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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 19, Issue 2


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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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Fernando Torres
  • IMECC/UNICAMP, R. Sérgio Buarque de Holanda, 651, Cidade Universitária “Zeferino Vaz”, 13083-859, Campinas, SP, Brazil
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  • Other articles by this author:
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Published Online: 2018-01-07 | DOI: https://doi.org/10.1515/advgeom-2017-0041


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

Published in Print: 2019-04-24

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.

Citation Information: Advances in Geometry, Volume 19, Issue 2, Pages 263–268, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0041.

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