Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method

TL;DR

This paper refines and simplifies the Dolgachev-Lehavi method for computing low-degree isogenies of genus 2 Jacobians, providing an efficient algorithm especially for the case when ell=3, useful in number theory.

Contribution

The paper presents a simplified, explicit, and efficient algorithm for constructing the curve X in genus 2 isogenies, improving upon the original Dolgachev-Lehavi approach.

Findings

Efficient algorithm for ell=3 case

Explicit construction of curve X

Simplified approach to Dolgachev-Lehavi method

Abstract

Let ell be a prime, and H a curve of genus 2 over a field k of characteristic not 2 or ell. If S is a maximal Weil-isotropic subgroup of Jac(H)[ell], then Jac(H)/S is isomorphic to the Jacobian of some (possibly reducible) curve X. We investigate the Dolgachev--Lehavi method for constructing the curve X, simplifying their approach and making it more explicit. The result, at least for ell=3, is an efficient and easily programmable algorithm suitable for number-theoretic calculations.