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Deformations of Maass forms


Authors: D. W. Farmer and S. Lemurell
Journal: Math. Comp. 74 (2005), 1967-1982
MSC (2000): Primary 11F03; Secondary 11F30
DOI: https://doi.org/10.1090/S0025-5718-05-01746-1
Published electronically: April 15, 2005
MathSciNet review: 2164106
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Abstract: We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface $S$ under deformation of the surface. Our calculations indicate that if the Teichmüller space of $S$ is not trivial, then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of $10^{8}$ smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.


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Additional Information

D. W. Farmer
Affiliation: American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94307
Email: farmer@aimath.org

S. Lemurell
Affiliation: Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Email: sj@math.chalmers.se

DOI: https://doi.org/10.1090/S0025-5718-05-01746-1
Keywords: Maass forms, deformations, Phillips-Sarnak conjecture, Teichmuller space
Received by editor(s): February 19, 2003
Received by editor(s) in revised form: April 30, 2004
Published electronically: April 15, 2005
Additional Notes: Research of the first author was supported in part by the National Science Foundation and the American Institute of Mathematics.
Research of the second author was supported in part by “Stiftelsen för internationalisering av högre utbildning och forskning” (STINT)
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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