On Chebyshev polynomials and maximal curves

Abstract

In [G-S-X] we have determined the genera of several subfields of the Hermitian function field H (it is well known that they also correspond to maximal curves). In order to have an explicit description of codes arising from curves one frequently needs that the curves (or their associated function fields) are explicitly given by equations. The subfields of H we are interested in here are among those appearing in [G-S-X, Theorem 5.4]. It will turn out that they can be described by using Chebyshev polynomials (see Remarks 4.4 and 5.2). Chebyshev polynomials are special cases of Dickson polynomials which have been intensively studied in connection with the theory of permutation polynomials over a finite field Fq, the main result being that a Dickson