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Frobenius nonclassicality of Fermat curves with respect to cubics

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Abstract

For Fermat curves F: aX n + bY n = Z n defined over F q , we establish necessary and sufficient conditions for F to be F q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F q -Frobenius nonclassical cases, we determine explicit formulas for the number N q (F) of F q -rational points on F. For the remaining Fermat curves, nice upper bounds for N q (F) are immediately given by the Stöhr–Voloch Theory.

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Authors and Affiliations

  1. Centro de Matemàtica, Computaçao e Cogniçao, Universidade Federal do ABC, Avenida dos Estados, 5001, CEP 09210-580, S˜ao Carlos SP, Brazil

    Nazar Arakelian

  2. Instituto de Ciências Matemàticas e de Computaçao, Universidade de S˜ao Paulo, Avenida Trabalhador S˜ao-Carlense, 400, CEP 13566-590, S˜ao Carlos SP, Brazil

    Herivelto Borges

Authors
  1. Nazar Arakelian
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  2. Herivelto Borges
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Correspondence to Nazar Arakelian.

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Arakelian, N., Borges, H. Frobenius nonclassicality of Fermat curves with respect to cubics. Isr. J. Math. 218, 273–297 (2017). https://doi.org/10.1007/s11856-017-1465-3

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  • DOI: https://doi.org/10.1007/s11856-017-1465-3

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