The local Langlands correspondence for GL_n in families

TL;DR

This paper extends the local Langlands correspondence to families of Galois representations over Noetherian rings, establishing existence, uniqueness, and a modified mod p version that better respects specialization.

Contribution

It introduces a framework for associating admissible modules to Galois representations over rings, proving existence and uniqueness, and defines a modified mod p correspondence with improved compatibility.

Findings

Existence of admissible modules over rings of integers of finite extensions of Q_p.

Uniqueness characterized by specific conditions.

A new modified mod p local Langlands correspondence with better specialization properties.

Abstract

Let E be a nonarchimedean local field with residue characteristic l, and suppose we have an n-dimensional representation of the absolute Galois group G_E of E over a reduced complete Noetherian local ring A with finite residue field k of characteristic p different from l. We consider the problem of associating to any such representation an admissible A[GL_n(E)]-module in a manner compatible with the local Langlands correspondence at characteristic zero points of Spec A. In particular we give a set of conditions that uniquely characterise such an A[GL_n(E)]-module if it exists, and show that such an A[GL_n(E)]-module always exists when A is the ring of integers of a finite extension of Q_p. We also use these results to define a "modified mod p local Langlands correspondence" that is more compatible with specialization of Galois representations than the mod p local Langlands correspondence of Vigneras.