On counting certain abelian varieties over finite fields

TL;DR

This paper provides explicit formulas for counting certain abelian varieties over finite fields, extending previous work on superspecial cases to supersingular cases and introducing new classifications for abelian varieties with additional structures.

Contribution

It offers an explicit formula for the size of isogeny classes of simple abelian surfaces with real Weil number, and generalizes the notion of genera and ideal complexes to abelian varieties with additional structures.

Findings

Explicit formula for isogeny class size of simple abelian surfaces

Extension of explicit calculations from superspecial to supersingular abelian surfaces

Introduction of genera and ideal complexes for abelian varieties with additional structures

Abstract

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation.