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

Volume 167 / 2015

Nazar Arakelian, Herivelto Borges 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.

Authors

  • Nazar ArakelianInstituto de Matemática, Estatística
    e Computaçõ Científica
    Universidade Estadual de Campinas
    Rua Sérgio Buarque de Holanda, 651
    CEP 13083-859, Campinas SP, Brazil
    e-mail
  • Herivelto BorgesInstituto de Ciências Matemáticas
    e de Computação
    Universidade de São Paulo
    Avenida Trabalhador São-carlense, 400
    CEP 13566-590, São Carlos SP, Brazil
    e-mail

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