On certain maximal hyperelliptic curves related to Chebyshev polynomials

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 $\text{cC}$ over a finite field ${\text{FF}}_{q^2}$ corresponding to the equation $y^2 = \varphi_{d}(x)$ismaximaloverthefinitefield$\FF_{q^2}$ofcardinality$q^2$\mathrm{.}Here$\varphi_{d}(x)$denotestheChebyshevpolynomialofdegree$d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2) \varphi_{d}(x)$,andalsofor$y^2=(x^2-4)\varphi_d(x)$.