Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Maaß cusp forms for large eigenvalues


Author: Holger Then
Journal: Math. Comp. 74 (2005), 363-381
MSC (2000): Primary 11F72, 11F30; Secondary 11F12, 11Yxx, 11-4, 81Q50
DOI: https://doi.org/10.1090/S0025-5718-04-01658-8
Published electronically: March 23, 2004
MathSciNet review: 2085897
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the numerical computation of Maaß cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed $r=40000$. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the $130$millionth eigenvalue.


References [Enhancements On Off] (What's this?)

  • Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
  • A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1971, pp. 1–25. MR 0337781
  • H. Avelin, On the deformation of cusp forms (Licentiate Thesis), UUDM report 2003:8 (Uppsala 2003).
  • Charles B. Balogh, Uniform asymptotic expansions of the modified Bessel function of the third kind of large imaginary order, Bull. Amer. Math. Soc. 72 (1966), 40–43. MR 188504, DOI https://doi.org/10.1090/S0002-9904-1966-11408-8
  • Charles B. Balogh, Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math. 15 (1967), 1315–1323. MR 222354, DOI https://doi.org/10.1137/0115114
  • M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083–2091. MR 489542
  • E. B. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69 (1992), no. 10, 1477–1480. MR 1178884, DOI https://doi.org/10.1103/PhysRevLett.69.1477
  • E. Bogomolny, F. Leyvraz, and C. Schmit, Distribution of eigenvalues for the modular group, Comm. Math. Phys. 176 (1996), no. 3, 577–617. MR 1376433
  • Jens Bolte, Some studies on arithmetical chaos in classical and quantum mechanics, Internat. J. Modern Phys. B 7 (1993), no. 27, 4451–4553. MR 1252073, DOI https://doi.org/10.1142/S0217979293003759
  • J. Bolte, G. Steil, and F. Steiner, Arithmetical chaos and violation of universality in energy level statistics, Phys. Rev. Lett. 69 (1992), no. 15, 2188–2191. MR 1184827, DOI https://doi.org/10.1103/PhysRevLett.69.2188
  • A. O. L. Atkin and B. J. Birch (eds.), Computers in number theory, Academic Press, London-New York, 1971. MR 0314733
  • P. Cartier, Analyse numérique d’un problème de valeurs propres a haute précision [application aux fonctions automorphes], preprint, IHES (1978).
  • C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Camb. Phil. Soc. 53 (1957), 599–611.
  • A. Csordás, R. Graham, and P. Szépfalusy, Level statistics of a noncompact cosmological billiard, Phys. Rev. A 44 (1991), 1491–1499.
  • V. V. Golovčanskiǐ and M. N. Smotrov, The first few eigenvalues of the Laplacian on the fundamental domain of the modular group, preprint, Far Eastern Scientific Center, Academy of Science USSR, Wladiwostok (1982) (Russian).
  • V. V. Golovchanskiĭ and M. N. Smotrov, Calculation of first Fourier coefficients of eigenfunctions of the Laplace operator on the fundamental domain of a modular group, Numerical methods in algebra and analysis (Russian), Akad. Nauk SSSR, Dal′nevostochn. Nauchn. Tsentr, Vladivostok, 1984, pp. 15–19, 85 (Russian). MR 858137
  • A. Gil, J. Segura, and N. M. Temme, Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion, CWI report MAS-R0205 (2002).
  • D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for ${\rm PSL}(2,\mathbf Z)$, Math. Comp. 61 (1993), no. 203, 245–267, S11–S16. MR 1199991, DOI https://doi.org/10.1090/S0025-5718-1993-1199991-8
  • H. Haas, Numerische Berechnung der Eigenwerte der Differentialgleichung $-{\Delta } u=\lambda y^{-2} u$ für ein unendliches Gebiet im $\mathbb {R}^2$, 1977, Diplomarbeit, Universität Heidelberg, Institut für Angewandte Mathematik.
  • D. A. Hejhal and B. Berg, Some new results concerning eigenvalues of the non-Euclidean Laplacian for $\operatorname {PSL}(2,\mathbb {Z})$, Tech. report 82-172, University of Minnesota, 1982.
  • H. Halberstam and C. Hooley (eds.), Recent progress in analytic number theory. Vol. 1, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. Papers from the Symposium held at Durham University, Durham, July 22–August 1, 1979. MR 637338
  • Dennis A. Hejhal, The Selberg trace formula for ${\rm PSL}(2,\,{\bf R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197
  • Dennis A. Hejhal, Eigenvalues of the Laplacian for ${\rm PSL}(2,{\bf Z})$: some new results and computational techniques, International Symposium in Memory of Hua Loo Keng, Vol. I (Beijing, 1988) Springer, Berlin, 1991, pp. 59–102. MR 1135805
  • Dennis A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992), no. 469, vi+165. MR 1106989, DOI https://doi.org/10.1090/memo/0469
  • Dennis A. Hejhal, On eigenvalues of the Laplacian for Hecke triangle groups, Zeta functions in geometry (Tokyo, 1990) Adv. Stud. Pure Math., vol. 21, Kinokuniya, Tokyo, 1992, pp. 359–408. MR 1210796, DOI https://doi.org/10.2969/aspm/02110359
  • 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 https://doi.org/10.1007/978-1-4612-1544-8_11
  • Dennis A. Hejhal and Barry N. Rackner, On the topography of Maass waveforms for ${\rm PSL}(2,{\bf Z})$, Experiment. Math. 1 (1992), no. 4, 275–305. MR 1257286
  • Dennis A. Hejhal and Andreas Strömbergsson, On quantum chaos and Maass waveforms of CM-type, Found. Phys. 31 (2001), no. 3, 519–533. Invited papers dedicated to Martin C. Gutzwiller, Part IV. MR 1839791, DOI https://doi.org/10.1023/A%3A1017521729782
  • Wolfgang Huntebrinker, Numerische Bestimmung von Eigenwerten des Laplace-Operators auf hyperbolischen Räumen mit adaptiven Finite-Element-Methoden, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 225, Universität Bonn, Mathematisches Institut, Bonn, 1991 (German). MR 1182153
  • 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
  • Tomio Kubota, Elementary theory of Eisenstein series, Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York-London-Sydney, 1973. MR 0429749
  • H. Maaß, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183.
  • H. Maass, Lectures on modular functions of one complex variable, 2nd ed., Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 29, Tata Institute of Fundamental Research, Bombay, 1983. With notes by Sunder Lal. MR 734485
  • Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004
  • Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
  • F. W. J. Olver, Some new asymptotic expansions for Bessel functions of large orders, Proc. Camb. Phil. Soc. 48 (1952), 414–427.
  • F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Phil. Trans. A 247 (1954), 328–368.
  • Walter Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II, Math. Ann. 167 (1966), 292–337; ibid. 168 (1966), 261–324 (German). MR 0243062, DOI https://doi.org/10.1007/BF01361556
  • 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
  • W. Schöbe, Eine an die Nicholsonformel anschließende asymptotische Entwicklung für Zylinderfunktionen, Acta. Math. 92 (1954), 265–307.
  • C. Schmit, Triangular billiards on the hyperbolic plane: Spectral properties, preprint, IPNO/TH 91-68 (1991).
  • A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47–87.
  • B. Selander and A. Strömbergsson, Sextic coverings of genus two which are branched at three points, UUDM report 2002:16 (Uppsala 2002).
  • H. M. Stark, Fourier coefficients of Maass waveforms, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 263–269. MR 803370
  • G. Steil, Über die Eigenwerte des Laplaceoperators und der Heckeoperatoren für $\operatorname {SL}(2,\mathbb {Z})$, 1992, Diplomarbeit, Universität Hamburg, II. Institut für Theoretische Physik.
  • G. Steil, Eigenvalues of the Laplacian and of the Hecke operators for $\operatorname {PSL}(2,\mathbb {Z})$, DESY report 94-28 (Hamburg 1994).
  • Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406
  • Alexei B. Venkov, Spectral theory of automorphic functions and its applications, Mathematics and its Applications (Soviet Series), vol. 51, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian by N. B. Lebedinskaya. MR 1135112
  • Marie-France Vignéras, Quelques remarques sur la conjecture $\lambda _{1}\geq {1\over 4}$, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 321–343 (French). MR 729180
  • G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1944.
  • Andrew M. Winkler, Cusp forms and Hecke groups, J. Reine Angew. Math. 386 (1988), 187–204. MR 936998, DOI https://doi.org/10.1515/crll.1988.386.187

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11F72, 11F30, 11F12, 11Yxx, 11-4, 81Q50

Retrieve articles in all journals with MSC (2000): 11F72, 11F30, 11F12, 11Yxx, 11-4, 81Q50


Additional Information

Holger Then
Affiliation: Abteilung Theoretische Physik, Universität Ulm, 89069 Ulm, Germany
MR Author ID: 742378
ORCID: 0000-0002-0368-639X
Email: holger.then@physik.uni-ulm.de

Keywords: Automorphic forms, spectral theory, computational number theory, Fourier coefficients, explicit machine computation, multiplicative number theory, Hecke operators, Ramanujan-Petersson conjecture, Sato-Tate conjecture, quantum chaos, Berry conjecture, approximation of special functions, modified Bessel function
Received by editor(s): November 26, 2002
Received by editor(s) in revised form: July 4, 2003
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society