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
Published electronically: March 23, 2004
MathSciNet review: 2085897
Full-text PDF

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?)

  • [AS65] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1965. MR 34:8606
  • [ASD71] 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. (1971), 1-25. MR 49:2550
  • [Ave03] H. Avelin, On the deformation of cusp forms (Licentiate Thesis), UUDM report 2003:8 (Uppsala 2003).
  • [Bal66] C. 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 32:5942
  • [Bal67] C. B. Balogh, Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math. 15 (1967), no. 5, 1315-1323. MR 36:5406
  • [Ber77] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), 2083-2091. MR 58:8961
  • [BGGS92] E. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69 (1992), 1477-1480. MR 93g:81025
  • [BLS96] E. Bogomolny, F. Leyvraz, and C. Schmit, Distribution of eigenvalues for the modular group, Comm. Math. Phys. 176 (1996), no. 3, 577-617. MR 97k:81052
  • [Bol93] J. Bolte, Some studies on arithmetical chaos in classical and quantum mechanics, Int. J. Mod. Phys. B 7 (1993), 4451-4553. MR 94j:81049
  • [BSS92] J. Bolte, G. Steil, and F. Steiner, Arithmetical chaos and violation of universality in energy level statistics, Phys. Rev. Lett. 69 (1992), 2188-2191. MR 93f:81039
  • [Car71] P. Cartier, Some numerical computations relating to automorphic functions, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, 1971, pp. 37-48. MR 47:3285
  • [Car78] P. Cartier, Analyse numérique d'un problème de valeurs propres a haute précision [application aux fonctions automorphes], preprint, IHES (1978).
  • [CFU57] C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Camb. Phil. Soc. 53 (1957), 599-611. MR 19:853a
  • [CGS91] A. Csordás, R. Graham, and P. Szépfalusy, Level statistics of a noncompact cosmological billiard, Phys. Rev. A 44 (1991), 1491-1499.
  • [GS82] V. V. Golovcanskii 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).
  • [GS84] V. V. Golovcanskii 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, Akad. Nauk SSSR, Dal. nevostochn. Nauchn. Tsentr, Vladivostok 85 (1984), 15-19 (Russian). MR 87i:11068
  • [GST02] 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).
  • [HA93] D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for $\operatorname{PSL}(2,\mathbb{Z})$, Math. Comp. 61 (1993), 245-267. MR 94a:11062
  • [Haa77] 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.
  • [HB82] 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.
  • [Hej81] D. A. Hejhal, Some observations concerning eigenvalues of the Laplacian and Dirichlet L-series, Recent Progress in Analytic Number Theory (H. Halberstam and C. Hooley, eds.), Academic Press, 1981, pp. 95-110. MR 82k:10005b
  • [Hej83] D. A. Hejhal, The Selberg trace formula for $\operatorname{PSL}(2,\mathbb{R})$, Lecture Notes in Math. 1001, Springer, 1983. MR 86e:11040
  • [Hej91] D. A. Hejhal, Eigenvalues for the Laplacian for $\operatorname{PSL}(2,\mathbb{Z})$: some new results and computational techniques, International Symposium in Memory of Hua Loo-Keng (S. Gong, Q. K. Lu, Y. Wang, and L. Yang, eds.), Springer, 1991, pp. 59-102. MR 92j:11048
  • [Hej92a] D. A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 469 (1992). MR 93f:11043
  • [Hej92b] D. A. Hejhal, On eigenvalues of the Laplacian for Hecke triangle groups, Zeta Functions in Geometry (N. Kurokawa and T. Sunada, eds.), vol. 21, Adv. Stud. Pure Math., 1992, pp. 359-408. MR 95f:11065
  • [Hej99] D. A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, Emerging applications of number theory (D. A. Hejhal, J. Friedman, M. C. Gutzwiller, and A. M. Odlyzko, eds.), IMA Series No. 109, Springer, 1999, pp. 291-315. MR 2000f:11063
  • [HR92] D. A. Hejhal and B. Rackner, On the topography of Maass waveforms for $\operatorname{PSL}(2,\mathbb{Z})$, Experiment. Math. 1 (1992), 275-305. MR 95f:11037
  • [HS01] D. A. Hejhal and A. Strömbergsson, On quantum chaos and Maass waveforms of CM-type, Found. Phys. 31 (2001), no. 3, 519-533. MR 2003k:81066
  • [Hun91] W. Huntebrinker, Numerische Bestimmung von Eigenwerten des Laplace-Operators auf hyperbolischen Räumen mit adaptiven Finite-Element-Methoden, Bonner Mathematische Schriften 225 (1991). MR 93f:65079
  • [Iwa95] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Revista Matemática Iberoamericana, 1995. MR 96f:11078
  • [Kub73] T. Kubota, Elementary Theory of Eisenstein Series, Kodansha, Tokyo and Halsted Press, 1973. MR 55:2759
  • [Maa49] H. Maaß, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141-183. MR 11:163c
  • [Maa64] H. Maaß, Lectures on Modular Functions of one Complex Variable, Tata Institute of Fundamental Research, 1964, Springer, Revised 1983. MR 85g:11034
  • [Miy89] T. Miyake, Modular forms, Springer, 1989. MR 90m:11062
  • [MOS66] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, 1966. MR 38:1291
  • [Olv52] F. W. J. Olver, Some new asymptotic expansions for Bessel functions of large orders, Proc. Camb. Phil. Soc. 48 (1952), 414-427. MR 14:45b
  • [Olv54] F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Phil. Trans. A 247 (1954), 328-368. MR 16:696a
  • [Roe66] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, Math. Ann. 167 (1966), 292-337 and 168 (1967), 261-324. MR 39:4386
  • [Sar95] P. Sarnak, Arithmetic quantum chaos, Israel Math. Conf. Proc. 8 (1995), 183-236. MR 96d:11059
  • [Sch54] W. Schöbe, Eine an die Nicholsonformel anschließende asymptotische Entwicklung für Zylinderfunktionen, Acta. Math. 92 (1954), 265-307. MR 16:696b
  • [Sch91] C. Schmit, Triangular billiards on the hyperbolic plane: Spectral properties, preprint, IPNO/TH 91-68 (1991).
  • [Sel56] 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. MR 19:531g
  • [SS02] B. Selander and A. Strömbergsson, Sextic coverings of genus two which are branched at three points, UUDM report 2002:16 (Uppsala 2002).
  • [Sta84] H. M. Stark, Fourier coefficients of Maass waveforms, Modular Forms (R. A. Rankin, ed.), Ellis Horwood, 1984, pp. 263-269. MR 87h:11128
  • [Ste92] 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.
  • [Ste94] G. Steil, Eigenvalues of the Laplacian and of the Hecke operators for $\operatorname{PSL}(2,\mathbb{Z})$, DESY report 94-28 (Hamburg 1994).
  • [Ter85] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, vol. 1, Springer, 1985. MR 87f:22010
  • [Ven90] A. B. Venkov, Spectral Theory of Automorphic Functions and Its Applications, Kluwer Academic Publishers, 1990. MR 93a:11046
  • [Vig83] M.-F. Vignéras, Quelques remarques sur la conjecture $\lambda_1\ge\frac{1}{4}$, Séminaire de Théorie des Nombres (M.-J. Bertin, ed.), Birkhäuser, 1983, pp. 321-343. MR 85c:11049
  • [Wat44] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1944. MR 6:64a
  • [Win88] A. M. Winkler, Cusp forms and Hecke groups, J. Reine Angew. Math. 386 (1988), 187-204. MR 90g:11067

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

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