Local Ihara's lemma and applications

TL;DR

This paper proves that non-degeneracy persistence holds for lattices from Lubin-Tate space cohomology, leading to new automorphic congruence constructions, extending Ihara's lemma in a local-global context.

Contribution

It establishes non-degeneracy persistence for certain lattices, generalizing Ihara's lemma and enabling new automorphic congruence results.

Findings

Non-degeneracy persistence holds for Lubin-Tate cohomology lattices.

A new construction of automorphic congruences is provided.

Extension of Ihara's lemma to a local-global setting.

Abstract

Persistence of non-degeneracy is a phenomenon which appears in the theory of $\overline{\mathbb Q}_l$-representations of the linear group: every irreducible submodule of the restriction to the mirabolic sub-representation of a non-degenerate irreducible representation is non-degenerate. This is not true anymore in general, if we look at the modulo $l$ reduction of some stable lattice. As in the Clozel-Harris-Taylor generalization of global Ihara's lemma, we show that this property, called non-degeneracy persistence and related to the notion of essentially absolutely irreducible and generic representations in the work of Emerton-Helm, remains true for lattices given by the cohomology of Lubin-Tate spaces. As an global application, we give a new construction of automorphic congruences in the Ribet spirit.