  The authors define an "anti-holomorphic" involution (or "real structure") on
an ordinary Abelian variety (defined over a finite field k) to be an involution
of the associated Deligne module (T,F,V) that exchanges F (the Frobenius) with
V (the Verschiebung). The definition extends to include principal polarizations
and certain level structures. The authors show there are finitely many
isomorphism classes in each dimension, and they give a formula for this number
which resembles the Kottwitz "counting formula" (for the number of principally
polarized Abelian varieties over k), where the symplectic group (in the
Kottwitz formula) has been replaced by the general linear group.

Number Theory