Computing (l, l)-isogenies in polynomial time on Jacobians of genus 2 curves

In this paper, we compute `-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in `, when ` is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4. Combined with the preceding algorithm, this gives a method to compute (`, `)-isogenies in polynomial time on Jacobians of genus 2 curves.