Hecke Eigensystems for (mod p) Modular Forms of PEL-type and Algebraic Modular Forms

TL;DR

This paper establishes a correspondence between Hecke eigensystems from (mod p) PEL-type modular forms and algebraic modular forms for inner forms of the same group, extending classical results to new group types.

Contribution

It generalizes Serre's and Ghitza's results to PEL-type modular forms of algebraic groups of type A or C, including explicit cases like unitary groups.

Findings

Hecke eigensystems from (mod p) PEL-type forms match those from algebraic modular forms for inner forms.

Explicit asymptotic formula for the number of eigensystems in the case of unitary groups.

Extension of classical results to broader classes of algebraic groups.

Abstract

We show that, under suitable assumptions, the systems of Hecke eigenvalues arising from (mod p) modular forms of PEL-type associated to an algebraic group G of type A or C coincide with the Hecke eigensystems arising from (mod p) algebraic modular forms for an inner form of G. We treat explicitly the case of unitary groups, and we give in this case an asymptotic formula for the number of eigensystems. The paper generalizes results of J-P. Serre for GL(2) and A. Ghitza for GSp(2g).