On the Dickson-Guralnick-Zieve curve

Abstract

The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field $\mathbb{F}_q$ arises naturally from the classical Dickson invariant of the projective linear group $PGL(3,\mathbb{F}_q)$. The DGZ curve is an (absolutely irreducible, singular) plane curve of degree $q^3-q^2$ and genus $\frac{1}{2}q(q-1)(q^3-2q-2)+1.$ In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse-Witt invariant is positive; the Fermat curve of degree $q-1$ is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over $\mathbb{F}_{q^3}$, the DGZ curve is optimal with respect the number of its $\mathbb{F}_{q^3}$-rational points.