : : : , k 1/, `k D `0, nk D ",

so that we transform (9.1) into (9.2). Moreover,

N def D max¹jni j, i D 0,1, : : : , kº N 0,

where N 0 is again given by the right-most side of (9.3) with D 1.

Let " D ˙1 and the points P E1 , : : : , P Er , P E0 are Z-linearly dependent. In this case

there exist integers ı, d 1, : : : , d r with ı 1 and gcd.ı, d 1, : : : , d r / D 1, such that

ıP E0 D d 1P E1 C C d r P Er C T 0.

Since the points P Ei (i

D 0,1, : : : , r) are known, we will consider the d i ’s as inte-

gers that are explicitly known. Applying the homomorphism l to the last displayedrelation, we get ı`0 d 1`1 C d r `r C s0

t0!1 .mod Z!1/, where2 l.T 0/ D s0=t0.

From this it follows that

ı`0 D d 1`1 C d r `r C s0

t0

!1 C ı0 !1,

where ı0 is an appropriate explicitly known integer. We note that the integers s0, t0

satisfy conditions similar to those in (4.5) with s0 and t0 in place of s and t , respec-

tively; in particular, t0jr0 and js0=t0j 1=2.

Now our linear form becomes (note that m00

Dm0 in this case)

L.P/ D .m0 C s

t C ı0

ı C s0

t0ı/!1 C .m1 C d 1

ı /`1 C C .mr C d r

ı /`r . (9.4)

Since t and t0 are divisors of r0, we can put

m0 C s

t C d 0

ı C s0

t0ı D n0

dr0

for some integer n0 and an appropriate positive divisor d of ı.3 We also put

k

Dr , ni

Dd mi

Cd i .i

D1, : : : , k/,

so that L.P/ is once again expressed by (9.2), and we compute now an upper bound

for max0ik jni j. Obviously, max1ik jni j dM C max1ik jd i j. For jn0j we

take into account that js=t j, js0=t0j 1=2 (cf. (4.5)) and we compute

jn0j D jd m0r0 C s

t d r0 C r0d 0 C s0

t0

r0j dr0jm0j C 1

2dr0 C r0d 0 C 1

2r0

dr0 max¹M , 1

2rM C 1º C 1

2.d C 1/r0 C jd 0jr0,

2 Cf. the discussion just before (4.5).

3 In general we can certainly take d D ı, but sometimes we can choose a smaller divisor of ı .




where, in the last inequality we have used (4.8). Therefore,

N def D max¹jni j, i D 0,1, : : : , kº N 0,

whereN 0 D max¹dM C max

1ikjd i j, dr0 max¹M , 1

2rM C1ºC 1

2.d C1/r0Cjd 0jr0º. (9.5)

Usually, in specific numerical applications it is not necessary to calculate N 0 from

(9.5); it is rather preferable to bound N in terms of M directly, using the expressions

of the ni ’s in terms of the mi ’s.

Conclusion. In all three cases described in the “bullets” above, we have N M

(obvious from the definitions of N and M ) and the upper bound N 0 of N given by

either (9.3) or (9.5), or otherwise, easily implies an upper bound N

˛M C

ˇ,

where ˛ and ˇ are explicit “small” positive constants. Thus, the linear form L.P/

involved in Theorems 5.2, 6.8, 7.1 and 8.7.2 can always be put in the form (9.2) and

for N D max1ik jni j an upper bound (9.6) holds for some explicitly computable

“small” positive constants ˛ and ˇ:

M N ˛M C ˇ. (9.6)

Remark. A remark concerning the points to which the ì ’s refer is necessary. Ac-

cording to Conclusions and remarks (2) on page 52, if ì D l.P i / and it happens that

P i 2

E1.Q/,4, then ì is the elliptic logarithm of the point P E

i C QE

2

, where, as

usually, QE2 D .e2, 0/. Therefore, if e2 62 Q, then e2 belongs to either a quadratic or

a cubic number field, hence this will be the case with the coordinates of P Ei C QE2 ,

as well; anyway, D 3. In the case that ì D `rC1 D l.P 0/, the coordinates of P E0belong to Q.

p a/ if we apply Theorem 6.8 or to Q.

p D1,

p D2/ if we apply Theorem

7.1. In the case of Theorem 6.8, if P E0 2 E1.R/, then l.P 0/ is the elliptic logarithm

of the point P E0 C QE2 , the coordinates of which belong to a number field of degree

D 6 (D ¤ 5). In the case of Theorem 7.1, P E0 2 E0.R/ and QE2 2 E.Q/, as

already noted on page 75. Therefore, the coordinates of P E0 C QE2 belong, anyway,

to Q.p D1,p D2/, so that D is a divisor of 4 in this case. If we apply Theorem 8.7.2

we cannot be so specific concerning the number field generated by the coordinates of

P E0 C QE2 in the case that P E0 2 E1.R/.

Our next task is to compute an explicit upper bound for M by using a very important

result of S. David, namely, Theorem 9.1.2. The following strategy will be followed.

Theorem 9.1.2 will provide us with a lower bound for L.P/ as a function of N –

hence, as a function of M as well – which we will combine with the upper bounds

of jL.P/j obtained in Theorems 5.2, 6.8, 7.1 and 8.7.2. The upper bounds given in

those theorems are again functions of M . However, as it will turn out, if M is “very

4 Necessarily then, f .X/ has three real roots e3 < e2 < e1.




large”, then the upper bound function is smaller than the lower bound function and

this contradiction will lead us to an upper bound for M .

We proceed now to our task. First, let us note that, for the application of Theo-

rem 9.1.2, we will need to compute a fundamental pair of periods .$ 1

, $ 2

/ as in

Lemma 9.1.1 below. By Theorem 3.4.1 and Section 3.4.4 we can explicitly calcu-

late a fundamental pair of periods .!01, !0

2/ with ! 01 D !1. From this pair it is possible

to compute the pair .$ 1, $ 2/.

Lemma 9.1.1. For the Weierstrass function associated to (4.1) there exists a fun-damental pair .$ 1, $ 2/ of periods with the property that, if we set Q D $ 2=$ 1 , thenthe following inequalities are satisfied:

=Q > 0, j Q j 1, j<Q j 1

2.

Proof. First a remark concerning notation: For any matrix M D . a bc d / 2 GL2.Z/

and any non-real 2 C we write, by definition,

M def D a C b

c C d .

Now we proceed to the proof. Let .!1, !02/ be a fundamental pair of periods for

obtained by Theorem 3.4.1, where, in the notation of that theorem, !02 D !2 in the case

of positive discriminant, and ! 02 D .!1 C !2/=2 in the opposite case. Let D !0

2=!1

(note that = > 0). By [45, Proposition 12.1], there exists

M 0 D

a0 b0

c0 d 0

2 SL2.Z/

such that M 0 satisfies both j<.M 0 /j 1

2 and jM 0 j 1. Actually, M 0 is a

“word” in SL2.Z/ generated by

S D

0 1

1 0

and T D

1 1

0 1

.

Put $ 1 D c0!

0

2 C d 0!1 and $ 2 D a0!

0

2 C b0!1. We have det M 0 D ˙1, therefore.$ 1, $ 2/ is a fundamental pair of periods. Moreover,

Q def D $ 2

$ 1D a0 C b0

c0 C d 0D M 0 ,

so that Q satisfies the requirements in the announcement of the lemma.

Theorem 9.1.2 below is one of the main tools for the application of Ellog in practice.

It is a rather straightforward consequence of an important deep result due to S. David

[12, Théorème 2.1], which, in turn, is the explicit version of a general important (ef-

fective but not explicit) result of N. Hirata-Kohno [17, Corollaire 2.16]).




Preparatory to Theorem 9.1.2. Referring to the short Weierstrass model E in (4.1),

we consider the linear form L.P / in (9.2) obtained as explained in the three “bullets”,

on pages 99 through 100, and do the following:

SetD D ŒK : Q,

where K is the least number field containing the coordinates of all points of E , the

elliptic logarithms of which are `1, : : : , `k , respectively; cf. the remark on page 101.

Set

hE D max¹1, h.1 : 4A : 4B/, h.jE /º,

where jE D 2833A3=.4A3 C 27B 2/ is the j -invariant of E and h. / means absolute

logarithmic height as defined in Section 2.3, especially page 17 ff.

Compute first a fundamental pair of periods .!1, !02/ and then a fundamental pair of

periods .$ 1, $ 2/, as explained in Lemma 9.1.1.

For i D 1, : : : , k compute Oh.P i / using a software package, e.g. PARI or MAGMA, if

this is possible; if not, compute an upper bound for Oh.P i / using Proposition 2.6.4.

Note that the routines of the above mentioned packages for the canonical height 5

can do the job always6 if the point has rational coordinates, but refuse7 to compute

the canonical height of a point with irrational coordinates. In such a case, we confine

ourselves to computing an upper bound for Oh.P 0/ using Proposition 2.6.4.

For i D 1, : : : , k compute `i , i.e. compute the elliptic logarithm of P i , if P Ei 2E0.R/ and the elliptic logarithm of P i C Q2, if P

Ei 2 E1.R/. According to the

remark on page 101, this means that, probably, we will have to compute the elliptic

logarithm of a point of E with non-rational coordinates. In this case, the routine

ellpointtoz of PARI, which accepts as input an elliptic curve over R, would do the

job, but not the routine EllipticLogarithm of MAGMA. Alternatively, it is not really

difficult to write a program for the algorithm on page 53, which would work with

points having real coordinates.8 Anyway, we should take care to adjust the output

by adding ˙!1, if necessary, so that the result falls in the interval .!1

2 ,

!1

2 , in the

case that the package we are using normalises elliptic logarithms otherwise.

Choose H 0, H 1, : : : , H k such that

H 0 max

²hE ,

3j!1j2

Dj$ 1j2 =Q

³,

H i max

²hE ,

3`2i

Dj$ 1j2 =Q , 3 Oh.P i /

³ .i D 1, : : : , k/.

5 ellheight of PARI and CanonicalHeight of MAGMA.6 Whatever that means for a machine!7 At least so far (early 2013).

8 For example, we wrote one on MAPLE.




Theorem 9.1.2 (Sinnou David). All points P E 2 E.Q/ for which L.P/ D 0 canbe explicitly calculated. More specifically, such points can result only in either of the following two cases: (1) If P is a torsion point, or (2) If the situation described in thethird “bullet”, page 100 , occurs, and .d , m

1, : : : , m

r/D

.1,

d 1

, : : : ,

d r

/. If L.P / ¤ 0 , then, either N c12 , where

c12 D max¹exp.e hE /, jdr0j, exp.H i =D/, i D 0, : : : , kº,

or jL.P/j > exp

c13.log N C c14/.log log N C c15/kC2

, (9.7)

where

c13 D 2.9 106kC12D2kC442.kC1/2

.k C 2/2k2C13kC23.3

k

YiD0

H i ,

c14 D 1 C log D ,c15 D 1 C hE C log D.

Proof. In the case of the first “bullet”, page 99, the relation L.P / D 0 means that

r0m1l.P 1/ C C r0mr l.P r / C n0!1 D 0. By Theorem 3.5.2 (c) it follows then that

m1 D D mr D 0, hence P is a torsion point.

In the case of the second “bullet”, page 99, the relation L.P / D 0 is impossible.

Indeed, in this case, L.P / D 0 means that r0m1l.P 1/ C C r0mr l.P r / ˙ r0l.P 0/ Cn0!1 D 0 and the points P E1 , : : : , P Er , P E0 are Z-linearly independent. Since the

coefficient of l.P 0/ is non-zero, this contradicts Theorem 3.5.2 (c).

In the case of the third “bullet”, page 100, L.P/ D 0 means (cf. (9.4)) that r0.d m1Cd 1/`1 C C r0.d mr C d r /`r C n0!1 D 0 and Theorem 3.5.2 (c) implies that mi Dd i =d for i D 1, : : : , r . In particular, d is a common divisor of d 1, : : : , d r and, since

gcd.d , d 1, : : : , d r / D 1, we conclude that d D 1 and mi D d i for i D 1, : : : , r .

If L.P/ ¤ 0, then the conclusion of the theorem results from an almost straightfor-

ward, though careful (see remarks below), application of [12, Théorème 2.1].

Remarks on the application of S. David’s theorem.

(1) In S. David’s theorem [12, Théorème 2.1] the k elliptic logarithms appearing in

the linear form may come from different elliptic curves defined over a number field

K . In our case, all elliptic curves coincide with the elliptic curve (4.1) defined over

Q. We remind, however, that some `i ’s may be elliptic logarithms of points with

non-rational coordinates, as noted on page 101.

(2) For the parameter E appearing in [12, Théorème 2.1] we chose E D e.

(3) A “detail” concerning canonical heights. According to our discussion in Chapter 2,

the canonical height in [12, Théorème 2.1], which on page 20 ff. we denoted by OhD ,

is, by Proposition 2.6.1, three times the canonical height defined by J. Silverman

in [45, section VIII.9], i.e. the canonical height that we adopt in this book. This

explains why in the definition of H i the canonical height Oh.P i / is multiplied by a

factor 3, which does not appear in S. David’s theorem.



Section 9.2 Computational remarks 105

Now we are ready state the theorem from which an explicit upper bound of M is

obtained.

Theorem 9.1.3. Let L.P / be the linear form appearing in any of Theorems 5.2, 6.8 ,7.1 or 8.7.2. Write L.P/ in the form (9.2) following the instructions in the appropriate“bullet” (page 99 ff.) and consider the constants ˛ and ˇ (cf. (9.6)) , as well as theconstants c12, c13, c14, c15 resulting from Theorem 9.1.2.

If L.P / ¤ 0 , then, either M c12 , or

M 2 c18c13.log.˛M Cˇ/Cc14/.log log.˛M Cˇ/Cc15/kC2C Cc18 log c16Cc17,

(9.8)

where,

.c16, c17, c18/ D 8<ˆ:

.23=2, 1

2 log ı, 1/ in case of Theorem 5.2

.4=p a, 12

log 3c7, 1/ in case of Theorem 6.8

..2V 0p

D1D2/1, 1

2c8, 1/ in case of Theorem 7.1

.c9=.1 C / , 1

2 c10, .c11=.2// in case of Theorem 8.7.2.

Proof. Let the linear form L.P / considered in any of the Theorems 5.2, 6.8, 7.1 or

8.7.2 and define N according to the appropriate “bullet”, page 99 ff., as the case may

be.

If M > c12, the left inequality (9.6) implies N > c12 and then Theorem 9.1.2

implies the lower bound (9.7) for j

L.P/j. In view of the right inequality (9.6), this

lower bound is also valid after replacing N by ˛M C ˇ . Then the resulting lower

bound, combined in a straightforward manner with the upper bound for jL.P/j already

obtained from Theorem 5.2 or 6.8 or 7.1 or 8.7.2, depending on the case, leads to the

inequality (9.8).

Concluding remark. In (9.8), the left-hand side tends to infinity faster than the right-

hand side. Therefore, if M is sufficiently large, then this relation cannot hold. Conse-

quently, if all constants in (9.8) are explicitly known, then an upper bound for M can

be explicitly calculated; cf. Chapter 4, immediately after (4.8).

9.2 Computational remarks

In sections 9.3, 9.4, 9.5 and 9.6 we show by concrete examples how to apply Theo-

rem 9.1.3 and the Concluding remark following it. In the present section we give some

practical information concerning our concrete numerical computations.

We make a combined use of MAGMA, PARI and a few rather simple codes based on

MAPLE written by the author. In all examples we started doing all calculations with a

precision of 100 decimal digits. With the exception of Section 9.4, in all other sections,

the numbers resulting from the calculations are of a size much less than 10100, so that




we consider them reliable. In Section 9.4, however, the value of the constant c13 turned

out to be considerably larger that 10100 and we repeated all the computations using a

200 decimal digits precision. All packages nowadays work with this and much larger

precision without the slightest problem. Anyway, in the text we write the results of

our computations correct to 16 decimal digits.

Computations related to elliptic curves are done using MAGMA and/or PARI. Given

the vector Œa1, a2, a3, a4, a6 of the coefficients of an elliptic curve (see [45, Chap-

ter III.1]), the routines EllipticCurve of MAGMA and ellinit of PARI create an object

e containing all necessary information to work with. Minimal models are calculated

by any of the routines ellminimalmodel of PARI or MinimalModel of MAGMA.

For the computation, even with a very high precision, of a fundamental pair of periods for our elliptic curves (or, rather, for the -functions associated to them) we

can easily write a simple code according to the instructions in Section 3.4.4. If one

does not feel comfortably with programming, any of the commands Periods.e/ of MAGMA or e.omega of PARI does the job.

Elliptic logarithms can be calculated by the algorithm on page 53, for which it

is not difficult to write a code. Alternatively, the routines ellpointtoz of PARI or

EllipticLogarithm of MAGMA do the same thing. We remind the reader that, if for

some point P we have P E 2 E1.Q/, then l.P / is the elliptic logarithm of P C Q2

(where QE2 D .e2, 0/) rather than the elliptic logarithm of P , which is not a real num-

ber; cf. Conclusions and remarks (1) on page 51. Probably, the point P E C QE2 has

non-rational coordinates; in this case, its coordinates belong to a quadratic or a cubic

number field. But computing the elliptic logarithm of such a point is not really a prob-lem, because the algorithm on page 53 works perfectly with real numbers. Also, the

routine ellpointtoz of PARI accepts input points on the elliptic curve with coordinates

of type “real”, but the routine EllipticLogarithm of MAGMA would complain. In a

symbolic calculation package like, for example, MAPLE, we can write a program for

calculating with points on an elliptic curve over a number field, by implementing the

Group law algorithm 2.3 of [45, Chapter III.2].9

The computation of the canonical height is quite a sophisticated task; see, for ex-

ample, [42, 43, 75, 76].10 For the needs of our examples we used either of the routines

CanonicalHeight of MAGMA or ellheight of PARI.

Warning! The definition of the canonical height given in Section 2.6 agrees with

the one given by Silverman [45, Chapter VIII]. The canonical height computed by the

above routines of PARI and MAGMA is twice as large, because the factor 1=2 in (2.36) is

missing from the definition of the canonical height adopted by PARI and MAGMA. Anal-

ogously, for the height pairing matrix, either of the routines HeightPairingMatrix or

ellheightmatrix of MAGMA or PARI, respectively, is used. Note that the eigenvalues

9 This is what the author has done.

10 The relevant bibliography is far from being exhausted by these references!



Section 9.3 Weierstrass equation example 107

of the height pairing matrix are not affected by the normalisation for the canonical

height that one may adopt.

Warning! The routines ellheight and ellheightmatrix of PARI demand that their

input data come from a minimal Weierstrass model.The extra point P 0 that makes its appearance possibly in Theorem 6.8 and, anyway, in

Theorems 7.1 and 8.7.2, has probably irrational coordinates and, in this case, neither

PARI nor MAGMA know how to compute its canonical height. Should we need to know

the precise value of Oh.P 0/, we have to implement for ourselves the algorithms of [43].

But for the application of Theorem 9.1.2 we only need a reasonably good upper bound

of Oh.P 0/, which we obtain from Proposition 2.6.4.

Computing the constants c12, : : : , c18 is a straightforward but quite boring task.

Therefore, we preferred to write a rather simple MAPLE code for their computation.

A considerably more sophisticated code running, for example, under MAGMA, whichwould receive much less data – only, say, the coefficients of C and E – and return the

numerical values of above ci ’s automatically, is possible. We did not attempt to write

such a code as we prefer a more transparent process which is more easily under our

control.

9.3 Weierstrass equation example

We use the notation, results etc. of Chapter 5.

In (5.1) let us take

C : g.u, v/ D 0, g.u, v/ D v2 C uv C v u3 u2 C 71u C 196.

Taking in (5.2)

., , , / D .1, 5=12, 1=2, 7=24/,

we obtain the model

E : y2 D x3 3409

48x 143623

864D f.x/.

The roots of f .X/ are e3 D 41=6 < e2 D 31=12 < e1 D 113=12. Note also that,

in Proposition 5.1,

ı D 12.

We will work with the elliptic curve E , two models of which are C and E, searching

for all points P on E such that P C 2 C.Z/. In order to apply Theorem 5.2, we need

that u.P / > 18. Also, we note that there are no points .u, v/ 2 C.R/ with u < 7.11,

therefore we first compute all P with integer u.P/ between 7 and 18; these are

exactly the following:

.u.P /, v.P// D.7, 1/, .7, 7/, .6, 5/, .6, 10/, .4, 5/, .4, 8/, .3, 1/

.9, 5/, .13, 43/, .13, 29/, .14, 50/, .14, 35/. (9.9)

From now it will be understood that P C

2 C.Z/ and u.P / > 18.




We compute E.Q/ Š Z2Z2ZZ, so that the rank r D 2. The torsion subgroup

consists of the points O and QEi D .ei , 0/ (i D 1, 2, 3). Thus,

Etors.Q/D h

QE1 , QE

2

i D h.113=12, 0/, .

31=12, 0/

i & r0

D2.

As generators of the Z Z part of the Mordell–Weil group we can take

P E1 D .79=12, 4/, P E2 D .67=12, 15=2/.

Let now g.u, v/ D 0, where u, v 2 Z and u > 18, and consider the point P C D.u, v/ 2 C.Z/. Then P E 2 E.Q/ and, specialising (4.2) to our case, we have

P E D m1P E1 C m2P E2 C T E ,

implying

L.P/ D l.P / D .m0 C s

2/ !1 C m1l.P 1/ C m2l.P 2/,

which is the specialisation of (4.6) to our case. By Theorem 9.1.2, L.P / D 0 implies

m1 D m2 D 0, i.e. P E 2 Etors.Q/ D ¹QE1 , QE

2 , QE1 C QE

2 ,Oº. Only for P DQ1, Q2 we obtain a point P C with integer coordinates, namely QC

1 D .9, 5/ and

QC 2 D .3, 1/, already listed in (9.9). Therefore, in what follows we assume that

L.P/ ¤ 0. The linear form L.P / falls under the scope of the first “bullet”, page 99,

so that

k D 2, d D 1, n0 D 2m0 C s, jsj 1, .n1, n2/ D .m1, m2/,

hence, N 0 D 2M C 3 in (9.3) and in (9.6),

.˛, ˇ/ D .2, 3/.

Also, all points involved in the linear form have rational coordinates, therefore, in the

notation of Preparatory to Theorem 9.1.2, page 103,

D D 1.

We also compute a fundamental pair of periods

!1 0.8394944402534986 !2 1.0599189920498769 i ,

noting that

D !2=!1 1.2625682091829191 i

satisfies = > 0, j j > 1 and j< j < 1

2; hence, in Lemma 9.1.1 we can take

.$ 1, $ 2, Q/ D .!1, !2, /.

We calculate the canonical heights

Oh.P 1/ 1.0547377027055837, Oh.P 2/ 0.3611721523399011



Section 9.3 Weierstrass equation example 109

and the height-pairing matrix

H

1.0547377027055837 0.2045874419576421

0.2045874419576421 0.3611721523399011 with least eigenvalue

0.6106414163983847.

Now we apply Proposition 2.6.3. In the notation of that proposition, as a curve D we

take the curve C of our example and, consequently, as and therein, we respectively

take 1 and 0. A straightforward calculation then gives

5.0391440734884212.

Now we need to compute l.P i / for i D 1, 2. Both P Ei ’s belong to E1.R/, therefore,

from Conclusions and remarks (1), page 51, l.P i / is the elliptic logarithm of the point

P E

i C QE2 which belongs to E0.R/; more specifically,

P E1 C QE2 D .61=6, 51=4/, P E2 C QE

2 D .173=12, 85=2/

and thus we compute

`1 D l.P 1/ 0.3588173091972342, `2 D l.P 2/ 0.2755755724720115.

Next we compute

c12

6.4272819599100395

1028, c13

9.1163192883638712

1073,

c14 D 1, c15 25.402522816413709.

The remaining constants appearing in Theorem 9.1.3 are

.c16, c17, c18/ D .23=2, 1

2 log 12, 1/.

Assuming M > c12, we insert into (9.8) the values of ˛, ˇ, , and c13, : : : , c18 that

we computed, and conclude that B .M/ > 0, where

B .M / D 9.11632 1073.log.2M C 3/ C 1/¹log.log.2M C 3// C 25.402523º4

C 7.32132 0.61064M 2.

But, for all M > 1.1 1041, we check that B .M / < 0, which implies that

M max¹c12, 1.1 1041º D 1.1 1041. (9.10)

We cannot obtain an upper bound for M essentially better than the above using The-

orem 9.1.3; indeed, we check that B .1.07 1041/ > 0 which shows that with “a little

smaller” bound for M we do not arrive at a contradiction. We will succeed to make

an impressive reduction of the huge upper bound (9.10) in Section 10.2.1 of the next

chapter.




9.4 Quartic equation example


In (6.1) and (6.2) let us take a D 5=2, b D 1=2, c D 1, d D 0, e D 1, so that

C : 0 D Q.u/ v2 D 5

2u4 C 1

2u3 C u2 C 1 v2.

By (6.3) we compute

A D 31

3, B D 685

108,

so that

E : y2 D f.x/ D x3 C Ax C B .

The approximate values of the roots of f .X/ are

e1 3.4860779869512022 > e2 0.6390559614483767> e3 2.8470220255028254.

We will work with the elliptic curve E , two models of which are C and E . The rank

of E is 3 and the torsion subgroup Etors.Q/ is trivial; thus

r D 3, r0 D 1.

The following points form a basis for E.Q/:

P E1 D .2=3, 1=2/, P E2 D .5=3, 5=2/, P E3 D .8=3, 3=2/.

Let now .u, v/ be an integer solution of the equation Q.u/ D v2

and P C

D .u, v/ 2C.Z/. Then P E 2 E.Q/ and, specialising (4.2) to our case, we have

P E D m1P E1 C m2P E2 C m3P E3

implying

l.P / D m0 !1 C m1l.P 1/ C m2l.P 2/ C m3l.P 3/,

which is the specialisation of (4.6) to our case. We also compute a fundamental pair

of periods

!1 1.3856089566101190, !2 1.5968599642739782 i ,

with

D !2=!1 1.1524607694372033 i

satisfying = > 0, j j > 1 and j< j < 1

2, hence, in Lemma 9.1.1 we can take

.$ 1, $ 2, Q / D .!1, !2, /.

The approximate values of the canonical heights are

Oh.P 1/ 0.55387253844458413, Oh.P 2/ 0.6064693463824573,

Oh.P 3/ 0.6518823598427176



Section 9.4 Quartic equation example 111

and the height pairing matrix is

H0@

0.55387253844458413 0.0609518603595160 0.1563221441147483

0.0609518603595160 0.6064693463824573 0.1436252586206832

0.1563221441147483 0.1436252586206832 0.6518823598427176

1A

with least eigenvalue

0.8549017536692952.

Now we apply Proposition 2.6.3. In the notation of that proposition, as a curve D

we take the minimal model of E which is D : y21 C y1 D x3

1 C x21 10x1 10.

The change of variables relating D and E is x1 D x 1

3, y1 D y 1

2, therefore

D 1, D 1

3 and from (2.41) we get

3.8640973065752320.

We need to compute l.P i / for i

D 1, 2, 3. All three points P Ei belong to E1.R/,

therefore, by Conclusions and remarks (1), page 51, it follows that l.P i / is the el-liptic logarithm of the point P Ei C QE

2 which belongs to E0.R/ but does not have

rational coordinates. Indeed, since f .X/ is irreducible over Q and QE2 D .e2, 0/, the

coordinates of the points P Ei CQE2 belong to the cubic number fieldQ.e2/. Actually,

P E1 C QE2 D

358 C 22e2 36e2

2 ,117173

18C 1258

3e2 656e2

2

P E2 C QE

2 D

1669

225 22

15e2 8

25e2

2 ,61477

2250C 472

75e2 532

125e2

2

P E3 C QE3 D 628

81 406

27e2 C 44

9e2

2 ,51653

486C 10642

81e2 1256

27e2

2

.

As noted in Section 9.2, the elliptic logarithm of points with irrational coordinates can

be computed without problem. Thus,

`1 D l.P 1/ 0.0551123268560820, `2 D l.P 2/ 0.3547306431725462,

`3 D l.P 3/ 0.5770693948785471.

We will search for all points P on E such that P C 2 C.Z/. We will apply The-

orem 6.8, which requires that ju.P/j be sufficiently large. Table 6.1 in Chapter 6 in-

dicates how to compute how large ju.P/j should be. Specialising that table to ourspecific example we get Table 9.1.

From Table 9.1 it follows that, with c7 D 3.498 in (6.15), Theorem 6.8 is certainly

applicable to all points P 2 E such that .u.P /, v.P// are integers with v.P/ > 0

and ju.P/j > 200. A quick computer search for points P with u.P / an integer of the

interval Œ200, 200 reveals the following points:

.u.P/, ˙v.P// D .130, 26701/, .4, 25/, .1, 2/, .0, 1/, .2, 7/.

Therefore, from now on we will assume that ju.P/j > 200. We have

L.P/ D l.P / ˙ l.P 0/.




Table 9.1. Parameters and auxiliary functions for the solution of the quartic elliptic equation

example (cf. Table 6.1).

Q.u/

D 5

2u4

C 1

2u3

Cu2

C1 Q.u/

D 5

2u4

1

2u3

Cu2

C1

D 1 D 1

x.u/ D u2 C 6 C 6 .Q.u//1=2

3u2 x.u/ D u2 C 6 C 6 .Q.u//1=2

3u2

u0 D 1 u0 D 17

u D 1 u D 17

u D 200, c7 D 3.498 u D 17, c7 D 3.496

P E0 D . 1

3Cp

10, 1

2/ P

E

0 D

1

3Cp

10, 1

2

cf. Lemma 6.1

and

(6.11), (6.8)

l.P 0/ l.P 0/ D l.P 0/

L.P/ D l.P / l.P 0/ L.P / D l.P / C l.P 0/

Using the routine IsLinearlyIndependent of MAGMA, we see that the points P E0 ,

P E1 , P E2 , P E3 are Z-linearly independent,11 so that we are in the situation described

in the second “bullet”, page 99. Therefore, in (9.2),

k D 4, d D 1, r0 D 1, .n0, n1, n2, n3/ D .m0, m1, m2, m3/, n4 D ˙1, `4 D `0.

In (9.3) N 0 D 3

2M C 3

2 hence, in (9.6),

.˛, ˇ/ D .3=2, 3=2/.

We compute

`0 D l.P 0/ 0.6737142222951107.

On applying Theorem 9.1.2, we will also need to compute Oh.P 0/. Since P E0 is not a

rational point we confine ourselves to a reasonably good upper bound of its canonical

height (see Section 9.2) which we obtain from Proposition 2.6.4. In the notation of

that proposition we take D : y2 C y D x3 C x 2 10x 10, which is, actually,

the minimal model of our elliptic curve. Then, P D0 D .p

10,0/. By Proposition 2.4.2,

h.p

10/ D .log 10/=2 and then, by Proposition 2.6.4,

Oh.P 0/ 4.12.

Finally, following the section Preparatory to Theorem 9.1.2 we see that

D D 6.

11 For a more transparent way to prove this fact see the end of this section.



Section 9.4 Quartic equation example 113

Indeed, in L.P / we have the elliptic logarithms of the points P Ei C QE2 , the coor-

dinates of which belong to Q.e2/, and the elliptic logarithm of the point P E0 , with

coordinates in Q.p

10/. Therefore, the least number field containing the coordinates

of all these points is Q.e2,

p 10/, the degree of which is 6.We also compute

c12 9.5771564156066246 1021, c13 5.6067135931158955 10165,

c14 2.7917594692280550, c15 21.4114872494028374.

According to the announcement of Theorem 9.1.3,

c16 2.5298221281347034, c17 1.1754018326322600, c18 D 1.


we computed, and conclude that B .M/ > 0, where

B .M / D 5.607 10165 ¹log.1.5M C 1.5/ C 2.79176º ¹log.log.1.5M C 1.5/ C 21.4115º6 C 5.96764 0.8549 M 2.

But, for all M 2.3 1088, we check that B .M / < 0, which implies that

jM j max¹c12, 2.3 1088º D 2.3 1088. (9.11)

We cannot obtain an upper bound for M essentially better than the above using Theo-

rem 9.1.3; indeed, we check that B .2.2 1088/ > 0 which shows that “a little smaller”

bound for M does not lead to a contradiction. A very big reduction of the huge upperbound (9.11) will be accomplished in Section 10.2.2 of the next chapter.

Z-linear independence of P E0 , P E1 , P E2 , P E3 . If these points were Z-linearly de-

pendent, then a positive integer d would exist, such that dP E0 2 E.Q/. We show that

this is impossible. Since P E0 D .1=3C , 1=2/, where Dp

10, our claim is obvious

for d D 1. Now we proceed by induction. Actually, we show something stronger:

If RE D .m C n , p C q/ 2 E.Q.//, m, n, p, q 2 Q , then RE C P E0 62 E.Q/.We prove this as follows. From f .m C n , p C q/ D 0 we obtain

108m3

C3240mn2

1116m

685

108p2

1080q2

D0

1080n3 C 324m2n 1116n 216pq D 0.(9.12)

A symbolic computation gives

RE C P E0 D

p10.m, n, p, q/ C p11.m, n, p, q/

d 1.m, n, p, q/ ,

p20.m, n, p, q/ C p21.m, n, p, q/

d 2.m, n, p, q/

,

where the pij ’s and d i ’s are specific polynomials with integer coefficients. We

have to show that p11.m, n, p, q/ and p21.m, n, p, q/ cannot both be zero. Suppose




the contrary and consider the two equations (9.12) along with the two equations

p11.m, n, p, q/ D 0 and p21.m, n, p, q/ D 0. Elimination of q from these four equa-

tions leads to three polynomial equations in m, n, p with integer coefficients. A new

elimination of p from the last three equations leads to two polynomial equations, say

gi .m, n/ D 0, i D 1, 2 with integer coefficients. Specifically,

g1.m, n/ D n8.9m2 C 30n2 31/6h1, g2.m, n/ D n8.9m2 C 30n2 31/3h2,

where hi D hi .m, n/ 2 ZŒm, n, i D 1, 2. The resultant of h1.m, n/ and h2.m, n/

with respect to m belongs to ZŒn and its only rational roots are n D 1,0, 1. For

these values of n, the equation h1.m, n/ D 0 has m D 1=3 as its only rational solution.

Inserting into (9.12) .m, n/ D .1=3, 0/ gives an obviously impossible system in .p, q/;

and inserting .m, n/ D .1=3, ˙1/ gives 27 108p2 1080q2 D 0 and pq D 0. If

p

D 0, then q2

D 1=40, impossible for rational q ; if q

D 0, then p

D ˙1=2 and

RE D .m C n , p C q/ D .1=3 ˙ , ˙1=2/ is not a point of E .It remains to check what happens if 9m2 C30n2 31 D 0. From the solution .m, n/ D.1=3, 1/ we find that all rational solutions to this equation are given by

m D 10t 2 60t 3

3.10t 2 C 3/ , n D 10t 2 C 2t 3

10t 2 C 3. (9.13)

From this, if we substitute for m and n in p11.m, n, p, q/ D 0 we get an equation

g.p, q, t / D 0; and if we substitute in (9.12) we get a system h.p, q, t / D 0, & pq D0. Of course, g and h are polynomials with integer coefficients.

If p D 0, then we have the equations g.0, q, t / D 0 and h.0, q, t / from which weeliminate q to obtain a polynomial in t with no rational solutions.

If q D 0, we work similarly but this time we obtain a polynomial in t , the only rational

root of which is t D 1=10. Inserting this value in (9.13) gives .m, n/ D .1=3, 1/.

But this means that RE D ˙P E0 and then RE C P E0 is either the zero point O,

which is excluded, or the point 2P E0 , which has no rational coordinates.

This completes the proof that no integer multiple of P E0 has rational coordinates,

without appealing to the routine IsLinearlyIndependent of MAGMA.

9.5 Simultaneous Pell equations example


In (7.1) we take

D1 D 11, A1 D 14, D2 D 7, A2 D 6.

We check for which V D 1, : : : , 1000 bothp

A1 C D1V 2 andp

A2 C D2V 2 are

integers; only V D 1, 5 have this property, giving us the solutions .U , V , W / D.5,1,1/, .17, 5, 13/. As a particular solution .U 0, V 0, W 0/ we choose

.U 0, V 0, W 0/ D .5,1,1/



Section 9.5 Simultaneous Pell equations example 115

and, from now on we assume that V > 1000. Consequently

p A1 C D1V 2 3.317 V ,

p A2 C D2V 2 2.646 V (9.14)

as required for the application of Theorem 7.1.By (7.12) and (7.13),

D C1, .a, b, c, d , e/ D .36, 120, 376, 280, 14/,

Q.u/ D 36u4 C 120u3 376u2 280u C 196

and in (7.17)

E : y2 D x3 C Ax C B D x3 326848

3x C 123412480

27.

As noted a few lines after (7.28), f .X/ has three rational roots, which are

e3 D 1048

3< e2 D 128

3< e1 D 920

3.

The functions u and v, defined in (7.18) and (7.19), are

u.V / D 5V p

14 C 11V 2

V Cp

6 C 7V 2,

v.V / D 4.7p 6

C7V 2

C82V

15p 14

C11V 2/

.V C p 6 C 7V 2/2 .

Plotting v.V / and Q.u.V // for V > 1000 we see that v.V / and Q.u.V // are positive,

as the second “bullet” in the announcement of Theorem 7.1 requires. Also, according

to (7.21) and (7.22),

x.u/ D 376

3C 28

p Q.u/ 280u C 392

u2

y.u/ D 1680u3

10528u2

11760u

C10976

C.784

280u/p Q.u/

u3 .

The functions V 7! x.V / and V 7! y.V / are defined in (7.23) by

x. V/ D .x ı u/.V /, y.V / D .y ı u/.V /.

An explicit expression of x.V / and y.V / is an easy matter for a symbolic computa-

tion package, like MAPLE, for example, but this is a too long expression to print here.

Besides, what we only need to check is (see third “bullet” in the announcement of The-

orem 7.1) that, in the interval .1000, C1/ the function V 7! y.V / is positive and

the function V 7! x.V / is strictly monotonous, with values exceeding e1. Plotting




these two functions for V > 1000 we see that this is indeed true. More precisely, the

function V 7! x. V/ is strictly decreasing and its limiting value is

4164

2300

p 11

C924

p 7

420

p 11

p 7

54 C 15p 11 >

920

3 D e1.

Finally, by (7.24) and (7.25) we compute P E0 D .x0, y0/, where

x0 D 1052

3C 44

p 7 C 140

p 11 C 20

p 11p

7

y0 D 8p

11p

7 .5p

7 Cp

11/.1 Cp

7/.5 Cp

11/.

Now we calculate the constant c8 following the guidelines of the paragraph before the

relation (7.32). From (7.31) we have

x. V/ D P 1.U , V , W /=P 2.U , V , W /

D 2.840U C 4592V C 392W 479V 2 C 130U V

C U 2 308V W C 140U W C 196W 2/=.U 5V /2.

We obtain an upper bound for P 1.U , V , W / of the form constant times V 2, where the

constant can be obtained using (9.14) and V 0.001V 2. We obtain a much better

bound of this shape if we consider the function

jP 1.p 14

C11V 2, V ,p 6

C7V 2/

jV 2

and observe that this is strictly decreasing in the interval .1000, C1/ taking a value

< 3503.22 at V D 1000. Working similarly for the denominator we see that

jP 2.p

14 C 11V 2, V ,p

6 C 7V 2/j=V 2 is a strictly increasing function with lim-

iting value < 2.834 as V ! C1. Hence, we can take

c8 D log.3503.22/.

Now it is time to look for the arithmetic-geometric “details” of the model E. Our

calculations have been made with a precision of 100 decimal digits; below we exhibit

the numerical values in 16 digits.

The free part of the group E.Q/ is generated by

P E1 D .1976=3, 14784/, P E2 D .1052=3, 3080/.

Also, Etors.Q/ Š Z2 Z2 is generated by QE1 D .e1, 0/ and QE

2 D .e2, 0/, so that

Etors.Q/ D h.920=3, 0/, .128, 0/i & r D 2, r0 D 2.

A fundamental pair of periods is

!1 0.1520305707483552, !2 D 0.1389333214997840 i



Section 9.5 Simultaneous Pell equations example 117

and, in the notation of Lemma 9.1.1, we take

$ 1 D !2, $ 2 D !1.

For the point P E D .x.V /, y.V // coming from the unknown integer solution.U , V , W / of (7.1), with V > 1000 and U , W 0, we write

P E D m1P E1 C m2P E2 C T E , T E 2 Etors.Q/,

so that (see immediately before relation (7.30)),

L.P/ D l.P / C l.P 0/ D

m0 C s

2

!1 C m1l.P 1/ C m2l.P 2/ l.P 0/.

In view of (7.27), 2P E

0 D .1052=3, 3080/ D P E

2 , so that we are in the situationdescribed in the third “bullet”, page 100. We compute that 2 l.P 0/ D l.P 2/, therefore,

L.P/ D .m0 C s

2/!1 Cm1`1 C.m2 1

2/`2, so that, in the notation of the third “bullet”,

page 100,

.d , d 1, d 2, ı0, s0=t0/ D .1,0,1,0,0=1/

and

k D 2, n0 D 2m0 C s, s 2 ¹1,0,1º, .n1, n2/ D .2m1, 2m2 1/.

Therefore (see also (4.8)), jn0j 2M 0 C 1 2.M C 1/ C 1 D 2M C 3 andmax1i2 jni j 2M C 1; hence, in (9.6) we can take

.˛, ˇ/ D .2, 3/.

Both points P E1 and P E2 belong to E0.R/, therefore their l-values are equal to their

respective elliptic logarithms. Thus, we calculate

`1 D l.P 1/ 0.0400052062019816, `2 D l.P 2/ 0.0606534846504600.

The canonical heights of P 1, P 2 are

Oh.P 1/ 0.7861708324460508, Oh.P 2/ 1.0456910246342577,

and the height pairing matrix is

H

0.7861708324460508 0.4317829329642320

0.4317829329642320 1.0456910246342577

with minimum eigenvalue

0.9301430899748535.




We apply Proposition 2.6.3 in order to compute . As a curve D we take the minimal

model of E , which is D : y21 D x3

1 x21 6809x1 C 73689. The change of vari-

ables .x1, y1/ D . 1

4x C 1

3, 1

8y/ transforms D into E , hence, D 1=2, D 1=3 and,

according to Proposition 2.6.3,

5.4867659523958290.

We also compute

c12 1.4100217348795374 1035, c13 1.6558700707656948 1074

c14 D 1, c15 30.7739853633026035.

By the definition of c16, c17 and c18 in Theorem 9.1.3, we obtain

c16 0.0569802882298189, c17 4.0807189122684443, c18 D 1.


we computed, and conclude that B .M / > 0, where

B .M / D 1.6559 1074.log.2M C 3/ C 1/.log.log.2M C 3// C 30.774/4

C 6.702435 0.9301430 M 2


jM j max¹c12, 1.7 1041º D 1.7 1041. (9.15)

Since B .1.6 1041/ > 0, the bound (9.15) is essentially the best upper bound that can

be obtained from Theorem 9.1.3; its reduction to an upper bound of manageable size

will be accomplished in Section 10.2.3 of the next chapter.

9.6 General elliptic equation: A quintic example

ExampleStep 7. (continued from the end of Step 6, page 96)We have already computed c9 D 0.11 and D 1=3 (see page 93), c10 D log.30000/and c11 D 3 at Step 6, page 96, and P E0 D .2169, 79947=2/ at Step 3, page 86.

Therefore, by the definition of c16, c17 and c18 we have

c16 D 0.0825, c17 D 1

2 log 3 C 2log10, c18 D 9

2.

The free part of E.Q/ is generated by the points

P E1 D .2439, 65853=2/, P E2 D .68292, 35626473=2/

and Etors.Q/ D ¹Oº. Moreover, P 0 D P 1 C P 2.



Section 9.6 General elliptic equation: A quintic example 119

A fundamental pair of periods is

!1 0.1173285064197781, !2 0.0586642532098890 C 0.0183162358182653i

and we take$ 1 D !1 C 2!2, $ 2 D !1 C !2,

Q D $ 2=$ 1 0.5 C 1.6014276566418661i .

We have P E0 2 E.Q/; actually, P E0 D P E1 C P E2 . At this point we note that f .X/

(right-hand side of (8.5)) has only one real root, namely,

e1 4898.1389881123821080

and, on calculating elliptic logarithms, we find out that l.P 0/

Dl.P 1/

Cl.P 2/. Because

of the relation P E0 D P E1 C P E2 , our example falls in the case of the third “bullet”,page 100. Therefore (cf. Theorem 8.7.2),

L.P/ D l.P / C "P l.P 0/ D m1l.P 1/ C m2l.P 2/ C m0!1 C "P l.P 1 C P 2/

D m1l.P 1/ C m2l.P 2/ C m0!1 C "P l.P 1/ C "P l.P 2/

D .m1 C "P /l.P 1/ C .m2 C "P /l.P 2/ C m0!1.

By Theorem 3.5.2 (c), L.P / ¤ 0, except if .m1, m2/ D ."P , "P /, i.e. only when

P E D ˙P E0 . This implies P D O, furnishing no solution to our Diophantine equa-

tion. Therefore, henceforth we assume L.P /

¤0 and we will apply Theorem 9.1.3 in

combination with Theorem 8.7.2.In the notation of the third “bullet”, page 100, we have r0 D 1, s=t D s0=t0 D 0=1,

.d , d 1, d 2, d 0/ D .1,1,1,0/, k D 2, n0 D m0 and ni D mi ˙ 1 (i D 1, 2). Therefore,

N D M 0 D max0i2 jmi j D M C 1, by (4.8). Hence, in (9.6) we can take

.˛, ˇ/ D .1, 1/.

Next, we compute the canonical heights of P 1, P 2,

Oh.P 1/ 0.2339011772221281, Oh.P 2/ 1.9162842498242523,

and the height pairing matrix

H

0.2339011772221281 0.0519655844031980

0.0519655844031980 1.9162842498242523

with minimum eigenvalue

0.4645951770663837.

Since f .X/ has only one real root, we have E.R/ D E0.R/ and, therefore, l.P i /

coincides with the elliptic logarithm of P i for i D 1, 2. Thus, we compute

`1 D l.P 1/ 0.0296373863107625, `2 D l.P 2/ 0.0038280584521748.




Next, we apply Proposition 2.6.3 in order to compute . As a curve D we take the

minimal model of E, which is D : y21 C y1 D x3

1 220323x1 C 41292202. The

change of variables .x1, y1/ D . 1

9x, 1

27y 1

2/ transforms D into E . Then, applying

Proposition 2.6.3, we compute

5.4600863256138232.

Finally, the values of the constants c12, c13, c14 and c15 are as follows:

c12 1.31931 1030, c13 1.042 1074, c14 D 1, c15 D 26.51416.

Now we have all data to apply Theorem 9.1.3. Inserting into (9.8) the values of the

constants ˛, ˇ, , and c13, : : : , c18 which we have already computed, we see that

B .M / > 0, where

B .M / D 4.689 1074 .log.M C 1/ C 1/ .log.log.M C 1// C 26.5142/4

0.6127 0.46459 M 2.


jM j max¹c12, 3.1 1041º D 3.1 1041. (9.16)

Since B .3 1041/ > 0, the bound (9.16) is essentially the best upper bound that can

be obtained from Theorem 9.1.3; its reduction to an upper bound of manageable size

will be accomplished in Section 10.2.4 of the next chapter.



Chapter 10

Reducing the bound obtained in Chapter 9

In this chapter we will show how one can reduce the upper bound for M , which was

obtained from the application of Theorem 9.1.3 and the Concluding remark following

it. In our examples, more specifically, in Sections 9.3–9.6 we obtained an upper bound

for M so large that there is no hope to check which points P “cover” solutions of the

Diophantine problem under consideration; cf. relevant discussion on page 55, below

(4.3). In the terminology therein, the Diophantine problems in our examples have been

solved effectively but not explicitly. In the present chapter we aim to show how onecan use the superficially “non-practical” huge upper bound for M in order to make an

enormous “jump” down to a new really small upper bound. Sounds strange but, still,

it is true! This new bound is so small that it allows one to check directly which points

P “cover” actual solutions and leads provably to the complete set of them.

We keep, of course, all notations etc. of Chapter 9; there we wrote L.P / for the

linear form (9.2). Let us consider the following slightly different linear form, which is

more appropriate for the purposes of the present chapter,

D

.P/D

dr0

!1

L.P/D

n0

Cn1 1

C Cnk k,

i D r0`i

!1

.i D 1, : : : , k/

(10.1)

The upper bound for jL.P/j given by (5.3), (6.15), (7.30) and (8.25) can be written

uniformly as jL.P/j c16 exp.c118 . C c17 M 2// (for c16, c17 and c18 see Theo-

rem 9.1.3). Moreover, by (9.6)),

N D max0ik

jni j ˛M C ˇ,

so that M 2 3N 2, for some explicit positive constant 3. It follows then that

jj 1 exp.2 4N 2/, (10.2)

where

1 D dr0c16

!1

, 2 D c118 . C c17/, 4 D c1

18 3. (10.3)

and c16, c17 and c18 are defined in Theorem 9.1.3.



122 Chapter 10 Reducing the bound obtained in Chapter 9

10.1 Reduction using the LLL-algorithm

In this section we put our problem in a more general setting, totally independent from

elliptic logarithms, as follows:Problem. Consider a linear form given by (10.1), where 1, : : : , k are known realnumbers and n0, n1, : : : , nk are unknown integers. Suppose first, that an upper bound

(10.2) holds for jj, where 1, 2 and 4 are explicit positive constants, and second,

that a huge upper bound for N def D max0ik jni j is known. Reduce this upper bound

to a manageable size.

We construct a lattice , which is a sublattice of ZkC1. We will write the vectors

of as columns .k C 1/ 1 and will denote them by bold letters. The lattice that

we will use in this section is generated by the column vectors of the matrix

M D

0BBBB@1 0 0

.

.

. . . .

.

.

....

0 1 0

ŒC 1 ŒC k C

1CCCCA ,

where C is a large positive integer which will be specified below. By Œx, where x 2 R,

we mean the integer resulting from x if we cut its decimal digits; thus Œ3.85 D 3

and Œ3.85 D 3. Formally, Œx D dxe if x < 0, and Œx D bxc if x 0.

We consider an LLL-reduced (ordered) basis .b0,b1, : : : , bk/ of in the sense of [27]. Roughly speaking, all vectors of an LLL-reduced basis have “approximately

equal lengths” and are “approximately orthogonal” to each other. We explain a lit-

tle more the important notion of an LLL-reduced basis, following [27, Section 1] (see

also [72, Chapter 3] and [9, Section 2.6]) although these explanations are not neces-

sary, in the strict sense of the word, for the applications of this book, in which we use

only some properties of the LLL-reduced basis.

We denote the usual euclidean length by j j. Let .b0 , b1 , : : : , bk

/ be the (ordered)

basis of RkC1 which we obtain if we apply to .b0, b1, : : : , bk/ the Gram–Schmidt

orthogonalisation process,

1

so that, for 0 j < i k,

bi D bi

i1Xj D1

ij bj , ij D

hbi , bj ijbj j2

,

where hi denotes the usual inner product. The fact that .b0, b1, : : : ,bk/ is an LLL-

reduced basis means, by definition, that

jij j 1=2 .0 j < i k/

1 b0 ,b1 , : : : ,bk is a basis of RkC1 but, in general, one cannot expect that it is a basis for .



Section 10.1 Reduction using the LLL-algorithm 123

and

jbi C i ,i1bi1j2 3

4jbi1

j2 .0 < i k/.

Starting from any basis of a given lattice – in our case, from the basis consisting of the

column vectors of M –, the LLL-algorithm of [27] computes an LLL-reduced basisvery effectively; in our case, this means that the number of arithmetic operations that

are needed are O..k C 1/4 log C /, in view of [27, Proposition 1.26].

LLL-reduced bases, in general, have a number of important properties, extremely

useful for various applications, among them simultaneous approximation [27, Propo-

sition 1.3] and factorisation of polynomials with rational coefficients [27, Sections 2

and 3]. The close connection of LLL-bases to the explicit solution of Diophantine

equations was described for the first time2 in [72]. Applications to the explicit resolu-

tion of various classes of Diophantine equations are found, for example, in3 [64] (Thue

equations), [65] (Thue–Mahler equations), [13] (norm form equations), [14] (“rela-

tive” Thue equations), [54, 15] (Weierstrass equations), [49] (Weierstrass equations

over number fields), [62] (quartic elliptic equations), [59] (general cubic equations),

[57] (elliptic binomial equations), [63] (simultaneous Pell equations), [56] (general el-

liptic equations), [48] (triangularly connected decomposable form equations), [47, 36]

(S -integral solutions of Weierstrass equations); see also the book [50] for a concise

overview of applications to Diophantine equations.

An important property of the LLL-bases that we will use in this section is the fol-

lowing: Once we know b0, we can have an explicit lower bound for the length of any

non-zero vector of . More specifically,

jxj 2k=2jb0j for every non-zero x 2 . (10.4)

Suppose now that we want to solve the inequality (10.2), with as in (10.1), where

.n0, n1, : : : , nk / 2 ZkC1 and jni j < B1.N / for i D 0, 1, : : : , k. Here we imagine

B1.N / as a “very large” integer.

We consider the following x 2 :

x def D M

0

BBBB@n1

.

.

.

nkn0

1

CCCCAD

0

BBBB@n1

.

.

.

nkQ

1

CCCCAD

0

BBBB@n1

.

.

.

nkn1ŒC 1 C C nkŒC k C n0C

1

CCCCA,

where Q D n1ŒC 1 C C nk ŒC k C n0C . For i D 1, : : : , k we have ŒC i DC i C i , for some i absolutely less than 1, hence, Q D C .n11 C C nkk/

and, consequently,

j Qj C jj C kN < C 1e24N 2 C kB1.N /.

2 To the best of the author’s knowledge.3 Here we list, indicatively, only papers dealing with broad classes of Diophantine equations; the list is

by no means exhaustive!




By (10.4), the above estimate of j Qj and (10.2) we have

2kjb0j2 k

XiD1

jni j2 C Q2 kB1.N /2 C .C 1e24N 2 C kB1.N//2.

From this and a few elementary calculations we obtain

4N 2 2 C log.1C / log¹q

2k jb0j2 kB1.N / kB1.N /º. (10.5)

In order that the left-hand side makes sense, the quantity inside the brackets must be

a positive real number, which is equivalent to 2kjb0j2 kB1.N /2 > k2B1.N /2.

This, in turn, is equivalent to 2kjb0j2 > .kB1.N / C 1

2/2 1

4 and, clearly, a sufficient

condition for the last inequality is

jb0

j> 2k=2.k

C 1

2/B1.N /. (10.6)

The question is: How can we guarantee the validity of the last condition? Now is the

moment for choosing C . Remember that the LLL-reduced basis of is “almost or-

thogonal”. Heuristically, this implies that the volume of the parallelepiped formed by

the vectors b0, b1, : : : ,bk – the fundamental volume of the lattice – is “approxi-

mately” equal toQk

iD0 jbi j. But the volume of is equal to the absolute value of the

determinant of M , which is C , henceQk

iD0 jbi j is of the size of C . On the other

hand, all vectors of an LLL-reduced basis have lengths of “same size”, hence we ex-

pect that jb0j is of the size of C 1=.kC1/. Therefore, if we had chosen the integer C

somewhat larger than

2k.kC1/=2 k C 1

2

kC1

B1.N /kC1, 013 (10.7)

it would be reasonable to expect that condition (10.6) is fulfilled; if not, we would try

a new choice for C , somewhat larger than the previous one; in practice this always

works. Summing up, we have proved the following proposition:

Proposition 10.1.1. Let D n0 C n1 1 C nk k , where 1, : : : , k are specificreal numbers and n0, n1, : : : , nk are unknown integers satisfying the two conditionsN D max0ik jni j B1.N / and jj 1 exp.2 4N 2/ , where we understand

that B1.N / is a specific very large integer and 1, 2, 4 specific positive real numbers.Choose a positive integer C of size (10.7) and consider the lattice defined at the beginning of this section. Compute an LLL-reduced basis .b0, b1, : : : ,bk/. If b0

satisfies (10.6) , then N satisfies (10.5).

The very important result of this proposition is that the new upper bound for N is

of the size of

1=24

2 C log 1 C .k C 1/

²k

2log2 C log

k C 1

2

C log B1.N /

³1=2

,

i.e. the size of the new upper bound for N is somewhat larger than p log.B1.N// !



Section 10.2 Examples 125

10.2 Examples

In this section we apply the reduction technique described in Section 10.1 to the lin-

ear form L.P / already obtained in each of Sections 9.3, 9.4, 9.5 and 9.6. In order tocompute an LLL-reduced basis for the lattice in each of our examples, we can use

e.g. either of the following:

The routine LLL of MAPLE with input data the column vectors of M .

The routine qflll of PARI with input data M and option 1, to declare that the

routine must view its data as integers. Actually, qflll returns the transformation

matrix from the reduced to the initial basis, so that the LLL-reduced basis is formed

by the column vectors of the matrix product M qflll.M /.

The routine LLL of MAGMA with input data the matrix MT , the transpose of the

matrix M .

As b0 we can take either the first row of the matrix LLL.MT / (MAGMA), or the first

column of the matrix M qflll.M / (PARI), or the first vector of the vector array

returned by LLL.Œc1, c2, c3/ (MAPLE), where c1, c2, c3 are the columns of M regarded

as vectors.

10.2.1 Weierstrass equation

We will make use of some of the numerical results that we obtained in Section 9.3. We

have k D 2, d D 1, r0 D 2, ˛ D 2, ˇ D 3 and we have computed all the parameters

involved in the definition of 1 and 2 (see (10.3)). Concerning 3, this must be chosen

so that M 2 3N 2 (cf. just above (10.2)). In our case, N ˛M C ˇ D 2M C 3,

hence M 2 .N 3/2=4 49

400N 2. For the right-most inequality it suffices to assume

that N 10, which certainly holds if M 10. Thus, 3 D 49

400 and now, in view of

(10.3), we can choose

1 D 6.73841, 2 D 6.2815, 4 D 0.0748.

In our case we have

D 2

!1

L.P/ D n0 C n1

2`1

!1

C n2

2`2

!1

D n0 C n1 1 C n2 2,

D n0 C n1.0.85484 : : : / C n2.0.65652 : : : /,

so that we have to compute !1, `1 D l.P 1/, `2 D l.P 2/ with a high precision. What is

the precision required for our computations? According to (9.10), B1.N / D 1.1 1041

and (10.7) suggests that the integer C be of the size of 1.664 10125. Let us be generous

and choose C D 10130. Since we must calculate ŒC i (i D 1, 2), the choice of C

forces us to make our computations with a precision of 130 decimal digits. For safety,




we do them with a precision of 150 decimal digits. Then ŒC is the integer resulting

from C if we truncate its decimal digits. In this way we calculate

ŒC 1

D 8548414188

0929152369„ ƒ‚ …130 digits

, ŒC 2

D 6565274509

9728533696„ ƒ‚ …130 digits

and consider the sublattice of Z3 generated by the columns of the matrix

M D0@ 1 0 0

0 1 0

ŒC 1 ŒC 2 C

1A .

Then we compute an LLL-reduced basis of . It turns out that

b0 D0@

5186369112644553909555354279275889624244640

12154461378581626599733559868313990698122965

8866073855682853407126296002425318856203365

1A

.

We see that jb0j 1.592 1043 which is of the size of C 1=3 2.1544347 1043, in

accordance with what we noted just above the relation (10.7).

It is straightforward to check that b0 satisfies the condition (10.6), therefore, by

Proposition 10.1.1, N satisfies (10.5), which implies that N 52, an extremely good

improvement! We set now B1.N / D 52 and repeat the process, by choosing C D 108,

so that the new matrix M is

M

D 0@1 0 0

0 1 0

85484141 65652745 100000000 1Aand the LLL-reduction to the columns of the above matrix furnishes us with

new b0 D0@ 117

166

167

1A .

We check that (10.6) is satisfied so that, by Proposition 10.1.1 we have a new, even

smaller, upper bound for N , obtained from (10.5). Indeed, the last relation implies that

N 17, hence also M 17.

Now we check which pointsP E D m1P 1 C m2P 2 C T , jm1j, jm2j 17, T 2 Etors.Q/

have the property that P C has integer coordinates. In other words, given a point P E D.x, y/ as above, we check whether or not the point P C D .u, v/ with4

u D x 5

12, v D y 1

2x 7

24has integer coordinates. A simple computer program can do the job very easily. Our

computational results are collected in Table 10.1.

4 See the transformation at the beginning of Section 9.3.




Table 10.1. All points P E D n1P E1 C n2P E2 C T E with P C D .u, v/ 2 ZZ.

n1, n2, T E P E D .x, y/ 12 5 P C D .u, v/

1,

1, O .293=12, 225=2/ .24, 100/

1, 0, O .79=12, 4/ .7, 1/

1,1, O .1733=12, 3465=2/ .144, 1805/

0, 2, O .161=12, 36/ .13, 29/

0, 1, O .67=12, 15=2/ .6, 5/

0,1, O .67=12, 15=2/ .6, 10/

0,2, O .161=12, 36/ .13, 43/

1, 1, O .1733=12, 3465=2/ .144, 1660/

1, 0, O .79=12,4/ .7, 7/

1,1, O .293=12,

225=2/ .24,

125/

1,0, .41=6, 0/ .3233=12, 4420/ .269, 4285/

0, 1, .41=6, 0/ .581=12, 663=2/ .48, 307/

0, 1, .41=6, 0/ .581=12, 663=2/ .48, 356/

1,0, .41=6, 0/ .3233=12, 4420/ .269, 4555/

1, 2, .31=12,0/ .2417=12, 2856/ .201, 2957/

0, 3, .31=12,0/ .73613=12, 960925=2/ .6134, 477395/

0, 1, .31=12,0/ .173=12, 85=2/ .14, 50/

0,0, .31=12, 0/ .31=12,0/ .3, 1/

0, 1, .

31=12,0/ .173=12, 85=2/ .14, 35/

0,3, .31=12, 0/ .73613=12, 960925=2/ .6134, 483530/

1,2, .31=12,0/ .2417=12, 2856/ .201, 2755/

1, 1, .113=12,0/ .269=12, 195=2/ .22, 109/

0, 1, .113=12,0/ .43=12, 13=2/ .4, 8/

0,0, .113=12,0/ .113=12,0/ .9, 5/

0, 1, .113=12,0/ .43=12, 13=2/ .4, 5/

1, 1, .113=12, 0/ .269=12, 195=2/ .22, 86/

Proposition 10.2.1. All integer solutions .u, v/ of the equation v2

Cuv

Cv

u3

u2 C 71u C 196 D 0 are those listed in the right-most column of Table 10.1.

10.2.2 Quartic equation

In Section 9.4 we saw that k D 4, d D 1, r0 D 1, ˛ D 3=2, ˇ D 3=2 and we computed

the parameters involved in the definition of 1 and 2. Further, N ˛M C ˇ D3

2.M C 1/, hence M 2 . 2

3N 1/2 289

900N 2, if we assume M 10 (so that also

N 10). Thus, 3 D 289=900 and, in view of (10.3), we can take

1 D 1.8256, 2 D 5.0395, 4 D 0.274518.




Choice of C : According to (9.11), B1.N / D 2.3 1088 and (10.7) suggest that the

integer C be of the size of 1.2162 10448. We choose C D 10450 and ask our computer

to work with a precision of 460 decimal digits.

The linear form to which we will apply the reduction process is

D 1

!1

L.P/ D n0 C n1

`1

!1

C n2

`2

!1

C n3

`3

!1

C n4

`0

!1

D n0 C n1 1 C n2 2 C n3 3 C n4 4

D n0 C n1.0.03977 : : : / C n2.0.25601 : : : /

C n3.0.41647 : : : / C n4.0.48622 : : : /.

In the present case is a sublattice of Z5 and

M D0BBBB@

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

ŒC 1 ŒC 2 ŒC 3 ŒC 4 C

1CCCCA ,

where

ŒC 1 D 3977480557 4553315116

„ ƒ‚ …449 digits

, ŒC 2 D 2560106453 4697144898

„ ƒ‚ …450 digits

ŒC 3

D

4164734877

6333939317„ ƒ‚ …450 digits

, ŒC 4

D

4862224793

9803530989„ ƒ‚ …450 digits

The first vector of the LLL-reduced basis is

b0 D

0B@ 82327895474728509176238354944130475886404355810644437312762526348821779131423108393917712

15471125794154247726983556666031672982938641013672184463414407364168745466146688236207559

111795263890064364945667733492720885363502478756971653430085800804576405203867180516532531

244350433420305070938288260659067994439387630719216219811147801739708020305288497923338381

491414271890840780959306036704856290884457656764343679435212935777287084290581917008213944

1CA,

with approximate length 5.664 1089, hence of the size of C 1=.kC1/ D 1090, as ex-

pected. We check thatb0 satisfies the condition (10.6), therefore, by Proposition 10.1.1,

N satisfies (10.5), which implies that N 55. We set now B1.N / D 55 and repeat

the process, by choosing C D 1016. We obtain

new b0 D

0BBBB@151

77

378

62

984

1CCCCAwhich satisfies (10.6) and Proposition 10.1.1 asserts that we can apply relation (10.5).

Doing so we obtain the new upper bound N 11, hence also M 11, and we check




which points

P E D m1P 1 C m2P 2 C m3P 3, jm1j, jm2j, jm3j 11

have the property that P C

has integer coordinates. In other words, we check for whichpoints P E D .x, y/ as above, the point P C D .u, v/ with5

.u, v/ D

4.3x C 2/

3.2y 1/,

216x3 C 216x2 59 108y2 C 108y

27.2y 1/2

(10.8)

has integer coordinates. Our computational results, obtained under the restriction y ¤1=2, are collected in Table 10.2.

Table 10.2. All points P E D n1P E1 C n2P E2 C n3P 3 with P C D .u, v/ 2 Z Z.

n1, n2, n3 12 5 P E

D .x, y/ P C

D .u, v/

1, 1, O .19=3, 27=2/ .1, 2/

1,0, 1 .43=12, 13=8/ .4, 25/

1,0,0 .2=3, 1=2/ .0, 1/

1,0,1 .13=3, 11=2/ .2, 7/

1,2,0 .1433=507, 2343=4394/ .130, 26701/

0, 2, 0 .262=75, 109=250/ .130, 26701/

0,0, 1 .8=3, 3=2/ .2, 7/

0,0,1 .

8=3, 3=2/ .

4,

25/

0,1,0 .5=3, 5=2/ .1, 2/

Does Table 10.2 miss any integer solution .u, v/?

First, in view of 2y 1 appearing in the denominators of (10.8), we must check

whether the rational point with y D 1=2, namely the point .2=3, 1=2/, corresponds

to an integer solution .u, v/. Viewing (10.8) as an equality in the function field Q.E/,

we see that the values of the functions u, v at the point .2=3, 1=2/ are 2=9 and

83=81, respectively.6 Actually, .u, v/ D .2=9,83=81/ is a point on C but does not

furnish an integer solution.

Second, because the birational transformation C 3 .u, v/ 7! .x, y/ 2 E is givenby7

.x, y/ D

6v C 6 C u2

3u2 ,

8v C 8 C 4u2 C u3

2u2

, (10.9)

we must consider separately the value u D 0. Obviously we have the solutions .u, v/ D.0, 1/, .0, 1/, only the first of which is listed in the table. The reason why the solution

5 See (6.5) and (6.6).6 This is not so straightforward. A justification of this claim is given at the end of this section; see

Singular values of the birational transformation.

7 See (6.4).




.u, v/ D .0, 1/ does not appear is because it corresponds to the zero point O 2 E , as

a computation in the function field Q.C / shows.8

We have thus proved the following proposition:

Proposition 10.2.2. All integer solutions of the equation 52 u4 C 1

2 u3 C u2 C 1 D v 2

are .u, v/ D .0, 1/ and those listed in the right-most column of Table 10.2.

Singular values of the birational transformation. We must check the finitely many

points .x, y/ 2 E, such that y is a zero of the denominator of the rational transforma-

tion E ! C separately, as to whether they correspond to a point on C which otherwise

we might miss. In our example, we have to check only the point .x, y/ D .2=3, 1=2/.

Below we explain why this point is mapped to the point .2=9,81=83/ 2 C , hence to

no integer solution of our Diophantine equation.

The finitely many points .u, v/

2C , such that u is an integer and, at the same time,

a zero of a denominator of the rational transformation C ! E can be trivially found.

In this sense, it is not really necessary to find out to which points .x, y/ 2 E such

points .u, v/ are mapped; nevertheless, below we discuss this issue as well.

First, we work in the function field Q.E/ viewing u and v in (10.8) as functions. In

view of y2 D x3.31=3/x685=108 we have y21=4 D .3xC2/.9x26x89/=27,

hence3x C 2

2y 1D 27

4 2y C 1

9x2 6x 89

and, therefore, by (10.8), the value of the function u at the point .2=3, 1=2/ is

u.2=3, 1=2/ D 2=9. Next, we consider the function v. By (10.8) we have

v D 8.3x 1/.3x C 2/2 27.2y 1/2

27.2y 1/2 D 8

27.3x 1/

3x C 2

2y 1

2

1,

from which we see that v.2=3, 1=2/ D 83=81. This proves that, in the birational

transformation from E to C , the point .x , y/ D .2=3, 1=2/ 2 E is mapped to the

point .u, v/ D .2=9,83=81/ 2 C .

Next, we will show that, in the inverse transformation from C to E, the point

.u, v/ D .0, 1/ 2 C is mapped to O 2 E.

We work in the function field Q.C /. By v2

1

D.5=2/u4

.1=2/u3

Cu2 we have

v C 1

u2 D 5u2 C u C 2

2.v 1/ .

Then, by this relation and (10.9),

x D 2v C 1

u2 C 1

3D 15u2 C 3u C 5 C v

3.v 1/ ,

y D 4

u v C 1

u2 C 2

uC 1

2D 2

u 5u2 C u C 2

v 1C 2

uC 1

2D 20u2 C 3u C 4 C uv C 4v

2u.v 1/ .

8 See below Singular values of the birational transformation.




Taking into account these expressions of x and y and writing projectively the functions

u, v and x, y, using u D U=W , v D V =W and x D X=Z , y D Y =Z, we can express

the rational map from the projective curve C to the projective curve E as follows:

.U : V : W / 7! 2.15U 2 C 3U W C 5W 2 C V W / U :

3.20U 2 C 3U W C 4W 2 C U V C 4V W / W :

6UW. V W /

,

from which we see that the point .0 : 1 : 1/ 2 C is mapped to the point .0 : 1 : 0/ DO 2 E.

10.2.3 System of simultaneous Pell equations

In Section 9.5 we saw that k D 2, d D 1, r0 D 2 and ˛ D 2, ˇ D 3, so that, as inSection 10.2.1, M 2 49

400N 2, if we assume M 10; hence, 3 D 49=400. In Section

9.5 we also computed the parameters involved in the definition of 1 and 2. Thus, in

view of (10.3), we can take

1 D 0.7496, 2 D 9.5675, 4 D 0.1139.

Choice of C : According to (9.5), B1.N / D 1.71041 and (10.7) suggest that the integer

C be of the size of 6.14125 10125. We choose C D 10126 and ask our computer to

work with a precision of 140 decimal digits.


D 1

!1

L.P/ D n0 C n1

`1

!1

C n2

`2

!1

D n0 C n1 1 C n2 2

D n0 C n1.0.52627 : : : / C n2.0.79791 : : : /.

The lattice is a sublattice of Z3 and

M

D 0@1 0 0

0 1 0

ŒC 1 ŒC 2 C 1A ,

where

ŒC 1 D 5262784452 0127775724„ ƒ‚ …126 digits

, ŒC 2 D 7979116877 4671848557„ ƒ‚ …126 digits


b0

D 0@1214390824826344155895547868148571092185

637000902668796003574070357246352983001516

641839507507851952420371897464942537470648 1A .




The vector b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N sat-

isfies (10.5), which implies that N 42. We set now B1.N / D 42 and repeat the

process, by choosing C D 108. We obtain

new b0 D0@ 96

37

192

1A .

The new b0 satisfies (10.6), hence we obtain the new upper bound N 14.

Given a point .x, y/ 2 E we obtain a solution .U , V , W / to the system of simulta-

neous Pell equations as follows:

From .x, y/ we obtain .u, v/ using (6.5) and (6.6), where is as in (7.12), not as in

Lemma 6.1.9

We use (7.15) to obtain the value of W and (7.16) to obtain V and U .

In this way we compute the following map .x, y/ 7! .U , V , W /:

U D 9.5220x2 4067520x C 642159360 132xy 46112y 15y2/

908119808 166320y 28404x2 27y2 C 54x3 ,

V D 3.3924x2 2197632x C 373490944 C 9y2 C 180xy 7680y/

908119808 166320y 28404x2 27y2 C 54x3 ,

W

D 3.8676x2 2203968x 208626944 C 1260xy 386400y C 9y2/

908119808 166320y 28404x2

27y2

C 54x3

.

Then we check which points P , where

P E D m1P 1 C m2P 2 C T , jm1j, jm2j 14, T 2 Etors.Q/,

have the property that P E D .x, y/ is not a zero of the denominator of the above

expressions of U , V , W and maps to an integer solution .U , V , W /. Our results are

summarised in Table 10.3.

The exceptional values of .x, y/. Now we must check the points .x, y/ 2E.Q/ which are zeros of the denominator q.x , y/

def

D 908119808 166320y 28404x2 27y2 C 54x3 in the expressions of U , V , W above Table 10.3. Comput-

ing Resy .q.x, y/, f.x/ y2/ we find out that it is the product of a quartic irreducible

polynomial in x times .3x 1052/2, hence the only points .x, y/ 2 E.Q/, for which

q.x, y/ D 0, are .1052=3, ˙3080/.

9 But see the comment just before Lemma 6.1.




Table 10.3. All points P E D n1P E1 C n2P E2 C T E mapping to .U , V , W / 2 Z3.

n1, n2, T E 12 3 P E D .x, y/ P C D .u, v/

1,0, O .1976=3,

14784/ .17, 5,

13/

0, 1, O .1052=3, 3080/ .5, 1, 1/

1,1, O .1304=3, 6272/ .17, 5, 13/

1, 0, .1048=3, 0/ .848=9, 101024=27/ .17, 5,13/

0,1, .1048=3, 0/ .1352=75, 202048=125/ .5,1,1/

1,1, .1048=3, 0/ .64=3, 2624/ .17, 5,13/

1,0, .128=3, 0/ .376=3, 4032/ .17,5,13/

0,1, .128=3, 0/ .880=3, 3360/ .5, 1, 1/

1,1, .128=3, 0/ .664=3, 4224/ .17, 5, 13/

1,0, .920=3, 0/ .2396=3, 20664/ .

17,

5,

13/

0,1, .920=3, 0/ .12728=3, 275520/ .5, 1,1/

1,1, .920=3, 0/ .4979=3, 66297/ .17,5,13/

We compute10 the Puiseux series of f .x/ y2 around x D 1052=3. This is equiv-

alent to computing the Puiseux series of f .t C 1052

3 / y2 around t D 0. We find

y1.t/ D 3080 211

5t C 2603

22000t 2 C O.t 3/

y2.t/ D 3080 C 211

5 t 2603

22000 t 2 C O.t 3/.

Inserting x D t C1052=3, y D yi .t/ (i D 1, 2) in the expressions of U , V and W and

writing the resulting expressions as series in t , which we denote by U i .t/, V i .t/, W i .t/,

we obtain

U 1.t/ D 18955

2603 55389191

948585260t 25665908007097

76050356898824000t 2 C O.t 3/

V 1.t/ D 4903

2603 3893357

189717052t 14906658779053

167310785177412800t 2 C O.t 3/

W 1.t/ D 11297

2603C 18587273

298126796t C 3883710777647

23901540739630400t 2 C O.t 3/

and

U 2.t / D 5 1

140t 27

616000t 2 C O.t 3/

V 2.t / D 1 1

308t 23

1355200t 2 C O.t 3/

10 All computations below were performed with MAPLE.




W 2.t/ D 1 1

44t C 139

1355200t 2 C O.t 3/.

Thus, the solutions .U , V , W / corresponding to .x, y/ D .1052=3, ˙3080/ are

.U 1

.0/, V 1

.0/, W 1

.0// D

26031.

18955,

4903, 11297/ and .U 2

.0/, V 2

.0/, W 2

.0//

D .5, 1, 1/, respectively. The second solution is not essentially new, while the

first, being a rational but not an integer solution to our system of Pell equations, is

rejected. Thus, we have the following proposition:

Proposition 10.2.3. All positive integer solutions of the simultaneous Pell equationsU 2 11V 2 D 14 , W 2 7V 2 D 6 are .U , V , W / D .5,1,1/, .17, 5, 13/.

10.2.4 General elliptic equation: A quintic example

In Section 9.6 we saw that k D 2, d D 1, r0 D 1 and ˛ D 1, ˇ D 1, so that M 2

81

100N 2, if we assume M 10; hence 3 D 49=400. We have already computed in

Section 9.6 the parameters involved in the definition of 1 and 2. Thus, in view of

(10.3), we can take

1 D 0.7032, 2 D 2.3588, 4 D 0.0836.

Choice of C : According to (9.6), B1.N / D 3.11041 and (10.7) suggest that the integer

C be of the size of 3.724 10126. We choose C D 10127 and work with a precision of

140 decimal digits.


D 1

!1

L.P/ D n0 C n1

`1

!1

C n2

`2

!1

D n0 C n1 1 C n2 2

D n0 C n1.0.25260 : : : / C n2.0.03262 : : : /.

The lattice is a sublattice of Z3 and

M D0

@1 0 0

0 1 0

ŒC 1 ŒC 2 C

1

A,

where

ŒC 1 D 2526017522 97484775902„ ƒ‚ …128 digits

,

ŒC 2 D 3262684039 70180087887„ ƒ‚ …127 digits

.


b0

D 0@2131974326945673073515451110976172071914863

763870092422929229257003034756651925844085

1984748517852306190196449030197544381066969 1A .




The vector b0 satisfies the condition (10.6), therefore, by Proposition 10.1.1, N sat-

isfies (10.5), which implies that N 48. We set now B1.N / D 48 and repeat the

process, by choosing C D 108. We obtain

new b0 D0@ 329

150

175

1A .

The new b0 satisfies (10.6), hence we obtain the new upper bound N 13.

We check, therefore, which points P , where

P E D m1P 1 C m2P 2, jm1j, jm2j 13,

have the property that P E

D .x, y/ maps via the transformation (8.7) to a point

.u, v/ 2 C with integer coordinates. The computer search does not include “singular”

points .x, y/ that are zeros of the denominators in (8.7). For “non-singular” points our

search is summarised in Table 10.4.

Table 10.4. All points P E D n1P E1 C n2P E2 with P C D .u, v/ 2 Z Z.

12 3 n1, n2 P E D .x, y/ P C D .u, v/

0, 1 .68292, 35626473=2/ .0, 0/

1,

2 .182145033=58564, 969834918357=14172488/ .243,

3/

1, 1 .2710, 81029=2/ .0, 0/

The exceptional values of .x, y/. The irreducible factors of the denominators in the

transformation (8.7) are x C 4365 and x 2169, hence we must check separately

the points .x, y/ D .4365, ˙315171=2/, .2169, ˙79947=2/ 2 E.Q/. Instead of

computing the Puiseux expansions of f .x/ y2 around x D 4365, we compute the

Puiseux series of f .t 4365/ y2 around t D 0. We find

y1.t/ D 315171

2 161784

1297 t C 1785281093

19636425657t 2 C O.t 3/

y2.t/ D 315171

2C 161784

1297t 1785281093

19636425657t 2 C O.t 3/.

Then, by (8.7),

U .t 4365, y1.t// D 13740972520149225

23841086865992C O.t/,

V .t

4365, y1.t//

D 6346275

313874 CO.t/,




hence the point .x, y/ D .4365, 315171=2/ 2 E is mapped to the point .u, v/ D.13740972520149225=23841086865992, 6346275=313874/ 2 C , with non-inte-

ger coordinates. We compute also

U.t 4365, y2.t// D 1226330361 t2 865647 t1 C 4923315231682209

C O.t/

V .t 4365, y2.t// D 35019 t1 C 16679

1297C O.t/,

which shows, if we write the rational map (8.7) projectively, that the projective point

.4365 : 315171=2 : 1/ 2 E is mapped to the projective point .1 : 0 : 0/, the

“point at infinity” of C , which does not furnish a solution to our Diophantine equation.

Similarly, working with the points .2169, ˙79947=2/, we find that .2169, 79947=2/

is mapped to the non-integer point

.19460428805563741=56309567282892, 1324909=238854/ 2 C ,

and .2169, 79947=2/ is mapped to the “point at infinity”.

We have, thus, proved the following proposition:

Proposition 10.2.4. The equation 3v5 C 3uv3 271uv 3u2 D 0 has .u, v/ D.0, 0/, .243, 3/ as its only integer solutions.



Chapter 11

S-integer solutions of Weierstrass equations

In this chapter we present a method for solving a Weierstrass equation in S -integers,

a term that we immediately explain.

Let an integer s 2 and S D ¹p1, : : : , ps1, 1º, where pi is a prime for i D1, : : : , s 1 and 1 is merely a symbol. We often characterise 1 as the infinite prime,

to distinguish it from the finite primes p1, : : : , ps1. The use of the infinity symbol 1will be justified below.

The systematic exposition of a method for the resolution of Weierstrass equationsin S -integers is due to N. Smart [47] and Peth˝ o–Zimmer–Gebel–Herrmann [36]; see

also page x in the Preface.

Let us agree that, for any non-zero rational number x, by denominator of x we

mean the denominator b 1 of the irreducible fraction a=b D x (a, b integers) and

by numerator of x we mean a. If the denominator of a rational prime x is divisible at

most by primes in ¹p1, : : : , ps1º, then we say that x is an S -integer . We will include

also 0 among the S -integers. The set of S -integers, which is obviously a subring of Q

containing Z, will be denoted by ZS . The assumption card.S / D s 2 is made in

order to ensure that S contains at least one finite prime; if this were not the case, thenZS coincides with Z and this chapter has nothing new to add to Chapter 5.

Throughout this chapter, with the exception of Section 11.4, where a specific exam-

ple is solved,

S D ¹p1, : : : , ps1, 1º, s 2

where p1, : : : , ps1 are finite primes. Our purpose is to present an explicit method for

computing C.ZS /, where

C : v2 C a1vu C a3 D u3 C a2u2 C a4u C a6, a1, a2, a3, a4, a6 2 Z (11.1)

is a model of an elliptic curve.

11.1 The formal group of C and p-adic elliptic logarithms

In this section we will study the curve C defined by (11.1) from the point of view of

its formal group. Our exposition is based mainly on [45, Chapter IV].

We fix a (rational) prime p and, as usual, we denote by Qp the field of p-adic

numbers (the completion of Q under the metric induced by the p-adic absolute value

j jp) and by Zp the ring of p-adic integers, i.e. those x 2 Qp with jxjp 1. The



138 Chapter 11 S -integer solutions of Weierstrass equations

(only) maximal ideal of Zp is pZp which, for simplicity, we will denote by M. The

natural homomorphism Zp ! Zp=M is denoted by t ! Qt . The residue field Zp=M

is isomorphic to Fp D Z=pZ.

The curve QC =F

p defined by v2

C Qa

1uv

C Qa

3v D

u3

C Qa

2u2

C Qa

4u

C Qa

6 is the

reduction of C mod p. The reduction mod p homomorphism

C.Qp/ 3 Q 7! QQ 2 QC .Fp/

is defined as follows. Any point Q 2 C .Qp/ can be written in projective coordinates

.U 0 : V 0 : W 0/ with all three coordinates in Zp and not all zero. Then QQ D . QU 0 : QV 0 :QW 0/ is a point belonging to QC .Fp/.

The curve QC =Fp may have singular points but, anyway, the subset of the non-

singular points of QC .Fp/, denoted by C ns.Fp/, is a group. We define

C 0.Qp/ D ¹Q 2 C.Qp/ : QQ 2 C ns.Fp/º,

which is a subgroup of C .Qp/, and the kernel of the reduction

C 1.Qp/ D ¹Q 2 C.Qp/ : QQ D QOº;

for the above we refer the reader to [45, beginning of Section VII.2 and Proposition

2.1]. Note that QO 2 C ns.Fp/, hence C 1.Fp/ C 0.Fp/.

In order to define the formal group associated to C =Qp we need the model of (11.1)

defined by

w D z3 C a1zw C a2z2w C a3w2 C a4zw2 C a6w3 def D h.z, w/. (11.2)

The birational transformation between (11.1) and (11.2) is

.u, v/ 7! .z, w/ Du

v, 1

v

.z, w/ 7! .u, v/ D

z

w, 1

w

.

With a few obvious exceptions, any point Q of our elliptic curve can be viewed either

as a pair .u.Q/, v.Q// satisfying equation (11.1) or as a pair .z.Q/, w.Q// satisfyingequation (11.2), with the two pairs related by the above birational transformation. We

characterise z.Q/ as the z-coordinate of Q.

Lemma 11.1.1. If Q 2 C.Q/ \ C 1.Qp/ , then z.Q/ 2 M.

Proof. By Lemma 1.2.2, the coordinates of Q are .U=W 2, V =W 3/, where U , V , W

are integers with gcd.U , W / D 1 D gcd.V , W /. Projectively, Q is the point

ŒU W , V , W 3 , hence, the assumption QQ D QO implies that W 0 .mod p/

and, consequently, U V 6 0 .mod p/. Then, z.Q/ D .U=W 2 / : .V=W 3/ DU W = V 2 M.



Section 11.1 The formal group of C and p-adic elliptic logarithms 139

If z 2 M and we define recursively

h1.z, w/ D h.z, w/ and hmC1.z, w/ D hm.z, h.z, w//,

then the limit w.z/ D limm!1

hm.z, 0/

is expressed as a convergent in Qp power series in z and w.z/ D h.z, w.z//; see [45,

Chapter IV, Proposition 1.1]. This makes it possible to express u and v as convergent

power series of Qp as follows:

u.z/ D z

w.z/ D z2 a1z1 a2 a3z .a4 C a1a3/z2 2 Qp (11.3)

v.z/ D 1

w.z/ D z3 C a1z2 C a2z1 C a3 C .a4 C a1a3/z 2 Qp.

The invariant differential1 then has the following expansion

!.z/ D du.z/

2v.z/ C a1u.z/ C a3

D .1 C a1z C .a21 C a2/z2 C .a3

1 C 2a1a2 C 2a3/z3

C .a41 C 3a2

1a2 C 6a1a3 C a22 C 2a4/z4 C /dz 2 Zpdz

(see [45, Section IV.1]). For z1, z2 2 M, a sum F .z1, z2/ is defined by means of a

p-adically convergent in M power series as follows:2

First, for z 2 M we define

i.z/ D u.z/

v.z/ C a1u.z/ C a3

D z

1 C a1z C a3w.z/ D z2 a1z1

z3 C 2a1z2 C 2 M.

(11.4)

Next we define

D .z1, z2/ D w.z2/ w.z1/

z2 z1

2 M, D .z1, z2/ D w.z1/ .z1, z2/z1 2 M,

then

z3 D z3.z1, z2/ D z1 z2 a1

Ca32

Ca2

C2a4

C3a62

1 C a2 C a42 C a63 2 M(the fact that , and z3 belong toM is not obvious at first glance; cf. [45, page 119])

and, finally,

F .z1, z2/ D i.z3.z1, z2//

D z1 C z2 a1z1z2 a2.z21 z2 C z1z2

2 /

C .2a3z31 z2 C .a1a2 3a3/z2

1 z22 C 2a3z1z3

2 / C 2 M.

1 [45, Chapter III].

2 [45, Section IV.1].




It is a fact3 that F .z1, z2/ D F .z2, z1/, F .z1,F .z2, z3// D F .F .z1, z2/, z3/ and

F .z, i.z// D 0, hence the operation .z1, z2/ 7! F .z1, z2/ makes M an abelian group

denoted OC .M/ or C ; this is the formal group of C . For every integer 1, OC .M /

is the subgroup of OC .M/ consisting of the elements of M .

A remarkable property of the formal group is that, for any points Q1, Q2 2 C.Q/\C 1.Qp/, we have4

z.Q1 C Q2/ D F .z.Q1/, z.Q2//. (11.5)

Next, a logarithmic and an exponential function are defined.

The (p-adic) logarithmic function on C D OC .M/ is defined5 by

logCz DZ

!.z/ D z C a1

2z2 C a2

1 C a2

3z3 C a3

1 C 2a1a2 C 2a3

4z4 C

D z C1X

kD2

k

k zk

2 M, (11.6)

with k 2 Z for every k. The above series is indeed convergent to an element of M;

this follows from the fact that, for every k 2 it is true that p.zk =k/ k , which is

an easy exercise to prove.

Let

D´

1 if p > 2

2 if p D 2. (11.7)

For z

2M the logarithmic series is convergent and log z

2M .

Since D 1 for p > 2, the above claim is already proven a few lines above; it suffices

therefore to check for p D 2, when D 2. In this case, if z 2 M2, then it is easily seen

that 2.zk=k/ k C 1 for every k 2, which is sufficient for the proof of the claim.

The (p-adic) exponential function on C D OC .M/ is formally defined6 as the unique

power series expCz satisfying

logC.expCz/ D expC.logCz/ D z. (11.8)

By [45, Chapter IV, Proposition 5.5], expCz D z C

P1kD2

kkŠ

z k , where k 2 Z for

every k . In analogy with the logarithmic function:

For z 2 M the exponential series

expCz D z C1X

kD2

k

kŠ z k , (11.9)

is convergent and expCz 2 M .

3 [45, page 120].4 Remember that, by Lemma 11.1.1, z .Qi / 2 M for i D 1,2.5 [45, Section IV.5 and Proposition 4.2].

6 [45, Section IV.5].




To prove this, it suffices to show that the value of p at each summand of the infinite

sum is at least . We have p.kt k=kŠ/ kp.t / p.kŠ/ k p.kŠ/. On the

other hand,

p.kŠ/ D kpC k

p2C < k

p C k

p2 C D k

p 1,

hence, p.zk=kŠ/ > k k

p1 k. 1

p1/. The observation that the right-most

side is k=2 if p 3 and k if p D 2 completes the proof of our claim.

In view of the above discussion, we also conclude that:

If z 2 M then (11.8) is, indeed, meaningful as a relation in M .

Let now OGa.M /, with as in (11.7), be the formal additive group, so that, by its

definition, “addition” in OGa.M

/ means usual addition in M

. An important fact isthat the function

logC : OC .M / ! OGa.M /

is a group isomorphism, the inverse isomorphism being the function expC .7 This means

in practice that, if z1, z2 2M , then

logCF .z1, z2/ D logCz1 C logCz2 and expC.z1 C z2/ D F .expCz1, expCz2/.

(11.10)

We are almost ready to give the definition of the p-adic elliptic logarithm for points

Q 2

C.Q/\

C 1.Qp/ which fulfil a certain condition. We give first a simple lemma

useful here and in the sequel.

Lemma 11.1.2.

(i) Let Q D .u.Q/, v.Q// 2 C.Q/ and a prime p such that p.u.Q// < 0. Then,p.u.Q// is an even number and p.z.Q// D 1

2p.u.Q// , hence jz.Q/jp D

ju.Q/j1=2p .

(ii) If Q D .u.Q/, v.Q// 2 C.Q/ and p.u.Q// < N for some positive N 2 Z ,then, also, p.u.Q// < N .

(iii) For i

D 1, 2 , let Qi

D .u.Qi /, v.Qi //

2 C.Q/ be such that p.u.Qi // <

N

for some positive integer N . Then, p.u.Q1 C Q2// < N .(iv) For i D 1, : : : , k , let Qi D .u.Qi /, v.Qi // 2 C.Q/ satisfy p.u.Qi // < N

for some positive integer N . Then, p.u.m1Q1 C C mkQk// < N for anyintegers m1, : : : , mk .

Proof. (i) In view of Lemma 1.2.2 we can write u.Q/ D U=W 2 and v.Q/ D V =W 3,

where U , V , W 2 Z and .U , W / D 1 D .V , W /. If p.u.Q// < 0, then, neces-

sarily, p.W / 1 and, consequently, p.U / D 0 D p.V /. But then, p.u.Q// D

7 [45, Theorem 6.4, Chapter IV].




2p.W / which is an even number 4. Moreover, z.Q/ D u.Q/=v.Q/ DW U = V , hence p.z.Q// D p.W / D 1

2p.u.Q//.

(ii) We have z.Q/ D i.z.Q//, where i.z/ is the series (11.4). By (i), p.z.Q// D

p.u.Q// > N=2 and now, if in (11.4) we put z

D z.Q/, it becomes clear that

p.i.z.Q/// > N=2, hence z.Q/ > N=2. On the other hand u.Q/ D u.Q/,

therefore, applying (i), with Q in place of Q,weget p.u.Q// D 2p.z.Q// <

N , as claimed.

(iii) As in the proof of (ii), above, we see that p.z.Qi // > N=2 for i D 1, 2. By

(11.5) we have z .Q1 C Q2/ D F .z.Q1/, z.Q2// and a look at the series F .z1, z2/

on page 139 suffices to convince one that p.F .z.Q1/, z.Q2/// > N=2, hence

p.z.Q1 C Q2// > N=2 and then, by (11.3),

p.u.Q1 C Q2// D 2p.z.Q1 C Q2// < N .

(iv) Straightforward combination of (ii) and (iii).

Suppose now that Q D .u.Q/, v.Q// is a point of C.Q/ with p.u.Q// < 0. Then,

in view of Lemma 11.1.2(i), p.u.Q// 2. In the case that p D 2 we will make the

stronger assumption that 2.u.Q// 4. In other words, we assume that p.u.Q// 2 , where is defined in (11.7). Then, by Lemma 11.1.2(i), p.z.Q// 2 , hence,

z.Q/ 2 M and, consequently, logCz.Q/ 2 M and expC.logCz.Q// D z.Q/.

Definition 11.1.3. Let p be a prime and define by (11.7). If Q 2 C.Q/ and

p.u.Q// 2 , then the p-adic elliptic logarithm of Q is, by definition,

lp.Q/ D logCz.Q/ 2 M .

Proposition 11.1.4. With p, Q as in Definition 11.1.3 we have jlp.Q/jp D jz.Q/jp . In particular, lp.Q/ ¤ 0.

Proof. By hypothesis and Lemma 11.1.2(i), we can write u.Q/ D u1=p2t and

v.Q/ D v1=p3t , where t and p.u1v1/ D 0. Then, z.Q/ D pt z1, with p.z1/ D0. By definition, lp.Q/ D logCz.Q/ D pt z1 CPk2

kk

pkt zk1 , where k 2 Z for all

k. Since p.z.Q// D t , it suffices to show that k t p.k/ > t for every k 2, which

is an easy exercise (if p D 2 we must take into account that t D 2). Finally, sincejlp.Q/jp D jz.Q/jp D pt , it follows that lp.Q/ ¤ 0.

Proposition 11.1.5. Let be defined by (11.7). If Q1, : : : , Qk 2 C.Q/ such that p.u.Qi // 2 for i D 1, : : : , k , and n1, : : : , nk are any integers, then

z.n1Q1 C C nkQk/ D expC.n1logCz.Q1/ C nklogCz.Qk//, (11.11)

or, equivalently,

lp.n1Q1 C C nk Qk/ D n1lp.Q1/ C nk lp.Qk/. (11.12)




Proof. The relation (11.11) is obtained by straightforward induction once we have

proved the following: If Q1, Q2 2 C.Q/ and p.u.Qi // 2 for i D 1, 2, then

z.Q1 C Q2/ D expC.logCz.Q1/ C logCz.Q2//. For the proof of this, let us put zi DlogCz.Q

i/ (i

D 1, 2). By our discussion just before Definition 11.1.3, z .Q

i/ 2 M ,

therefore, zi 2 M . Then,

expC.logCz.Q1/ C logCz.Q2// D expC.z1 C z2/ D F .expCz1, expCz2/,

the right-most equality being implied by (11.10). We have F .expCz1, expCz2/ DF .expC.logCz.Q1//, expC.logCz.Q2/// D F .z.Q1/, z.Q2// D z.Q1 C Q2/, by

(11.5), as claimed.

Applying logC to the relation expC.logCz.Q1/ C logCz.Q2// D z.Q1 C Q2/, we

obtain logCz.Q1/ C logCz.Q2/ D logCz.Q1 C Q2/, which is equivalent to lp.Q1/ Clp

.Q2

/D

lp

.Q1 C

Q2

/.

In our applications to Diophantine equations, points Q 2 C.Q/ will be involved

which may not fulfil the condition p.u.Q// 2 necessary for both Definition

11.1.3 and relation (11.11). We overcome this complication using the fact that, for

any point Q 2 C .Q/, there exists a positive integer m such that p.u.mQ// 2 ;

see Proposition 11.1.7. First we prove a lemma.

Lemma 11.1.6. Let Q D .u.Q/, v.Q// 2 C.Q/ and let p be a prime such that p.u.Q// < 0.

(i) If m 2 Z and p.m/ 1

2p.u.Q// , then p.u.mQ// 2p.u.Q//.

(ii) For any N 2 Z there exists an integer n such that p.u.nQ// < N .

Proof. (i) By a1, : : : , a6 in this proof we mean a1, a2, a3, a4, a6. We will make use of

various statements of Exercise 3.7 in [45], dealing with the so-called division polyno-mials k 2 ZŒu, a1, : : : , a6 (k D 1, 2, : : :) and the related polynomials k and !k ,

again in 2 ZŒu, a1, : : : , a6 . With the aid of these polynomials we can express the

.u, v/-coordinates of the point kQ as

kQ D k.u.Q//k.u.Q//2

, !k .u.Q//k.u.Q//3

.

For the point Q in the announcement of the lemma, let us put for simplicity u.Q/ D u.

Then, we have

m.u/ D um2 C .lower degree terms/ 2 ZŒa1, : : : , a6, u

m.u/2 D m2um21 C .lower degree terms/ 2 ZŒa1, : : : , a6, u.

It follows that p.um2

/ is strictly less than the value of p at any other term of m.u/;

therefore p.m.u// D m2p.u/.




Now we estimate p.m.u/2/. The value of p at any term different from the leading

is at least .m2 2/p.u/, while

p.m2um21/

D.m2

1/p.u/

C2p.m/

.m2

1/p.u/

p.u/

D.m2

2/p.u/.

Therefore, p.m.u/2/ .m2 2/p.u/ and, consequently,

p.u.mQ// D p.m.u// p.m.u/2/ m2p.u/ .m2 2/p.u/ D 2p.u/.

(ii) For N 0 we choose n D m, where m 2 Z is as in (i). Next, let N < 0. In

view of (i) we can choose m1 2 Z such that p.u.m1Q// 2p.u.Q//. We put

Q1 D m1Q and apply once again (i) with Q1 in place of Q, finding m2 2 Z such that

p.u.m2Q1// 2p.u.Q1//. But then, p.m1m2Q/ 22p.u.Q//. Proceeding this

way we find integers m1, m2, : : : , mk such that p.m1m2 mk Q/ 2kp.u.Q//.

For sufficiently large k the right-hand side is < N and we take nD

m1m2 mk .

Proposition 11.1.7. Let p be a prime and define by (11.7). For every point P 2C.Q/ there exists a positive integer n such that p.u.nP // 2 . Consequently, byDefinition 11.1.3 , the p-adic elliptic logarithm l.nP / is meaningful.

Proof. First we show that there exists a positive integer n0, such that n0 QP D QO in the

curve QC =Fp.

If the curve QC =Fp is non-singular – hence it is an elliptic curve8 – we can take

n0 D j QC .Fp/j.If the curve QC =Fp has singular points,

9

then we turn to [45, Corollary 6.2, ChapterVII], according to which the subgroup C 0.Fp/ of C.Fp/ is of finite index, say k. Then

kP 2 C.Fp/, hence, fkP 2 C ns.Fp/. Since the reduction map is group homomorphism,

we obtain then k QP 2 C ns.Fp/. If ` is the order of the finite group C ns.Fp/, then

l.k QP / D QO, hence n0 QP D QO with n0 D `k.

Put Q D n0P , so that QQ D QO. This means that p.u.Q// < 0. By Lemma

11.1.6, there exists a positive integer m such that p.u.mQ// 2p.u.Q//. By

Lemma 11.1.2(i), p.u.Q// 2, hence p.u.mQ// 4 2 .

11.2 Points with coordinates in ZS

In this and the following sections we will continue the study of the elliptic curve C

defined by (11.1). For our numerical computations of p-adic elliptic logarithms, when

p ¤ 1, we will make use of the routine pAdicEllipticLogarithm of MAGMA. This

routine works with minimal models of elliptic curves, therefore, at this point we im-

pose the further condition that C be a minimal model. This condition is not really

restrictive, at least in principle. Indeed, let C 0 : g0.u0, v0/ D 0 be a minimal model

8 In other words, if p is a prime of good reduction for C .

9 In other words, if p is a prime of bad reduction for C .



Section 11.2 Points with coordinates inZS 145

for C . The coordinates .u0, v0/ are related to .u, v/ by .u0, v0/ D .2u C , 3v C2u C /, where , , , are appropriate rational numbers. If S 0 is the set of finite

primes dividing the product of the denominators of , , , , then, clearly, in order

to compute all S -integer points .u, v/ on C , it suffices to compute all S [

S 0

-integer

points .u0, v0/ on the minimal model C 0.

Along with the model C we will need to consider a short Weierstrass model

E : y2 D f.x/ def D x3 C Ax C B, A, B 2 Z (11.13)

of the same elliptic curve, which we will denote byE . Obviously, everything in Section

11.1 applies if in place of the model (11.1) we have the model (11.13).

A change of variable from C to E is of the form

u D 2x C , v D 3y C 2x C , (11.14)

with , , , appropriate rational numbers. The numbers , , and can be chosen

subject to the additional property that, in the inverse transformation

.x, y/ D .2u 2, 3v 3u C 3. //

D .0 2u C 0, 0 3v C 0u C 0/, (11.15)

the coefficients 0, 0, 0, 0 are integers; in particular 1 2 Z. For example, with

D 1

6, D a2

1 C 4a2

12, D a1

2, D a3

1 C 4a1a2 12a3

24

we transform equation (11.1) into the equation (11.13), with

A D 27a41 216a2

1a2 C 1296a4 432a22 C 648a1a3,

B D 3456a32 C 648a4

1a2 7776a1a2a3 C 46656a6 C 11664a23 15552a4a2

3888a4a21 1944a3

1a3 C 2592a21a2

2 C 54a61

and

.x, y/ D .36u C 3a21 C 12a2, 216v C 108a1u C 108a3/.

In specific numerical examples, we may find “better” values for , , and .

Since in the inverse transformation from .u, v/ to .x, y/ the coefficients are integers,

we see that, if .u, v/ 2 C.ZS /, then .x, y/ 2 E.ZS /.

Notation. From now on, P will denote a typical point of E – the elliptic curve, two

models of which are C and E – such that P C 2 C.ZS /, where S is the set of primes

defined at the beginning of this chapter.

In accordance with Section 11.1 we will write

z.P C / D u.P/

v.P/ , w.P C / D 1

v.P/,

z.P E /

D

x.P/

y.P/

, w.P E /

D

1

y.P/

.




As always, we denote the roots of f.X/ by e1, e2, e3, where e1 is real and e3 < e2 <

e1 if all three roots are real.

With , , , chosen as described immediately after (11.15) – in particular 1 2Z – and as in (11.7), we define, for every prime q

2S ,

bq D min¹1 2 C 2q./, q./, 2q ./, 2

3q. /º if q ¤ 1

b1 D´

2max¹je1j, je3jº if e2, e3 2 R2je1j if e2, e3 62 R

.

Note that, for q ¤ 1 we have bq < 0 because 1 2 Z and 2 ¹1, 2º.

The two lemmas below are technical and will permit us to define a certain effec-

tively computable10 finite subset S of C.ZS /; all points of C.ZS / outside S “behave

properly” so that we can apply the theory of elliptic logarithms to the explicit compu-

tation of C.ZS /. The proofs of the lemmas are somewhat long, but elementary and not

difficult at all. We note that the condition 1 2 C 2q./ for q ¤ 1 is not actually

necessary for the proofs; it is only in Theorem 11.2.6 that we will need it.

Lemma 11.2.1.

(i) If q 2 ¹p1, : : : , ps1º and jx.P/jq > q2q./bq , then

ju.P/jq D jj2q jx.P/jq , jv.P/jq D jj3

q jy.P/jq ,

jz.P E /jq D jjq jz.P C /jq and jz.P E /jq D jx.P/j1=2q .

Moreover, both lq.P C / and lq.P E / exist and jlq.P E /jq D jjqjlq.P C /jq .(ii) If jx.P/j > b1 , then x.P / > 0 , hence x.P / > b1. Moreover, P E 2 E0.R/

and

0 <

Z 1x.P/

dtp f.t/

4p

2 x.P/1=2. (11.16)

Proof. (i) As in Lemma 11.1.2(i), we write u.P/ D U=W 2 and v.P/ D V =W 3 ,

where U , V , W are integers and .U , W /

D1

D.V , W /. By hypothesis, q.x.P // >

2q./ bq , hence q.2x.P // < bq q./ and then

q.u.P // D q.2x.P/ C / D q.2x.P// D 2q./ C q.x.P // < bq < 0.

This implies that q.W / > 0 and q. U V / D 0, so that we can write u.P/ D u1=q2ˇ ,

v.P/ D v1=q3ˇ , where ˇ 1 and q.u1v1/ D 0. Then, q.u.P // D 2ˇ and

q.v.P // D 3ˇ. We now turn to (11.15), where 0 D 1, 0 D 2, 0 D3 and 0 D 3. /.

10 Explicitly computable, in practice, we hope!




We have q.x.P// D q.0 2u.P/ C 0/ and q.0 2u.P // < q.0/. Indeed, the last

inequality is equivalent to 2q./ Cq.u.P // < q./2q./ which is true since,

by hypothesis, q .u.P // < bq . Therefore,

q.x.P// D q.0 2u.P // D 2q./ C q.u.P // D 2q./ 2ˇ

and

ju.P/jq D q2ˇ D q2q./ qq.x.P// D jj2q jx.P/jq .

Analogously, q.y.P // D q.0 3v.P/ C 0u.P/ C 0/ and q.0 3v.P// is strictly

less than both q.0u.P // and q.0/, because of the relation q.u.P // < bq . For ex-

ample, q.0 3v.P// < q.0u.P // is equivalent to 3q./ C q.v.P // < q ./ 3q./ C q.u.P //, hence equivalent to ˇ > q./, since q.u.P // D 2ˇ

and q.v.P //

D 3ˇ. We see that the last inequality is true on replacing ˇ by

q.u.P //=2.We thus conclude that

q.y.P // D q.0 3v.P// D 3q./ C q.v.P // D 3q./ 3ˇ

and

jv.P/jq D q3ˇ D q3q./ qq.y.P// D jj3q jy.P/jq ,

so that

jz.P C /

jq

D

ju.P/jq

jv.P/jq D

jj2q jx.P/jq

jj3

q jy.P/jq D j

j1q

jz.P E /

jq .

Finally, from ju.P/jq D jj2q jx.P/jq we obtain 2q./ C q.x.P// D

q.u.P// < bq ; hence, q .x.P// < bq 2q./ 0. Therefore x.P / D x0=q2t ,

y.P/ D y0=q3t , where t 1 and q.x0y0/ D 0. Consequently, jx.P/jq D q2t and

jz.P E /jq D jx.P/=y.P/jq D jqt x0=y0jq D qt D jx.P/j1=2q .

Next we show that lq.P C / and lq.P E / are meaningful. By hypothesis, jx.P/jq >

q2q./bq , which is equivalent to q.x.P// C 2q./ < bq . By the definition of bq

we have bq q./, therefore, q.u.P // D q.2x.P/ C / D q.2x.P//, be-

cause q.2x.P// D 2q./ C q.x.P// < bq q./. Thus, q.u.P// < bq 1 2 C 2q./ 1 2 (remember that

1

2 Z), hence q.u.P // 2 and we are allowed to define lp.P C / according to Definition 11.1.3. Also, we saw

a few lines above that q.x.P// < bq 2q./ and, by the definition of bq , the

right-hand side is 2 C 1; hence, q.x.P// 2 and, once again, Definition

11.1.3 allows us to define lq.P E / as well. Now, using Proposition 11.1.4, and the re-

lation jz.P E /jq D jjqjz.P C /jq that we have already proved, we have jlq.P E /jq Djz.P E /jq D jjqjz.P C /jq D jjqjlq.P C /jq, as claimed.

(ii) For simplicity in the notation, let us put x.P/ D x. Note that, if e2, e3 62 R, then

x e1; if e2, e3 2 R, then either x e1, or e3 x e2. By hypothesis, jxj > b1.

First we show that x > 0. In the opposite case we would have x < b1.




If e2, e3 62 R, this means that x < 2je1j e1, which is impossible.

If e2, e3 2 R, the inequality x < b1 means that x < 2max¹je1j, je3jº and we

distinguish two cases. If je1j je3j, then x < 2je1j 2je3j, hence je3j e3 x <

2je

1j 2je

3j, which is impossible. If

je

1j <

je

3j, then x <

2je

3j, hence

je3j e3 x < 2je3j, again impossible.

Now we know that x > b1, in view of our discussion above. We will prove that

f.x/ > 1

8x3.

Let first e2, e3 2 R. From e3 < e2 < e1 we see that je2j max¹je1j, je3jº and now,

from x > b1, we obtain x > 2jei j 2ei for i D 1, 2, 3. It follows that x ei > 1

2x

(i D 1, 2, 3) and f .x/ D .x e1/.x e2/.x e3/ > 1

8x3.

Next, consider the case e2, e3 62 R. As before, we have x e1 > 1

2x and we will

show that .x e2/.x e3/ > 1

4x2. For this purpose we put e2 D t C wi with t , w 2 R

and w ¤

0, so that e3

Dt

wi and e1

D 2t . Then .x

e2/.x

e3/

D.x

t /2

Cw2

and it suffices to show that .x t /2 14

x2, which is implied by x t 12

x. The last

inequality is true because x > 2je3j > 2jt j 2t which we write as x t > 1

2x.

From x.P / D x > b1 > 2je1j je1j, we conclude in particular that P 2 E0.R/.

Note also that the real function t 7! f.t/ is strictly increasing, so that f .t/ > 1

8t 3 for

t x.P/. Therefore, for any X > x.P /, we have

0 <

Z X

x.P/

dt

p f.t/

Z X

x.P/23=2t3=2dt D 4

p 2 .x.P/1=2 X 1=2/.

Letting X ! C1, we obtain (11.16).

Lemma 11.2.2. For q 2 S we define

P q D´

¹P C 2 C.ZS / : maxp2S jx.P/jp D jx.P/jq q2q./bq º if q ¤ 1¹P C 2 C.ZS / : maxp2S jx.P/jp D jx.P/j1 b1º if q D 1 .

Then, each set P q is finite and, in principle, can be explicitly calculated.

Proof. Since the coordinates of P C are S -integers and the coefficients 0, 0, 0 and 0

in (11.15) are integers, it follows that the coordinates of P E are also S -integers. Then,

let us write x.P/ D x0=.p˛11 p˛s1

s1 /, where, x0 2 Z and, for every i D 1, : : : , s1,

it is true that ˛i 0 and the prime pi does divide x0 except, possibly, if ˛i D 0.

(i) First, let q ¤ 1, so that q D pi0 2 ¹p1, : : : , ps1º.

We put ˛ D ˛i0. Applying the first statement of Lemma 11.2.3 below, with x.P /

in place of x, we conclude that jx.P/jq 1 and then, necessarily, q.x.P // D ˛,

hence jx.P/jq D q˛. In view of the hypothesis then, it follows that ˛ 2q./bq . If

the right-hand side is < 0, then P q D ;; therefore, let us suppose that 2q./ bq 0. Then, for every i D 1, : : : , s 1, we have, jx.P/jpi jx.P/jq D q˛, by the

maximality of jx.P/jq. On the other hand, jx.P/jpi is D p˛ii , if ˛i 1, and 1, if




˛i D 0. In any case, the relation jx.P/jpi jx.P/jq implies that

˛i max

°0, .2q ./ bq / log q

log pi

± (i D 1, : : : , s 1).

Thus, all exponents ˛i (i D 1, : : : , s 1) are bounded by explicit bounds. Finally,jx.P/j jx.P/jq D q˛, hence jx0j q˛ Qs1iD1 p

˛ii , which provides us with an ex-

plicit upper bound for x0. Existence of explicit upper bounds for the numerator and the

denominator of jx.P/j means that we have an explicit upper bound for the Weil height

of the point P E . Searching for all rational points of an elliptic curve with bounded

Weil height is an effectively solvable problem, provided that the bound of the height

is not very large; a bound for the Weil height of around 16 would require about an

hour of computational time, using the routine ratpoints of M. Stoll. One can also

use the routine Points of MAGMA. From the bounded set of points P E 2 E.ZS / that

we compute this way, we easily recover the set P q by means of (11.14).

(ii) Next, let q D 1.

Let P C 2 P 1. Now, for every prime pi (i D 1, : : : , s 1), we have jx.P/jpi jx.P/j b1. Since jx.P/jpi is equal to p

˛ii if ˛i > 0 and 1 if ˛i D 0, it follows

that, anyway, ˛i log b1= log pi . Thus, the numerator and denominator of x.P/ are

bounded by explicit bounds hence, as already noted in the case q ¤ 1, finding all

possibilities for P E is an effectively solvable problem. As noted in the case q ¤ 1,

computing P 1 is a trivial matter then.

Notation. From now on and until the end of the chapter, we will consider the generic

point P C

D .u.P /, v.P// 2 C.ZS / X P , where

P D[

p2S

P p, (11.17)

and the sets P p are as in Lemma 11.2.2.

For the corresponding point P E D .x.P/, y.P// 2 E.ZS / we follow the notation,

assumptions etc. of Chapter 4 and write11

P E D m1P E1 C mr P Er C T E , M D max1ir

jmi j. (11.18)

Our main purpose is to obtain an upper bound for M . We will need to investigateWeil heights of points with S -integer coordinates. First, a few general remarks have

their place.

If x is a non-zero rational number and p is any (finite) prime, p divides the numer-

ator of x if jxjp < 1, while p divides the denominator of x if jxjp > 1. Therefore, for

any S -integer x, the formula (2.20) of the absolute logarithmic height of x becomes

h.x/ DXp2S

log max¹1, jxjpº for x 2 ZS .

11 Cf. relation (4.2); all relations (4.5) through (4.8) are valid.




Also, the product formula (2.10) specialised to a non-zero rational x , takes the formQp jxjp D 1, where we understand that p runs through all (finite) primes and 1. In

the particular case that x is an S -integer we have

1 D Yp2S

jxjp Yp62S

jxjp Yp2S

jxjp ,

where the right-most inequality holds because, if p 62 S , then jxjp 1, with strict

inequality if p divides the numerator of x. The inequality above, in particular, proves

the following simple but very useful lemma:

Lemma 11.2.3. Let x be any S -integer and suppose that for some q 2 S we havejxjq D maxp2S jxjp. Then jxjq 1 and h.x/ s log jxjq .

Remarks.

(1) The isogeny of C and E and relation (11.18) imply that

P C D m1P C 1 C mr P C

r C T C . (11.19)

(2) Combining Proposition 11.1.7 and Lemma 11.1.6 we see that for every q 2 S X¹1º we can choose a positive integer tq satisfying q.u.tqP i // < bq for every

i D 1, : : : , r . Then, q.x.P// D q.u.P // 2q./ (cf. displayed relation at the

beginning of the proof of Lemma 11.2.1(i)). Therefore, q .x.P // < bq 2q ./,

which is equivalent to jx.P/jq > q2q./bq . Consequently, by Lemma 11.2.1(i),

we conclude that, for every i D 1, : : : , r , the p-adic elliptic logarithms lq.P C i /

and lq.P E

i / are meaningful and satisfy jlq.P E

/jq D jjqjlq .P C

/jq .We will need to be more selective about our choice of tq because of our application

of Theorem 11.2.5, which gives an explicit lower bound for a linear form in p-adic

elliptic logarithms. In that theorem the number of elliptic logarithms involved in the

linear form has an essential effect on the size of the bound; the smaller their number,

the better the lower bound. Therefore, we will take tq with the extra property of being

a multiple of the order of the Q-torsion subgroup, so that tqT is the zero point for

every torsion point T . We have to impose one further condition on tq : The original

paper [19] of N. Hirata-Kohno, of which Theorem 11.2.5 is an immediate corollary,

requires, for convergence reasons, that jlq.tq P E

i /jq < qq

, where q is defined inTheorem 11.2.5. Using Lemma 11.2.1(i), Proposition 11.1.4, and Lemma 11.1.2(i),

we see that the last condition can be written as

qq > jjqjlq.tqP C i /jq D jjqjz.tqP C

i /jq D jjqju.tqP C i /j1=2

q ,

and this is equivalent to q.tqP C i / < 2q./ 2q . Summing up we have the follow-

ing instructions for choosing tq :

Choice of tq. If q ¤ 1 we choose the positive integer tq such that (i) q.tqP C i / <

min¹bq , 2q./ 2qº for every i D 1, : : : , r , and (ii) tq T C D O for every torsion

point T C

2 C.Q/.




Theorem 11.2.4. Let P C 2 C.ZS / XP , where P is defined in (11.17). Let q 2 S besuch that jx.P/jq D maxp2S jx.P/jp and choose tq as indicated above. Define

Lq.P E

/

def

D ´tq

lq.P E / if q

¤ 1l.P / if q D 1 and Lq.P C

/

def

D tq lq.P C

/ if q ¤ 1 ,

where, in case q D 1 , the map l is that of Theorem 3.5.2 , and the integer m0 ,satisfying jm0j 1

2rM , is chosen according to the discussion in Chapter 4 between

relations (4.4) and (4.8).Then, the above definitions are meaningful, i.e. the q-adic elliptic logarithms

lq.P C / and lq.P E / are meaningful, and

Lq.P C / D m1lq.tq P C 1 / C C mr lq.tq P C

r / if q ¤ 1

Lq .P E / D ´m1lq.tqP E

1 / C C mr lq.tqP E

r / if q ¤ 1m1l.P 1/ C C mr l.P r / C l.T / C m0 !1 if q D 1.

Moreover,jLq.P E /jq c19 exp.c20M 2/, (11.20)

where

c19 D´

jtqjq e=s if q ¤ 12p

2e=s if q D 1 , c20 D =s

with , defined in Propositions 2.6.2 and 2.6.3 , respectively.

In the case that q ¤ 1 we also havejLq.P C /jq D j1jq jLq.P E /jq . (11.21)

Proof. Let P C 2 C.ZS / X P . We distinguish two cases.

(i) Let q ¤ 1. Since P C 62 P q, we have jx.P/jq > q2q./bq , which is equiv-

alent to q.x.P // C 2q. / < bq . By the definition of bq we have bq q./,

therefore, q .u.P // D q.2x.P/ C / D q.2x.P//, because q.2x.P// D2q./ C q.x.P// < bq q./. Thus, q.u.P// < bq 1 2 C 2q./ 1 2 (remember that 1 2 Z), hence q.u.P // 2 and we are allowed

to define lq.P

C

/ according to Definition 11.1.3. Also, we saw a few lines abovethat q.x.P// < bq 2q./ and, by the definition of bq , the right-hand side is

2 C 1; hence, q.x.P// 2 and, once again, Definition 11.1.3 allows us to

define lq.P E /, as well.

The relation (11.18) implies tqP E D m1.tqP E1 /C mr .tqP Er /, hence, by Propo-

sition 11.1.5 (in particular, relation (11.12))

Lq.P E / D tq lq.P E / D lq.tqP E / D m1lq.tqP E1 / C mr lq.tqP Er /.

Analogously, in view of (11.19), we obtain

Lq.P C

/ D tq lq.P C

/ D lq.tqP C

/ D m1lq.tqP C

1 / C mr lq.tqP C

r /.




Then, jLq.P E /jq D jtqjq jlq.P E /jq D jtqjq jz.P E /jq , by Proposition 11.1.4 and,

analogously, jLq.P C /jq D jtqjq jlq.P C /jq D jtqjq jz.P C /jq . By Lemma 11.2.1,

jz.P C /jq D j1jq jz.P E /jq , which proves the relation (11.21).

Now we estimate j

Lq

.P E /jq

. We saw above that q

.u.P// < bq

, therefore, by

Lemma 11.2.1(i), jz.P E /jq D jx.P/j1=2q . Thus, finally,

jLq.P E /jq D jtqjq jx.P/j1=2q jtq jq eh.x.P //=2s , (11.22)

where the right-most inequality is implied by Lemma 11.2.3. Using Propositions 2.6.3

and 2.6.2 we obtain successively,

1

2sh.x.P // 1

s. Oh.P // 1

s. M 2/,

therefore by (11.22),

jLq.P E /jq jtqjqe=s e.=s/M 2 ,

as claimed.

(ii) Let q D 1. The hypothesis P C 62 P 1 implies jx.P/j > b1 so that, by relation

(3.32) of Chapter 3 and Lemma 11.2.1(ii), we have

jL1.P E /j1 D jl.P /j D 1

2

Z C1x.P/

dtp f.t/

2p

2 x.P/1=2.

By Lemma 11.2.3 we have h.x.P //

s logjx.P/

j Ds log x.P/. Then,

log x.P/1=2 1

2sh.x.P //

1

s. Oh.P // (by relation (5.1) of Proposition 2.6.3)

1

s. M 2/ (by Proposition 2.6.2).

If we combine this with the above displayed upper bound of jl.P /j we obtain jl.P /j <

2p

2e=s e.=s/M 2 , as claimed.

Finally, by definition, L1.P E / D l.P / and, according to Chapter 4, relations (4.4)

through (4.8),

l.P / D m1l.P 1/ C C mr l.P r / C l.T / C m0 !1,

where m0 is an appropriate integer satisfying jm0j 1

2rM C 1.

When we solve an elliptic Diophantine equation over Z, Theorem 9.1.3 was the ba-

sic tool for obtaining a first, very large, upper bound for M . Now that we are interested

in solutions over ZS , the analogous tool is Theorem 11.2.6, below, which is heavily

based on the following very important theorem.




Theorem 11.2.5 (N. Hirata-Kohno). For 1 ¤ q 2 S consider the linear formLq.P E / of Theorem 11.2.4 and define the following:

h

Dlog max

¹jA

j,

jB

jº,

q D´3 if q D 2

1=.q 1/ if q > 2.

ai D max¹1, h, t 2q Oh.P i /º .i D 1, : : : , r/,

H D .qq max1ir

jlq.tqP Ei /jq /1, d D max¹1, 1= log H º

and g D max¹1, h, log a1, : : : ,log ar , log d º.

If M eg , then jLq.P E /jq exp.c23 log M /, (11.23)

where

c23 D 24r 2C3r .r C 1/2r 2C9rC4d 2rC2grC1.log H /

rYiD1

ai .

Proof. This theorem is an almost straightforward specialisation to our case of N. Hira-

ta-Kohno’s result [19] which gives an effective lower bound for the p-adic absolute

value of non-vanishing linear forms in p-adic elliptic logarithms. In the notation of that

result, ui D lq.tqP E

i / and expp.ui / D .x.P i / : y.P i / : 1/ (projectively). The non-vanishing of our linear form follows from its definition: Lq.P E / D tq lq.P E / ¤ 0

by Proposition 11.1.4.

Theorem 11.2.6. With the assumptions and notations of Theorem 11.2.4 , the follow-ing holds:

Let q D 1. Referring to the model E in (11.13) , follow the instructions in “Prepara-

tory to Theorem 9.1.2 ”, page 103 to the linear form L1.P E / in order to computehE and H 0, H 1, : : : H r , having in mind that, in the notation of the instructions,

L.P/ D L1.P E

/ and in (9.2) , k D r , d D 1 , ni D mi for i D 0, 1, : : : , r ,`i D l.P i / for i D 1, : : : , r . Also, let and be as in Propositions 2.6.2 , and 2.6.3 ,respectively. Then, either

˛M C ˇ c12def D max¹exp.e hE /, jr0j, exp.H i /, i D 0, : : : , rº,

where 12

˛ D r0 max¹1, 1

2rº, ˇ D 3

2r0,

12 We remind that r0 is the lcm of the orders of non-zero points of Etors.Q/.




or

M 2 < 3

2s log2 C C c13s.log.˛M C ˇ/ C 1/.log log.˛M C ˇ/ C 1 C hE /rC2,

(11.24)

wherec13 D 2.9 106rC1242.rC1/2

.r C 2/2r 2C13rC23.3

rYiD0

H i .

Let q ¤ 1. If M eg , then

M 2 s log.jtqjq/ C C s c23 log M , (11.25)

where , are as in the case q D 1 , tq is chosen as described just before theannouncement of Theorem 11.2.4 , s is the cardinality of S , and c23 is defined inTheorem 11.2.5.

Proof. Let q D 1. Firstly, we claim that, if ˛M C ˇ > c12, then

L1.P E / > expc13.log.˛M C ˇ/ C 1/.log log.˛M C ˇ/ C 1 C hE /rC2

,

(11.26)

where c12, c13 are those in the statement of the theorem. Our claim follows easily from

Theorem 9.1.2 applied to the linear form L.P/ D L1.P E /. Indeed, as already noted

in the announcement of the present theorem, in the notation of both Theorem 9.1.2

and relation (9.2), k D r , d D 1, ni D mi for i D 0,1, : : : , r , and `i D l.P i / for

i D 1, : : : , r ; also, for the N appearing in Theorem 9.1.2, we have N ˛M C ˇ,according to the relation (9.6). For the explicit determination of ˛ and ˇ we refer to

the first “bullet” on page 99 and we ascertain that, indeed, ˛ and ˇ are those in the

statement of the present theorem.

Now, (11.26) results from a straightforward application of Theorem 9.1.2, specialised

to the present situation. It suffices to check (trivially) that c14 D 1, c15 D 1 C hE and

c12, c13 are those in the statement of the present theorem.

On the other hand, by Theorem 11.2.4, L1.P E / c19 exp.c20M 2/ with c19 D23=2e=s and c20 D =s. The combination of the last inequality with (11.26) immedi-

ately proves (11.24) in the case that ˛M

Cˇ > c12.

Next, let q ¤ 1. Then, the relation (11.25) follows from a straightforward combi-

nation of the bounds (11.20) and (11.23).

Remark. Obviously, both relations (11.24) and (11.25) imply an upper bound for M .

11.3 The p-adic reduction

In this section, we assume that the assumptions of Theorem 11.2.6 hold for a certain

finite prime q 2 S , which we will keep fixed until the end of the section. The reduction

method that we expose is essentially due to B. M. M. de Weger [72].



Section 11.3 The p-adic reduction 155

For reasons explained at the very beginning of Section 11.2, we choose to work

with the linear form Lq.P C /, rather than with Lq.P E /. On the other hand, the elliptic

curve in N. Hirata-Kohno’s Theorem [19] is defined by a short Weierstrass model E

with integer coefficients. Therefore we will have to pass from the model C to the

model E and vice versa; in our transition from one model to the other, the basic tool

is Lemma 11.2.1.

Simplifying the notation, we rewrite the linear form Lq.P C / as follows:

L def D Lq.P C / D m1 1 C C mk r , i D lq .tqP C

i / .i D 1, : : : , r/.

By (11.20) and (11.21) it follows immediately that

q.L/

c21

Cc22M 2, c21

D q./

log c19

log q

, c22

D c20

log q

. (11.27)

Forgetting now the specific meaning of the i ’s, except that they all belong to M ,

and the specific values of c21 and c22, except that c22 > 0, we will solve the following

computational problem:

Problem. Let 0, 1, : : : , r 2 M and consider the linear form

L D m1 1 C C mr r ,

where m1, : : : , mr 2 Z. Let M D max1ir jmi j and assume the following:

(i) M M q for some specific large bound M q .

(ii) q.L/ c21 C c22M 2, for some explicit parameters c21 and c22 > 0.

(iii) (Without loss of generality) q. r / D min1ir q. i /.

Under the assumptions (i)–(iii), find an upper bound of M considerably smaller

than M q.

For any 2 Zq and any integer n > 1 we denote by .n/ the n-th rational approxi-

mation of . Thus, .n/

2Z and q. .n/

/ > n. In other words, if

D P1iD0 ai qi ,

where ai is an integer with 0 ai < p for every i 0, then .n/ D PniD0 ai qi .

First we consider the case r 2. We put

i D i

r.i D 1, : : : , r 1/,

r L D ƒ D r1XiD1

mi i C mr , ƒ.n/ D r1XiD1

mi .n/i C mr .

Note that, for i D 1, : : : , r 1 we have i 2 Zq and, consequently, .n/i 2 Z.




For every integer n 1 we consider the lattice .n/ generated by the column vectors

of the matrix

Z.n/ D 0BBBB@1 0 0

.

.. . . ....

.

..

0 1 0

.n/1

.n/r1 qn

1CCCCA .

Proposition 11.3.1.

q .ƒ/ n ,

0BB@

m1

.

.

.

mr

1CCA

2 .n/.

Proof. Suppose first that q.ƒ/ n. From q. i .n/i / > n for every i D 1, : : : , r ,

we see that q.ƒ ƒ.n// n, hence q.ƒ.n// n. Since ƒ.n/ 2 Z, it follows that

ƒ.n/ D Pr1iD1 mi

.n/i C mr D xr qn for some xr 2 Z. Consequently,

Z.n/

0BBBB@

m1

.

.

.

mr1

xr

1CCCCA

D

0BBBB@

m1

.

.

.

mr1

mr

1CCCCA

,

which shows that the right-hand side belongs to .n/.

Conversely, if 0BB@m1

.

.

.

mr

1CCA 2 .n/,

then, there exist x1, : : : , xr1, xr 2 Z such that

Z.n/

0BBBB@x1

.

.

.

xr1

xr

1CCCCA D

0BBBB@m1

.

.

.

mr1

mr

1CCCCA .

It follows that mi D xi for i D 1, : : : , r 1 and mr DPr1

iD1 xi .n/i C xr qn. Conse-

quently,

ƒ.n/

D

r1

XiD1

xi .n/i

C.

r1

XiD1

xi .n/i

Cxr qn/

Dxr qn,



Section 11.3 The p-adic reduction 157

which shows that q.ƒ.n// n. Since q.ƒ ƒ.n// n, this proves that

q.ƒ/ n.

Theorem 11.3.2. With the assumptions of the Problem at the beginning of this sec-tion, the following is true. Let b1 be the first vector of an LLL-reduced basis of thelattice Z.n/. If jb1j > 2r=2

p r M q then

M 2 n 1 C q. r / c21

c22

.

Proof. Suppose that M 2 > .n 1 C q. r / c21/=c22. Then,

q.ƒ/ D q.L/ q. r / c21 C c22M 2 q. r / > n 1,

hence q.ƒ/ n. Then, by Proposition 11.3.1,0BB@m1

.

.

.

mr

1CCA 2 .n/.

But the norm of every point of .n/ is at least 2r=2jb1j, by [27, Proposition (1.11)];

therefore

p r M q m21 C C m2r 2r=2

jb1j > 2r=2

.2r=2p r M q/ D p rM q ,

which contradicts the assumption M M q .

Now we consider the case r D 1. In this case we have the following situation: L Dm1 1, M D jm1j and jLjq < qc21c22jm1j

2

. Since m1 2 Z, we have jm1j1 jm1jq ,

therefore,

jm1j1 jm1jq D jLjqj 1j1q < qq.1/c21c22jm1j

2

,

or, equivalently,

log jm1j > log q .c21 q. 1/ C c22jm1j2/.

This inequality, obviously, implies an upper bound for jm1j. Since we expect that the

positive parameter c22 is not extremely small, the upper bound that we obtain is so

small that no further reduction of it is really necessary.

Remark. How large should we choose n when we construct the matrix Z.n/, in order

that the condition jb1j > 2r=2p

r M q of Theorem 11.3.2 be fulfilled? Heuristically, we

argue as follows. Since the r vectors of the LLL-reduced basis have “almost equal”

lengths and are “almost orthogonal” to each other, the volume of the parallelepiped

formed by them is approximately jb1jr

. On the other hand, this volume is equal to




the volume formed by the initial basis of the lattice which, in turn is equal to the

determinant of Z.n/. But this is qn, therefore we expect that jb1j is of the size of qn=r .

Since we want that jb1j satisfies jb1j > 2r=2p

r M q , we must choose n so that

n &r log.2r=2p rM q

log q

'. (11.28)

11.4 Example

We will compute all S -integral points on

C : v2 C v D u3 7u C 6, (11.29)

for S D ¹2,3,5,7, 1º. Note that C is a minimal model. The rank over Q is 3 and the

torsion subgroup is trivial. A Mordell–Weil basis is

P C 1 D .1, 0/, P C

2 D .2, 0/, P C 3 D .0, 3/

and an isomorphic to C short Weierstrass model is

E : y2 D x3 112x C 400,

obtained by means of the transformation

.u, v/ D . 1

4x,

1

8y 1

2/.

Thus, in the notation of (11.14),

D 1

2, D 0, D 0, D 1

2

and

P E1 D .4, 4/, P E2 D .8, 4/, P E3 D .0, 20/.

In Proposition 2.6.3, in place of the curve D we take our present curve C and

we compute 2.87085138. Also, applying Proposition 2.6.2, we obtain 0.48599751, therefore c20 D 3=5. We need and for calculating the parameters

c19 (depending on the prime q 2 S ) of Theorem 11.2.6.

First we compute all points P C 2 P ; cf. Lemma 11.2.2. We compute b1 24.09944 < 24.1 and .b2, b3, b5, b7/ D .5, 1, 1, 1/. Following the steps of the

constructive proof of Lemma 11.2.2, it is easy to see that, if P C 2 P , then the height

of x.P/ (maximum of the numerator and denominator of jx.P/j) is less than 121462.

Using the routine Points of MAGMA, we explicitly compute 274 points only 119 of

which have S -integral coordinates; thus P is a subset of an explicit set P consisting

of 119 S -integral points.



Section 11.4 Example 159

Now we assume that P C 62 P so that we can apply Theorem 11.2.6. We de-

note by q that prime in S for which jx.P/jq D maxp2S jx.P/jp and we distin-

guish cases according to q D 2,3,5,7, 1. In the case that q ¤ 1, the routine

pAdicEllipticLogarithm of MAGMA is used for computing the lq

-values.

q D 2. The instructions for the choice of tq on page 150 lead us to t2 D 80, so that

L2.P C / D m1 l2.80P C 1 / C m2 l2.80P C

2 / C m3 l2.80P C 3 /.

Theorem 11.2.6, in particular relation (11.25), provides us with the upper bound

M M 2 D 1029.

Now we reduce this large upper bound M 2. We need first to calculate the parameters

c21 and c22 according to (11.27); we actually need an approximation (not a high one)

from below:

c21 3.17164739, c22 0.865617.

According to (11.28) we must work with a 2-adic precision at least O.2316/. We pre-fer to work with a somewhat larger precision, choosing n D 320 in order to construct

the matrix Z.n/. A first low precision computation shows that 2.l2.80P C i // D 6 for

i D 1, 2 and 2.l2.80P C 3 // D 5. Therefore we put i D l2.80P C

i / for i D 1,2,3,

so that i D i = 3 for i D 1, 2. Using MAGMA we compute

.320/1 D 224782 369609„ ƒ‚ …

96 digits

2, .320/2 D 163742 694465„ ƒ‚ …

96 digits

2.

In the notation of Section 11.3 we have

Z.320/ D0@ 1 0 0

0 1 0

.320/1

.320/2 2320

1Aand we compute an LLL-reduced basis for the lattice .320/ generated by the vector-

columns of Z.320/.13 The first vector of the reduced basis is

b1 D0

@28345891354399044132619085347326

87234034300570141766432578346775

7183933776826462841800336346962

1

A.

Its length is approximately 9.2004 1031, “slightly larger” than 23=2p 3 M 2

4.89898 1031, hence b1 satisfies the condition of Theorem 11.3.2. Consequently,

according to the theorem, M 2 .321 C 2 c21/=c22 < 370, therefore M 19.

Repeating the reduction process with M 2 and 2-adic precision O.222/, we reduce

the upper bound for M further, finding M 2 D 4.

q D 3. The instructions for the choice of tq on page 150 lead us to t3 D 7, so that

L3.P C / D

P3iD1 mi l3.7P C

i /.

13 See the “bullets” on page 125.




Theorem 11.2.6, in particular relation (11.25), provides us with an upper bound M M 3 D 1028.

For the reduction of M 3 we calculate c21 0.52263230 and c22 0.546143.

We compute 3.l3.7P C i // D 1 for i D 1, 3 and 3.l3.7P C 2 // D 2. Therefore weput i D l3.7P i / for i D 1, 2, 3, so that i D i = 3 for i D 1, 2. Then we work

as in the case q D 2. With 3-adic precision O.3189/, we reduce the upper bound for

M , finding M M 3 D 18. One further reduction step with precision O.315/ leads

to M M 3 D 5.

q D 5. Now we choose t5 D 10, so that L5.P C / D P3iD1 mi l5.10P C

i /.

The relation (11.25) implies upper bound M M 5 D 1028.


We compute 5.l5.10P C i // D 2 for i D 1, 3 and 5.l5.10P C 2 // D 3. Therefore weput i D l5.10P C

i / for i D 1, 2, 3, so that i D i = 3 for i D 1, 2. Then we work

as in the case q D 2. With 5-adic precision O.5128/, we reduce the upper bound

for M , finding M M 5 D 18. One further reduction step with precision O.511/

results to M M 5 D 5.

q D 7. Now we choose t7 D 6, so that L7.P C / D P3iD1 mi l7.6P C

i /.

The relation (11.25) implies the upper bound M M 7 D 1027.


We compute 7.l7.6P C i // D 2 for i D 1, 2 and 7.l7.6P C 3 // D 3. Therefore we put 1 D l7.6P C

3 /, 2 D l7.6P C 2 / and 3 D l7.6P C

1 /, so that i D i = 3 for i D 1, 2.

Working as before with 7-adic precision O.7104/, we find in the first reduction step

M M 7 D 18 and one further reduction step with precision O.79/ results to

M M 7 D 5.

q D 1. According to Theorem 11.2.6, we work with the linear form

L1.P E / D m1l.P 1/ C m2l.P 2/ C m3l.P 3/ C m0!1,

where, as always, !1 is the least positive real period for the corresponding Weier-

strass function. We calculate !1 1.03792199 and, in the notation of Lemma9.1.1, we take $ 1 0.74027413, $ 2 D !1, so that Q 1.40207788i . Fur-

ther, we compute (note that, in our example, r0 D 1, hence ˛ D 3=2 D ˇ) c12 3.99858611020, c13 6.651560410110, c19 5.02231532 and c20 0.09719950.

According to Theorem 11.2.6, if M > c12 then it satisfies the inequality (11.24)

and an easy computation shows that this is not possible if M 2.6 1060, hence

we obtain the upper bound M M 1 D 2.6 1060. We reduce M 1 using an LLL-

reduction process completely analogous to that in the examples of Section 10.2,

therefore we do not think it worthwhile to give any numerical details. We only say




that, in the first reduction step, the upper bound falls to M 1 D 51 and the next step

reduces it further to M 1 D 13. As expected, all 119 points in P (see the beginning

of this section) are rediscovered by our search, while 9 points from those found do

not belong to P .

Summing up, we conclude that M max¹M 2, M 3, M 5, M 7, M 1º D 13 and we

check all points P C D m1P C 1 C m2P C

2 C m3P C 3 with M D max1i3 jmi jº for

S -integrality. The results of our search is summarised in Table 11.1.

Table 11.1. All S -integral points P C D m1P C 1 C m2P C

2 C m3P C 3 on the curve (11.29),

S D ¹2,3,5,7, 1º.

m1, m2, m3 P C

P C

4, 2, 0 1541761

153664 , 1882859327

60236288 1541761

153664 ,

1822623039

60236288

4,0,1 11948

3969 , 176534

250047

11948

3969 , 73513

250047

3, 2, 0

6169

6561, 641312

531441

6169

6561,

109871

531441

3, 2, 2

2759

1024,

60819

32768

2759

1024, 93587

32768

3, 1, 1

391

25 ,

7564

125

391

25 , 7689

125

3, 1, 3

19849

8100 , 1787743

729000 19849

8100 ,

1058743

729000 3, 0, 1 4537

36 , 305425

216 4537

36 , 305641

216 3,1,0

47

256,

9191

4096

47

256, 132871

4096

3,1,1 .816, 23309/ .816, 23310/

2, 3, 2

207331217

4096 ,

2985362173625

262144

207331217

4096 , 2985362435769

262144

2, 3, 2

6142

81 ,

480700

729

6142

81 , 481429

729

2, 2, 0 .406, 8181/ .406, 8180/

2,

2, 1

13

49

,

1147

343 13

49

, 804

3432, 2, 4 848961

3136 ,

782099809

175616

848961

3136 , 782275425

175616

2, 1, 0

33

16,

17

64

33

16, 81

64

2, 1, 1

4

9,

35

27

4

9, 62

27

2, 1, 2 .93, 897/ .93, 896/

continued on next page




continued from previous page

m1, m2, m3 P C P C

2, 0, 1 43

49 ,

132

343 43

49 , 475

3432,0,0 .14, 52/ .14, 51/

2,0,1 7

4, 33

8

7

4,

25

8

2,0,2 . 106

49 ,

209

343/

106

49 , 552

343

2,1, 1

69

25, 329

125

69

25,

204

125

2,1,0

25

9 ,

64

27

25

9 , 91

27

2,1,1 24

25,

18

125 24

25,

143

1252,1,2 625

64 , 15351

512

625

64 ,

14839

512

2,1,3

33304

15625, 7596119

1953125

33304

15625,

5642994

1953125

2,2,0

221

49 , 2967

343

221

49 ,

2624

343

2, 3,1

8159

2916,

233461

157464

8159

2916, 390925

157464

1, 3, 0

1343

576 , 50399

13824

1343

576 ,

36575

13824

1, 2, 0

26

25, 24

125 26

25, 101

1251, 2, 1 254 , 119

8 254 , 111

8 1, 2, 2

66

25, 377

125

66

25,

252

125

1, 1, 1

31

9 , 143

27

31

9 ,

116

27

1, 1, 0 .3, 0/ .3, 1/

1, 1, 1 .4, 6/ .4, 7/

1, 1, 2

17

16, 25

64

17

16, 39

64

1, 0, 2 151

64 ,

1333

512 151

64 , 1845

512 1,0, 1 .8,21/ .8, 22/

1,0,0 .1, 1/ .1, 0/

1,0,1 .3, 4/ .3, 3/

1,0,2 26

9 ,

28

27

26

9 , 55

27

continued on next page




continued from previous page

m1, m2, m3 P C P C

1,1, 2 58

81 , 1288

729 58

81 ,

559

7291,1, 1

9

4, 15

8

9

4,

7

8

1,1,0 .2, 3/ .2, 4/

1,1,1 .11, 35/ .11, 36/

1,2, 1 .52, 374/ .52, 375/

1,2,0

9

16, 133

64

9

16,

69

64

1,2,1 172

81 ,

1079

729 172

81 , 350

7291, 3, 3 338776

225 , 197184149

3375

338776

225 ,

197180774

3375

1, 3, 1

1219

625 , 9828

15625

1219

625 , 5797

15625

1,3,1

13961

100 ,

1648791

1000

13961

100 , 1649791

1000

0, 3, 2

1793

256 ,

68991

4096

1793

256 , 73087

4096

0, 2, 1

68

49,

1079

343

68

49, 1422

343

0, 2, 0 .21, 95/ .21, 96/

0, 2, 1 79 , 44

27 79 , 17

270, 2, 2

114

49 , 720

343

114

49 ,

377

343

0, 1, 2 .342, 6324/ .342, 6325/

0, 1, 1

1

4, 21

8

1

4,

13

8

0, 1, 0 .2, 1/ .2, 0/

0, 1, 1 .1, 3/ .1, 4/

0, 1, 2 .37, 224/ .37, 225/

0,0, 3 1525

2401, 438957

117649

1525

2401,

321308

117649

0, 0, 2

49

25, 32

125

49

25, 93

125

0,0, 1 .0, 2/ .0, 3/



List of symbols

On our compiling of the list we adopted the following rules. 1

Mainly (but not exclusively) symbols introduced in this book are listed. For the

definition or explanation of a symbol, the second column gives reference to either of

the following: Definition (Def.), Fact (Fact), Lemma (Lemma), page (p.), Proposition(Prop.), Relation (Eq.), Section (Sec.), Theorem (Thm.).

Symbols are listed by chapters. In each chapter, only the first occurrence of the sym-

bol, where this is defined, is listed, except if its meaning is specialised subsequently. Asymbol common to at least two chapters is listed in the chapter of its first occurrence,

except if its meaning/context changes in a subsequent chapter.

In the third column the meaning of the symbol very briefly and/or the context in

which the symbol is used.

Symbol Reference Meaning and/or context

Chapter 1 Elliptic Curves and Equations

g D 0 Eq. (1.2) homogenisation of g D 0 U ij , V ij Def. 1.1.1 birational transformation between curves

C i , C j

C , E p. 4 birationally equivalent models usually repre-

senting elliptic curve with E short Weierstrass

model

X , Y , U , V Eqs. (1.3), (1.4) birational transformation between curves

E, C

P C , P E p. 4 ff. representatives of point P on models C , E

.x.P/, y.P// coordinates of point P on model E

.u.P/, v.P// coordinates of point P on model C

a1, a2, a3, a4, a6 Eq. (1.6) coefficients of Weierstrass equation

Etors Thm. 1.2.1 torsion subgroup of E

r rank of elliptic curve

P 1, : : : , P r Mordell–Weil basis

P Eq. (1.7) generic point of elliptic curve

T generic torsion point of elliptic curve

1 We were not able, however, to be absolutely consistent with them.



166 List of symbols


Chapter 2 Heights

p Sec. 2.1 exponent of prime ideal p

p p. 17 exponent of rational prime p

eL=K .P/ Eq. (2.1) ramification index of prime ideal P

f L=K .P/ above Eq. (2.3) degree of prime ideal P

d L=K .P/ Eq. (2.7) local degree of L=K at prime ideal P

j jp Eq. (2.9) p-adic absolute value

NL=K ./ Eqs. (2.4), (2.5) ideal norm with respect of L=K

H K ./ Sec. 2.3 K -height of projective point

H K ,fin./ finite-prime factor of K -height

H K ,1./ infinite-prime factor of K -height

H K ,fin.P / Sec. 2.4 finite-prime height of polynomial P

H./ Def. 2.3.1 absolute height of projective point

h./ absolute logarithmic height

h.P/ Sec. 2.5 naive or Weil height of point P

hD.P / height of point P used by S. David

Oh.P/ Sec. 2.6 canonical height of point P OhD.P / Prop. 2.6.1 canonical height used by S. David

H Eq. (2.39) height pairing matrix

Prop. 2.6.2 least eigenvalue of H, , , , Prop. 2.6.3 difference of naive from canonical height

Chapter 3 Short Weierstrass equation over C

.!1, !2/ Sec. 3.1 fundamental pair of periods for the Weierstrass

-function

g2, g3 parameters of the Weierstrass -function

e1, e2, e3 Eq. (3.6) 4e3i g2ei g3 D 0 (i D 1,2,3)

Fact 3.1.2 discriminant of the Weierstrass -function

below Fact 3.1.2 D !2=!1

A, B Sec. 3.2 coefficients of Weierstrass Eq. over Cg2, g3 parameters of the Weierstrass -function g2 D

4A, g3 D 4B

Fact 3.2.1 group isomorphism from a fundamental paral-

lelogram to E.C/

Sec. 3.3 isomorphism : E.C/ ! C=ƒ; inverse of above



List of symbols 167


Def. 3.3.2 elliptic logarithm; as above with values on a

specified fundamental parallelogram

!1 Sec. 3.4 least positive real period of when g2, g3 2 R!2 period 2 iRwith least positive imaginary part,

when g2, g3 2 R.!1, !2/ p. 37 fundamental pair of periods when g2, g3 2 R,

> 0

.1, 2/ fundamental pair of periods when g2, g3 2 R,

< 0

M.a, b/ Sec. 3.4.4 arithmetic-geometric mean of a, b

Sec. 3.5 of Sec. 3.3 restricted to E0.R/

"P Thm. 3.5.1 "P D 1 or 1 according to whether y.P/ is> 0 or 0

l0 Thm. 3.5.2 isomorphism E0.R/ ! R=Z!1

l Eq. (3.31) two-to-one epimorphism E.R/ ! R=Z!1

Chapter 4 General exposition of Ellog

A, B; E Eq. (4.1) elliptic curve E : y 2 D x3CAxCB A, B 2 Q

m1, : : : , mr Eq. (4.2) coefficients of generic point P with respect to

a Mordell–Weil basis P 1, : : : , P r

M below Eq. (4.2) M D max 1ir jmi j

m0 Eq. (4.4) “adjusting summand” m0!1

M 0 Eq. (4.7) M 0 max 0ir jmi jr0 p. 54 lcm of the orders of torsion points

e1, e2, e3 p. 55 roots of short Weierstrass equation

Qi Eq. (3.7) Qi D .ei , 0/ (i D 1,2,3)

s=t p. 56 order of the generic torsion point

L.P/ linear form for the generic point P “covering”

a generic solution to the Diophantine problem

Chapter 5 Ellog applied to Weierstrass equation

C Eq. (5.1) elliptic curve related to the problem of thischapter

, , , ı Prop. 5.1 difference of naive from canonical height

E Thm. 5.2 short Weierstrass model birationally equiva-

lent to C

L.P/ linear form for the point P “covering” a

generic solution to the problem of this chapter



168 List of symbols


Chapter 6 Ellog applied to quartic elliptic equation

C Eq. (6.1) elliptic curve related to the problem of this

chapter

Q, a, b, c, d , e Eq. (6.2) quartic elliptic equation

A, B Eq. (6.3) coefficients of corresponding short Weierstrass

model

E Eq. (4.1) E : y2 D x3 C Ax C B

X ,Y Eq. (6.4) birational transformation from model C to

model E

U ,V Eqs. (6.5), (6.6) birational transformation from model E to

model C

Lemma 6.1 technical parameterx0 Eq. (6.8) 2e

p a C c=3

u0 Lemma 6.3 technical parameter

u Prop. 6.4

x.u/ Eq. (6.9) auxiliary function; see Eq. (6.10)

P 0 Eq. (6.11) auxiliary point 2 E.Q.p

a//

u, c7 Thm. 6.8 technical parameters

L.P/ linear form for the point P “covering” a

generic solution to the problem of this chapter

Q, , x0, u0 Table 6.1 parameters as above in the case u < 0

x.u/, u, P 0, u

c7, L.P/

Chapter 7 Ellog applied to simultaneous Pell equations

A1, D1; A2, D2 Eq. (7.1) coefficients of the two Pell equations

.U 0, V 0, W 0/ below Eq. (7.1) known rational solution

Q Eq. (7.10) quartic polynomial related to the system of Pell

equations

Eq. (7.12) technical parameter

a, b, c, d , e Eq. (7.13) coefficients of Q, above

A, B Eq. (6.3) specialised to the above a, b, c, d , e

E Eq. (7.17) E : y2 D x3 C Ax C B

u1 p. 72, item (a) limV !C1 u.V /

u.V /, v.V / Eqs. (7.18), (7.19) v.V /2 D Q.u.V //

x.u/, y.u/ Eqs. (7.21), (7.22) auxiliary functions satisfying y.u/2 D x.u/3 CAx.u/ C B



List of symbols 169


x.V /, y.V / Eq. (7.23) x.V / D x ı u.V /, y.V / D y ı u.V /

x0, y0 Eqs. (7.24), (7.25) coordinates of auxiliary point P E0P E0 Eq. (7.26) P E0 D .x0, y0/

1, 2, 3 p. 75 roots of X 3CAX CB; permutation of e1, e2, e3

V Thm. 7.1 technical parameter

L.P/ linear form for the generic point P “covering”

a generic solution to the problem of this chapter

c8 Eq. (7.30) technical parameter

Chapter 8 Ellog applied to general elliptic equation


chapter

E Prop. 8.1.1 short Weierstrass model birationally equiva-

lent to C

X ,Y ; U ,V Eq. (8.3) ff. birational transformation between C and short

Weierstrass model E

vi Fact 8.2.1(a) Puiseux series: g.u, vi .u// D 0

˛k,i , i , i coefficients and parameters related to vi , above

B0 Fact 8.2.1(d) 8i , vi .u/ converges for juj > B0

x, y Prop. 8.3.2 functions of u such that .x.u/, y.u// is a point

on E

B1 in the interval .B1, C1/ x, y have certainproperties

P E0 Def. 8.3.3 auxiliary point involved in Prop. 8.4.1

B2 Prop. 8.4.2 technical parameter; we require u B2 B1 B0

c9, technical parameters

B3, c10, c11 Prop. 8.7.1 upper bound for h.x.P// in terms of

log ju.P/jL.P/ Thm. 8.7.2 linear form for the generic point P “covering”

a generic solution to the problem of this chapter

Chapter 9 Computing effective bound for M

L.P/ Eq. (9.2) uniform expression for the L.P/’s of all pre-

vious chapters

`0, `1, : : : p. 103 `i D l.P i /

" Sec. 9.1 " 2 ¹1,0,1º as explained below Eq. (9.1)

m00



170 List of symbols


k, n0, n1, : : : , nk , d Eq. (9.2) related to the coefficients of L.P /

“bullets” p. 99 ff. distinguish three cases for linear forms in ellip-

tic logarithms

N pp. 99–100 max0ik jni j; depends on “bullet”

ı third “bullet” p. 100 case when points involved in L.P/ are linearly

dependent

˛, ˇ Eq. (9.6) N ˛M C ˇ

$ 1, $ 2, Q Lemma 9.1.1 normalised fundamental pair of periods

K , D, jE , hE p. 103 Preparatory to Thm. 9.1.2

H 0, H 1, H 2 : : :

c12, c13, c14, c15 Thm. 9.1.2 involved in the lower bound of M

c16, c17, c18 Thm. 9.1.3 involved in condition for M leading to upper

bound of M

Chapter 10 Reduction of M

D .P/ Eq. (10.1) linear form rewritten for the reduction process

1, : : : , k below Eq. (10.1) is a linear form in the i ’s

n0, n1, : : : , nk Eq. (10.1) integers: D n0 CPkiD1 ni i

N above Eq. (10.2) N D max0ik jni j3 involved in the upper bound of jj1, 2, 4 Eqs. (10.2), (10.3) involved in the upper bound of

j

jb0 Sec. 10.1 first vector of LLL-basis

B1.N / below Eq. (10.4) upper bound for N

Chapter 11 Resolution in S -integers of Weierstrass

equation

1 beginning of Ch. 11 the “infinite” prime

S set of primes including 1s cardinality of S , above


chapterz.Q/ Sec. 11.1 z-parameter of point Q

!.z/ z-expansion of invariant differential

OC .M/ or C p. 140 formal group of C

logC Eq. (11.6) p-adic logarithmic function

expC Eq. (11.9) p-adic exponential function

Eq. (11.7) p.z/ : condition for existence of

logC.z/,expC.z/



List of symbols 171


lp Definition 11.1.3 p-adic elliptic logarithm

E Eq. (11.13) short Weierstrass model birationally equiva-

lent to C

, , , Eq. (11.14) ff. parametersrevisited in the contextof this chap-

ter

bq , b1 p. 146 technical parameters in Sec. 11.2

P q Lemma 11.2.2 set of “exceptional” points to which Lemma

11.2.1 does not apply

P Eq. (11.17) set of “exceptional” points to which

Thm. 11.2.4 does not apply

tq p. 150 tqP i satisfies certain crucial conditions for ev-

ery i D 1, : : : , rLq./ Thm. 11.2.4 linear form in q-adic elliptic logs

c19, c20 parameters involved in upper bound

h, q , H , d , g Thm. 11.2.5 technical parameters

a1, : : : , ar

c23 involved in the lower bound of jLq .P E /jq

˛, ˇ, c12, c13 Thm. 11.2.6 parametersrevisitedin thecontext ofthischap-

ter

c21, c22 Eq. (11.27) constants involved in p-adic reduction

M q Problem at p. 155 upper bound for M when jx.P/jq Dmaxp2S jx.P/jp, see Thm. 11.2.4

.n/ p. 155 for 2 Qp : .n/ 2 Q and p. .n// n

ƒ linear form to which p-adic reduction is ap-

plied

Z.n/, .n/ above Prop. 11.3.1 matrix and lattice involved in p-adic reduction

b1 Thm. 11.3.2 p-adic reduction: first vector of LLL-basis



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Index

absolute value, 11–13

archimedean, 12

non-archimedean, 11

p-adic, 137

addition

of points, 6, 7

AGM, 44–45algorithm

basis reduction, x

Coates’s, 83

LLL, x, xi, 122–134

van Hoeij’s, 79

Zagier’s, 52

Arithmetica, vii, viii

arithmetic-geometric mean, 44

basis

LLL-reduced, 122–134, 157

Mordell-Weil, 7

birational

equivalence, 2

over, 3

transformation, 2–5, 7–8

Cauchy’s rule, 91–96

conjugacy class, 81

curve, 1

defined over, 1elliptic, 3–7

degree

local

of an extension, 11

of an ideal, 10

dehomogenisation, 2

denominator, 137

Diophantus, vii–ix

discriminant, 6, 30

elliptic

integral

of first kind, 88

Elliptic Logarithm Method, 54

Ellog, vii, ix, x, xii, 54

embedding

complex, 11real, 11

equation

algebraic, 1

Diophantine

elliptic, vii, 3–7

elliptic

general, 78

Pell, 69

projective, 1

quartic, 60Thue

quartic, 69

Weierstrass, 6, 31, 36, 57

short, 6

equivalent

absolute values, 11

models

birationally, 2–5

exponent

of prime ideal, 9

formula

addition

for , 33

duplication

for , 33

integral, 36

product, 12, 14, 16, 18, 150

function

elliptic, 29



178 Index

exponential

p-adic, 140

logarithmic

p-adic, 140

of Weierstrass, 29–49function-field, 2

genus, 3

group

formal, 137, 140

height

absolute

of projective point, 15–17

absolute logarithmic

of algebraic number, 15–19

of projective point, 15

canonical, 23–28, 106

naive, 20

Néron-Tate, 24

of polynomial, 18

over Q, 17

Weil, 20, 149

homogenisation, 1

infinitypoint at, 2

solution at, 2

integer

p-adic, 137

S -, x, 137–161

invariant differential, 139

kernel

of reduction, 138

lemma

Gauss’s, 18

linear form

in elliptic logarithms, vii–x, 56

linear form L.P/, 58, 66, 76, 97–99, 104,

105, 121

logarithm

elliptic, 35, 51, 52, 101, 106

computation of, 52–53, 106

p-adic

elliptic, xii, 142

matrix

height-pairing, 25, 106

method

chord-tangent, vii, 6

elliptic logarithm, vii

model

of curve, 1

projective, 1

quartic elliptic, 60

Weierstrass, 6, 57

minimal, 27, 107, 144

short, 5–6

norm

absolute, 9

ideal, 10

number

p-adic, 137

numerator, 137

pairing

Néron-Tate, 25

Weil, 25

parallelogram

fundamental, 29, 38, 40

period-, 29, 38, 40

period lattice, 29

periods, 29

computation of, 44–47

explicit expressions, 41–44

fundamental pair of, xii, 29, 37, 44,

102, 106

place, 11

archimedean, 11

non-archimedean, 11point

at infinity, 6

of a model, 1

zero, 6

polynomial

division-, 143

prime

finite, 137

infinite, 12, 107

Puiseux series, 80



Index 179

ramification index, 10

reduction

mod p, 138

p-adic, 154, 158

reduction methodde Weger’s, 154

resolution

effective, ix, 55

explicit, x, 55

routine

CanonicalHeight, 103, 106

ellheight, 103, 106, 107

ellheightmatrix, 106, 107

ellinit, 106

EllipticCurve, 106

EllipticLogarithm, 103, 106

ellminimalmodel, 106

ellpointtoz, 103, 106

HeightPairingMatrix, 106

MinimalModel, 106

omega, 106

pAdicEllipticLogarithm, 144, 159

Periods, 106

Points, 60, 69, 149, 158

puiseux, 81

ratpoints, 60, 149

software

MAGMA, xiii, 105–107

MAPLE, xiii, 105–107

PARI, xiii, 105–107

SAGE, xiii

sum

of points, 6

theorem

S. David’s, ix–xi, 20, 102, 104

N. Hirata-Kohno’s, x, xii, 153

implicit function, 80

Lutz-Nagell, 7

B. Mazur’s, 7

Mordell-Weil, 6

z-coordinate, 138

elliptic diophantine equations

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