Real structures on ordinary Abelian varieties

TL;DR

This paper introduces a new notion of real structures on ordinary Abelian varieties over finite fields, extending the concept to include polarizations and level structures, and provides a counting formula for their isomorphism classes.

Contribution

It defines anti-holomorphic involutions as real structures on Abelian varieties and derives a finite classification with a novel counting formula involving the general linear group.

Findings

Finite isomorphism classes in each dimension

Derived a counting formula similar to Kottwitz's

Extended the concept to include polarizations and level structures

Abstract

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the number of principally polarized Abelian varieties over k), where the symplectic group (in the Kottwitz formula) has been replaced by the general linear group.