Elementary number theory and elliptic curves (web draft, June 2003)

251 Pages·1.589 MB·English

by  Stein W.

i
Hello,
Thank youverymuchforlooking atmy book,Elementary Number Theory
and Elliptic Curves.
This book is slated for publication in Springer-Verlag’s Undergraduate
Texts in Mathematics (UTM) series. Since this book is yet another un-
dergraduate book on number theory, I want it to be difierent in that it is
hopefully concise, timely, and takes the reader to one frontier of modern
number theory (elliptic curves). Incidentally, after Springer publishes this
book, I’m assured that I will be allowed to continue to make an electronic
version available for free online.
I also don’t want this book to contain any mistakes or annoying ways of
explaining things. That’s where you come in. Please look through some of
the book, any part that interests you, and tell me what annoys you. Give
me any constructive criticism; in the interest of giving under-
graduates a better-quality book, I’m thick skinned. What should I
have described but didn’t? Who should I have referenced? What obvious
example did I miss? What helpful diagram could I have given? What is
incomprehensible? What did I forget to deflne? What couldn’t you flnd in
the index?
Note: In part III on computation, only the introduction and the chapter
on Maple are completely flnished.
Thanks!
William Stein
Department of Mathematics
Harvard University
[email protected]
http://modular.fas.harvard.edu
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Elementary Number Theory
&
Elliptic Curves
William Stein
June 26, 2003
ii
To my wife, Clarita Lefthand.
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Contents
1 Preface 3
2 Introduction 5
2.1 Elementary Number Theory . . . . . . . . . . . . . . . . . . 5
2.2 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Notation and Conventions . . . . . . . . . . . . . . . . . . . 9
I Elementary Number Theory 10
3 Primes and Congruences 11
3.1 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The Sequence of Prime Numbers . . . . . . . . . . . . . . . 17
3.3 Congruences Modulo n . . . . . . . . . . . . . . . . . . . . . 22
3.4 The Chinese Remainder Theorem . . . . . . . . . . . . . . . 27
3.5 Quickly Computing Inverses and Huge Powers. . . . . . . . 30
4 Public-Key Cryptography 37
4.1 The Di–e-Hellman Key Exchange . . . . . . . . . . . . . . 37
4.2 The RSA Cryptosystem . . . . . . . . . . . . . . . . . . . . 42
4.3 Attacking RSA . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 The Structure of (Z=p) 51
£
5.1 Polynomials over Z=p . . . . . . . . . . . . . . . . . . . . . 51
iv Contents
5.2 Existence of Primitive Roots . . . . . . . . . . . . . . . . . 52
5.3 Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . 54
6 Quadratic Reciprocity 57
6.1 Statement of the Quadratic Reciprocity Law . . . . . . . . 57
6.2 Euler’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3 First Proof of Quadratic Reciprocity . . . . . . . . . . . . . 61
6.4 A Proof of Quadratic Reciprocity Using Gauss Sums . . . . 66
6.5 How To Find Square Roots . . . . . . . . . . . . . . . . . . 70
7 Continued Fractions 73
7.1 Finite Continued Fractions . . . . . . . . . . . . . . . . . . 74
7.2 Inflnite Continued Fractions . . . . . . . . . . . . . . . . . . 78
7.3 The Continued Fraction of e . . . . . . . . . . . . . . . . . . 83
7.4 Quadratic Irrationals . . . . . . . . . . . . . . . . . . . . . . 85
7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 p-adic Numbers 99
8.1 The N-adic Numbers . . . . . . . . . . . . . . . . . . . . . . 99
8.2 The 10-adic Numbers . . . . . . . . . . . . . . . . . . . . . 101
8.3 The Field of p-adic Numbers . . . . . . . . . . . . . . . . . 102
8.4 The Topology of Q (is Weird) . . . . . . . . . . . . . . . . 103
N
8.5 The Local-to-Global Principle of Hasse and Minkowski . . . 103
9 Binary Quadratic Forms and Ideal Class Groups 107
9.1 Sums of Two Squares . . . . . . . . . . . . . . . . . . . . . 107
9.2 Binary Quadratic Forms . . . . . . . . . . . . . . . . . . . . 111
9.3 Reduction Theory . . . . . . . . . . . . . . . . . . . . . . . 116
9.4 Class Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.5 Correspondence Between Binary Quadratic Forms and Ideals 119
II Elliptic Curves 128
10 Introduction to Elliptic Curves 129
10.1 Elliptic Curves Over the Complex Numbers . . . . . . . . . 129
10.2 The Group Structure on an Elliptic Curve . . . . . . . . . . 134
10.3 Rational Points . . . . . . . . . . . . . . . . . . . . . . . . . 142
11 Algorithmic Applications of Elliptic Curves 151
11.1 Elliptic Curves Over Finite Fields. . . . . . . . . . . . . . . 151
11.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.3 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12 Modular Forms and Elliptic Curves 165
12.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . 165
Contents 1
12.2 Modular Elliptic Curves . . . . . . . . . . . . . . . . . . . . 169
12.3 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . 170
13 The Birch and Swinnerton-Dyer Conjecture 175
13.1 The Congruent Number Problem . . . . . . . . . . . . . . . 175
13.2 The Birch and Swinnerton-Dyer Conjecture . . . . . . . . . 178
13.3 Computing L(E;s) with a Computer . . . . . . . . . . . . . 179
13.4 A Rationality Theorem . . . . . . . . . . . . . . . . . . . . 180
13.5 A Way to Approximate the Analytic Rank. . . . . . . . . . 181
III Computing 183
14 Introduction 185
14.1 Some Assertions About Primes . . . . . . . . . . . . . . . . 185
14.2 Some Tools for Computing . . . . . . . . . . . . . . . . . . 189
15 MAGMA 191
15.1 Elementary Number Theory . . . . . . . . . . . . . . . . . . 191
15.2 Documentation . . . . . . . . . . . . . . . . . . . . . . . . . 191
15.3 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . 192
15.4 Programming Magma . . . . . . . . . . . . . . . . . . . . . 194
15.5 Getting Comfortable . . . . . . . . . . . . . . . . . . . . . . 195
16 Maple 201
16.1 Elementary Number Theory . . . . . . . . . . . . . . . . . . 201
16.2 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . 206
17 Mathematica 215
17.1 Elementary Number Theory . . . . . . . . . . . . . . . . . . 215
17.2 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 215
18 PARI 217
18.1 Getting Started with PARI . . . . . . . . . . . . . . . . . . 217
18.2 Pari Programming . . . . . . . . . . . . . . . . . . . . . . . 219
18.3 Computing with Elliptic Curves . . . . . . . . . . . . . . . . 223
19 Other Computational Tools 231
19.1 Hand Calculators . . . . . . . . . . . . . . . . . . . . . . . . 231
References 233
Index 238
2 Contents
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1
Preface
This is a textbook about classical number theory and modern elliptic
curves. Part I discusses elementary topics such as primes, factorization,
continued fractions, and quadratic forms, in the context of cryptography,
computation, and deep open research problems. The second part is about
ellipticcurves,theirapplicationstoalgorithmicproblems,andtheirconnec-
tions with problems in number theory such as Fermat’s Last Theorem, the
CongruentNumberProblem,andtheConjectureofBirchandSwinnerton-
Dyer.
The goal of part I is to give the reader a solid foundation in the stan-
dard topics of elementary number theory. In contrast, the goal of part II
is to convey the central importance of elliptic curves in modern number
theory and give a feeling for the big open problems about them without
becoming overwhelmed by technical details. Part III describes how to use
several standard mathematics programs to do computations with many of
the objects described in this book.
The intended audience is a strong undergraduate with some familiarity
with abstract algebra (rings, flelds, and flnite abelian groups), who has
not necessarily seen any number theory. For the elliptic curves part of the
book, some prior exposure to complex analysis would be useful but is not
necessary.
This book grew out of an undergraduate course that the author taught
at Harvard University in 2001 and 2002.
Acknowledgment.IwouldliketothankLawrenceCabusoraforcarefully
reading the flrst draft of this book and making many helpful comments.
4 1. Preface
Brian Conrad made clarifying comments on the flrst 30 pages, which I’ve
included. Noam Elkies made many comments about the chapter on p-adic
numbers, Section 3.2, and many other parts of the book. I would also like
to thank the students of my Math 124 course at Harvard during the Fall
of 2001 and 2002, who provided the flrst audience for this book, as well
as David Savitt for conversations. Hendrik Lenstra made helpful remarks
about how to present his factorization algorithm.
SethKleinermanwrotetheflrstversionofSection7.3andExercise7.14.
Peopleofieringcorrectionsandcommentsviaemail:GeorgeStephanides,
Kevin Stern, Heidi L. Williams.
1. Peter Hawthorne (discussions about algebra; helped write ...)
2. Seth Kleinerman (e; flnding many typos)
I also found LATEX, xflg, MAGMA, PARI, and Emacs to be extremely
helpful in the preparation of this manuscript.
Part I of this book grew out of a course based on Davenport’s [22], so in
someplaceswefollow[22]closely.Thereareafewpictures(inparticular,of
Di–eandHellman)thatwereswipedfromotherbookswithoutpermission;
this was fair use for lecture notes during a course, but not for a textbook,
so this will have to be remedied.