2011-11-11

Towards the Breuil-Schneider Conjecture

Claus Sorensen, Princeton University

The p-adic Langlands program tries to link p-adic Hodge theory with p-adic functional analysis. The former is concerned with Galois representations coming from geometry, and one of the goals is to associate unitary representations of GL(n) on p-adic Banach spaces, in a natural way. One has a fairly complete understanding of this for GL(2) over the p-adics, but for other groups the anticipated correspondence remains rather mysterious. The Breuil-Schneider conjecture is a somewhat coarse (but precise) first approximation, asserting merely that an invariant norm exists. We believe we can prove this in many cases. For instance, for generalized Steinberg representations (a case mentioned explicitly in the Breuil-Schneider paper).This case will be emphasized in the talk. It turns out it can be rephrased purely group-theoretically, and in fact we prove the analogue for any reductive group, by global means. That is, we embed the situation in an automorphic representation, using trace formula techniques, and construct the desired norm by making use of p-adic modular forms.