Abstract
We show that the generalized Giulietti-Korchmáros curve defined over \(\mathbb{F}_{q^{2n} }\), for n ≥ 3 odd and q ≥ 3, is not a Galois subcover of the Hermitian curve over \(\mathbb{F}_{q^{2n} }\). This answers a question raised by Garcia, Güneri and Stichtenoth.
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M. Abdón, J. Bezerra and L. Quoos. Further examples of maximal curves. J. Pure Appl. Algebra, 213(6) (2009), 1192–1196.
M. Abdón and A. Garcia. On a characterization of certain maximal curves. Finite Fields Appl., 10(2) (2004), 133–158.
M. Abdón and F. Torres. On maximal curves in characteristic two. Manuscripta Math., 99(1) (1999), 39–53.
A. Cossidente, G. Korchmáros and F. Torres. On curves covered by the Hermitian curve. J. Algebra, 216(1) (1999), 56–76.
A. Cossidente, G. Korchmáros and F. Torres. Curves of large genus covered by the Hermitian curve. Comm. Algebra, 28(10) (2000), 4707–4728.
I.M. Duursma. Two-point coordinate rings for GK-curves. IEEE Trans. Inform. Theory, 57(2) (2011), 593–600.
R. Fuhrmann, A. Garcia and F. Torres. On maximal curves. J. Number Theory, 67(1) (1997), 29–51.
R. Fuhrmann and F. Torres. The genus of curves over finite fields with many rational points. Manuscripta Math., 89(1) (1996), 103–106.
A. Garcia. Curves over finite fields attaining the Hasse-Weil upper bound. European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., Birkh.user, Basel, 202 (2001), 199–205.
A. Garcia, C. Güneri and H. Stichtenoth. A generalization of the Giulietti-Korchmáros maximal curve. Adv. Geom., 10(3) (2010), 427–434.
A. Garcia and H. Stichtenoth. A maximal curve which is not a Galois subcover of the Hermitian curve. Bull. Braz. Math. Soc. (N.S.), 37(1) (2006), 139–152.
A. Garcia, H. Stichtenoth and C.-P. Xing. On subfields of the Hermitian function field. Compositio Math., 120(2) (2000), 137–170.
M. Giulietti and G. Korchmáros. A new family of maximal curves over a finite field. Math. Ann., 343(1) (2009), 229–245.
D.R. Hughes and F.C. Piper. Projective planes. Springer-Verlag, New York, Graduate Texts in Mathematics, 6 (1973).
Y. Ihara. Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3) (1981), 721–724.
G. Korchmáros and F. Torres. Embedding of a maximal curve in a Hermitian variety. Compositio Math., 128(1) (2001), 95–113.
G. Korchmáros and F. Torres. On the genus of a maximal curve. Math. Ann., 323(3) (2002), 589–608.
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(16) (1987), 729–732.
V. Landazuri and G.M. Seitz. On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra, 32 (1974), 418–443.
H.-W. Leopoldt. über die Automorphismengruppe des Fermatk.rpers. J. Number Theory, 56(2) (1996), 256–282.
H.-G. Rück and H. Stichtenoth. A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math., 457 (1994), 185–188.
J.-P. Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42.
J.-P. Serre. Local fields. Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979, Translated from the French by Marvin Jay Greenberg.
H. Stichtenoth. Über die Automorphismengruppe eines algebraischen Funktionenk.rpers von Primzahlcharakteristik. I, II. Arch. Math. (Basel), 24 (1973), 527–544, 615–631.
H. Stichtenoth. Algebraic function fields and codes, second ed., Graduate Texts in Mathematics, Springer-Verlag, Berlin, 254 (2009).
H. Stichtenoth and C.P. Xing. The genus of maximal function fields over finite fields. Manuscripta Math., 86(2) (1995), 217–224.
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Duursma, I., Mak, KH. On maximal curves which are not Galois subcovers of the Hermitian curve. Bull Braz Math Soc, New Series 43, 453–465 (2012). https://doi.org/10.1007/s00574-012-0022-2
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DOI: https://doi.org/10.1007/s00574-012-0022-2