Conjectures in Arithmetic Algebraic Geometry
A Survey
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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 P...
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.
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