Arithmetic Algebraic Geometry

Gerard van der Geer, Frans Oort, Jozef Steenbrink (auth.), G. van der Geer, F. Oort, J. Steenbrink (eds.)0817635130, 9780817635138, 3764335130

Arithmetic algebraic geometry is in a fascinating stage of growth, providing a rich variety of applications of new tools to both old and new problems. Representative of these recent developments is the notion of Arakelov geometry, a way of “completing” a variety over the ring of integers of a number field by adding fibres over the Archimedean places. Another is the appearance of the relations between arithmetic geometry and Nevanlinna theory, or more precisely between diophantine approximation theory and the value distribution theory of holomorphic maps.

Inspired by these exciting developments, the editors organized a meeting at Texel in 1989 and invited a number of mathematicians to write papers for this volume. Some of these papers were presented at the meeting; others arose from the discussions that took place. They were all chosen for their quality and relevance to the application of algebraic geometry to arithmetic problems.

Topics include: arithmetic surfaces, Chjerm functors, modular curves and modular varieties, elliptic curves, Kolyvagin’s work, K-theory and Galois representations. Besides the research papers, there is a letter of Parshin and a paper of Zagier with is interpretations of the Birch-Swinnerton-Dyer Conjecture.

Research mathematicians and graduate students in algebraic geometry and number theory will find a valuable and lively view of the field in this state-of-the-art selection.

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Table of contents :

Content:
Front Matter….Pages i-x
Introduction….Pages 1-2
Well-Adjusted Models for Curves over Dedekind Rings….Pages 3-24
On the Manin constants of modular elliptic curves….Pages 25-39
The action of monodromy on torsion points of Jacobians….Pages 41-49
An exceptional isomorphism between modular varieties….Pages 51-74
Chern Functors….Pages 75-152
Curves of genus 2 covering elliptic curves and an arithmetical application….Pages 153-176
Jacobians with complex multiplication….Pages 177-192
Familles de courbes hyperelliptiques � multiplications réelles….Pages 193-208
Séries de kronecker et Fonctions L des Puissances Symétriques de Courbes Elliptiques sur Q….Pages 209-245
Hyperelliptic supersingular curves….Pages 247-284
Letter to Don Zagier by A.N. Parshin….Pages 285-292
The old subvariety of J o (pq)….Pages 293-307
Kolyvagin’s System of Gauss Sums….Pages 309-324
The exponents of the groups of points on the reductions of an elliptic curve….Pages 325-335
The Generalized De Rham-Witt Complex and Congruence Differential Equations….Pages 337-358
Arithmetic discriminants and quadratic points on curves….Pages 359-376
The Birch-Swinnerton-Dyer Conjecture from a Naive Point of View….Pages 377-389
Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory of Fields….Pages 391-430
Finiteness theorems for dimensions of irreducible λ-adic representations….Pages 431-444
Back Matter….Pages 445-446