Computing endomorphism rings of abelian varieties of dimension two

TL;DR

This paper introduces a subexponential algorithm for computing endomorphism rings of ordinary abelian surfaces over finite fields, extending previous methods for elliptic curves and demonstrating practical efficiency.

Contribution

It generalizes a method for elliptic curves to abelian surfaces, providing a new algorithm with promising practical performance despite theoretical assumptions.

Findings

Algorithm performs well in practice

Handles previously intractable cases

Based on assumptions for correctness and complexity

Abstract

Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.