On maximal curves that are not quotients of the Hermitian curve

For each prime power ź the plane curve X ź with equation Y ź 2 - ź + 1 = X ź 2 - X is maximal over F ź 6 . Garcia and Stichtenoth in 2006 proved that X 3 is not Galois covered by the Hermitian curve and raised the same question for X ź with ź 3 ; in this paper we show that X ź is not Galois covered by the Hermitian curve for any ź 3 . Analogously, Duursma and Mak proved that the generalized GK curve C ź n over F ź 2 n is not a quotient of the Hermitian curve for ź 2 and n ź 5 , leaving the case ź = 2 open; here we show that C 2 n is not Galois covered by the Hermitian curve over F 2 2 n for n ź 5 .

[1]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[2]  Luciane Quoos,et al.  Further examples of maximal curves , 2009 .

[3]  Henning Stichtenoth,et al.  Algebraic function fields over finite fields with many rational places , 1995, IEEE Trans. Inf. Theory.

[4]  Arnaldo Garcia,et al.  Curves over Finite Fields Attaining the Hasse-Weil Upper Bound , 2001 .

[5]  Ciro Ciliberto,et al.  Applications of algebraic geometry to coding theory, physics and computation , 2001 .

[6]  G. Geer.  Coding Theory and Algebraic Curves Over Finite Fields , 2001 .

[7]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[8]  P. Hall.  A Note on Soluble Groups , 1928 .

[9]  C. Xing,et al.  On Subfields of the Hermitian Function Field , 2000, Compositio Mathematica.

[10]  Arnaldo Garcia.  On Curves with Many Rational Points over Finite Fields , 2002 .

[11]  Gábor Korchmáros,et al.  A new family of maximal curves over a finite field , 2007, 0711.0445.

[12]  W. Kantor,et al.  2-Transitive groups in which the stabilizer of two points is cyclic☆ , 1972 .

[13]  W. Burnside,et al.  Theory of Groups of Finite Order , 1909 .

[14]  Fernando Torres,et al.  ON WEIERSTRASS POINTS AND OPTIMAL CURVES , 2008 .

[15]  F. Torres,et al.  Algebraic Curves over Finite Fields , 1991 .

[16]  G. M.,et al.  Projective Geometry , 1938, Nature.

[17]  Gerard van der Geer,et al.  Curves over Finite Fields and Codes , 2001 .

[18]  The Automorphism Groups of a Family of Maximal Curves , 2011, 1105.3952.

[19]  Antonio Cossidente,et al.  United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency Curves of Large Genus Covered by the Hermitian Curve , 2022 .

[20]  A maximal curve which is not a Galois subcover of the Hermitian curve , 2006 .

[21]  R. W. Hartley,et al.  Determination of the Ternary Collineation Groups Whose Coefficients Lie in the GF(2 n ) , 1925 .

[22]  P. Dembowski.  Finite geometries , 1997 .

[23]  On maximal curves which are not Galois subcovers of the Hermitian curve , 2010, 1012.2068.

[24]  Antonio Cossidente,et al.  On Curves Covered by the Hermitian Curve , 1999 .

[25]  B. Huppert.  Endliche Gruppen I , 1967 .

[26]  A. N. Parshin.  Algebraic curves over functional fields with a finite field of constants , 1974 .

[27]  G. Miller.  Linear Groups with an Exposition of the Galois Field Theory , 1902 .

[28]  H. Weyl.  Permutation Groups , 2022 .

[29]  Hans Zassenhaus,et al.  Über endliche Fastkörper , 1935 .

[30]  L. Dickson,et al.  Linear Groups: With an Exposition of the Galois Field Theory , 1902 .

[31]  Henning Stichtenoth,et al.  A generalization of the Giulietti-Korchmaros maximal curve , 2010 .

[32]  D. M. Clark.  Theory of Groups , 2012 .

[33]  Howard H. Mitchell,et al.  Determination of the ordinary and modular ternary linear groups , 1911 .

[34]  F. Torres.  ON UNRAMIFIED COVERINGS OF MAXIMAL CURVES by Arnaldo Garcia & , 2010 .

[35]  Henning Stichtenoth,et al.  The automorphism group of the generalized Giulietti–Korchmáros function field , 2013 .