A local Weyl's law, the angular distribution and multiplicity of cusp forms on product spaces
Authors:
Jonathan Huntley and David Tepper
Journal:
Trans. Amer. Math. Soc. 330 (1992), 97110
MSC:
Primary 11F72; Secondary 11F55
MathSciNet review:
1053114
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Abstract: Let be a finite volume symmetric space with the product of half planes. Let be the Laplacian on the th half plane, and assume that we have a cusp form , so we have for . Let and let with . Letting , we let denote the dimension of the space of cusp forms with eigenvalue . More generally, let denote the number of independent eigenfunctions such that the associated to an eigenfunction is inside the ball of radius , centered at . We will define a function , which is generally equal to a linear sum of products of the . We prove the following theorems. Theorem 1. Theorem 2.
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 H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Differential Geom. 11 (1982), 239255. MR 664496 (83m:58079)
 [D2]
 , Eigenvalue estimates for certain noncompact manifolds, Michigan Math. J. 31 (1984), 349357. MR 767614 (86d:58120)
 [E1]
 I. Efrat, Selberg trace formulas, rigidity and Weyl's law, Ph.D. Thesis, New York Univ., 1981.
 [E2]
 , The Selberg trace formula for , Mem. Amer. Math. Soc., No. 359 (1987).
 [Hu]
 J. Huntley, Spectral multiplicity on products of hyperbolic spaces, Proc. Amer. Math. Soc. 111 (1991), 112. MR 1031667 (91d:11055)
 [Ku]
 S. Kudle, Relations between automorphic forms produced by theta functions, Lecture Notes in Math., vol. 627, SpringerVerlag, New York, 1977. MR 0480343 (58:516)
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 P. Lax and R. Philips, Scattering theory for automorphic forms, Princeton Univ. Press, Princeton, N.J., 1974.
 [M]
 W. Müller, The trace class conjecture in the theory of automorphic forms, preprint.
 [Ro]
 W. Roelcke, Über die Wellengeleichung bei Grenzkreisgruppen erster Art, S. B. Heidelberger Akad. Wiss. Math.Nat. Kl. 1953/1955 (1956), 159267. MR 0081967 (18:476d)
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 A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemann spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 4787. MR 0088511 (19:531g)
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 J. Serre, Abelian adic representations and elliptic curves, Benjamin, New York, 1968. MR 0263823 (41:8422)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719921053114X
PII:
S 00029947(1992)1053114X
Article copyright:
© Copyright 1992 American Mathematical Society
