Frobenius lifts and point counting for smooth curves
Abstract
We describe an algorithm for computing the zeta function of a proper, smooth curve over a finite field k of characteristic p, when the curve is given together with some auxiliary data, including a lift C to the valuation ring in a finite extension of Qp. The algorithm is denominator-free if the ramification is at most p. Our method computes the matrix of the action of a semilinear Frobenius on the first de Rham cohomology group of the curve by means of Poincaré duality, using cup products that can be computed from local expansions of a globally defined lift of Frobenius. Its complexity is softly cubic in the field degree for (general) smooth, planar curve, for which we work out our general estimates in more detail. We make explicit how to compute a suitable basis of the first de Rham cohomology group of C, base on 1-forms with `locally integrable polar parts', in both the general case and when the curve is smooth planar. We show the crystalline Frobenius preserves the first de Rham cohomology group of C if the ramification is at most p, improving upon known results. In an appendix we prove a well-known formula for the cup product, and a formula by Serre, on the first de Rham cohomology group for a curve in characteristic zero, for which no reference seems to exist.
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BibTeX
@misc{besser2026frobenius,
title = {{Frobenius lifts and point counting for smooth curves}},
author = {Amnon Besser and Rob de Jeu and Pengju Guan and Muxi Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30236},
eprint = {2609.30236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30236},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Frobenius lifts and point counting for smooth curves
AU - Amnon Besser
AU - Rob de Jeu
AU - Pengju Guan
AU - Muxi Li
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30236
UR - https://arxiv.org/abs/2609.30236
AB - We describe an algorithm for computing the zeta function of a proper, smooth curve over a finite field k of characteristic p, when the curve is given together with some auxiliary data, including a lift C to the valuation ring in a finite extension of Q_p. The algorithm is denominator-free if the ramification is at most p. Our method computes the matrix of the action of a semilinear Frobenius on the first de Rham cohomology group of the curve by means of Poincaré duality, using cup products that can be computed from local expansions of a globally defined lift of Frobenius. Its complexity is softly cubic in the field degree for (general) smooth, planar curve, for which we work out our general estimates in more detail. We make explicit how to compute a suitable basis of the first de Rham cohomology group of C, base on 1-forms with `locally integrable polar parts', in both the general case and when the curve is smooth planar. We show the crystalline Frobenius preserves the first de Rham cohomology group of C if the ramification is at most p, improving upon known results. In an appendix we prove a well-known formula for the cup product, and a formula by Serre, on the first de Rham cohomology group for a curve in characteristic zero, for which no reference seems to exist.
ER -