Abelian varieties and Neron models

DMS-0302043 Lorenzini，Dino J.摘要标题：阿贝尔族与农隆模型算术几何中的主要问题之一是确定有理系数多项式方程组的有理解集。研究这种解集最成功的技巧之一是将模为素数p的方程化简，并研究后一组方程的解集。结果表明，在许多情况下，可以描述模为p的方程的一种典范约化方法.当A/K是阿贝尔变种时，这种典范约化被称为A/K的N-隆隆模型.这种约化是Lorenzini在本提案中的前三个研究项目中的研究对象.例如，连接到任何农隆模型的是一个有限群，称为组件群。当约化是纯可加性时，人们猜测，一旦维度g固定，这个群就只有有限多的可能性。洛伦齐尼提出的研究将更多地阐明这一猜想，以及当模为小素数p的约化不好时出现的其他特殊现象。几个世纪以来，人类一直痴迷于求解丢番图方程，以生活在公元3世纪的希腊数学家Diophantus的名字命名。丢番图方程的领域在现代世界中变得更加重要，因为它在包括加密在内的各种领域中都有应用。从希腊时代开始，数学家就开发出复杂的工具来帮助解方程。这位调查员已经开发了一些这样的工具，目前正在为这一领域做出进一步的贡献。

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.