Local langlands correspondence in rigid families - Johansson, C; Newton, J; Sorensen, C (Mathematical Sciences Publishers, 2020)
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 GLn 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 GLn.