The moduli space for elliptic curves (or Abelian varieties) is a complex manifold which admits complex conjugation. The "real locus" turns out to be the moduli space for "real" elliptic curves (or "real" abelian varieties). Is it possible to make sense of these statements over a finite field?
Perhaps.

Over a finite field there is a large class of Abelian varieties that are called "ordinary". For this class, at least, there is a very interesting way to answer this question.

This talk represents joint work with Yung-sheng Tai, Haverford College. Speaker(s): Mark Goresky (Institute for Advanced Study, Princeton)