Topics in Arithmetic Algebraic Geometry

9820909张守武教授将继续他的算术几何研究，重点是代数点的几何和模形式的算术。 这项研究是基于他以前对Bogomolov猜想和Gross-Zagier公式的工作。 在代数点方面，张将研究“大点的散点性”的一个交换品种的子品种和“算术Kodaira-Spencer类”的算术表面。 在模形式方面，他将尝试证明一个“Maass形式的Gross-Zagier型公式”，并以正特征完成“志村曲线上的Atkin-Lehner理论”。这项研究福尔斯数论的一个现代领域，即算术几何，其中几何方法被用来解决数论问题。 数论的历史根源在于对自然数的研究。 它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。 然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

9820909Professor Shou-Wu Zhang will continues his studies in arithmetic geometry, with focus on the geometry of algebraic points and the arithmetic of modular forms. This research is based on his previous work on the Bogomolov conjecture and the Gross-Zagier formula. On the side of algebraic points, Zhang will study "the scatteredness of big points" on a subvariety of an abelian variety and "the arithmetic Kodaira-Spencer class" for arithmetic surfaces. On the modular form side, Zhang will try to prove a "Gross-Zagier type formula for Maass forms" and to finish his work on "the Atkin-Lehner theory on Shimura curves" in positive characteristic.This research falls into a modern area of number theory known as arithmetic geometry in which geometric methods are used to solve problems in number theory. Number theory has its historical roots in the study of the natural numbers. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.