Patrick Allen - Urbana-Champaign

Title: Elliptic curves and automorphic forms

Abstract: Elliptic curves are some of the simplest examples of algebraic varieties that carry a group structure. The interplay between this group structure and Diophantine problems has led to some of the most interesting questions in number theory. It is a remarkable fact that in order to study many arithmetic properties of elliptic curves, it seems necessary to relate them to them objects much more analytic in nature called automorphic forms. For elliptic curves defined over the rationals, a proof of many cases of this correspondence was the subject of the famous work of Wiles that implied Fermat's Last Theorem. The Langlands philosophy predicts that such a correspondence should exist for any finite field extension of the rationals, but things become more mysterious because a certain connection to algebraic geometry disappears. I will discuss some solutions to this problem over so-called CM fields, such as the field of Gaussian numbers. Different parts of this are joint work with Caraiani, Calegari, Gee, Helm, Khare, Le Hung, Newton, Scholze, Taylor, and Thorne.