Hilbert Modular Forms and Iwasawa Theory

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ISBN: 9780198571025 | 019857102X
Cover: Hardcover
Copyright: 8/10/2006

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The 1995 work of Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt. This book, authored by a leading researcher, describes the striking applications that have been found for this technique. In the book, the deformation theoretic techniques of Wiles-Taylor are first generalized to Hilbert modular forms (following Fujiwara's treatment), and some applications found by the author are then discussed. With many exercises and open questions given, this text is ideal for researchers and graduate students entering this research area.

1 Introduction

(69)

1.1 Classical Iwasawa theory

(17)

1.1.1 Galois theoretic interpretation of the class group

(3)

1.1.2 The Iwasawa algebra as a deformation ring

(1)

1.1.3 Pseudo-representations

(4)

1.1.4 Two-dimensional universal deformations

(2)

1.2 Selmer groups

(12)

1.2.1 Deligne's rationality conjecture

(7)

1.2.2 Ordinary Galois representations

(2)

1.2.3 Greenberg's Selmer groups

(1)

1.2.4 Selmer groups with general coefficients

(2)

1.3 Deformation and adjoint square Selmer groups

(16)

1.3.1 Nearly ordinary deformation rings

(3)

1.3.2 Adjoint square Selmer groups and differentials

(6)

1.3.3 Universal deformation rings are noetherian

(2)

1.3.4 Elliptic modularity at a glance

(4)

1.4 Iwasawa theory for deformation rings

(12)

1.4.1 Galois action on deformation rings

(2)

1.4.2 Control of adjoint square Selmer groups

(7)

1.4.3 -adic forms

(3)

1.5 Adjoint square -invariants

(11)

1.5.1 Balanced Selmer groups

(2)

1.5.2 Greenberg's -invariant

(3)

1.5.3 Proof of Theorem 1.80

(3)

2 Automorphic forms on inner forms of GL(2)

(92)

2.1 Quaternion algebras over a number field

(10)

2.1.1 Quaternion algebras

(4)

2.1.2 Orders of quaternion algebras

(6)

2.2 A short review of algebraic geometry

(9)

2.2.1 Affine schemes

(4)

2.2.2 Affine algebraic groups

(2)

2.2.3 Schemes

(2)

2.3 Automorphic forms on quaternion algebras

(34)

2.3.1 Arithmetic quotients

(3)

2.3.2 Archimedean Hilbert modular forms

(5)

2.3.3 Hilbert modular forms with integral coefficients

104

(5)

2.3.4 Duality and Hecke algebras

109

(1)

2.3.5 Quaternionic automorphic forms

110

(4)

2.3.6 The Jacquet–Langlands correspondence

114

(3)

2.3.7 Local representations of GL(2)

117

(8)

2.3.8 Modular Galois representations

125

(4)

2.4 The integral Jacquet–Langlands correspondence

129

(10)

2.4.1 Classical Hecke operators

129

(3)

2.4.2 Hecke algebras

132

(2)

2.4.3 Cohomological correspondences

134

(4)

2.4.4 Eichler–Shimura isomorphisms

138

(1)

2.5 Theta series

139

(14)

2.5.1 Quaternionic theta series

139

(2)

2.5.2 Siegel's theta series

141

(6)

2.5.3 Transformation formulas

147

(3)

2.5.4 Theta series of imaginary quadratic fields

150

(3)

2.6 The basis problem of Eichler

153

(9)

2.6.1 The elliptic Jacquet–Langlands correspondence

156

(2)

2.6.2 Eichler's integral correspondence

158

(4)

3 Hecke algebras as Galois deformation rings

162

(124)

3.1 Hecke algebras

163

(20)

3.1.1 Automorphic forms on definite quaternions

163

(4)

3.1.2 Hecke operators

167

(1)

3.1.3 Inner products

168

(6)

3.1.4 Ordinary Hecke algebras

174

(6)

3.1.5 Automorphic forms of higher weight

180

(3)

3.2 Galois deformation

183

(62)

3.2.1 Minimal deformation problems

183

(4)

3.2.2 Tangent spaces of local deformation functors

187

(2)

3.2.3 Taylor–Wiles systems

189

(11)

3.2.4 Hecke algebras are universal

200

(10)

3.2.5 Flat deformations

210

(3)

3.2.6 Freeness over the Hecke algebra

213

(4)

3.2.7 Hilbert modular basis problems

217

(13)

3.2.8 Locally cyclotomic deformation

230

(3)

3.2.9 Locally cyclotomic Hecke algebras

233

(10)

3.2.10 Global deformation over a p-adic field

243

(2)

3.3 Base change

245

(6)

3.3.1 p-Ordinary Jacquet–Langlands correspondence

245

(1)

3.3.2 Base fields of odd degree

246

(1)

3.3.3 Automorphic base change

247

(1)

3.3.4 Galois base change

248

(3)

3.4 r-invariants of Hilbert modular forms

251

(35)

3.4.1 Statement of the result

251

(5)

3.4.2 Deformation without monodromy conditions

256

(6)

3.4.3 Selmer groups of induced representations

262

(3)

3.4.4 -invariant of induced representations

265

(9)

3.4.5 Adjoint square Selmer groups and differentials

274

(5)

3.4.6 Proof of Theorem 3.73

279

(4)

3.4.7 Logarithm of the universal norm

283

(3)

4 Geometric modular forms

286

(67)

4.1 Modular curves

286

(10)

4.1.1 Modular curves and elliptic curves

286

(1)

4.1.2 Arithmetic Weierstrass theory

287

(2)

4.1.3 Moduli of level N

289

(2)

4.1.4 Toric action

291

(1)

4.1.5 Compactification

292

(2)

4.1.6 Action of an adele group

294

(2)

4.2 Hilbert AVRM moduli

296

(27)

4.2.1 Abelian variety with real multiplication

296

(4)

4.2.2 AVRM moduli with level structure

300

(3)

4.2.3 Classical Hilbert modular forms

303

(4)

4.2.4 Toroidal compactification

307

(4)

4.2.5 Tate AVRM

311

(2)

4.2.6 Hasse invariant

313

(2)

4.2.7 Geometric Hilbert modular forms

315

(2)

4.2.8 p-Adic Hilbert modular forms

317

(2)

4.2.9 Hecke operators

319

(4)

4.3 Hilbert modular Shimura varieties

323

(18)

4.3.1 Abelian varieties up to isogenies

324

(6)

4.3.2 Finite level structure

330

(2)

4.3.3 Modular varieties of level Γ0( )

332

(1)

4.3.4 Isogeny action

332

(2)

4.3.5 Reciprocity law at CM points

334

(1)

4.3.6 Hilbert modular Igusa towers

334

(2)

4.3.7 Finite level Hecke algebra

336

(1)

4.3.8 q-Expansion

337

(1)

4.3.9 Universal Hecke algebras

338

(3)

4.4 Exceptional zeros and extension

341

(12)

4.4.1 Λ-adic automorphic representations

343

(4)

4.4.2 Extensions of automorphic representations

347

(4)

4.4.3 Extensions of Galois representations

351

(2)

5 Modular Iwasawa theory

353

(34)

5.1 The cyclotomic tower of deformation rings

353

(13)

5.1.1 Control of deformation rings

354

(1)

5.1.2 Kähler differentials as Iwasawa modules

355

(8)

5.1.3 Dimension of Rinfinity

363

(3)

5.2 Adjoint square exceptional zeros

366

(11)

5.2.1 Order of exceptional zeros

367

(8)

5.2.2 Base change of Selmer groups

375

(2)

5.3 Torsion of Iwasawa modules for CM fields

377

(10)

5.3.1 Ordinary CM fields and their Iwasawa modules

377

(2)

5.3.2 Anticyclotomic Iwasawa modules

379

(4)

5.3.3 The -invariant of CM fields

383

(4)

References

387

(10)

Symbol Index

397

(2)

Statement Index

399

(2)

Subject Index

401

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Hilbert Modular Forms and Iwasawa Theory

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