Heegner modules and elliptic curves:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Series: | Lecture Notes in Mathematics
1849 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 517 S. |
| ISBN: | 3540222901 |
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MARC
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| 100 | 1 | |a Brown, Martin L. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Heegner modules and elliptic curves |c M. L. Brown |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a X, 517 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture Notes in Mathematics |v 1849 | |
| 650 | 4 | |a Elliptische Kurve - Heegner-Punkt - Drinfeld-Modul | |
| 650 | 4 | |a Algebraic fields | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Homology theory | |
| 650 | 0 | 7 | |a Elliptische Kurve |0 (DE-588)4014487-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Drinfeld-Modul |0 (DE-588)4132653-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Heegner-Punkt |0 (DE-588)4791286-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Heegner-Punkt |0 (DE-588)4791286-8 |D s |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Introduction 1
1.1 Statement of the Tate conjecture 2
1.2 The Drinfeld modular curve X0Drin(/) 3
1.3 Analogue for F of the Shimura Taniyama Weil conjecture .... 3
1.4 Drinfeld Heegner points 4
1.5 Heegner sheaves 4
1.6 Hecke operators 4
1.7 Bruhat Tits buildings with complex multiplication 5
1.8 Bruhat Tits buildings with complex multiplication and
Drinfeld Heegner points 6
1.9 Classification of Bruhat Tits buildings with complex
multiplication 6
1.10 The Heegner module of a galois representation 7
1.11 Cohomology of the Heegner module 8
1.12 The Tate conjecture and the Heegner module 8
1.13 Statement of the main result on the Tate conjecture 9
1.14 Heegner points on the classical modular curve Xo(N)/Q 10
1.15 Prerequisites and guide 10
2 Preliminaries 13
2.1 Notation 13
2.2 Orders in quadratic field extensions 14
2.3 Ring class fields 18
2.4 The Drinfeld moduli schemes Y%lin(I), X$rin(I), Mf 23
2.5 Complex multiplication of rank 2 Drinfeld modules 26
3 Bruhat Tits trees with complex multiplication 31
3.1 The Bruhat Tits building A for SL2 of a discretely valued field 32
3.2 Lattices in quadratic extensions: elementary results 33
3.3 Lattices in quadratic extensions: discrete valuation rings 36
3.4 Proofs of the propositions of §3.3 38
VIII Contents
3.5 Explicit formulae for the conductor Exp(A) 41
3.6 Bruhat Tits trees with complex multiplication 45
3.7 The standard metric and Bruhat Tits trees with complex
multiplication 46
3.8 Classification of Bruhat Tits trees with complex multiplication 57
3.9 Lattices in quadratic extensions: Dedekind domains 62
3.10 The global Bruhat Tits net 65
3.11 Bruhat Tits nets with complex multiplication 69
4 Heegner sheaves 75
4.1 Drinfeld Heegner points, Heegner morphisms 75
4.2 Construction of Drinfeld Heegner points 76
4.3 Notation for Drinfeld Heegner points 78
4.4 Galois action on Drinfeld Heegner points 79
4.5 Hecke operators on X®rin(I) and the Bruhat Tits net
AA(SL2(F)) 80
4.6 Drinfeld Heegner points and Hecke operators 86
4.7 Elliptic curves and Drinfeld modular curves 96
4.8 Drinfeld Heegner points and elliptic curves 97
4.9 Heegner sheaves 99
5 The Heegner module 105
5.1 Group rings of finite abelian groups 106
5.2 The group rings Ac of Picard groups 109
5.3 The Heegner module of a galois representation Ill
5.4 Cech galois cohomology 121
5.5 Group rings and Cech cohomology 144
5.6 Group cohomology; Kolyvagin elements 169
5.7 Basic properties of the Heegner module 195
5.8 Proofs of the propositions of §5.7 198
5.9 Faithful flatness of the Heegner module 207
5.10 Proofs of the results of §5.9 209
5.11 The Heegner module as a Heegner sheaf 221
6 Cohomology of the Heegner module 223
6.1 General notation 224
6.2 Exact sequences and preliminary lemmas 224
6.3 Cohomology of the Heegner module: vanishing cohomology . . . 232
6.4 The main lemma 232
6.5 Proof of lemma 6.4.3 234
6.6 The main proposition 241
6.7 The submodule Jc,z,s 245
6.8 The kernel of E 273
6.9 Proofs 277
Contents IX
6.10 Galois invariants of the Heegner module:
S is an infinitesimal trait 294
6.11 Proof of theorem 6.10.7 298
7 Finiteness of Tate Shafarevich groups 329
7.1 Quasi modules 330
7.2 Igusa s theorem 339
7.3 Consequences of Igusa s theorem 341
7.4 Proof of proposition 7.3.6 352
7.5 Preliminaries 357
7.6 Statement of the main result and historical remarks 357
7.7 Tate Shafarevich groups 363
7.8 Proof that theorem 7.7.5 implies theorem 7.6.5 365
7.9 The Selmer group 366
7.10 The set V of prime numbers 367
7.11 Frobenius elements and the set XV of divisors 369
7.12 The Heegner module attached to E/F 371
7.13 Galois invariants of the Heegner module and the map r 372
7.14 The cohomology classes 7(c), S(c) 382
7.15 Tate Poitou local duality 402
7.16 Application of Tate Poitou duality 404
7.17 Equivariant Pontrjagin duality 406
7.18 Proof of theorem 7.7.5 412
7.19 Comments and errata for [Br2] 433
Appendix A. Rigid analytic modular forms 435
A.I Basic definitions 435
A.2 The Tate algebra 436
A.3 Affinoid spaces, rigid analytic spaces 437
A.4 Etale cohomology of rigid analytic spaces 440
A.5 The space f2d 444
A.6 The moduli scheme Mf 445
A.7 An analytic description of Mf 448
A.8 Rigid analytic modular forms 450
A.9 Analytic modular forms 451
2
A. 10 (/ expansions at the cusps of Mj 452
A. 11 Eisenstein series 454
A.12 Hecke operators 456
A. 13 Elliptic curves over F and modular forms 458
Appendix B. Automorphic forms and elliptic curves over
function fields 461
B.I The Bruhat Tits building for PGL over a local field 461
B.2 The building map on Qd 464
B.3 Fibres of the building map on i?2 466
X Contents
B.4 Structures of level H on Drinfeld modules;
the moduli scheme Mfj 468
B.5 Action of arithmetic subgroups of GL2(F) on T 471
B.6 Cohomology of fl 1 and harmonic cochains 479
B.7 Cohomology of the moduli space Mjj 483
B.8 Harmonic cochains and the special representation of GI^Foo) 488
B.9 Automorphic forms and the main theorem 492
B.10 The Langlands correspondence tt — t(tt) 495
B.ll Elliptic curves as images of Drinfeld modular curves 497
B. 12 The Langlands conjecture for GLn over function fields
(according to Lafforgue) 504
References 507
Index 511
|
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| indexdate | 2024-07-09T19:59:06Z |
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| spelling | Brown, Martin L. Verfasser aut Heegner modules and elliptic curves M. L. Brown Berlin [u.a.] Springer 2004 X, 517 S. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Mathematics 1849 Elliptische Kurve - Heegner-Punkt - Drinfeld-Modul Algebraic fields Curves, Elliptic Homology theory Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Drinfeld-Modul (DE-588)4132653-2 gnd rswk-swf Heegner-Punkt (DE-588)4791286-8 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 s Heegner-Punkt (DE-588)4791286-8 s Drinfeld-Modul (DE-588)4132653-2 s DE-604 Lecture Notes in Mathematics 1849 (DE-604)BV000676446 1849 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012849836&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brown, Martin L. Heegner modules and elliptic curves Lecture Notes in Mathematics Elliptische Kurve - Heegner-Punkt - Drinfeld-Modul Algebraic fields Curves, Elliptic Homology theory Elliptische Kurve (DE-588)4014487-2 gnd Drinfeld-Modul (DE-588)4132653-2 gnd Heegner-Punkt (DE-588)4791286-8 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4132653-2 (DE-588)4791286-8 |
| title | Heegner modules and elliptic curves |
| title_auth | Heegner modules and elliptic curves |
| title_exact_search | Heegner modules and elliptic curves |
| title_full | Heegner modules and elliptic curves M. L. Brown |
| title_fullStr | Heegner modules and elliptic curves M. L. Brown |
| title_full_unstemmed | Heegner modules and elliptic curves M. L. Brown |
| title_short | Heegner modules and elliptic curves |
| title_sort | heegner modules and elliptic curves |
| topic | Elliptische Kurve - Heegner-Punkt - Drinfeld-Modul Algebraic fields Curves, Elliptic Homology theory Elliptische Kurve (DE-588)4014487-2 gnd Drinfeld-Modul (DE-588)4132653-2 gnd Heegner-Punkt (DE-588)4791286-8 gnd |
| topic_facet | Elliptische Kurve - Heegner-Punkt - Drinfeld-Modul Algebraic fields Curves, Elliptic Homology theory Elliptische Kurve Drinfeld-Modul Heegner-Punkt |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012849836&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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