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) = xℓ(xm+1).
Communicated by: G. Korchmáros
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.
References
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