Fast change of level and applications to isogenies

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