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Modular Forms and Fermat's Last Theorem 1997. Corr. 2nd Printing ed. [Kõva köide]

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  • Formaat: Hardback, 601 pages, 15 figures
  • Ilmumisaeg: 14-Jan-2000
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387946098
  • ISBN-13: 9780387946092
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Modular Forms and Fermat's Last Theorem 1997. Corr. 2nd Printing ed.Suurem pilt
  • Formaat: Hardback, 601 pages, 15 figures
  • Ilmumisaeg: 14-Jan-2000
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387946098
  • ISBN-13: 9780387946092
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780387946092.html
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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
Preface v(8) Contributors xiii(4) Schedule of Lectures xvii(2) Introduction xix CHAPTER I An Overview of the Proof of Fermats Last Theorem 1(16) GLENN STEVENS
1. A remarkable elliptic curve 2(1)
2. Galois representations 3(4)
3. A remarkable Galois representation 7(1)
4. Modular Galois representations 7(2)
5. The Modularity Conjecture and Wiless Theorem 9(1)
6. The proof of Fermats Last Theorem 10(1)
7. The proof of Wiless Theorem 10(5) References 15(2) CHAPTER II A Survey of the Arithmetic Theory of Elliptic Curves 17(24) JOSEPH H. SILVERMAN
1. Basic definitions 17(1)
2. The group law 18(1)
3. Singular cubics 18(1)
4. Isogenies 19(1)
5. The endomorphism ring 19(1)
6. Torsion points 20(1)
7. Galois representations attached to E 20(1)
8. The Weil pairing 21(1)
9. Elliptic curves over finite fields 22(2)
10. Elliptic curves over C and elliptic funcitons 24(2)
11. The formal group of an elliptic curve 26(1)
12. Elliptic curves over local fields 27(2)
13. The Selmer and Shafarevich-Tate groups 29(2)
14. Discriminants, conductors, and L-series 31(2)
15. Duality theory 33(1)
16. Rational torsion and the image of Galois 34(1)
17. Tate curves 34(1)
18. Heights and descent 35(2)
19. The conjecture of Birch and Swinnerton-Dyer 37(1)
20. Complex multiplication 37(2)
21. Integral points 39(1) References 40(1) CHAPTER III Modular Curves, Hecke Correspondences, L-Functions 41(60) DAVID F. ROHRLICH
1. Modular curves 41(20)
2. The Hecke correspondences 61(12)
3. L-functions 73(26) References 99(2) CHAPTER IV Galois Cohomology 101(20) LAWRENCE C. WASHINGTON
1. H(0), H(1), and H(2) 101(4)
2. Preliminary results 105(2)
3. Local Tate duality 107(1)
4. Extensions and deformations 108(3)
5. Generalized Selmer groups 111(2)
6. Local conditions 113(1)
7. Conditions at p 114(3)
8. Proof of theorem 2 117(3) References 120(1) CHAPTER V Finite Flat Group Schemes 121(34) JOHN TATE Introduction 121(1)
1. Group objects in a category 122(3)
2. Group schemes. Examples 125(7)
3. Finite flat group schemes; passage to quotient 132(14)
4. Raynauds results on commutative p-group schemes 146(8) References 154(1) CHAPTER VI Three Lectures on the Modularity of XXXE, 3 and the Langlands Reciprocity Conjecture 155(54) STEPHEN GELBART Lecture I. The modularity of XXXE, 3 and automorphic representations of weight one 156(20)
1. The modularity of XXXE, 3 157(7)
2. Automorphic representations of weight one 164(12) Lecture II. The Langlands program: Some results and methods 176(16)
3. The local Langlands correspondence for GL(2) 176(3)
4. The Langlands reciprocity conjecture (LRC) 179(3)
5. The Langlands functoriality principle theory and results 182(10) Lecture III. Proof of the Langlands-Tunnell theorem 192(12)
6. Base change theory 192(5)
7. Application to Artins conjecture 197(7) References 204(5) CHAPTER VII Serres Conjectures 209(34) BAS EDIXHOVEN
1. Serres conjecture: statement and results 209(13)
2. The cases we need 222(2)
3. Weight two, trivial character and square free level 224(6)
4. Dealing with the Langlands-Tunnell form 230(9) References 239(4) CHAPTER VIII An Introduction to the Deformation Theory of Galois Representations 243(70) BARRY MAZUR
Chapter I. Galois representations 246(5)
Chapter II. Group representations 251(8)
Chapter III. The deformation theory for Galois representations 259(8)
Chapter IV. Functors and representability 267(17)
Chapter V. Zariski tangent spaces and deformation problems subject to conditions 284(10)
Chapter VI. Back to Galois representations 294(15) References 309(4) CHAPTER IX Explicit Construction of Universal Deformation Rings 313(14) BART DE SMIT HENDRIK W. LENSTRA, JR.
1. Introduction 313(1)
2. Main results 314(3)
3. Lifting homomorphisms to matrix groups 317(1)
4. The condition of absolute irreducibility 318(2)
5. Projective limits 320(3)
6. Restrictions on deformations 323(1)
7. Relaxing the absolute irreducibility condition 324(2) References 326(1) CHAPTER X Hecke Algebras and the Gorenstein Property 327(16) JACQUES TILOUINE
1. The Gorenstein property 328(2)
2. Hecke algebras 330(1)
3. The main theorem 331(3)
4. Strategy of the proof of theorem 3.4 334(1)
5. Sketch of the proof 335(5) Appendix 340(1) References 341(2) CHAPTER XI Criteria for Complete Intersections 343(14) BART DE SMIT KARL RUBIN RENE SCHOOF Introduction 343(2)
1. Preliminaries 345(2)
2. Complete intersections 347(3)
3. Proof of Criterion I 350(3)
4. Proof of Criterion II 353(2) Bibliography 355(2) CHAPTER XII XXX-adic Modular Deformations and Wiless Main Conjecture 357(16) FRED DIAMOND KENNETH A. RIBET
1. Introduction 357(1)
2. Strategy 358(1)
3. The Main Conjecture 359(4)
4. Reduction to the case XXX = 0 363(7)
5. Epilogue 370(1) Bibliography 370(3) CHAPTER XIII The Flat Deformation Functor 373(48) BRIAN CONRAD Introduction 373(1)
0. Notation 374(1)
1. Motivation and flat representations 375(19)
2. Defining the functor 394(3)
3. Local Galois cohomology and deformation theory 397(9)
4. Fontaines approach to finite flat group schemes 406(6)
5. Applications to flat deformations 412(6) References 418(3) CHAPTER XIV Hecke Rings and Universal Deformation Rings 421(26) EHUD DE SHALIT
1. Introduction 421(3)
2. An outline of the proof 424(8)
3. Proof of proposition 10 - On the structure of the Hecke algebra 432(4)
4. Proof of proposition 11 - On the structure of the universal deformation ring 436(6)
5. Conclusion of the proof: Some group theory 442(2) Bibliography 444(3) CHAPTER XV Explicit Families of Elliptic Curves with Prescribed Mod N Representations 447(16) ALICE SILVERBERG Introduction 447(1) Part
1. Elliptic curves with the same mod N representation 448(6)
1. Modular curves and elliptic modular surfaces of level N 448(1)
2. Twists of Y(N) and W(N) 449(1)
3. Model for W when N = 3, 4, or 5 450(1)
4. Level 4 451(3) Part
2. Explicit families of modular elliptic curves 454(7)
5. Modular j invariants 454(1)
6. Semistable reduction 455(1)
7. Mod 4 representations 456(1)
8. Torsion subgroups 457(4) References 461(2) CHAPTER XVI Modularity of Mod 5 Representations 463(12) KARL RUBIN Introduction 463(2)
1. Preliminaries: Group theory 465(1)
2. Preliminaries: Modular curves 466(4)
3. Proof of the irreducibility theorem (Theorem 1) 470(1)
4. Proof of the modularity theorem (Theorem 2) 470(1)
5. Mod 5 representations and elliptic curves 471(2) References 473(2) CHAPTER XVII An Extension of Wiles Results 475(24) FRED DIAMOND
1. Introduction 475(1)
2. Local representations mod XXX 476(4)
3. Minimally ramified liftings 480(1)
4. Universal deformation rings 481(1)
5. Hecke algebras 482(1)
6. The main results 483(1)
7. Sketch of proof 484(4) References 488(3) APPENDIX TO CHAPTER XVII Classification of XXXE, XXX by the j Invariant of E 491(8) FRED DIAMOND KENNETH KRAMER CHAPTER XVIII Class Field Theory and the First Case of Fermats Last Theorem 499(6) HENDRIK W. LENSTRA, JR. PETER STEVENHAGEN CHAPTER XIX Remarks on the History of Fermats Last Theorem 1844 to 1984 505(22) MICHAEL ROSEN Introduction 507(1)
1. Fermats last theorem for polynomials 507(1)
2. Kummers work on cyclotomic fields 508(5)
3. Fermats last theorem for regular primes and certain other cases 513(4)
4. The structure of the p-class group 517(4)
5. Suggested readings 521(1) Appendix A: Kummer congruence and Hilberts theorem 94 522(2) Bibliography 524(3) CHAPTER XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves 527(22) GERHARD FREY
1. Conjectures 527(13)
2. The generic case 540(2)
3. K = Q 542(6) References 548(1) CHAPTER XXI Wiles Theorem and the Arithmetic of Elliptic Curves 549(24) HENRI DARMON
1. Prelude: plane conics, Fermat and Gauss 549(3)
2. Elliptic curves and Wiles theorem 552(5)
3. The special values of L(E Q, s) at s = 1 557(6)
4. The Birch and Swinnerton-Dyer conjecture 563(3) References 566(7) Index 573