Computing gluing and splitting \((\ell ,\ell )\)-isogenies

We give algorithms to compute \((\ell ,\ell )\) $(\ell ,\ell )$ -isogenies between Jacobians of genus 2 curves and products of elliptic curves in time of \({\tilde{O}}(\ell ^2)\) $\tilde{O}({\ell }^2)$ basic field operations, where \(\ell \) $\ell$ is an odd prime different from the characteristic of the field. The method relies on the notion of a normal Weil set of Weil functions due to Shepherd-Barron, and works for the case of computing \((\ell ,\ell )\) $(\ell ,\ell )$ -isogenies between Jacobians of genus 2 curves.