Abstract Let k be a global field and A₊ be its ring of adèles. Let be a prime number and fix a field isomorphism from C to {{Q}_ }. Let ₁, ₂ be cuspidal automorphic representations of GL₍ (A₊) 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 ₁, ₕ ₂ {{Q}_ } and ₂, ₕ ₂ {{Q}_ } are unramified and their Satake parameters are integral and congruent mod, are the local components ₁, ₖ ₂ {{Q}_ } and ₂, ₖ ₂ {{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 ₁, ₂, the answer is yes. We also give a simple proof when k is a function field.
Matringe et al. (Mon,) studied this question.