Topics in arithmetic algebraic geometry

张志平在其先前关于Gross-Zagier公式和Arakelov理论的工作基础上，提出了在算术代数几何方面的几个课题，特别是自同构形式和代数曲线方面的研究。这些课题包括证明Shimura变量的Gross-Zagier型公式；研究其在Beilinson-Bloch猜想和Tate猜想中的应用；研究了与Caporaso， Harris， and Mazur的一致性猜想有关的正则子群的体积；利用度量带图研究正则坐标及其在曲线模空间上寻找指数函数或j函数类似物的Belyi-Grothendieck理论的算法应用；研究投影线减去四个点的覆盖模，构建与GL相关的伽罗瓦表示(3)。在过去的几十年里，数论、自同构表示、代数几何的研究在许多方面都取得了迅速的进展。在不同的关键时刻，每个学科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，并提出新的鼓舞人心的问题。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，但电子通信和机器人设计的安全性已经以一种基本的方式与许多这些几何和算术问题相关。本提案主要针对几种数学环境中出现的几何和算术问题，旨在发展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，连同各种免费提供的笔记已经张贴在他的网站上，将为希望学习算术几何一些基本主题的研究生提供有用的参考。

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.