Contents -- Preface to the second edition -- Preface -- 1. An Algebro-Geometric Tool Box -- 1.1 Sheaves -- 1.1.1 Sheaves and Presheaves -- 1.1.2 Sheafication -- 1.1.3 Sheaf Kernel and Cokernel -- 1.2 Schemes -- 1.2.1 Local Ringed Spaces -- 1.2.2 Schemes as Local Ringed Spaces -- 1.2.3 Sheaves over Schemes -- 1.2.4 Topological Properties of Schemes -- 1.3 Projective Schemes -- 1.3.1 Graded Rings -- 1.3.2 Functor Proj -- 1.3.3 Sheaves on Projective Schemes -- 1.4 Categories and Functors -- 1.4.1 Categories -- 1.4.2 Functors -- 1.4.3 Schemes as Functors -- 1.4.4 Abelian Categories -- 1.5 Applications of the Key-Lemma -- 1.5.1 Sheaf of Differential Forms on Schemes -- 1.5.2 Fiber Products -- 1.5.3 Inverse Image of Sheaves -- 1.5.4 Affine Schemes -- 1.5.5 Morphisms into a Projective Space -- 1.6 Group Schemes -- 1.6.1 Group Schemes as Functors -- 1.6.2 Kernel and Cokernel -- 1.6.3 Bialgebras -- 1.6.4 Locally Free Groups -- 1.6.5 Schematic Representations -- 1.7 Cartier Duality -- 1.7.1 Duality of Bialgebras -- 1.7.2 Duality of Locally Free Groups -- 1.8 Quotients by a Group Scheme -- 1.8.1 Naive Quotients -- 1.8.2 Categorical Quotients -- 1.8.3 Geometric Quotients -- 1.9 Morphisms -- 1.9.1 Topological Definitions -- 1.9.2 Diffeo-Geometric Definitions -- 1.9.3 Applications -- 1.10 Cohomology of Coherent Sheaves -- 1.10.1 Coherent Cohomology -- 1.10.2 Summary of Known Facts -- 1.10.3 Cohomological Dimension -- 1.11 Descent -- 1.11.1 Covering Data -- 1.11.2 Descent Data -- 1.11.3 Descent of Schemes -- 1.12 Barsotti-Tate Groups -- 1.12.1 p-Divisible Abelian Sheaf -- Exercise -- 1.12.2 Connected- Etale Exact Sequence -- 1.12.3 Ordinary Barsotti-Tate Group -- 1.13 Formal Scheme -- 1.13.1 Open Subschemes as Functors -- Exercises -- 1.13.2 Examples of Formal Schemes -- 1.13.3 Deformation Functors -- 1.13.4 Connected Formal Groups -- 2. Elliptic Curves.

2.1 Curves and Divisors -- 2.1.1 Cartier Divisors -- 2.1.2 Serre-Grothendieck Duality -- 2.1.3 Riemann-Roch Theorem -- 2.1.4 Relative Riemann-Roch Theorem -- 2.2 Elliptic Curves -- 2.2.1 Definition -- 2.2.2 Abel's Theorem -- 2.2.3 Holomorphic Differentials -- 2.2.4 Taylor Expansion of Differentials -- 2.2.5 Weierstrass Equations of Elliptic Curves -- 2.2.6 Moduli of Weierstrass Type -- 2.3 Geometric Modular Forms of Level 1 -- 2.3.1 Functorial Definition -- 2.3.2 Coarse Moduli Scheme -- 2.3.3 Fields of Moduli -- 2.4 Elliptic Curves over C -- 2.4.1 Topological Fundamental Groups -- 2.4.2 Classical Weierstrass Theory -- 2.4.3 Complex Modular Forms -- 2.5 Elliptic Curves over p-Adic Fields -- 2.5.1 Power Series Identities -- 2.5.2 Universal Tate Curves -- 2.5.3 Etale Covering of Tate Curves -- 2.6 Level Structures -- 2.6.1 Isogenies -- 2.6.2 Level N Moduli Problems -- 2.6.3 Generality of Elliptic Curves -- 2.6.4 Proof of Theorem 2.6.8 -- Exercise -- 2.6.5 Geometric Modular Forms of Level N -- 2.7 L-Functions of Elliptic Curves -- 2.7.1 L-Functions over Finite Fields -- 2.7.2 Hasse-Weil L-Function -- 2.8 Regularity -- 2.8.1 Regular Rings -- 2.8.2 Regular Moduli Varieties -- 2.9 p-Ordinary Moduli Problems -- 2.9.1 The Hasse Invariant -- 2.9.2 Ordinary Moduli of p-Power Level -- 2.9.3 Irreducibility of p-Ordinary Moduli -- 2.9.4 Moduli Problem of r0 and r1 Type -- 2.9.5 Moduli Problem of r0(p) and r1(p) Type -- Exercises -- 2.10 Deformation of Elliptic Curves -- 2.10.1 A Theorem of Drinfeld -- 2.10.2 A Theorem of Serre-Tate -- 2.10.3 Deformation of an Ordinary Elliptic Curve -- 3. Geometric Modular Forms -- 3.1 Integrality -- 3.1.1 Spaces of Modular Forms -- 3.1.2 Horizontal Control Theorem -- 3.2 Vertical Control Theorem -- 3.2.1 False Modular Forms -- Exercises -- 3.2.2 p-Adic Modular Forms -- 3.2.3 Hecke Operators.

3.2.4 Families of p-Adic Modular Forms -- 3.2.5 Horizontal Control of p-Power Level -- 3.2.6 Control of Hecke algebra -- 3.2.7 Irreducible Components and Analytic Families -- 3.3 Action of GL(2) on Modular Forms -- 3.3.1 Action of GL2(Z/NZ) -- 3.3.2 Action of GL2( Z) -- 4. Jacobians and Galois Representations -- 4.1 Jacobians of Stable Curves -- 4.1.1 Non-Singular Curves -- 4.1.2 Union of Two Curves -- 4.1.3 Functorial Properties of Jacobians -- Exercises -- 4.1.4 Self-Duality of Jacobian Schemes -- 4.1.5 Generality on Abelian Schemes -- 4.1.6 Endomorphism of Abelian Schemes -- Exercises -- 4.1.7 l-Adic Galois Representations -- 4.2 Modular Galois Representations -- 4.2.1 Hecke Correspondences -- 4.2.2 Galois Representations on Modular Jacobians -- 4.2.3 Ramification at the Level -- Exercises -- 4.2.4 Ramification of p-Adic Representations at p -- 4.2.5 Modular Galois Representations of Higher Weight -- Exercise -- 4.3 Fullness of Big Galois Representations -- 4.3.1 Big I-adic Galois Representations -- 4.3.2 Ramification of I-adic Galois Representations -- 4.3.3 Lie Algebras over p-Adic Ring -- 4.3.4 Lie Algebras of p-Profinite Subgroups of SL(2) -- 4.3.5 Lie Algebra and Lie Group over Zp -- Exercise -- 4.3.6 Arithmetic Galois Characters -- 4.3.7 Fullness of Modular Galois Representation -- 4.3.8 Fullness of Elliptic Curves -- 4.3.9 Fullness of Lie Algebra over A -- 4.3.10 Fullness of I-Adic Galois Representation -- 4.3.11 Basic Subgroups -- 4.3.12 Proof of Theorem 4.3.4 -- 5. Modularity Problems -- 5.1 Induced and Extended Galois Representations -- 5.1.1 Induction and Extension -- Exercises -- 5.1.2 Automorphic Induction -- Exercises -- 5.1.3 Artin Representations -- Exercise -- 5.2 Some Other Solutions -- 5.2.1 A Theorem of Wiles -- 5.2.2 Modularity of Extended Galois Representations -- 5.2.3 Elliptic Q-Curves -- Exercise.

5.2.4 Shimura-Taniyama Conjecture -- 5.3 Modularity of Abelian Q-Varieties -- 5.3.1 Abelian F-varieties of GL(2)-type -- 5.3.2 Endomorphism Algebras of Abelian F-varieties -- 5.3.3 Application to Abelian Q-Varieties -- 5.3.4 Abelian Varieties with Real Multiplication -- Bibliography -- List of Symbols -- Statement Index -- Index.