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

Saeed Tafazolian
  • Corresponding author
  • School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Teheran, Iran
  • Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran
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/ Fernando Torres
  • IMECC/UNICAMP, R. Sérgio Buarque de Holanda, 651, Cidade Universitária “Zeferino Vaz”, 13083-859, Campinas, SP, Brazil
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Published Online: 2018-01-07 | DOI: https://doi.org/10.1515/advgeom-2017-0041

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).

Keywords: Finite field; maximal curve; Weierstrass semigroup; Kummer extension

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

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About the article


Received: 2017-01-18

Revised: 2017-06-02

Published Online: 2018-01-07


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


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0041.

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