Local Ihara’s Lemma and Applications

Abstract

Abstract Persistence of nondegeneracy is a phenomenon that 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 nondegenerate irreducible representation is nondegenerate. 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 nondegeneracy persistence and related to the notion of essentially absolutely irreducible and generic representations in the work of Emerton and Helm, remains true for lattices given by the cohomology of Lubin–Tate spaces. As a global application, we give a new construction of automorphic congruences in the Ribet spirit.