#### 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