L-Functions of Algebraic Varieties Over Finite Fields

万大庆（UC Irvine 9970417）教授将继续他在特征为p的有限域上由代数几何产生的L函数的p进的研究。本研究的中心问题是1973年Dwork关于单位根f晶体的L函数的p进延延的猜想。万教授将试图证实这一猜想的普遍有效性。他还将研究该猜想的算术结果，包括Gouvea-Mazur猜想的高维推广，某些几何l函数的零点，以及与卡拉比-丘变种家族产生的神秘镜像映射的联系。这是一个关于数论和代数几何的课题。数论是数学最古老的分支之一，它研究的是可以用最简单的计数数（1、2、3、4、5）精确表达的数字。在本世纪下半叶，现代几何对数论产生了巨大的影响，引入了新方法和新问题。在过去的30年里，代数几何的发展与数论的发展并驾齐驱。万教授将研究在这种几何的最简单的数学基础上定义的几何对象，即有限域。它们同时是代数几何中最简单和最有趣的对象。它们在通信、安全和其他令人惊讶的领域得到了非常实际的应用。

Abstract Daquing Wan UC Irvine 9970417 Professor Wan will continue his study of the p-adic of L-functions arising from algebraic geometry over a finite field of characteristic p. The central problem in this study is a 1973 conjecture of Dwork about the p-adic continuation of the L function of a unit root F-chrystal. Prof. Wan will try to establish the general validity of this conjecture. He will also investigate the arithmetic consequences of the conjecture including higher dimensional generalizations of the Gouvea-Mazur conjecture, and the zeros of certain geometric L-functions, and the connections with the mysterious mirror map that arises from a family of Calabi-Yau varieties. This is a project in number theory and algebraic geometry. One of mathematics oldest branches, number theory is the study of numbers that can be expressed exactly in terms of the simplest counting numbers, 1, 2, 3, 4, 5. In the second half of this century, modern geometry has had a tremendous impact on number theory introducing new methods and new problems. The development of algebraic geometry has paralleled advances in number theory for the past 30 years. Professor Wan will study geometric objects defined over the simplest possible mathematical basis for such a geometry, a finite field. These are simultaneously the simplest and most intriguing objects in algebraic geometry. They have found very practical application in communications, security, and other surprising areas.