Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 509–527 | Cite as

Weil representations associated with finite quadratic modules

  • Fredrik Strömberg


To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of \(\mathrm{Mp}_{2}(\mathbb Z )\), the metaplectic cover of the modular group. This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients. This result completes earlier work by Scheithauer in the case when the representation factors through \(\mathrm{SL}_{2}(\mathbb Z )\). Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer.


Weil representation Metaplectic group Finite quadratic module 

Mathematics Subject Classification (2000)

11F27 20C25 



I would like to thank Nils Scheithauer for clarifying details of [15], Nils-Peter Skoruppa for sharing thoughts about Weil representations in general, and the manuscript [19], in particular. I would also like to thank Stephan Ehlen for his extensive assistance with proof reading the manuscript and in simplifying Theorem 4.1.


  1. 1.
    Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1998)Google Scholar
  2. 2.
    Borcherds, R.E.: Reflection groups of Lorentzian lattices. Duke Math. J. 104(2), 319–366 (2000). doi: 10.1215/S0012-7094-00-10424-3
  3. 3.
    Bruinier, J.H., Strömberg, F.: Computation of harmonic weak Maass forms. Exp. Math. 21(2), 117–131 (2011)CrossRefGoogle Scholar
  4. 4.
    Cassels, J.W.S.: Rational Quadratic Forms. London Mathematical Society Monographs, vol. 13. Academic Press Inc., London (1978)Google Scholar
  5. 5.
    Conway, J.H., Sloane, N.J.A.: On the classification of integral quadratic forms. In: Conway, J.H., Sloane, N.J.A. (eds.) Sphere Packings, Lattices and Groups, pp. 352–405. Die Grundlehren der mathematischen Wissenschaften, vol. 290, Springer, New York (1999)Google Scholar
  6. 6.
    Ebeling, W.: Lattices and Codes. Friedrich Vieweg & Sohn, Braunschweig (1994)CrossRefMATHGoogle Scholar
  7. 7.
    Eichler, M.: Introduction to the Theory of Algebraic Numbers and Functions. Pure and Applied Mathematics, vol. 23. Academic Press, New York (1966)Google Scholar
  8. 8.
    Gelbart, S.: Weil’s Representation and the Spectrum of the Metaplectic Group. Lecture Notes in Mathematics, vol. 530. Springer, Berlin (1976)Google Scholar
  9. 9.
    Kloosterman, H.D.: The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups. I, II. Ann. Math. (2) 47, 317–375, 376–447 (1946)Google Scholar
  10. 10.
    Kubota, T.: Topological covering of SL(2) over a local field. J. Math. Soc. Jpn. 19, 114–121 (1967)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Maass, H.: Lectures on Modular Functions of One Complex Variable, 2nd edn. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 29. Tata Institute of Fundamental Research, Bombay (1983)Google Scholar
  12. 12.
    Milnor, J., Husemoller, D.: Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Springer, New York (1973)CrossRefGoogle Scholar
  13. 13.
    Pfetzer, W.: Die Wirkung der Modulsubstitutionen auf mehrafache Thetareihen zu quadratischen Formen ungerader Variablenzahl. Arch. Math. 4, 448–454 (1953)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ryan, N., Skoruppa, N., Strömberg, F.: Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two. Math. Comput. 81(280), 2361–2376 (2012)CrossRefMATHGoogle Scholar
  15. 15.
    Scheithauer, N.R.: The Weil representation of \({\rm SL}_2({\mathbb{Z}})\) and some applications. Int. Math. Res. Not. 8, 1488–1545 (2009)Google Scholar
  16. 16.
    Schoeneberg, B.: Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen. Math. Ann. 116(1), 511–523 (1939)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shimura, G.: On modular forms of half integral weight. Ann. Math. 97, 440–481 (1973)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58, 83–126 (1975)MathSciNetMATHGoogle Scholar
  19. 19.
    Skoruppa, N.-P.: Finite quadratic modules, Weil representations and vector valued modular forms. (2013, preprint)Google Scholar
  20. 20.
    Skoruppa, N.P.: Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts. Bonner Math. Schriften, no. 159. University of Bonn, Bonn (1985)Google Scholar
  21. 21.
    Stein, W., et al.: Sage mathematics software (Version 5.3). The Sage Development Team (2012).
  22. 22.
    Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUK

Personalised recommendations