Abelian surfaces in Hesse form and explicit isogeny formulas

TL;DR

This paper introduces a novel method for computing (3,3)-isogenies between abelian surfaces using models in projective space and theta structures, simplifying the formulas and exploring generalizations.

Contribution

It presents a new approach leveraging theta structures in projective space to derive explicit isogeny formulas for abelian surfaces, with potential extensions to higher dimensions.

Findings

Explicit formulas for (3,3)-isogenies using theta models

Connection with the Burkhardt quartic threefold

Potential for generalizations to higher dimensions

Abstract

We develop a new method for the computation of $(3,3)$-isogenies between principally polarized abelian surfaces. The idea is to work with models in $\mathbb{P}^8$ induced by a symmetric level-$3$ theta structure. In this setting, the action of three-torsion points is linear, and the isogeny formulas can be described in a simple way as the composition of easy-to-evaluate maps. In the description of these formulas, the relation with the Burkhardt quartic threefold plays an important role. Furthermore, we discuss generalizations of the idea to higher dimensions as well as different isogeny degrees.