American Institute of Mathematical Sciences

Advanced
February  2016, 10(1): 151-162. doi: 10.3934/amc.2016.10.151

Further results on fibre products of Kummer covers and curves with many points over finite fields

Ferruh Özbudak 1, , Burcu Gülmez Temür 2, and  Oǧuz Yayla 3,

1. 

Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Dumlupnar Bulvar, 06800, Ankara, Turkey
2. 

Department of Mathematics, Atlm University, Incek, Golbas, 06836, Ankara, Turkey
3. 

Department of Mathematics, Hacettepe University, Beytepe, 06800, Ankara, Turkey

Received  December 2014 Revised  December 2015 Published  March 2016

We study fibre products of an arbitrary number of Kummer covers of the projective line over $\mathbb{F}_q$ under suitable weak assumptions. If $q-1 = r^a$ for some prime $r$, then we completely determine the number of rational points over a rational point of the projective line. Using this result we obtain explicit examples of fibre products of three Kummer covers supplying new entries for the current table of curves with many points (http://www.manypoints.org, October 31 2015).
Keywords: fibre products, Kummer covers, rational points, Curves with many points over finite fields, algebraic function fields..
Mathematics Subject Classification: Primary: 11G20, 14G15; Secondary: 14H2.
Citation: Ferruh Özbudak, Burcu Gülmez Temür, Oǧuz Yayla. Further results on fibre products of Kummer covers and curves with many points over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 151-162. doi: 10.3934/amc.2016.10.151
References:
[1]

A. Garcia and A. Garzon, On Kummer covers with many rational points over finite fields, J. Pure Appl. Algebra, 185 (2003), 177-192. doi: 10.1016/S0022-4049(03)00110-5.  Google Scholar

[2]

G. van der Geer and M. van der Vlugt, Tables of curves with many points, Math. Comput., 69 (2000), 797-810. doi: 10.1090/S0025-5718-99-01143-6.  Google Scholar

[3]

J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd edition, The Clarendon Press, New York, 1998.  Google Scholar

[4]

J. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves over a Finite Field, Princeton Univ. Press, Princeton, 2008.  Google Scholar

[5]

B. Huppert and N. Blackburn, Finite Groups II Springer-Verlag, New York, 1981.  Google Scholar

[6]

M. Q. Kawakita, Kummer curves and their fibre products with many rational points, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 55-64.  Google Scholar

[7]

H. Niederreiter and C. Xing, Rational Points on Curves over Finite Fields, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9781107325951.  Google Scholar

[8]

H. Niederreiter and C. Xing, Algebraic Geometry in Coding Theory and Cryptography, Princeton Univ. Press, Princeton, 2009.  Google Scholar

[9]

F. Özbudak and H. Stichtenoth, Curves with many points and configurations of hyperplanes over finite fields, Finite Fields Appl., 5 (1999), 436-449. doi: 10.1006/ffta.1999.0262.  Google Scholar

[10]

F. Özbudak and B. G. Temür, Finite number of fibre products of Kummer covers and curves with many points over finite fields, Des. Codes Crypt., 70 (2014), 385-404. doi: 10.1007/s10623-012-9706-2.  Google Scholar

[11]

H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin, 1993.  Google Scholar

[12]

M. A. Tsfasman, S. G. Vlădut and D. Nogin, Algebraic Geometric Codes: Basic Notions, Amer. Math. Soc., Providence, 2007. doi: 10.1090/surv/139.  Google Scholar

[13]

, Manypoints-Table of Curves with Many Points,, available online at , ().   Google Scholar

show all references

References:
[1]

A. Garcia and A. Garzon, On Kummer covers with many rational points over finite fields, J. Pure Appl. Algebra, 185 (2003), 177-192. doi: 10.1016/S0022-4049(03)00110-5.  Google Scholar

[2]

G. van der Geer and M. van der Vlugt, Tables of curves with many points, Math. Comput., 69 (2000), 797-810. doi: 10.1090/S0025-5718-99-01143-6.  Google Scholar

[3]

J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd edition, The Clarendon Press, New York, 1998.  Google Scholar

[4]

J. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves over a Finite Field, Princeton Univ. Press, Princeton, 2008.  Google Scholar

[5]

B. Huppert and N. Blackburn, Finite Groups II Springer-Verlag, New York, 1981.  Google Scholar

[6]

M. Q. Kawakita, Kummer curves and their fibre products with many rational points, Appl. Algebra Engrg. Comm. Comput., 14 (2003), 55-64.  Google Scholar

[7]

H. Niederreiter and C. Xing, Rational Points on Curves over Finite Fields, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9781107325951.  Google Scholar

[8]

H. Niederreiter and C. Xing, Algebraic Geometry in Coding Theory and Cryptography, Princeton Univ. Press, Princeton, 2009.  Google Scholar

[9]

F. Özbudak and H. Stichtenoth, Curves with many points and configurations of hyperplanes over finite fields, Finite Fields Appl., 5 (1999), 436-449. doi: 10.1006/ffta.1999.0262.  Google Scholar

[10]

F. Özbudak and B. G. Temür, Finite number of fibre products of Kummer covers and curves with many points over finite fields, Des. Codes Crypt., 70 (2014), 385-404. doi: 10.1007/s10623-012-9706-2.  Google Scholar

[11]

H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin, 1993.  Google Scholar

[12]

M. A. Tsfasman, S. G. Vlădut and D. Nogin, Algebraic Geometric Codes: Basic Notions, Amer. Math. Soc., Providence, 2007. doi: 10.1090/surv/139.  Google Scholar

[13]

, Manypoints-Table of Curves with Many Points,, available online at , ().   Google Scholar

[1]

Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010
[2]

Ryutaroh Matsumoto. Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields. Advances in Mathematics of Communications, 2019, 13 (1) : 1-10. doi: 10.3934/amc.2019001
[3]

Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107
[4]

Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289-292. doi: 10.3934/amc.2017020
[5]

Joseph H. Silverman. Local-global aspects of (hyper)elliptic curves over (in)finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 101-114. doi: 10.3934/amc.2010.4.101
[6]

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
[7]

Marc Briane. Isotropic realizability of electric fields around critical points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353
[8]

Josep M. Miret, Jordi Pujolàs, Anna Rio. Explicit 2-power torsion of genus 2 curves over finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 155-168. doi: 10.3934/amc.2010.4.155
[9]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755
[10]

Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[11]

Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203
[12]

Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022
[13]

Peter Birkner, Nicolas Thériault. Efficient halving for genus 3 curves over binary fields. Advances in Mathematics of Communications, 2010, 4 (1) : 23-47. doi: 10.3934/amc.2010.4.23
[14]

Xiao-Song Yang. Index sums of isolated singular points of positive vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 1033-1039. doi: 10.3934/dcds.2009.25.1033
[15]

Fatma-Zohra Benahmed, Kenza Guenda, Aicha Batoul, Thomas Aaron Gulliver. Some new constructions of isodual and LCD codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 281-296. doi: 10.3934/amc.2019019
[16]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[17]

Nian Li, Qiaoyu Hu. A conjecture on permutation trinomials over finite fields of characteristic two. Advances in Mathematics of Communications, 2019, 13 (3) : 505-512. doi: 10.3934/amc.2019031
[18]

Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1
[19]

Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080
[20]

László Mérai, Igor E. Shparlinski. Unlikely intersections over finite fields: Polynomial orbits in small subgroups. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1065-1073. doi: 10.3934/dcds.2020070

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (148)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy