Wilfred W. J. Hulsbergen

Conjectures in Arithmetic Algebraic Geometry (eBook, PDF)

A Survey

• Format: PDF

Produktdetails

• Verlag: Vieweg+Teubner Verlag
• Seitenzahl: 240
• Erscheinungstermin: 9. März 2013
• Englisch
• ISBN-13: 9783322854667
• Artikelnr.: 53085309

Inhaltsangabe

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.