A p-adic quasi-quadratic point counting algorithm

TL;DR

This paper introduces a quasi-quadratic time algorithm for counting rational points on the Jacobian of generic ordinary hyperelliptic curves over finite fields, generalizing existing methods and using theta constants.

Contribution

It presents a novel algorithm that extends AGM and canonical lifting methods for point counting on hyperelliptic Jacobians, with improved complexity.

Findings

Algorithm runs in O(n^{2+o(1)}) time and O(n^2) space.

Generalizes AGM and canonical lifting methods.

Supports the existence of a quasi-quadratic time algorithm for generic ordinary abelian varieties.

Abstract

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.