Local Langlands correspondence in rigid families

Christian Johansson, James Newton and Claus Sorensen

Vol. 304 (2020), No. 1, 65–102

DOI: 10.2140/pjm.2020.304.65

Abstract

We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. We do this by interpolating the local Langlands correspondence for ${GL}_n$ across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Chenevier and Scholze used in Scholze’s proof of the local Langlands conjecture for ${GL}_n$.

Keywords

Eigenvarieties, p-adic automorphic forms, Galois representations, local Langlands correspondence

Mathematical Subject Classification 2010

Primary: 11F70

Secondary: 11F80

Milestones

Received: 12 October 2018

Revised: 15 July 2019

Accepted: 1 August 2019

Published: 18 January 2020