Kyoto Journal of Mathematics

A theta expression of the Hilbert modular functions for 5 via the periods of K3 surfaces

Atsuhira Nagano

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Abstract

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for Q(5). We construct the period mapping for a family F={S(X,Y)} of K3 surfaces with 2 complex parameters X and Y. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for Q(5). Moreover, we show an explicit expression of this inverse correspondence by theta constants.

Article information

Source
Kyoto J. Math., Volume 53, Number 4 (2013), 815-843.

Dates
First available in Project Euclid: 21 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1385042735

Digital Object Identifier
doi:10.1215/21562261-2366102

Mathematical Reviews number (MathSciNet)
MR3160602

Zentralblatt MATH identifier
1285.14042

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 33C05: Classical hypergeometric functions, $_2F_1$

Citation

Nagano, Atsuhira. A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces. Kyoto J. Math. 53 (2013), no. 4, 815--843. doi:10.1215/21562261-2366102. https://projecteuclid.org/euclid.kjm/1385042735


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References

  • [Gr] P. Griffiths, Periods of integrals on algebraic varieties, III: Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180.
  • [Gu] K.-B. Gundlach, Die Bestimmung der Funktionen zur Hirbertschen Modulgruppe des Zahlkörpers $\mathbb{Q}(\sqrt{5})$, Math. Ann. 152 (1963), 226–256.
  • [H] F. Hirzebruch, “The ring of Hilbert modular forms for real quadratic fields of small discriminant” in Modular Functions of One Variable, VI (Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, 287–323.
  • [Kl] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Tauber, Leipzig, 1884.
  • [Ko] K. Kodaira, On compact analytic surfaces II, Ann. of Math. (2) 77 (1963), 563–626; III, 78 (1963), 1–40.
  • [KKN] R. Kobayashi, K. Kushibiki and I. Naruki, Polygons and Hilbert modular groups, Tohoku Math. J. (2) 41 (1989), 633–646.
  • [M] R. Müller, Hilbertsche Modulformen und Modulfunctionen zu $\mathbb{Q}(\sqrt{5})$, Arch. Math. (Basel) 45 (1985), 239–251.
  • [N1] A. Nagano, Period differential equations for the families of $K3$ surfaces with two parameters derived from the reflexive polytopes, Kyushu J. Math. 66 (2012), 193–244.
  • [N2] A. Nagano, Double integrals on a weighted projective plane and the Hilbert modular functions for $Y\mathbb{Q}(Y\sqrt{5})$, preprint, 2013.
  • [Sa] T. Sato, Uniformizing differential equations of several Hilbert modular orbifolds, Math. Ann. 291 (1991), 179–189.
  • [Sh] H. Shiga, One Attempt to the K3 modular function, III, Technical Report, Chiba University, 2010.