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arXiv · 1306.5102

Frobenius lifts and point counting for smooth curves

Amnon Besser·François-Renaud Escriva·Rob de Jeu
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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.

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Amnon Besser, François-Renaud Escriva, Rob de Jeu. 2013-06-21. Frobenius lifts and point counting for smooth curves. https://arxiv.org/abs/1306.5102

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