Parallel computation of maass cusp forms using mathematica - Chan, Kar Tim (2013)
Spectral studies on the eigenfunctions of Laplace-Beltrami operator on a cusp hyperbolic surface are known to contain both continuous and discrete eigenvalues. While the continuous eigenvalues are known analytically, whose eigenfunctions are usually spanned by the Eisenstein series, it is more subtle to solve for the discrete ones which can only be found numerically where the eigenfunctions are described by the Maass cusp forms. The main aim of this research is to compute the discrete eigenvalues and visualize the eigenfunctions for the modular group, commutator subgroup and principal congruence subgroup of level two in a parallel computing environment using GridMathematica software. Our parallel programme comprises of two important parts namely the pullback algorithm and also the Maass cusp form algorithm. The latter is developed using an adapted algorithm of Hejhal and Then which is based on implicit automorphy and finite Fourier series. This algorithm applies to the computation of Maass cusp forms on Fuchsian group whose fundamental domain has only one cusp namely the modular group and the commutator subgroup. Special attention is given to the computation of eigenvalues for the modular group because this part is intended to serve as the basis for further development of computation for the more complex surfaces. This parallel programme is further modified using a generalized Hejhal’s algorithm to cater for fundamental domain that has several cusps namely the principal congruence subgroup of level two. To facilitate the complete pullback process of this group, a point locater algorithm is developed. In this work, we present three different pullback algorithms for the surfaces we considered and carefully integrate them into our Maass cusp form algorithm. With it, we manage to compute 190 eigenvalues for the modular group where 111 belong to the odd class and 79 belong to the even class. The computational accuracy of the eigenvalues is expected to be accurate at least up to nine decimal places since the tolerance for the bisection module is set as 10−10. For the commutator subgroup, we manage to compute 104 eigenvalues where 52 belong to the odd class and 52 belong to the even class. For the principal congruence subgroup of level two, 20 lower lying eigenvalues are computed. From these eigenvalues, 11 belong to the odd class and nine belong to the even class. The tolerance of the bisection module for these two subgroups are set as 10−9 and 10−6 respectively. As such, the computational accuracy of the eigenvalues are expected to be accurate at least up to eight for the former and five decimal places for the latter. Eigenvalues from these surfaces are checked using selected procedures such as y independent solution, automorphy condition, Hecke relation and RamanujanPetersson conjecture for their authenticity. Later, we visualize the eigenstates of selected eigenvalues from each surface using GridMathematica. Some features that appear in the plots are explained. We have also compared the performance of parallel programming and normal programming here in order to justify the feasibility and advantages of using the parallel version of commercially available software for complex computations of Maass cusp forms. We find that the parallel programming is about 5.75 times faster than the normal programming while its efficiency is capped at 0.443.