Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
The topic of this study is the computation of Maass cusp form, i.e. the eigenfunctions of the hyperbolic Laplace-Beltrami operator on punctured surfaces namely singly punctured two-torus and triply punctured two-sphere. Punctured surfaces are surfaces with points removed or located infinitely far...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2009
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| Online Access: | http://psasir.upm.edu.my/id/eprint/7575/1/ABS_---__IPM_2009_8.pdf http://psasir.upm.edu.my/id/eprint/7575/ |
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| Summary: | The topic of this study is the computation of Maass cusp form, i.e. the eigenfunctions
of the hyperbolic Laplace-Beltrami operator on punctured surfaces namely singly
punctured two-torus and triply punctured two-sphere. Punctured surfaces are surfaces
with points removed or located infinitely far away and they have complex
topological and geometrical properties. The presence of the punctures or cusps
means that there is a continuous spectrum as well as the discrete one. This work
focuses on the discrete part of computational nature.
Hejhal developed an algorithm to compute Maass cusp form on triangle groups. The
algorithm of Hejhal is based on automorphy condition and also applies to the
computation of the Maass cusp forms on Fuchsian group whose the fundamental
domain has exactly one cusp. In this work the method due to Hejhal was recalled and extended for computation of
Maass cusp on singly punctured two-torus which still has one cusp but a nonzero
genus. The algorithm was modified further to carry out the computation for the
surfaces with three cusps i.e. triply punctured two-sphere. All the computations were
implemented in Mathematica and built in a way accessible to any one with an
introductory knowledge in Mathematica.
The results of the study are the first low-lying eigenvalues, examples of Fourier
coefficients and graphic plots of Maass cusp forms each for modular group, singly
punctured two-torus and triply punctured two-sphere. The eigenvalues and the
Fourier coefficients were computed with the desired accuracy. Some comparisons
between singly punctured two-torus and triply punctured two-sphere are also
presented. |
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