Number Theory III

I Some Qualitative Diophantine Statements.- §1. Basic Geometric Notions.- §2. The Canonical Class and the Genus.- §3. The Special Set.- §4. Abelian Varieties.- §5. Algebraic Equivalence and the Néron-Severi Group.- §6. Subvarieties of Abelian and Semiabelian Varieties.- §7. Hilbert Irreducibility.- II Heights and Rational Points.- §1. The Height for Rational Numbers and Rational Functions.- §2. The Height in Finite Extensions.- §3. The Height on Varieties and Divisor Classes.- §4. Bound for the Height of Algebraic Points.- III Abelian Varieties.- §0. Basic Facts About Algebraic Families and Néron Models.- §1, The Height as a Quadratic Function.- §2. Algebraic Families of Heights.- §3. Torsion Points and the l-Adic Representations.- §4. Principal Homogeneous Spaces and Infinite Descents.- §5. The Birch-Swinnerton-Dyer Conjecture.- §6. The Case of Elliptic Curves Over Q.- IV Faltings’ Finiteness Theorems on Abelian Varieties and Curves.- §1. Torelli’s Theorem.- §2. The Shafarevich Conjecture.- §3. The l-Adic Representations and Semisimplicity.- §4. The Finiteness of Certain l-Adic Representations. Finiteness I Implies Finiteness II.- §5. The Faltings Height and Isogenies: Finiteness I.- §6. The Masser-Wustholz Approach to Finiteness I.- V Modular Curves Over Q.- §1. Basic Definitions.- §2. Mazur’s Theorems.- §3. Modular Elliptic Curves and Fermat’s Last Theorem.- §4. Application to Pythagorean Triples.- §5. Modular Elliptic Curves of Rank 1.- VI The Geometric Case of Mordell’s Conjecture.- §0. Basic Geometric Facts.- §1. The Function Field Case and Its Canonical Sheaf.- §2. Grauert’s Construction and Vojta’s Inequality.- §3. Parshin’s Method with (?;2x/y).- §4. Manin’s Method with Connections.- §5. Characteristic p and Voloch’s Theorem.- VII Arakelov Theory.- §1. Admissible Metrics Over C.- §2. Arakelov Intersections.- §3. Higher Dimensional Arakelov Theory.- VIII Diophantine Problems and Complex Geometry.- §1. Definitions of Hyperbolicity.- §2. Chern Form and Curvature.- §3. Parshin’s Hyperbolic Method.- §4. Hyperbolic Imbeddings and Noguchi’s Theorems.- §5. Nevanlinna Theory.- IX Weil Functions. Integral Points and Diophantine Approximations.- §1. Weil Functions and Heights.- §2. The Theorems of Roth and Schmidt.- §3. Integral Points.- §4. Vojta’s Conjectures.- §5. Connection with Hyperbolicity.- §6. From Thue-Siegel to Vojta and Faltings.- §7. Diophantine Approximation on Toruses.- X Existence of (Many) Rational Points.- §1. Forms in Many Variables.- §2. The Brauer Group of a Variety and Manin’s Obstruction.- §3. Local Specialization Principle.- §4. Anti-Canonical Varieties and Rational Points.