Topics in arithmetic algebraic geometry

摘要奖DMS-0700322 Zhang PI建议工作的几个主题有关算术代数几何，特别是自守形式和代数曲线的基础上，他以前的工作对格罗斯-扎吉尔公式和Arakelov理论。这些课题包括证明Shimura簇的一个Gross-Zagier型公式，研究它在Beilinson-Bloch猜想和Tate猜想中的应用，研究与Caporaso，Harris和Mazur的一致性猜想有关的典型子群的体积，研究与该猜想有关的典型子群的体积。利用度量化带状图研究正则坐标，并利用Belyi-Grothendieck理论，它涉及在曲线的模空间上寻找指数或J函数的模拟;研究射影直线负四点覆盖的模，构造与GL（3）有关的Galois表示。在过去的几十年里，数论的研究，自守表示，代数几何是推进在许多方面的速度很快。在各个关键点上，每个主题都大量借鉴了相邻领域的最新进展，以实现突破，解决长期存在的问题，并提出新的启发性问题。PI参与的所有主题都旨在加强这些联系。虽然最初的动机来自纯数学，但电子电信和机器人设计的安全性已经与许多几何和算术问题密切相关。该建议侧重于在几个数学环境中出现的几何和算术问题，旨在开发一些新的理论方法，并将其应用于具体问题。 此外，PI建议继续他长期以来监督本科生和研究生水平的传统。他还将完成两本书，连同分类免费提供的笔记已经张贴在他的网站上，将提供有用的参考研究生谁希望了解一些基本的主题算术几何。

Abstract for the award DMS-0700322 of ZhangThe PI proposes to work on several topics related to arithmetic algebraic geometry, in particular to automorphic forms and algebraic curves based on his previous work on the Gross-Zagier formula and Arakelov theory. These topics include to prove a Gross-Zagier type formula for Shimura varieties; to study its applications to the Beilinson-Bloch conjecture and the Tate conjecture; to study volume of canonical subgroups which is related to the uniformity conjecture of Caporaso, Harris, and Mazur; to study canonical coordinates using metrized Ribbon graph and its arithmetic application using Belyi-Grothendieck theory which is related to find an analog of exponential or j-function on the moduli space of curves; to study moduli of coverings of projective line minus four points related to construct Galois representations related to GL(3).In the past few decades, research in number theory, automorphic representation, algebraic geometry is advancing at a rapid rate on many fronts. At various crucial points, each subject has drawn heavily on recent progress in the adjoining fields to bring about breakthroughs, solving longstanding problems, and raising new inspiring questions. All topics in which the PI is engaged have the aim of strengthening these connections. Though the initial motivation comes from within pure mathematics, security of electronic telecommunications and robot design have come to be related in an essential way to many of these geometric and arithmetic problems. This proposal focuses on both geometric and arithmetic questions that arise in several mathematical settings, aiming to develop some new theoretical methods and to apply them to specific problems. In addition, the PI proposes to continue his long-standing tradition of supervising undergraduate and graduate levels. He will also complete two books that, together with assorted freely available notes already posted on his web site, will provide useful references for graduate students who wish to learn about some fundamental topics in arithmetic geometry.