## Modular Forms and Fermat’s Last TheoremGary Cornell, Joseph H. Silverman, Glenn Stevens This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. The purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi- stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. Contributors to this volume include: B. Conrad, H. Darmon, E. de Shalit, B. de Smit, F. Diamond, S.J. Edixhoven, G. Frey, S. Gelbart, K. Kramer, H.W. Lenstra, Jr., B. Mazur, K. Ribet, D.E. Rohrlich, M. Rosen, K. Rubin, R. Schoof, A. Silverberg, J.H. Silverman, P. Stevenhagen, G. Stevens, J. Tate, J. Tilouine, and L. Washington. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable |

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### Contents

V | 1 |

VI | 2 |

VII | 3 |

VIII | 7 |

X | 9 |

XI | 10 |

XIII | 15 |

XIV | 17 |

XCV | 330 |

XCVI | 331 |

XCVII | 334 |

XCVIII | 335 |

XCIX | 340 |

C | 341 |

CI | 343 |

CIII | 345 |

XVI | 18 |

XVIII | 19 |

XX | 20 |

XXII | 21 |

XXIII | 22 |

XXIV | 24 |

XXV | 26 |

XXVI | 27 |

XXVII | 29 |

XXVIII | 31 |

XXIX | 33 |

XXX | 34 |

XXXII | 35 |

XXXIII | 37 |

XXXV | 39 |

XXXVI | 40 |

XXXVII | 41 |

XXXIX | 61 |

XL | 73 |

XLI | 99 |

XLII | 101 |

XLIV | 105 |

XLV | 107 |

XLVI | 108 |

XLVII | 111 |

XLVIII | 113 |

XLIX | 114 |

L | 117 |

LI | 120 |

LII | 121 |

LIV | 122 |

LV | 125 |

LVI | 132 |

LVII | 146 |

LVIII | 154 |

LIX | 155 |

LX | 156 |

LXI | 157 |

LXII | 164 |

LXIII | 175 |

LXIV | 176 |

LXV | 179 |

LXVI | 182 |

LXVII | 192 |

LXVIII | 197 |

LXIX | 204 |

LXX | 209 |

LXXII | 222 |

LXXIII | 224 |

LXXIV | 230 |

LXXV | 239 |

LXXVI | 243 |

LXXVII | 246 |

LXXVIII | 251 |

LXXIX | 259 |

LXXX | 267 |

LXXXI | 284 |

LXXXII | 294 |

LXXXIII | 309 |

LXXXIV | 313 |

LXXXVI | 314 |

LXXXVII | 317 |

LXXXVIII | 318 |

LXXXIX | 320 |

XC | 323 |

XCI | 324 |

XCII | 326 |

XCIII | 327 |

XCIV | 328 |

CIV | 348 |

CV | 350 |

CVI | 353 |

CVII | 355 |

CVIII | 357 |

CX | 358 |

CXI | 359 |

CXII | 363 |

CXIII | 370 |

CXV | 373 |

CXVII | 374 |

CXVIII | 375 |

CXIX | 394 |

CXX | 397 |

CXXI | 406 |

CXXII | 412 |

CXXIII | 418 |

CXXIV | 421 |

CXXV | 424 |

CXXVI | 432 |

CXXVII | 436 |

CXXVIII | 442 |

CXXIX | 444 |

CXXX | 447 |

CXXXII | 448 |

CXXXIII | 449 |

CXXXIV | 450 |

CXXXV | 451 |

CXXXVI | 454 |

CXXXVII | 455 |

CXXXVIII | 456 |

CXXXIX | 457 |

CXL | 461 |

CXLI | 463 |

CXLIII | 465 |

CXLIV | 466 |

CXLV | 470 |

CXLVII | 471 |

CXLVIII | 473 |

CXLIX | 475 |

CLI | 476 |

CLII | 480 |

CLIII | 481 |

CLIV | 482 |

CLV | 483 |

CLVI | 484 |

CLVII | 488 |

CLVIII | 491 |

CLIX | 499 |

CLX | 505 |

CLXII | 507 |

CLXIII | 508 |

CLXIV | 513 |

CLXV | 517 |

CLXVI | 521 |

CLXVII | 522 |

524 | |

CLXIX | 527 |

CLXXI | 540 |

CLXXII | 542 |

CLXXIII | 548 |

CLXXIV | 549 |

CLXXVI | 552 |

CLXXVII | 557 |

CLXXVIII | 563 |

CLXXIX | 566 |

571 | |

### Common terms and phrases

A-algebra A-module absolutely irreducible artinian assume automorphic base change character cocycle coefficient coefficient-A-algebra coefficient-ring cohomology commutative condition conductor Conjecture Corollary corresponding curve over Q cusp cuspidal representation cyclic cyclotomic defined deformation problem denote diagram dividing eigenform element equation equivalent exact sequence extension Fermat's Last Theorem fiber finite flat group fixed flat group scheme follows formula free of rank function field functor Galois representations given gives GK,S group scheme Hecke algebra hence homomorphism implies induced integer isogeny isomorphism kernel L-function Langlands Lemma Math matrix maximal ideal Mazur modular curves Modular elliptic curves modular forms modular of type module morphism multiplicative newform noetherian number field p-adic prime number properties Proposition prove quadratic quotient ramified reduction residue field restriction result Ribet satisfies semistable Serre space subgroup Suppose surjective Tate theory topological trivial universal deformation ring unramified Wiles