Skip to main content
Download PDF

On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms

Journal of Mathematical Sciences volume 193, pages 1–7 (2013)Cite this article

Transformation formulas of theta-series with harmonic polynomials of one-class quadratic forms under Hecke operators are interpreted as a result of interaction of the standard representation of the symplectic Hecke-Shimura ring on theta-series with the natural representation of the orthogonal Hecke-Shimura ring on the same theta-series considered as invariants of quadratic forms. Properties of the interaction rnaps and their relations to the action of Hecke operators are considered. Bibliography: 9 titles.

References

  1. A. N, Andrianov, “The action of Hecke operators on nonhomogeneous theta series,” Mat. Sb., 131, No. 3, 275–292 (1986).

    Google Scholar 

  2. A. N. Andrianov, “Symmetries of harmonic theta-functions of integral quadratic forms,” Usp. Mat. Nauk, 50, No. 4, 3–44 (1995).

    MathSciNet  Google Scholar 

  3. A. N, Andrianov, “Harmonic theta-functions and Hecke operators,” Algebra Analiz, 8, No. 5, 1–31 (1996).

  4. A. N. Andrianov, “Around the Ikeda lift,” Functs. Anal. Prilozh., 35, No. 2, 70–74 (2001).

    MathSciNet  Article  Google Scholar 

  5. A. N. Andrianov, “On zeta functions of orthogonal groups of single-class positive definite quadratic forms,” Algebra Analiz, 17, No. 4, 3–41 (2005).

    MathSciNet  Google Scholar 

  6. A. N. Andrianov, “Zeta functions of orthogonal groups of integral positive definite quadratic forms,” Usp. Mat. Nauk, 61, No. 6, 3–44 (2006).

    MathSciNet  Article  Google Scholar 

  7. A. N. Andrianov, “Interaction sums and action of Hecke operators on theta-series,” Acta Arithm., 140, No. 3, 271–304 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  8. J. W. S. Cassels, Rational Quadratic Forms, Academic Press (1978).

  9. A. N. Andrianov, “On interaction of Hecke-Shimura rings: symplectic versus orthogonal,” Acta Arithm., 155, No. 1, 97–107 (2012).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    A. N. Andrianov

Authors
  1. A. N. Andrianov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Andrianov.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 5–17.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Andrianov, A.N. On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms. J Math Sci 193, 1–7 (2013). https://doi.org/10.1007/s10958-013-1428-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1428-0

Keywords

  • Quadratic Form
  • Standard Representation
  • Natural Representation
  • Transformation Formula
  • Harmonic Polynomial