Frobenius nonclassicality of generalized Fermat curves with respect to conics

Abstract

The effective application of the St\"ohr-Voloch theory for the linear system of plane curves of a fixed degree to bound the number of rational points of a family of plane curves defined over $\mathbb{F}_q$ requires the characterization of the $\mathbb{F}_q$-Frobenius nonclassical curves in the family. In this paper, we provide necessary and sufficient conditions for certain generalized Fermat curves $\mathcal{F}$ defined over $\mathbb{F}_q$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the Frobenius classical cases, we obtain nice bounds for the number $N_q(\mathcal{F})$ of rational points of $\mathcal{F}$ via St\"ohr-Voloch theory, whereas in the Frobenius nonclassical cases, we derive explicit formulas for $N_q(\mathcal{F})$.