Shimura Varieties and Abelian Varieties

Harvard University, Cambridge MA

Investigators

Abstract

The award supports the principal investigator's research in arithmetic geometry, a branch of mathematics that studies integer solutions of polynomial equations, also called “rational points.” Arithmetic geometry 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 "Abelian varieties" and "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 long-standing conjectures. The project provides training opportunities for graduate students. This project concerns problems in and applications of the arithmetic of Abelian varieties and Shimura varieties, the latter being generalizations of the moduli space of abelian varieties. The first goal of the project is to show a new kind of Northcott property for the isogeny class of an abelian variety over a number field. Namely that, up to isomorphism, there are only finitely many abelian varieties of bounded height in the isogeny class. The second goal of the project is 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 principal investigator in some cases, that the isogeny class of every mod p contains the reduction of a special point. These results can be used to give a spectral interpretation of the Hasse-Weil zeta function of a Shimura variety, following a program of Langlands. 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.