On Counting Certain Abelian Varieties Over Finite Fields

Abstract

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [32,31,33,34], 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 √ 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 idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of [40] 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.

On the cyclicity of the rational points group of abelian varieties over finite fields

“…Another interesting related topic is counting isomorphism classes of abelian varieties defined over a fixed finite field. In the 1-dimensional case, this corresponds to compute class numbers of some orders in quadratic number fields, and in the general case, this was treated for example (very recently) in [23]. Isomorphism classes can be grouped in isogeny classes.…”