Topics in arithmetic algebraic geometry

张氏DMS-0700322奖的提要派建议在他之前关于格罗斯-扎吉尔公式和阿拉克洛夫理论的工作的基础上，研究与算术代数几何有关的几个主题，特别是自同构形和代数曲线。这些主题包括证明Shimura簇的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标准子群的体积；利用度量化带图研究标准坐标及其利用Belyi-Grothendieck理论的算法应用；研究与GL(3)有关的射影直线减去四个点的覆盖模.在过去的几十年里，数论、自同构表示、代数几何等方面的研究在许多方面都取得了很大的进展.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。国际和平研究所参与的所有专题都旨在加强这些联系。虽然最初的动机来自于纯粹的数学，但电子通信和机器人设计的安全已经以一种重要的方式与许多这样的几何和算术问题联系在一起。这项建议集中在几何和算术问题，在几个数学背景下出现，旨在发展一些新的理论方法，并将它们应用于具体的问题。此外，私隐专员公署建议延续其一贯的督导本科生和研究生程度的传统。他还将完成两本书，连同他网站上已经张贴的各种免费提供的笔记，将为希望学习算术几何基础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.