On certain maximal curves related to Chebyshev polynomials

This paper studies curves defined using Chebyshev polynomials ${\phi }_d\left(x\right)$ over finite fields. Given the hyperelliptic curve $C$ corresponding to the equation $v^2={\phi }_d\left(u\right)$, the prime powers $q\equiv 3\mathrm{mod}4$ are determined such that ${\phi }_d\left(x\right)$ is separable and $C$ is maximal over $F_{q^2}$. This extends a result from [30] that treats the special cases $2|d$ as well as d a prime number. In particular a proof of [30, Conjecture 1.7] is presented. Moreover, we give a complete description of the pairs $(d,q)$ such that the projective closure of the plane curve defined by $v^d={\phi }_d\left(u\right)$ is smooth and maximal over $F_{q^2}$.

A number of analogous maximality results are discussed.