Topics in arithmetic algebraic geometry

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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.