Curves with many points and configurations of hyperplanes over finite fields
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 applications (Ankara 1997), Lect. Notes Pure Appl. Math. 204, 227-259 (1999; Zbl 0930.11042)]. Fibre products of certain Artin-Schreier curves have already been considered by \textit{G. van der Geer} and \textit{M. van der Vlugt} [J. Comb. Theory, Ser. A 70, 337-348 (1995; Zbl 0824.94025)] who also found by this method curves with many rational points.
scientific article; zbMATH DE number 1369833
scientific article