Fearless Symmetry: Exposing the Hidden Patterns of Numbers - New Edition

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 for a 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