Abelian varieties over finite fields as basic abelian varieties

TL;DR

This paper demonstrates that basic abelian varieties with additional structures over algebraically closed fields of characteristic p are isogenous to those over finite fields, and connects categories of abelian varieties over finite fields with basic abelian varieties to derive a new mass formula.

Contribution

It establishes a link between basic abelian varieties over algebraically closed fields and those over finite fields, introducing a new mass formula for polarized abelian surfaces.

Findings

Any basic abelian variety with additional structures over an algebraically closed field of characteristic p is isogenous to one over a finite field.

The category of abelian varieties over finite fields embeds into the category of basic abelian varieties with endomorphism structures.

A new mass formula for finite orbits of polarized abelian surfaces over finite fields is derived.

Abstract

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.