Computing $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves

Abstract

© 2014 American Mathematical Society. In this paper, we compute ℓ-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in ℓ, when ℓ is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In the 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 (ℓ, ℓ)-isogenies in polynomial time on Jacobians of genus 2 curves.