Primes of the Form X^2 + Ny^2: Fermat, Class Field Theory, and Complex Multiplication, With Solutions

551 Pages·2022·7.537 MB·English

by  David A. Cox

P F x2 + ny2
RIMES OF THE ORM
F , C F T ,
ERMAT LASS IELD HEORY
C M .
AND OMPLEX ULTIPLICATION
T E S
HIRD DITION WITH OLUTIONS
D A. C
AVID OX
W R L
ITH CONTRIBUTIONS BY OGER IPSETT
AMS CHELSEA PUBLISHING
P f x2 + ny2
rimes of the orm
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P f x2 + ny2
rimes of the orm
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AMS CHELSEA PUBLISHING
2020 Mathematics Subject Classification. Primary 11A41; Secondary 11F11, 11R11,
11R16, 11R18, 11R37, 11Y11.
For additional informationand updates on this book, visit
www.ams.org/bookpages/chel-387
Library of Congress Cataloging-in-Publication Data
Names: Cox,DavidA.,author. |Lipsett,Roger,1950-other.
Title: Primesoftheformx2+ny2: Fermat,classfieldtheory,andcomplexmultiplication. Third
EditionwithSolutions/DavidA.Cox;withcontributionsbyRogerLipsett.
Othertitles: Primesoftheformpequalsx2 plusny2
Description: Third edition. | Providence, Rhode Island : AMS Chelsea Publishing/American
Mathematical Society, [2022] | Includes bibliographical references and index. | Summary:
“The goalof the new edition of Primes of the Form x2+ny2 is to make this wonderful part
ofnumbertheoryavailabletoreadersinaformespeciallysuitedtoself-study,mainlybecause
completesolutionstoallexercisesareincluded”–Providedbypublisher.
Identifiers: LCCN2022025796|ISBN9781470470289(paperbackacid-freepaper)|9781470471835
(ebook)
Subjects: LCSH:Numbers,Prime. |Mathematics. |AMS:Numbertheory–Elementarynumber
theory–Primes. |Numbertheory–Discontinuousgroupsandautomorphicforms–Holomor-
phic modular forms of integral weight. | Number theory – Algebraic number theory: global
fields – Quadratic extensions. | Number theory – Algebraic number theory: global fields –
Cubic and quartic extensions. | Number theory – Algebraic number theory: global fields –
Cyclotomicextensions. |Numbertheory–Algebraicnumbertheory: globalfields–Classfield
theory. |Numbertheory–Computationalnumbertheory–Primality.
Classification: LCCQA246.C692022|DDC512.7/23–dc23/eng20220829
LCrecordavailableathttps://lccn.loc.gov/2022025796
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Contents
Preface ix
First Edition ix
Second Edition ix
Third Edition with Solutions x
Notation xiii
Introduction 1
Chapter 1. From Fermat to Gauss 7
§1. Fermat, Euler and Quadratic Reciprocity 7
A. Fermat 7
B. Euler 8
C. p=x2+ny2 and Quadratic Reciprocity 11
D. Beyond Quadratic Reciprocity 16
E. Exercises 17
§2. Lagrange, Legendre and Quadratic Forms 20
A. Quadratic Forms 20
B. p=x2+ny2 and Quadratic Forms 25
C. Elementary Genus Theory 27
D. Lagrange and Legendre 31
E. Exercises 35
§3. Gauss, Composition and Genera 38
A. Composition and the Class Group 38
B. Genus Theory 43
C. p=x2+ny2 and Euler’s Convenient Numbers 48
D. Disquisitiones Arithmeticae 51
E. Exercises 53
§4. Cubic and Biquadratic Reciprocity 60
A. Z[ω] and Cubic Reciprocity 60
B. Z[i] and Biquadratic Reciprocity 65
C. Gauss and Higher Reciprocity 67
D. Exercises 71
Chapter 2. Class Field Theory 77
§5. The Hilbert Class Field and p=x2+ny2 77
A. Number Fields 77
B. Quadratic Fields 81
C. The Hilbert Class Field 83
v
vi CONTENTS
D. Solution of p=x2+ny2 for Infinitely Many n 86
E. Exercises 91
§6. The Hilbert Class Field and Genus Theory 95
A. Genus Theory for Field Discriminants 95
B. Applications to the Hilbert Class Field 100
C. Exercises 101
§7. Orders in Imaginary Quadratic Fields 104
A. Orders in Quadratic Fields 105
B. Orders and Quadratic Forms 108
C. Ideals Prime to the Conductor 113
D. The Class Number 115
E. Exercises 118
§8. Class Field Theory and the Cˇebotarev Density Theorem 125
A. The Theorems of Class Field Theory 126
B. The Cˇebotarev Density Theorem 133
C. Norms and Ideles 136
D. Exercises 137
§9. Ring Class Fields and p=x2+ny2 141
A. Solution of p=x2+ny2 for√All n √ 142
B. The Ring Class Fields of Z[ −27] and Z[ −64] 145
C. Primes Represented by Positive Definite Quadratic Forms 148
D. Ring Class Fields and Generalized Dihedral Extensions 150
E. Exercises 152
Chapter 3. Complex Multiplication 157
§10. Elliptic Functions and Complex Multiplication 157
A. Elliptic Functions and the Weierstrass ℘-Function 157
B. The j-Invariant of a Lattice 162
C. Complex Multiplication 164
D. Exercises 170
§11. Modular Functions and Ring Class Fields 173
A. The j-Function 173
B. Modular Functions for Γ (m) 177
0
C. The Modular Equation Φ (X,Y) 181
m
D. Complex Multiplication and Ring Class Fields 185
E. Exercises 190
§12. Modular Functions and Singular j-Invariants 195
A. The Cube Root of the j-Function 195
B. The Weber Functions 201
C. j-Invariants of Orders of Cl√ass Number 1 205
D. Weber’s Computation of j( −14) 207
E. Imaginary Quadratic Fields of Class Number 1 213
F. Exercises 216
§13. The Class Equation 225
A. Computing the Class Equation 225
B. Computing the Modular Equation 231
C. Theorems of Deuring, Gross and Zagier 235
D. Exercises 238
CONTENTS vii
Chapter 4. Additional Topics 243
§14. Elliptic Curves 243
A. Elliptic Curves and Weierstrass Equations 243
B. Complex Multiplication and Elliptic Curves 246
C. Elliptic Curves over Finite Fields 249
D. Elliptic Curve Primality Tests 255
E. Exercises 261
§15. Shimura Reciprocity 265
A. Modular Functions 265
B. The Shimura Reciprocity Theorem 269
C. Extended Ring Class Fields 272
D. Shimura Reciprocity for Extended Ring Class Fields 274
E. Shimura Reciprocity for Ring Class Fields 278
F. Class Field Theory 284
G. Exercises 291
Solutions by Roger Lipsett and David Cox 297
Solutions to Exercises in §1 298
Solutions to Exercises in §2 306
Solutions to Exercises in §3 320
Solutions to Exercises in §4 340
Solutions to Exercises in §5 354
Solutions to Exercises in §6 367
Solutions to Exercises in §7 378
Solutions to Exercises in §8 397
Solutions to Exercises in §9 406
Solutions to Exercises in §10 421
Solutions to Exercises in §11 432
Solutions to Exercises in §12 445
Solutions to Exercises in §13 473
Solutions to Exercises in §14 486
Solutions to Exercises in §15 496
References 517
Further Reading 523
Index 525

Preface
First Edition
Several years ago, while reading Weil’s Number Theory: An Approach Through
History, I noticed a conjecture of Euler concerning primes of the form x2+14y2.
That same week I picked up Cohn’s A Classical Invitation to Algebraic Numbers
and Class Fields and saw the same example treated from the point of view of the
Hilbert class field. The coincidence made it clear that something interesting was
going on, and this book is my attempt to tell the story of this wonderful part of
mathematics.
I am an algebraic geometer by training, and number theory has always been
more of an avocation than a profession for me. This will help explain some of the
curiousomissionsinthebook. Theremayalsobeerrorsofhistoryorattribution(for
whichItakefullresponsibility), anddoubtlesssomeoftheproofscanbeimproved.
Corrections and comments are welcome!
I would like to thank my colleagues in the number theory seminars of Okla-
homaStateUniversityandtheFiveColleges(AmherstCollege,HampshireCollege,
MountHolyokeCollege,SmithCollegeandtheUniversityofMassachusetts)forthe
opportunitytopresentmaterialfromthisbookinpreliminaryform. Specialthanks
go to Dan Flath and Peter Norman for their comments on earlier versions of the
manuscript. IalsothankthereferencelibrariansatAmherstCollegeandOklahoma
Slate University for their help in obtaining books through interlibrary loan.
Amherst, Massachusetts
August 1989
Second Edition
The philosophy of the second edition is to preserve as much of the original text as
possible. The major changes are:
• A new §15 on Shimura Reciprocity has been added, based on work of Peter
Stevenhagen and Alice Gee [53,54,126] and Bumkyo Cho [22].
• The fifteen sections are now organized into four chapters:
– The original §§1–13, which present a complete solution of p=x2+ny2, now
constitute Chapters 1, 2 and 3.
– The new Chapter 4 consists of the original §14 (on elliptic curves) and the
new §15 (on Shimura Reciprocity).
• An “Additional References” section has been added to supplement the original
references [1]–[112]. This section is divided into five parts:
– The first part consists of references [A1]–[A24] that are cited in the text.
These references (by no means complete) provide updates to the book.
ix