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On the curve Yn = X(Xm + 1) over finite fields

Saeed Tafazolian and Fernando Torres
From the journal Advances in Geometry

Abstract

Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = x(xm+1).

MSC 2010: 11G20; 11M38; 14G15; 14H25

  1. Communicated by: G. Korchmáros

  2. Funding: The authors were in part supported respectively by IPM grant No. 93140117, and by CNPq-Brazil grant 308326/2014-8.

Acknowledgements

We thank the referee for pointing out Remark 2.6 and for valuable comments and suggestions which led to improve the exposition of the paper.

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Received: 2017-01-18
Revised: 2017-06-02
Published Online: 2018-01-07
Published in Print: 2019-04-24

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