Advertisement

On the Construction of Class Fields by Picard Modular Forms

Chapter
Part of the Progress in Mathematics book series (PM, volume 260)

Abstract

The goal of this article is to construct modular functions living on the complex ball of dimension two such that the values in special points — similar to the elliptic modular function — generate class fields. For this purpose we are well prepared by the papers [5] and [6]. The first one classifies the moduli space of abelian 3-folds with a multiplication by ℚ(i) of type (2, 1) as projective surface. Via Jacobians we connect this Shimura surface with the moduli space of a family of curves of Shimura equation type. Thus we are able to continue the construction of the inverse period map of the family by theta constants given in [6]. Knowing the action of the modular group we reach a modular function j by modular forms with respect to the congruence subgroup of level (1 + i) of the full Picard modular group of Gauß numbers. If τ is the period of a (Jacobian of a) curve with complex multiplication the corresponding moduli field is generated over the rational numbers by j(τ). Hence the values in CM points of this function generate abelian extensions of the associated reflex field.

Keywords

Please provide some keywords 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Deligne, G. D. Mostow, Monodromy Of Hypergeometric Functions And Non-Lattice Integral Monodromy, Publ. Math. IHES 63 (1986), 5–88.zbMATHMathSciNetGoogle Scholar
  2. [2]
    R.-P. Holzapfel, Geometry and Arithmetic around Euler partial differential equations, Dt. Verlag d. Wiss./Reidel Publ. Comp., Berlin-Dordrecht, 1986.Google Scholar
  3. [3]
    R.-P. Holzapfel, Hierarchies of endomorphism algebras of abelian varieties corresponding to Picard modular surfaces, Schriftenreihe Komplexe Mannigfaltigkeiten 190, Univ. Erlangen, 1994.Google Scholar
  4. [4]
    R.-P. Holzapfel, A. Piñeiro, N. Vladov, Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle, HU preprint Nr. 98-15, 1998.Google Scholar
  5. [5]
    R.-P. Holzapfel, N. Vladov, Quadric-line configurations degenerating plane Picard-Einstein metrics I–II, Sitzungsber. d. Berliner Math. Ges., Jahrgänge 1997–2000, Berlin 2001, 79–142.Google Scholar
  6. [6]
    K. Matsumoto, On Modular Functions in 2 Variables Attached to a Family of Hyperelliptic Curves of Genus 3, Sc. Norm. Sup. Pisa 16 no. 4 (1989), 557–578.zbMATHGoogle Scholar
  7. [7]
    H. Shiga, On the representation of the Picard modular function by ϑ constants I–II, Publ. R.I.M.S. Kyoto Univ. 24 (1988), 311–360.zbMATHMathSciNetGoogle Scholar
  8. [8]
    H. Shiga, J. Wolfart, Criteria for complex multiplication and transcendence properties of automorphic functions, J. reine angew. Math. 463 (1995), 1–25.zbMATHMathSciNetGoogle Scholar
  9. [9]
    G. Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. Math. 78 no. 1 (1963), 149–192.CrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Shimura, Y. Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory, Publ. Math. Soc. Japan 6, Tokyo, 1961.Google Scholar
  11. [11]
    M. Yoshida, Fuchsian Differential Equations, Vieweg, Braunschweig-Wiesbaden, 1987zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-University of BerlinBerlinGermany

Personalised recommendations