Computing isogenies from modular equations in genus 2

Given two l-isogenous elliptic curves, a well-known algorithm of Elkies uses modular polynomials to compute this isogeny explicitly. In this work, we generalize his ideas to Jacobians of genus 2 curves. Our algorithms works for both l-isogenies and (in the RM case) cyclic isogenies, and uses Siegel or Hilbert type modular equations respectively. This has applications for point counting in genus 2: SEA-style methods are now available.