Fast change of level and applications to isogenies

Let $$(A, {{\mathscr {L}}}, \Theta _n)$$ be a dimension g abelian variety together with level n theta structure over field k of odd characteristic. We thus denote by $$(\theta ^{\Theta _{{\mathscr {L}}}}_{i})_{(\mathbb {Z}/ \mathbb {Z})^g} \in \Gamma (A, {L}}})$$ the associated standard basis. For positive integer $$\ell $$ relatively prime to and characteristic k, we study change algorithms which allow one compute n$$ functions _i^{\Theta _{{{\mathscr {L}}^\ell }}}(x))_{i (\mathbb \ell {Z})^g}$$ from knowledge {L}}}}_i(x))_{(\mathbb {Z}/n or vice versa. The classical duplication formulas are an example algorithm go 2n. main result this paper states that there exists in $$O(n^g ^{2g})$$ operations k. derive isogeny $$f : A \rightarrow B$$ $$K \subset A[\ell ]$$ isotropic for Weil pairing computes f(x) $$x A(k)$$ $$O((n\ell )^g)$$ remark computation is quasi-linear complexity size K.