Curves with many points and configurations of hyperplanes over finite fields
type: Item
instance of: scholarly article
title: Curves with many points and configurations of hyperplanes over finite fields
zbMATH Open document ID: 0953.11021
author: Ferruh Özbudak
author: Henning Stichtenoth
published in: Finite Fields and their Applications
publication date: 2000-02-10T00:00:00Z
full work available at URL: https://hdl.handle.net/11511/40935
review text: This paper presents a method to construct curves over finite fields whose number of rational points is close to the bound obtained by Osterlé via optimization of Serre-Weil explicit formulae [see \textit{J.-P. Serre}, Rational points on curves over finite fields, Notes by F. Gouvea of lectures at Harvard University (1985)]. Let \(q\) be a prime power, \(m\), \(r\) integers such that \(m,r\geq 2\), and \(\gcd(q,m)=1\). Let \(w_1,\ldots,w_r\) be a \(\mathbb F_q\)-base of \(\mathbb F_{q^r}\). For \(\alpha=\sum_{i=1}^r\alpha_iw_i\in\mathbb F_{q^r}\) set \(\Psi(\alpha):=(\alpha_1,\ldots,\alpha_r)\). Then via \(\Psi\) the set \(S_{a,b}:=\{\alpha\in\mathbb F_{q^r}: \text{Tr}(a\alpha)+b=0\}\), \(a\in \mathbb F_{q^r}^*\), \(b\in \mathbb F_q\), can be identified with the hyperplane in \(\mathbb F_q^r\), namely \(H_{a,b}: \sum_{i=1}^r\text{Tr}(aw_i)X_i+b=0\). A sequence of hyperplanes in \(\mathbb F_q^r\), \(H_1,\ldots,H_s\), is called admissible whenever \(H_{i+1}\not\subseteq \cup_{j=1}^i H_j\) for \(i=1,\ldots,s-1\). Now let \(\mathcal X\) be the fibre product of the following \(s\) Kummer curves: \(y_i^m=\text{Tr}(a_ix)+b_i\), where \(m\) is a divisor of \((q^r-1)/(q-1)\), and \(a_i\in \mathbb F_{q^r}^*\), \(b_i\in\mathbb F_q\) for each \(i\). The main result of the paper is that \(\mathcal X\) is defined over \(\mathbb F_{q^r}\) whenever \(H_{a_1,b_1},\ldots,H_{a_s,b_s}\) is admissible. In this case one can compute the number of \(\mathbb F_{q^r}\)-rational points of \(\mathcal X\) as well as its genus. Then by choosing sequences of hyperplanes suitably, one improves in many cases the tables constructed by \textit{G. van der Geer} and \textit{M. van der Vlugt} [Math. Comput. 69, 797-810 (2000)]. The authors' approach subsumes previous work of \textit{S. A. Stepanov} [Finite fields and applications (Glasgow 1995), Lond. Math. Soc. Lect. Note Ser. 233, 355-378 (1996; Zbl 0868.11057), Discrete Math. Appl. 7, No. 1, 77-88 (1997; Zbl 0904.94026) and Number theory and its appli…
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 1369833
zbMATH Keywords: Hasse-Weil-Serre bound
zbMATH Keywords: curves with many rational points
zbMATH Keywords: fibre product of Kummer curves
zbMATH Keywords: arrangements of hyperplanes
MaRDI profile type: Publication
OpenAlex ID: W1974658245
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DOI: 10.1006/FFTA.1999.0262
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description: scientific article; zbMATH DE number 1369833