Number Theory III

I Some Qualitative Diophantine Statements.- x1. Basic Geometric Notions.- x2. The Canonical Class and the Genus.- x3. The Special Set.- x4. Abelian Varieties.- x5. Algebraic Equivalence and the Neron-Severi Group.- x6. Subvarieties of Abelian and Semiabelian Varieties.- x7. Hilbert Irreducibility.- II Heights and Rational Points.- x1. The Height for Rational Numbers and Rational Functions.- x2. The Height in Finite Extensions.- x3. The Height on Varieties and Divisor Classes.- x4. Bound for the Height of Algebraic Points.- III Abelian Varieties.- x0. Basic Facts About Algebraic Families and Neron Models.- x1, The Height as a Quadratic Function.- x2. Algebraic Families of Heights.- x3. Torsion Points and the l-Adic Representations.- x4. Principal Homogeneous Spaces and Infinite Descents.- x5. The Birch-Swinnerton-Dyer Conjecture.- x6. The Case of Elliptic Curves Over Q.- IV Faltings' Finiteness Theorems on Abelian Varieties and Curves.- x1. Torelli's Theorem.- x2. The Shafarevich Conjecture.- x3. The l-Adic Representations and Semisimplicity.- x4. The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II.- x5. The Faltings Height and Isogenies: Finiteness I.- x6. The Masser-Wustholz Approach to Finiteness I.- V Modular Curves Over Q.- x1. Basic Definitions.- x2. Mazur's Theorems.- x3. Modular Elliptic Curves and Fermat's Last Theorem.- x4. Application to Pythagorean Triples.- x5. Modular Elliptic Curves of Rank 1.- VI The Geometric Case of Mordell's Conjecture.- x0. Basic Geometric Facts.- x1. The Function Field Case and Its Canonical Sheaf.- x2. Grauert's Construction and Vojta's Inequality.- x3. Parshin's Method with (?;2x/y).- x4. Manin's Method with Connections.- x5. Characteristic p and Voloch's Theorem.- VII Arakelov Theory.- x1. Admissible Metrics Over C.- x2. Arakelov Intersections.- x3. Higher Dimensional Arakelov Theory.- VIII Diophantine Problems and Complex Geometry.- x1. Definitions of Hyperbolicity.- x2. Chern Form and Curvature.- x3. Parshin's Hyperbolic Method.- x4. Hyperbolic Imbeddings and Noguchi's Theorems.- x5. Nevanlinna Theory.- IX Weil Functions. Integral Points and Diophantine Approximations.- x1. Weil Functions and Heights.- x2. The Theorems of Roth and Schmidt.- x3. Integral Points.- x4. Vojta's Conjectures.- x5. Connection with Hyperbolicity.- x6. From Thue-Siegel to Vojta and Faltings.- x7. Diophantine Approximation on Toruses.- X Existence of (Many) Rational Points.- x1. Forms in Many Variables.- x2. The Brauer Group of a Variety and Manin's Obstruction.- x3. Local Specialization Principle.- x4. Anti-Canonical Varieties and Rational Points.