The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field F-q arises naturally from the classical Dickson invariant of the projective linear group PGL(3, F-q). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q^3 - q^2 and genus 1/2q(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 F-q^3, the DGZ curve is optimal with respect the number of its F-q^3-rational points
On the Dickson–Guralnick–Zieve curve
Giulietti, Massimo;Timpanella, Marco
2019
Abstract
The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field F-q arises naturally from the classical Dickson invariant of the projective linear group PGL(3, F-q). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q^3 - q^2 and genus 1/2q(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 F-q^3, the DGZ curve is optimal with respect the number of its F-q^3-rational pointsI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
