Number theory, representation theory, and arithmetic geometry

Princeton University, Princeton NJ

Investigators

Abstract

This project is part of an ongoing research program. It revolves around understanding how various fundamental algebraic objects occurring in mathematics---such as rings, class groups, algebraic curves and varieties, and maps relating such structures---are parametrized. Our goals in understanding these parametrizations are at least fourfold: 1) to describe how these fundamental algebraic objects are distributed with respect to their most basic invariants; 2) to discover new invariants of these objects, and their applications; 3) to develop efficient and practical algorithms for performing computations with these algebraic objects; and, perhaps most importantly, 4) to discover and understand how various seemingly different algebraic structures are in fact closely related to each other. Such parametrizations have been already used by the PI over the past few years to obtain precise information on the distribution of number fields with respect to basic invariants such as discriminant and class number divisibility. Applying refined counting methods to these parametrizations has led, for example, to a proof of the first known case of the Cohen-Lenstra-Martinet class number heuristics for higher degree number fields, and other theorems of this nature are forthcoming. In more recent joint works with Arul Shankar, Wei Ho, and Dick Gross, theorems on the average sizes of Selmer groups in families of elliptic curves and hyperelliptic curves have been obtained, with applications to understanding the distributions of rational points on such curves. We expect several further analogous results on rational points, divisors, and other data of this kind for algebraic curves and surfaces in the near future. The primary purpose of applying for this grant is the support of graduate students. Many of the projects described above will involve graduate (and undergraduate) students in an essential way. All these projects involve very fundamental questions in number theory, and are an excellent way to introduce and bring students into the subject. Many other mathematicians from various institutions will also be involved in these projects, which will thus increase interactions among researchers from many different but related subjects.