Geometric Modular Forms and Elliptic Curves

by Hida, Haruzo

• ISBN13: 9789814368643
• ISBN10: 9814368644
• Edition: 2nd
• Format: Hardcover
• Copyright: 2012-01-31
• Publisher: World Scientific Pub Co Inc

Summary

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the ShimuraTaniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.In this new second edition, a detailed description of BarsottiTate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to 'big' ?-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian ?-varieties and ?-curves).

Table of Contents

Preface to the second edition	p. v
Preface	p. vii
An Algebro-Geometric Tool Box	p. 1
Sheaves	p. 1
Sheaves and Presheaves	p. 1
Sheafication	p. 3
Sheaf Kernel and Cokernel	p. 4
Schemes	p. 5
Local Panged Spaces	p. 5
Schemes as Local Ringed Spaces	p. 8
Sheaves over Schemes	p. 9
Topological Properties of Schemes	p. 11
Projective Schemes	p. 13
Graded Rings	p. 13
Functor Proj	p. 13
Sheaves on Projective Schemes	p. 16
Categories and Functors	p. 20
Categories	p. 20
Functors	p. 22
Schemes as Functors	p. 23
Abelian Categories	p. 26
Applications of the Key-Lemma	p. 28
Sheaf of Differential Forms on Schemes	p. 29
Fiber Products	p. 32
Inverse Image of Sheaves	p. 33
Affine Schemes	p. 35
Morphisms into a Projective Space	p. 37
Group Schemes	p. 38
Group Schemes as Functors	p. 38
Kernel and Cokernel	p. 39
Bialgebras	p. 40
Locally Free Groups	p. 42
Schematic Representations	p. 44
Cartier Duality	p. 45
Duality of Bialgebras	p. 45
Duality of Locally Free Groups	p. 47
Quotients by a Group Scheme	p. 50
Naive Quotients	p. 50
Categorical Quotients	p. 52
Geometric Quotients	p. 154
Morphisms	p. 62
Topological Definitions	p. 62
Diffeo-Geometric Definitions	p. 67
Applications	p. 69
Cohomology of Coherent Sheaves	p. 73
Coherent Cohomology	p. 73
Summary of Known Facts	p. 77
Cohomological Dimension	p. 78
Descent	p. 82
Covering Data	p. 82
Descent Data	p. 83
Descent of Schemes	p. 85
Baxsotti-Tate Groups	p. 88
p-Divisible Abelian Sheaf	p. 88
Connected-Étale Exact Sequence	p. 92
Ordinary Barsotti-Tate Group	p. 93
Formal Scheme	p. 95
Open Subschemes as Functors	p. 96
Examples of Formal Schemes	p. 97
Deformation Functors	p. 101
Connected Formal Groups	p. 102
Elliptic Curves	p. 105
Curves and Divisors	p. 105
Cartier Divisors	p. 105
Serre-Grothendieck Duality	p. 108
Riemann-Roch Theorem	p. 114
Relative Riemann-Roch Theorem	p. 119
Elliptic Curves	p. 122
Definition	p. 122
Abel's Theorem	p. 123
Holomorphic Differentials	p. 125
Taylor Expansion of Differentials	p. 126
Weierstrass Equations of Elliptic Curves	p. 127
Moduli of Weierstrass Type	p. 130
Geometric Modular Forms of Level 1	p. 134
Functorial Definition	p. 134
Coarse Moduli Scheme	p. 136
Fields of Moduli	p. 138
Elliptic Curves over C	p. 139
Topological Fundamental Groups	p. 140
Classical Weierstrass Theory	p. 142
Complex Modular Forms	p. 143
Elliptic Curves over p-Adic Fields	p. 145
Power Series Identities	p. 145
Universal Tate Curves	p. 148
Etale Covering of Tate Curves	p. 153
Level Structures	p. 155
Isogenies	p. 155
Level N Moduli Problems	p. 157
Generality of Elliptic Curves	p. 163
Proof of Theorem 2.6.8	p. 165
Geometric Modular Forms of Level N	p. 168
L-Functions of Elliptic Curves	p. 173
L-Functions over Finite Fields	p. 173
Hasse-Weil L-Function	p. 176
Regularity	p. 180
Regular Rings	p. 180
Regular Moduli Varieties	p. 183
p-Ordinary Moduli Problems	p. 189
The Hasse Invariant	p. 189
Ordinary Moduli of p-Power Level	p. 193
Irreducibility of p-Ordinary Moduli	p. 195
Moduli Problem of ¿0 and ¿1 Type	p. 196
Moduli Problem of ¿0(p) and ¿ 1(p) Type	p. 198
Deformation of Elliptic Curves	p. 209
A Theorem of Drinfeld	p. 209
A Theorem of Serre-Tate	p. 211
Deformation of an Ordinary Elliptic Curve	p. 214
Geometric Modular Forms	p. 223
Integrality	p. 223
Spaces of Modular Forms	p. 223
Horizontal Control Theorem	p. 236
Vertical Control Theorem	p. 238
False Modular Forms	p. 240
p-Adic Modular Forms	p. 252
Hecke Operators	p. 257
Families of p-Adic Modular Forms	p. 266
Horizontal Control of p-Power Level	p. 271
Control of Hecke algebra	p. 273
Irreducible Components and Analytic Families	p. 275
Action of GL(2) on Modular Forms	p. 276
Action of GL2(Z/NZ)	p. 276
Action of GL2(Z)	p. 280
Jacobians and Galois Representations	p. 287
Jacobians of Stable Curves	p. 287
Non-Singular Curves	p. 287
Union of Two Curves	p. 295
Functorial Properties of Jacobians	p. 298
Self-Duality of Jacobian Schemes	p. 302
Generality on Abelian Schemes	p. 304
Endomorphism of Abelian Schemes	p. 313
l-Adic Galois Representations	p. 318
Modular Galois Representations	p. 322
Hecke Correspondences	p. 323
Galois Representations on Modular Jacobians	p. 326
Ramification at the Level	p. 330
Ramification of p-Adic Representations at p	p. 335
Modular Galois Representations of Higher Weight	p. 337
Fullness of Big Galois Representations	p. 342
Big I-adic Galois Representations	p. 344
Ramification of I-adic Galois Representations	p. 345
Lie Algebras over p-Adic Ring	p. 346
Lie Algebras of p-Profinite Subgroups of SL(2)	p. 348
Lie Algebra and Lie Group over Zp	p. 355
Arithmetic Galois Characters	p. 359
Fullness of Modular Galois Representation	p. 361
Fullness of Elliptic Curves	p. 365
Fullness of Lie Algebra over ¿	p. 368
Fullness of I-Adic Galois Representation	p. 371
Basic Subgroups	p. 373
Proof of Theorem 4.3.4	p. 380
Modularity Problems	p. 383
Induced and Extended Galois Representations	p. 384
Induction and Extension	p. 385
Automorphic Induction	p. 392
Artin Representations	p. 395
Some Other Solutions	p. 402
A Theorem of Wiles	p. 402
Modularity of Extended Galois Representations	p. 404
Elliptic Q-Curves	p. 406
Shimura-Taniyama Conjecture	p. 413
Modularity of Abelian QVarieties	p. 416
Abelian F-varieties of GL(2)-type	p. 417
Endomorphism Algebras of Abelian F-varieties	p. 424
Application to Abelian Q-Varieties	p. 425
Abelian Varieties with Real Multiplication	p. 432
Bibliography	p. 437
List of Symbols	p. 447
Statement Index	p. 449
Index	p. 451
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