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Overview
"All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different—by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can followsystematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat's Last Theorem."—Peter Galison, Harvard University
Editorial Reviews
Nature  Timothy Gowers
The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before.SIAM Review  William M. McGovern
The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject.Mathematical Reviews  Lindsay N. Childs
To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program.From the Publisher
"The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before."—Timothy Gowers, Nature"The authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields."—
Science News
"The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject."—William M. McGovern, SIAM Review
"Unique. . . . [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics."—Lindsay N. Childs, Mathematical Reviews
"To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program."—Lindsay N. Childs, MathSciNet
Nature
The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before.— Timothy Gowers
Science News
The authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields.SIAM Review
The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject.— William M. McGovern
Mathematical Reviews
Unique. . . . [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics.— Lindsay N. Childs
MathSciNet
To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program.— Lindsay N. Childs
Product Details
Meet the Author
Avner Ash is professor of mathematics at Boston College and the coauthor of "Smooth Compactification of Locally Symmetric Varieties". Robert Gross is associate professor of mathematics at Boston College.
Read an Excerpt
Fearless Symmetry
Exposing the Hidden Patterns of NumbersBy Avner Ash Robert Gross
Princeton University Press
Copyright © 2006 Princeton University PressAll right reserved.
ISBN: 0691124922
Chapter One
REPRESENTATIONSRoad Map
To start our journey, we discuss the basic concept of representation from a formal point of view. This is the key concept underlying the numbertheoretic methods of Galois representations that are our goal. To flesh out the abstract formalism, we go through an example: The ordinary act of counting can be viewed as a representation of sets. So we give (or review) mathematical definitions of sets, functions, morphisms, and representations, which will be with us for the whole book.
The Bare Notion of Representation
Before we narrow our focus to mathematical concepts, we start by discussing the general concept of a representation. In philosophy, the concept of one thing representing or misrepresenting another thing is a central concern. The distinction between truth and appearance, the thinginitself and its representation, is a keynote of philosophy. It plays a critical role in the works of such figures as Plato, Kant, Schopenhauer, and Nietzsche. Generally speaking, for these philosophers the "appearance" of something is thought to be an impediment or veil, which we wish to penetrate through to thereality acting behind it. But in mathematics, matters stand somewhat differently.
Consider, in an abstract way, the relationship that occurs when one thing represents another. Say B represents A. We have three terms that stand together in some kind of relationship: A, B, and the fact that B represents A. We can call this fact X. It is important to remember that, in a representation, the three terms A, B, and X are usually distinct.
For example, A may be a citizen of Massachusetts, B her state representative, and X the legal fact that B represents A by voting in the legislature on her behalf. Or, to jump ahead, A may be an abstract group, B a group of matrices, and X a morphism from A to B. (We will define these terms later.)
It can happen, though, that A = B. For instance, B may be said to (also) represent herself in the state legislature. Or A may be a group of matrices and B the same group of matrices. But whether A = B or A ‡ B, we call these relationships "representations." Note that the fact of representation, X, is always going to be different from A and B, because A and B are objects and X is a fact of representation.
Now, what would be a good picture of A, B, and X? We can view X as an arrow going from A to B. This captures the oneway quality of the relationship, showing that B is representing A, not vice versa:
A [right arrow] [B.sup.2]
We can abstract even further, if we do not want to name A and B and we just want to visualize their relationship. We can picture them with dots. Then the picture of a representation becomes
[right arrow]
which is the ultimate in abstraction. The dots are just placeholders for the names of the objects. The two dots can stand for two different objects or the same object. The dot or object from which the arrow emanates is called the source of that arrow, and the dot or object to which the arrow goes is called the target of that arrow.
In normal life, if A represents B, B and A can be very different kinds of things. For instance, a flag can represent a country, a slogan on a Tshirt can represent an idea, and a mental image can represent a beloved person. In mathematics, the situation is different. All the mathematical entities we encounter or invent are considered to be on the same plane and have the same degree and type of reality or ideality: They are all mathematical entities.
What are representations used for? They explain one thing by means of another. The object we want to understand is the "thing": the thinginitself, the source. The object that we know quite a bit about already, to which we compare the source via a representation, we call the standard object. It is the site of appearance, the target.
Our conventions might not correspond to your expectations. The target, the object at the head of the arrow, is the piece of the picture that we understand better. We will derive information about the source by using properties of both the arrow and the target.
An Example: Counting
We look at the simplest possible example, one that goes back to prehistory: counting. Suppose we have a sack of potatoes or a flock of sheep. We want to know how many potatoes or sheep we have.
This is a much more sophisticated question than knowing whether they are the same in number as another sack of potatoes or another flock of sheep. We start with the less sophisticated question. Suppose we want to know whether the flock of sheep being herded home this evening is the same size as the herd we let out to the pasture in the morning. In the morning, we put a small pebble in our pouch for each sheep as it went out of the fold. Now we take a pebble out of the pouch as each sheep returns to the fold.
We were careful to make sure the pouch was empty in the morning before we began, and careful not to put anything in or take anything out during the day. So if the pouch becomes empty exactly as the last sheep comes in, we are happy. A mathematician says that we have demonstrated the existence of a onetoone correspondence from the sheep in the morning to the sheep in the evening.
To make this mathematically precise, we make two definitions:
DEFINITION: A set is a collection of things, which are called the elements of the set.
For example, the collection of all odd numbers is a set, and the odd number 3 is an element of that set.
DEFINITION: A onetoone correspondence from a set A to a set B is a rule that associates to each element in A exactly one element in B, in such a way that each element in B gets used exactly once, and for exactly one element in A.
Digression: Definitions
A mathematician uses the term "definition" in a way that might be surprising to nonmathematicians. The Oxford English Dictionary defines "definition" as "a precise statement of the essential nature of a thing." Mathematicians agree that a definition should be "precise," but we are not so sure about capturing the "essential nature." Our definition of onetoone correspondence above will let you recognize a onetoone correspondence if one is shown to you. Suppose that A is the set {red, blue, green} and B is the set {1, 2, 3}. Then a onetoone correspondence between the two sets is given by
red [right arrow] 1 blue [right arrow] 2 green [right arrow] 3
You can check that this associates to each element of the set A a different element of the set B, and that each element of the set B is used once.
Our definition of onetoone correspondence, however, does not tell you the "essential nature" of a onetoone correspondence. We have given you no clue why you should care about onetoone correspondences, nor does our definition tell you how to make a onetoone correspondence.
Even when a mathematical definition technically has all of the properties listed by the OED, it often strikes nonmathematicians as unusual. A mathematical definition can redefine a commonly used word to mean something else. For example, mathematicians refer to "simple" groups, which are in fact not particularly simple. They define the words "tree" and "quiver" in ways that have nothing to do with oaks and arrows.
Sometimes a mathematician defines an object in terms of its properties, and only then proves that an object with these properties exists. Here is an example: The greatest common divisor of two positive integers a and b can be defined to be a positive number d so that:
1. d divides a.
2. d divides b.
3. If c is any other number that divides both a and b, then c divides d.
With this definition, it is not obvious that the greatest common divisor exists, because there might not be any number d that satisfies all three properties. So right after making the definition, it should be proved that a number with the properties outlined actually exists.
Counting (Continued)
In our example, each pebble corresponded to one sheep in the morning and one sheep in the afternoon. This sets up the rule that associates to each morning sheep the afternoon sheep that shared its pebble. This rule is a onetoone correspondence under the conditions of our story.
But we do not need to know any set theory, nor what a onetoone correspondence is, to count sheep in this way. In fact, we do not even need to know how to count! In a book about Sicily in the 1950s (Dolci, 1959), a young shepherd boy was interviewed who did not know how to count:
I can't count, but even when I was a long way away, I could see if one of my goats was missing. I knew every goat in my herdit was a big herd, but I could tell every one of them apart. I could tell what kid belonged to what mother The master used to count them to see if they were all there, but I knew they were all there without counting them.
You can see that for the shepherd boy, counting was not necessary. Nor is it required if we want to sell our flock for one dollar a sheep. We just pair up the dollars and the sheep. And in the case of two sacks of potatoes, we can take one potato out of each sack and throw the pair of potatoes over our shoulders. We repeat until one sack is empty. If the other sack is also empty, we have confirmed that there were the same number of potatoes in each sack to begin with.
Counting Viewed as a Representation
But if there are thousands of potatoes, or if we want to keep a record, or tell someone far away how many sheep we have, something else needs to be done, involving languagein this case, mathematical language. The flock of sheep is our "thing," our source object. For a target, we need a standard object that we know how to count in a standard way. This is the series of counting words, for example, in English, "one, two, three, ..." As each sheep enters the fold, we count it with the next word in the series, and the last counting word that we utter is the number of sheep.
Again we have made a onetoone correspondence, but this time with a standard object, so we have something to write home about. The folks at home have the same standard, so they will know how to interpret our report. (If we report our result to people who do not know the English counting words, they will not know how many sheep we have.)
In the case of the two sacks of potatoes, if we use the tossingoverourshoulders method, when we are done we will know whether the sacks contained the same number of potatoes or not, but the place will be strewn with potatoes and we will not know what that number is. If instead we use counting words, we can count the potatoes one sack at a time, neatly, and then compare the answers.
The Definition of a Representation
A onetoone correspondence is an example of a function and of a morphism. We will be using these terms throughout this book. We will take a stab at defining them now, and refine and amplify the definitions as we continue.
DEFINITION: A function from a set A to a set B is a rule that assigns to each element in A an element of B. If f is the name of the function and a is an element of A, then we write f (a) to mean the element of B that is assigned to a. A function f is often written as f : A [right arrow] B.
DEFINITION: A morphism is a function from A to B that captures at least part of the essential nature of the set A in its image in B.
We must be intentionally vague in this chapter about the way that a morphism "captures the essential nature" of A, mostly because it depends on the nature of the entities A and B. When we use the word "morphism" later in the book, our source A and target B will both be groups. After we have defined "group" in chapter 2, we will revisit the idea of a "morphism of groups" in chapter 12.
Some people may think "morphism" is an ugly word, but it is the standard mathematical term for this concept. The longer word "homomorphism" is also used, but we will stick with the shorter version. It derives from the Greek word for "form," and we view the "essential nature" captured by a morphism as the "form" of A.
There are many kinds of functions, but the most useful ones for us are the morphisms from a source to a wellunderstood standard target. We will call this a representation. It is implicit that the target we choose is one that we know a lot about, so that from our knowledge that there is a morphism, and better yet our knowledge of some additional properties of the morphism, we can obtain new knowledge about the source object.
DEFINITION: A representation is a morphism from a source object to a standard target object.
Counting and Inequalities as Representations
Going back to the counting example, we think about finite setsfor example, {sun, earth, moon, Jupiter} or {1, Kremlin, p} or any set that contains a finite number of items. This collection of finite sets contains the special sets {1}, {1, 2}, {1, 2, 3}, and so on. In the context of counting, given any two finite sets A and B, a morphism is a onetoone correspondence from A to B. A representation in this case is a morphism from the source (a given finite set, e.g., the set of sheep in your flock) to the target, which must be one of the special sets {1}, {1, 2}, {1, 2, 3}, and so on. The special property that we demand of the morphisms in the context of counting is that they should be onetoone correspondences. For example, if you have a flock of exactly three sheep for your source, a representation of that flock must have {1, 2, 3} as its target. Thus, the "essential nature" of the source that is preserved by the morphism, in this context, is the number of elements it contains.
There are a lot of possible morphismsn! to be exact, where n is the number of elements in the source and target. When we are counting the number of elements in a set, we do not actually care about which morphism we grasp onto. But there is no choice about the target: it is {1, 2, 3, ... , n} if and only if n is the number of elements in our source.
We could alter the counting process, and stipulate that a morphism be a onetoone correspondence from the source to a subset of the target. But that would allow us to count the three oranges on our desk as "19, 3, 55," for example, which is useless.
Or is it? If that is our count, then we know that there are three oranges, because {19, 3, 55} is a set of three numbers. But how do we know how many numbers are in the set {19, 3, 55}? We still have to count them, so this technique has not helped us.
Suppose that we require the count to go in order of size. Then the above example is invalid, but "3, 19, 55" is valid. As always, knowing the last number in the count is the point. In this case, we would then know that the source has at most 55 objects. This leads to the concept of less than or equal. We could now generate the science of inequalities by using this kind of morphism.
(Continues...)
Table of Contents
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
ZEquations 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 MuchNeeded 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: GraecoLatin 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 ZPolynomial 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 TWODIMENSIONAL REPRESENTATIONS 200
Roots of Unity 200
How Frobq Acts on Roots of Unity 202
OneDimensional Galois Representations 204
TwoDimensional Galois Representations Arising from the pTorsion Points of an Elliptic Curve 205
How Frob_{q} Acts on pTorsion Points 207
The 2Torsion 209
An Example 209
Another Example 211
Yet Another Example 212
The Proof 214
CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216
Simultaneous Eigenelements 217
The ZVariety 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 TaylorWiles Did 252
Generalized Fermat Equations 254
What Henri Darmon and Loyc 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