Symmetric compatibility of theta structures, with applications to change of level and isogenies
Antoine Dequay, David Lubicz
原文摘要
Let $m,n,d >1$ be integers such that $n=md$ and let $(B, \mathscr{M}, Θ\_{\mathscr{M}})$ be a marked abelian variety of level $m$ over a field of odd characteristic. We study the theta structures of level $n$ for $\mathscr{M}^d$, and the theta structures on the abelian varieties $d$-isogenous to $B$, which are compatible with $Θ\_{\mathscr{M}}$. When $d$ is odd and coprime to $m$, they correspond to the isotropic subgroups of $B[n]$, and this correspondence underlies the known algorithms in theta coordinates. In general, this is no longer true. We introduce the notion of symmetric compatibility of a torsion point with a symmetric level subgroup, and we show that a subgroup of $B[n]$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^g$ comes from a compatible theta structure if and only if it is isotropic and all its points are symmetric compatible. We extend to every $d$ Mumford's notion of compatible pairs of theta structures, defined by Mumford for $d=2$, and prove that it is equivalent to our notion, which can be tested on theta coordinates; we also describe exactly the remaining ambiguity, which explains the sign choices met when the duplication formula is used to change the level. As applications, we obtain change of level and isogeny algorithms without assuming that $d$ is odd and coprime to $m$, using $O(n^{3g} \log(d))$ and $O(n^{2g} \log(d))$ operations in the base field respectively. In particular, they give change of level algorithms for even $d$, and $2$-isogenies in level $4$ theta coordinates with an explicit control of the theta structure of the codomain.
cs.SC · math.NT