Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms

Abstract

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$.