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

2014/06/06 by Sorensen, Claus
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1406.1830

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 π_Σ(⋅), as currently being developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod ℓ Galois representation r, which is modular of full level (and small weight), and a finite set of places Σ -- none of which divide ℓ. Then π_Σ(r) exists, and has a global realization as a natural module of algebraic modular forms, where r is the universal Σ-deformation of r. This is unconditional for n=2, where Ihara's lemma is an almost trivial consequence of the strong approximation theorem.

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