Wilfred W. J. Hulsbergen

Conjectures in Arithmetic Algebraic Geometry

A Survey

• Broschiertes Buch

Produktbeschreibung

In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math ematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to introduce L functions, the main, motivation being the calculation of class numbers. In partic ular, Kummer showed that the class numbers of cyclotomic fields play a decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirichlet had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by properties of L-functions. Twentieth century number theory, class field theory and algebraic geome try only strengthen the nineteenth century number theorists's view. We just mention the work of E. H~cke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generalization of Dirichlet's L-functions with a generalization of class field theory to non-abelian Galois extensions of number fields in mind.

Produktdetails

• Aspects of Mathematics 18
• Verlag: Vieweg+Teubner / Vieweg+Teubner Verlag
• Artikelnr. des Verlages: 978-3-663-09507-1
• 2. Aufl.
• Seitenzahl: 256
• Erscheinungstermin: 3. Oktober 2013
• Englisch
• Abmessung: 244mm x 170mm x 15mm
• Gewicht: 449g
• ISBN-13: 9783663095071
• ISBN-10: 366309507X
• Artikelnr.: 39578642

Autorenporträt

Dr. Wilfried Hulsbergen is teaching at the KMA, Breda,Niederlande.

Inhaltsangabe

1 The zero-dimensional case: number fields.- 2 The one-dimensional case: elliptic curves.- 3 The general formalism of L-functions, Deligne cohomology and Poincaré duality theories.- 4 Riemann-Roch, K-theory and motivic cohomology.- 5 Regulators, Deligne's conjecture and Beilinson's first conjecture.- 6 Beilinson's second conjecture.- 7 Arithmetic intersections and Beilinson's third conjecture.- 8 Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps.- 9 Mixed realizations, mixed motives and Hodge and Tate conjectures for singular varieties.- 10 Examples and Results.- 11 The Bloch-Kato conjecture.

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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 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.