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Advances in Geometry

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Weierstrass points on Kummer extensions

Miriam Abdón
  • Corresponding author
  • IME, Univ. Federal Fluminense, Rua M√°rio Santos Braga s/n, CEP 24020-140, Niter√≥i, Brazil
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/ Herivelto Borges
  • Instituto de Ci√™ncias Matem√°ticas e de Computa√ß√£o, Universidade de S√£o Paulo, Av. Trabalhador S√£o Carlense, CEP 13560-970, S√£o Carlos, Brazil
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/ Luciane Quoos
  • Instituto de Matem√°tica, Universidade Federal do Rio de Janeiro, Cidade Universit√°ria, CEP 21941-909, Rio de Janeiro, Brazil
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Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/advgeom-2018-0021

Abstract

For Kummer extensions given by ym = f(x), we discuss conditions for an integer to be a Weierstrass gap at a place P. In the case of fully ramified places, the conditions are necessary and sufficient. As a consequence, we extend independent results of several authors. Moreover, we show that if the Kummer extension is ūĚĒĹq2-maximal and f(x) ‚ąą ūĚĒĹq2[x] has at least two roots with the same multiplicity őĽ coprime to m, then m divides 2(q + 1). Under the extra condition that either m or the multiplicity of a third root of f(x) is odd, we conclude that m divides q + 1.

Keywords: Weierstrass points; maximal curves; Kummer extensions

MSC 2010: 14H55; 11R58; 14Hxx

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About the article


Received: 2017-02-09

Revised: 2017-07-10

Published Online: 2018-07-20


Funding: The second author was partially supported by FAPESP-Brazil grant 2017/04681-3.


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0021.

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