Judith Ludwig will speak on

A quotient of the Lubin-Tate tower

Abstract: In this talk we explain how to construct the quotient of an infinite-level Lubin-Tate space by the Borel subgroup B(Q_p) of upper triangular matrices in GL(2,Q_p) as a perfectoid space. The motivation for this is as follows. Scholze recently constructed a candidate for the mod p Jacquet-Langlands correspondence and the mod p local Langlands correspondence for GL(n,F), F/Q_p finite. Given a smooth admissible representation \pi of GL(n,F), the candidate for these correspondences is given by the etale cohomology groups of the adic projective space P^{n-1} with coefficients in a sheaf F_\pi that one constructs from \pi. The finer properties of this candidate remain mysterious.

As an application of the quotient construction one can show that in the case of n=2,F=Q_p, and \pi an irreducible principal series representation or a twist of the Steinberg representation, the cohomology H^i_et(P^1,F_\pi) is concentrated in degree one.