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

TL;DR

This paper demonstrates that prime-to-p Hecke eigenvalues from automorphic forms mod p for Hodge type groups are equivalent to those from algebraic modular forms mod p of an inner form, revealing a deep connection between these automorphic objects.

Contribution

It establishes an equivalence between Hecke eigensystems of automorphic forms of Hodge type and algebraic modular forms of an inner form mod p.

Findings

Hecke eigensystems of automorphic forms mod p match those of algebraic modular forms of an inner form.

The result applies to automorphic forms associated with algebraic groups of Hodge type.

The work connects automorphic forms and algebraic modular forms through their Hecke eigenvalues.

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 $G$.