×

The number of points on an algebraic curve over a finite field. (English) Zbl 1144.14021

Hilton, Anthony (ed.) et al., Surveys in combinatorics 2007. Papers from the 21st British combinatorial conference, Reading, UK, July 8–13, 2007. Cambridge: Cambridge University Press (ISBN 978-0-521-69823-8/pbk). London Mathematical Society Lecture Note Series 346, 175-200 (2007).
The authors survey existing results on curves with many rational points over a finite field. The relevance of such curves comes from the fact that they are useful for the construction of error-correcting codes. In addition, a section about the number of rational points on singular curves has been included.
If \(C\) is a non-singular projective curve of genus \(g\) over a finite field \(\mathbb F_q\) with \(q\) elements, then one says that \(C\) is maximal if \(\# C(\mathbb F_q)=q+1+2g \sqrt{q}\). In fact, the latter value is a uniform upper bound, called the Weil bound, for the number of \(\mathbb F_q\)-rational points on any curve of genus \(g\). Obviously, for a maximal curve to exist over \(\mathbb F_q\) the cardinality \(q\) has to be a square. Serre has sharpened the Weil bound to \(q+1+g \lfloor 2 \sqrt{q} \rfloor\). One is interested in the question whether a maximal curve of a given genus \(g\) over a given finite field \(\mathbb F_q\) exists. If this is the case, then one would like to have an explicit model of such a maximal curve. Otherwise, one may ask for an optimal curve of genus \(g\) over \(\mathbb F_q\), which is supposed to have a maximal number of \(\mathbb F_q\)-rational points where the maximum is taken over all non-singular projective curves over \(\mathbb F_q\) of fixed genus \(g\). If \(N(g,q)\) denotes the number of \(\mathbb F_q\)-rational points of an optimal curve of genus \(g\) over \(\mathbb F_q\) then Drinfeld and Vladut proved that \(\text{limsup}_{g \rightarrow \infty} \frac{N(g,q)}{g} \leq \sqrt{q}-1\). Note, that the existence of the Weil bound trivially implies that the above limit is smaller or equal \(2\sqrt{q}\).
In the first part of the article the authors recall some well known results about maximal curves. For example, Stichtenoth and Rück proved that a maximal curve of genus \(g=\frac{1}{2} q(q-1)\) over \(\mathbb F_{q^2}\) exists and is necessarily isomorphic to the Hermitian curve \(y^{q}+y=x^{q+1}\). We remark, that in genus \(g>\frac{1}{2} q(q-1)\) there are no maximal curves over \(\mathbb F_{q^2}\). It was pointed out by Serre that subfields of a function field of a maximal curve belong to maximal curves. As a consequence, Hermitian curves give rise to many other examples of maximal curves. Some of them are given in the text.
Among others, a classical result of Stöhr and Voloch and its consequences are discussed. The Theorem of Stöhr and Voloch reads as follows. Suppose that one is given an irreducible, non-singular and projective curve of genus \(g\) over a finite field \(\mathbb F_q\) with \(q\) elements. If there exists on this curve a base-point-free linear system of degree \(d\), dimension \(n\) and Frobenius order-sequence \(\nu_0, \ldots, \nu_{n-1}\), then the number of \(\mathbb F_q\)-rational points on the curve is bounded by \(\big( (\nu_1+\ldots+\nu_{n-1})(2g-2)+(q+n)d \big)/n\). The authors give several corollaries of the theorem of Stöhr and Voloch.
A remarkable point about this article is that it contains a lot of examples. There are listings and tables which feature data related to optimal curves, maximal curves of large genus and families of curves containing maximal curves. The reader has to be aware of the fact that the tables are not complete. At some point in this article it is claimed in the text that a complete classification of maximal curves was out of reach.
Complementing the results about non-singular curves, the authors provide bounds for the number of rational points on possibly singular curves over \(\mathbb F_q\). These results are stated without proof. The reader can imagine that one may obtain these bounds by blowing up the finitely many singularities of a given curve and taking into account the type of the singularities. The results of this section are illustrated by some singular examples.
For further reading we recommend the books [J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic curves over a finite field. Princeton Series in Applied Mathematics. (Princeton), NJ: Princeton University Press. (2008; Zbl 1200.11042); W. Lütkebohmert, Codierungstheorie. (Braunschweig): Vieweg. (2003; Zbl 1043.94001); H. Stichtenoth, Algebraic function fields and codes. Universitext. (Berlin): Springer-Verlag. (1993; Zbl 0816.14011)].
For the entire collection see [Zbl 1117.05001].
Reviewer: Robert Carls (Ulm)

MSC:

14G15 Finite ground fields in algebraic geometry
14G50 Applications to coding theory and cryptography of arithmetic geometry
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14H25 Arithmetic ground fields for curves
PDFBibTeX XMLCite