Abelian varieties and Neron models

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

DMS-0302043 Lorenzini, Dino J. Abstract Title: Abelian varieties and Neron models One of the main problems in arithmetic geometry is the determination of the set of rational solutions of a system of polynomial equations with rational coefficients. One of the most successful techniques in the study of such a set of solutions is to reduce the equations modulo a prime p and to study the set of solutions of the latter system of equations. It turns out that, in many situations, it is possible to describe a canonical way of reducing the equations modulo p. When A/K is an abelian variety, this canonical reduction is called the N\'eron model of A/K. This reduction is the object of study in Lorenzini's first three research projects in this proposal. For instance, attached to any Neron model is a finite group called the group of components. When the reduction is purely additive, it is conjectured that there are only finitely many possibilities for this group once the dimension g is fixed. Lorenzini's proposed research will shed more light on this conjecture and on other special phenomena that arise when the reduction modulo a small prime p is not `good.' For centuries, human beings have been fascinated with solving Diophantine equations, named after the Greek mathematician Diophantus who lived in the third century AD. The field of Diophantine equations has taken added significance in the modern world as it finds applications in a variety of areas, including encryption. Since the time of the Greeks, mathematicians have developed sophisticated tools to aid in solving equations. This investigator has developed some such tools, and is currently working on further contributions to this field.