Number Theory and Related Fields

Harvard University, Cambridge MA

Investigators

Abstract

The Principal Investigator proposes to work on two distinct but related projects. The first project is a study of Selmer groups and Mordell-Weil groups of elliptic curves over towers of number fields, and the second is to resolve certain deformational problems related to automorphic forms. One of the aims of the first project is to prove that Selmer rank grows at least as fast as would be predicted by heuristics governing the signs of expected functional equations. One of the aims of the second is to understand the parameter spaces of automorphic forms that are usually referred to as "eigenvarieties." The first project pursued by the principal investigator is the fundamental arithmetic problem of understanding the structure of triples of algebraic numbers that are solutions of given homogeneous polynomials of degree three in three variables, this cubic case having an extraordinary amount of structure and playing a pivotal role in the larger project of understanding the arithmetic of polynomial equations in general. This type of number theory was critical, for example, in the establishment of Fermat's Last Theorem (work of Wiles and Taylor-Wiles) over a dozen years ago, and - indeed - becomes ever more powerful and continues to be crucial for a wide range of applications, for example in cryptography. The second project proposed by the Principal Investigator has its historical origin in classical work of Ramanujan, that dealt with the arithmetic properties of the Fourier coefficients of modular forms. Ramanujan unearthed striking congruences that on the one hand contain important number theoretic information, and on the other suggest that a mysterious coherence underlies a large assortment of basic arithmetic phenomena; one of the goals of modern number theory is to expand this, and use its power for a range of applications. 1