Progress in Mathematics

By: Geer, Oort, Steenbrink

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Hardcover | 1 December 1990

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460 Pages

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24.13 x 15.88 x 3.18

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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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Participants
Contributors
Introduction
Well-Adjusted Models for Curves over Dedekind Rings
On the Manin Constants of Modular Elliptic Curves
The Action of Monogromy on Torsion Points of Jacobians
An Exceptional Isomorphism between Modular Varieties
Chern Functors
Curves of Genus 2Covering Elliptic Curves and an Arithmetical Application
Jacobians with Complex Multiplication
Familles de Courbes Hyperelliptiques ß Nultiplications RFelles
SFries de Kronecker et Fonctions L des Puissances SymFtriques de Courbes Elliptiques sur Q
Hyperelliptic Supersingular Curves
Letter to Don Zagier
The Old Subvariety of J(pq)
Loyvagin's System of Gauss Sums
The Exponents of the Groups of Points on the Reductions of an Elliptic Curve
The Generalized De Rham-Witt Complex and Congruence Differential Equations
Arithmetic Discriminants and Quadratic Points on Curves
The Birch-Swinnerton-Dyer Conjecture from a Nanve Point of View
Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory of Fields
Finiteness Theorems for Dimensions of Irreducible lambda-adic Representations
Table of Contents provided by Publisher. All Rights Reserved.

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