Computing the endomorphism ring of an ordinary abelian surface over a finite field

TL;DR

This paper introduces a subexponential algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field, extending previous methods from elliptic curves to higher dimensions.

Contribution

It generalizes an existing elliptic curve algorithm to abelian surfaces and reduces reliance on multiple heuristic assumptions, improving theoretical robustness.

Findings

Algorithm is subexponential in complexity.

Avoids multiple heuristic assumptions used in previous methods.

Outputs an ideal in the maximal totally real subfield of the endomorphism algebra.

Abstract

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.