On the Dickson–Guralnick–Zieve curve

Abstract

The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field Fq arises naturally from the classical Dickson invariant of the projective linear group P GL(3, Fq). 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 − 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.

MRD codes with maximum idealizers

“…Very recently, Giulietti, Korchmáros and Timpanella [21] also investigated this curve and they called it the Dickson-Guralnick-Zieve curve after the work [23] by Guralnick and Zieve, see also [4]. They can show that this curve is absolutely irreducible [21,Proposition 4.7] and the genus of V is g q = 1 2 q(q − 1)(q 3 − 2q − 2) + 1 [21,Theorem 4.10]. Moreover, by Lemmas 4.5 and 4.6 in [21], each singular point of V has a unique branch centered on it, which means the number R q n of the F q nplaces of the associated function field of V equals the number of F q n -rational points of V (for further details see [24,Chapter 4]).…”

“…They can show that this curve is absolutely irreducible [21,Proposition 4.7] and the genus of V is g q = 1 2 q(q − 1)(q 3 − 2q − 2) + 1 [21,Theorem 4.10]. Moreover, by Lemmas 4.5 and 4.6 in [21], each singular point of V has a unique branch centered on it, which means the number R q n of the F q nplaces of the associated function field of V equals the number of F q n -rational points of V (for further details see [24,Chapter 4]). By the Hasse-Weil Theorem, one gets…”

“…Hence there are q 2 − q points at infinity. Very recently, Giulietti, Korchmáros and Timpanella [21] also investigated this curve and they called it the Dickson-Guralnick-Zieve curve after the work [23] by Guralnick and Zieve, see also [4]. They can show that this curve is absolutely irreducible [21,Proposition 4.7] and the genus of V is g q = 1 2 q(q − 1)(q 3 − 2q − 2) + 1 [21,Theorem 4.10].…”

Galois points for double-Frobenius nonclassical curves

“…This is impossible. 6. The case where n − m = 2 Assume that n − m = 2, (m, q) = (1, 2) and P ∈ (P 2 \ F ) ∪ Sing(F ) is a Galois point.…”

“…In conclusion, it follows that if ∆(F ) = ∅, then m = 1 and ∆(F ) ⊂ P 2 (F q ) ⊂ P 2 \ F . (see [1, p.542] or [6,Remark 1]). In this case, it is known that the claim follows ([2,…”