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

Abstract

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface A A over  Q \mathbb {Q} with geometric endomorphism ring equal to  Z \mathbb {Z} , computes all the other p.p. abelian surfaces over  Q \mathbb {Q} that are isogenous to A 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 \numprint{1440894} isogeny classes of Jacobians of genus 2 curves.