Galois Representations and Automorphic Forms

In this talk, I will describe a construction of a geometric realisation of a p-adic Jacquet-Langlands correspondence for certain forms of GL(2) over a totally real field. The construction makes use of the completed cohomology of Shimura curves, and a study of the bad reduction of Shimura curves due to Rajaei (generalising work of Ribet for GL(2) over the rational numbers). Along the way I will also describe a p-adic analogue of Mazur's principle in this setting.

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F_1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic spaces and formal schemes, and is related to logarithmic and tropical geometry. I'll explain what analytic spaces over F_1 are, and will describe non-Archimedean and complex analytic spaces which are obtained from them.

A well known result of Coleman says that p-adic overconvergent (ellitpic) eigenforms of small slope are actually classical modular forms. Now consider an overconvergent p-adic Hilbert eigenform F for a totally real field L. When p is totally split in L, Sasaki has proved a similar result on the classicality of F. In this talk, I will explain how to treat the case when L is a quadratic real field and p is inert in L.

http://math.ias.edu/files/seminars/GriffithsTwoTalks.pdf

http://math.ias.edu/files/seminars/GriffithsTwoTalks.pdf

Let G be a connected reductive group over Q such that G(R) has discrete series representations. I will report on some statistical results on the Satake parameters (w.r.t. Sato-Tate distributions) and low-lying zeros of L-functions for families of automorphic representations of G(A). This is a joint work with Nicolas Templier.