The "moduli space" for principally polarized complex n dimensional Abelian varieties with real structure (that is, anti-holomorphic involution) may be identified with a certain locally symmetric space for the group GL(n) over the real numbers.  Is it possible to make sense of the points of this space over a finite field?  This talk describes joint work with Yung-sheng Tai (Haverford College) in which we propose an approach to this question for ordinary Abelian varieties.