On certain maximal curves related to Chebyshev polynomials

This paper studies curves defined using Chebyshev polynomials phi d ( x ) over finite fields. Given the hyperelliptic curve C corresponding to the equation v 2 = phi d ( u ), the prime powers q equivalent to 3 mod 4 are determined such that phi d ( x ) 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 ( u ) is smooth and maximal over F q 2 . A number of analogous maximality results are discussed. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).