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The Ramanujan Journal

, Volume 47, Issue 3, pp 475–499 | Cite as

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

  • Lynne H. Walling
Article
  • 56 Downloads

Abstract

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of \(\Gamma _0(\mathcal N )\). To each of these, we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level \(\mathcal N \) and Dirichlet character \(\chi _{_L}\) modulo \(\mathcal N \) as a linear combination of Siegel Eisenstein series.

Keywords

Theta series Quadratic forms Eisenstein series Siegel modular forms 

Mathematics Subject Classification

Primary 11F46 11F11 

References

  1. 1.
    Andrianov, A.N.: Quadratic Forms and Hecke Operators. Springer, Berlin (1987)CrossRefGoogle Scholar
  2. 2.
    Gerstein, L.: Basic Quadratic Forms. Graduate Studies in Mathematics, vol. 90. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  3. 3.
    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, Berlin (1987)zbMATHGoogle Scholar
  4. 4.
    Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. Ann. Math. 36, 527–606 (1935)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Walling, L.H.: Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character. Int. J. Number Theory 13(02), 325–370 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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