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.