On Modular Rigidity for GLn

Let k be a global field and Aₖ be its ring of adèles. Let be a prime number and fix a field isomorphism from C to Q_. Let Π ₁, Π ₂ be cuspidal automorphic representations of GLₙ(Aₖ) for some integer n 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 such that, for all v∉ S, the local components Π 1,v ⊗ _C Q_ and Π 2,v ⊗ _C Q_ are unramified and their Satake parameters are integral and congruent mod , are the local components Π 1,w ⊗ _C Q_ and Π 2,w ⊗ _C Q_ integral, and do their reductions mod share an irreducible factor for all non-Archimedean places w not dividing ? We show that, under certain conditions on Π ₁, Π ₂, the answer is yes. We also give a simple proof when k is a function field.