On the Dickson–Guralnick–Zieve curve.

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 P G L (3 , F q). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q (q − 1) (q 3 − 2 q − 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. [ABSTRACT FROM AUTHOR]