Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

TL;DR

This paper presents an efficient algorithm to compute all isogenous principally polarized abelian surfaces over the rationals for a given surface with trivial geometric endomorphism ring, using Galois representations and algebraic methods.

Contribution

It introduces a novel algorithm combining Galois representation techniques and algebraic methods to identify isogeny classes of abelian surfaces over ield.

Findings

Successfully applied to over 1.4 million isogeny classes

Demonstrates practical efficiency of the algorithm

Provides a comprehensive computational approach for abelian surface classification

Abstract

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1 440 894 isogeny classes of Jacobians of genus 2 curves.