Computing change of level and isogenies between abelian varieties

Let $m,n,d>1$ be integers such that $n=md$. In this paper, we present an efficient change of level algorithm that takes as input $(B,\mathcal{M},{\mathrm{\Theta }}_{\mathcal{M}})$ a marked abelian variety of level $m$ over the base field $k$ of odd characteristic and returns $(B,{\mathcal{M}}^d,{\mathrm{\Theta }}_{{\mathcal{M}}^d})$ a marked abelian variety of level $n$ at the expense of $O(m^g{d}^{2g})$ operations in $k$. A similar algorithm allows to compute $d$-isogenies: from $(B,\mathcal{M},{\mathrm{\Theta }}_{\mathcal{M}})$ a marked abelian variety of level $m$, $K\subset B[d]$ isotropic for the Weil pairing isomorphic to $(\mathbb{Z}\mathrm{/}d\mathbb{Z}{)}^g$ defined over $k$, the isogeny algorithm returns $(A,\mathcal{L},{\mathrm{\Theta }}_{\mathcal{L}})$ of level $m$ such that $A=B\mathrm{/}K$ with $O(m^g{d}^g)$ operations in $k$. Our algorithms extend previous known results in the case that $d\wedge m=1$ and $d$ odd. In this paper, we lift theses restrictions. We use the same general approach as in the literature in conjunction with the notion of symmetric compatible that we introduce, study and link to previous results of Mumford. For practical computation, most of the time $m$ is $2$ or $4$ so that our algorithms allows in particular to compute $2^e$-isogenies which are important for the theory of theta functions but also for computational applications such as isogeny based cryptography.