Arithmetic Geometry and Applications

点击翻译按钮获取中文摘要

The award supports the principal investigator's research in arithmetic geometry. Arithmetic geometry is a branch of mathematics that studies integer solutions of polynomial equations and has played a central role in solving many outstanding problems in number theory, such as Fermat's Last Theorem and the Mordell conjecture concerning the number of rational points on a curve. The main objects of study in this research project are called "Shimura varieties," the study of which is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of far-reaching and influential conjecturesThe project concerns problems in and applications of the arithmetic of Shimura varieties, which are generalizations of the moduli space of abelian varieties. In the first part of the project the PI plans to study the structure of the cohomology of Shimura varieties, and the structure of their mod p points. Specifically, there is a conjecture, proved by the PI in some cases, that the isogeny class of every mod p point contains the reduction of a special point. These results can be used to study the Hasse-Weil zeta function of a Shimura variety, following a program of Langlands. Another part of the project aims to study an analogue of the Mumford-Tate conjecture in crystalline cohomology; this amounts to an algebraisation problem for certain formal cycles on a Shimura variety. Finally, the PI plans to apply techniques from the deformation theory of finite flag group schemes to study Hilbert's 13th problem, which asks for the minimal m for which the solution of a general polynomial of degree n can be written as a composite of functions of m variables.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.