On the Theory of Maass Wave Forms

Description

Preface Acknowledgements Contents List of Symbols Introductory Roadmap Chapter 1 – A Short Introduction to Modular Forms Chapter 2 – Period Polynomials Chapter 3 – Maass Wave Forms of Real Weight Chapter 4 – Families of Maass Cusp Forms, L-Series, and Eichler Integrals Chapter 5 – Period Functions Chapter 6 – Continued Fractions and the Transfer Operator Approach Chapter 7 – Weak Harmonic Maass Wave Forms 1 A Brief Introduction to Modular Forms 1.1 Notations and Some Simple Results 1.2 The Full Modular Group and Its Subgroups 1.2.1 The Full Modular Group 1.2.2 Fundamental Domains for the Full Modular Group 1.2.3 Fundamental Domains for Subgroups of (1) 1.2.4 Cuspidal Points 1.3 Modular Forms – Properties and Examples 1.3.1 Introduction – Modular Forms 1.3.2 Multiplier Systems 1.3.3 Modular Forms 1.3.4 Eisenstein Series 1.3.5 The Discriminant Function (z) 1.3.6 Petersson Scalar Product and the Hilbert Space of Cusp Forms 1.4 Holomorphic Poincaré Series 1.4.1 Further Remarks on Modular Forms 1.5 Theta Series 1.5.1 Jacobi Theta Function 1.5.2 Theta Series of Index m 1.5.3 Theta Series and Sums of Squares 1.6 Hecke Operators 1.6.1 The Slash Operator 1.6.2 Hecke Operators of Index n N 1.6.3 Eigenforms 1.7 L-Functions Associated with Cusp Forms 1.7.1 The Mellin Transform 1.7.2 L-Series Associated with Cusp Forms on (1) 1.7.3 L-Functions of Hecke Eigenforms 1.8 Concluding Remarks 2 Period Polynomials 2.1 Various Ways to Period Polynomials 2.1.1 Period Polynomials via Integral Transformations 2.1.2 Eichler Integrals and Period Polynomials 2.1.3 The (k-1)-Fold Antiderivative 2.1.4 An Integral Representation of the Eichler Integral for Cusp Forms 2.1.5 An Integral Representation of the Eichler Integral for Entire Forms 2.1.6 The Niebur Integral Representation for Entire Forms 2.2 The Eichler Cohomology Group 2.3 The Eichler Cohomology Theorem 2.3.1 Injectivity of the Map μ 2.3.2 Surjectivity of the Map μ 2.4 Hecke Operators and Period Polynomials 2.5 Concluding Remarks 3 Maass Wave Forms of Real Weight 3.1 Multiplier Systems and Unitary Automorphic Factors 3.2 The Differential Operators k and Ek 3.2.1 Whittaker Functions 3.2.2 Whittaker-Fourier Expansions on (1) 3.2.3 Whittaker-Fourier Expansions for (1) 3.2.4 Maass Operators Ek and the Fourier-WhittakerExpansion 3.3 Maass Wave Forms of Real Weight 3.4 Examples of Maass Wave Forms 3.4.1 The Embedding of Classical Modular Forms 3.4.2 Non-holomorphic Eisenstein Series for the Full Modular Group 3.4.3 Maass Cusp Forms Derived from the Dedekindη-Function 3.5 Scalar Product and the Hilbert Space of Cusp Forms 3.5.1 Definition of the Petersson Scalar Product 3.5.2 Properties of the Petersson Scalar Product 3.5.3 The Petersson Scalar Product and Maass Cusp Forms 3.6 Hecke Operators on Maass Cusp Forms 3.7 The Laplace Operator and Its Self-Adjoint FriedrichsExtension 3.7.1 The Friedrichs Extension of 0 for the Full Modular Group 3.7.2 The Friedrichs Extension of k 3.8 Selberg's Conjecture 3.9 Concluding Remarks 4 Families of Maass Cusp Forms, L-Series, and Eichler Integrals 4.1 Families of Maass Cusp Forms on the Full Modular Group 4.2 L-Functions Associated with Maass Cusp Forms 4.2.1 L-Functions for Sk,v(to.(1),ν)to. 4.2.2 A Simpler Representation of Families of Cusp Forms 4.2.3 D-Functions for Sk,v(to.(1),ν)to. 4.3 Nearly Periodic Functions and Maass Cusp Forms 4.3.1 D-Functions Giving Rise to Nearly Periodic Functions 4.3.2 Nearly Periodic Functions Giving Rise to D-Functions 4.3.3 Nearly Periodic Functions via Integral Transforms 4.3.3.1 The R-Function 4.3.3.2 The Maass-Selberg Differential Form 4.3.3.3 Everything Combined 4.3.3.4 Nearly Periodic Functions via Integral Transformations 4.3.4 Nearly Periodic Functions and Eichler Integrals 4.4 Concluding Remarks 5 Period Functions 5.1 Period Functions Associated with Maass Cusp Forms 5.1.1 Period Functions and Nearly Periodic Functions 5.1.2 Period Functions via Integral Transformations 5.1.3 Period Functions via Nearly Periodic Functions Given by an Integral Transform 5.1.4 Period Functions and Families of Maass Cusp Forms 5.1.5 Period Functions and Period Polynomials 5.1.6 Some Remarks on Period Functions 5.2 A Visit to the Semi-Analytic Cohomology Group 5.2.1 The Spaces Vν∞, Vνω, and Vνω*,∞ 5.2.2 Mixed Parabolic Cohomology Group H1par(to.(1), Vνω, Vνω*,∞)to. 5.2.3 Zagier's Cohomology Theorem 5.3 Hecke Operators on Period Functions 5.3.1 On Farey Sequences 5.3.2 Left Neighbor Sequences 5.3.3 Paths in the Upper Half-Plane 5.3.4 Hecke Operators for Period Functions for (1) 5.4 Concluding Remarks 6 Continued Fractions and the Transfer Operator Approach 6.1 The Artin Billiard and Continued Fractions 6.1.1 Continued Fractions and the Gauss Map 6.1.2 GL2( Z ) and the Artin Billiard 6.1.2.1 The Group GL2( Z ) 6.1.2.2 GL2( Z ) and Möbius Transformations 6.1.2.3 The Artin Billiard 6.1.3 Geodesics on the Artin Billiard 6.1.3.1 GL2( Z )-Equivalent Geodesics 6.1.3.2 GL2( Z )-Equivalent Geodesics and Continued Fractions 6.1.4 Continued Fractions, Geodesics, and the Full Modular Group 6.1.5 Closed Geodesics Under SL2( Z ) and Periodic Continued Fraction Expansions 6.2 Ruelle's Transfer Operator Approach for the Gauss Map 6.2.1 Motivation of Dynamical Zeta Functions and the Associated Transfer Operator 6.2.2 The Transfer Operator for the Gauss Map 6.2.3 Mayer's Transfer Operator 6.3 The Transfer Operator and Period Functions on (1) 6.3.1 Eigenfunctions of the Transfer Operator and Solutions of a Three-Term Equation 6.3.2 Solutions of a Three-Term Equation and PeriodFunctions 6.4 The Transfer Operator and the Selberg Zeta Function 6.4.1 Geodesics on H, SL2( Z ) Periodic Orbits, and Continued Fractions 6.4.2 The Dynamical Zeta Function by Ruelle and the Selberg Zeta Function 6.5 Concluding Remarks 7 Weak Harmonic Maass Wave Forms 7.1 Weak Harmonic Maass Wave Forms 7.2 Mock Modular Forms 7.2.1 Pure Mock Modular Forms 7.2.2 Mock Jacobi Forms 7.2.3 General Mock Modular Forms 7.3 L-Functions of Weakly Harmonic Maass Wave Forms 7.3.1 Regularized L-Series for S!k,1(to.(1))to. 7.3.2 Regularized L-Series for Hk(to.(1))to. 7.4 Periods and Weakly Harmonic Maass Wave Forms 7.4.1 The Mock Modular Period Function 7.4.2 Period Functions via Formal Eichler Integrals 7.5 Concluding Remarks A Background Material A.1 Chinese Remainder Theorem – Special Case A.2 Phragmén-Lindelöf principle A.3 Stirling's Estimate of the -Function A.4 Cauchy-Riemann Equations A.5 Cauchy's Integral Theorem A.6 Rouché's Theorem A.7 The Beta Function and -Function A.8 Unique Continuation Principle For Holomorphic Functions A.9 Whittaker Functions A.10 Generalized Hypergeometric Functions A.11 Poisson Summation Formula A.12 Liouville's Theorem A.13 Residue Theorem A.14 Geometric Series A.15 Continued Fractions A.16 Hurwitz Zeta Function A.17 Incomplete -Function A.18 Properties of Linear Fractional Transformations A.19 Nuclear Operator A.20 Geodesics on the Upper Half-Plane (Poincaré Model of a Hyperbolic Plane) References Symbol Index Index