Local Ihara's lemma and applications

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.

References in corpus (5)

• Théorie de Lubin-Tate non abélienne l-entière
• Un cas simple de correspondance de Jacquet-Langlands modulo l
• Congruences automorphes et torsion dans la cohomologie d'un système local d'Harris-Taylor
• Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces
• $p$-stabilization in higher dimension