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The curve Yn = Xℓ(Xm + 1) over finite fields II

  • Saeed Tafazolian EMAIL logo and Fernando Torres
Published/Copyright: June 24, 2021
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Advances in Geometry
From the journal Advances in Geometry Volume 21 Issue 3

Abstract

Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type yn=xℓ(xm+1). New results are obtained via certain subcovers of the nonsingular model of vN=ut2−u where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.

Keywords: Finite field; maximal curve; Kummer extension
MSC 2010: 11G20; 11M38; 14G15; 14H25

Funding statement: The authors were in part supported respectively by FAPESP/SP-Brazil (Grant 2017/19190-5) and by CNPq-Brazil (Grant 310623/2017-0)

Acknowledgements

The authors hearty thank the referee for her/ his valuable suggestions that helped them to improve the manuscript. In addition, F. Torres would like to thank P. Chacón for her constant support and friendship.

  1. Communicated by: G. Korchmáros

References

[1] M. Abdón, J. Bezerra, L. Quoos, Further examples of maximal curves. J. Pure Appl. Algebra 213 (2009), 1192–1196. MR2498810 Zbl 1166.11017Search in Google Scholar

[2] M. Abdón, H. Borges, L. Quoos, Weierstrass points on Kummer extensions. Adv. Geom. 19 (2019), 323–333. MR3982570 Zbl 1420.14070Search in Google Scholar

[3] N. Arakelian, S. Tafazolian, F. Torres, On the spectrum for the genera of maximal curves over small fields. Adv. Math. Commun. 12 (2018), 143–149. MR3808220 Zbl 1414.94949Search in Google Scholar

[4] F. Dalla Volta, M. Montanucci, G. Zini, On the classification problem for the genera of quotients of the Hermitian curve. Comm. Algebra 47 (2019), 4889–4909. MR4019313 Zbl 07122744Search in Google Scholar

[5] I. Duursma, K.-H. Mak, On maximal curves which are not Galois subcovers of the Hermitian curve. Bull. Braz. Math. Soc. N.S.) 43 (2012), 453–465. MR3024066 Zbl 1307.11076Search in Google Scholar

[6] R. Fuhrmann, A. Garcia, F. Torres, On maximal curves. J. Number Theory 67 (1997), 29–51. MR1485426 Zbl 0914.11036Search in Google Scholar

[7] A. Garcia, C. Güneri, H. Stichtenoth, A generalization of the Giulietti–Korchmáros maximal curve. Adv. Geom. 10 (2010), 427–434. MR2660419 Zbl 1196.14023Search in Google Scholar

[8] A. Garcia, H. Stichtenoth, A maximal curve which is not a Galois subcover of the Hermitian curve. Bull. Braz. Math. Soc. N.S.) 37 (2006), 139–152. MR2223491 Zbl 1118.14033Search in Google Scholar

[9] A. Garcia, H. Stichtenoth, C.-P. Xing, On subfields of the Hermitian function field. Compositio Math. 120 (2000), 137–170. MR1739176 Zbl 0990.11040Search in Google Scholar

[10] M. Giulietti, J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Curves covered by the Hermitian curve. Finite Fields Appl. 12 (2006), 539–564. MR2257083 Zbl 1218.11064Search in Google Scholar

[11] M. Giulietti, G. Korchmáros, A new family of maximal curves over a finite field. Math. Ann. 343 (2009), 229–245. MR2448446 Zbl 1160.14016Search in Google Scholar

[12] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic curves over a finite field. Princeton Univ. Press 2008. MR2386879 Zbl 1200.11042Search in Google Scholar

[13] C. Hu, S. Yang, Multi-point codes over Kummer extensions. Des. Codes Cryptogr. 86 (2018), 211–230. MR3742842 Zbl 1391.14063Search in Google Scholar

[14] N. E. Hurt, Many rational points, volume 564 of Mathematics and its Applications. Kluwer 2003. MR2042828 Zbl 1072.11042Search in Google Scholar

[15] G. Lachaud, Sommes d’Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 729–732. MR920053 Zbl 0639.14013Search in Google Scholar

[16] R. Lidl, H. Niederreiter, Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, Reading, MA 1983. MR746963 Zbl 0554.12010Search in Google Scholar

[17] M. Montanucci, G. Zini, On the spectrum of genera of quotients of the Hermitian curve. Comm. Algebra 46 (2018), 4739–4776. MR3864261 Zbl 06959292Search in Google Scholar

[18] H.-G. Rück, H. Stichtenoth, A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math. 457 (1994), 185–188. MR1305281 Zbl 0802.11053Search in Google Scholar

[19] S. A. Stepanov, Arithmetic of algebraic curves. Consultants Bureau, New York 1994. MR1321599 Zbl 0862.11036Search in Google Scholar

[20] H. Stichtenoth, A note on Hermitian codes over GF(q2). Coding techniques and coding theory. IEEE Trans. Inform. Theory 34 (1988), 1345–1348. MR987682 Zbl 0665.94015Search in Google Scholar

[21] H. Stichtenoth, Algebraic function fields and codes. Springer 2009. MR2464941 Zbl 1155.14022Search in Google Scholar

[22] S. Tafazolian, F. Torres, On maximal curves of Fermat type. Adv. Geom. 13 (2013), 613–617. MR3181538 Zbl 1384.11076Search in Google Scholar

[23] S. Tafazolian, F. Torres, On the curve yn = xm + x over finite fields. J. Number Theory 145 (2014), 51–66. MR3253292 Zbl 1377.11075Search in Google Scholar

[24] S. Tafazolian, F. Torres, On the curve Yn = Xℓ(Xm + 1) over finite fields. Adv. Geom. 19 (2019), 263–268. MR3940924 Zbl 07072823Search in Google Scholar

[25] S. Yang, C. Hu, Weierstrass semigroups from Kummer extensions. Finite Fields Appl. 45 (2017), 264–284. MR3631364 Zbl 1366.14032Search in Google Scholar
Received: 2019-09-24
Revised: 2020-01-06
Published Online: 2021-06-24
Published in Print: 2021-07-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On vector bundles over reducible curves with a node
  3. Inscribed rectangle coincidences
  4. 𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve
  5. Tetrahedral cages for unit discs
  6. When is M0,n(ℙ1,1) a Mori dream space?
  7. Rank-one isometries of CAT(0) cube complexes and their centralisers
  8. Wall-crossing in genus-zero hybrid theory
  9. The curve Yn = Xℓ(Xm + 1) over finite fields II
  10. Uniqueness results for bodies of constant width in the hyperbolic plane
  11. A note on large Kakeya sets
  12. On static manifolds and related critical spaces with cyclic parallel Ricci tensor
  13. The motivic Igusa zeta function of a space monomial curve with a plane semigroup
  14. Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator
https://doi.org/10.1515/advgeom-2021-0017
Keywords for this article
Finite field; maximal curve; Kummer extension

Articles in the same Issue

  1. Frontmatter
  2. On vector bundles over reducible curves with a node
  3. Inscribed rectangle coincidences
  4. 𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve
  5. Tetrahedral cages for unit discs
  6. When is M0,n(ℙ1,1) a Mori dream space?
  7. Rank-one isometries of CAT(0) cube complexes and their centralisers
  8. Wall-crossing in genus-zero hybrid theory
  9. The curve Yn = Xℓ(Xm + 1) over finite fields II
  10. Uniqueness results for bodies of constant width in the hyperbolic plane
  11. A note on large Kakeya sets
  12. On static manifolds and related critical spaces with cyclic parallel Ricci tensor
  13. The motivic Igusa zeta function of a space monomial curve with a plane semigroup
  14. Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator
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