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Library of Congress Cataloging-in-Publication Data:

Cox, David A.

Primes of the form x² + ny²: Fermat, class field theory, and complex multiplication/David A. Cox. — Second edition, pages cm

Originally published: Primes of the form x² + ny², 1989.

Includes bibliographical references and index.

ISBN 978-1-118-39018-4 (cloth)

1. Numbers, Prime. 2. Mathematics. I. Title. II. Title: Primes of the form x² + ny².

QA246.C69 2013

512.7’23—dc23

2013000406

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PREFACE TO THE 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 curious omissions in the book. There may also be errors of history or attribution (for which I take full responsibility), and doubtless some of the proofs can be improved. Corrections and comments are welcome!

I would like to thank my colleagues in the number theory seminars of Oklahoma State University and the Five Colleges (Amherst College, Hampshire College, Mount Holyoke College, Smith College and the University of Massachusetts) for the opportunity to present material from this book in preliminary form. Special thanks go to Dan Flath and Peter Norman for their comments on earlier versions of the manuscript. I also thank the reference librarians at Amherst College and Oklahoma Slate University for their help in obtaining books through interlibrary loan.

David A. Cox

Amherst, Massachsusetts
August 1989

PREFACE TO THE 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 [A 10, A11, A23] and Bumkyo Cho [A6].

• 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 One, Two and Three.

– The new Chapter Four 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.

– The remaining four parts give some references (also not complete) for further reading that are relevant to the topics covered in Chapters One, Two, Three and Four.

• The expanded Notation section now includes all notation used in the book. Specialized notation is listed according to the page where it first appears.

The other changes to the text are very minor, mostly to enhance clarity, improve formatting, and simplify some of the proofs. One exception is the addition of new exercises: at the end of §12, Exercise 12.31 shows how Ramanujan could have derived Weber's formula for images (thanks to Heng Huat Chan), and at the end of §14, Exercise 14.24 gives an elliptic curve primality test for Mersenne numbers due to Dick Gross [A 12] (thanks to Alice Silverberg).

The web site for the book includes typographical errors and a link to supplementary exercises for §§1–3 written by Jeffrey Stopple. The URL of the web site is

http://www.cs.amherst.edu/~dac/primes.html

I would like to thank the following people for the errors they found in the first edition and for the suggestions they made: Michael Baake, Dominique Bemardi, Jeff Beyerl, Reinier Bröker, Tony Feng, Nicholas Gavrielides, Lee Goswik, Christian Guenther, Shiv Gupta, Kazuo Hata, Yves Hellegouarach, Norm Hurt, Tim Hutchinson, Trevor Hyde, Maurice Kostas, Susumu Kuninaga, Franz Lemmermeyer, Joseph Lipman, Mario Magioladitis, David May, Stephen Mildenhall, Takashi Ono, Frans Oort, Alf van der Poorten, Jerry Shurman, Alice Silverberg, Neil Sloane, Steve Swanson, Cihangir Tezcan, Satoshi Tomabechi, Fan Xingyuan and Noriko Yui.

Please let me know if you find any errors in the new edition!

My hope is that the second edition of Primes of the Form x² + ny² will help bring this wonderful part of number theory to a new audience of students and researchers.

David A. Cox

Amherst, Massachsusetts
November 2012

NOTATION

The following standard notation will be used throughout the book.

Notation for Chapter One

Notation for Chapter Two

Notation for Chapter Three

Notation for Chapter Four

INTRODUCTION

Most first courses in number theory or abstract algebra prove a theorem of Fermat which states that for an odd prime p,

This is only the first of many related results that appear in Fermat's works. For example, Fermat also states that if p is an odd prime, then

These facts are lovely in their own right, but they also make one curious to know what happens for primes of the form x² + 5y², x² + 6y², etc. This leads to the basic question of the whole book, which we formulate as follows:

Basic Question 0.1. Given a positive integer n, which primes p can be expressed in the form

where x and y are integers?

We will answer this question completely, and along the way we will encounter some remarkably rich areas of number theory. The first steps will be easy, involving only quadratic reciprocity and the elementary theory of quadratic forms in two variables over images. These methods work nicely in the special cases considered above by Fermat. Using genus theory and cubic and biquadratic reciprocity, we can treat some more cases, but elementary methods fail to solve the problem in general. To proceed further, we need class field theory. This provides an abstract solution to the problem, but doesn't give explicit criteria for a particular choice of n in x² + ny². The final step uses modular functions and complex multiplication to show that for a given n, there is an algorithm for answering our question of when p = x² + ny².

This book has several goals. The first, to answer the basic question, has already been stated. A second goal is to bridge the gap between elementary number theory and class field theory. Although our basic question is simple enough to be stated in any beginning course in number theory, we will see that its solution is intimately bound up with higher reciprocity laws and class field theory. A related goal is to provide a well-motivated introduction to the classical formulation of class field theory. This will be done by carefully stating the basic theorems and illustrating their power in various concrete situations.

Let us summarize the contents of the book in more detail. We begin in Chapter One with the more elementary approaches to the problem, using the works of Fermat, Euler, Lagrange, Legendre and Gauss as a guide. In §1, we will give Euler's proofs of the above theorems of Fermat for primes of the form x² + y², x² + 2y² and x² + 3y², and we will see what led Euler to discover quadratic reciprocity. We will also discuss the conjectures Euler made concerning p = x² + ny² for n > 3. Some of these conjectures, such as

(0.2) images

are similar to Fermat's theorems, while others, like

are quite unexpected. For later purposes, note that this conjecture can be written in the following form:

(0.3) images

In §2, we will study Lagrange’s theory of positive definite quadratic forms. After introducing the basic concepts of reduced form and class number, we will develop an elementary form of genus theory which will enable us to prove (0 .2 ) and similar theorems. Unfortunately, for cases like (0.3), genus theory can only prove the partial result that

(0.4) images

The problem is that x² + 27y² and 4x² + 2xy + 7y² lie in the same genus and hence can't be separated by simple congruences. We will also discuss Legendre's tentative attempts at a theory of composition.

While the ideas of genus theory and composition were already present in the works of Lagrange and Legendre, the real depth of these theories wasn't revealed until Gauss came along. In §3 we will present some basic results in Gauss' Disquisitiones Arithmeticae, and in particular we will study the remarkable relationship between genus theory and composition. But for our purposes, the real breakthrough came when Gauss used cubic reciprocity to prove Euler's conjecture (0.3) concerning p = x² + 27y². In §4 we will give a careful statement of cubic reciprocity, and we will explain how it can be used to prove (0.3). Similarly, biquadratic reciprocity can be used to answer our question for x² + 64y². We will see that Gauss clearly recognized the role of higher reciprocity laws in separating forms of the same genus. This section will also begin our study of algebraic integers, for in order to state cubic and biquadratic reciprocity, we must first understand the arithmetic of the rings images[e2πi/3] and images[i].

To go further requires class field theory, which is the topic of Chapter Two. We will begin in §5 with the Hilbert class field, which is the maximal unramified Abelian extension of a given number field. This will enable us to prove the following general result:

Theorem 0.5. Let n = 1,2 mod 4 be a positive squarefree integer. Then there is an irreducible polynomial fn(x) ∈ images[x] such that for a prime p dividing neither n nor the discriminant of fn (x),

While the statement of Theorem 0.5 is elementary, the polynomial fn(x) is quite sophisticated: it is the minimal polynomial of a primitive element of the Hilbert class field L of images.

As an example of this theorem, we will study the case n = 14. We will show that the Hilbert class field of images is L = K(α), where images. By Theorem 0.5, this will show that for an odd prime p,

(0.6) images

which answers our basic question for x² + 14y². The Hilbert class field will also enable us in §6 to give new proofs of the main theorems of genus theory.

The theory sketched so far is very nice, but there are some gaps in it. The most obvious is that the above results for x² + 27y² and x² + 14y² ((0.3) and (0.6) respectively) both follow the same format, but (0.3) does not follow from Theorem 0.5, for n = 27 is not squarefree. There should be a unified theorem that works for all positive n, yet the proof of Theorem 0.5 breaks down for general n because images is not in general the full ring of integers in images.

The goal of §§7–9 is to show that Theorem 0.5 holds for all positive integers n. This, in fact, is the main theorem of the whole book. In §7 we will study the rings images for general n, which leads to the concept of an order in an imaginary quadratic field. In §8 we will summarize the main theorems of class field theory and the Čebotarev Density Theorem, and in §9 we will introduce a generalization of the Hilbert class field called the ring class field, which is a certain (possibly ramified) Abelian extension of images determined by the order images. Then, in Theorem 9.2, we will use the Artin Reciprocity Theorem to show that Theorem 0.5 holds for all n > 0, where the polynomial fn(x) is now the minimal polynomial of a primitive element of the above ring class field. To give a concrete example of what this means, we will apply Theorem 9.2 to the case x² + 27y², which will give us a class field theory proof of (0.3). In §§8 and 9 we will also discuss how class field theory is related to higher reciprocity theorems.

The major drawback to the theory presented in §9 is that it is not constructive: for a given n > 0, we have no idea how to find the polynomial fn(x). From (0.3) and (0.6), we know f₂₇(x) and f₁₄(x), but the methods used in these examples hardly generalize. Chapter Three will use the theory of complex multiplication to remedy this situation. In §10 we will study elliptic functions and introduce the idea of complex multiplication, and then in §11 we will discuss modular functions for the group Γ₀(m) and show that the j-function can be used to generate ring class fields. As an example of the wonderful formulas that can be proved, in §12 we will give Weber's computation that

These methods will enable us to prove the Baker–Heegner–Stark Theorem on imaginary quadratic fields of class number 1. In §13 of the book we will discuss the class equation, which is the minimal polynomial of j(images). We will learn how to compute the class equation, which will lead to a constructive solution of p = x² + ny². We will then describe some work by Deuring and by Gross and Zagier. In 1946 Deuring proved a result about the difference of singular j-invariants, which implies an especially elegant version of our main theorem, and drawing on Deuring's work, Gross and Zagier discovered yet more remarkable properties of the class equation.

The first three chapters of the book present a complete solution to the problem of when p = x² + ny². In Chapter Four, we pursue two additional topics, elliptic curves in § 14 and Shimura reciprocity in § 15, that give a more modem approach to the study of complex multiplication. We also include applications to primality testing in §14. The new § 15 discusses ideles and the field of modular functions, and replaces certain pretty but ad-hoc arguments used in §12 with a more systematic treatment based on Shimura reciprocity. We also give an unexpected application to p = x² + ny².

Number theory is usually taught at three levels, as an undergraduate course, a beginning graduate course, or a more advanced graduate course. These levels correspond roughly to the first three chapters of the book. Chapter One requires only beginning number theory (up to quadratic reciprocity) and a semester of abstract algebra. Since the proofs of quadratic, cubic and biquadratic reciprocity are omitted, this book would be best suited as a supplementary text in a beginning course. For Chapter Two, the reader should know Galois theory and some basic facts about algebraic number theory (these are reviewed in §5), but no previous exposure to class field theory is assumed. The theorems of class field theory are stated without proof, so that this book would be most useful as a supplement to the topics covered in a first graduate course. Chapter Three requires a knowledge of complex analysis, but otherwise it is self-contained. (Brief but complete accounts of the Weierstrass images-function and modular functions are included in §§10 and 11.) This portion of the book should be suitable for use in a graduate seminar. The same is true for Chapter Four.

There are exercises at the end of each section, many of which consist of working out the details of arguments sketched in the text. Readers learning this material for the first time should find the exercises to be useful, while more sophisticated readers may skip them without loss of continuity.

Many important (and relevant) topics are not covered in the book. An obvious omission in Chapter One concerns forms such as x² – 2y², which were certainly considered by Fermat and Euler. Questions of this sort lead to Pell's equation and the class field theory of real quadratic fields. We have also ignored the problem of representing arbitrary integers, not just primes, by quadratic forms, and there are interesting questions to ask about the number of such representations (this material is covered in Grosswald's book [47]). In Chapter Two we give a classical formulation of class field theory, with only a brief mention of adeles and ideles. A more modem treatment can be found in Neukirch [80] or Weil [104] (see also the new §15). We also do not do justice to the use of analytic methods in number theory. For a nice introduction in the case of quadratic fields, see Zagier [111]. Our treatment of elliptic curves in Chapter Four is rather incomplete. See Husemoller [58], Knapp [A 14] or Silverman [93] for the basic theory, while more advanced topics are covered by Lang [73], Shimura [90] and Silverman [A21]. At a more elemenary level, there is the wonderful book [A22] by Silverman and Tate.

There are many books which touch on the number theory encountered in studying the problem of representing primes by x² + ny². Four books that we particularly recommend are Cohn's A Classical Invitation to Algebraic Numbers and Class Fields [19], Lang's Elliptic Functions [73], Scharlau and Opolka's From Fermat to Minkowski [86], and Weil's Number Theory: An Approach Through History [106]. These books, as well as others to be found in the References, open up an extraordinarily rich area of mathematics. The purpose of this book is to reveal some of this richness and to encourage the reader to learn more about it.

Notes on the Second Edition

The original text of the book consisted of §§1–14. For the second edition, we added the new §15 on Shimura reciprocity described above.

As a supplement to the references for the first edition, a new section Additional References has been added. The new references cited in the text are indicated with a leading “A” (e.g., the references Knapp [A14], Silverman [A21], and Silverman and Tate [A22] given above). This section also contains suggestions for further reading for the four chapters.

CHAPTER ONE

FROM FERMAT TO GAUSS

§1. FERMAT, EULER AND QUADRATIC RECIPROCITY

In this section we will discuss primes of the form x² + ny², where n is a fixed positive integer. Our starting point will be the three theorems of Fermat for odd primes p

(1.1) images

mentioned in the introduction. The goals of §1 are to prove (1.1) and, more importantly, to get a sense of what’s involved in studying the equation p = x² + ny² when n > 0 is arbitrary. This last question was best answered by Euler, who spent 40 years proving Fermat’s theorems and thinking about how they can be generalized. Our exposition will follow some of Euler’s papers closely, both in the theorems proved and in the examples studied. We will see that Euler’s strategy for proving (1.1) was one of the primary things that led him to discover quadratic reciprocity, and we will also discuss some of his remarkable conjectures concerning p = x² + ny² for n > 3. These conjectures touch on quadratic forms, composition, genus theory, cubic and biquadratic reciprocity, and will keep us busy for the rest of the chapter.

A. Fermat

Fermat’s first mention of p = x² + y² occurs in a 1640 letter to Mersenne [35, Vol. II, p. 212], while p = x² + 2y² and p = x² + 3y² come later, first appearing in a 1654 letter to Pascal [35, Vol. II, pp. 310–314]. Although no proofs are given in these letters, Fermat states the results as theorems. Writing to Digby in 1658, he repeats these assertions in the following form:

Every prime number which surpasses by one a multiple of four is composed of two squares. Examples are 5, 13, 17, 29, 37, 41, etc.

Every prime number which surpasses by one a multiple of three is composed of a square and the triple of another square. Examples are 7, 13, 19, 31, 37, 43, etc.

Every prime number which surpasses by one or three a multiple of eight is composed of a square and the double of another square. Examples are 3, 11, 17, 19, 41, 43, etc.

Fermat adds that he has solid proofs—“firmissimis demonstralibus” [35, Vol. II, pp. 402–408 (Latin), Vol. Ill, pp. 314–319 (French)].

The theorems (1.1) are only part of the work that Fermat did with x² + ny². For example, concerning x² + y², Fermat knew that a positive integer N is the sum of two squares if and only if the quotient of N by its largest square factor is a product of primes congruent to 1 modulo 4 [35, Vol. Ill, Obs. 26, pp. 256–257], and he knew the number of different ways N can be so represented [35, Vol. Ill, Obs. 7, pp. 243–246]. Fermat also studied forms beyond x² + y², x² + 2y² and x² + 3y². For example, in the 1658 letter to Digby quoted above, Fermat makes the following conjecture about x² + 5y², which he admits he can’t prove:

If two primes, which end in 3 or 7 and surpass by three a multiple of four, are multiplied, then their product will be composed of a square and the quintuple of another square.

Examples are the numbers 3, 7, 23, 43, 47, 67, etc. Take two of them, for example 7 and 23; their product 161 is composed of a square and the quintuple of another square. Namely 81, a square, and the quintuple of 16 equal 161.

Fermat’s condition on the primes is simply that they be congruent to 3 or 7 modulo 20. In §2 we will present Lagrange’s proof of this conjecture, which uses ideas from genus theory and the composition of forms.

Fermat’s proofs used the method of infinite descent, but that’s often all he said. As an example, here is Fermat’s description of his proof for p = x² + y² [35, Vol. II, p. 432]:

If an arbitrarily chosen prime number, which surpasses by one a multiple of four, is not a sum of two squares, then there is a prime number of the same form, less than the given one, and then yet a third still less, etc., descending infinitely until you arrive at the number 5, which is the least of all of this nature, from which it would follow was not the sum of two squares. From this one must infer, by deduction of the impossible, that all numbers of this form are consequently composed of two squares.

This explains the philosophy of infinite descent, but doesn’t tell us how to produce the required lesser prime. We have only one complete proof by Fermat. It occurs in one of his marginal notes (the area of a right triangle with integral sides cannot be an integral square [35, Vol. Ill, Obs. 45, pp. 271–272]—for once the margin was big enough!). The methods of this proof (see Weil [106, p. 77] or Edwards [31, pp. 10–14] for modem expositions) do not apply to our case, so that we are still in the dark. An analysis of Fermat’s approach to infinite descent appears in Bussotti [A5]. Weil’s book [106] makes a careful study of Fermat’s letters and marginal notes, and with some hints from Euler, he reconstructs some of Fermat’s proofs. Weil’s arguments are quite convincing, but we won’t go into them here. For the present, we prefer to leave things as Euler found them, i.e., wonderful theorems but no proofs.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

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