Abelian varieties and Neron models

点击翻译按钮获取中文摘要

DMS-0302043Lorenzini, Dino J.AbstractTitle: Abelian varieties and Neron modelsOne of the main problems in arithmetic geometry is the determinationof the set of rational solutions of a system of polynomial equations with rational coefficients. One of the most successful techniquesin the study of such a set of solutions is to reduce the equations moduloa 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 wayof reducing the equations modulo p. When A/Kis an abelian variety, this canonical reduction is called the N\'eron modelof 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 modelis a finite group called the group of components. When the reduction is purelyadditive, it is conjectured that there are only finitely many possibilities for this group once the dimension g is fixed.Lorenzini's proposed research willshed more light on this conjecture and on other specialphenomena that arise when the reduction modulo a small prime pis not `good.'For centuries, human beings have been fascinated with solving Diophantine equations, named after the Greek mathematician Diophantus who livedin the third century AD. The field of Diophantine equationshas 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 developedsophisticated tools to aid in solving equations. This investigator has developed some such tools, and is currently working on further contributions to this field.