Frobenius nonclassicality of Fermat curves with respect to cubics

TL;DR

This paper characterizes when Fermat curves over finite fields are Frobenius nonclassical with respect to cubics, providing explicit formulas for rational points in these cases and bounds otherwise.

Contribution

It establishes necessary and sufficient conditions for Frobenius nonclassicality of Fermat curves with respect to cubics and derives explicit formulas for their rational points.

Findings

Necessary and sufficient conditions for Frobenius nonclassicality.

Explicit formulas for the number of rational points.

Upper bounds for rational points in non-nonclassical cases.

Abstract

For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, we establish necessary and sufficient conditions for $\mathcal{F}$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane cubics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine explicit formulas for the number $N_q(\mathcal{F})$ of $\mathbb{F}_q$-rational points on $\mathcal{F}$. For the remaining Fermat curves, nice upper bounds for $N_q(\mathcal{F})$ are immediately given by the St\"ohr-Voloch Theory.