Conjectures in arithmetic algebraic geometry

Inhaltsverzeichnis1 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.