HILBERT MODULAR FORMS AND IWASAWA THEORY
CONTENTS INTRODUCTION CLASSICAL IWASAWA THEORY GALOIS THEORETIC INTERPRETATION OF THE CLASS GROUP THE IWASAWA ALGEBRA AS A DEFORMATION RING PSEUDO-REPRESENTATIONS TWO-DIMENSIONAL UNIVERSAL DEFORMATIONS SELMER GROUPS DELIGNE’S RATIONALITY CONJECTURE ORDINARY GALOIS REPRESENTATIONS GREENBERG’S SELMER GROUPS SELMER GROUPS WITH GENERAL COE?CIENTS DEFORMATION AND ADJOINT SQUARE SELMER GROUPS NEARLY ORDINARY DEFORMATION RINGS ADJOINT SQUARE SELMER GROUPS AND DI?ERENTIALS UNIVERSAL DEFORMATION RINGS ARE NOETHERIAN ELLIPTIC MODULARITY AT A GLANCE IWASAWA THEORY FOR DEFORMATION RINGS GALOIS ACTION ON DEFORMATION RINGS CONTROL OF ADJOINT SQUARE SELMER GROUPS ?-ADIC FORMS ADJOINT SQUARE L-INVARIANTS BALANCED SELMER GROUPS GREENBERG’S L-INVARIANT AUTOMORPHIC FORMS ON INNER FORMS OF GL(2) QUATERNION ALGEBRAS OVER A NUMBER ?ELD QUATERNION ALGEBRAS ORDERS OF QUATERNION ALGEBRAS A SHORT REVIEW OF ALGEBRAIC GEOMETRY A?NE SCHEMES A?NE ALGEBRAIC GROUPS SCHEMES AUTOMORPHIC FORMS ON QUATERNION ALGEBRAS ARITHMETIC QUOTIENTS ARCHIMEDEAN HILBERT MODULAR FORMS HILBERT MODULAR FORMS WITH INTEGRAL COE?CIENTS DUALITY AND HECKE ALGEBRAS QUATERNIONIC AUTOMORPHIC FORMS THE JACQUET–LANGLANDS CORRESPONDENCE LOCAL REPRESENTATIONS OF GL(2) MODULAR GALOIS REPRESENTATIONS THE INTEGRAL JACQUET–LANGLANDS CORRESPONDENCE CLASSICAL HECKE OPERATORS HECKE ALGEBRAS COHOMOLOGICAL CORRESPONDENCES EICHLER–SHIMURA ISOMORPHISMS THETA SERIES QUATERNIONIC THETA SERIES SIEGEL’S THETA SERIES TRANSFORMATION FORMULAS THETA SERIES OF IMAGINARY QUADRATIC ?ELDS THE BASIS PROBLEM OF EICHLER THE ELLIPTIC JACQUET–LANGLANDS CORRESPONDENCE EICHLER’S INTEGRAL CORRESPONDENCE HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS AUTOMORPHIC FORMS ON DE?NITE QUATERNIONS HECKE OPERATORS INNER PRODUCTS ORDINARY HECKE ALGEBRAS AUTOMORPHIC FORMS OF HIGHER WEIGHT GALOIS DEFORMATION MINIMAL DEFORMATION PROBLEMS TANGENT SPACES OF LOCAL DEFORMATION FUNCTORS TAYLOR–WILES SYSTEMS HECKE ALGEBRAS ARE UNIVERSAL FLAT DEFORMATIONS FREENESS OVER THE HECKE ALGEBRA HILBERT MODULAR BASIS PROBLEMS LOCALLY CYCLOTOMIC DEFORMATION LOCALLY CYCLOTOMIC HECKE ALGEBRAS 3210 GLOBAL DEFORMATION OVER A P-ADIC ?ELD BASE CHANGE P-ORDINARY JACQUET–LANGLANDS CORRESPONDENCE BASE ?ELDS OF ODD DEGREE AUTOMORPHIC BASE CHANGE GALOIS BASE CHANGE L-INVARIANTS OF HILBERT MODULAR FORMS STATEMENT OF THE RESULT DEFORMATION WITHOUT MONODROMY CONDITIONS SELMER GROUPS OF INDUCED REPRESENTATIONS L-INVARIANT OF INDUCED REPRESENTATIONS LOGARITHM OF THE UNIVERSAL NORM GEOMETRIC MODULAR FORMS MODULAR CURVES MODULAR CURVES AND ELLIPTIC CURVES ARITHMETIC WEIERSTRASS THEORY MODULI OF LEVEL N TORIC ACTION COMPACTI?CATION ACTION OF AN ADELE GROUP HILBERT AVRM MODULI ABELIAN VARIETY WITH REAL MULTIPLICATION AVRM MODULI WITH LEVEL STRUCTURE CLASSICAL HILBERT MODULAR FORMS TOROIDAL COMPACTI?CATION TATE AVRM HASSE INVARIANT GEOMETRIC HILBERT MODULAR FORMS P-ADIC HILBERT MODULAR FORMS HILBERT MODULAR SHIMURA VARIETIES ABELIAN VARIETIES UP TO ISOGENIES FINITE LEVEL STRUCTURE MODULAR VARIETIES OF LEVEL ?0(N) ISOGENY ACTION RECIPROCITY LAW AT CM POINTS HILBERT MODULAR IGUSA TOWERS FINITE LEVEL HECKE ALGEBRA Q-EXPANSION UNIVERSAL HECKE ALGEBRAS EXCEPTIONAL ZEROS AND EXTENSION ?-ADIC AUTOMORPHIC REPRESENTATIONS EXTENSIONS OF AUTOMORPHIC REPRESENTATIONS EXTENSIONS OF GALOIS REPRESENTATIONS MODULAR IWASAWA THEORY THE CYCLOTOMIC TOWER OF DEFORMATION RINGS CONTROL OF DEFORMATION RINGS K¨AHLER DI?ERENTIALS AS IWASAWA MODULES DIMENSION OF R? ADJOINT SQUARE EXCEPTIONAL ZEROS ORDER OF EXCEPTIONAL ZEROS BASE CHANGE OF SELMER GROUPS TORSION OF IWASAWA MODULES FOR CM ?ELDS ORDINARY CM ?ELDS AND THEIR IWASAWA MODULES ANTICYCLOTOMIC IWASAWA MODULES THE L-INVARIANT OF CM ?ELDS REFERENCES