zbMATH — the first resource for mathematics

Further examples of maximal curves. (English) Zbl 1166.11017
A smooth, projective and geometrically irreducible curve \(C\) of genus \(g\) over a finite field \(\mathbb F_{q^{2}}\) is called maximal over \(\mathbb F_{q^{2}}\) if \(\# C(\mathbb F_{q^{2}})\) attains the Hasse-Weil upper bound, i.e., \(\# C(\mathbb F_{q^{2}})=q^{2}+1+2gq\). For such curves, one necessarily has \(g\leq q(q-1)/2\). Further, there exists a unique maximal curve \(\mathcal H\) over \(\mathbb F_{q^{2}}\), with affine model \(y^{q}+y=x^{q+1}\) and called the Hermitian curve, such that \(g\) has the maximal value \(q(q-1)/2\).
This paper studies some classes of maximal curves and discusses whether or not the curves studied are Galois-covered by \(\mathcal H\). First, the authors study the nonsingular models of the curves \(C(q,n): y^{q^{2}}-y=x^{\frac{q^{n}+1}{q+1}}\) defined over \(\mathbb F_{q^{2n}}\), where \(n\geq 3\) is an odd integer. If \(\ell\) is a prime, the \(\mathbb F_{\ell^{6}}\)-curves \(C(\ell,3)\) were previously studied by A. García and H. Stichtenoth. They showed that these curves are maximal over \(\mathbb F_{\ell^{6}}\) of genus \((\ell^{2}-1)(\ell^{2}-\ell)/2\). An interesting observation is the following: the curve \(C(3,3)\) is not Galois-covered by the Hermitian curve \(\mathcal H\), in contrast to the case of \(C(2,3)\).
In this paper the curves \(C(q,n)\) are shown to be maximal for any \(q\). Further, it is shown that the curves \(C(2,n)\) are Galois-covered by \(\mathcal H\). The authors also discuss the family of maximal curves \(C_{b}\) defined over \(\mathbb F_{q^{2n}}\) by the affine equation \(y^{N}=-x^{b}(x+1), 1\leq b\leq N-1\), where \(N\) is an odd divisor of \(q^{n}+1\) and \((N,b)=(N,b+1)=1\).

11G20 Curves over finite and local fields
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
Full Text: DOI
[1] Abdón, M.; Garcia, A., On a characterization of certain maximal curves, Finite fields appl., 10, 133-158, (2004) · Zbl 1046.11045
[2] Fuhrmann, R.; Garcia, A.; Torres, F., On maximal curves, J. number theory, 67, 29-51, (1997) · Zbl 0914.11036
[3] Abdón, M.; Torres, F., On maximal curves in characteristic two, Manuscripta math., 99, 35-53, (1999) · Zbl 0931.11022
[4] Cossidente, A.; Korchmáros, G.; Torres, F., On curves covered by the Hermitian curve, J. algebra, 216, 56-76, (1999) · Zbl 1054.14026
[5] Korchmáros, G.; Torres, F., Embedding of a maximal curve in a Hermitian variety, Compositio math., 128, 95-113, (2001) · Zbl 1024.11044
[6] Goppa, V.D., ()
[7] Ihara, Y., Some remarks on the number of rational points of algebraic curves over finite fields, J. fac. sci. Tokyo, 28, 721-724, (1981) · Zbl 0509.14019
[8] Rück, H.G.; Stichtenoth, H., A characterization of Hermitian function fields over finite fields, J. reine angew. math., 457, 185-188, (1994) · Zbl 0802.11053
[9] Fuhrmann, R.; Torres, F., The genus of curves over finite fields with many rational points, Manuscripta math., 89, 103-106, (1996) · Zbl 0857.11032
[10] Cossidente, A.; Korchmáros, G.; Torres, F., On curves of large genus covered by the Hermitian curve, Comm. algebra, 28, 4707-4728, (2000) · Zbl 0974.11031
[11] Garcia, A.; Stichtenoth, H.; Xing, C.P., On subfields of the Hermitian function field, Compositio math., 120, 137-170, (2000) · Zbl 0990.11040
[12] Abdón, M.; Quoos, L., On the genera of subfields of the Hermitian function field, Finite fields appl., 10, 271-284, (2004) · Zbl 1052.11045
[13] Garcia, A.; Stichtenoth, H., A maximal curve which is not a Galois subcover of the Hermitian curve, Bull. braz. math. soc., 37, 139-152, (2006) · Zbl 1118.14033
[14] A. Garcia, F. Torres, On unramified coverings of maximal curves, in: Proceedings AGCT-10, Sémin. Congr. (in press) · Zbl 1270.11061
[15] M. Giulietti, G. Korchmáros, A new family of maximal curves over a finite field. arXiv:0711.0445v1 · Zbl 1160.14016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.