Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1704.04661

A bootstrap for the number of $\mathbb{F}_{q^r}$-rational points on a curve over $\mathbb{F}_q$

Santiago Molina·Narcís Sayols·Sebastià Xambó-Descamps
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note we present a fast algorithm that finds for any $r$ the number $N_r$ of $\mathbb{F}_{q^r}$ rational points on a smooth absolutely irreducible curve $C$ defined over $\mathbb{F}_{q}$ assuming that we know $N_1,\cdots,N_g$, where $g$ is the genus of $C$. The proof of its validity is given in detail and its working are illustrated with several examples. In an Appendix we list the Python function in which we have implemented the algorithm together with other routines used in the examples.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Santiago Molina, Narcís Sayols, Sebastià Xambó-Descamps. 2017-04-15. A bootstrap for the number of $\mathbb{F}_{q^r}$-rational points on a curve over $\mathbb{F}_q$. https://arxiv.org/abs/1704.04661

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗