Shimura Varieties and the Bernstein center

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The principal investigator will study an emerging connection between the bad reduction of Shimura varieties and the Bernstein center of an associated p-adic group. Shimura varieties form one of the main testing grounds for conjectures of Langlands on the calculation of Hasse-Weil zeta functions of algebraic varieties over number fields in terms of automorphic L-functions. The zeta function is defined as a product of local zeta functions, over all prime ideals in the field of definition. At primes of good reduction, the approach of Langlands and Kottwitz has been completed, thanks in part to the recent proof of the Langlands-Shelstad conjecture ("fundamental lemma") due to Ngo, Laumon, Waldspurger, and others. When singularities exist in the reduction modulo a prime ideal, geometric and representation-theoretic difficulties arise, which the PI will investigate. An important role will be played by some new "fundamental lemmas" which were not predicted by Langlands and which are formulated using Bernstein's decomposition of the category of smooth representations of a p-adic group. Another ingredient will be a decomposition of nearby cycles in a manner parallel to Bernstein's decomposition, and geometric arguments showing these nearby cycles give rise to test functions in the Bernstein center of the associated p-adic group. Many deep results in number theory involve the expression of a purely arithmetic object (such as a zeta function) in terms of a purely analytic object (such as an L-function). The latter are functions of a complex variable originating in the classical theory of modular forms, which are functions on the complex upper-half plane satisfying very stringent symmetry conditions. Several famous conjectures (e.g. the Birch and Swinnerton-Dyer conjecture, a Clay Math foundation Millennium Problem) postulate relations between values of zeta functions and arithmetic invariants. Shimura varieties form an important class of objects where links between arithmetic and analysis can be fully carried out. Progress in their study often has impact on other central questions in automorphic forms: a recent example is the proof due to M. Harris and R. Taylor of the local Langlands conjecture for general linear groups over p-adic fields. The PI will seek to further our understanding of Shimura varieties and other aspects of the Langlands program.