Local Langlands correspondence in rigid families

TL;DR

This paper proves local-global compatibility for a p-adic eigenvariety of a unitary group at split primes, extending results to non-classical points and employing Scholze's techniques for the local Langlands conjecture.

Contribution

It extends local-global compatibility results to non-classical points on the eigenvariety using Scholze's methods, interpolating the local Langlands correspondence across the family.

Findings

Compatibility holds at all points, including non-classical ones.

Interpolation of local Langlands correspondence across eigenvariety.

Employs Scholze's approach to advance the conjecture.

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. The novelty is we allow non-classical points, possibly non-\'{e}tale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.