Effective computation of Maass cusp forms. (English) Zbl 1154.11018
Let \(\mathbb{H}\) be the hyperbolic upper half-plane and \(\Gamma\) a co-finite Fuchsian group. A Maass cusp form is a cuspidal eigenfunction of the Laplace-Beltrami operator, \(\Delta\), on \(\Gamma\backslash\mathbb{H}.\) Computations of Maass cusp forms has received an increasing popularity in recent years [see e.g. N. Strömberg, Uppsala Dissertations in Mathematics 39. Uppsala: Uppsala Univ., Department of Mathematics. (2005)], H. Then [Math. Comput. 74, No. 249, 363–381 (2005; Zbl 1112.11029)], D. W. Farmer and S. Lemurell [Math. Comput. 74, No. 252, 1967–1982 (2005; Zbl 1079.11024)] and H. Avelin [Math. Comput. 76, No. 257, 361–384 (2007; Zbl 1114.11049)] for a selection of the most recent results) and although the methods employed in the cited publications differ they all have one property in common, namely that they are heuristical in nature.
The paper under review presents a framework in which rigorous computations of Maass cusp forms can be performed, at least in the case when \(\Gamma\) is a subgroup of \(\text{PSL}_{2}\left(\mathbb{Z}\right)\). The main results of the paper are stated for \(\Gamma=\text{PSL}_{2}\left(\mathbb{Z}\right)\) but generalizations to congruence subgroups should be relatively straightforward, even if containing tedious and non-trivial computations. A brief sketch of the method is as follows: First use a heuristic method, e.g. Hejhal’s algorithm [cf. e.g. D. A. Hejhal, Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15–26, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 109, 291–315 (1999; Zbl 0982.11029)]) to obtain a set of tentative Maass cusp form data. From this data one constructs a quasimode and the final step is to show that the quasimode is close to a true eigenfunction.
As examples of applications of the techniques developed in the paper, they give a list of the ten first eigenvalues of \(\Delta\) on \(\text{PSL}_{2}\left(\mathbb{Z}\right) \backslash \mathbb{H}\) and state as a theorem that these values are correct to 100 decimal places.
The effectiveness of the computations is also addressed. The central theorem of the paper, Thm. 3.9, tells us that there is an algorithm which can certify a given (tentative) Maass cusp form in polynomial time{ }in the parameters involved.
In the last section of the paper, algebraicity of Fourier coefficients and eigenvalues of Maass cusp forms on \(\text{PSL}\left(2,\mathbb{Z}\right)\backslash\mathbb{H}\) is investigated. In agreement with previous conjectures the results indicate that the eigenvalues are transcendental. As an example of the results obtained we have that none of the first ten eigenvalues is the zero of a polynomial of degree less than or equal to \(10\) with integer coefficients which are smaller than or equal to \(10^{7}\) in absolute value.
Inspired by the relations between eigenvalues and Fourier coefficients of Maass forms of CM-type a particular form of algebraic relations between Fourier coefficients and eigenvalues is also investigated and the result is also here in the negative, no relation (up to some bounds of the involved polynomials) was found.
The paper under review presents a framework in which rigorous computations of Maass cusp forms can be performed, at least in the case when \(\Gamma\) is a subgroup of \(\text{PSL}_{2}\left(\mathbb{Z}\right)\). The main results of the paper are stated for \(\Gamma=\text{PSL}_{2}\left(\mathbb{Z}\right)\) but generalizations to congruence subgroups should be relatively straightforward, even if containing tedious and non-trivial computations. A brief sketch of the method is as follows: First use a heuristic method, e.g. Hejhal’s algorithm [cf. e.g. D. A. Hejhal, Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15–26, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 109, 291–315 (1999; Zbl 0982.11029)]) to obtain a set of tentative Maass cusp form data. From this data one constructs a quasimode and the final step is to show that the quasimode is close to a true eigenfunction.
As examples of applications of the techniques developed in the paper, they give a list of the ten first eigenvalues of \(\Delta\) on \(\text{PSL}_{2}\left(\mathbb{Z}\right) \backslash \mathbb{H}\) and state as a theorem that these values are correct to 100 decimal places.
The effectiveness of the computations is also addressed. The central theorem of the paper, Thm. 3.9, tells us that there is an algorithm which can certify a given (tentative) Maass cusp form in polynomial time{ }in the parameters involved.
In the last section of the paper, algebraicity of Fourier coefficients and eigenvalues of Maass cusp forms on \(\text{PSL}\left(2,\mathbb{Z}\right)\backslash\mathbb{H}\) is investigated. In agreement with previous conjectures the results indicate that the eigenvalues are transcendental. As an example of the results obtained we have that none of the first ten eigenvalues is the zero of a polynomial of degree less than or equal to \(10\) with integer coefficients which are smaller than or equal to \(10^{7}\) in absolute value.
Inspired by the relations between eigenvalues and Fourier coefficients of Maass forms of CM-type a particular form of algebraic relations between Fourier coefficients and eigenvalues is also investigated and the result is also here in the negative, no relation (up to some bounds of the involved polynomials) was found.
Reviewer: Fredrik Strömberg (Darmstadt)
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11F30 | Fourier coefficients of automorphic forms |
| 11Y35 | Analytic computations |
