The local Langlands correspondence in families and Ihara's lemma for 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.