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.

"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

• 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 Frobp 184 An Example of Computing Frobenius Elements 185 Frobp and Factoring Polynomials modulo p 186 Appendix: The Official Definition of the Bad Primes fora Galois Representation 188 Appendix: The Official Definition of "Unramified" and Frobp 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 Frobq 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 x2-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 Etale 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 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

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.

"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

• 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 Frobp 184 An Example of Computing Frobenius Elements 185 Frobp and Factoring Polynomials modulo p 186 Appendix: The Official Definition of the Bad Primes fora Galois Representation 188 Appendix: The Official Definition of "Unramified" and Frobp 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 Frobq 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 x2-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 Etale 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 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

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.

"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

• 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 Frobp 184 An Example of Computing Frobenius Elements 185 Frobp and Factoring Polynomials modulo p 186 Appendix: The Official Definition of the Bad Primes fora Galois Representation 188 Appendix: The Official Definition of "Unramified" and Frobp 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 Frobq 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 x2-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 Etale 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 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