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Fearless Symmetry by Robert Gross, Avner Ash

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xiv CONTENTS
Linearization 228
´
Etale Cohomology 229
Conjectures about
´
Etale Cohomology 231
C
HAPTER 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
C
HAPTER 22. FERMATS 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
¨
ıc Merel Did 255
Prospects for Solving the Generalized Fermat Equations 256
C
HAPTER 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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