Computing low-degree isogenies in genus 2 with the 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.