On certain maximal hyperelliptic curves related to Chebyshev polynomials

Abstract

In this talk, we study some families of maximal curves arisingfrom Chebyshev polynomials.  Our main interest in this talk is to characterizethe pairs (q, d) such that the hyperelliptic curve  over a finite fieldq2correspond-ing to the equationy2=φd(u) is maximal over the finite fieldq2of cardinalityq2.  Hereφd(u) denotes the Chebyshev polynomial of degreed.  We consider thesame question for the curves corresponding toy2= (x±2)φd(x), and also fory2= (x2−4)φd(x).  This is a joint work with Jaap Top