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

We show that, under suitable assumptions, the Hecke eigensystems arising from (mod p) modular forms of PEL type associated to an algebraic group $G/\mathbb Q$ of type A or C, coincide with the Hecke eigensystems arising from (mod p) algebraic modular forms associated to an inner form I of G. The form I has to be compact modulo center at infinity and satisfies: $I_{\mathbb Q_v}=G_{\mathbb Q_v}$ for $v\ne p,\ \infty$. In order to realize the Hecke correspondence, we first adapt a result of Rapoport and Zink to obtain an explicit uniformization of the superspecial locus of PEL Shimura varieties over $\bar {\mathbb F}_p$ (such a uniformization in the general context of PEL Shimura varieties was first described by C.-F. Yu). We then compare (mod p) modular forms with superspecial modular forms by exploiting the Hermitian $\mathbb F_{p^2}$-structure of the cotangent spaces of the p-divisible groups of the superspecial points. We study in detail the case in which G is of type A, and we prove that the number $\mathcal N$ of (mod p) Hecke eigensystems arising from unitary modular forms for a fixed quadratic imaginary field, having signature (r,s), prime-to-p level N≥3 and variable weight satisfies the estimate $\mathcal N\in O(p^{g^2+g+1-rs})$ for p large, where g=r+s. The paper generalizes results of J-P. Serre for GL(2) and Ghitza for GSp(2g) (g>1).