The local Langlands correspondence in families and Ihara's lemma for U(n)

TL;DR

This paper reformulates the Ihara's lemma for unitary groups using the local Langlands correspondence in families, linking Galois representations, automorphic forms, and deformation theory.

Contribution

It provides a new formulation of Ihara's lemma in terms of the local Langlands correspondence in families for $U(n)$, extending known results for $n=2$ to higher dimensions.

Findings

Unconditional existence of $ ilde{ ho}_{ar{r}}( ext{mod } ext{ell})$ for $n=2$.

Reformulation of Ihara's lemma in terms of local Langlands correspondence.

Connection between Galois deformations and automorphic forms in the context of $U(n)$.

Abstract

The goal of this paper is to reformulate the conjectural "Ihara lemma" for $U(n)$ in terms of the local Langlands correspondence in families $\tilde{\pi}_{\Sigma}(\cdot)$, as currently being developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod $\ell$ Galois representation $\bar{r}$, which is modular of full level (and small weight), and a finite set of places $\Sigma$ -- none of which divide $\ell$. Then $\tilde{\pi}_{\Sigma}(r)$ exists, and has a global realization as a natural module of algebraic modular forms, where $r$ is the universal $\Sigma$-deformation of $\bar{r}$. This is unconditional for $n=2$, where Ihara's lemma is an almost trivial consequence of the strong approximation theorem.