Families of Automorphic Forms with Prescribed Local Behavior

Cornell University, Ithaca NY

Investigators

Abstract

The goal of the research project is to develop a quantitative theory of families with prescribed local behavior. Generally speaking, the idea is that one can study a mathematical object: a variety, a representation, or an L-function, by deforming it in families. A family is a set of global objects that share some common local features. Mathematicians predict that families have equidistribution properties in the sense that their local components should vary in a uniform way within their prescribed space. Building proofs of such properties is a milestone in our mathematical ability to work with these objects. Additionally this project will provide research training opportunities for graduate students. Families are crucial even if one is a priori interested in a single automorphic form. Arthur conjectured in 1988 that if an automorphic representation has a local component that belongs to a supercuspidal L-packet, then it is tempered. Over function fields, the PI proposes to establish with Sawin this conjecture for monomial geometric supercuspidal (mgs) packets, that is for those packets that arise from compact-induction of characters and remain supercuspidal after unramified base change. The method relies on combining the l-adic geometry of the moduli space of G-bundles on a curve and on trace formulas. 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.