Advertisement

The Ramanujan Journal

, Volume 46, Issue 1, pp 201–227 | Cite as

Icosahedral invariants and a construction of class fields via periods of K3 surfaces

  • Atsuhira Nagano
Article
  • 83 Downloads

Abstract

In the theory of complex multiplication, it is important to construct class fields over CM fields. In this paper, we consider explicit K3 surfaces parametrized by Klein’s icosahedral invariants. Via the periods and the Shioda–Inose structures of K3 surfaces, the special values of icosahedral invariants generate class fields over quartic CM fields. Moreover, we give an explicit expression of the canonical model of the Shimura variety for the simplest case via the periods of K3 surfaces.

Keywords

Class fields K3 surfaces Shimura varieties Abelian varieties Complex multiplication Hilbert modular functions Quartic fields 

Mathematics Subject Classification

Primary 11G45 Secondary 14J28 14G35 11F46 11G15 11R16 

Notes

Acknowledgements

The author would like to thank Professor Hironori Shiga for helpful advice and valuable suggestions, and also to Professor Kimio Ueno for kind encouragement.

References

  1. 1.
    Clinger, A., Doran, C.: Lattice polarized \(K3\) surfaces and Siegel modular forms. Adv. Math. 231, 172–212 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Elkies, N., Kumar, A.: \(K3\) Surfaces and equations for Hilbert modular surfaces. Algebra Number Theory 8, 2297–2411 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fuertes, Y., González-Diez, G.: Fields of moduli and definition of hyperelliptic covers. Arch. Math. 86(5), 398–408 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gundlach, K.: Die Bestimmung der Funktionen zur Hilbertschen Modulgruppe des Zahlköpers \(Q(\sqrt{5})\). Math. Ann. 152, 226–256 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gunji, K.: Defining equations of the universal abelian surfaces with level three structure. Manuscr. Math. 119, 61–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hardy, K., Hudson, R.H., Richmann, D., Williams, K.S., Holtz, N.M.: Calculation of class numbers of imaginary cyclic quartic fields. Math. Comput. 49, 615–620 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hardy, K., Hudson, R.H., Richmann, D., Williams, K.S.: Determination of all imaginary cyclic quartic fields with class number 2. Trans. Am. Math. 311, 1–55 (1989)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hashimoto, K., Nagano, A., Ueda, K.: Modular surfaces associated with toric K3 surfaces (2014). arXiv:1403.5818
  9. 9.
    Hirzebruch, F.: Lecture Notes in Mathematics. The ring of Hilbert modular forms for real quadratic fields of small discriminant, vol. 627. Springer, Berlin (1977)Google Scholar
  10. 10.
    Huard, J.G., Spearman, B.K., Williams, K.S.: Integral basis for quartic fields with quadratic subfields. J. Number Theory 51, 87–102 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hudson, R.H., Williams, K.S.: The integers of a cyclic quartic field. Rocky Mt. J. Math. 20, 145–150 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Igusa, J.: Arithmetic variety of moduli for genus two. Ann. Math. 72(3), 612–649 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Tauber (1884)Google Scholar
  14. 14.
    Kumar, A.: K3 surfaces associated to curves of genus two. Int. Math. Res. Not. 16 (2008). ArticleID: rnm165Google Scholar
  15. 15.
    Lauter, K., Yang, T.H.: Computing genus 2 curves from invariants on the Hilbert moduli space. J. Number Theory 131, 936–958 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Müller, R.: Hilbertsche Modulformen und modulfunktionen zu \(\mathbb{Q}(\sqrt{5})\). Arch. Math. 45, 239–251 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mumford, D.: On the equations defining abelian varieties I. Invent. Math. 1, 287–354 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Murabayashi, N., Umegaki, A.: Determination of all \(\mathbb{Q}\)- rational CM-points in the moduli space of principally polarized abelian surfaces. J. Algebra 235, 267–274 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nagano, A.: A theta expression of the Hilbert modular functions for \(\sqrt{5}\) via period of K3 surfaces. Kyoto J. Math. 53(4), 815–843 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nagano, A.: Double integrals on a weighted projective plane and the Hilbert modular functions for \(\mathbb{Q}(\sqrt{5})\). Acta Arith. 167, 327–345 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nagano, A.: Icosahedral invariants and Shimura curves. J. Theor. Nom. Bordeaux (2017)Google Scholar
  22. 22.
    Nagano, A., Shiga, H.: Modular map for the family of abelian surfaces via elliptic K3 surfaces. Math. Nachr. 288, 89–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nagano, A., Shiga, H.: To the Hilbert class field from the hypergeometric modular function. J. Number Theory 165, 408–430 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shimura, G.: On the theory of automorphic functions. Ann. Math. 70, 101–144 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shimura, G.: On purely transcendental fields of automorphic functions of several variables. Osaka J. Math. 1, 1–14 (1963)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math. 85, 58–159 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Shimura, G.: On abelian varieties with complex multiplication. Proc. Lond. Math. Soc. 34(3), 65–86 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions. Princeton University Press, Princeton (1997)zbMATHGoogle Scholar
  29. 29.
    van der Geer, G.: Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  30. 30.
    Weil, A.: On the theory of complex multiplication. In: Proceedings of International Symposium Algebraic Number Theory, pp. 9–22 (1955)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations