#### Vol. 304, No. 1, 2020

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Local Langlands correspondence in rigid families

### Christian Johansson, James Newton and Claus Sorensen

Vol. 304 (2020), No. 1, 65–102
##### 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
Primary: 11F70
Secondary: 11F80
##### Milestones
Received: 12 October 2018
Revised: 15 July 2019
Accepted: 1 August 2019
Published: 18 January 2020
##### Authors
 Christian Johansson Department of Mathematical Sciences Chalmers University of Technology and the University of Gothenburg SE-412 96 Gothenburg Sweden James Newton Department of Mathematics King’s College London Strand London United Kingdom Claus Sorensen Department of Mathematics University of California, San Diego La Jolla, CA United States