On the distribution of eigenvalues of Maass forms on certain moonshine groups

Jorgenson, Jay; Smajlović, Lejla; Then, Holger

doi:10.1090/S0025-5718-2014-02823-8

On the distribution of eigenvalues of Maass forms on certain moonshine groups
HTML articles powered by AMS MathViewer

by Jay Jorgenson, Lejla Smajlović and Holger Then PDF

Math. Comp. 83 (2014), 3039-3070 Request permission

Abstract:

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma _0(N)^+$, where $N$ is a positive, square-free integer. After we prove that $\Gamma _0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an “average” Weyl’s law for the distribution of eigenvalues of Maass forms, from which we prove the “classical” Weyl’s law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\Gamma _0(5)^+$ than for $\Gamma _0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl’s laws. In addition, we employ Hejhal’s algorithm, together with recently developed refinements from [H. Then, Computing large sets of consecutive Maass forms, in preparation], and numerically determine the first $3557$ eigenvalues of $\Gamma _0(5)^+$ and the first $12474$ eigenvalues of $\Gamma _0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

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
M. V. Berry and M. Tabor, Closed orbits and the regular bound spectrum, Proc. Roy. Soc. London Ser. A 349 (1976), no. 1656, 101–123. MR 471721, DOI 10.1098/rspa.1976.0062
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 10.1103/PhysRevLett.69.1477
Oriol Bohigas, Marie-Joya Giannoni, and Charles Schmit, Spectral fluctuations, random matrix theories and chaotic motion, Stochastic processes in classical and quantum systems (Ascona, 1985) Lecture Notes in Phys., vol. 262, Springer, Berlin, 1986, pp. 118–138. MR 870168, DOI 10.1007/3540171665_{5}9
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 10.1103/PhysRevLett.69.2188
Andrew R. Booker, Turing and the Riemann hypothesis, Notices Amer. Math. Soc. 53 (2006), no. 10, 1208–1211. MR 2263990
Andrew R. Booker and Andreas Strömbergsson, Numerical computations with the trace formula and the Selberg eigenvalue conjecture, J. Reine Angew. Math. 607 (2007), 113–161. MR 2338122, DOI 10.1515/CRELLE.2007.047
A. R. Booker and A. Strömbergsson, Theoretical and practical aspects of Maass form computations, in preparation.
Andrew R. Booker, Andreas Strömbergsson, and Akshay Venkatesh, Effective computation of Maass cusp forms, Int. Math. Res. Not. , posted on (2006), Art. ID 71281, 34. MR 2249995, DOI 10.1155/IMRN/2006/71281
John Conway, John McKay, and Abdellah Sebbar, On the discrete groups of Moonshine, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2233–2240. MR 2052398, DOI 10.1090/S0002-9939-04-07421-0
C. J. Cummins and T. Gannon, Modular equations and the genus zero property of moonshine functions, Invent. Math. 129 (1997), no. 3, 413–443. MR 1465329, DOI 10.1007/s002220050167
C. J. Cummins, Congruence subgroups of groups commensurable with $\textrm {PSL}(2,\Bbb Z)$ of genus 0 and 1, Experiment. Math. 13 (2004), no. 3, 361–382. MR 2103333, DOI 10.1080/10586458.2004.10504547
C. J. Cummins, Fundamental domains for genus-zero and genus-one congruence subgroups, LMS J. Comput. Math. 13 (2010), 222–245. MR 2671533, DOI 10.1112/S1461157008000041
J. S. Friedman, J. Jorgenson, and J. Kramer, An effective bound for the Huber constant for cofinite Fuchsian groups, Math. Comp. 80 (2011), no. 274, 1163–1196. MR 2772118, DOI 10.1090/S0025-5718-2010-02430-5
Terry Gannon, Moonshine beyond the Monster, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2006. The bridge connecting algebra, modular forms and physics. MR 2257727, DOI 10.1017/CBO9780511535116
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
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, 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
Heinz Helling, Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z. 92 (1966), 269–280 (German). MR 228437, DOI 10.1007/BF01112194
M. N. Huxley, Scattering matrices for congruence subgroups, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 141–156. MR 803366
Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR 1942691, DOI 10.1090/gsm/053
Jay Jorgenson and Jürg Kramer, On the error term of the prime geodesic theorem, Forum Math. 14 (2002), no. 6, 901–913. MR 1932525, DOI 10.1515/form.2002.040
J. Jorgenson and L. Smajlović, On the distribution of zeros of the derivative of the Selberg’s zeta function associated to finite volume Riemann surfaces, arXiv:1302.5928.
Mong-Lung Lang, The signature of $\Gamma ^+_0(n)$, J. Algebra 241 (2001), no. 1, 146–185. MR 1838848, DOI 10.1006/jabr.2001.8756
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183 (German). MR 31519, DOI 10.1007/BF01329622
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
Peter Sarnak, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441–478. MR 1997348, DOI 10.1090/S0273-0979-03-00991-1
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Fredrik Strömberg, Maass waveforms on $(\Gamma _0(N),\chi )$ (computational aspects), Hyperbolic geometry and applications in quantum chaos and cosmology, London Math. Soc. Lecture Note Ser., vol. 397, Cambridge Univ. Press, Cambridge, 2012, pp. 187–228. MR 2895234
A. Strömbergsson, A pullback algorithm for general (cofinite) Fuchsian groups, (2000), http://www2.math.uu.se/~astrombe/papers/pullback.ps.
Holger Then, Maass cusp forms for large eigenvalues, Math. Comp. 74 (2005), no. 249, 363–381. MR 2085897, DOI 10.1090/S0025-5718-04-01658-8
H. Then, Computing large sets of consecutive Maass forms, in preparation.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953), 99–117. MR 55785, DOI 10.1112/plms/s3-3.1.99