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Geometric & Functional Analysis GAFA

, Volume 9, Issue 6, pp 1128–1155 | Cite as

A Spectral Correspondence for Maaß Waveforms

  • J. Bolte
  • S. Johansson

Abstract.

Let \( {\cal O}^1 \) be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order \( {\cal O} \) in an indefinite quaternion division algebra over \( {\Bbb Q} \). Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for \( {\cal O}^1 \) and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maaß newforms for the Hecke congruence group \( \Gamma_0(d) \), when d is the discriminant of the maximal order \( {\cal O} \). We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for \( {\cal O}^1 \) and the newforms of level d, respectively.

Keywords

Division Algebra Trace Formula Maximal Order Fuchsian Group Congruence Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • J. Bolte
    • 1
  • S. Johansson
    • 2
  1. 1.Abteilung Theoretische Physik, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany, e-mail: bol@physik.uni-ulm.deDE
  2. 2.Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden, e-mail: sj@math.chalmers.seSE

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