Conjectures in Arithmetic Algebraic Geometry

Inhaltsverzeichnis

• 1 The zero-dimensional case: number fields.
• 1.1 Class Numbers.
• 1.2 Dirichlet L-Functions.
• 1.3 The Class Number Formula.
• 1.4 Abelian Number Fields.
• 1.5 Non-abelian Number Fields and Artin L-Functions.
• 2 The one-dimensional case: elliptic curves.
• 2.1 General Features of Elliptic Curves.
• 2.2 Varieties over Finite Fields.
• 2.3 L-Functions of Elliptic Curves.
• 2.4 Complex Multiplication and Modular Elliptic Curves.
• 2.5 Arithmetic of Elliptic Curves.
• 2.6 The Tate-Shafarevich Group.
• 2.7 Curves of Higher Genus.
• 2.8 Appendix.
• 3 The general formalism of L-functions, Deligne cohomology and Poincaré duality theories.
• 3.1 The Standard Conjectures.
• 3.2 Deligne-Beilinson Cohomology.
• 3.3 Deligne Homology.
• 3.4 Poincaré Duality Theories.
• 4 Riemann-Roch, K-theory and motivic cohomology.
• 4.1 Grothendieck-Riemann-Roch.
• 4.2 Adams Operations.
• 4.3 Riemann-Roch for Singular Varieties.
• 4.4 Higher Algebraic K-Theory.
• 4.5 Adams Operations in Higher Algebraic K-Theory.
• 4.6 Chern Classes in Higher Algebraic K-Theory.
• 4.7 Gillet’s Riemann-Roch Theorem.
• 4.8 Motivic Cohomology.
• 5 Regulators, Deligne’s conjecture and Beilinson’s first conjecture.
• 5.1 Borel’s Regulator.
• 5.2 Beilinson’s Regulator.
• 5.3 Special Cases and Zagier’s Conjecture.
• 5.4 Riemann Surfaces.
• 5.5 Models over Spec(Z).
• 5.6 Deligne’s Conjecture.
• 5.7 Beilinson’s First Conjecture.
• 6 Beilinson’s second conjecture.
• 6.1 Beilinson’s Second Conjecture.
• 6.2 Hilbert Modular Surfaces.
• 7 Arithmetic intersections and Beilinson’s third conjecture.
• 7.1 The Intersection Pairing.
• 7.2 Beilinson’s Third Conjecture.
• 8 Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps.
• 8.1 The Hodge Conjecture.
• 8.2 Absolute Hodge Cohomology.
• 8.3 Geometric Interpretation.
• 8.4Abel-Jacobi Maps.
• 8.5 The Tate Conjecture.
• 8.6 Absolute Hodge Cycles.
• 8.7 Motives.
• 8.8 Grothendieck’s Conjectures.
• 8.9 Motives and Cohomology.
• 9 Mixed realizations, mixed motives and Hodge and Tate conjectures for singular varieties.
• 9.1 Tate Modules.
• 9.2 Mixed Realizations.
• 9.3 Weights.
• 9.4 Hodge and Tate Conjectures.
• 9.5 The Homological Regulator.
• 10 Examples and Results.
• 10.1 B & S-D revisited.
• 10.2 Deligne’s Conjecture.
• 10.3 Artin and Dirichlet Motives.
• 10.4 Modular Curves.
• 10.5 Other Modular Examples.
• 10.6 Linear Varieties.