Abelian varieties and Neron models

阿贝尔簇和 Neron 模型

• 批准号: 0302043
• 负责人: Dino Lorenzini
• 金额: --
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0302043&HistoricalAwards=false
• 关键词: Abelian; varieties; Neron; models

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.

算术几何中的一个主要问题是确定有理系数多项式方程组的有理解集。在研究这样一组解的过程中，最成功的方法之一是将方程简化为模a素数p，并研究 后一个方程组的解。 事实证明，在许多情况下，有可能描述一种规范的方式来减少方程模p。当A/K是一个阿贝尔簇时，这种规范的减少被称为A/K的N\'eron模型。这种减少是洛伦齐尼的前三个研究项目在这个建议的研究对象。例如，附加到任何Neron模型的是一个有限群，称为组件群。当约简是纯可加的时，证明了当维数g固定时，这个群只有100多种可能性。Lorenzini提出的研究将对这个猜想以及当约简模为小素数p时出现的其他特殊现象有更多的解释。“几个世纪以来，人类一直着迷于求解丢番图方程，这个方程是以生活在公元三世纪的希腊数学家丢番图命名的。丢番图方程在现代世界中具有更重要的意义，因为它在包括加密在内的各种领域都有应用。从古希腊时代起，数学家们就发展出了复杂的工具来帮助解方程。本研究员开发了一些此类工具，目前正在努力为这一领域做出进一步贡献。