Advertisement

Research in Number Theory

, 5:27 | Cite as

Counting rational points on Kummer surfaces

  • Andreas MalmendierEmail author
  • Yih Sung
Research
  • 17 Downloads

Abstract

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin’s unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.

Keywords

Kummer surface Manin principle Rational points 

Mathematics Subject Classification

14D0x 14J28 33C65 

Notes

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)Google Scholar
  2. 2.
    Aspinwall, P.S.: \(K3\) Surfaces and String Duality. Fields, Strings and Duality (Boulder, CO, 1996), pp. 421–540. World Scientific Publishing, River Edge (1997)Google Scholar
  3. 3.
    Bailey, W.N.: The generating function of jacobi polynomials. J. Lond. Math. Soc. 13(1), 8–12 (1938).  https://doi.org/10.1112/jlms/s1-13.1.8 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bailey, W.N.: On two manuscripts by Bishop Barnes. Quart. J. Math. Oxford Ser. (2) 10, 236–240 (1959).  https://doi.org/10.1093/qmath/10.1.236 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barman, R., Kalita, G.: Certain values of Gaussian hypergeometric series and a family of algebraic curves. Int. J. Number Theory 8(4), 945–961 (2012).  https://doi.org/10.1142/S179304211250056X MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brieskorn, E.: Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, German. Math. Ann. 178, 255–270 (1968).  https://doi.org/10.1007/BF01352140 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields. II, Calabi-Yau varieties and mirror symmetry, (Toronto, ON, 2001), Fields Inst. Commun., vol. 38, American Mathematical Society, Providence, pp. 121–157, (2003)Google Scholar
  8. 8.
    Clemens, C. Herbert: A Scrapbook of Complex Curve Theory, 2nd ed. Graduate Studies in Mathematics, vol. 55, American Mathematical Society, Providence (2003)Google Scholar
  9. 9.
    Clingher, A., Doran, C.F., Malmendier, A.: Special function identities from superelliptic Kummer varieties. Asian J. Math. 21(5), 909–951 (2017).  https://doi.org/10.4310/AJM.2017.v21.n5.a6 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dolgachev, I.V.: Mirror symmetry for lattice polarized \(K3\) surfaces. J. Math. Sci. 81(3), 2599–2630 (1996).  https://doi.org/10.1007/BF02362332 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, vol. I, II. McGraw-Hill Book Company Inc, New York-Toronto-London (1953)zbMATHGoogle Scholar
  12. 12.
    Griffin, E., Malmendier, A.: Jacobian elliptic Kummer surfaces and special function identities. Commun. Number Theory Phys. 12(1), 97–125 (2018).  https://doi.org/10.4310/CNTP.2018.v12.n1.a4 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978). Pure and Applied MathematicszbMATHGoogle Scholar
  14. 14.
    Igusa, J.-I.: Class number of a definite quaternion with prime discriminant. Proc. Natl. Acad. Sci. USA 44, 312–314 (1958)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Koblitz, N.: The number of points on certain families of hypersurfaces over finite fields. Compos. Math. 48(1), 3–23 (1983)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kodaira, K.: On the structure of compact complex analytic surfaces. I, II. Proc. Nat. Acad. Sci. USA 50, 218–221 (1963). ibid., 51, 1963, 1100–1104CrossRefGoogle Scholar
  17. 17.
    Kuwata, M., Shioda, T.: Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface. Algebraic geometry in East Asia-Hanoi 2005, Advanced Studies in Pure Mathematics, vol. 50, Mathematical Society Japan, Tokyo , pp. 177–215 (2008)Google Scholar
  18. 18.
    Malmendier, A.: Kummer surfaces associated with Seiberg–Witten curves. J. Geom. Phys. 62(1), 107–123 (2012).  https://doi.org/10.1016/j.geomphys.2011.09.010 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Malmendier, A., Morrison, D.R.: K3 surfaces, modular forms, and non-geometric heterotic compactifications. Lett. Math. Phys. 105(8), 1085–1118 (2015).  https://doi.org/10.1007/s11005-015-0773-y MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Oguiso, K.: On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Japan 41(4), 651–680 (1989).  https://doi.org/10.2969/jmsj/04140651 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Oguiso, K., Shioda, T.: The Mordell–Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40(1), 83–99 (1991)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sasaki, T., Yoshida, M.: Linear differential equations in two variables of rank four. I. Math. Ann. 282(1), 69–93 (1988).  https://doi.org/10.1007/BF01457013 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sasaki, T., Yoshida, M.: Linear differential equations in two variables of rank four. II. The uniformizing equation of a Hilbert modular orbifold. Math. Ann. 282(1), 95–111 (1988).  https://doi.org/10.1007/BF01457014 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Salerno, A.: Counting points over finite fields and hypergeometric functions. Funct. Approx. Comment. Math. 49(1), 137–157 (2013).  https://doi.org/10.7169/facm/2013.49.1.9 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schmickler-Hirzebruch, Ulrike: Elliptische Flächen über \({\rm P}_1\) C mit drei Ausnahmefasern und die hypergeometrische Differentialgleichung. Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 33, Universität Münster, Mathematisches Institut, Münster, (1985) (German)Google Scholar
  26. 26.
    Sung, Y.: Rational points over finite fields on a family of higher genus curves and hypergeometric functions. Taiwan. J. Math. 21(1), 55–79 (2017).  https://doi.org/10.11650/tjm.21.2017.7724 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vidūnas, R.: Specialization of Appell’s functions to univariate hypergeometric functions. J. Math. Anal. Appl. 355(1), 145–163 (2009).  https://doi.org/10.1016/j.jmaa.2009.01.047 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vega, M.: Valentina, hypergeometric functions over finite fields and their relations to algebraic curves. Int. J. Number Theory 7(8), 2171–2195 (2011).  https://doi.org/10.1142/S1793042111004976 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yui, N.: Arithmetic of certain Calabi-Yau varieties and mirror symmetry. Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Mathematical Series, vol. 9, American Mathematical Society, Providence, pp. 507–569 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations