Topics in Arithmetic Algebraic Geometry

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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.