Deformation of Γ₀(5)-cusp forms

Avelin, Helen

doi:10.1090/S0025-5718-06-01911-9

Deformation of $\Gamma _0(5)$-cusp forms
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by Helen Avelin;

Math. Comp. 76 (2007), 361-384

DOI: https://doi.org/10.1090/S0025-5718-06-01911-9

Published electronically: October 4, 2006

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Abstract:

We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods, we study the spectrum of the Laplace–Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of $\Gamma _0(5)$-cusp forms is presented.

References

A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
H. Avelin, Research announcement on the deformation of cusp forms, U.U.D.M. Report 2002:26, Uppsala, 21 pp., 2002.
—, Computer programs and data, 2003, http://www.math.uu.se/staff/pages/?uname=helen.
—, On the deformation of cusp forms, U.U.D.M. Report 2003:8, Uppsala, 85 pp., 2003.
—, Some computations used to prove that ${\Gamma }_{a,r}$ is a Fuchsian group, 2004, Maple-file http://www.math.uu.se/staff/pages/?uname=helen.
—, Zeros and coefficients, 2004, http://www.math.uu.se/staff/pages/?uname=helen.
—, Values of taylor coefficients, 2005, http://www.math.uu.se/staff/pages/?uname=helen.
Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
A. Booker, A. Strömbergsson, and A. Venkatesh, Effective computations with Maass forms, http://www.math.uu.se/~astrombe/papers.html, preprint, 2005.
Yves Colin de Verdière, Pseudo-laplaciens. I, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xiii, 275–286 (French, with English summary). MR 688031, DOI 10.5802/aif.890
Yves Colin de Verdière, Pseudo-laplaciens. II, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 87–113 (French). MR 699488, DOI 10.5802/aif.917
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779
K. G. Valēēv, Linear differential equations with sinusoidal coefficients and stationary retardations of the argument, Qualitative methods in the theory of non-linear vibrations (Proc. Internat. Sympos. Non-linear Vibrations, Vol. II, 1961) Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1961, pp. 100–119 (Russian, with English summary). MR 164106
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
Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,\,\textbf {R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197, DOI 10.1007/BFb0061302
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
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
Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
Bernard Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971), 219–230. MR 297997, DOI 10.1016/S0001-8708(71)80003-8
M. H. Millington, On cycloidal subgroups of the modular group, Proc. London Math. Soc. (3) 19 (1969), 164–176. MR 248232, DOI 10.1112/plms/s3-19.1.164
Toshitsune Miyake, On automorphic forms on $\textrm {GL}_{2}$ and Hecke operators, Ann. of Math. (2) 94 (1971), 174–189. MR 299559, DOI 10.2307/1970741
Toshitsune Miyake, Modular forms, Reprint of the first 1989 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda. MR 2194815
J. Marklof and A. Strömbergsson, Equidistribution of Kronecker sequences along closed horocycles, Geom. Funct. Anal. 13 (2003), no. 6, 1239–1280. MR 2033838, DOI 10.1007/s00039-003-0445-4
Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797–809. MR 961614, DOI 10.1090/S0002-9947-1988-0961614-2
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
Morten S. Risager, Asymptotic densities of Maass newforms, J. Number Theory 109 (2004), no. 1, 96–119. MR 2098479, DOI 10.1016/j.jnt.2004.06.007
Anthony Ralston and Philip Rabinowitz, A first course in numerical analysis, 2nd ed., Dover Publications, Inc., Mineola, NY, 2001. MR 1828483
Peter Sarnak, Arithmetic quantum chaos, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 183–236. MR 1321639
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
—, Göttingen Lectures on Harmonic Analysis, in Collected Papers, vol. 1, Springer-Verlag, 1989, pp. 626–674.
G. Steil, Eigenvalues of the Laplacian and of the Hecke Operators for $PSL(2,{\mathbb Z})$, DESY report 94-028, 1994.
F. Strömberg, Computations of Maass waveforms in non-congruence subgroups of the modular group, in Computational aspects of Maass waveforms, Ph.D. Thesis, Uppsala University, Uppsala, Sweden, 2005, pp. 107–123.
A. Strömbergsson, An application of an explicit trace formula to a well-known spectral correspondence on quaternion groups, 2000, Chapter 3 of Ph.D. Thesis, http://www.math.uu.se/~astrombe/papers.html.
—, A pullback algorithm for general (cofinite) Fuchsian groups., 2000, http://www.math.uu.se/~astrombe/papers.html, technical note.
Andreas Strömbergsson, Some remarks on a spectral correspondence for Maass waveforms, Internat. Math. Res. Notices 10 (2001), 505–517. MR 1832538, DOI 10.1155/S1073792801000265
H. Then, Arithmetic quantum chaos of Maass waveforms, Ulm Report ULM-TP/03-4, Ulm, 30 pp., 2003.
A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172 (Russian). MR 665585
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944. MR 10746