UNIVERSITI PUTRA MALAYSIA COMPUTATION OF MAASS...

UNIVERSITI PUTRA MALAYSIA

COMPUTATION OF MAASS CUSP FORMS ON SINGLY PUNCTURED TWO-TORUS AND TRIPLY PUNCTURED TWO-SPHERE USING

MATHEMATICA

ABUBAKER AHMED MOHAMED SIDDIG

IPM 2009 8

COMPUTATION OF MAASS CUSP FORMS ON SINGLY PUNCTURED

TWO-TORUS AND TRIPLY PUNCTURED TWO-SPHERE USING MATHEMATICA

By


Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in

Fulfilment of the Requirements for the Degree of Doctor of Philosophy

October 2009

DEDICATION

To my parents, my brothers , my sisters and my wife

Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfillment of the requirement for the degree of Doctor of Philosophy

COMPUTATION OF MAASS CUSP FORMS ON SINGLY PUNCTURED TWO-TORUS AND TRIPLY PUNCTURED TWO-SPHERE USING

MATHEMATICA

By


October 2009

Chairman : Associate Professor Hishamuddin Zainuddin, PhD

Institute : Institute For Mathematical Research

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.

ii

iii

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.

Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai

memenuhi keperluan untuk ijazah Doktor Falsafah

KOMPUTASI FUNGSI BENTUK JURING MAASS DI ATAS 2-TORUS BERJURING TUNGGAL DAN 2 SFERA BERJURING TIGA MENGGUNAKAN

MATHEMATICA

Oleh


Oktober 2009

Pengerusi : Profesor Madya Hishamuddin Zainuddin, PhD

Institut : Institut Penyelidikan Matematik

Topik kajian ini adalah pengiraan fungsi bentuk juring Maass, yang merupakan fungsi

eigen operator hiperbolik Laplace-Beltrami di atas permukaan berjuring iaitu 2- torus

berjuring tunggal dan 2-sfera berjuring tiga. Permukaan berjuring adalah permukaan

dengan beberapa titik diasingkan atau diletakkan pada jarak infiniti dan permukaan

seumpama ini mempunyai ciri topologi dan geometri yang kompleks. Kewujudan juring

bermakna terdapat kedua-dua spektrum yang selanjar dan yang diskrit. Hejhal telah

membangunkan satu algoritma untuk mengira fungsi bentuk juring Maass untuk

kumpulan segi tiga berdasarkan syarat automorfi dan turut dapat digunakan untuk

mengira fungsi bentuk juring Maass untuk kumpulan Fuchsian, yang domain asasnya

mempunyai satu juring.

Dalam kajian ini, kaedah penyelesaian Hejhal diguna kembali dan diperluaskan kepada

pengiraan fungsi bentuk juring Maass bagi 2-torus berjuring tunggal yang masih

iv

v

mempunyai satu juring tetapi bergenus bukan sifar. Algoritma ini kemudian diubah

seterusnya untuk membuat pengiraan bagi permukaan yang mempunyai tiga juring, iaitu

2 sfera berjuring tiga. Semua pengiraan dibuat menggunakan Mathematica dan dibangun

dengan cara yang boleh digunakan oleh sesiapa sahaja yang mempunyai pengetahuan

awal Mathematica.

Hasil peyelidikan ini adalah nilai eigen paras rendah, contoh pekali Fourier dan plot

grafik fungsi bentuk juring Maass untuk kumpulan modular, 2-torus berjuring tunggal

dan 2-sfera berjuring tiga. Nilai eigen dan pekali Fourier telah dikira mengikut kejituan

yang diingini. Perbandingan juga dibuat antara 2-torus berjuring tunggal dengan 2 sfera

berjuring tiga.

ACKNOWLEDGEMENT

In the name of Allah, the Beneficent, the Most Merciful. Praises and thanks be to

Allah, The Lord of the Worlds, and peace be upon the last messenger, Mohammed,

his Family an Companions.

This thesis would not exist without the support and generosity of my advisor,

Assoc. Prof. Dr. Hishamuddin Zainuddin. I have learned a great deal from working

with him, and it is a pleasure to thank him now for making it such a pleasant

experience; many of the ideas that have been developed here are due to him. I am

most grateful to Holger Then for his friendly advice and valuable discussions in

carrying this numerical work. I am also grateful to Prof. Adem Kilickman and Assoc.

Prof. Dr. Jumiah for their support. I would like to thank all my dear friends. I would

rather not list their names here, but merely trust that they know who I am referring to

should they ever read this.

My sincere appreciation goes to my parents, brothers and sisters and my wife for all

their years of unconditional love and support, without which I would not be here

now. I am obliged to Hassan Eltayeb for encouraging me to use the LaTeX as

platform to write this manuscript. I am indebted to my friend Ustaz Kamaruddin bin

Dauod for his prayer and support, may Allah bless him.

This work is supported by the Malaysian Ministry of Science, Technology and

Innovations (MOSTI), under the project no. 06-01-04-SF0256. I would like to thank

the Institute of Advanced Technology, Universiti Putra Malaysia where the project is

initiated.

vi

vii

Finally I pray to Almighty Allah to reward all those who contributed, either directly

or indirectly, to the success of this work.

I certify that a Thesis Examination Committee has met on 23 October 2009 to conduct the final examination of Abubaker Ahmed Mohamed Siddig on his thesis entitled " COMPUTATION OF MAASS CUSP FORMS ON SINGLY PUNCTURED TWO-TORUS AND TRIPLY PUNCTURED TWO-SPHERE USING MATHEMATICA " in accordance with the Universities and University Colleges Act 1971 and the Constitution of the Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. The Committee recommends that the student be awarded the (Doctor of Philosophy). Members of the Examination Committee are as follows: Abdul Halim Shaari, PhD Professor Faculty of Science Universiti Putra Malaysia (Chairman) Zuriati Ahmad Zukarnain, PhD Lecturer Faculty of Computer Science and Information Technology Universiti Putra Malaysia (Internal Examiner) Zainal Abidin Talib, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Internal Examiner) Kurunathan a/l Ratnavelu, PhD Professor Institute Sains Matematik Universiti Malaya (External Examiner)

______________________ Bujang Kim Huat, PhD Professor/Deputy Dean School of Graduate Studies Universiti Putra Malaysia Date:

viii

This thesis submitted to the Senate of Universiti Putra Malaysia and has been accepted as fulfillment of the requirement for the degree of Doctor of Philosophy. The members of Committee are as follows: Hishamuddin Zainuddin, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Chairman) Adem Kilicman, PhD Professor Faculty of Science Universiti Putra Malaysia (Member) Jumiah Hassan, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Member)

_________________________________ HASANAH MOHD. GHAZALI, PhD Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date:

ix

DECLARATION

I hereby declare that the thesis is based on my original work except for quotations and citations which have been duly acknowledged. I also declare that it has not been previously or concurrently submitted for any other degree at UPM or other institutions. _______________________________________ ABUBAKER AHMED MOHAMED SIDDIG

Date: 9 November 2009

x

TABLE OF CONTENTS PageDEDICATION ABSTRACT ABSTRAK ACKNOWLEDGEMENTS APPROVAL DECLARATION LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS

iiivi

viivii

xxivxvxx

CHAPTER 1 INTRODUCTION 1 1.1 Introduction 1 1.3 Objectives of the Research 4 1.4 Organization of the Thesis 4

2 LITERATURE REVIEW 6 2.1 Punctured Surfaces 6 2.2 Mass Waveforms 10 2.3 Computation of Maas waveforms 11 2.3.1 Published Works Concerning Hejhal’s Algorithm 14 2.4 Conclusion 18

3 TOPOLOGY OF THE SURFACES 19 3.1 Hyperbolic Geometry 19 3.2 Fuchsian Group 21 3.3 The Modular Group and Congruence Subgroups 23 3.3.1 The Modular Group 23 3.3.2 Congruence Subgroups 26 3.4 Construction of the Modular Surfaces 27 3.4.1 Construction of Riemann Surfaces Using Graphs 27 3.5 The Fundamental Domain of Singly Punctured Two-Torus 33 3.6 The Fundamental Domain of Triply Punctured Two-Sphere 37

4 INTRODUCTION TO MAASS WAVEFORMS 43 4.1 Quantum Mechanics 43 4.2 Solving the Laplacian 45 4.3 Maass Cusp Forms 46 4.3.1 Automorphic Form 46 4.3.2 The Fourier Series 47 4.3.3 Definition of Maass Cusp Forms 49

xi

4.3.3 The Eisenstein Series 52

5 COMPUTATIONAL ASPECTS 54 5.1 Introduction 54 5.2 The K-Bessel Function 56 5.3 The Algorithm 57 5.3.1 Computing Cusp Forms 57 5.3.2 Finding the Eigenvalues 61 5.4 The Pullback Algorithm 63 5.4.1 The Pullback for SL(2,Ζ) 64 5.4.2 The Pullback for Γ’ 64 5.4.3 The Pullback for Γ(2) 67

6 COMPUTATION OF MAASS CUSP FORMS ON MODULAR

GROUP 70

6.1 Numerical Implementation 71 6.1.2 The Parameters 71 6.1.2 K-Bessel Test 72 6.2 Computational Results From Mathematica 73 6.3 Pictures of The Maass Cusp Form 83 6.4 Conclusion 89

7 MAASS CUSP FORMS ON SINGLY PUNCTURED TWO-TORUS 93 7.1 The Eigenequation and the Linear System 93 7.2 Numerical Results 96 7.21 The Parameters 96 7.3 Computational Results 97 7.3.1 The Experiments 99 7.3.1.1 Locating the Eigenvalues 99 7.3.2 Pictures of Maass Forms on Γ’ 103 7.4 Conclusion 105

8 MAASS CUSP FORMS ON TRIPLY PUNCTURED TWO-

SPHERE 116

8.1 Introduction 116 8.2 The Fundamental Domain of Γ(2) 117 8.3 The Algorithm 119 8.4 Numerical Results 130 8.4.1 The Parameters 130 8.4.2 Computational Results 130 8.4.3 The Experiments 132 8.4.3.1 Locating the Eigenvalues 133 8.4.4 Remarks on the Bessel Function 139 8.4.5 Pictures of Maass Forms on Γ(2) 139 8.5 Conclusion 146

xii

9 DISCUSSION AND CONCLUSION 147 9.1 Discussion 147 9.2 Conclusion 150 9.2.1 Summary of the Contributions 151 9.3 Future Works 152 REFERENCES 154 APPENDICES 159 BIODATA OF STUDENT 251 LIST OF PUBLICATIONS 252

xiii

LIST OF TABLES Figure Page

5.1 Shows the words and their equivalent string. 67

6.1 Comparison of the Bessel function values. 72

6.2 r-Values for the even Maass cusp form for the interval

I=[0,50]. 81

6.3 r-Values for the odd Maass cusp form for the interval I=[0,50]. 82

6.4 Fourier coefficients for the second three lowest Maass cusp

forms. 83

6.5 Fourier coefficients for the second three lowest Maass cusp

forms. 83

6.6 Time to depict Representative Maass cusp forms. 90

7.1 r-values for the lowest fifteen odd and even Maass cusp forms

103

7.2 Time to depict Representative Maass cusp forms, The pictures plotted with 10 Fourier coefficients

111

7.3 Fourier coefficients for the three lowest odd Maass cusp forms 112

7.4 Fourier coefficients for the three lowest even Maass cusp

forms 113

7.5 Fourier coefficients for the old form Maasss cusp forms

114

8.1 r-values for the lowest eleven odd and even Maass cusp

forms 133

8.2 Fourier coefficients for the three lowest odd Maass cusp forms 136

8.3 Fourier coefficients for the three lowest even Maass cusp

forms 137

8.4 Time to depict Representative Maass cusp forms, the pictures

plotted with 15 Fourier coefficients 140

xiv

LIST OF FIGURES

Figure Page

3.1 The upper half plane, the geodesic are semi circles and straight lines

20

3.2 Fundamental domain F for the modular group 22

3.3 Fundamental domain F for the modular group 24

3.4 The ideal hyperbolic triangle with 3 vertices at ∞ 29

3.5 The marked ideal hyperbolic triangle with orientation order

29

3.6 The gluing of two triangle to construct the singly punctured torus

32

3.7 The gluing of two triangles to construct the triply punctured sphere

32

3.8 Transformation of A and B give the geometry of simple

torus

34

3.9 Transformation of A and B give the geometry of torus with one puncture

34

3.10 The fundamental domain of singly punctured two-torus at point which cover six adjacent copies of fundamental domain

z i= ∞35

3.11 The fundamental domain with the transformation A, B and C which identify the boundaries

36

3.12 Transformation of A and B give the spherical geometry with 3 cusps

39

3.13 Fundamental domain of triply punctured sphere, punctures at z=i∞, z=0 and at z=1.

41

3.14 A subdomain corresponds to the mapping from z=0 z=i∞. 41

3.15 A subdomain corresponds to the mapping from z=1 z=i∞. 42

5.1 The amplitude of the K-Bessel function gets exponentially small when r increases

57

5.2 shows a pullback procedure 64

xv

Figure Page

6.1 Plot of gm shows the small steps size and that interval is small to search for the eigenvalues

75

6.2 Plot of gm 's shows the small steps size and that interval is

small to search for the eigenvalues 75

6.3 Plot of gm's shows the interval with high probability to find

eigenvalue for the even Maass form for the range r=10 to r= 20.

76

6.4 Plot of gm 's shows the interval with high probability to

find eigenvalue for the even Maass form for the range r=20 to r= 30

77

6.5 Plot of gm's shows the small steps size and small interval

I=[40.54,40.7] contains two close eigenvalues.

77


I=[47.9265,47.9267] contains eigenvalue correspond to r=47.92655833060.

78


eigenvalue for the odd Maass form for the range r=10 to r= 20.

79


eigenvalue for the even Maass form for the range r=20 to r= 30.

80

6.9 Plot of gm's shows the interval with high probability to find eigenvalue for the odd Maass form for the range r=10 to r=20.

80

6.10 Picture of the even Maass cusp form corresponding to

r=13.77975135189 85


r=17.73856338105 85


r=19.42348147082 86


r=21.31579590020 86


r=9.533695261353 87

xvi

Figure Page

6.15 Picture of the even Maass cusp form corresponding to r=12.17300832467

87


88


r=16.13807317152

88


89

6.19 Picture of the odd Maass cusp form corresponding to r=49.99122049300

89


eigenvalue for the even Maass form for the range r=12 to r=14

99

7.2 Plot of gm's shows the interval with high probability to find eigenvalue for the even Maass form for the range r=7.55 to r=7.60

100

7.3 Plot of gm's shows close eigenvalues in the interval I=[6.04,6.13]

100

7.4 Plot of gm's shows close eigenvalues in the interval I=[12.20,12.40]

101

7.5 Plot of gm's shows the interval that contains the r-value for

the odd old form for corresponding to r=9.533695 101

7.6 Plot of gm's shows the interval that contains the r-value for

the odd old form for corresponding to r=12.173008 102

7.7 Picture of the odd old form Maass cusp corresponding to

r=9.533695298433306 105

7.8 Picture of the odd old form Maass cusp corresponding to

r=12.173008328676225. 105

7.9 Picture of the even old form Maass cusp corresponding to

r=13.779751359678958 106

7.10 Picture of the odd Maass cusp form corresponding to

r=5.838327413797378 106

xvii

Figure Page


r=6.094677346944810 107


r=6.547398298978807

107

7.13 Picture of the odd Maass cusp form corresponding to r=7.199489030838013

108


r=8.003627711534501

108


109


109


r=6.496506319046022 110


r=7.576405676454307 110


r=8.386322660446167 111

8.1 The fundamental domain F I corresponds to the cusp at ∞. 117

8.2 The fundamental domain F S corresponds to the cusp at 0. 117

8.3 The fundamental domain

1STF−

corresponds to the cusp at 1.

118

8.4 Plot of gm's shows the candidate intervals with high

probability contain eigenvalue. 133


I= [4.464, 4.47] contains an eigenvalue. 134


I= [4.834, 4.840] contains an eigenvalue. 134

8.7 Plot of gm's shows the candidate intervals in $

I= [1.0, 3.0] with high probability contain eigenvalue. 135

8.8 Plot of gm's shows the small steps size and small interval I=

[2.84, 2.850] contains an eigenvalue. 135

xviii

Figure Page

8.9 Plot of gm's shows the small steps size and small interval I=[3.59,3.6] contains an eigenvalue.

138

8.10 Picture of the lowest Maass cusp form with an eigenvalue

corresponding to r=2.0921046352 140

8.11 Picture of the odd Maass cusp form with an eigenvalue




xix



8.14 Picture of the even Maass cusp form with an eigenvalue








8.18 Picture of the odd old form with an eigenvalue


8.19 Picture of the even old form with an eigenvalue


xx

LIST OF ABBREVIATIONS Ζ Integers

Ρ Real number

Θ Rational number

Χ Complex number

∞ Infinity

Η The upper half- plane

Γ The Modular group

( )2Γ Principal congruence subgroup of level 2

G Finite 3-regular graph

Ο Orientation

SO(G, Ο) Finite-area Riemann surface

SO(G, Ο) Closed Riemann surface

LHT Left hand turn

χ Euler Characteristic

( )NΓ Principal congruence subgroup of level N

Δ Hyperbolic Laplace operator

λ Eigenvalue

Kir K-Bessel function with imaginary order

SL(2, Ζ) Special Linear Two-Dimensioanl of integers

SL(2, Ρ) Special Linear Two-Dimensioanl of reals

PSL(2, Ζ) Projective Special Linear Two-Dimensioanl of integers

F Fundamental domain

xx

iF∞ Fundamental domain with a cusp at z=i∞

0F Fundamental domain with a cusp at z=0

1F Fundamental domain with a cusp at z=1

IF Subdomain with a cusp at z=i∞

SF Subdomain with a cusp at z=0

1−STF Subdomain with a cusp at z=1

xxi

LIST OF PUBLICATIONS JOURNALS Abubaker Ahmed Mohamed Siddig and Hishamuddin Zainuddin, Computation Of Maass Cusp Forms On Modular Group In Mathematica. International Journal of Pure and Applied Mathematics (IJPAM), vol. 54, No 2 (2009) Hishamuddin Zainuddin, Ahmed Bouketir, Abubaker Ahmed Mohamed Siddig, F.P. Zen & Nurisya Mohd Shah Group-Theoretic Quantization on Triply-Punctured Two-Sphere Jurnal Fizik Malaysia 27 (2006) 135-139. CONFERENCES A.A. Siddig} and H. Zainuddin, Computation of Maass Cusp Forms on Triply Punctured Two-Sphere with Mathematica. ICREM4, 21-23 October 2009, Kuala Lumpur. Hishamuddin Zainuddin and Abubaker A. M. Siddig, Maass Cusp Forms on Singly Punctured Two-Torus and Triply Punctured Two-Sphere. Submitted to Quantum Theory and Symmetries 6, July 2009, University of Kentucky, USA. A.A.M. Siddig, N.M. Shah and H. Zainuddin, Maass Cusp Forms ON Singly Punctured Two-Torus, AIP Proceeding of International Meeting on Frontiers of Physics, Jan 2009, Genting, Malaysia. A.A. Siddig & H. Zainuddin Punctured Spheres & Regular Graphs: Relationships and Method of Construction PERFIK 2004, 5-7 October 2004, Kuala Lumpur. H. Zainuddin, A. Bouketir, A.A.M. Siddig, N.M. Shah & F.P. Zen Group Theoretic Quantization on Triply Punctured Two-Sphere International Meeting on Frontiers of Physics, 25-29 July 2005, Mines Beach Resort, Seri Kembangan. Technology. MANUSCRIPT UNDER PREPRATION Abubaker Ahmed Mohamed Siddig and Hishamuddin Zainuddin, A Numerical Study of Eigenvalues of the Hyperbolic Laplacian for Singly Punctured Two-torus. Abubaker A. M. Siddig and Hishamuddin Zainuddin, Maass Cusp Forms on Triply Punctured Two-Sphere.

252

Date post:	17-Mar-2019
Category:	Documents
Upload:	phungdung
View:	213 times
Download:	0 times

UNIVERSITI PUTRA MALAYSIA COMPUTATION OF MAASS...

Documents