Conjectures in Arithmetic Algebraic Geometry: A Survey [Paperback]

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.- 2.8.1 B & S-D for Abelian Varieties.- 2.8.2 Blochs Version of B & S-D.- 2.8.3 1-Motives, Mixed Motives and B & S-D.- 3 The general formalism of L-functions, Deligne cohomology and Poincar? duality theories.- 3.1 The Standard Conjectures.- 3.2 Deligne-Beilinson Cohomology.- 3.3 Deligne Homology.- 3.4 Poincar? 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 Gillets Riemann-Roch Theorem.- 4.8 Motivic Cohomology.- 5 Regulators, Delignes conjecture and Beilinsons first conjecture.- 5.1 Borels Regulator.- 5.2 Beilinsons Regulator.- 5.3 Special Cases and Zagiers Conjecture.- 5.4 Riemann Surfaces.- 5.5 Models over Spec(Z).- 5.6 Delignes Conjecture.- 5.7 Beilinsons First Conjecture.- 6 Beilinsons second conjecture.- 6.1 Beilinsons Second Conjecture.- 6.2 Hilbert Modular Surfaces.- 7 Arithmetic intersections and Beilinsons third conjecture.- 7.1 The Intersection Pairing.- 7.2 Beilinsons 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.- 8.4 Abel-Jacobi Maps.- 8.5 The Tate Conjecture.- 8.6 Absolute Hodge Cycles.- 8.7 Motives.- 8.8 Grothendiecl3ã