p-adic Analysis Compared with Real

http://dx.doi.org/10.1090/stml/037

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STUDENT MATHEMATICAL LIBRARY Volume 37

p-adic Analysis Compared with Real

Svetlana Katok

American Mathematical Society Mathematics Advanced Study Semesters

Editorial Board Gerald B. Folland Brad Osgood Robin Forman (Chair) Michael S tarb i rd

2000 Mathematics Subject Classification. P r i m a r y 11-01 , 26E30, 12Jxx.

For addi t ional information and upda t e s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s t m l - 3 7

Library of Congress Cataloging-in-Publicat ion D a t a

Katok, Svetlana. P-adic analysis compared with real / Svetlana Katok.

p. cm — (Student mathematical library, ISSN 1520-9121 ; v. 37) Includes bibliographical references and index. ISBN 978-0-8218-4220-1 (alk. paper)

1. p-adic analysis. I. Title. QA241 .K367 2007 512.7'4—dc22 2006047983

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Contents

Foreword: MASS and REU at Penn State University ix

Preface xi

Chapter 1. Arithmetic of the p-adic Numbers 1

§1.1. From Q to R; the concept of completion 2

Exercises 1-8 5

§1.2. Normed fields 6

Exercises 9-16 14

§1.3. Construction of the completion of a normed field 15

Exercises 17-19 19

§1.4. The field of p-adic numbers Qp 19

Exercises 20-25 26

§1.5. Arithmetical operations in Qp 27

Exercises 26-31 30

§1.6. The p-adic expansion of rational numbers 30

Exercises 32-34 33

§1.7. HenseFs Lemma and congruences 33

Exercises 35-44 38

§1.8. Algebraic properties of p-adic integers 39

VI Contents

§1.9. Metrics and norms on the rational numbers.

Ostrowski's Theorem 43

Exercises 45-46 47

§1.10. A digression: what about Q^ if g is not a prime? 47

Exercises 47-50 50

Chapter 2. The Topology of Qp vs. the Topology of E 53

§2.1. Elementary topological properties 53

Exercises 51-53 60

§2.2. Cantor sets 60

Exercises 54-65 68

§2.3. Euclidean models of Zp 69

Exercises 66-68 73

Chapter 3. Elementary Analysis in Qp 75

§3.1. Sequences and series 75

Exercises 69-73 80

§3.2. p-adic power series 80

Exercises 74-78 86

§3.3. Can a p-adic power series be analytically continued? 87

§3.4. Some elementary functions 89

Exercises 79-81 92

§3.5. Further properties of p-adic exponential and

logarithm 92

§3.6. Zeros of p-adic power series 98

Exercises 82-83 102

Chapter 4. p-adic Functions 103

§4.1. Locally constant functions 103

Exercises 84-87 107

§4.2. Continuous and uniformly continuous functions 108

Exercises 88-90 112

§4.3. Points of discontinuity and the Baire Category Theorem 112

Contents VII

Exercises 91-96

§4.4. Differentiability of p-adic functions

§4.5. Isometries of Qp

Exercises 97-100

§4.6. Interpolation

Exercises 101-103

Answers, Hints, and Solutions for Selected Exercises

Bibliography

Index

115

116

121

123

123

134

135

149

151

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Foreword: MASS and REU at Penn State University

This is the second book in the new collection published jointly by the American Mathematical Society and the MASS (Mathematics Advanced Study Semesters) program as a part of the Student Math­ematical Library series. The books in the collection are based on lec­ture notes for advanced undergraduate topics courses taught at the MASS and/or Penn State summer REU (Research Experiences for Undergraduates). Each book presents a self-contained exposition of a nonstandard mathematical topic, often related to current research areas, accessible to undergraduate students familiar with an equiva­lent of two years of standard college mathematics and suitable as a text for an upper division undergraduate course.

Started in 1996, MASS is a semester-long program for advanced undergraduate students from across the USA. The program's curricu­lum amounts to sixteen credit hours. It includes three core courses from the general areas of algebra/number theory, geometry/topology and analysis/dynamical systems, custom designed every year; an in­terdisciplinary seminar; and a special colloquium. In addition, ev­ery participant completes three research projects, one for each core course. The participants are fully immersed into mathematics, and

ix

x Foreword: MASS and REU at Penn State University

this, as well as intensive interaction among the students, usually leads to a dramatic increase in their mathematical enthusiasm and achieve­ment. The program is unique for its kind in the United States.

The summer mathematical REU program is formally indepen­dent of MASS, but there is a significant interaction between the two: about half of the REU participants stay for the MASS semester in the fall. This makes it possible to offer research projects that re­quire more than seven weeks (the length of the REU program) for completion. The summer program includes the MASS Fest, a two to three day conference at the end of the REU at which the partici­pants present their research and that also serves as a MASS alumni reunion. A nonstandard feature of the Penn State REU is that, along with research projects, the participants are taught one or two intense topics courses.

Detailed information about the MASS and REU programs at Penn State can be found on the website www.math.psu.edu/mass.

Preface

This book is a result of the MASS course "Real and p-adic analysis" that I gave in the MASS program in the fall of 2000. The notes were first published in MASS Selecta [12], and a Russian translation of a revised version appeared in [7]. The present text is further revised and expanded.

The choice of the topic was motivated by the internal beauty of the subject of p-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart.

There are several pedagogical advantages of this approach. Both real and p-adic numbers are obtained from the rationals by a proce­dure called completion, which can be applied to any metric space, by using different distances on the rationals: the usual Euclidean dis­tance for the reals and a new p-adic distance for each prime p, for the p-adics. The p-adic distance satisfies the "strong triangle inequality" that causes surprising properties of p-adic numbers and leads to in­teresting deviations from the classical real analysis much like how the renunciation of the fifth postulate of Euclid's Elements, the axiom of parallels, leads to non-Euclidean geometry. Similarities, on the other hand, arise when the fact does not depend on the "strong triangle inequality", and in these cases the same proof works in the real and

xi

Xll Preface

p-adic cases. Analysis of the differences and similarities helps the students to better understand the proofs in both contexts.

The material covered in this book appears in several classical and recent sources [2, 4, 8, 9, 10, 11, 14, 16] but either remains on an elementary level with more emphasis on number theory than on analysis or quickly leads to matters way too sophisticated for the undergraduate students. My only contribution was to choose the ap­propriate material and to present it in the proper context, simplifying the proofs in some cases.

I included several topics from real analysis and elementary topol­ogy which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectiveness of isometries of com­pact metric spaces). They enhanced the students' understanding of real analysis and intertwined the real and p-adic contexts of the course. In [15], a standard reference for real analysis, many of these topics appear only in exercises. Basic algebraic notions are discussed only briefly, and the reader is referred to any standard text in abstract algebra, e.g. [6].

The course entailed a large number of exercises (with emphasis on proofs) which appear in this book. Particularly, some parts of proofs in the main body of the text are relegated to exercises. While it may create some difficulties for uninterrupted reading, it helps students' deeper understanding of the subject and makes this book particu­larly appropriate for self-study. A few harder exercises are marked with a *. Answers, hints, and solutions for most of the exercises are included at the end of the book.

Besides solving the homework exercises, the students in the MASS program were asked to give presentations on additional topics. These presentations, some of which were quite advanced, included the fol­lowing: "The Signum function"; "Euclidean models of the p-adic in­tegers"; "Interpolation series and the p-adic Weierstrass theorem"; "Finite extensions of p-adic numbers and p-adic circles"; "Isometries on the p-adic integers"; "p-adic solenoid"; "The X-adic norm of power series"; "Equiareal triangulations"; "A graphical model of the Peano

Preface xin

curve via the Cantor set"; "Revised harmonic series". The first three of these topics are now included in the book.

Svetlana Katok

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Bibliography

1. R. Bojanic, A simple proof of Mahler's theorem on approximation of functions of a p-adic variable by polynomials, J. Number Th. 6 (1974), 412-415.

2. Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York, 1966.

3. J.W.S. Cassels, Local Fields, Cambridge University Press, Cambridge, 1986.

4. F. Q. Gouvea, p-adic Numbers: An Introduction, Springer-Verlag, Berlin, Heidelberg, New York, Second Edition, Universitext, 2000.

5. H. Hasse, Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

6. I. N. Herstein, Topics in Algebra, 2nd edition, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1975.

7. S. Katok, p-adic analysis in comparison with real, translation into Rus­sian, MCCME Press, Moscow, 2004.

8. A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Func­tional Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1982.

9. A. A. Kirillov, Chto Takoe Chislo?, Sovremennaia Matematika dlia Studentov, Nauka, Moscow, 1993 (in Russian).

10. N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions, Springer-Verlag, Berlin, Heidelberg, New York, Graduate Texts in Mathematics, 1984.

11. K. Mahler, p-adic Numbers and Their Functions, Cambridge University Press, 1973.

149

150 Bibliography

12. MASS Selecta: Teaching and Learning Advanced Undergradute Math­ematics, American Mathematical Society, Providence, 2003.

13. C. Pugh, Real Mathematical Analysis, Springer, Undergraduate Texts in Mathematics, 2002.

14. A. M. Robert, A Course in p-adic Analysis, Springer-Ver lag, Berlin, Heidelberg, New York, 2000.

15. W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill Book Company, New York, 1976.

16. W. H. Schikoff, Ultrametric Calculus, An Introduction to p-adic Anal­ysis, Cambridge Studies in Adv. Math. 4, Cambridge University Press, 1984.

17. J. P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973.

Index

B(a,r), see also open ball B(a, r) , see also closed ball expp, see also p-adic exponential Fv, 113 Gs, 113 lnp, see also p-adic logarithm N, see also natural numbers Q, see also rational numbers Qp, see also p-adic numbers M, see also real numbers Z, see also integers Zp, see also p-adic integers Z* , see also p-adic units

absolute value, 3

Baire Category Theorem, 114 Bolzano-Weierstrass Theorem, 26

canonical p-adic expansion, 24 canonical form of a p-adic number, see

also canonical p-adic expansion Cantor diagonal process, 62 Cantor set, 60

middle thirds, 64 Cantor's Theorem, 56 capacity, 122 closed ball, 4 commutative ring, 6, 39 Completion Theorem, 2 congruent modulo pn, 34 convex hull, 72

derivative, 84 Dirichlet function, 112 distance

induced by the norm, 7 distance function, see also metric

equivalent Cauchy sequences, 2, 16 metrics, 9 norms, 9

Euclidean distance, 3 Euclidean models of Zp, 69 Euclidean norm, see also absolute value Euler's Theorem, 42

fat Cantor set, 64 Fermat's Little Theorem, 43 field, 6

additive group of, 6 multiplicative group of, 6

first difference quotient, 119 function

characteristic, 103 continuous, 63, 103 differentiable, 84 locally constant, 104 oscillation of, 113 strictly differentiable, 119 uniformly continuous, 103

group abelian, 6

Hadamar's formula, 81 Hasse-Minkowski Theorem, 47 Heine-Borel Theorem, 58 Hensel's Lemma, 34 Holder condition, 117 homeomorphism, 63

151

152 Index

ideal, 16, 39 maximal, 39

integers, 12 integral domain, 39 interpolation

coefficient, 126 series, 126

isometry, 63, 95 isomorphic

fields, 47

Legendre symbol, 144 Local-to-Global principle, 46

map continuous, 63 continuous at the point, 63 open, 63 uniformly continuous, 63

Mean Value Theorem, 116 metric, 2

ultra-metric, 11 metric space, 2

compact, 57 complete, 2 completion of, 2 separable, 53 sequentially compact, 57 ultra-metric, 11

natural numbers, 7 nested sequence of sets, 60 norm, 6

Archimedean, 11 Archimedean property of, 12 non-Archimedean, 11 trivial, 7

normed field, 3

open ball, 4 Ostrowski's Theorem, 43

p-adic binomial series, 83 p-adic exponential, 90 p-adic integers, 25 p-adic logarithm, 89 p-adic numbers, 21 p-adic order, 20 p-adic units, 29 p-adic valuation, see also p-adic order power series

formal, 80 radius of convergence of, 81

pseudo-constant, 117 pseudo-norm, 48

rational numbers, 19 real numbers, 2 reduction modulo p, 40 Riemann function, 112 Rollers Theorem, 118 root of unity, 41

primitive, 41

sequence bounded, 8 Cauchy, 2, 8 convergent, 8 null, 8

series convergent, 76 convergent absolutely, 76 convergent unconditionally, 77

set boundary point of, 55 closed, 4 compact, 57 connected, 59 of type Ta, 113 of type 05, 113 open, 4 perfect, 62 sequentially compact, 57

Sierpinski gasket, 71 signum function, 41 sphere in Qp, 54 step function, 105

order of, 105 Strassman's Theorem, 98 strong triangle inequality, 11 strongest wins, 13, see also strong tri­

angle inequality subset

dense, 64 nowhere dense, 64

topological space, 54 connected, 59 disconnected, 59 locally compact, 53 totally disconnected, 59

topology, 54 induced, 54

total order, 51 triangle inequality, 6

strong, 11

ultra-metric space, 11

Weierstrass Approximation Theorem, 132

quadratic nonresidue modulo p, 37 quadratic residue modulo p, 37

zero divisors, 6

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