Lectures on Hilbert modular varieties and modular forms:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
AMS
2002
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| Series: | CRM Monograph series
14 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 270 S. graph. Darst. |
| ISBN: | 082181995X |
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MARC
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| 100 | 1 | |a Goren, Eyal Z. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Lectures on Hilbert modular varieties and modular forms |c Eyal Z. Goren |
| 264 | 1 | |a Providence, RI |b AMS |c 2002 | |
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| 650 | 4 | |a Formes modulaires | |
| 650 | 4 | |a Modules, Théorie des | |
| 650 | 4 | |a Variétés abéliennes | |
| 650 | 4 | |a Abelian varieties | |
| 650 | 4 | |a Forms, Modular | |
| 650 | 4 | |a Moduli theory | |
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| adam_text | Titel: Lectures on Hilbert modular varieties and modular forms
Autor: Goren, Eyal Z
Jahr: 2002
Contents
Introduction 1
Chapter 1. Tori and Abelian Varieties 5
1. Algebraic groups 5
2. Moduli problems 15
3. Moduli of elliptic curves 18
4. Modular forms 22
5. Some examples of modular forms 26
5.1. Level 1 26
5.2. Higher level 27
6. Abelian varieties over C 34
6.1. The Appell-Humbert Theorem 38
6.2. The dual abelian variety 42
Chapter 2. Complex Abelian Varieties with Real Multiplication and Hilbert
Modular Forms 49
1. Algebraic preliminaries 50
2. Complex abelian varieties with real multiplication 51
2.1. Complex and rational representations 52
2.2. Construction of families of abelian varieties with real multiplication 53
3. Hilbert modular forms 63
4. Construction of Hilbert modular forms 68
4.1. Eisenstein series 68
4.2. Other methods of constructing modular forms 70
5. More on the diagonal curve 72
5.1. Modular interpretation 72
6. Siegel s formula 74
Chapter 3. Abelian Varieties with Recti Multiplication over General Fields 77
1. Abelian varieties over a general field 78
1.1. The dual abelian variety 79
1.2. Examples 82
1.3. Abelian schemes 83
2. Finite Heisenberg groups 84
3. Honda-Tate Theorem 90
4. Ordinary abelian varieties and Serre-Tate coordinates 93
4.1. Ordinary abelian varieties 93
4.2. Serre-Tate coordinates 94
5. Abelian varieties with recti multiplication over a general field 97
6. Irreducibility of the moduli space of -level structure 101
vii
viii CONTENTS
6.1. Moduli spaces for full and p;v-level ^
6.2. Deligne s description of ordinary abelian varieties 104
6.3. Irreducibility *95
Chapter 4. p-adic Elliptic Modular Forms 109
1. Introduction 199
1.1. p-adic L-functions H9
1.2. Deformation of Galois representations 113
2. Congruences between modular forms mod p 117
3. Operators and systems of eigenvalues 124
3.1. A highbrow view of modular forms in characteristic p 124
3.2. Operators - -. ¦ 126
3.3. Filtration and systems of eigenvalues v 130
3.4. Congruences mod pm i v; 132
4. Serre s p-adic modular forms and p-adic zeta functions 134
5. A geometric approach to congruences 137
5.1. The Hasse invariant . •* - ,r 139
5.2. The kernel of the ^-expansion map 140
5.3. Operators revisited 1 ; 142
6. p-adic elliptic modular forms . 146
6.1. Test objects and overconvergent forms 146
6.2. g-expansion for p-adic modular forms ,, , 149
6.3. Standing assumptions and their raison d etre 150
6.4. The case when p is nilpotent * 151
6.5. The case of r a unit 153
6.6. Katz s expansion 155
6.7. Properties of g-expansions of p-adic modular forms 158
7. The ring of divided congruences 159
Chapter 5. p-adic Hilbert Modular Forms 167
1. Algebraic Hilbert modular forms 168
2. Tate objects and the g-expansion 171
3. Hasse invariants 176
3.1. Definition and main properties of partial Hasse invariants 176
3.2. Further properties 180
4. The kernel of the g-expansion map 181
5. Applications 184
6. p-adic Hilbert modular forms 186
6.1. Test objects and overconvergent forms 186
6.2. g-expansion for p-adic modular forms 188
6.3. The case when p is nilpotent or r is a unit 189
6.4. Katz s expansion 190
6.5. Properties of g-expansions of p-adic modular forms 191
Chapter 6. Deformation Theory of Abelian Varieties 193
1. Fine moduli schemes 193
2. Proof of the Serre-Tate Theorem 197
3. Deformation of p-divisible groups I99
4. Commutative smooth formal groups 201
CONTENTS ix
4.1. Curves 202
4.2. Formal groups 203
4.3. Operators on £(£) 204
5. Modules over Cart (if) 207
6. The case of Q-algebras 209
6.1. Digression on Witt vectors 209
7. Formal groups in characteristic p 212
8. Classification of p-divisible groups in characteristic p,
Newton polygons and types 213
9. Mid-way summary 220
10. Displays 221
10.1. Basics 222
10.2. Examples 225
10.3. The main result 226
11. The universal display and applications to abelian varieties 226
11.1. Base change and deformations 226
11.2. Universal display 226
11.3. Endomorphisms conditions 227
11.4. The local structure of Wr and Hasse invariants 229
11.5. Polarization conditions 231
Appendix A. Group Schemes 235
1. Some definitions 235
2. Digression on Frobenius and Verschiebung 237
3. Important examples 238
3.1. The multiplicative group Gm 238
3.2. The roots of unity pjv 238
3.3. The additive group Ga 239
3.4. The group apr 239
3.5. The group GLn 239
3.6. A non-commutative group scheme of rank p2 240
3.7. The constant group scheme V 240
3.8. Etale group schemes 242
3.9. The p-torsion group scheme A p 242
4. The basic exact sequence 244
5. Group schemes over a perfect field of characteristic p 245
6. The a-group 248
Appendix B. Calculating with Cusps 251
1. ra(A0 -level 252
2. r0 (AO-level 253
3. r( AO-level 254
Bibliography 255
Notation Index 263
Index ; 267
|
| any_adam_object | 1 |
| author | Goren, Eyal Z. |
| author_facet | Goren, Eyal Z. |
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| author_sort | Goren, Eyal Z. |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 SK 240 |
| ctrlnum | (OCoLC)47893507 (DE-599)BVBBV014198528 |
| dewey-full | 516.3/53 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/53 |
| dewey-search | 516.3/53 |
| dewey-sort | 3516.3 253 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014198528 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:59:26Z |
| institution | BVB |
| isbn | 082181995X |
| language | English |
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| physical | IX, 270 S. graph. Darst. |
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| series2 | CRM Monograph series |
| spelling | Goren, Eyal Z. Verfasser aut Lectures on Hilbert modular varieties and modular forms Eyal Z. Goren Providence, RI AMS 2002 IX, 270 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier CRM Monograph series 14 Formes modulaires Modules, Théorie des Variétés abéliennes Abelian varieties Forms, Modular Moduli theory p-adische Modulform (DE-588)4197571-6 gnd rswk-swf Abelsche Mannigfaltigkeit (DE-588)4140992-9 gnd rswk-swf Hilbertsche Modulform (DE-588)4159855-6 gnd rswk-swf Modulraum (DE-588)4183462-8 gnd rswk-swf Abelsche Mannigfaltigkeit (DE-588)4140992-9 s Hilbertsche Modulform (DE-588)4159855-6 s p-adische Modulform (DE-588)4197571-6 s Modulraum (DE-588)4183462-8 s DE-604 CRM Monograph series 14 (DE-604)BV007928388 14 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009733151&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Goren, Eyal Z. Lectures on Hilbert modular varieties and modular forms CRM Monograph series Formes modulaires Modules, Théorie des Variétés abéliennes Abelian varieties Forms, Modular Moduli theory p-adische Modulform (DE-588)4197571-6 gnd Abelsche Mannigfaltigkeit (DE-588)4140992-9 gnd Hilbertsche Modulform (DE-588)4159855-6 gnd Modulraum (DE-588)4183462-8 gnd |
| subject_GND | (DE-588)4197571-6 (DE-588)4140992-9 (DE-588)4159855-6 (DE-588)4183462-8 |
| title | Lectures on Hilbert modular varieties and modular forms |
| title_auth | Lectures on Hilbert modular varieties and modular forms |
| title_exact_search | Lectures on Hilbert modular varieties and modular forms |
| title_full | Lectures on Hilbert modular varieties and modular forms Eyal Z. Goren |
| title_fullStr | Lectures on Hilbert modular varieties and modular forms Eyal Z. Goren |
| title_full_unstemmed | Lectures on Hilbert modular varieties and modular forms Eyal Z. Goren |
| title_short | Lectures on Hilbert modular varieties and modular forms |
| title_sort | lectures on hilbert modular varieties and modular forms |
| topic | Formes modulaires Modules, Théorie des Variétés abéliennes Abelian varieties Forms, Modular Moduli theory p-adische Modulform (DE-588)4197571-6 gnd Abelsche Mannigfaltigkeit (DE-588)4140992-9 gnd Hilbertsche Modulform (DE-588)4159855-6 gnd Modulraum (DE-588)4183462-8 gnd |
| topic_facet | Formes modulaires Modules, Théorie des Variétés abéliennes Abelian varieties Forms, Modular Moduli theory p-adische Modulform Abelsche Mannigfaltigkeit Hilbertsche Modulform Modulraum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009733151&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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