computational aspects of diophantine equations

Computational Aspects of Diophantine Equations

Conference handbook

February 15-19, 2016

University of Salzburg

Contents

Welcome 3

General Information 4

Conference Location . . . . . . . . 4Registration . . . . . . . . . . . . . 4Conference Opening . . . . . . . . 4Technical Equipment of the Lec-

ture Room . . . . . . . . . . . 4Internet Access during Conference . 4Lunch and Dinner . . . . . . . . . 5Co�ee Breaks . . . . . . . . . . . . 5Conference Picture . . . . . . . . . 5Local Transportation . . . . . . . . 5Social Program . . . . . . . . . . . 5Information about the Conference

Venue Salzburg . . . . . . . . 6Information about the University

of Salzburg . . . . . . . . . . 6

Program 7

Summary . . . . . . . . . . . . . . 7Monday . . . . . . . . . . . . . . . 7Tuesday . . . . . . . . . . . . . . . 8Wednesday . . . . . . . . . . . . . 9Thursday . . . . . . . . . . . . . . 9Friday . . . . . . . . . . . . . . . . 10

Abstracts 11

András Bazsó . . . . . . . . . . . . 11Michael A. Bennett . . . . . . . . . 11Attila Bérczes . . . . . . . . . . . . 11Csanád Bertók . . . . . . . . . . . 12Nicolas Billerey . . . . . . . . . . . 13

Yuri F. Bilu . . . . . . . . . . . . . 13Laura Capuano . . . . . . . . . . . 13Andrej Dujella . . . . . . . . . . . 14Bernadette Faye . . . . . . . . . . . 14Christopher Frei . . . . . . . . . . 14Maciej Gawron . . . . . . . . . . . 15Eva Goedhart . . . . . . . . . . . . 17Krisztián Gueth . . . . . . . . . . . 17Kálmán Gy®ry . . . . . . . . . . . 17Lajos Hajdu . . . . . . . . . . . . . 18Christoph Hutle . . . . . . . . . . . 18Angelos Koutsianas . . . . . . . . . 19Dijana Kreso . . . . . . . . . . . . 19Florian Luca . . . . . . . . . . . . 19Takafumi Miyazaki . . . . . . . . . 20Filip Najman . . . . . . . . . . . . 20Roland Paulin . . . . . . . . . . . . 21Attila Peth® . . . . . . . . . . . . . 21László Remete . . . . . . . . . . . . 22Ivan Soldo . . . . . . . . . . . . . . 23Michael Stoll . . . . . . . . . . . . 23Tímea Szabó . . . . . . . . . . . . 23László Szalay . . . . . . . . . . . . 24Márton Szikszai . . . . . . . . . . . 25Petra Tadi¢ . . . . . . . . . . . . . 26Alain Togbé . . . . . . . . . . . . . 26Maciej Ulas . . . . . . . . . . . . . 26Nóra Varga . . . . . . . . . . . . . 27Francesco Veneziano . . . . . . . . 27Martin Widmer . . . . . . . . . . . 27Gisbert Wüstholz . . . . . . . . . . 28

List of participants 30

2

Welcome

The Department of Mathematics and the local organizers welcome you at the conferenceAlgorithmic Aspects of Diophantine Equation, February 15-19, 2016 at the University ofSalzburg. We hope that you will have a very fruitful and enjoyable time in Salzburg.

The aim of this conference is to bring together mathematicians from number theoryespecially working on Diophantine equations, Diophantine analysis or Diophantine geom-etry and to exchange new results and techniques in this area. But, also contributionson applications of Diophantine equations to cryptography, numeration systems, etc. arewelcome. We hope that an active interaction between those areas will take place. Inparticular, the main topics of this conference are gathered around the computational as-pects of Diophantine equations and include: e�ective solutions to Diophantine equations,automatic solutions to Diophantine equations, e�ective results from Diophantine approx-imation, heights in Diophantine analysis, e�ective Diophantine geometry, counting pointson varieties.

The invited plenary speakers are Michael A. Bennett (Vancouver), Yuri Bilu (Bor-deaux), Andrej Dujella (Zagreb), Kálmán Gy®ry (Debrecen), Michael Stoll (Bayreuth),Martin Widmer, and Gisbert Wüstholz (Zurich). The list of participants includes over�fty scientists coming from Austria (Graz, Linz, Salzburg), Canada (Vancouver), Croa-tia (Osijek, Pula, Zagreb), France (Aubiére, Bordeaux), Germany (Bayreuth), Hungary(Debrecen, Sopron, Szombathely), Italy (Pisa), Japan (Maebashi, Wakayama), Poland(Krakow), Senegal (Dakar), Serbia (Belgrade), South Africa (Johannesburg), Switzerland(Basel, Zurich), United Kingdom (London, Warwick), and from the USA (Northamp-ton/MA, Westville/IN).

The scienti�c program starts on Monday, February 15, 2016 at 9:45 in HS402; itcontinues until Friday, February 19 at noon. The conference language is English.

We thank the University of Salzburg, in particular the Department of Mathematics,and the Austrian Science Fund (FWF grants P24801-N26 and P24574-N26) for �nancialsupport.

The OrganizersClemens Fuchs

István PinkVolker Ziegler

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General Information

Conference Location

The conference takes place in the lecture room HS402 in the Faculty of Natural Scienceof the University of Salzburg. The address is:

Hellbrunner Str. 345020 SalzburgAustria

The Department of Mathematics can be contacted by phone at +43(662)80445300.

Registration

The registration takes place in front of the conference lecture room HS402 on Monday,February 15, from 9:00-10:00 a.m.

Conference Opening

The conference will be opened on Monday, February 15, at 9:45 in HS402. The inauguraladdress will be given by:

Prof. Dr. Fatima Ferreira-BrizaVice-Rector for Research, University of Salzburg

The scienti�c program starts on Monday, February 15 at 10:00 with the plenary talkof Michael A. Bennett.

Technical Equipment of the Lecture Room

The lecture room is equipped with PC and beamer as well as with large blackboards.Technical assistants are present in the breaks to help you transfer your �le to the PC.

Internet Access during Conference

WLAN is available at the campus.

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Lunch and Dinner

A list of restaurants and supermarkets available in the close vicinity of the conferencelocation is being handed to you with the conference folder.

Co�ee Breaks

Refreshments are served free of charge to participants on:

Monday 11:00-11:30 and 15:30-16:00Tuesday 10:00-10:30 and 15:30-16:00Wednesday, 10:00-10:30Thursday, 10:00-10:30 and 15:30-16:00Friday, 10:00-10:30

In addition, the university cafeteria serves co�ee and snacks.

Conference Picture

The conference picture will be taken on Wednesday at 10:00 immediately after the plenarytalk of Martin Widmer.

Local Transportation

To reach the conference location, take bus 3 or 8 from the city center to stop �Faistauer-gasse�; you can also leave one stop earlier (at �Akademiestraÿe�) or one stop later (at�Jose�au�). For more information, please visit the web page:

https://www.salzburg-ag.at/verkehr/obus/

Social Program

Registration is necessary for the city tour as well as for the conference dinner.

The city tour takes place on Wednesday afternoon. We meet at 14:00 in front of theconference lecture room. The tour takes 2,5-3 hours and will be given in English.

The conference dinner takes place at the restaurant STERNBRÄU (�Jedermannstube�,Griesgasse 23, phone +43(662)842140) at 18:30. The restaurant can be reached on foot,or by bus 8 (leave at stop �F.-Hanusch Platz�).

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Information about the Conference Venue Salzburg

Salzburg is the capital of Salzburg. For more information about its history and culture,please visit the web page:

http://www.salzburg.info

Information about the University of Salzburg

The Paris-Lodron University of Salzburg was founded in 1622. For more informationabout the university, please visit the web page:

http://www.uni-salzburg.at

For information on the Department of Mathematics of the University of Salzburg,please visit the web page:

http://www.uni-salzburg.at/mathematik

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Program

Summary

Time Monday Tuesday Wednesday Thursday Friday

09:00-10:00 Registration andOpening (9:45)

Bilu Widmer WüstholzPaulin

09:30-10:00 Faye10:00-10:30

BennettCo�ee Break Co�ee Break Co�ee Break Co�ee Break

10:30-11:00 Luca Szabó Frei Goedhart11:00-11:30 Co�ee Break Varga Remete Kreso Szalay11:30-12:00 Billerey Gawron Miyazaki Tadi¢12:00-12:30 Koutsianas Bertók Veneziano12:30-14:00 Lunch Break14:00-14:30

Stoll Dujella

City Tour

Gy®ry14:30-15:0015:00-15:30 Capuano Soldo Bérczes15:30-16:00 Co�ee Break Co�ee Break Co�ee Break16:00-16:30 Najman Togbe Peth®16:30-17:00 Ulas Hutle Hajdu17:00-17:30 Szikszai Gueth Bazsó

Monday

Time Speaker Title of the talk

09:00-09:45 Registration09:45-10:00 Opening10:00-10:50 Michael A. Bennett Elliptic curves of conductor p2

11:00-11:30 Co�ee break

11:30-11:55 Nicolas Billerey Sum of two S-units via Frey-Hellegouarch curves12:00-12:25 Angelos Koutsianas Elliptic Curves with Good Reduction Outside S

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12:30-14:00 Lunch break

14:00-14:50 Michael Stoll The Generalized Fermat Equation x2 + y3 = z11

15:00-15:25 Laura Capuano Unlikely Intersections in certain families of abelianvarieties and the polynomial Pell equation

15:30-16:00 Co�ee Break

16:00-16:25 Filip Najman Mordell-Weil groups of elliptic curves over number�elds

16:30-16:55 Maciej Ulas On representing coordinates of points on ellipticcurves by quadratic forms

17:00-17:25 Márton Szikszai Perfect powers in products of terms of an EDS

Tuesday


9:00-09:50 Yuri Bilu Subgroups of Class Groups and the AbsoluteChevalley-Weil Theorem

10:00-10:30 Co�ee break

10:30-10:55 Florian Luca One the Diophantine equation |τ(n!)| = m!

11:00-11:25 Nóra Varga Equal values of combinatorial numbers11:30-11:55 Maciej Gawron On Decompositions of quadrinomials and related Dio-

phantine equations


14:00-14:50 Andrej Dujella There are in�nitely many rational Diophantine sextu-ples

15:00-15:25 Ivan Soldo Diophantine triples in the ring of integers of thequadratic �eld Q(

√t), t > 0

15:30-16:00 Co�ee break

16:00-16:25 Alain Togbé Diophantine triples of Fibonacci Numbers16:30-16:55 Christoph Hutle Diophantine Triples with Values in k-generalized Fi-

bonacci Sequences17:00-17:25 Krisztián Gueth Diophantine triples in Lucas-Lehmer sequences

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Wednesday


09:00-09:50 Martin Widmer Weakly admissible lattices, primitive lattice points,and Diophantine approximation

10:00-10:30 Co�ee break

10:30-10:55 Tímea Szabó Power integral bases in quartic �elds and quarticextensions

11:00-11:25 László Remete Power integral bases in pure quartic number �elds:an application of binomial Thue equations

11:30-11:55 Takafumi Miyazaki A polynomial-exponential equation related to theRamanujan-Nagell equation

12:00-12:25 Csanád Bertók On the equation Un = 2a + 3b + 5c

12:30-14:00 Lunch break14:00-16:30 City Tour18:30- Conference dinner

Thursday


09:00-09:50 Gisbert Wüstholz Billiard on the triaxial ellipsoid

10:00-10:30 Co�ee break

10:30-10:55 Christopher Frei Rational points on smooth cubic surfaces11:00-11:25 Dijana Kreso Diophantine equations and monodromy groups11:30-11:55 Petra Tadi¢ A criterion for injectivity of the specialization ho-

momorphism of elliptic curves and its applications12:00-12:25 Francesco Veneziano Rational points on explicit families of curves


14:00-14:50 Kálmán Gy®ry E�ective results for Diophantine equations over�nitely generated domains

15:00-15:25 Attila Bérczes Arithmetic and geometric progressions in the solu-tion set of Diophantine equations

15:30-16:00 Co�ee break

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16:00-16:25 Attila Peth® On a Binet-type formula for nearly linear recursivesequences and its applications

16:30-16:55 Lajos Hajdu Power values of sums of products of consecutiveintegers

17:00-17:25 András Bazsó On the coe�cients of polynomials related to (al-ternating) power sums of arithmetic progressions

Friday


09:00-09:25 Roland Paulin Asymptotic Expansion of the Zeros of the Lerch ZetaFunction

09:30-09:55 Bernadette Faye Repdigits as Euler Functions of Lucas Numbers

10:00-10:30 Co�ee break

10:30-10:55 Eva Goedhart The Diophantine equation (a2cxk − 1)(b2czk − 1) =

(abczk − 1)2

11:00-11:25 László Szalay The diophantine equation F |k|x = F|l|y

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Abstracts

On the coe�cients of polynomials related to (alternating) power sums of

arithmetic progressions

András Bazsó

University of Debrecen, [email protected]

Coauthors: István Mez®

Let a 6= 0, b, k > 0 be integers with gcd(a, b) = 1. In the talk we study the coe�cientsof the polynomials which, at positive integer values n, give, respectively, the sum or thealternating sum of the k-th powers of the n-term arithmetic progression b, a + b, 2a +

b, . . . , (n− 1)a+ b. Joint work with István Mez®.

Elliptic curves of conductor p2

Michael A. Bennett

University of British Columbia, Vancouver, [email protected]

We sketch an approach to �nding all elliptic curves of conductor p2 for prime p

that is computationally reasonably e�cient. There are some nice connections to solv-ing parametrized families of Thue equations and simplest cubic �elds.

Arithmetic and geometric progressions in the solution set of Diophantine

equations

Attila Bérczes


In 2004 Bérczes and Peth® started to investigate the arithmetic progressions in thesolution set of norm form equations. Since then the investigation of special progressionsappearing in the solution set of diophantine equations has resulted in a series of interestingresults.

Bérczes and Peth®, Bérczes, Peth® and Ziegler and later Bazsó determined all arith-metic progressions forming solutions of some parametric families of norm form equations.

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Arithmetic progressions in the solution set of Pell equations were investigated by Peth®and Ziegler, and by Dujella, Peth® and Tadic.

In this talk a survey on these results will be presented, along with new results obtainedby Bérczes and Ziegler on geometric progressions in the solution set of Pell-equations. Fur-ther, Bérczes and Ziegler also determined all geometric progressions consisting of elementsof a given Lucas sequence.

On the equation Un = 2a + 3b + 5c

Csanád Bertók


Coauthors: Lajos Hajdu, István Pink, Zsolt Rábai

In the talk, �rst we propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential Diophantine equations. Namely, consider the equation

a1bα1111 · · · b

α1l1l + . . .+ akb

αk1k1 · · · b

αklkl = c (1)

in non-negative integers α11, . . . , α1l, . . . , αk1, . . . , αkl, where ai, bij, are non-zero integersfor every i = 1, . . . , k and j = 1, . . . , l, and c is an integer. Our conjecture is that ifthe equation above has no solutions, then there exists an integer m ≥ 2 such that thecongruence

a1bα1111 · · · b

α1l1l + . . .+ akb

αk1k1 · · · b

αklkl ≡ c (mod m) (2)

has no solutions in non-negative integers αij, i = 1, . . . , k, j = 1, . . . , l.

In the talk we present a result showing that in a sense, the conjecture is valid for"almost all" equations. Further, based upon the conjecture we propose a general methodfor the solution of exponential Diophantine equations, relying on a generalization of aresult of Erd®s, Pomerance and Schmutz concerning Carmichael's λ function.

Finally, we illustrate that our method works not only in Z, but also in the ring ofintegers of Q(α) (where α is a real algebraic number) by generalizing a result of D.Marques and A. Togbé and solving a problem of F. Luca and S. G. Sanchez. Let Un =

A · Un−1 + B · Un−2 (n ≥ 2) with A,B ∈ Z and initial terms U0, U1 ∈ Z be a binarysequence. If a, b, c are non-negative integers, then we give all solutions of the equations

Un = 2a + 3b,

Un = 2a + 3b + 5c,

in the case when (A,B, U0, U1) = (1, 1, 0, 1), (1, 1, 2, 1), (2, 1, 0, 1), (2, 1, 2, 2).

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Sum of two S-units via Frey-Hellegouarch curves

Nicolas Billerey

Université Clermont - Ferrand 2 - Blaise Pascal, [email protected]

Coauthors: Michael A. Bennett

In this talk we shall describe a method for explicitly �nding all perfect powers thatcan be expressed as the sum of two S-units, where S is a �xed �nite set a primes. Ourapproach, which is based on the modularity of some attached Galois representations,allows us to explicitly solve the problem for some small sets of primes such as S = 2, 3

and S = 3, 5, 7. This is a joint work with Michael A. Bennett.

Subgroups of Class Groups and the Absolute Chevalley-Weil Theorem

Yuri F. Bilu

Université de Bordeaux, [email protected]

The following conjecture is widely believed to be true: given a �nite abelian group G,a number �eld K and an integer d > 1, there exist in�nitely many extensions L/K ofdegree d such that the class group of L contains G as a subgroup. I will speak on someold and recent results on this conjecture, in particular, on my joint work with J. Gillibertin course.

Unlikely Intersections in certain families of abelian varieties and the

polynomial Pell equation

Laura Capuano

Scuola Normale Superiore di Pisa, [email protected]

Coauthors: Fabrizio Barroero

Given n independent points on the Legendre family of elliptic curves of equationY 2 = X(X − 1)(X − c) with coordinates algebraic over Q(c), we will see that there are atmost �nitely many specializations of c such that two independent relations hold betweenthe n points on the specialized curve. This �ts in the framework of the so-called UnlikelyIntersections. We will see a higher-dimensional analogue of this result and explain howit applies to the problem of studying the solvability of the (almost-)Pell equation inpolynomials. This is joint work with Fabrizio Barroero.

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There are in�nitely many rational Diophantine sextuples

Andrej Dujella

University of Zagreb, [email protected]

Coauthors: Matija Kazalicki, Miljen Miki¢, Márton Szikszai

A rational Diophantine m-tuple is a set of m nonzero rationals such that the productof any two of them increased by 1 is a perfect square. The �rst rational Diophantinequadruple was found by Diophantus, while Euler proved that there are in�nitely manyrational Diophantine quintuples. In 1999, Gibbs found the �rst example of a rationalDiophantine sextuple. In this talk, we describe construction of in�nitely many rationalDiophantine sextuples. This is joint work with Matija Kazalicki, Miljen Miki¢ and MártonSzikszai.

Repdigits as Euler functions of Lucas numbers

Bernadette Faye

UCAD, Dakar, [email protected]

Let ϕ(m) be the Euler function of the positive integer m. Various Diophantine equationsinvolving the Euler function of members of Fibonacci or Lucas numbers have been in-vestigated during the past years. In the current paper, we prove some results about thestructure of all Lucas numbers whose Euler function is a repdigit in base 10. In otherswords, we look at the Diophantine equation

ϕ(Ln) = d

(10m − 1

9

)d ∈ {1, . . . , 9}. (3)

Numbers as the ones appearing in the right�hand side of equation (3) are called rep-digitsin base 10, since their base 10 representation is the string dd · · · d︸︷︷︸

m times

.

Here, using linear forms in logarithms and the Baker-Davenport reduction method,we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2 , wherep3|10p−1 − 1.

Rational points on smooth cubic surfaces

Christopher Frei

Graz University of Technology, [email protected]

Coauthors: Efthymios Sofos

Manin's conjecture predicts an asymptotic formula for the number of rational pointsof bounded height on smooth cubic surfaces over number �elds. For a large class of cubic

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surfaces with a conic bundle structure, we prove that the asymptotic lower bound impliedby Manin's conjecture has the correct order of magnitude. As a consequence, the lowerbound has the correct order of magnitude for all smooth cubic surfaces after a smallextension of the base �eld. The central technical tool is the analysis of divisor sums ofcertain binary forms. This is joint work with Efthymios Sofos (Leiden).

On decompositions of quadrinomials and related Diophantine equations

Maciej Gawron

Jagiellonian University, Krakow, [email protected]

In paper [9] Schinzel, Pintér and Péter give an ine�cient criterion for the Diophantineequation of the form

axm + bxn + c = dyp + eq,

where a, b, c, d, e rationals, ab 6= 0 6= de, m > n > 0, p > q > 0, gcd(m,n) = 1, gcd(p, q) =1, and m, p ≥ 3 to have in�nitely many integer solutions.

In the later paper Schinzel [10] dropped the assumption gcd(m,n) = 1, gcd(p, q) = 1

and gives a necessary and su�cient condition for such equation to have in�nitely manyinteger solutions.

In the recent paper Kreso [6] proved the �nitness of integral solutions for the equation

a1xn1 + a2x

n2 + . . .+ alxnl + al+1 = b1y

m1 + b2ym2 ,

where l ≥ 2 and m1 > m2, n1 > n2 > . . . > nl are �xed positive integers satisfyinggcd(m1,m2) = 1, gcd(n1, n2, . . . , nl) = 1, a1, a2, . . . , al, al+1, b1, b2 are non-zero rationals,except for possibly al+1. With n1 ≥ 3,m1 ≥ 2l(l − 1) and (n1, n2) 6= (m1,m2).

All the mentioned results relies on Bilu-Tichy Theorem [1], and theorems concerningdecompositions of trinomials [2] as main ingredients. No such results for the equationsinvolving at least three non-zero coe�cients at positive powers on both sides are knownmainly because we have no results concerning decompositions of lacunary polynomialswith more than three non-zero terms [6]. Some partial results in this direction are givenin [7].

In this note we describe all possible decompositions of quadrinomials. In the sequelwe use Bilu-Tichy theorem to prove the following generalizations of Schinzel and Kresoresults. More precisely, we prove the following

Theorem. Let f(x) = Axn1 + Bxn2 + Cxn3 + D, g(x) = Exm1 + Fxm2 + Gxm3 + H

with f, g ∈ Q[x], n1 > n2 > n3 > 0, m1 > m2 > m3 > 0, and gcd(n1, n2, n3) = 1,gcd(m1,m2,m3) = 1, (m1,m2,m3) 6= (n1, n2, n3), ABC 6= 0, EFG 6= 0 and n1,m1 ≥ 9.Then the equation

f(x) = g(y)

has only �nitely many integer solutions.

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Theorem. Let l ≥ 4 and n1 > n2 > . . . > nl > 0, m1 > m2 > m3 > 0 be positiveintegers. Let

f(x) = A1xn1 + A2x

n2 + . . .+ Alxnl + Al+1 and g(x) = Exm1 + Fxm2 +Gxm3

be polynomials with rational coe�cients such that gcd(n1, n2, . . . , nl) = 1, gcd(m1,m2,m3) =

1, A1A2 . . . Al 6= 0, EFG 6= 0 and m1 ≥ 2l(l − 1), n1 ≥ 4. Then the equation

f(x) = g(y)

has only �nitely many integer solutions.

Our results are ine�ective as we use Theorem of Bilu and Tichy which relies on classicaltheorem of Siegel on integral points.

References

[1] Y.F. Bilu, R.F. Tichy, The Diophantine equation f (x) = g(y), Acta Arith. 95 (2000),no. 3, 261-288.

[2] M. Fried, A. Schinzel, Reducibility of quadrinomials, Acta Arith. 21 (1972), 153-171[3] M. Gawron, On decompositions of quadrinomials and related Diophantine equations,

preprint arXiv:1512.02817v1[4] I.M. Gessel, G. Viennot, Binomial determinants, paths, and hook length formulae,

Advances in Math. 58 (1985), 300-321[5] G. Hajós, The solution of Problem 41, (Hungarian). Mat. Lapok, 4, (1953), 40-41[6] D. Kreso, On common values of lacunary polynomials at integer points, New York

J. Math. 21 (2015) 987-1001[7] D. Kreso, R.F. Tichy, Functional composition of polynomials: inde- composability,

Diophantine equations and lacunary polynomials, arXiv:1503.05401[8] R. C. Mason, Diophantine Equations over Function Fields, L. M. S. Lecture Notes

No. 96, Cambridge UP, (1984).[9] G. Péter, Á. Pintér, A. Schinzel, On equal values of trinomials, Monatsh. Math.

162 (2011), no. 3, 313-320.[10] A. Schinzel, Equal values of trinomials revisited. Tr. Mat. Inst. Steklova 276

(2012), Teoriya Chisel, Algebra i Analiz, 255-261; translation in Proc. Steklov Inst.Math. 276 (2012), no. 1, 250-256.

[11] C.L. Siegel, Uber einige Anwendungen Diophantischer Approximationen, Abh. Preuss.Akad. Wiss. Phys.-Math. Kl., 1929, Nr. 1; also: Gesammelte Abhandlungen, Band1, 209-266.

[12] W. W. Stothers, Polynomial identities and hauptmoduln, Quarterly J. Math. Ox-ford, 2, 32, (1981)

[13] U. Zannier, On composite lacunary polynomials and the proof of a conjecture ofSchinzel Invent. Math. 174 (2008), no. 1, 127-138

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The Diophantine equation (a2cxk − 1)(b2czk − 1) = (abczk − 1)2

Eva Goedhart

Smith College, Northampton, [email protected]

Coauthors: Helen G. Grundman

For a, b, c, k ∈ Z+ with k ≥ 7, the equation

(a2cxk − 1)(b2cyk − 1) = (abczk − 1)2

has no integer solutions x, y, z > 1 with a2xk 6= b2yk. I will dscuss the proof of this resultwhich uses standard results on continued fractions and a Diophantine approximationtheorem due to M.A. Bennett. This is joint work with Helen G. Grundman.

Diophantine triples in Lucas-Lehmer sequences

Krisztián Gueth University of West Hungary, Savaria Campus, [email protected]

In this lecture, I talk about the non-existence of diophantine triples linked to a givenlinear recurrence of order four. A diophantine m-tuple means m distinct positive integerssuch that the product of any two of them is one less then a square of an integer. Weconsider this problem for the terms of a Lucas-Lehmer recurrence sequence instead of thesquare numbers, in the case of m = 3. I proved that there are no integers 0 < a < b < c

such that ab+ 1, ac+ 1 and bc+ 1 all are terms of this sequence.

E�ective results for Diophantine equations over �nitely generated domains

Kalman Gy®ry


Recently Evertse and the speaker, partly with Bérczes, have extended the e�ectivetheory of Diophantine equations over number �elds to the case of equations over arbitrary�nitely generated domains over Z (which may contain transcendental elements, too.) Inthe �rst part of the lecture, the most important e�ective results over number �elds willbe formulated in qualitative form. Then we give a survey of their generalizations over�nitely generated domains. Finally, the method of proofs will be brie�y discussed.

17

Power values of sums of products of consecutive integers

Lajos Hajdu


For k = 0, 1, 2, . . . put

fk(x) =k∑i=0

i∏j=0

(x+ j).

In the talk we consider the Diophantine equation

fk(x) = yn (4)

in integers x, y, k, n with k ≥ 0 and n ≥ 2. Without loss of generality, we shall assumethat n is a prime. We mention that equation (4) is closely related to several classicalproblems and results.

In the talk we present a general �niteness result concerning (4). Further, we provideall solutions to this equation for k ≤ 10. In our proofs we combine several tools andtechniques, including Baker's method, local arguments, Runge's method, and a methodof Gebel, Peth®, Zimmer and Stroeker, Tzanakis to �nd integer points on elliptic curves.

Diophantine Triples with Values in k-generalized Fibonacci Sequences

Christoph Hutle

Paris Lodron University of Salzburg, [email protected]

One of the oldest problems in number theory is the question of Diophantus, which isabout constructing sets of rationals or integers with the property that the product of anytwo of its distinct elements plus 1 is square. Recently, several variations of this problemhave been investigated. The problem of �nding bounds on the size m for Diophantinem-tuples with values in linear recurrences is one such variation.

It was shown by Fuchs, Hutle, Irmak, Luca and Szalay in 2015 (to appear in Math.Slovaca), that for the Tribonacci sequence {Tn}n≥0 given by T0 = T1 = 0, T2 = 1 andTn+3 = Tn+2 + Tn+1 + Tn for all n ≥ 0, there exist only �nitely many Diophantine tripleswith values in {Tn}n≥0.

In this talk, we will consider the k-generalized Fibonacci sequence given for some k ≥ 2

by F (k)0 = . . . = F

(k)k−2 = 0, F

(k)k−1 = 1 and

F(k)n+k = F

(k)n+k−1 + · · ·+ F (k)

n

for all n ≥ 0. Improving the previous result, we show that there are only �nitely manyDiophantine triples with values in {F (k)

n }n≥0.The proof is not constructive, since it is based on a version of the Subspace Theorem,

one of the most important results in Diophantine approximation.

18

Elliptic Curves with Good Reduction Outside S

Angelos Koutsianas

University of Warwick, United [email protected]

In this talk, we will present a new algorithm that �nds all elliptic curves E over anumber �eld K with good reduction outside a �nite set of primes S in K by solvingS-unit equations.

Diophantine equations and monodromy groups

Dijana Kreso

Graz University of Technology, [email protected]

Coauthors: Robert F. Tichy

Jointly with Robert Tichy, I have studied Diophantine equations of type f(x) = g(y),where f and g have at least two distinct critical points and equal critical values at atmost two distinct critical points. Our results cover and generalize several results in theliterature on the �niteness of integral solutions of such equations. In so doing, we analyzethe properties of the monodromy groups of such polynomials. We show that if f hascoe�cients in a �eld of characteristic 0, at least two distinct critical points and all distinctcritical values, then the monodromy group of f is a doubly transitive permutation group.This in particular means that f can not be represented as a composition of lower degreepolynomials. We further show that if f has at least two distinct critical points and nothree distinct critical points with equal critical values, and f(x) = g(h(x)) with g, h withcoe�cients in K and deg g > 1, then either deg h ≤ 2, or f is of special type. In the lattercase, in particular, f has no three simple critical points, nor �ve distinct critical points.

On the Diophantine equation |τ(n!)| = m!

Florian Luca

University of the Witwatersrand, Johannesburg, South [email protected]

Coauthors: Jhon Jairo Bravo Grijalba

Let τ(n) be the Ramanujan function given by

q( ∞∏n=1

(1− qn))24

=∑n≥1

τ(n)qn (|q| < 1).

In 2006, jointly with I. E. Shparlinski, we proved that the Diophantine equation |τ(n!)| =m! has only �nitely many positive integer solutions (n,m). In my talk, I take this onestep further and report that the only such solutions are (n,m) = (1, 1), (2, 4). The proofuses linear forms in logarithms. This is joint work with Jhon Jairo Bravo Grijalba fromUniversidad del Cauca, Colombia.

19

A polynomial-exponential equation related to the Ramanujan-Nagell

equation

Takafumi Miyazaki

Gunma University, [email protected]

Let us consider the equation

x2 +Dm = pn in positive integers x,m, n (1)

where p is a �xed prime and D > 1 is a �xed integer not divisible by p. This is ageneralization of the Ramanujan-Nagell equation for (p,D) = (2, 7) with m = 1. Manyworks on equation (1) concern to bound its number of solutions (x,m, n). In this direction,the de�nitive results in the case of m = 1 are already obtained by F. Beukers (1990) forp = 2, and by Y. Bugeaud and T.N. Shorey (2001) for odd p. In 2001, Bugeaud completelydescribed the solutions of equation (1) for p = 2, and also showed that equation (1) hasat most two solutions when p is odd and D > 2, except for some speci�c pairs of (p,D).After his work, P. Yuan and Y. Hu (2005) succeeded in dealing with those exceptionalcases. The combination of their works implies the following result:

Proposition. Let p be odd and D > 2. Then equation (1) has at most two solutions,unless (p,D) = (5, 4). The solutions in this exceptional case are given by (x,m, n) =

(1, 1, 1), (3, 2, 2), (11, 1, 3).

In this talk, we consider equation (1) when D takes a concrete form in terms of p. In sucha case, we show that all solutions x,m, n can be bounded by an e�ectively computableconstant depending only on p. Moreover, we solve the equation completely under someconditions on p. The motivation for this study is to apply a classical lemma obtained byV.A. Dem'janenko (1965) which is used in the study of an unsolved problem concerningprimitive Pythagorean triples.

Mordell-Weil groups of elliptic curves over number �elds

Filip Najman

University of Zagreb, [email protected]

The Mordell-Weil group E(K) ofK-rational points of an elliptic curve E over a number�eld K, is a �nitely generated abelian group and hence isomorphic to the direct productof its torsion subgroup and Zr, where r is the rank of E/K.

In this talk we will consider the question of what this group can be over number�elds of certain type, e.g. over all number �elds of degree d or over a �xed number �eld.After surveying known results, both old and new, about torsion groups, we will showthat prescribing the torsion over number �elds (as opposed to over Q!) can force various

20

properties on the elliptic curve. For instance, all elliptic curves with points of order 13

or 18 over quadratic �elds have to have even rank and elliptic curves with points of order16 over quadratic �elds are base changes of elliptic curves de�ned over Q. We show thatthese properties arise from the geometry of the corresponding modular curves.

Asymptotic Expansion of the Zeros of the Lerch Zeta Function

Roland Paulin

Paris Lodron University of Salzburg, [email protected]

We discuss the asymptotic behavior of the zeros s of the Lerch zeta function L(λ, α, s) =∑∞n=0

e2πiλn

(n+α)swhen z = e2πiλ → 0. In particular we de�ne recursively a series expansion,

and we also give an explicit, non-recursive formula for the coe�cients. The obtainedresults considerably strengthen some of the statements of a 1975 paper of Fornberg andKölbig. The methods used are quite general, and can be applied to describe the asymp-totic behavior of the zeros of a wide class of general Dirichlet series.

On a Binet-type formula for nearly linear recursive sequences and its

applications

Attila Peth®


Coauthors: Shigeki Akiyama, Jan-Hendrik Evertse

This talk is based on a joint work with Shigeki Akiyama, Tsukuba University, Japan,and Jan-Hendrik Evertse, Universiteit Leiden, The Netherlands.

We de�ne a nearly linear recursive sequence (an) and give a Binet-type formula. Weprove that the �uctuation of a linear recursive sequences can be extremely large, then weanalyze the distance of a nlrs to a naturally chosen lrs.

We investigate the zero multiplicity and common terms of nlrs's. We show for two nlrs,having single dominating roots, which are algebraic and multiplicatively independent,that the indices of consecutive common terms grow exponentially. We prove that theSkolem-Lech-Mahler theorem, does not hold generally for nlrs with at least two dominatingroots with equal absolute values. This implies that unbounded nlrs with multiplicativelyindependent characteristic roots exist with in�nitely many common values.

21

Power integral bases in pure quartic number �elds: an application of

binomial Thue equations

László Remete


A number �eld K of degree n is monogene if there exist an integer ϑ ∈ K suchthat ZK = Z[ϑ], that is {1, ϑ, ϑ2, . . . , ϑn−1} is an integral basis (so called power integralbasis) in K.

We consider the problem of monogenity and generators of power integral bases in purequartic �elds K = Q( 4

√m) where m is a square free integer with m ≡ 2, 3 (mod 4). Set

α = 4√m. For 1 < m < 107 we determine all generators

ϑ = a+ xα + yα2 + zα3

of power integral bases of K where a, x, y, z ∈ Z with

max(|x|, |y|, |z|) < 101000.

To obtain this result we calculated solutions (x, y) with max(|x|, |y|) < 10500 of binomialThue equations of type

x4 −my4 = ±1

for 2 ≤ m ≤ 107. This extensive calculation was performed on a supercomputer. Weextended this calculation also to exponents n = 3, 4, 5, 7, 11, 13, 17, 19, 23, 29.

We generalized these results also to the relative case. Let d be one of

d = 3, 7, 11, 19, 43, 67, 163

and let L = Q(i√d). Let m ≡ 2, 3(mod 4), assume (d,m) = 1 and set α = 4

√m.

For 1 < m ≤ 5000 we calculate all generators ϑ = A + Xα + Y α2 + Zα3 of relativepower integral bases of K over L with A,X, Y, Z ∈ ZL with max(|X|, |Y |, |Z|) < 10500.To prove this later result we solved binomial Thue equations in the relative case, overimaginary quadratic �elds.

We also proved that these octic �elds K does not admit any generators of (absolute)power integral bases of the form

ϑ = A+ ε(Xα+ Y α2 + Zα3)

where A,X, Y, Z ∈ ZL, ε a unit in L and

max(|X|, |Y |, |Z|) < 10500.

22

Diophantine triples in the ring of integers of the quadratic �eld Q(√−t), t > 0

Ivan Soldo

University of Osijek, [email protected]

Let R be a commutative ring and z ∈ R. A set {a1, a2, . . . , am} in R such that ai 6= 0,i = 1, . . . ,m, ai 6= aj and aiaj + z is a square in R for all 1 ≤ i < j ≤ m is called aDiophantine m-tuple with the property D(z), or simply a D(z)-m-tuple in the ring R.We study D(−1)-triples of the form {1, b, c} in the ring Z[

√−t], t > 0, for positive integer

b such that b is a prime, twice prime and twice prime squared. We prove that in thosecases c has to be an integer. By using that result we obtained some results about theexistence of D(−1)-quadruples in certain ring of integers Z[

√−t] of the quadratic �eld

Q(√−t), t > 0.

The Generalized Fermat Equation x2 + y3 = z11

Michael Stoll

Bayreuth University, [email protected]

Coauthors: Nuno Freitas, Bartosz Naskrecki

Generalizing Fermat's original problem, equations of the form xp+yq = zr, to be solvedin coprime integers, have been quite intensively studied. It is conjectured that there areonly �nitely many solutions in total for all triples (p, q, r) such that 1/p + 1/q + 1/r <

1 (the `hyperbolic case'). The case (p, q) = (2, 3) is of special interest, since severalsolutions are known. To solve it completely in the hyperbolic case, one can restrictto r = 8, 9, 10, 15, 25 or a prime ≥ 7. The cases r = 7, 8, 9, 10, 15 have been dealtwith by various authors. In joint work with Nuno Freitas and Bartosz Naskrecki, weare now able to solve the case r = 11 and prove that the only solutions (up to signs)are (x, y, z) = (1, 0, 1), (0, 1, 1), (1,−1, 0), (3,−2, 1). We use Frey curves to reduce theproblem to the determination of the sets of rational points satisfying certain conditionson certain twists of the modular curve X(11). A study of local properties of mod-11Galois representations cuts down the number of twists to be considered. The main newingredient is the use of the `Selmer group Chabauty' techniques developed recently by thespeaker to �nish the determination of the relevant rational points.

Power integral bases in quartic �elds and quartic extensions

Tímea Szabó


The existence of power integral bases is a classical topic in algebraic number theory. Itis well known that if a number �eld admits a power integral basis of type (1, θ, . . . , θn−1)

23

then up to equivalence it admits only �nitely many of them. There is an extensiveliterature of calculating power integral bases in special algebraic number �elds. Thisproblem is equivalent to solving diophantine equations, so called index form equationsThere are e�cient algorithms for calculating power integral bases in lower degree (≤ 6) andin special higher degree (6, 8, 9) number �elds. The problem of power integral bases wasalso considered in relative extensions. Algorithms for calculating relative power integralbases were given in relative cubic and in relative quartic extensions. It is an especiallydelicate problem if we solve the index form equation not only in a speci�c number �eldbut in an in�nite parametric family of number �elds, where the index form equation isgiven in a parametric form. Such results are known in certain parametric families of cubic,quartic and quintic number �elds. Similar results for calculating relative power integralbases in in�nite parametric families of relative extensions were not known before.In this talk we present the resolution of the index form equations in two families oftotally complex biquadratic �elds depending on two parameters and prove that up toequivalence, they admit only one generator of power integral bases. Note that these arethe �rst families of number �elds with two parameters where all generators of powerintegral bases determined.In the second half of my talk considering in�nite parametric families of octic �elds, thatare quartic extensions of quadratic �elds, we describe all relative power integral bases ofthe octic �elds over the quadratic sub�elds and then we check if there exist correspondinggenerators of absolute power integral bases.

The diophantine equation F[k]x = F

[`]y

László Szalay

University of West Hungary, Sopron, [email protected]

Let {Fn} denote the sequence of Fibonacci numbers de�ned by F0 = 0, F1 = 1, andFn = Fn−1 + Fn−2 for n ≥ 2. The notion of hyper-Fibonacci numbers F [k]

n (n ≥ 0, k ≥ 0)was introduced by Dil and Mez® [1] as follows. Let F [0]

n = Fn, F[k]0 = 0, and

F [k]n = F

[k]n−1 + F [k−1]

n , kn > 0.

Considering the title equation, we have the following theorems.

Theorem. Given the positive integer `, the equation F [k]x = F

[`]y has �nitely many solu-

tions in the nonnegative integers x, y and k < `, which are e�ectively computable.

Theorem. Beside the trivial solutions, the equation F [k]x = F

[`]y possesses only the solu-

tions

(k, `, x, y) = (0, 11, 14, 4), (0, 16, 16, 4), (1, 2, 4, 3), (1, 7, 12, 5), (1, 20, 11, 3), (5)

(2, 8, 6, 3), (2, 11, 7, 3), (2, 33, 11, 3), (4, 6, 5, 4), (4, 12, 5, 3), (6, 12, 4, 3).

if 0 ≤ k < ` ≤ 50.

The trivial solutions are considerd as

24

• F [k]0 = 0 = F

[`]0 ,

• F [k]1 = 1 = F

[`]1 ,

• F [0]2 = 1 = F

[`]1 ,

• F [k]x = F

[F

[k]−1x

]2 .

We conjecture, that (5) and the trivial solutions give all the solutions to F [k]x = F

[`]y .

References

[1] Dil, A. � Mez®, I., A symmetric algorithm for hyperharmonic and Fibonacci num-bers, Appl. Math. Comp., 206 (2008), 942-951.

Perfect powers in products of terms of an EDS

Márton Szikszai


Let E be a non-singular elliptic curve and P ∈ E(Q) be a point of in�nite order. Writethe multiples of P as

nP =

(AnB2n

,CnB2n

)with An, Bn, Cn ∈ Z, Bn > 0 and (AnCn, Bn) = 1. The sequence B = (Bn)

∞n=1 is called

an elliptic divisibility sequence, an important class of non-linear recurrences. Let ` ≥ 2.Set

P`(B) = {i : Bi is `−th power}

and

N` = #P`(B), M` = maxP`(B).

Consider the diophantine equation

BmBm+d . . . Bm+(k−1)d = y`

in unknown positive integersm, d, k, y with (m, d) = 1 and k ≥ 2. In this talk we show thatthe above equation admits �nitely many solutions. Further, if P`(B) is given explicitly,then we prove that for every solution (m, d, k, y) we have max(m, d, k, y) ≤ C(N`,M`),where C is an e�ective constant depending on N` and M` only. We combine several tools,including the arithmetic properties of B and bounds concerning the greatest prime divisorand the number of prime divisors of blocks of consecutive terms of arithmetic progressions.

25

A criterion for injectivity of the specialization homomorphism of elliptic

curves and its applications

Petra Tadi¢

Juraj Dobrila University of Pula, [email protected]

Let K be a number �eld. Let E be a nonconstant elliptic curve over K(t) with at leastone nontrivial K(t)-rational 2-torsion point. By the Silverman specialization theorem thespecialization homomorphism t 7→ t0 is injective for all but �nitely many t0 ∈ K. For Kof class number one, Gusi¢ and Tadi¢ obtained a method for �nding t0 ∈ K for which thecorresponding specialization homomorphism is injective. In principal they extended themethod to arbitrary number �elds K.

This method can be used to calculate the rank of elliptic curves of the form as above,by looking at the structure of a chosen specialized curve. This method has been used byGusi¢, Dujella, Peral and Tadi¢ to calculate exactly the rank of some record high rankelliptic curves over Q(t).

Diophantine triples of Fibonacci Numbers

Alain Togbé

Purdue University North Central, Westville, [email protected]

Coauthors: Bo He, Florian Luca

Let Fm be the mth Fibonacci number. In this talk, We will prove that if F2nFk + 1

and F2n+2Fk + 1 are both perfect square, then k = 2n + 4, for n ≥ 1 or k = 2n − 2, forn ≥ 2, except when n = 2 case in which we can additionally have k = 1. This talk isbased on a joint paper with B. He and F. Luca.

On representing coordinates of points on elliptic curves by quadratic forms

Maciej Ulas

Jagiellonian University, Krakow, [email protected]

Coauthors: Andrew Bremner

Given an elliptic quartic of type Y 2 = f(X) representing an elliptic curve of positiverank over Q, we investigate the question of when the Y -coordinate can be representedby a quadratic form of type ap2 + bq2. In particular, we give examples of equations ofsurfaces of type c0 + c1x + c2x

2 + c3x3 + c4x

4 = (ap2 + bq2)2, a, b, c ∈ Q where we candeduce the existence of in�nitely many rational points. We also investigate surfaces oftype Y 2 = f(ap2 + bq2) where the polynomial f is of degree 3. Joint work with AndrewBremner.

26

Equal values of combinatorial numbers

Nóra Varga


Coauthors: Lajos Hajdu, Ákos Pintér, Szabolcs Tengely

Let k,m be integers with k ≥ 3 and m ≥ 3 and denote by

fk,m(X) =X(X + 1) · · · (X + k − 2)((m− 2)X + k + 2−m)

k!

the Xth �gurate number with parameters k and m. In this talk some e�ective �nitenessstatements for the general equation

fk,m(x) = f2,n(y) (6)

in integers x and y are presented. Further, it is proved that the unique solution of theequation

fk,k+2(x) = f2,4(y) (7)

in integers k ≥ 5, x ≥ k − 2 and y ≥ 1 is (k, x, y) = (5, 47, 3290). (The talk is based on ajoint work with Lajos Hajdu, Ákos Pintér and Szabolcs Tengely).

Rational points on explicit families of curves

Francesco Veneziano

University of Baselfrancesco.veneziano-at-unibas.ch

Coauthors: Sara Checcoli, Evelina Viada.

I will present a method, of easy application, to compute the rational points on curvesin some products of elliptic curves. We prove some explicit and very sharp estimates forthe height of such rational points. The bounds are so good that we can implement acomputer search. I will present several explicit examples in which this has been done. Allresults are in collaboration with Sara Checcoli and Evelina Viada.

Weakly admissible lattices, primitive lattice points, and Diophantine

approximation

Martin Widmer

Royal Holloway University of London, United [email protected]

After surveying some impressive results of Skriganov on counting lattice points inaligned boxes for weakly admissible lattices we present some new counting results in amore general framework. Our error estimates are inferior to Skriganov's regarding the

27

dependence on the volume of the box but superior regarding the dependence on thelattice. It is this improvement that allows for counting results for primitive lattice points.When time permits we will also discuss applications to Diophantine approximation byprimitive points as studied by Chalk and Erd®s and more recently by Dani, Laurent, andNogueira.

Billiard on the triaxial ellipsoid

Gisbert Wüstholz

ETH Zürich, [email protected]

Coauthors: Ronaldo Garcia

The triaxial ellipsoid seems on the �rst sight an extremely trivial object. Topologicallyit is just a sphere and there is not much to say about it. However at least since Jacobi weknow that this is not the case. Jacobi has shown in his wonderful monograph VorlesungenÜber Dynamik, Gesammelte Werke 8, how rich the mathematics around the ellipsoid is.One of the key object is the pencil of confocal ellipsoids

x2

a− λ+

y2

b− λ+

z2

c− λ= 1

with 0 < a < b < c and λ ∈ R. Jacobi introduced the famous elliptic coordinateswhich are nowadays indispensable for geodesists and earth scientists. It is further of greatimportance for studying the di�erential geometry of the ellipsoid. As an outcome of thesestudies a large variety of elliptic and hyperelliptic curves became visible and with thatthe theory of dynamical and Hamiltonian systems entered. One of the foci were the so-called principal curvature lines which appear as the intersection of two such ellipsoids inthe pencil. Much more complicated were the genus two hyperelliptic curves which areintimately related to the geodesics on the ellipsoid which turn out to have a very strangetopological property. In particular they are lines on the ellipsoid which in general windaround without getting closed in �nite time. In a joint project with Ronaldo Garcia welooked at the problem of characterizing closed geodesics in terms of the geometry of theassociated abelian surface which is the Jacobian of some genus 2 hyperelliptic curve. Toour knowledge so far no curvature line which takes a non-zero angle with respect to oneof the principal curvature lines has been shown to be not closed. It is expected they arenot closed in general so that a billiard ball turning around along these lines never comesback. Both of these problems lead to abelian integrals, in the curvature lines case to anelliptic integral of the third kind and in the geodesic situation to an abelian integral ofthe �rst kind and the questions can be answered by solving two of the problems of Th.Schneider about the transcendence of such integrals. In both cases one has to determinethe structure of the endomorphism algebra of special commutative algebraic groups. Thisis not so di�cult (surprise-surprise!) in the case of geodesics. In the case of the curvature

28

lines the algebraic group is an extension of an elliptic curve by a product of additive andmultiplicative groups and here things become much more complicated.

29

List of participants

1. Cristoph Aistleitner Linz Austria2. András Bazsó Debrecen Hungary3. Michael A. Bennett Vancouver Canada4. Attila Bérczes Debrecen Hungary5. Csanád Bertók Debrecen Hungary6. Nicolas Billerey Aubière France7. Yuri F. Bilu Bordeaux France8. Laura Capuano Pisa Italy9. Kwok Chi Chim Graz Austria10. Andrej Dujella Zagreb Croatia11. Bernadette Faye Dakar Senegal12. Christofer Frei Graz Austria13. Clemens Fuchs Salzburg Austria14. Stevan Gaiovi¢ Belgrade Serbia15. Maciej Gawron Krakow Poland16. Eva Goedhart Northampton USA17. Krisztián Gueth Szombathely Hungary18. Kálmán Gy®ry Debrecen Hungary19. Lajos Hajdu Debrecen Hungary20. Peter Hellekalek Salzburg Austria21. Markus Hittmaier Salzburg Austria22. Christoph Hutle Salzburg Austria23. Christina Karolus Salzburg Austria24. Hidetaka Kitayama Wakayama Japan25. Angelos Koutsianas Warwick United Kingdom26. Dijana Kreso Graz Austria27. Pierre Lezowski Aubière France28. Florian Luca Johannesburg South Africa29. Antoine Marnat Graz Austria30. Takafumi Miyazaki Maebashi Japan31. Filip Najman Zagreb Croatia

30

32. Roland Paulin Salzburg Austria33. Attila Peth® Debrecen Hungary34. István Pink Salzburg/Debrecen Austria/Hungary35. Ákos Pintér Debrecen Hungary36. László Remete Debrecen Hungary37. Adrian Scheerer Graz Austria38. Ivan Soldo Osijek Croatia39. Michael Stoll Bayreuth Germany40. Tímea Szabó Debrecen Hungary41. László Szalay Sopron Hungary42. Márton Szikszai Debrecen Hungary43. Petra Tadi¢ Pula Croatia44. Robert F. Tichy Graz Austria45. Niclas Technau Graz Austria46. Alain Togbé Westville USA47. Maciej Ulas Krakow Poland48. Nóra Varga Debrecen Hungary49. Francesco Veneziano Basel Switzerland50. Martin Widmer London United Kingdom51. Gisbert Wüstholz Zürich Switzerland52. Volker Ziegler Salzburg Austria

31

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34

Restaurants nearby

Zentrum Herrnau

http://www.zentrumherrnau.at/

Alpbenstraÿe 48Mo.-Fr.: 09:00-19:00Sa.: 09:00-18:00SPAR (supermarket)Hofer (supermarket)dm (drugstore)pharmacycash machinewashing

Basic Bio Supermarket

Alpenstraÿe 75Mo.-Fr.: 08:00-19:00Sa.: 08:00-18:00

Merkur Supermarket

Otto-Holzbauer-Straÿe 1 (behind Motel One)Mo.-Thursday: 07:30-19:30Fr.: 07:00-20:00Sa.: 07:00-18:00

Die Shopping Arena

http://www.dieshoppingarena.at/

Alpenstraÿe 107Mo.-Fr.: 09:00-19:00Sa.: 09:00-18:00dm (drugstore)Eurospar (Supermarket)Libro (stationery)Müller (drugstore)cash mashine

35

computational aspects of diophantine equations

Documents