Deformations of Maass forms
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- by D. W. Farmer and S. Lemurell PDF
- Math. Comp. 74 (2005), 1967-1982 Request permission
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.References
- H. Avelin, Research Announcement on the deformation of cusp forms, U.U.D.M. Report 2002:26, Uppsala Univ.
- Harvey Cohn, A numerical survey of the reduction of modular curve genus by Fricke’s involutions, Number theory (New York, 1989/1990) Springer, New York, 1991, pp. 85–104. MR 1124636
- Yves Colin de Verdière, Pseudo-laplaciens. I, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xiii, 275–286 (French, with English summary). MR 688031
- Yves Colin de Verdière, Pseudo-laplaciens. II, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 87–113 (French). MR 699488
- D. Farmer and S. Lemurell, in preparation.
- Dennis A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992), no. 469, vi+165. MR 1106989, DOI 10.1090/memo/0469
- Dennis A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, Emerging applications of number theory (Minneapolis, MN, 1996) IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 291–315. MR 1691537, DOI 10.1007/978-1-4612-1544-8_{1}1
- D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for $\textrm {PSL}(2,\mathbf Z)$, Math. Comp. 61 (1993), no. 203, 245–267, S11–S16. MR 1199991, DOI 10.1090/S0025-5718-1993-1199991-8
- Henryk Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matemática Iberoamericana. [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1995. MR 1325466
- Wenzhi Luo, Nonvanishing of $L$-values and the Weyl law, Ann. of Math. (2) 154 (2001), no. 2, 477–502. MR 1865978, DOI 10.2307/3062104
- Yiannis N. Petridis, Perturbation of scattering poles for hyperbolic surfaces and central values of $L$-series, Duke Math. J. 103 (2000), no. 1, 101–130. MR 1758241, DOI 10.1215/S0012-7094-00-10316-X
- R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $\textrm {PSL}(2,\textbf {R})$, Invent. Math. 80 (1985), no. 2, 339–364. MR 788414, DOI 10.1007/BF01388610
- R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), no. 6, 853–866. MR 812352, DOI 10.1002/cpa.3160380614
- R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), no. 1, 1–32. MR 1127079, DOI 10.1090/S0894-0347-1992-1127079-X
- Peter Sarnak, On cusp forms, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 393–407. MR 853570, DOI 10.1090/conm/053/853570
- P. Sarnak, On cusp forms. II, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 237–250. MR 1159118
- Kisao Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612. MR 398991, DOI 10.2969/jmsj/02740600
- Scott A. Wolpert, Disappearance of cusp forms in special families, Ann. of Math. (2) 139 (1994), no. 2, 239–291. MR 1274093, DOI 10.2307/2946582
Additional Information
- D. W. Farmer
- Affiliation: American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94307
- MR Author ID: 341467
- Email: farmer@aimath.org
- S. Lemurell
- Affiliation: Chalmers University of Technology, SE-412 96 Göteborg, Sweden
- Email: sj@math.chalmers.se
- 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) - © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 74 (2005), 1967-1982
- MSC (2000): Primary 11F03; Secondary 11F30
- DOI: https://doi.org/10.1090/S0025-5718-05-01746-1
- MathSciNet review: 2164106