Ihara's lemma and level rising in higher dimension

TL;DR

This paper extends Ihara's lemma to higher dimensions for certain similitude groups, enabling new level raising results in the context of unramified Hecke algebras, which has implications for automorphic forms and Galois representations.

Contribution

It proves new instances of a generalized Ihara's lemma in higher dimensions, specifically for non pseudo Eisenstein maximal ideals, leading to level raising results.

Findings

Proved new cases of generalized Ihara's lemma in higher dimension.

Established a level raising theorem for certain automorphic forms.

Connected the lemma to properties of unramified Hecke algebras.

Abstract

A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara's lemma which is used to rise the modularity property between some congruent galoisian representations. In their work on Sato-Tate, Clozel-Harris-Taylor proposed a generalization of the Ihara's lemma in higher dimension for some similitude groups. The main aim of this paper is then to prove some new instances of this generalized Ihara's lemma by considering some particular non pseudo Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level rising statement.