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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.
Sommario
Well-Adjusted Models for Curves over Dedekind Rings.- On the Manin Constants of Modular Elliptic Curves.- The Action of Monodromy on Torsion Points of Jacobians.- An Exceptional Isomorphism between Modular Varieties.- Chern Functors.- Curves of Genus 2 Covering Elliptic Curves and an Arithmetical Application.- Jacobians with Complex Multiplication.- Familles de Courbes Hyperelliptiques à Multiplications Réelles.- Séries de Kronecker et Fonctions L des Puissances Symétriques de Courbes Elliptiques sur Q.- Hyperelliptic Supersingular Curves.- Letter to Don Zagier.- The Old Subvariety of J0(pq).- Kolyvagin'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 Naive Point of View.- Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory of Fields.- Finiteness Theorems for Dimensions of Irreducible ?-adic Representations.
