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Publisher:

Princeton University Press

Publication Date:

2006

Number of Pages:

271

Format:

Hardcover

Price:

24.95

ISBN:

0691124922

Category:

General

[Reviewed by , on ]

Darren Glass

07/3/2006

It would be fair to say that the recent explosion of math books for popular audiences began with the publication of several books on Fermat’s Last Theorem in the mid-1990s including (but not limited to) Simon Singh’s *Fermat’s Enigma* and Amir Aczel’s *Fermat’s Last Theorem*. This seemed natural as it was a topic that was on the cutting edge of modern mathematics while still being an easy-to-state problem that most middle school students could understand. Several years later there was a rash of books on the Riemann Hypothesis (including *Prime Obsession* by John Derbyshire, *The Music of the Primes* by Marcus du Sautoy and Karl Sabbagh’s *The Riemann Hypothesis*), which seemed another natural choice due to its mathematical depth yet (somewhat) accesible topics.

Unfortunately, such problems are few and far between, and most pop math books are left with a choice between topics that are not as cutting edge to mathematicians or topics that are not as easily accessible to the lay reader. I have often been impressed with the boldness of authors’ attempts to describe serious mathematics to a lay audience, and only somewhat less impressed with the quality of the results. But I thought that the game was over when I read Leonard Wapner’s *The Pea and The Sun*, a pop-math exposition of the Banach-Tarski Theorem. Certainly this was as mathematically serious as anyone would attempt to get in a book for the common reader. Right?

You can imagine my shock several weeks ago when the intrepid editor of Read This! and *MAA Reviews* sent me an email asking if I was interested in reviewing “a new book about the Langlands program intended for a popular audience”. For those readers who are unfamiliar with the Langlands program, it is a series of interconnected theorems and conjectures which connect number theory to representation theory. One way to think about it is as a massive extension of the theorems of quadratic reciprocity that we learn about in an elementary number theory course, but this extension deals with generalized reciprocity laws involving L-functions associated to Galois groups and their representations. The Langlands program has been largely shown to be correct in the case of local fields and function fields (work on the latter won Laffourge the Fields medal in 2002), but there are still many open questions in the case of number fields, and it remains a very active field of research in which it is not yet clear what the exact conjectures should be, let alone if they are true. In other words, this is seriously deep mathematics. I would be very excited to see a well-written book on the Langlands program which is accessible to algebraists, let alone accessible to the general public, so you can imagine the skepticism that I felt at reviewing this book, but my curiosity got the best of me and I quickly agreed to do it.

The book, *Fearless Symmetry: Exposing the Hidden Patterns of Numbers* by Avner Ash and Robert Gross, showed up in the mail several days later, and I read it very quickly. That was three weeks ago, and I still am not sure what I think of the book.

Let’s start by discussing what is covered in the book. The first part of the book is dedicated to “Algebraic Preliminaries” — the authors begin by defining what a representation is and then move on to introduce the idea of groups, motivated by symmetries of spheres but also discussing in detail the theory of permutations, modular arithmetic, and complex numbers. (Yes, you read that right. They define group representations *before* defining what groups are. And they define an abstract group 20 pages before introducing concepts such as prime numbers and imaginary numbers. And I have to admit that it works. But more on that later.) They then go on to define varieties as the set of solutions of polynomial equations, and give a large number of examples of how equations work and how the solution set looks different depending on what “number system” you are working in. This leads the authors to consider the question “When is –1 a square?” and look at what numbers are square numbers mod p for different primes, which involves discussing quadratic reciprocity. This first part of the book is very chatty and accessible — it reminds me of Joe Silverman’s *A Friendly Introduction to Number Theory* in its ability to explain the beginnings of deep mathematics in a language that wouldn’t scare off many readers.

The second part of the book steps things up a notch, starting off by discussing symmetries of the roots of polynomials and the absolute Galois group of the rational numbers. This part of the book also includes chapters on elliptic curves, matrices, matrix groups, and matrix representations and a more mathematically sophisticated look at group representations than was found in the first part of the book. This includes discussions of character theory, conjugacy classes, and the inverse Galois problem. The authors also introduce the concept of the Frobenius morphism, a specific element (well, sort of) of the absolute Galois group of the rational numbers associated to a specific prime number which will be important in defining generalized reciprocity laws. While this is a highly technical beast to define, the authors give a series of definitions of increasing precision (and complexity) so that the reader can get the level of technical difficulty appropriate to their background and taste for technical details.

As is likely clear from the list of topics, this part of the book is somewhat less accessible than the first part, and the authors start to go back and forth between mathematical precision and ease of understanding. While they do not make all of the same choices that I would when it comes to this tradeoff, I think that anyone who is willing to work at it (and able to keep many definitions in their head at the same time, as they often fly at the reader with enormous speed) could learn quite a bit from reading this section. I also think that this part of the book would make excellent supplemental reading in an abstract algebra or even linear algebra course, to give a less rigorous but more ‘big picture’ view of a number of topics covered in those courses. But this part definitely feels more tailored to a mathematical audience than to civilians.

The third and final part of the book is dedicated to reciprocity laws. Ash and Gross begin by defining reciprocity laws, relying heavily on the metaphor of a black box depending on a group representation into which one inputs a series of prime numbers and gets from the box the traces of the Frobenius elements. These black boxes will often contain information about modular forms or torsion points on elliptic curves or cohomology classes or other concepts that may or may not be beyond the scope of the book, and it is what you learn from these black boxes that gives you a reciprocity law. If my two sentence paraphrasing is unclear, I suggest you read Ash and Gross’s fleshing out of this metaphor as it is a very nice introduction to the idea of reciprocity laws. (For those readers who want an even deeper level of understanding, I wholeheartedly recommend BF Wyman’s 1972 article in the *American Mathematical Monthly* entitled “What is a Reciprocity Law” available through JSTOR for more details). Ash and Gross then go on to give many examples of reciprocity laws, including a discussion of why the previously discussed quadratic reciprocity law qualifies. The last chapters of the book discuss a “machine” for making Galois representations, etale cohomology, the modularity conjecture, and some applications of these laws to solving polynomial equations, including a full chapter on the proof of Fermat’s last theorem given by Andrew Wiles. While this discussion is far briefer than those in the aforementioned books by Singh and Azcel it is also much deeper mathematically.

I must confess that the third part of this book is a mystery to me. While I think that a dedicated reader with little mathematical background could get a lot out of the first two thirds of the book, it is hard for me to imagine them sticking around through this third section — while Ash and Gross do a good job of defining their terms and sweeping the technical details (not to mention the proofs) under the metaphorical rug, there are still far more definitions and ideas floating around for me to imagine anyone other than a mathematician reading. In some ways I think that the ideal reader for this third section would be a mathematician who works in a different area of mathematics (and yes, I’m looking at most of you reading this review) but I worry that, with so many of those details left under the rug, many mathematicians would be left unsatisfied and wanting more details or a deeper explanation of what is really happening. Furthermore, those readers probably would not stick around that far in the book if they started in the beginning, as the early chapters on topics such as modular arithmetic and imaginary numbers might be too elementary.

There is much to admire in *Fearless Symmetry*: the style of writing is engrossing and the authors do an admirable job of explaining deep concepts in a way that is accessible more often than not. I am not sure that the book is wholly successful in its goals — in part because it is still not clear to me what the goals of the book really are — but I am glad that I spent some time with this book, and I hope that more writers will undertake this type of challenge in the future. I think that it is beneficial to the mathematical community to have this kind of a book exist both from an expository point of view and from a public-relations point of view. But when our editor emails me to ask to review a pop-math book on open conjectures about noncommutative C* algebras, I think I’ll pass.

Darren Glass (dglass@gettysburg.edu) is an Assistant Professor at Gettysburg College.

PART ONE: ALGEBRAIC PRELIMINARIES

CHAPTER 1. REPRESENTATIONS 3

The Bare NotionofRepresentation 3

An Example: Counting 5

Digression: Definitions 6

Counting (Continued)7

Counting Viewed as a Representation 8

The Definition of a Representation 9

Counting and Inequalities as Representations 10

Summary 11

CHAPTER 2. GROUPS 13

The Group of Rotations of a Sphere 14

The General Concept of "Group" 17

In Praise of Mathematical Idealization 18

Digression: Lie Groups 19

CHAPTER 3. PERMUTATIONS 21

The abc of Permutations 21

Permutations in General 25

Cycles 26

Digression: Mathematics and Society 29

CHAPTER 4. MODULAR ARITHMETIC 31

Cyclical Time 31

Congruences 33

Arithmetic Modulo a Prime 36

Modular Arithmetic and Group Theory 39

Modular Arithmetic and Solutions of Equations 41

CHAPTER 5. COMPLEX NUMBERS 42

Overture to Complex Numbers 42

Complex Arithmetic 44

Complex Numbers and Solving Equations 47

Digression: Theorem 47

Algebraic Closure 47

CHAPTER 6. EQUATIONS AND VARIETIES 49

The Logic of Equality 50

The History of Equations 50

Z-Equations 52

Vari eti es 54

Systems of Equations 56

Equivalent Descriptions of the Same Variety 58

Finding Roots of Polynomials 61

Are There General Methods for Finding Solutions to Systems of Polynomial Equations? 62

Deeper Understanding Is Desirable 65

CHAPTER 7. QUADRATIC RECIPROCITY 67

The Simplest Polynomial Equations 67

When is -1 aSquaremodp? 69

The Legendre Symbol 71

Digression: Notation Guides Thinking 72

Multiplicativity of the Legendre Symbol 73

When Is 2 a Square mod p?74

When Is 3 a Square mod p?75

When Is 5 a Square mod p? (Will This Go On Forever?) 76

The Law of Quadratic Reciprocity 78

Examples of Quadratic Reciprocity 80

PART TWO. GALOIS THEORY AND REPRESENTATIONS

CHAPTER 8. GALOIS THEORY 87

Polynomials and Their Roots 88

The Field of Algebraic Numbers Q ^{alg} 89

The Absolute Galois Group of Q Defined 92

A Conversation with s: A Playlet in Three Short Scenes 93

Digression: Symmetry 96

How Elements of G Behave 96

Why Is G a Group? 101

Summary 101

CHAPTER 9. ELLIPTIC CURVES 103

Elliptic Curves Are "Group Varieties" 103

An Example 104

The Group Law on an Elliptic Curve 107

A Much-Needed Example 108

Digression: What Is So Great about Elliptic Curves? 109

The Congruent Number Problem 110

Torsion and the Galois Group 111

CHAPTER 10. MATRICES 114

Matrices and Matrix Representations 114

Matrices and Their Entries 115

Matrix Multiplication 117

Linear Algebra 120

Digression: Graeco-Latin Squares 122

CHAPTER 11. GROUPS OF MATRICES 124

Square Matrices 124

Matrix Inverses 126

The General Linear Group of Invertible Matrices 129

The Group GL(2, Z) 130

Solving Matrix Equations 132

CHAPTER 12. GROUP REPRESENTATIONS 135

Morphisms of Groups 135

A4, Symmetries of a Tetrahedron 139

Representations of A4 142

Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves 146

CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL 149

The Field Generated by a Z-Polynomial 149

Examples 151

Digression: The Inverse Galois Problem 154

Two More Things 155

CHAPTER 14. THE RESTRICTION MORPHISM 157

The BigPicture andthe Little Pictures 157

Basic Facts about the Restriction Morphism 159

Examples 161

CHAPTER 15. THE GREEKS HAD A NAME FOR IT 162

Traces 163

Conjugacy Classes 165

Examples of Characters 166

How the Character of a Representation Determines the Representation 171

Prelude to the Next Chapter 175

Digression: A Fact about Rotations of the Sphere 175

CHAPTER 16. FROBENIUS 177

Something for Nothing 177

Good Prime, Bad Prime 179

Algebraic Integers, Discriminants, and Norms 180

A Working Definition of Frob_{p} 184

An Example of Computing Frobenius Elements 185

Frobp and Factoring Polynomials modulo p 186

Appendix: The Official Definition of the Bad Primes for a Galois Representation 188

Appendix: The Official Definition of "Unramified" and Frob_{p} 189

PART THREE. RECIPROCITY LAWS

CHAPTER 17. RECIPROCITY LAWS 193

The List of Traces of Frobenius 193

Black Boxes 195

Weak and Strong Reciprocity Laws 196

Digression: Conjecture 197

Kinds of Black Boxes 199

CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS 200

Roots of Unity 200

How Frobq Acts on Roots of Unity 202

One-Dimensional Galois Representations 204

Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve 205

How Frob_{q} Acts on p-Torsion Points 207

The 2-Torsion 209

An Example 209

Another Example 211

Yet Another Example 212

The Proof 214

CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216

Simultaneous Eigenelements 217

The Z-Variety x^{2}-W 218

A Weak Reciprocity Law 220

A Strong Reciprocity Law 221

A Derivation of Quadratic Reciprocity 222

CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS 225

Vector Spaces and Linear Actions of Groups 225

Linearization 228

´ Etale Cohomology 229

Conjectures about Étale Cohomology 231

CHAPTER 21. A LAST LOOK AT RECIPROCITY 233

What Is Mathematics? 233

Reciprocity 235

Modular Forms 236

Review of Reciprocity Laws 239

A Physical Analogy 240

CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 242

The Three Pieces of the Proof 243

Frey Curves 244

The Modularity Conjecture 245

Lowering the Level 247

Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves 249

Bring on the Reciprocity Laws 250

What Wiles and Taylor-Wiles Did 252

Generalized Fermat Equations 254

What Henri Darmon and Lo¨ýc Merel Did 255

Prospects for Solving the Generalized Fermat Equations 256

CHAPTER 23. RETROSPECT 257

Topics Covered 257

Back to Solving Equations 258

Digression: Why Do Math? 260

The Congruent Number Problem 261

Peering Past the Frontier 263

Bibliography 265

Index 269

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