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

Nazar Arakelian¹ ; Herivelto Borges²

¹ Instituto de Matemática, Estatística e Computaçõ Científica Universidade Estadual de Campinas Rua Sérgio Buarque de Holanda, 651 CEP 13083-859, Campinas SP, Brazil
² Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo Avenida Trabalhador São-carlense, 400 CEP 13566-590, São Carlos SP, Brazil

Acta Arithmetica, Tome 167 (2015) no. 1, pp. 43-66.

Résumé

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.

Détail

DOI : 10.4064/aa167-1-3

Keywords: each integer geq present family curves mathbb q frobenius nonclassical respect linear system plane curves degree necessary sufficient conditions curves mathbb q frobenius nonclassical respect linear system conics mathbb q frobenius nonclassical cases determine exact number mathbb q rational points remaining cases upper bound number mathbb q rational points follow voloch theory