A p-Adic Quasi-Quadratic Time Point Counting Algorithm

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 $\mathbb F_q$ of cardinality q with time complexity O(n^2+o(1)) and space complexity O(n²), 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^νp where ν>0 is an integer and $p={\mathrm char}(\mathbb F_q)> 2$. 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.