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
[1] M. Abdón, H. Borges, L. Quoos, Weierstrass points on Kummer extensions. arXiv 1308.2203v3 [math.AG]Search in Google Scholar
[2] N. Arakelian, S. Tafazolian, F. Torres, On the spectrum for the genera of maximal curves over small fields.To appear in Adv. Math. Commun.10.3934/amc.2018009Search in Google Scholar
[3] R. Fuhrmann, A. Garcia, F. Torres, On maximal curves. J. Number Theory67 (1997), 29–51. MR1485426 Zbl 0914.1103610.1006/jnth.1997.2148Search in Google Scholar
[4] A. Garcia, H. Stichtenoth, C.-P. Xing, On subfields of the Hermitian function field. Compositio Math. 120 (2000), 137–170. MR1739176 Zbl 0990.1104010.1023/A:1001736016924Search in Google Scholar
[5] G. van der Geer, E. W. Howe, K. E. Lauter, C. Ritzenthaler, Tables of Curves with Many Points, Korteweg–de Vries Instituut, Universiteit van Amsterdam 2009, www.manypoints.orgSearch in Google Scholar
[6] 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 1218.1106410.1016/j.ffa.2004.10.003Search in Google Scholar
[7] M. Giulietti, G. Korchmáros, A new family of maximal curves over a finite field. Math. Ann. 343 (2009), 229–245. MR2448446 Zbl 1160.1401610.1007/s00208-008-0270-zSearch in Google Scholar
[8] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic curves over a finite field. Princeton Univ. Press 2008. MR2386879 Zbl 1200.1104210.1515/9781400847419Search in Google Scholar
[9] N. E. Hurt, Many rational points, volume 564 of Mathematics and its Applications. Kluwer 2003. MR2042828 Zbl 1072.1104210.1007/978-94-017-0251-5Search in Google Scholar
[10] A. Kazemifard, A. R. Naghipour, S. Tafazolian, A note on superspecial and maximal curves. Bull. Iranian Math. Soc. 39 (2013), 405–413. MR3095333 Zbl 1298.11057Search in Google Scholar
[11] G. Korchmáros, F. Torres, On the genus of a maximal curve. Math. Ann. 323(2002), 589–608. MR1923698 Zbl 1018.1102910.1007/s002080200316Search in Google Scholar
[12] 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
[13] R. Lidl, H. Niederreiter, Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publ. Co., Reading, MA 1983. MR746963 Zbl 0554.12010Search in Google Scholar
[14] 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
[15] J. Serre, Résumé des cours de 1983–1984. Ann. Collége de France (1984), 79–83.Search in Google Scholar
[16] S. A. Stepanov, Arithmetic of algebraic curves. Consultants Bureau, New York 1994. MR1321599 Zbl 0862.11036Search in Google Scholar
[17] H. Stichtenoth, Algebraic function fields and codes. Springer 2009. MR2464941 Zbl 1155.1402210.1007/978-3-540-76878-4Search in Google Scholar
[18] S. Tafazolian, A. Teherán-Herrera, F. Torres, Further examples of maximal curves which cannot be covered by the Hermitian curve. J. Pure Appl. Algebra220 (2016), 1122–1132. MR3414410 Zbl 0650697710.1016/j.jpaa.2015.08.010Search in Google Scholar
[19] S. Tafazolian, F. Torres, On maximal curves of Fermat type. Adv. Geom. 13 (2013), 613–617. MR3181538 Zbl 0622923510.1515/advgeom-2013-0002Search in Google Scholar
[20] S. Tafazolian, F. Torres, On the curve yn = xm + x over finite fields. J. Number Theory145 (2014), 51–66. MR3253292 Zbl 0634397310.1016/j.jnt.2014.05.019Search in Google Scholar
[21] J. Tate, Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134–144. MR0206004 Zbl 0147.2030310.1007/BF01404549Search in Google Scholar
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