Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree
Volume 167 / 2015
Acta Arithmetica 167 (2015), 43-66
MSC: Primary 11G20, 14H05; Secondary 14Hxx.
DOI: 10.4064/aa167-1-3
Abstract
For each integer $s \geq 1$, we present a family of curves that are $\mathbb {F}_q$-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 $\mathbb {F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the $\mathbb {F}_q$-Frobenius nonclassical cases, we determine the exact number of $\mathbb {F}_q$-rational points. In the remaining cases, an upper bound for the number of $\mathbb {F}_q$-rational points will follow from Stöhr–Voloch theory.
