On the Dickson-Guralnick-Zieve curve

TL;DR

The paper investigates the properties of the Dickson-Guralnick-Zieve (DGZ) curve over finite fields, revealing its large automorphism group, relation to Fermat curves, and optimality in rational points among similar curves.

Contribution

This work characterizes the DGZ curve's automorphism group, its connection as a quotient of Fermat curves, and establishes its optimality regarding rational points over finite fields.

Findings

DGZ curve has a large automorphism group relative to its genus.

The Fermat curve of degree q-1 is a quotient of the DGZ curve.

DGZ curve is optimal in the number of rational points among similar curves.

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.