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 under deformation of the surface. Our calculations indicate that if the Teichmüller space of 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 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.