Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

TL;DR

This paper provides a categorical framework and algorithms for classifying abelian varieties over finite fields with commutative endomorphism rings, enabling the computation of isomorphism classes and revealing new exotic examples.

Contribution

It introduces a categorical description linking abelian varieties to fractional ideals and develops algorithms for classifying these varieties over finite fields.

Findings

Categorical description of abelian varieties via fractional ideals.

Effective algorithms for computing isomorphism classes.

Discovery of new examples with exotic patterns.

Abstract

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a fractional $W\otimes_{\mathbb Z_p} \mathbb Z_p[\pi,q/\pi]$-ideal, with $\pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified extension of $\mathbb Q_p$ of degree $a$. The latter ideal should be compatible at $p$ with the former and stable under the action of a semilinear Frobenius (and Verschiebung) operator; it will be the Dieudonn\'e module of the corresponding abelian variety. Using this categorical description we create effective algorithms to compute isomorphism classes of these objects and we produce many new examples exhibiting exotic patterns.