On certain maximal hyperelliptic curves related to Chebyshev polynomials

Abstract

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2={\phi }_d(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here ${\phi }_d(x)$ denotes the Chebyshev polynomial of degree $d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2){\phi }_d(x)$, and also for $y^2=(x^2-4){\phi }_d(x)$.