On modular rigidity for ${\mathrm{GL}}_n$

Abstract

Let $k$ be a global field and ${\mathbb{A}}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to ${\overline{\mathbb{Q}}}_{\ell }$. Let ${\hat{I}}_1$ and ${\hat{I}}_2$ be cuspidal automorphic representations of ${\mathrm{GL}}_n({\mathbb{A}}_k)$ for some integer $n\ge 1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components ${\hat{I}}_{1,v}{\otimes }_{\mathbb{C}}{\overline{\mathbb{Q}}}_{\ell }$ and ${\hat{I}}_{2,v}{\otimes }_{\mathbb{C}}{\overline{\mathbb{Q}}}_{\ell }$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components ${\hat{I}}_{1,w}{\otimes }_{\mathbb{C}}{\overline{\mathbb{Q}}}_{\ell }$ and ${\hat{I}}_{2,w}{\otimes }_{\mathbb{C}}{\overline{\mathbb{Q}}}_{\ell }$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on ${\hat{I}}_1$ and ${\hat{I}}_2$, the answer is yes. We also give a simple proof when $k$ is a function field.