Frobenius lifts and point counting for smooth curves
Algebraic Geometry
2013-06-24 v1 Number Theory
Abstract
We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.
Cite
@article{arxiv.1306.5102,
title = {Frobenius lifts and point counting for smooth curves},
author = {Amnon Besser and François-Renaud Escriva and Rob de Jeu},
journal= {arXiv preprint arXiv:1306.5102},
year = {2013}
}
Comments
29 pages, 2 figures