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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Main Author:	Mohamed Siddig, Abubaker Ahmed
Format:	Thesis
Language:	English English
Published:	2009
Subjects:	Geometrical drawing
Online Access:	http://psasir.upm.edu.my/id/eprint/7575/1/ABS_---__IPM_2009_8.pdf
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spelling	my-upm-ir.75752013-05-27T07:35:42Z Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica 2009 Mohamed Siddig, Abubaker Ahmed 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. Geometrical drawing 2009 Thesis http://psasir.upm.edu.my/id/eprint/7575/ http://psasir.upm.edu.my/id/eprint/7575/1/ABS_---__IPM_2009_8.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Geometrical drawing Institute : Institute for Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Geometrical drawing
spellingShingle	Geometrical drawing Mohamed Siddig, Abubaker Ahmed Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
description	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.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Mohamed Siddig, Abubaker Ahmed
author_facet	Mohamed Siddig, Abubaker Ahmed
author_sort	Mohamed Siddig, Abubaker Ahmed
title	Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
title_short	Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
title_full	Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
title_fullStr	Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
title_full_unstemmed	Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica
title_sort	computation of maass cusp forms on singly punctured two-torus and triply punctured two-sphere using mathematica
granting_institution	Universiti Putra Malaysia
granting_department	Institute : Institute for Mathematical Research
publishDate	2009
url	http://psasir.upm.edu.my/id/eprint/7575/1/ABS_---__IPM_2009_8.pdf
_version_	1747810703569321984

Computation of Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere Using Mathematica

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