Skip to main content
Log in

A finiteness theorem for subgroups of SP (4,Z)

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This paper proves that there are only finitely many subgroupsH of finite index in Sp(4,Z) such that coresponding quotient\(\mathcal{H}/H\) of the Siegel upper half space of rank two is not of general type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. O. Bolza, “On binary sextics with linear transformations into themselves,”Amer. J. Math.,10, 70 (1888).

    MathSciNet  Google Scholar 

  2. H. Clemens, J. Kollár, and S. Mori, “Higher dimensional complex geometry”,Asterisque,166 (1988).

  3. D. Eisenbud and M. Hochster, “A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions,”J. Algebra 58, No. 1, 157–161 (1979).

    MathSciNet  Google Scholar 

  4. K. O’Grady, “On the Kodaira dimension of moduli spaces of abelian surfaces,”Compos. Math.,72, No. 2, 121–163 (1989).

    MathSciNet  Google Scholar 

  5. V. Gritsenko, “Modular forms and moduli spaces of abelian andK3 surfaces,”Algebra Analiz,6, No. 6, 65–102 (1994).

    MATH  MathSciNet  Google Scholar 

  6. G. van der Geer, “Note on abelian schemes of level three,”Math. Ann.,278, Nos. 1–4, 401–408 (1987).

    MATH  MathSciNet  Google Scholar 

  7. G. van der Geer, “On the geometry of a Siegel modular threefold,”Math. Ann.,260, No. 3, 317–350 (1982).

    MATH  MathSciNet  Google Scholar 

  8. W. P. Hammond, “On the graded ring of Siegel modular forms of genus two,”Amer. J. Math.,87, No. 2, 502–506 (1965).

    MATH  MathSciNet  Google Scholar 

  9. K. Hulek and G. K. Sankaran, “The Kodaira dimension of certain moduli spaces of abelian surfaces,”Compos. Math.,90, No. 1, 1–35 (1994).

    MathSciNet  Google Scholar 

  10. J.-I. Igusa, “A desingularization problem in the theory of Siegel modular functions,”Math. Ann.,168, 228–260 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Lee, S. Weintraub, “An interesting algebraic variety,”Math. Intelligencer 8, No. 1, 34–39 (1986).

    MathSciNet  Google Scholar 

  12. M. Reid, “Canonical threefolds,” in:Géometrie Algébrique, Angers. Sijthoff & Noordhoff (1980), pp. 273–310.

    Google Scholar 

  13. B. Teissier,Monômes, Volumes et Multiplicités. Introduction à la théorie des Singularités, II, Hermann, Paris (1988).

    Google Scholar 

  14. J. G. Thompson, “A finiteness theorem for subgroups of PSl(2,R) which are commensurable with PSl(2,Z),” In:Proc. Symp. Pure Math.,37 (1980), pp. 533–550.

  15. T. Yamazaki, “On Siegel modular forms of degree two,”Amer J. Math.,98, No. 1, 39–53 (1976).

    MATH  MathSciNet  Google Scholar 

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. vol. 56. Algebraic Geometry-9, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borisov, L.A. A finiteness theorem for subgroups of SP (4,Z). J Math Sci 94, 1073–1099 (1999). https://doi.org/10.1007/BF02367249

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02367249

Keywords

Navigation