Hilbert modular forms and Iwasawa theory

Includes index

Includes bibliography (p. 387-396) and references

1.I Classical lwasawa theory 2 1.1.1 Galois theoretic interpretation of the class group 9 1.1.2 The Iwasawa algebra as a deformation ring 12 1.1.3 Pseudo-representations 13 t.1.4 Two-dimensional universal deformations 17 1.2 Sehner groups 19 1.2.1 Deligne's rationality conjecture 19 1.2.2 Ordinary Galois representations 26 1.2.3 Gre nberg's Selmer groups 28 1.2,1 Selmer groups with general coefficients 29 1.3 Deformation and adjoint square Selmer groups 31 1.3.1 Nearly ordinary deformation rings 32 1.3.2 Adjoint square Selmer groups and differentials 35 1.3.3 Universal deformation rings are noetherian 41 13.14 Elliptic modularity at a glance 43 1.4 Iwasawa theory for deformation rings 47 1.4.1 Galois action on deformation rings 47 1.4.2 Control of adjoint square Selmer groups 49 1.4.3 A-adic forms 56 1.5 Adjoint square £-invariants 59 1.5.1 Balanced Selmer groups 62 1.5.2 Greenberg's £-invariant 64 1.5.3 Proof of Theoremrn 1.80 67 2 Automorphic forms on inner forms of GL(2) 70 2.1 Quaternion algebras over a number field 76 2.1.1 Quaternion algebras 76 2,1.2 Orders of quaternion algebras 80 2.2 A short review of algebraic geometry 86 .2.21. Affine schemes 87 2,2.2 Affine algebraic groups 91 2,2.3 Schemes 93 2.3 Automorphic forms on quaternion algebras 95 2.3.1 Arithmetic quotients 96 2.3.2 Archimedean Hilbert modular forms 99 2.3.3 Hilbert modular forms with integral coefficients 104 2.3.4 Duality and Hecke algebras 109 2.3.5 Quaternionic automorphic forms 110 2.3.6 The Jacquet-Langlands correspondence 114 2.3.7 Local representations of GL(2) 117 2.3.8 oduIlar Galois representations 125 2.4 The integral Jacquet-Langlands correspondence 129 2.41 Classical Hecke operators 129 2.4.2 Hecke algebras 132 2.4.3 Cohomnological correspondences 134 2.4.4 Eiehler-Shimurar isomorphisms 138 2.5 Theta series 139 2.5.1 Quaternionic theta series 139 2.5.2 Siegefs theta series 141 2.5.3 Transformation formulas 147 2.54 Theta series of imaginary quadratic fields 150 2.6 The basis problem of Eichler 153 2.6.1 The elliptic Jacquet-Langlands correspondence 156 2.6.2 Eichler's integral correspondence 158 3 Hecke algebras as Galois deformation rings 162 3.1 Hecke algebras 163 3.1.1 Automorphic forms on definite quaternions 163 3.1.2 Hecke operators 167 3.1.3 Inner products 168 3.1.4 Ordinary Hecke algebras 174 3.1.5 Autonorphic forms of higher weight 180 3.2 Galois deformation 183 3.2.1 Minimal deformation problems 183 3.2.2 Tangent spaces of local deformation functors 187 3.2.3 Taylor-Wiles systems 189 3.2.4 Hecke algebras are universal 200 3.2.5 Flat deforma tions 210 3.2.6 Freeness over the Hecke algebra 213 3.2.7 Hilbert modular basis problems 17 32.8 Locally cyclotomic deformation 230 3.2.9 Locallv cyclotomic Hecke algebras 233 3.2.10 Global deformation over a p-adic field 243 3.3 Base change 245 3 3.1 p-Ordinary Jacquet-Langlands correspondence 245 .3.2 Base fields of odd degree 246 3.33 Automorphic base change 247 33.4 bGa;lois base change 248 3.4 invarants of Hilbert modular forms 251 3.4.1 Statement of the result 251 3.4.2 Deformation without monodromy conditions 256 3.4.3 Selmer groups of induced representations 262 3.4.4 £-invariant of induced representations 265 3.4.5 Adjoint square Selmer groups and differentials 274 3.4.6 Proof of Theorem 3.73 279 3.4.7 Logarithm of the universal norm 283 4 Geometric modular forms 286 4.1 Modular curves 286 4.11 Modular curves and elliptic curves 286 S1.2 Arithmetic VWeierstrass theory 287 4.13 Moduli of level N 289 41.4 Toric action 291 4.1.5 Compactification 292 4.1 6 Action of an adele group 294 4.2 lilbert AVRM moduli 296 4.2,1 Abelian variety with real multiplication 296 "4.2.2 AVRM moduli with level structure 300 4.2.3 Classical Hilbert modular forms 303 4.2.4 Toroidal compactification 307 4.2.5 Tate AVRM 311 4.2.6 Hasse invariant 313 4.2.7 Geometric Hilbert modular forms 315 4.28 r-Adic Hilbert modular forms 317 4.2.9 Hecke operators 319 4i3 Hilbert modular Shimura varieties 323 4.3.1 Abelian varieties up to isogenies 324 4.3.2 Finite level structure 330 4ý.33 Modular varieties of level F0(91) 332 4,3.4 Isogeny action 332 4.3.5 Reciprocity law at CM points 334 4.3.6 Hilbert modular Igusa towers 334 4.3.7 Finite level Hecke algebra 336 43.18 q-Expansion 337 4.3,9 tUniversal Heeck algebras 338 .4 Exceptional zeros and extension 341 4.41 A-adic automorphic representations 343 4.4.2 Extensions of automorphic representations 347 .4.3 Extensions of Galois representations 351 5 Modular Iwasawa theory 353 5.1 The cyclotomic tower of deformation rings 353 5.1.1 Control of deformation rings 354 5.1.2 KHiler differentials as Iwasawa modules 355 5.1.3 Dimension of 7, 363 5.2 Adjoint square exceptional zeros 366 5.2.1 Order of exceptional zeros 367 512.2 Base change of Selmer groups 375 5.3 Torsion of Iwasawa modules for CM fields 377 5.3.1 Ordinary CM fields and their Iwasawa modules 377 5.3.2 Anticyclotomic Iwasawa modules 379 5.3.3 The L-invariant of CM fields 383.

Describing the applications found for the Wiles and Taylor technique, this book generalizes the deformation theoretic techniques of Wiles-Taylor to Hilbert modular forms (following Fujiwara's treatment), and also discusses applications found by the author.