Computing isogenies from modular equations in genus two

Abstract

Consider two genus 2 curves over a field whose Jacobians are linked by an isogeny of known type: either an ℓ-isogeny or, in the real multiplication case, an isogeny with cyclic kernel. We present a completely algebraic algorithm to compute this isogeny using modular equations of either Siegel or Hilbert type. An essential step of independent interest is to construct an explicit Kodaira–Spencer isomorphism for principally polarized abelian surfaces.