On maximal curves related to Chebyshev polynomials
Let \(\mathcal{C}\) be a (projective, nonsingular, geometrically irreducible, algebraic) curve of genus \(g\) defined over the finite field \(\mathbb{F}_{q^{2}}\), where \(q\) is a power of a prime \(p\). \(\mathcal{C}\) is called \(\mathbb{F}_{q^{2}}\)-maximal if its number of \(\mathbb{F}_{q^{2}}\)-rational points \(\#\mathcal{C}(\mathbb{F}_{q^{2}})\) meets the Hasse-Weil upper bound, that is, \[ \#\mathcal{C}(\mathbb{F}_{q^{2}})=q^{2}+1+2gq. \] Maximal curves have interesting and remarkable applications for instance in coding theory, cryptography, and finite geometry; see [\textit{J. W. P. Hirschfeld} et al., Algebraic curves over a finite field. Princeton, NJ: Princeton University Press (2008; Zbl 1200.11042)] and [\textit{N. E. Hurt}, Many rational points. Coding theory and algebraic geometry. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1072.11042)]. In this article, continuing the work started in [\textit{A. Garcia} and \textit{H. Stichtenoth}, Acta Arith. 90, No. 4, 301--311 (1999; Zbl 0933.11032)], and considering suitable dominant \(\mathbb{F}_{q^{2}}\)-coverings of curves, the authors find further examples of Kummer \(\mathbb{F}_{q^{2}}\)-maximal curves described by \[ v^{N}=F(u), \] where \(F(u)\) is a polynomial (not always separable) related to a Chebyshev polynomial. In some cases, the genus of the corresponding curve is also computed.
scientific article; zbMATH DE number 6882138
scientific article