Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Abstract

Abstract. For each integer s ≥ 1, we present a family of curves that are F q -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s = 2, we give necessary and sufficient conditions for such curves to be F q -Frobenius nonclassical with respect to the linear system of conics. In the F q -Frobenius nonclassical cases, we determine the exact number of F q -rational points. In the remaining cases, an upper bound for the number of F q -rational points will follow from Stöhr-Voloch theory.

On the Dickson–Guralnick–Zieve curve

“…Frobenius non-classical curves are somewhat rare; see [3,21]. In some cases, they have many points over F q ; see [1,4,10,21]. Also, they are closely related to univariate polynomials with minimal values sets, see [2].…”

“…In Section 5, we look inside the action and the ramification groups of a Sylow p-subgroup of G. The results collected are used in Section 6 to find the genus of C which is g = 1 2 q(q − 1)(q 3 − 2q − 2) + 1, and show that the quotient curve of C arising from a Sylow p-subgroup of G is isomorphic to the Fermat curve F q−1 of equation x q−1 + y q−1 + z q−1 = 0.…”