On the spectrum for the genera of maximal curves over small fields

Nazar Arakelian; Saeed Tafazolian; Fernando Torres

doi:10.3934/amc.2018009

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February 2018, 12(1): 143-149. doi: 10.3934/amc.2018009

On the spectrum for the genera of maximal curves over small fields

Nazar Arakelian , Saeed Tafazolian ^2,3, and Fernando Torres ^4,

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

Received October 2016 Published March 2018

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper [11], we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7≤ q≤ 16$. In particular, by using a result in Kudo and Harashita (2016) paper [22], the set $\mathbf{M}(7^2)$ is completely determined.

Keywords: Finite field, Hasse-Weil bound, Stöhr-Voloch theory, maximal curve.

Mathematics Subject Classification: Primary: 11G20, 11M38, 14H05, 14G15; Secondary: 14HXX.

Citation: Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009

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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