On Modular Rigidity for GLn

Abstract Let $k$ be a global field and ${\mathbb{A}}_{k}$ be its ring of adèles. Let $\ell $ be a prime number and fix a field isomorphism from ${\mathbb{C}}$ to ${\overline{{\mathbb{Q}}}_\ell }$. Let $\Pi _{1}$, $\Pi _{2}$ be cuspidal automorphic representations of $\mathrm{GL}_{n}({\mathbb{A}}_{k})$ for some integer $n \geqslant 1$. 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 $\Pi _{1,v} \otimes _{{\mathbb{C}}} {\overline{{\mathbb{Q}}}_\ell }$ and $\Pi _{2,v} \otimes _{{\mathbb{C}}} {\overline{{\mathbb{Q}}}_\ell }$ are unramified and their Satake parameters are integral and congruent mod $\ell $, are the local components $\Pi _{1,w} \otimes _{{\mathbb{C}}} {\overline{{\mathbb{Q}}}_\ell }$ and $\Pi _{2,w} \otimes _{{\mathbb{C}}} {\overline{{\mathbb{Q}}}_\ell }$ 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 $\Pi _{1}$, $\Pi _{2}$, the answer is yes. We also give a simple proof when $k$ is a function field.