Number theory volume II analytic and modern tools

Modular Functions and Dirichlet Series in Number Theory.. Thecentral although not unique theme is the solution of Diophantine equa-tions, i.e., equations or systems of polynomial equatio

Graduate Texts in Mathematics 240

Editorial Board

S AxlerK.A Ribet

Graduate Texts in Mathematics

1 T AKEUTI /Z ARING Introduction to

Axiomatic Set Theory 2nd ed.

2 O XTOBY Measure and Category 2nd ed.

3 S CHAEFER Topological Vector Spaces.

2nd ed.

4 H ILTON /S TAMMBACH A Course in

Homological Algebra 2nd ed.

5 M AC L ANE Categories for the Working

Mathematician 2nd ed.

6 H UGHES /P IPER Projective Planes.

7 J.-P S ERRE A Course in Arithmetic.

8 T AKEUTI /Z ARING Axiomatic Set Theory.

9 H UMPHREYS Introduction to Lie

Algebras and Representation Theory.

10 C OHEN A Course in Simple Homotopy

Theory.

11 C ONWAY Functions of One Complex

Variable I 2nd ed.

12 B EALS Advanced Mathematical Analysis.

13 A NDERSON /F ULLER Rings and

Categories of Modules 2nd ed.

14 G OLUBITSKY /G UILLEMIN Stable

Mappings and Their Singularities.

15 B ERBERIAN Lectures in Functional

Analysis and Operator Theory.

16 W INTER The Structure of Fields.

17 R OSENBLATT Random Processes 2nd ed.

18 H ALMOS Measure Theory.

19 H ALMOS A Hilbert Space Problem

Book 2nd ed.

20 H USEMOLLER Fibre Bundles 3rd ed.

21 H UMPHREYS Linear Algebraic Groups.

22 B ARNES /M ACK An Algebraic

Introduction to Mathematical Logic.

23 G REUB Linear Algebra 4th ed.

24 H OLMES Geometric Functional

Analysis and Its Applications.

25 H EWITT /S TROMBERG Real and Abstract

Analysis.

26 M ANES Algebraic Theories.

27 K ELLEY General Topology.

28 Z ARISKI /S AMUEL Commutative

Algebra Vol I.

29 Z ARISKI /S AMUEL Commutative

Algebra Vol II.

30 J ACOBSON Lectures in Abstract Algebra

I Basic Concepts.

31 J ACOBSON Lectures in Abstract Algebra

II Linear Algebra.

32 J ACOBSON Lectures in Abstract Algebra

III Theory of Fields and Galois

Theory.

33 H IRSCH Differential Topology.

34 S PITZER Principles of Random Walk 2nd ed.

35 A LEXANDER /W ERMER Several Complex Variables and Banach Algebras 3rd ed.

36 K ELLEY /N AMIOKA et al Linear Topological Spaces.

37 M ONK Mathematical Logic.

38 G RAUERT /F RITZSCHE Several Complex Variables.

39 A RVESON An Invitation to C* -Algebras.

40 K EMENY /S NELL /K NAPP Denumerable Markov Chains 2nd ed.

41 A POSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed.

42 J.-P S ERRE Linear Representations of Finite Groups.

43 G ILLMAN /J ERISON Rings of Continuous Functions.

44 K ENDIG Elementary Algebraic Geometry.

45 L OÈVE Probability Theory I 4th ed.

46 L OÈVE Probability Theory II 4th ed.

47 M OISE Geometric Topology in Dimensions 2 and 3.

48 S ACHS /W U General Relativity for Mathematicians.

49 G RUENBERG /W EIR Linear Geometry 2nd ed.

50 E DWARDS Fermat's Last Theorem.

51 K LINGENBERG A Course in Differential Geometry.

52 H ARTSHORNE Algebraic Geometry.

53 M ANIN A Course in Mathematical Logic.

54 G RAVER /W ATKINS Combinatorics with Emphasis on the Theory of Graphs.

55 B ROWN /P EARCY Introduction to Operator Theory I: Elements of Functional Analysis.

56 M ASSEY Algebraic Topology: An Introduction.

57 C ROWELL /F OX Introduction to Knot Theory.

58 K OBLITZ p-adic Numbers, p-adic

Analysis, and Zeta-Functions 2nd ed.

59 L ANG Cyclotomic Fields.

60 A RNOLD Mathematical Methods in Classical Mechanics 2nd ed.

61 W HITEHEAD Elements of Homotopy Theory.

62 K ARGAPOLOV /M ERIZJAKOV Fundamentals of the Theory of Groups.

63 B OLLOBAS Graph Theory.

(continued after index)

Henri Cohen

Number Theory Volume II:

Analytic and Modern Tools

Mathematics Department Mathematics Department

San Francisco State University University of California at BerkeleySan Francisco, CA 94132 Berkeley, CA 94720-3840

axler@sfsu.edu ribet@math.berkeley.edu

Mathematics Subject Classification (2000): 11-xx 11-01 11Dxx 11Rxx 11Sxx

Library of Congress Control Number: 2007925737

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This book deals with several aspects of what is now called “explicit numbertheory,” not including the essential algorithmic aspects, which are for themost part covered by two other books of the author [Coh0] and [Coh1] Thecentral (although not unique) theme is the solution of Diophantine equa-tions, i.e., equations or systems of polynomial equations that must be solved

in integers, rational numbers, or more generally in algebraic numbers Thistheme is in particular the central motivation for the modern theory of arith-metic algebraic geometry We will consider it through three of its most basicaspects

The first is the local aspect: the invention of p-adic numbers and their

generalizations by K Hensel was a major breakthrough, enabling in particularthe simultaneous treatment of congruences modulo prime powers But more

importantly, one can do analysis in p-adic fields, and this goes much further than the simple definition of p-adic numbers The local study of equations

is usually not very difficult We start by looking at solutions in finite fields,

where important theorems such as the Weil bounds and Deligne’s theorem

on the Weil conjectures come into play We then lift these solutions to local solutions using Hensel lifting.

The second aspect is the global aspect: the use of number fields, and

in particular of class groups and unit groups Although local considerationscan give a considerable amount of information on Diophantine problems,the “local-to-global” principles are unfortunately rather rare, and we willsee many examples of failure Concerning the global aspect, we will firstrequire as a prerequisite of the reader that he or she be familiar with thestandard basic theory of number fields, up to and including the finiteness ofthe class group and Dirichlet’s structure theorem for the unit group This can

be found in many textbooks such as [Sam] and [Marc] Second, and this isless standard, we will always assume that we have at our disposal a computeralgebra system (CAS) that is able to compute rings of integers, class and unitgroups, generators of principal ideals, and related objects Such CAS are nowvery common, for instance Kash, magma, and Pari/GP, to cite the most useful

in algebraic number theory

The third aspect is the theory of zeta and L-functions This can be ered a unifying theme3 for the whole subject, and it embodies in a beautifulway the local and global aspects of Diophantine problems Indeed, these func-tions are defined through the local aspects of the problems, but their analyticbehavior is intimately linked to the global aspects A first example is given bythe Dedekind zeta function of a number field, which is defined only through

consid-the splitting behavior of consid-the primes, but whose leading term at s = 0 contains

at the same time explicit information on the unit rank, the class number, theregulator, and the number of roots of unity of the number field A secondvery important example, which is one of the most beautiful and importantconjectures in the whole of number theory (and perhaps of the whole of math-ematics), the Birch and Swinnerton-Dyer conjecture, says that the behavior

at s = 1 of the L-function of an elliptic curve defined overQ contains at thesame time explicit information on the rank of the group of rational points

on the curve, on the regulator, and on the order of the torsion group of thegroup of rational points, in complete analogy with the case of the Dedekind

zeta function In addition to the purely analytical problems, the theory of

L-functions contains beautiful results (and conjectures) on special values, of

which Euler’s formula

n11/n2= π2/6 is a special case.

This book can be considered as having four main parts The first part givesthe tools necessary for Diophantine problems: equations over finite fields,number fields, and finally local fields such as p-adic fields (Chapters 1, 2, 3,

4, and part of Chapter 5) The emphasis will be mainly on the theory ofp-adic fields (Chapter 4), since the reader probably has less familiarity withthese Note that we will consider function fields only in Chapter 7, as a toolfor proving Hasse’s theorem on elliptic curves An important tool that we willintroduce at the end of Chapter 3 is the theory of the Stickelberger ideal overcyclotomic fields, together with the important applications to the Eisensteinreciprocity law, and the Davenport–Hasse relations Through Eisenstein reci-procity this theory will enable us to prove Wieferich’s criterion for the firstcase of Fermat’s last theorem (FLT), and it will also be an essential tool inthe proof of Catalan’s conjecture given in Chapter 16

The second part is a study of certain basic Diophantine equations orsystems of equations (Chapters 5, 6, 7, and 8) It should be stressed thateven though a number of general techniques are available, each Diophantineequation poses a new problem, and it is difficult to know in advance whether

it will be easy to solve Even without mentioning families of Diophantine

equations such as FLT, the congruent number problem, or Catalan’s equation,all of which will be stated below, proving for instance that a specific equation

such as x3+ y5= z7 with x, y coprime integers has no solution with xyz = 0

seems presently out of reach, although it has been proved (based on a deeptheorem of Faltings) that there are only finitely many solutions; see [Dar-Gra]

3 Expression due to Don Zagier

and Chapter 14 Note also that it has been shown by Yu Matiyasevich (after

a considerable amount of work by other authors) in answer to Hilbert’s tenthproblem that there cannot exist a general algorithm for solving Diophantineequations

The third part (Chapters 9, 10, and 11) deals with the detailed study

of analytic objects linked to algebraic number theory: Bernoulli

polynomi-als and numbers, the gamma function, and zeta and L-functions of Dirichlet characters, which are the simplest types of L-functions In Chapter 11 we also study p-adic analogues of the gamma, zeta, and L-functions, which have

come to play an important role in number theory, and in particular the Gross–

Koblitz formula for Morita’s p-adic gamma function In particular, we will

see that this formula leads to remarkably simple proofs of Stickelberger’s

con-gruence and the Hasse–Davenport product relation More general L-functions such as Hecke L-functions for Gr¨ ossencharacters, Artin L-functions for Galois representations, or L-functions attached to modular forms, elliptic curves, or

higher-dimensional objects are mentioned in several places, but a systematicexposition of their properties would be beyond the scope of this book.Much more sophisticated techniques have been brought to bear on thesubject of Diophantine equations, and it is impossible to be exhaustive Be-cause the author is not an expert in most of these techniques, they are notstudied in the first three parts of the book However, considering their impor-tance, I have asked a number of much more knowledgeable people to write

a few chapters on these techniques, and I have written two myself, and thisforms the fourth and last part of the book (Chapters 12 to 16) These chap-ters have a different flavor from the rest of the book: they are in general notself-contained, are of a higher mathematical sophistication than the rest, andusually have no exercises Chapter 12, written by Yann Bugeaud, GuillaumeHanrot, and Maurice Mignotte, deals with the applications of Baker’s explicitresults on linear forms in logarithms of algebraic numbers, which permit thesolution of a large class of Diophantine equations such as Thue equationsand norm form equations, and includes some recent spectacular successes.Paradoxically, the similar problems on elliptic curves are considerably lesstechnical, and are studied in detail in Section 8.7 Chapter 13, written bySylvain Duquesne, deals with the search for rational points on curves of genusgreater than or equal to 2, restricting for simplicity to the case of hyperellipticcurves of genus 2 (the case of genus 0—in other words, of quadratic forms—istreated in Chapters 5 and 6, and the case of genus 1, essentially of ellipticcurves, is treated in Chapters 7 and 8) Chapter 14, written by the author,

deals with the so-called super-Fermat equation x p + y q = z r, on which severalmethods have been used, including ordinary algebraic number theory, classi-cal invariant theory, rational points on higher genus curves, and Ribet–Wilestype methods The only proofs that are included are those coming from alge-braic number theory Chapter 15, written by Samir Siksek, deals with the use

of Galois representations, and in particular of Ribet’s level-lowering theorem

and Wiles’s and Taylor–Wiles’s theorem proving the modularity conjecture.The main application is to equations of “abc” type, in other words, equations

of the form a + b + c = 0 with a, b, and c highly composite, the “easiest”

application of this method being the proof of FLT The author of this chapterhas tried to hide all the sophisticated mathematics and to present the method

as a black box that can be used without completely understanding the derlying theory Finally, Chapter 16, also written by the author, gives thecomplete proof of Catalan’s conjecture by P Mih˘ailescu It is entirely based

un-on notes of Yu Bilu, R Schoof, and especially of J Bo´echat and M Mischler,and the only reason that it is not self-contained is that it will be necessary toassume the validity of an important theorem of F Thaine on the annihilator

of the plus part of the class group of cyclotomic fields

Warnings

Since mathematical conventions and notation are not the same from onemathematical culture to the next, I have decided to use systematically un-ambiguous terminology, and when the notations clash, the French notation.Here are the most important:

– We will systematically say that a is strictly greater than b, or greater than

or equal to b (or b is strictly less than a, or less than or equal to a), although the English terminology a is greater than b means in fact one of the two

(I don’t remember which one, and that is one of the main reasons I refuse

to use it) and the French terminology means the other Similarly, positiveand negative are ambiguous (does it include the number 0)? Even though

the expression “x is nonnegative” is slightly ambiguous, it is useful, and I

will allow myself to use it, with the meaning x 0

– Although we will almost never deal with noncommutative fields (which is

a contradiction in terms since in principle the word field implies tativity), we will usually not use the word field alone Either we will writeexplicitly commutative (or noncommutative) field, or we will deal with spe-cific classes of fields, such as finite fields, p-adic fields, local fields, numberfields, etc., for which commutativity is clear Note that the “proper” way

commu-in English-language texts to talk about noncommutative fields is to callthem either skew fields or division algebras In any case this will not be anissue since the only appearances of skew fields will be in Chapter 2, where

we will prove that finite division algebras are commutative, and in Chapter

7 about endomorphism rings of elliptic curves over finite fields

– The GCD (respectively the LCM) of two integers can be denoted by (a, b) (respectively by [a, b]), but to avoid ambiguities, I will systematically use the explicit notation gcd(a, b) (respectively lcm(a, b)), and similarly when

more than two integers are involved

– An open interval with endpoints a and b is denoted by (a, b) in the glish literature, and by ]a, b[ in the French literature I will use the French notation, and similarly for half-open intervals (a, b] and [a, b), which I will denote by ]a, b] and [a, b[ Although it is impossible to change such a well-

En-entrenched notation, I urge my English-speaking readers to realize the

dreadful ambiguity of the notation (a, b), which can mean either the dered pair (a, b), the GCD of a and b, the inner product of a and b, or the

or-open interval

– The trigonometric functions sec(x) and csc(x) do not exist in France, so

I will not use them The functions tan(x), cot(x), cosh(x), sinh(x), and tanh(x) are denoted respectively by tg(x), cotg(x), ch(x), sh(x), and th(x)

in France, but for once to bow to the majority I will use the English names

– (s) and (s) denote the real and imaginary parts of the complex number

s, the typography coming from the standard TEX macros.

Notation

In addition to the standard notation of number theory we will use the lowing notation

fol-– We will often use the practical self-explanatory notation Z>0, Z0, Z<0,

Z0, and generalizations thereof, which avoid using excessive verbiage Onthe other hand, I prefer not to use the notationN (for Z0, or is itZ>0?)

– If a and b are nonzero integers, we write gcd(a, b ∞) for the limit of the

ultimately constant sequence gcd(a, b n ) as n → ∞ We have of course

gcd(a, b ∞) =

p |gcd(a,b) p v p (a) , and a/ gcd(a, b ∞ ) is the largest divisor of a coprime to b.

– If n is a nonzero integer and d | n, we write dn if gcd(d, n/d) = 1 Note

that this is not the same thing as the condition d2 n, except if d is prime.

– Capital italic letters such as K and L will usually denote number fields.

– Capital calligraphic letters such asK and L will denote general p-adic fields

(for specific ones, we write for instance Kp)

– Letters such asE and F will always denote finite fields

– The letterZ indexed by a capital italic or calligraphic letter such as ZK,

ZL, ZK, etc., will always denote the ring of integers of the correspondingfield

– Capital italic letters such as A, B, C, G, H, S, T , U , V , W , or lowercase italic letters such as f , g, h, will usually denote polynomials or formal power series with coefficients in some base ring or field The coefficient of degree m

of these polynomials or power series will be denoted by the corresponding

letter indexed by m, such as A m , B m, etc Thus we will always write (for

instance) A(X) = A d X d + A d −1 X d −1+· · ·+A0, so that the ith elementary

symmetric function of the roots is equal to (−1) i A d −i /A d

Acknowledgments

A large part of the material on local fields has been taken with little changefrom the remarkable book by Cassels [Cas1], and also from unpublished notes

of Jaulent written in 1994 For p-adic analysis, I have also liberally borrowed

from work of Robert, in particular his superb GTM volume [Rob1] For part ofthe material on elliptic curves I have borrowed from another excellent book byCassels [Cas2], as well as the treatises of Cremona and Silverman [Cre2], [Sil1],[Sil2], and the introductory book by Silverman–Tate [Sil-Tat] I have alsoborrowed from the classical books by Borevich–Shafarevich [Bor-Sha], Serre[Ser1], Ireland–Rosen [Ire-Ros], and Washington [Was] I would like to thank

my former students K Belabas, C Delaunay, S Duquesne, and D Simon,who have helped me to write specific sections, and my colleagues J.-F Jaulentand J Martinet for answering many questions in algebraic number theory Iwould also like to thank M Bennett, J Cremona, A Kraus, and F Rodriguez-Villegas for valuable comments on parts of this book I would especially like

to thank D Bernardi for his thorough rereading of the first ten chapters

of the manuscript, which enabled me to remove a large number of errors,mathematical or otherwise Finally, I would like to thank my copyeditor,who was very helpful and who did an absolutely remarkable job

It is unavoidable that there still remain errors, typographical or otherwise,and the author would like to hear about them Please send e-mail to

Henri.Cohen@math.u-bordeaux1.frLists of known errors for the author’s books including the present one can

be obtained on the author’s home page at the URL

http://www.math.u-bordeaux1.fr/~cohen/

Table of Contents

Volume I

Preface . v

1 Introduction to Diophantine Equations . 1

1.1 Introduction 1

1.1.1 Examples of Diophantine Problems 1

1.1.2 Local Methods 4

1.1.3 Dimensions 6

1.2 Exercises for Chapter 1 8

Part I Tools 2 Abelian Groups, Lattices, and Finite Fields 11

2.1 Finitely Generated Abelian Groups 11

2.1.1 Basic Results 11

2.1.2 Description of Subgroups 16

2.1.3 Characters of Finite Abelian Groups 17

2.1.4 The Groups (Z/mZ) ∗ 20

2.1.5 Dirichlet Characters 25

2.1.6 Gauss Sums 30

2.2 The Quadratic Reciprocity Law 33

2.2.1 The Basic Quadratic Reciprocity Law 33

2.2.2 Consequences of the Basic Quadratic Reciprocity Law 36 2.2.3 Gauss’s Lemma and Quadratic Reciprocity 39

2.2.4 Real Primitive Characters 43

2.2.5 The Sign of the Quadratic Gauss Sum 45

2.3 Lattices and the Geometry of Numbers 50

2.3.1 Definitions 50

2.3.2 Hermite’s Inequality 53

2.3.3 LLL-Reduced Bases 55

2.3.4 The LLL Algorithms 58

2.3.5 Approximation of Linear Forms 60

2.3.6 Minkowski’s Convex Body Theorem 63

2.4 Basic Properties of Finite Fields 65

2.4.1 General Properties of Finite Fields 65

2.4.2 Galois Theory of Finite Fields 69

2.4.3 Polynomials over Finite Fields 71

2.5 Bounds for the Number of Solutions in Finite Fields 72

2.5.1 The Chevalley–Warning Theorem 72

2.5.2 Gauss Sums for Finite Fields 73

2.5.3 Jacobi Sums for Finite Fields 79

2.5.4 The Jacobi Sums J (χ1, χ2) 82

2.5.5 The Number of Solutions of Diagonal Equations 87

2.5.6 The Weil Bounds 90

2.5.7 The Weil Conjectures (Deligne’s Theorem) 92

2.6 Exercises for Chapter 2 93

3 Basic Algebraic Number Theory 101

3.1 Field-Theoretic Algebraic Number Theory 101

3.1.1 Galois Theory 101

3.1.2 Number Fields 106

3.1.3 Examples 108

3.1.4 Characteristic Polynomial, Norm, Trace 109

3.1.5 Noether’s Lemma 110

3.1.6 The Basic Theorem of Kummer Theory 111

3.1.7 Examples of the Use of Kummer Theory 114

3.1.8 Artin–Schreier Theory 115

3.2 The Normal Basis Theorem 117

3.2.1 Linear Independence and Hilbert’s Theorem 90 117

3.2.2 The Normal Basis Theorem in the Cyclic Case 119

3.2.3 Additive Polynomials 120

3.2.4 Algebraic Independence of Homomorphisms 121

3.2.5 The Normal Basis Theorem 123

3.3 Ring-Theoretic Algebraic Number Theory 124

3.3.1 Gauss’s Lemma on Polynomials 124

3.3.2 Algebraic Integers 125

3.3.3 Ring of Integers and Discriminant 128

3.3.4 Ideals and Units 130

3.3.5 Decomposition of Primes and Ramification 132

3.3.6 Galois Properties of Prime Decomposition 134

3.4 Quadratic Fields 136

3.4.1 Field-Theoretic and Basic Ring-Theoretic Properties 136

3.4.2 Results and Conjectures on Class and Unit Groups 138

3.5 Cyclotomic Fields 140

3.5.1 Cyclotomic Polynomials 140

3.5.2 Field-Theoretic Properties ofQ(ζ n) 144

3.5.3 Ring-Theoretic Properties 146

3.5.4 The Totally Real Subfield ofQ(ζ ) 148

3.6 Stickelberger’s Theorem 150

3.6.1 Introduction and Algebraic Setting 150

3.6.2 Instantiation of Gauss Sums 151

3.6.3 Prime Ideal Decomposition of Gauss Sums 154

3.6.4 The Stickelberger Ideal 160

3.6.5 Diagonalization of the Stickelberger Element 163

3.6.6 The Eisenstein Reciprocity Law 165

3.7 The Hasse–Davenport Relations 170

3.7.1 Distribution Formulas 171

3.7.2 The Hasse–Davenport Relations 173

3.7.3 The Zeta Function of a Diagonal Hypersurface 177

3.8 Exercises for Chapter 3 179

4. p-adic Fields 183

4.1 Absolute Values and Completions 183

4.1.1 Absolute Values 183

4.1.2 Archimedean Absolute Values 184

4.1.3 Non-Archimedean and Ultrametric Absolute Values 188

4.1.4 Ostrowski’s Theorem and the Product Formula 190

4.1.5 Completions 192

4.1.6 Completions of a Number Field 195

4.1.7 Hensel’s Lemmas 199

4.2 Analytic Functions in p-adic Fields 205

4.2.1 Elementary Properties 205

4.2.2 Examples of Analytic Functions 208

4.2.3 Application of the Artin–Hasse Exponential 217

4.2.4 Mahler Expansions 220

4.3 Additive and Multiplicative Structures 224

4.3.1 Concrete Approach 224

4.3.2 Basic Reductions 225

4.3.3 Study of the Groups U i 229

4.3.4 Study of the Group U1 231

4.3.5 The Group K ∗ /K ∗2 234

4.4 Extensions of p-adic Fields 235

4.4.1 Preliminaries on Local Field Norms 235

4.4.2 Krasner’s Lemma 238

4.4.3 General Results on Extensions 239

4.4.4 Applications of the Cohomology of Cyclic Groups 242

4.4.5 Characterization of Unramified Extensions 249

4.4.6 Properties of Unramified Extensions 251

4.4.7 Totally Ramified Extensions 253

4.4.8 Analytic Representations of pth Roots of Unity 254

4.4.9 Factorizations in Number Fields 258

4.4.10 Existence of the Field Cp 260

4.4.11 Some Analysis inC 263

4.5 The Theorems of Strassmann and Weierstrass 266

4.5.1 Strassmann’s Theorem 266

4.5.2 Krasner Analytic Functions 267

4.5.3 The Weierstrass Preparation Theorem 270

4.5.4 Applications of Strassmann’s Theorem 272

4.6 Exercises for Chapter 4 275

5 Quadratic Forms and Local–Global Principles 285

5.1 Basic Results on Quadratic Forms 285

5.1.1 Basic Properties of Quadratic Modules 286

5.1.2 Contiguous Bases and Witt’s Theorem 288

5.1.3 Translations into Results on Quadratic Forms 291

5.2 Quadratic Forms over Finite and Local Fields 294

5.2.1 Quadratic Forms over Finite Fields 294

5.2.2 Definition of the Local Hilbert Symbol 295

5.2.3 Main Properties of the Local Hilbert Symbol 296

5.2.4 Quadratic Forms overQp 300

5.3 Quadratic Forms overQ 303

5.3.1 Global Properties of the Hilbert Symbol 303

5.3.2 Statement of the Hasse–Minkowski Theorem 305

5.3.3 The Hasse–Minkowski Theorem for n 2 306

5.3.4 The Hasse–Minkowski Theorem for n = 3 307

5.3.5 The Hasse–Minkowski Theorem for n = 4 308

5.3.6 The Hasse–Minkowski Theorem for n 5 309

5.4 Consequences of the Hasse–Minkowski Theorem 310

5.4.1 General Results 310

5.4.2 A Result of Davenport and Cassels 311

5.4.3 Universal Quadratic Forms 312

5.4.4 Sums of Squares 314

5.5 The Hasse Norm Principle 318

5.6 The Hasse Principle for Powers 321

5.6.1 A General Theorem on Powers 321

5.6.2 The Hasse Principle for Powers 324

5.7 Some Counterexamples to the Hasse Principle 326

5.8 Exercises for Chapter 5 329

Part II Diophantine Equations 6 Some Diophantine Equations 335

6.1 Introduction 335

6.1.1 The Use of Finite Fields 335

6.1.2 Local Methods 337

6.1.3 Global Methods 337

6.2 Diophantine Equations of Degree 1 339

6.3 Diophantine Equations of Degree 2 341

6.3.1 The General Homogeneous Equation 341

6.3.2 The Homogeneous Ternary Quadratic Equation 343

6.3.3 Computing a Particular Solution 347

6.3.4 Examples of Homogeneous Ternary Equations 352

6.3.5 The Pell–Fermat Equation x2− Dy2= N 354

6.4 Diophantine Equations of Degree 3 357

6.4.1 Introduction 358

6.4.2 The Equation ax p + by p + cz p= 0: Local Solubility 359

6.4.3 The Equation ax p + by p + cz p= 0: Number Fields 362

6.4.4 The Equation ax p + by p + cz p= 0: Hyperelliptic Curves 368

6.4.5 The Equation x3+ y3+ cz3= 0 373

6.4.6 Sums of Two or More Cubes 376

6.4.7 Skolem’s Equations x3+ dy3= 1 385

6.4.8 Special Cases of Skolem’s Equations 386

6.4.9 The Equations y2= x3± 1 in Rational Numbers 387

6.5 The Equations ax4+ by4+ cz2= 0 and ax6+ by3+ cz2= 0 389 6.5.1 The Equation ax4+ by4+ cz2= 0: Local Solubility 389

6.5.2 The Equations x4± y4= z2 and x4+ 2y4= z2 391

6.5.3 The Equation ax4+ by4+ cz2= 0: Elliptic Curves 392

6.5.4 The Equation ax4+ by4+ cz2= 0: Special Cases 393

6.5.5 The Equation ax6+ by3+ cz2= 0 396

6.6 The Fermat Quartics x4+ y4= cz4 397

6.6.1 Local Solubility 398

6.6.2 Global Solubility: Factoring over Number Fields 400

6.6.3 Global Solubility: Coverings of Elliptic Curves 407

6.6.4 Conclusion, and a Small Table 409

6.7 The Equation y2= x n + t 410

6.7.1 General Results 411

6.7.2 The Case p = 3 414

6.7.3 The Case p = 5 416

6.7.4 Application of the Bilu–Hanrot–Voutier Theorem 417

6.7.5 Special Cases with Fixed t 418

6.7.6 The Equations ty2+ 1 = 4x p and y2+ y + 1 = 3x p 420

6.8 Linear Recurring Sequences 421

6.8.1 Squares in the Fibonacci and Lucas Sequences 421

6.8.2 The Square Pyramid Problem 424

6.9 Fermat’s “Last Theorem” x n + y n = z n 427

6.9.1 Introduction 427

6.9.2 General Prime n: The First Case 428

6.9.3 Congruence Criteria 429

6.9.4 The Criteria of Wendt and Germain 430

6.9.5 Kummer’s Criterion: Regular Primes 431

6.9.6 The Criteria of Furtw¨angler and Wieferich 434

6.9.7 General Prime n: The Second Case 435

6.10 An Example of Runge’s Method 439

6.11 First Results on Catalan’s Equation 442

6.11.1 Introduction 442

6.11.2 The Theorems of Nagell and Ko Chao 444

6.11.3 Some Lemmas on Binomial Series 446

6.11.4 Proof of Cassels’s Theorem 6.11.5 447

6.12 Congruent Numbers 450

6.12.1 Reduction to an Elliptic Curve 451

6.12.2 The Use of the Birch and Swinnerton-Dyer Conjecture 452 6.12.3 Tunnell’s Theorem 453

6.13 Some Unsolved Diophantine Problems 455

6.14 Exercises for Chapter 6 456

7 Elliptic Curves 465

7.1 Introduction and Definitions 465

7.1.1 Introduction 465

7.1.2 Weierstrass Equations 465

7.1.3 Degenerate Elliptic Curves 467

7.1.4 The Group Law 470

7.1.5 Isogenies 472

7.2 Transformations into Weierstrass Form 474

7.2.1 Statement of the Problem 474

7.2.2 Transformation of the Intersection of Two Quadrics 475

7.2.3 Transformation of a Hyperelliptic Quartic 476

7.2.4 Transformation of a General Nonsingular Cubic 477

7.2.5 Example: The Diophantine Equation x2+ y4= 2z4 480

7.3 Elliptic Curves over C, R, k(T ), F q , and Kp 482

7.3.1 Elliptic Curves overC 482

7.3.2 Elliptic Curves overR 484

7.3.3 Elliptic Curves over k(T ) 486

7.3.4 Elliptic Curves overFq 494

7.3.5 Constant Elliptic Curves over R[[T ]]: Formal Groups 500

7.3.6 Reduction of Elliptic Curves over Kp 505

7.3.7 The p-adic Filtration for Elliptic Curves over Kp 507

7.4 Exercises for Chapter 7 512

8 Diophantine Aspects of Elliptic Curves 517

8.1 Elliptic Curves over Q 517

8.1.1 Introduction 517

8.1.2 Basic Results and Conjectures 518

8.1.3 Computing the Torsion Subgroup 524

8.1.4 Computing the Mordell–Weil Group 528

8.1.5 The Na¨ıve and Canonical Heights 529

8.2 Description of 2-Descent with Rational 2-Torsion 532

8.2.1 The Fundamental 2-Isogeny 532

8.2.2 Description of the Image of φ 534

8.2.3 The Fundamental 2-Descent Map 535

8.2.4 Practical Use of 2-Descent with 2-Isogenies 538

8.2.5 Examples of 2-Descent using 2-Isogenies 542

8.2.6 An Example of Second Descent 546

8.3 Description of General 2-Descent 548

8.3.1 The Fundamental 2-Descent Map 548

8.3.2 The T -Selmer Group of a Number Field 550

8.3.3 Description of the Image of α 552

8.3.4 Practical Use of 2-Descent in the General Case 554

8.3.5 Examples of General 2-Descent 555

8.4 Description of 3-Descent with Rational 3-Torsion Subgroup 557

8.4.1 Rational 3-Torsion Subgroups 557

8.4.2 The Fundamental 3-Isogeny 558

8.4.3 Description of the Image of φ 560

8.4.4 The Fundamental 3-Descent Map 563

8.5 The Use of L(E, s) 564

8.5.1 Introduction 564

8.5.2 The Case of Complex Multiplication 565

8.5.3 Numerical Computation of L (r) (E, 1) 572

8.5.4 Computation of Γr (1, x) for Small x 575

8.5.5 Computation of Γr (1, x) for Large x 580

8.5.6 The Famous Curve y2+ y = x3− 7x + 6 582

8.6 The Heegner Point Method 584

8.6.1 Introduction and the Modular Parametrization 584

8.6.2 Heegner Points and Complex Multiplication 586

8.6.3 The Use of the Theorem of Gross–Zagier 589

8.6.4 Practical Use of the Heegner Point Method 591

8.6.5 Improvements to the Basic Algorithm, in Brief 596

8.6.6 A Complete Example 598

8.7 Computation of Integral Points 600

8.7.1 Introduction 600

8.7.2 An Upper Bound for the Elliptic Logarithm on E(Z) 601 8.7.3 Lower Bounds for Linear Forms in Elliptic Logarithms 603 8.7.4 A Complete Example 605

8.8 Exercises for Chapter 8 607

Bibliography 615

Index of Notation 625

Index of Names 633

General Index 639

Volume II

Preface . v

Part III Analytic Tools 9 Bernoulli Polynomials and the Gamma Function . 3

9.1 Bernoulli Numbers and Polynomials 3

9.1.1 Generating Functions for Bernoulli Polynomials 3

9.1.2 Further Recurrences for Bernoulli Polynomials 10

9.1.3 Computing a Single Bernoulli Number 14

9.1.4 Bernoulli Polynomials and Fourier Series 16

9.2 Analytic Applications of Bernoulli Polynomials 19

9.2.1 Asymptotic Expansions 19

9.2.2 The Euler–MacLaurin Summation Formula 21

9.2.3 The Remainder Term and the Constant Term 25

9.2.4 Euler–MacLaurin and the Laplace Transform 27

9.2.5 Basic Applications of the Euler–MacLaurin Formula 31

9.3 Applications to Numerical Integration 35

9.3.1 Standard Euler–MacLaurin Numerical Integration 36

9.3.2 The Basic Tanh-Sinh Numerical Integration Method 37

9.3.3 General Doubly Exponential Numerical Integration 39

9.4 χ-Bernoulli Numbers, Polynomials, and Functions 43

9.4.1 χ-Bernoulli Numbers and Polynomials 43

9.4.2 χ-Bernoulli Functions 46

9.4.3 The χ-Euler–MacLaurin Summation Formula 50

9.5 Arithmetic Properties of Bernoulli Numbers 52

9.5.1 χ-Power Sums 52

9.5.2 The Generalized Clausen–von Staudt Congruence 61

9.5.3 The Voronoi Congruence 64

9.5.4 The Kummer Congruences 67

9.5.5 The Almkvist–Meurman Theorem 70

9.6 The Real and Complex Gamma Functions 71

9.6.1 The Hurwitz Zeta Function 71

9.6.2 Definition of the Gamma Function 77

9.6.3 Preliminary Results for the Study of Γ(s) 81

9.6.4 Properties of the Gamma Function 84

9.6.5 Specific Properties of the Function ψ(s) 95

9.6.6 Fourier Expansions of ζ(s, x) and log(Γ(x)) 100

9.7 Integral Transforms 103

9.7.1 Generalities on Integral Transforms 104

9.7.2 The Fourier Transform 104

9.7.3 The Mellin Transform 107

9.7.4 The Laplace Transform 108

9.8 Bessel Functions 109

9.8.1 Definitions 109

9.8.2 Integral Representations and Applications 113

9.9 Exercises for Chapter 9 118

10 Dirichlet Series andL-Functions 151

10.1 Arithmetic Functions and Dirichlet Series 151

10.1.1 Operations on Arithmetic Functions 152

10.1.2 Multiplicative Functions 154

10.1.3 Some Classical Arithmetical Functions 155

10.1.4 Numerical Dirichlet Series 160

10.2 The Analytic Theory of L-Series 162

10.2.1 Simple Approaches to Analytic Continuation 163

10.2.2 The Use of the Hurwitz Zeta Function ζ(s, x) 168

10.2.3 The Functional Equation for the Theta Function 169

10.2.4 The Functional Equation for Dirichlet L-Functions 172

10.2.5 Generalized Poisson Summation Formulas 177

10.2.6 Voronoi’s Error Term in the Circle Problem 182

10.3 Special Values of Dirichlet L-Functions 186

10.3.1 Basic Results on Special Values 186

10.3.2 Special Values of L-Functions and Modular Forms 193

10.3.3 The P´olya–Vinogradov Inequality 198

10.3.4 Bounds and Averages for L(χ, 1) 200

10.3.5 Expansions of ζ(s) Around s = k ∈ Z1 205

10.3.6 Numerical Computation of Euler Products and Sums 208 10.4 Epstein Zeta Functions 210

10.4.1 The Nonholomorphic Eisenstein Series G(τ, s) 211

10.4.2 The Kronecker Limit Formula 213

10.5 Dirichlet Series Linked to Number Fields 216

10.5.1 The Dedekind Zeta Function ζ K (s) 216

10.5.2 The Dedekind Zeta Function of Quadratic Fields 219

10.5.3 Applications of the Kronecker Limit Formula 223

10.5.4 The Dedekind Zeta Function of Cyclotomic Fields 230

10.5.5 The Nonvanishing of L(χ, 1) 235

10.5.6 Application to Primes in Arithmetic Progression 237

10.5.7 Conjectures on Dirichlet L-Functions 238

10.6 Science Fiction on L-Functions 239

10.6.1 Local L-Functions 239

10.6.2 Global L-Functions 241

10.7 The Prime Number Theorem 245

10.7.1 Estimates for ζ(s) 246

10.7.2 Newman’s Proof 250

10.7.3 Iwaniec’s Proof 254

10.8 Exercises for Chapter 10 258

11. p-adic Gamma and L-Functions 275

11.1 Generalities on p-adic Functions 275

11.1.1 Methods for Constructing p-adic Functions 275

11.1.2 A Brief Study of Volkenborn Integrals 276

11.2 The p-adic Hurwitz Zeta Functions 280

11.2.1 Teichm¨uller Extensions and Characters onZp 280

11.2.2 The p-adic Hurwitz Zeta Function for x ∈ CZ p 281

11.2.3 The Function ζ p (s, x) Around s = 1 288

11.2.4 The p-adic Hurwitz Zeta Function for x ∈ Z p 290

11.3 p-adic L-Functions 300

11.3.1 Dirichlet Characters in the p-adic Context 300

11.3.2 Definition and Basic Properties of p-adic L-Functions 301 11.3.3 p-adic L-Functions at Positive Integers 305

11.3.4 χ-Power Sums Involving p-adic Logarithms 310

11.3.5 The Function L p (χ, s) Around s = 1 317

11.4 Applications of p-adic L-Functions 319

11.4.1 Integrality and Parity of L-Function Values 319

11.4.2 Bernoulli Numbers and Regular Primes 324

11.4.3 Strengthening of the Almkvist–Meurman Theorem 326

11.5 p-adic Log Gamma Functions 329

11.5.1 Diamond’s p-adic Log Gamma Function 330

11.5.2 Morita’s p-adic Log Gamma Function 336

11.5.3 Computation of some p-adic Logarithms 346

11.5.4 Computation of Limits of some Logarithmic Sums 356

11.5.5 Explicit Formulas for ψ p (r/m) and ψ p (χ, r/m) 359

11.5.6 Application to the Value of L p (χ, 1) 361

11.6 Morita’s p-adic Gamma Function 364

11.6.1 Introduction 364

11.6.2 Definitions and Basic Results 365

11.6.3 Main Properties of the p-adic Gamma Function 369

11.6.4 Mahler–Dwork Expansions Linked to Γp (x) 375

11.6.5 Power Series Expansions Linked to Γp (x) 378

11.6.6 The Jacobstahl–Kazandzidis Congruence 380

11.7 The Gross–Koblitz Formula and Applications 383

11.7.1 Statement and Proof of the Gross–Koblitz Formula 383

11.7.2 Application to L  p (χ, 0) 389

11.7.3 Application to the Stickelberger Congruence 390

11.7.4 Application to the Hasse–Davenport Product Relation 392 11.8 Exercises for Chapter 11 395

Part IV Modern Tools

12 Applications of Linear Forms in Logarithms 411

12.1 Introduction 41112.1.1 Lower Bounds 41112.1.2 Applications to Diophantine Equations and Problems 41312.1.3 A List of Applications 41412.2 A Lower Bound for|2 m − 3 n | 415

12.3 Lower Bounds for the Trace of α n 41812.4 Pure Powers in Binary Recurrent Sequences 42012.5 Greatest Prime Factors of Terms of Some Recurrent Se-quences 42112.6 Greatest Prime Factors of Values of Integer Polynomials 422

12.7 The Diophantine Equation ax n − by n = c 423

12.8 Simultaneous Pell Equations 42412.8.1 General Strategy 42412.8.2 An Example in Detail 42512.8.3 A General Algorithm 42612.9 Catalan’s Equation 42812.10 Thue Equations 43012.10.1 The Main Theorem 43012.10.2 Algorithmic Aspects 43212.11 Other Classical Diophantine Equations 43612.12 A Few Words on the Non-Archimedean Case 439

13 Rational Points on Higher-Genus Curves 441

13.1 Introduction 44113.2 The Jacobian 44213.2.1 Functions on Curves 44313.2.2 Divisors 44413.2.3 Rational Divisors 44513.2.4 The Group Law: Cantor’s Algorithm 44613.2.5 The Group Law: The Geometric Point of View 44813.3 Rational Points on Hyperelliptic Curves 44913.3.1 The Method of Demyanenko–Manin 44913.3.2 The Method of Chabauty–Coleman 45213.3.3 Explicit Chabauty According to Flynn 45313.3.4 When Chabauty Fails 45513.3.5 Elliptic Curve Chabauty 45613.3.6 A Complete Example 459

14 The Super-Fermat Equation 463

14.1 Preliminary Reductions 463

14.2 The Dihedral Cases (2, 2, r) 465 14.2.1 The Equation x2− y2= z r 465

14.2.2 The Equation x2+ y2= z r 466

14.2.3 The Equations x2+ 3y2= z3 and x2+ 3y2= 4z3 466

14.3 The Tetrahedral Case (2, 3, 3) 467 14.3.1 The Equation x3+ y3= z2 467

14.5.2 The Icosahedral Case (2, 3, 5) 479

14.6 The Parabolic and Hyperbolic Cases 48114.6.1 The Parabolic Case 48114.6.2 General Results in the Hyperbolic Case 482

15 The Modular Approach to Diophantine Equations 495

15.1 Newforms 49515.1.1 Introduction and Necessary Software Tools 49515.1.2 Newforms 49615.1.3 Rational Newforms and Elliptic Curves 49715.2 Ribet’s Level-Lowering Theorem 49815.2.1 Definition of “Arises From” 49815.2.2 Ribet’s Level-Lowering Theorem 49915.2.3 Absence of Isogenies 50115.2.4 How to use Ribet’s Theorem 50315.3 Fermat’s Last Theorem and Similar Equations 50315.3.1 A Generalization of FLT 504

15.3.2 E Arises from a Curve with Complex Multiplication 505

15.3.3 End of the Proof of Theorem 15.3.1 506

15.3.4 The Equation x2= y p+ 2r z p for p  7 and r  2 507

15.3.5 The Equation x2= y p + z p for p 7 50915.4 An Occasional Bound for the Exponent 50915.5 An Example of Serre–Mazur–Kraus 51115.6 The Method of Kraus 51415.7 “Predicting Exponents of Constants” 517

15.7.1 The Diophantine Equation x2− 2 = y p 51715.7.2 Application to the SMK Equation 52115.8 Recipes for Some Ternary Diophantine Equations 522

15.8.1 Recipes for Signature (p, p, p) 523 15.8.2 Recipes for Signature (p, p, 2) 524 15.8.3 Recipes for Signature (p, p, 3) 526

16 Catalan’s Equation 529

16.1 Mih˘ailescu’s First Two Theorems 52916.1.1 The First Theorem: Double Wieferich Pairs 530

16.1.2 The Equation (x p − 1)/(x − 1) = py q 53216.1.3 Mih˘ailescu’s Second Theorem: p | h −

q and q | h −

p 53616.2 The + and− Subspaces and the Group S 537

16.2.1 The + and− Subspaces 538

Bibliography 561

Index of Notation 571

Index of Names 579

General Index 585

Part III

Analytic Tools

9 Bernoulli Polynomials and the Gamma

Function

We now begin our study of analytic methods in number theory This is of

course a vast subject, but we will not deal with what is usually called

“an-alytic number theory,” but with the methods that are related to the study

of L-functions, which we will study in the next chapter This essentially

in-volves Bernoulli numbers and polynomials, the Euler–MacLaurin summationformula, and the gamma function and related functions

9.1 Bernoulli Numbers and Polynomials

9.1.1 Generating Functions for Bernoulli Polynomials

We start by recalling some properties of Bernoulli numbers and polynomials

Definition 9.1.1. We define the Bernoulli polynomials B k (x) and their

ex-ponential generating function E(t, x) by

The first few polynomials are B0(x) = 1, B1(x) = x − 1/2, B2(x) =

x2− x + 1/6, and B3(x) = x3− 3x2

/2 + x/2 Note that most of the results

that we give in this section for Bernoulli polynomials also apply to Bernoulli numbers by specializing to 0 the variable x.

The reader will notice as we go along that more natural numbers would

be B k /k instead of B k However, it is impossible to change a definition that

is centuries old

Proposition 9.1.2. We have the following properties:

(1) B  k (x) = kB k −1 (x).

(2) B k (x) is a monic polynomial of degree k.

(3) For k = 1 we have B k (1) = B k (0) = B k , while for k = 1 we have

B1(1) = 1/2 = B1(0) + 1 In other words, if we set δ k,1 = 1 if k = 1 and

δ = 0 otherwise, we have B (1) = B + δ

B j x k −j

Proof All these results are immediate consequences of the definition: (1)

is equivalent to ∂E(t,x) ∂x = tE(t, x), (2) follows by induction, (3) is equivalent

to E(t, 1) − E(t, 0) = t, (4) to the fact that E(t, 0) + t/2 = (t/2) cotanh(t/2)

is an even function, and (5) by formal multiplication of the power series for

It is immediate to check that (1) and (3) together with B0(x) = 1 in fact

characterize Bernoulli polynomials (Exercise 1).

In addition to the initial values B0 = 1 and B1 = −1/2, the first few

nonzero values are B2 = 1/6, B4=−1/30, B6= 1/42, B8=−1/30, B10=

5/66, B12 =−691/2730, B14 = 7/6, B16 =−3617/510 For instance, every

time that you meet the (prime) number 691, you must immediately think of

the Bernoulli number B12

Further immediate properties of Bernoulli polynomials are the following



j=0



k j

the trivial identities E(t, x + 1) = E(t, x) + te tx , E( −t, −x) = e t E(t, x) = E(t, x)+te tx , E( −t, 1−x) = E(t, x), E(t, x+y) = e ty E(t, x), (e t −1)E(t, x) =

te tx, and

0j<N E(N t, x + j/N ) = N E(t, N x). 

Note that the formula for B k(−x) generalizes the fact that B k = 0 for

k  3 odd Like all formulas involving j/N for 0  j < N, the last formula

is called the distribution formula for Bernoulli polynomials.

Bernoulli numbers and polynomials are by definition Taylor coefficients

of certain power series Thus they occur in the Taylor expansion of a number

of classical functions, as follows

Proposition 9.1.4. We have the following Taylor series expansions with radii of convergence R indicated in parentheses:

Corollary 9.1.5. We have

B k

12

Proof For the first formula we note that te t/2 /(e t − 1) = (t/2)/ sinh(t/2),

and the second statement follows from the vanishing of B k for k 3 odd and

the fact that B k (1) = B k (0) for k = 1 

Definition 9.1.6. We define the tangent numbers T k for k  0 by

T k = 2k+1(2k+1 − 1) B k+1

k + 1 .

Thus tanh(t) =

k1T 2k −1 t 2k −1 /(2k − 1)! and similarly for tan(t) We

have T0=−1, T 2k = 0 for k  1, and the first few values of T k for k odd are

and in particular T 2k −1 ∈ Z for all k  1.

Proof This immediately follows from the identity cosh(t) tanh(t) =

sinh(t), and the details are left to the reader 

The fact that T 2k −1 ∈ Z also follows from the Clausen–von Staudt

theo-rem that we will prove below (Exercise 59)

Definition 9.1.8. We define the Euler numbers E k for k  0 by setting

E 2k+1 = 0 for k  0 and

E 2k=−4 2k+1 B 2k+1 (1/4)

2k + 1 . The first few values are E0= 1, E2=−1, E4= 5, E6=−61, E8= 1385,

so once again if you meet the prime 61 in a computation, you may suspect

that it comes from E6

Proof Multiplying the identity 1/(e t + 1) = 1/(e t − 1) − 2/(e 2t − 1) given

above by e t/2 and replacing t by 2t, we obtain

Since cosh(t) is an even function, we first deduce that 2 2k B 2k (1/4) =

B 2k (1/2), and since B 2k(1− x) = B 2k (x), we obtain the first formula thermore, since B 2k+1 (1/2) = 0 for k 0 by the above corollary, we have



E 2j = 0 for k > 0 ,

and in particular E 2k ∈ Z for all k.

Proof This immediately follows from the identity cosh(t)(1/ cosh(t)) = 1.

It also follows from the second formula of Proposition 9.1.14 below applied to

x = y = 1/4 We thus have E 2k=−0j<k

2k

2j



E 2j, from which we deduce

by induction that E 2k is an integer for all k 

Remark. Although Bernoulli numbers satisfy the recurrencek −1

0, which is very similar to the one for E k if we replace k by 2k and B j by 0

when j > 1 is odd, the main difference is that this recurrence leads to

Interestingly enough, although the natural generating function for Bernoulli

polynomials is the exponential generating function E(t, x) =

k0B k (x)t k = S(1/t, x)/t, but the corresponding

formulas would be slightly more complicated

It is easy to check that the series S(t, x) does not converge for any value

of t, but as a formal power series it makes sense, and we will also see that

even though the series is divergent we can assign to it a specific value Note,

however, that in Chapter 11 we will see that it converges for all p-adic values

of t such that |t| > 1, and that S(t, x) = ψ p  (t − x + 1) (which follows

immediately from Proposition 11.5.2 (2)), to be compared with Corollary9.1.13, which is formally identical

Proposition 9.1.11. We have

(t − x)2 , S(t, x + 1) = S(t, x) + 1

(t − x)2 , S(t − y, x) = S(t, x + y) , and in particular

S(t − 1, x) = −S(−t, −x) = S(t, x) + 1

(t − x)2 and S(t, x) = S(t − x, 0)

Proof Using the formula for B n(−x) mentioned above we have

(t − x)2 ,

proving the first formula, and the second follows similarly from the formula

for B k (x + 1) (or from the first and the formula for B k(1− x)).

For the third, we use the formula for B k (x + y), which gives

Proof The first statement is clear by expanding E(u, x) as a power series

Proof From Corollary 9.6.43 below we have

so the result follows from the proposition 

See also Theorems 9.6.48 and 9.6.49 for continued fraction expansions of

S(t, x).

9.1.2 Further Recurrences for Bernoulli Polynomials

There are a great many useful recurrences for Bernoulli numbers and nomials We begin with the following

poly-Proposition 9.1.14. For k  0 we have

B k+1 (x + y) + ( −1) k −1 B k+1 (x − y) − 2y k+1

Proof We could give a proof of the first formula directly from the

generat-ing function, as we did for Proposition 9.1.3 It is however instructive to give

an alternative proof After all, if we integrate with respect to x the formula for B k (x + y) given in Proposition 9.1.3 and use B j+1  (x) = (j + 1)B j (x), we obtain the result up to addition of a function of y, which is not easy to de-

termine This approach almost never works What does almost always work

is to use trivial transformations of binomial coefficients Here we note that

from which the first formula follows by dividing by k and changing j into

j + 1 and k into k + 1 The other two formulas follow by computing the sum

and difference of the first formula applied to y and to −y 

Corollary 9.1.15. For k  0 We have



B 2j+1 (x) 2j + 1 =



B 2j (x) 2j =

x 2k + (x − 1) 2k

2k + 1 , k

B 2j+1 (x) 2j + 1 =



B 2j (x) 2j =

x 2k+1 − (x − 1) 2k+1

2k + 2 , k

Proof These formulas are obtained by suitable specializations to y = 1

or y = 1/2 of the formulas of the proposition 

Remarks. (1) If we want formulas involving B j (x) itself instead of B j (x)/j,

we simply differentiate with respect to x the formulas of the proposition

and of the corollary We can of course differentiate several times

In-versely, if we want formulas involving B j (x)/(j(j + 1)) for instance, we

must in principle integrate the given formulas, but as explained above thiswill not give the constant term, so we simply use as above the relation

for j 1; see Exercise 23

(2) Since B k (x + 1) and B k(1−x) have simple expressions in terms of B k (x),

if we want to specialize again the above formulas (or their derivatives),

we may as well restrict to 0 x  1/2 Using the formulas B k (0) = B k,

B k (1/2) = −(1−1/2 k −1 )B k , B 2k (1/4) = −(1/2 2k)(1−1/2 2k −1 )B 2k, andthe analogous formulas for B 2k (1/3) and B 2k (1/6) given by Exercise 10,

we obtain in this way a very large number of recurrence relations forBernoulli numbers We can obtain even more such relations by replacing

directly x and y in the formulas of Proposition 9.1.14, for instance x =

y = 1/4 in the third formula We also obtain the standard relation for

Euler numbers given in Corollary 9.1.10 by choosing x = y = 1/4 in the

second formula It is to be noted, however, that all these formulas have

approximately k terms; in other words, they express B 2k in terms of all

the B 2j for 1 j < k We are going to see that we can reduce this by a

factor of 2

The second type of recurrence that we are going to study is not wellknown, although it is essentially due to Seidel in 1877, and Lucas soon af-terward It has the advantage of having half as many terms in the sum,and smaller binomial coefficients I thank my colleague C Batut for havingpointed it out to me

Proposition 9.1.16. For any k and m inZ0 we have

m j



m j

x k (x − 1) k

(−1) k+1 (4k + 2)2k

k (x − 1) k

let D = d/dt be the differentiation operator with respect to t, and let I be

the identity operator We begin with the following lemma

Lemma 9.1.17. With the above notation we have

(e t D m (D + I) k − D k

(D − I) m

)F x (t) = x k (x − 1) m

e xt Proof For simplicity write F x instead of F x (t) Leibniz’s rule can be writ-

ten in operator notation

proving the lemma after multiplication by e t  Proof of the proposition Denote by G[0] (not G(0)) the constant term in

a Laurent series G(t) Taking constant terms in the lemma we obtain

+ (−1) j+1



m j

where the second sum starts at j = 1, since for j = 0 the binomial coefficients

cancel Furthermore, we have

t m (t − 1) k

dt

= (−1) k

1 0

Corollary 9.1.18. For any k and m inZ0 we have

max(k,m) k

j

+ (−1) k+m



m j



B k+m+1 −j

k + m + 1 − j =

(−1) m+1 (k + m + 1)k+m  ,

j=0



k j

+ (−1) k+m+1



m j

(−1) k+1 (4k + 2)2k

Proof The first four formulas follow by taking x = 0 in the proposition

and using the formulas for the odd Bernoulli numbers The replacement of(−1) j by±(−1) k+m and the fact that we begin at j = 0 removes the special

cases The details are left to the reader The last formula is obtained by

taking x = 1/4 in the last formula of the proposition 

A restatement of the fourth formula is the following:

Corollary 9.1.19. For k  2 we have

(k + 1)(2k + 1)

k/2

j=1 (2k − 2j + 1)

Thus, as mentioned above, we obtain a recurrence giving B 2k as a linear

combination of the preceding B 2k −2j , but only those with 2k − 2j  2k/2 ,

hence half as many as the formulas obtained using the more standard rences Furthermore, the coefficients of the linear combinations are smaller

recur-binomial coefficients since (forgetting the simple factor (2k − 2j + 1)) they

have the formk+1

j

instead of2k

j



9.1.3 Computing a Single Bernoulli Number

If we want to compute a table of Bernoulli numbers up to a desired limit, the above recurrence or others are suitable But if we want to compute a single value of a Bernoulli number B for k even, computing all the preceding B

up to B k using recurrences is a waste of time and space since there exist

more efficient direct methods We assume of course that k is even The first method is based on a direct formula for B k given in Exercise 26 The secondmethod is based on two results that we shall prove below (Corollary 9.1.21and Theorem 9.5.14) One is the well-known formula

which gives a very precise asymptotic estimate on the size of B k The other is

the Clausen–von Staudt congruence, which gives the exact denominator D k

of the rational number B k:

(p −1)|k

p ,

where the product is over prime numbers p such that (p − 1) | k It is thus

sufficient to compute an approximation A k to D k B ksuch that|A k −D k B k | <

1/2, and the numerator of B k will then be equal to the nearest integer to A k.This indeed gives a very efficient method to compute an individual value of

B k

Note that the implementation of this method should be done with care

We first compute the denominator D k and k! in a na¨ıve way We must then estimate the number of decimal digits d with which to perform the computa- tion, and the number N of terms to take in the zeta series A cursory analysis

shows that one can take

d = 3 + d1/ log(10) N = 1 + exp((d1− log(k − 1))/(k − 1)) ,

where

d1= log(D k ) + (k + 1/2) log(k) − k(log(2π) + 1) + log(2π)/2 + log(2) + 0.1

Thanks to Stirling’s formula the reader will recognize that d1 is close to

log(D k |B k |), and the 3+ and 1+ are safety precautions Note that the above

computations should be done to the lowest possible accuracy, since at this

point we only want integers d and N

The computation of π can be done using many different methods, but

since anyway you will have to use a CAS for the multiprecision operations,

this is always built in Of course π k is computed using a binary poweringmethod

When k is large all this takes only a small fraction of the time, almost all the time being spent in the computation of ζ(k) =

m1m −k to the desired

number of decimal digits d Note that since k is large, ζ(k) is very close to

1 Once again there are several methods to do this computation, but in theauthor’s opinion the best method is as follows First, instead of computing

the series ζ(k), we compute the Euler product 1/ζ(k) =

P truncated to the accuracy d  Indeed, contrary to most computations in

numerical analysis, here we need absolute and not relative accuracy Although

this is a technical remark it can in itself gain a factor of 3 or 4

Note that when suitably implemented the above method is so efficientthat it can even be faster than the method using recurrences for computing

a table.

To give an idea of the speed, on a Pentium 4 at 3 Ghz the computation of

B10000requires 33 seconds using the formula of Exercise 26, but only 0.3

sec-onds using the above method The computation of all Bernoulli numbers up

to B5000requires 205 seconds using the standard recurrences given above and

26 seconds using the above method that computes each number individually,

which is indeed considerably faster

9.1.4 Bernoulli Polynomials and Fourier Series

In this section we give a direct link between Bernoulli polynomials and tain Fourier series This will later be useful for computing special values of

cer-Dirichlet L-functions (see Section 10.3).

It is important to compute the Fourier series corresponding to the

func-tions B k (x) for k  1 (for k = 0 it is trivial), more precisely to the functions obtained by extending by periodicity of period 1 the kth Bernoulli polyno- mial on the interval [0, 1[ We will denote by {x} the fractional part of x,

in other words the unique real number in [0, 1[ such that x − {x} ∈ Z, i.e.,

k({x}) is evidently periodic of period 1 The

Proof (1) and (2) Since B n (1) = B n (0) for n = 1, the function B n({x})

is piecewise C ∞ and continuous for n 2, with simple discontinuities at the

integers if n = 1 If n 2 we thus have

B n (t)e −2iπkt dt

For n = 1, the same formula is valid for x / ∈ Z, and for x ∈ Z we must replace

B1({x}) by (B1(1− ) + B1(0+))/2 = 0.

Using the definitions and the formulas B n  (x) = nB n −1 (x) and B n(1) =

B n (0) for n = 1, by integration by parts we obtain for k = 0

as soon as n  1 Thus, with the above interpretation for x ∈ Z when n = 1,

we obtain that for n 1 we have

=− log1− e 2iπx=− log(2| sin(πx)|) ,

Corollary 9.1.21. For n  1 we have

to B5000requires 205 seconds using the standard recurrences given above and

26 seconds using the above method that computes each number. .. |), and the 3+ and 1+ are safety precautions Note that the above

computations should be done to the lowest possible accuracy, since at this

point we only want integers d and