Skip to main content
Log in

Defining equations of the universal abelian surfaces with level three structure

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

In this paper we write down the defining equations of abelian surface with level three structures explicitly, and we see that their coefficients are given by Siegel modular forms of level three with some characters.

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. Andrianov, A.N.: Quadratic forms and Hecke operators. Glundl. math. Wiss. 286, Springer-Verlag, 1987

  2. Barth, W.: Quadratic equations for level-3 abelian surfaces. Abelian varieties (Egloffstein, 1993), 1-18, de Gruyter, Berlin, 1995

  3. Birkenhake, Ch., Lange, H.: Cubic theta relations. J. reine. angew. Math. 407, 167–177 (1990)

    MATH  MathSciNet  Google Scholar 

  4. Freitag, E., Salvati Manni, R.: The Burkhardt Group and Modular Forms. Transform. Groups 9 (1), 25–45 (2004)

    Article  MathSciNet  Google Scholar 

  5. Freitag, E., Salvati Manni, R.: The Burkhardt Group and Modular Forms II. Transform. Groups 9 (3), 237–256 (2004)

    MathSciNet  Google Scholar 

  6. Gunji, K.: On the graded ring of Siegel modular forms of degree 2, level 3. J. Math. Soc. Japan 56 (2), 375–403 (2004)

    MathSciNet  Google Scholar 

  7. Kempf, G.: Projective coordinate rings of abelian varieties. In: Algebraic Analysis, Geometry and Number Theory. J. Igusa (ed.), The John Hopkins Press, 1989, pp. 225–236

  8. Klingen, H., Introductory lectures on Siegel modular forms. Cambridge Std. in Adv. Math. 20, Cambridge Univ. Press, Cambridge, 1990

  9. Koizumi, S.: Theta relations and projective normality of abelian varieties. Am. J. Math. 98, 865–889 (1976)

    MATH  MathSciNet  Google Scholar 

  10. Koizumi, S.: The equations defining abelian varieties and modular functions. Math. Ann. 242, 127–145 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lange, H., Birkenhake, Ch.: Complex Abelian Varieties. Glundl. Math. Wiss. 302, Springer-Verlag, 1992

  12. Mumford, D.: On the equations defining abelian varieties I. Invent. Math. 1, 287–354 (1967)

    Article  MathSciNet  Google Scholar 

  13. Mumford, D.: Varieties defined by quadratic equations. Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 29–100

  14. Sekiguchi, T.: On the cubics defining abelian varieties. J. Math. Soc. Japan 30 (4), 701–721 (1978)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Keiichi Gunji.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gunji, K. Defining equations of the universal abelian surfaces with level three structure. manuscripta math. 119, 61–96 (2006). https://doi.org/10.1007/s00229-005-0606-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-005-0606-x

Keywords

Mathematics Subject Classification (2000)

Navigation