American Institute of Mathematical Sciences

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

 1 Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Dumlupnar Bulvar, 06800, Ankara, Turkey 2 Department of Mathematics, Atlm University, Incek, Golbas, 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).
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

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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
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