  In this article we give an algorithm for the computation of the number of
rational points on the Jacobian variety of a generic ordinary hyperelliptic
curve defined over a finite field of cardinality $q$ with time complexity
$O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter
complexity estimate the genus and the characteristic are assumed as fixed. Our
algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and
the canonical lifting method of T. Satoh. We canonically lift a certain
arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of
theta constants. The theta null values are computed with respect to a
semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer
and $p=\mathrm{char}(\F_q)>2$. The results of this paper suggest a global
positive answer to the question whether there exists a quasi-quadratic time
algorithm for the computation of the number of rational points on a generic
ordinary abelian variety defined over a finite field.
