D-modules on rigid analytic spaces

We give an overview of the theory of \wideparen{\mathcal{D}}-modules on rigid analytic spaces and its applications to admissible locally analytic representations of $p$-adic Lie groups.Comment: to appear in the Proceedings of the IC

A trace formula approach to control theorems for overconvergent automorphic forms

We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.Comment: 25 pages. Main results strengthened, higher level version include

Zero-loci of Brauer group elements on semi-simple algebraic groups

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.Comment: 35 pages. Added more details and fixed typos. Final versio