Frobenius nonclassicality of generalized Fermat curves with respect to conics

TL;DR

This paper characterizes when certain generalized Fermat curves over finite fields are Frobenius nonclassical with respect to conics, providing bounds and explicit formulas for their rational points.

Contribution

It offers necessary and sufficient conditions for Frobenius nonclassicality of generalized Fermat curves with respect to conics, advancing the application of St"ohr-Voloch theory.

Findings

Derived bounds for the number of rational points in Frobenius classical cases.

Obtained explicit formulas for the number of rational points in Frobenius nonclassical cases.

Characterized Frobenius nonclassicality conditions for generalized Fermat curves.

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 $N_q(\mathcal{F})$.