×

Eigenvalues of the Laplacian for Hecke triangle groups. (English) Zbl 0746.11025

Mem. Am. Math. Soc. 469, 165 p. (1992).
The work under review splits into two parts I, II which shed new light on the computational aspects of the eigenvalue problem for automorphic forms on the upper half plane \(H\).
The object of part I is to report on a series of experiments aimed at computing the eigenvalues of the hyperbolic Laplacian for some Hecke triangle groups \({\mathfrak G}(2\cos\pi /N)\). The numerical results support the (still unproven) Phillips-Sarnak conjecture. Since the odd portion of \(L^ 2({\mathfrak G}(2\cos\pi /N)\backslash H)\) is discrete, the Phillips- Sarnak conjecture claims here: For \(N\neq 3,4,6\), the Hecke group \({\mathfrak G}(2\cos\pi /N)\) admits no even cusp forms. The contrast between even/odd and arithmetic/nonarithmetic in the numerical results is striking. To illustrate various aspects of the computation, 17 examples are discussed in some detail. Besides the usual eigenvalues and eigenfunctions the author considers the related concept of pseudo cusp form (with weak singularities). The numerical results suggest that pseudo cusp forms with a single logarithmic singularity at \(\exp(\pi i/N)\) may be characterized by an extremal property. The author expects that the space of pseudo cusp forms may well enjoy a richer structure than previously thought. – The numerical strategy is basically one of achieving high probability (as opposed to rigorous and error-controlled algorithms). The numerical procedure is explained, the program used is given and the possible traps are discussed. Part I is the first part of a projected series of papers by the author dealing with computational aspects of the eigenvalue problem for the Hecke groups.
Part II of the Memoir under review is a second printing of the author’s paper [Eigenvalues of the Laplacian for \(PSL(2,\mathbb{Z})\): some new results and computational techniques. Int. Symp. in Memory of Hua Loo-Keng, Vol. 1, 59-102 (1991)]. The general remarks pertaining to part I apply equally well to part II. Recent work in quantum chaos and the Riemann zeta function suggests that it may be interesting to compute the eigenvalues and eigenfunctions of the Laplacian for \(PSL(2,\mathbb{Z})\) for much larger values of \(R\) (\(\lambda = {1\over 4}+R^ 2\)) than the previous bound which is approximately 25. A new program for the Bessel functions \(K_{iR}(X)\) developed by E. Bombieri and the author works satisfactorily well up to \(R=75000\) or more and this enables the author to extend the previous calculations considerably: It is now possible to sample \(\lambda_ n\) all the way out to \(\lambda=250 000\) or so. The paper contains tables of the even/odd eigenvalues less than \(R=50\), the Fourier coefficients of the even/odd eigenfunctions less than 35, the even eigenvalues around \(R=125\) and some associated Fourier coefficients, the even eigenvalues around \(R=250\) and some corresponding Fourier coefficients, and the even eigenvalues around \(R=500\) and some corresponding Fourier coefficients. The results of the author’s experiments are that the barrier with the \(K\)-Bessel functions now is pushed far ahead and that the computational difficulties are more in line with linear algebra.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F30 Fourier coefficients of automorphic forms
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
11-04 Software, source code, etc. for problems pertaining to number theory
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
35P15 Estimates of eigenvalues in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI