Maass Waveforms
Introduction and definitions
Let \mathcal{H} be the hyperbolic upper half-plane with metric ds=\frac{1}{y}|dz| and measure d\mu = \frac{dxdy}{y^{2}}, \Gamma \subseteq PSL(2,\mathbb{R}) a (cofinite) Fuchsian group with \mathcal{M}=\mathcal{H} / \Gamma and \chi: \Gamma \mapsto S^{1} a character of \Gamma. A Maass waveform for (\Gamma,\chi) is a function \phi: \mathcal{H} \mapsto \mathbb{C} with the following properties:
\phi(\gamma z) = \chi(\gamma) \phi(z), \forall \gamma \in \Gamma,
(\Delta + \lambda) \phi = 0,
\int _{ \mathcal{H} / \Gamma } | \phi |^{2} d\mu < \infty
where \Delta = y^{2} \left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right) is the Laplace-Beltrami operator corresponding to the hyperbolic metric given above. Note that \Delta is an elliptic differential operator so \phi is automatically real analytic. Furthermore \Delta admits a self-adjoint extension to the space L^2(\mathcal{M}). Thus the eigenvalue \lambda=s(1-s)=1/4+R^2 is non-negative, meaning that s \in \frac{1}{2}+i\mathbb{R} \cup [0,1].
Historically the Maass waveforms were first introduced by Maass in [1,2] and subsequently studied by several authors. Most prominent of those were A.Selberg [3] who developed his celebrated "Selberg Trace Formula" (see below) which he used to show that there exist an infinite number of Maass waveforms for the modular group.
Spectral Decomposition
It is known that the spectrum of the (self-adjoint extension to L^{2}(\mathcal{M}) of the) Laplacian on \mathcal{M} consists of a continuous spectrum [0,\infty) with the Multiplicity equal to the number of open cusps, \kappa_0 together with a discrete spectrum embedded in the same interval. Here a cusp v of \Gamma is said to be "open" for (\Gamma,\chi) if v(T)=1 for T\in \Gamma_v, the stabilizer subgroup of v in \Gamma.
The space L^{2}(\mathcal{M}) can be decomposed into \Delta-invariant subspaces corresponding to the spectrum of \Delta.
where {\varphi_{n}} denotes the set of all Maass waveforms counted with multiplicity. Here (\Delta+\lambda_{n})\varphi_{n}=0 and 0<\lambda_1<\cdots \rightarrow \infty. The continuous part of the spectrum corresponds to the space $\mathcal{E} spanned by Eisenstein series (eigenpackets).
Physical Interpretation and Applications
A Maass waveform (as defined above), i.e. an eigenfunction of the Laplacian which is square-integrable can be thought of as the (time-independent) distribution function of a quantum mechanical particle moving freely on the surface \mathcal{M}.
- Knowing a large consecutive set of Maass waveforms for groups acting in hyperbolic three space allows to compute the cosmic microwave background temperature fluctuations of hyperbolic universes [10].
The Selberg Trace Formula
The Selberg Eigenvalue Conjecture
Computational Aspects of Maass waveforms
The only known explicit examples of Maass waveforms comes are realized as lifts of Hecke Grössen characters and were introduced already by Maass in e.g. [2]. Sometimes these Forms are denoted "CM forms" or "CM-like forms" in analogy with the classical/holomorphic case. See also the paper by Hejhal and Strömbergsson [4]. All other examples are computed numerically.
There are some "main" algorithms for finding Maass waveforms
- the "implicit automorphy" algorithm in the version by Hejhal generalized by Strömberg to subgroups of the modular group and arbitrary weight and multiplier system (character). This algorithm also produces Fourier coefficients (Hecke eigenvalues). Implementationa are done in Fortran 77/95.
Online: Program that computes with Maass waveforms on Hecke congruence subgroups and real Dirichlet characters. http://www.mathematik.tu-darmstadt.de/~stroemberg/prog/
the "implicit automorphy" algorithm in the version by Then [8,9] that is based on the main idea of Hejhal's algorithm [7], but is structured differently and is highly optimized with respect to speed and finite computer resources. The algorithm computes Maass cusp forms for finite, but non-compact groups acting in hyperbolic n-space (n\ge2). Implemented in C.
- another kind of automorphy algorithm by Farmer-Lemurell. Implementation in Mathematika.
- trace formula algorithm by Booker-Strömbergsson. Implemented in Pari.
Available Data Sets (online)
- Booker-Strömbergsson:
- First 2000 eigenvalues on PSL(2,Z) (rigorously computed)
- Small eigenvalues on Gamma_0(N) with Dirichlet character and N large?.
Online: Some ev/coeff to high accuracy on PSL(2,Z): http://www.math.uu.se/~astrombe/emaass/emaass.html
- Farmer-Lemurell: Lots of eigenvalues on Gamma_0(N) for small to moderate N.
Online: nothing yet.
- Hejhal: Eigenvalues on PSL(2,Z) and Hecke triangle groups G_q (q=4,5,6,7 at least).
- Strömberg:
- Eigenvalues on Gamma_0(N) for N up to 109. With Dirichlet character.
- Larger sets for certain N, e.g. 5.
- Fourier coefficients of forms on e.g. N=5 with the quadratic Dirichlet character.
- Eigenvalues and Fourier coefficients on PSL(2,Z) with arbitrary real weight and eta multiplier.
- Eigenvalues/coefficients for Gamma_0(4) weight 1/2, 3/2 with theta multiplier.
Online: Eigenvalues on Gamma_0(N) for N up to 30 and R up to 10. http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg/research/data/.
Online: Miscellaneous data/pictures of Maass waveforms. See e.g. http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg/research/gallery/.
- Then:
- More than 50000 eigenvalues corresponding to Maass cusp forms on PSL(2,Z). (About 110 eigenvalues are still missing in order to make the list of the 50000 eigenvalues to be a consecutive list. The first 2000 of these eigenvalues agree within an accuracy of at least 10 digits after the decimal point with the eigenvalues found by Booker-Strömbergsson. The accuracy of the expansion coefficients of the corresponding Maass cusp forms has been checked independently using Hecke operators).
A few eigenvalues corresponding to Maass cusp forms on PSL(2,Z) that exceed the 130millionth
eigenvalue. (Accuracy of expansion coefficients checked with Hecke operators). - First 13950 eigenvalues on PSL(2,Z[i]). (Consecutive list of eigenvalues. Accuracy of expansion coefficients checked using Hecke operators).
Some more eigenvalues on PSL(2,\cal O), where \cal O is a ring of complex integers.
Online: http://www.staff.uni-oldenburg.de/holger.then/downloads/
References
[1] Maass, H., Über automorphe Funktionen von mehreren Veränderlichen und die Bestimmung von Dirichletschen Reihen durch Funktionalgleichungen. Ber. Math.-Tagung Tübingen 1946 (1946), 100--102 (1947).
[2] Maass, H., Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., vol. 121, pp. 141--183, (1949).
[3] Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), vol. 20, no. MR0088511 (19,531g), pp. 47--87, (1956) http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=259031&r=24
[4] Hejhal, Dennis A.(S-UPPS); Strömbergsson, Andreas(S-UPPS)On quantum chaos and Maass waveforms of CM-type. (English summary) Invited papers dedicated to Martin C. Gutzwiller, Part IV. Found. Phys. 31(2001),no. 3, 519--533.
[5] Strömberg, F., Maass waveforms on $(Gamma0(N),\chi)$ (Computational aspects).
[6] Strömberg, F., Computation of Maass waveforms with non-trivial multiplier system.
[7] Hejhal, D. A., On eigenfunctions of the Laplacian for Hecke triangle groups. In Emerging Applications in Number Theory (Hejhal, D. A., Friedman, J., Gutzwiller, M. C., and Odlyzko, A. M., eds.), IMA Series No. 109, Springer, 1999, pp, 291--315.
[8] Then, H., Maass cusp forms for large eigenvalues, Math. Comp. 74, 363--381 (2004).
[9] Then, H., Arithmetic quantum chaos of Maass waveforms. In Frontiers in Number Theory, Physics, and Geometry I (Cartier, P., Julia, B., Moussa, P., and Vanhove, P., eds.), Springer, 2006, pp 183--212.
[10] Aurich, R., Steiner, F., and Then, H., Numerical computation of Maass waveforms and an application to cosmology. To appear in Lecture Notes in Physics, Springer.