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We show that, under suitable assumptions, the Hecke eigensystems arising from ( mod   p ) modular forms of PEL type associated to an algebraic groupof 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:for. 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(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-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 numberof ( 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 estimatefor  p  large, where  g = r + s . The paper generalizes results of J-P. Serre for  GL (2) and Ghitza for  GSp (2 g ) ( g &gt;1).

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