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On the spectrum for the genera of maximal curves over small fields
1. | Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580, Santo André, SP , Brazil |
2. | School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran |
3. | Dept. of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave, P.O. Box: 15875-4413, Tehran, Iran |
4. | Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas, R. Sérgio Buarque de Holanda 651, Cidade Universitária "Zeferino Vaz", 13083-859, Campinas, SP, Brazil |
Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper [
References:
[1] |
manYPoints-Table of Curves with Many Points, available at www.manypoints.org |
[2] |
M. Abdón and F. Torres,
Maximal curves in characteristic two, Manuscripta Math., 99 (1999), 39-53.
|
[3] |
A. Cossidente, G. Korchmáros and F. Torres,
On curves covered by the Hermitian curve, J. Algebra, 216 (1999), 56-76.
|
[4] |
Y. Danisman and M. Ozdemir,
On the genus spectrum of maximal curves over finite fields, J. Discrete Math. Sci. Crypt., 18 (2015), 513-529.
|
[5] |
I. Duursma and K. H. Mak,
On maximal curves which are not Galois subcovers of the Hermitian curve, Bull. Braz. Math. Soc. New Series, 43 (2012), 453-465.
|
[6] |
S. Fanali and M. Giulietti,
On some open problems on maximal curves, Des. Codes Crypt., 56 (2010), 131-139.
|
[7] |
S. Fanali, M. Giulietti and I. Platoni,
On maximal curves over finite fields of small order, Adv. Math. Commun., 6 (2012), 107-120.
|
[8] |
R. Fuhrmann, A. Garcia and F. Torres,
On maximal curves, J. Number Theory, 67 (1997), 29-51.
|
[9] |
R. Fuhrmann and F. Torres,
The genus of curves over finite fields with many rational points, Manuscripta Math., 89 (1996), 103-106.
|
[10] |
A. Garcia and H. Stichtenoth,
A maximal curve which is not a Galois subcover of the Hermitian curve, Bulletin Braz. Math. Soc., 37 (2006), 1-14.
|
[11] |
A. Garcia, H. Stichtenoth and C. P. Xing,
On subfields of the Hermitian function field, Compositio Math., 120 (2000), 137-170.
|
[12] |
M. Giulietti and G. Korchmáros,
A new family of maximal curves over a finite field, Math. Ann., 343 (2009), 229-245.
|
[13] |
M. Giulietti, M. Montanucci, L. Quoos and G. Zini,
The automorphism group of some Galois covers of the Suzuki and Ree curves,
preprint, arXiv: 1609.09343 |
[14] |
M. Giulietti, M. Montanucci and G. Zini,
On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl., 41 (2016), 71-78.
|
[15] |
M. Giulietti, L. Quoos and G. Zini,
Maximal curves from subcovers of the GK-curve, J. Pure Appl. Algebra, 220 (2016), 3372-3383.
|
[16] |
J. W. P. Hirschfeld, G. Korchmáros and F. Torres,
Algebraic Curves over Finite Fields,
Princeton Univ. Press, 2008. |
[17] |
E. W. Howe, Quickly constructing curves of genus 4 with many points, in Frobenius Distributions: Sato-Tate and Lang-Trotter Conjectures, Amer. Math. Soc., Providence, 2016,
149–173. |
[18] |
N. E. Hurt,
Many Rational Points, Coding Theory and Algebraic Geometry,
Kluwer Acad. Publ., The Netherlands, 2003. |
[19] |
I. Ihara,
Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo Sec. Ia, 28 (1981), 721-724.
|
[20] |
G. Korchmáros and F. Torres,
Embedding of a maximal curve in a Hermitian variety, Compositio Math., 128 (2001), 95-113.
|
[21] |
G. Korchmáros and F. Torres,
On the genus of a maximal curve, Math. Ann., 323 (2002), 589-608.
|
[22] |
M. Kudo and S. Harashita,
Superspecial curves of genus 4 in small characteristic, Finite Fields Appl., 45 (2017), 131-169.
|
[23] |
M. Montanucci and G. Zini,
On the spectrum of genera of quotients of the Hermitian curve,
preprint, arXiv: 1703.10592 |
[24] |
M. Montanucci and G. Zini,
Some Ree and Suzuki curves are not Galois covered by the Hermitian curve, Finite Fields Appl., 48 (2017), 175-195.
|
[25] |
H. G. Rück and H. Stichtenoth,
A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math., 457 (1994), 185-188.
|
[26] |
H. Stichtenoth,
Algebraic Function Fields and Codes, 2nd edition,
Springer-Verlag, New York, 2009. |
[27] |
K. O. Stöhr and J. F. Voloch,
Weierstrass points and curves over finite fields, Proc. London Math. Soc., 52 (1986), 1-19.
|
[28] |
S. Tafazolian, A. Teherán-Herrera and F. Torres,
Further examples of maximal curves which cannot be covered by the Hermitian curve, J. Pure Appl. Algebra, 220 (2016), 1122-1132.
|
[29] |
S. Tafazolian and F. Torres,
On the curve $y^n=x^m+x$ over finite fields, J. Number Theory, 145 (2014), 51-66.
|
[30] |
C. P. Xing and H. Stichtenoth,
The genus of maximal functions fields, Manuscripta Math., 86 (1995), 217-224.
|
show all references
References:
[1] |
manYPoints-Table of Curves with Many Points, available at www.manypoints.org |
[2] |
M. Abdón and F. Torres,
Maximal curves in characteristic two, Manuscripta Math., 99 (1999), 39-53.
|
[3] |
A. Cossidente, G. Korchmáros and F. Torres,
On curves covered by the Hermitian curve, J. Algebra, 216 (1999), 56-76.
|
[4] |
Y. Danisman and M. Ozdemir,
On the genus spectrum of maximal curves over finite fields, J. Discrete Math. Sci. Crypt., 18 (2015), 513-529.
|
[5] |
I. Duursma and K. H. Mak,
On maximal curves which are not Galois subcovers of the Hermitian curve, Bull. Braz. Math. Soc. New Series, 43 (2012), 453-465.
|
[6] |
S. Fanali and M. Giulietti,
On some open problems on maximal curves, Des. Codes Crypt., 56 (2010), 131-139.
|
[7] |
S. Fanali, M. Giulietti and I. Platoni,
On maximal curves over finite fields of small order, Adv. Math. Commun., 6 (2012), 107-120.
|
[8] |
R. Fuhrmann, A. Garcia and F. Torres,
On maximal curves, J. Number Theory, 67 (1997), 29-51.
|
[9] |
R. Fuhrmann and F. Torres,
The genus of curves over finite fields with many rational points, Manuscripta Math., 89 (1996), 103-106.
|
[10] |
A. Garcia and H. Stichtenoth,
A maximal curve which is not a Galois subcover of the Hermitian curve, Bulletin Braz. Math. Soc., 37 (2006), 1-14.
|
[11] |
A. Garcia, H. Stichtenoth and C. P. Xing,
On subfields of the Hermitian function field, Compositio Math., 120 (2000), 137-170.
|
[12] |
M. Giulietti and G. Korchmáros,
A new family of maximal curves over a finite field, Math. Ann., 343 (2009), 229-245.
|
[13] |
M. Giulietti, M. Montanucci, L. Quoos and G. Zini,
The automorphism group of some Galois covers of the Suzuki and Ree curves,
preprint, arXiv: 1609.09343 |
[14] |
M. Giulietti, M. Montanucci and G. Zini,
On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl., 41 (2016), 71-78.
|
[15] |
M. Giulietti, L. Quoos and G. Zini,
Maximal curves from subcovers of the GK-curve, J. Pure Appl. Algebra, 220 (2016), 3372-3383.
|
[16] |
J. W. P. Hirschfeld, G. Korchmáros and F. Torres,
Algebraic Curves over Finite Fields,
Princeton Univ. Press, 2008. |
[17] |
E. W. Howe, Quickly constructing curves of genus 4 with many points, in Frobenius Distributions: Sato-Tate and Lang-Trotter Conjectures, Amer. Math. Soc., Providence, 2016,
149–173. |
[18] |
N. E. Hurt,
Many Rational Points, Coding Theory and Algebraic Geometry,
Kluwer Acad. Publ., The Netherlands, 2003. |
[19] |
I. Ihara,
Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo Sec. Ia, 28 (1981), 721-724.
|
[20] |
G. Korchmáros and F. Torres,
Embedding of a maximal curve in a Hermitian variety, Compositio Math., 128 (2001), 95-113.
|
[21] |
G. Korchmáros and F. Torres,
On the genus of a maximal curve, Math. Ann., 323 (2002), 589-608.
|
[22] |
M. Kudo and S. Harashita,
Superspecial curves of genus 4 in small characteristic, Finite Fields Appl., 45 (2017), 131-169.
|
[23] |
M. Montanucci and G. Zini,
On the spectrum of genera of quotients of the Hermitian curve,
preprint, arXiv: 1703.10592 |
[24] |
M. Montanucci and G. Zini,
Some Ree and Suzuki curves are not Galois covered by the Hermitian curve, Finite Fields Appl., 48 (2017), 175-195.
|
[25] |
H. G. Rück and H. Stichtenoth,
A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math., 457 (1994), 185-188.
|
[26] |
H. Stichtenoth,
Algebraic Function Fields and Codes, 2nd edition,
Springer-Verlag, New York, 2009. |
[27] |
K. O. Stöhr and J. F. Voloch,
Weierstrass points and curves over finite fields, Proc. London Math. Soc., 52 (1986), 1-19.
|
[28] |
S. Tafazolian, A. Teherán-Herrera and F. Torres,
Further examples of maximal curves which cannot be covered by the Hermitian curve, J. Pure Appl. Algebra, 220 (2016), 1122-1132.
|
[29] |
S. Tafazolian and F. Torres,
On the curve $y^n=x^m+x$ over finite fields, J. Number Theory, 145 (2014), 51-66.
|
[30] |
C. P. Xing and H. Stichtenoth,
The genus of maximal functions fields, Manuscripta Math., 86 (1995), 217-224.
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