Contents -- 1 Introduction -- 1.1 Classical Iwasawa theory -- 1.1.1 Galois theoretic interpretation of the class group -- 1.1.2 The Iwasawa algebra as a deformation ring -- 1.1.3 Pseudo-representations -- 1.1.4 Two-dimensional universal deformations -- 1.2 Selmer groups -- 1.2.1 Deligne's rationality conjecture -- 1.2.2 Ordinary Galois representations -- 1.2.3 Greenberg's Selmer groups -- 1.2.4 Selmer groups with general coefficients -- 1.3 Deformation and adjoint square Selmer groups -- 1.3.1 Nearly ordinary deformation rings -- 1.3.2 Adjoint square Selmer groups and differentials -- 1.3.3 Universal deformation rings are noetherian -- 1.3.4 Elliptic modularity at a glance -- 1.4 Iwasawa theory for deformation rings -- 1.4.1 Galois action on deformation rings -- 1.4.2 Control of adjoint square Selmer groups -- 1.4.3 Λ-adic forms -- 1.5 Adjoint square L-invariants -- 1.5.1 Balanced Selmer groups -- 1.5.2 Greenberg's L-invariant -- 1.5.3 Proof of Theorem 1.80 -- 2 Automorphic forms on inner forms of GL(2) -- 2.1 Quaternion algebras over a number field -- 2.1.1 Quaternion algebras -- 2.1.2 Orders of quaternion algebras -- 2.2 A short review of algebraic geometry -- 2.2.1 Affine schemes -- 2.2.2 Affine algebraic groups -- 2.2.3 Schemes -- 2.3 Automorphic forms on quaternion algebras -- 2.3.1 Arithmetic quotients -- 2.3.2 Archimedean Hilbert modular forms -- 2.3.3 Hilbert modular forms with integral coefficients -- 2.3.4 Duality and Hecke algebras -- 2.3.5 Quaternionic automorphic forms -- 2.3.6 The Jacquet-Langlands correspondence -- 2.3.7 Local representations of GL(2) -- 2.3.8 Modular Galois representations -- 2.4 The integral Jacquet-Langlands correspondence -- 2.4.1 Classical Hecke operators -- 2.4.2 Hecke algebras -- 2.4.3 Cohomological correspondences -- 2.4.4 Eichler-Shimura isomorphisms -- 2.5 Theta series -- 2.5.1 Quaternionic theta series.

2.5.2 Siegel's theta series -- 2.5.3 Transformation formulas -- 2.5.4 Theta series of imaginary quadratic fields -- 2.6 The basis problem of Eichler -- 2.6.1 The elliptic Jacquet-Langlands correspondence -- 2.6.2 Eichler's integral correspondence -- 3 Hecke algebras as Galois deformation rings -- 3.1 Hecke algebras -- 3.1.1 Automorphic forms on definite quaternions -- 3.1.2 Hecke operators -- 3.1.3 Inner products -- 3.1.4 Ordinary Hecke algebras -- 3.1.5 Automorphic forms of higher weight -- 3.2 Galois deformation -- 3.2.1 Minimal deformation problems -- 3.2.2 Tangent spaces of local deformation functors -- 3.2.3 Taylor-Wiles systems -- 3.2.4 Hecke algebras are universal -- 3.2.5 Flat deformations -- 3.2.6 Freeness over the Hecke algebra -- 3.2.7 Hilbert modular basis problems -- 3.2.8 Locally cyclotomic deformation -- 3.2.9 Locally cyclotomic Hecke algebras -- 3.2.10 Global deformation over a p-adic field -- 3.3 Base change -- 3.3.1 p-Ordinary Jacquet-Langlands correspondence -- 3.3.2 Base fields of odd degree -- 3.3.3 Automorphic base change -- 3.3.4 Galois base change -- 3.4 L-invariants of Hilbert modular forms -- 3.4.1 Statement of the result -- 3.4.2 Deformation without monodromy conditions -- 3.4.3 Selmer groups of induced representations -- 3.4.4 L-invariant of induced representations -- 3.4.5 Adjoint square Selmer groups and differentials -- 3.4.6 Proof of Theorem 3.73 -- 3.4.7 Logarithm of the universal norm -- 4 Geometric modular forms -- 4.1 Modular curves -- 4.1.1 Modular curves and elliptic curves -- 4.1.2 Arithmetic Weierstrass theory -- 4.1.3 Moduli of level N -- 4.1.4 Toric action -- 4.1.5 Compactification -- 4.1.6 Action of an adele group -- 4.2 Hilbert AVRM moduli -- 4.2.1 Abelian variety with real multiplication -- 4.2.2 AVRM moduli with level structure -- 4.2.3 Classical Hilbert modular forms -- 4.2.4 Toroidal compactification.

4.2.5 Tate AVRM -- 4.2.6 Hasse invariant -- 4.2.7 Geometric Hilbert modular forms -- 4.2.8 p-Adic Hilbert modular forms -- 4.2.9 Hecke operators -- 4.3 Hilbert modular Shimura varieties -- 4.3.1 Abelian varieties up to isogenies -- 4.3.2 Finite level structure -- 4.3.3 Modular varieties of level Γ[sub(0)](n) -- 4.3.4 Isogeny action -- 4.3.5 Reciprocity law at CM points -- 4.3.6 Hilbert modular Igusa towers -- 4.3.7 Finite level Hecke algebra -- 4.3.8 q-Expansion -- 4.3.9 Universal Hecke algebras -- 4.4 Exceptional zeros and extension -- 4.4.1 Λ-adic automorphic representations -- 4.4.2 Extensions of automorphic representations -- 4.4.3 Extensions of Galois representations -- 5 Modular Iwasawa theory -- 5.1 The cyclotomic tower of deformation rings -- 5.1.1 Control of deformation rings -- 5.1.2 Kähler differentials as Iwasawa modules -- 5.1.3 Dimension of R[sub(∞)] -- 5.2 Adjoint square exceptional zeros -- 5.2.1 Order of exceptional zeros -- 5.2.2 Base change of Selmer groups -- 5.3 Torsion of Iwasawa modules for CM fields -- 5.3.1 Ordinary CM fields and their Iwasawa modules -- 5.3.2 Anticyclotomic Iwasawa modules -- 5.3.3 The L-invariant of CM fields -- References -- Symbol Index -- Statement Index -- Subject Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W.