Elliptic Curves and Cohomological Automorphic Forms over CM Fields

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The study of symmetry pervades mathematics. In number theory, interesting symmetries exist among algebraic numbers, numbers which are roots of polynomials with integer coefficients. These symmetries underpin a surprising connection, known as Langlands reciprocity, between arithmetic, geometry, and analysis. The resulting bridges constructed between seemingly disparate areas bring powerful analytic and algebraic tools to bear on arithmetic questions. This project aims to establish new cases Langlands reciprocity, and to apply the resulting tools to questions in number theory. Proving automorphy of Galois representations is an important theme in modern algebraic number theory, and is currently the only known technique that establishes many conjectural properties of arithmetic L-functions. Part of this project aims to establish automorphy of many elliptic curves over imaginary quadratic fields, or more generally CM number fields. The second part of this project aims to refine our knowledge of local-global compatibility in the Langlands program over CM fields. This finer compatibility will then be applied to the study of adjoint Selmer groups in the third part of this project, establishing cases of conjectures of Bloch-Kato and Perrin-Riou, as well as having applications to a program of Venkatesh. 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.