Computational Aspects of Mass Waveforms

The topic of this thesis is computation of Mass waveforms, and we consider a number of different cases: Congruence subgroups of the modular group and Dirichlet characters (chapter 1); congruence subgroups and general multiplier systems and real weight (chapter 2); and noncongruence subgroups (chapter 3). In each case we first discuss the necessary theoretical background. We then outline the algorithm and display some of the results obtained by it.

Contents

Introduction
1 Mass Waveforms on Hecke Congruence subgroups with Dirichlet
characters
(Computational Aspects)
1.1 General Deﬁnitions and Notation
1.1.1 A Brief Introduction to Fuchsian Groups
1.1.2 Hecke Congruence Groups
1.1.3 Introduction to Dirichlet Characters
1.1.4 A Brief Introduction to Mass Waveforms
1.2 Some Structural Theory of M(Γ0(N),χ,λ)
1.2.1 The Conjugation Operator
1.2.2 The Fourier Series
1.2.3 Involutions and Normalizers
1.2.4 The Reﬂection Operator
1.2.5 Complete Symmetrization
1.2.6 Hecke Operators
1.2.7 Oldforms and Newforms
1.2.8 The Cusp Normalizing Maps as Normalizers of (Γ0(N),χ)
1.3 Computational Aspects
1.3.1 Introduction
1.3.2 Phase 1
1.3.3 Phase 2
1.3.4 Remarks on the Performance of the Algorith
1.4 Results
1.4.1 Eigenvalues
1.4.1 Eigenvalues
1.4.2 Lowest Eigenvalues
1.4.3 Fourier Coefﬁcients
2 Computation of Maass Waveforms with Non-trivial Multiplier Sys-tems
2.1 Introduction
2.2 Multiplier Systems
2.2.1 Introduction
2.2.2 The η multiplier system
2.2.3 The θ Multiplier System
2.2.4 Further properties of the multiplier systems
2.3 Maass Waveforms
2.3.1 Decomposition of the discrete spectrum
2.4 Operators
2.4.1 Conjugation and reﬂection
2.4.2 The involution ωN
2.4.3 The operator σ4
2.4.4 Maass operators
2.4.5 Maass operators and the symmetry about k = 6
2.4.6 Hecke operators
2.4.7 Lifts at weight 1 and Fourier coefﬁcients
2.5 Oldforms
2.5.1 Γ0(N) with N prime and η-multiplier
2.6 The Eisenstein serie
2.6.1 Weight 0
2.7 Half integer weight
2.7.1 The Shimura correspondence
2.8 Some Computational Remarks
2.9 Numerical results
2.9.1 Varying weight
2.9.2 Small weights
2.9.3 Lifts at weight 1
2.9.4 Half integer weight
3 Computation of Maass waveforms on Non-congruence Subgroups of the Modular Group
3.1 Introductio
3.2 Subgroups of the Modular Group
3.2.1 Permutations and subgroups of PSL(2,Z)
3.2.2 Congruence subgroups
3.3 Numerical implementation
3.4 Numerical Result
3.4.1 Subgroups
3.4.2 Newforms
3.4.3 Fourier coefﬁcients
4 An Algorithm for Whittaker’s W-function
4.1 Introduction
4.2 Presentation of the Algorithm
4.3 The path
4.4 The integrand
4.5 Evaluation of the Integrand
4.5.1 The power serie
4.5.2 Recursion
4.5.3 Asymptotic series
4.5.4 Comparison
4.6 Evaluation of the Integrals
4.7 Remarks on the K-Bessel function
4.8 Numerical veriﬁcation
Swedish Summary
Acknowledgements
References

Author: Strömberg, Fredrik

Source: Uppsala University Library