Preface Chapter 0. Introduction Chapter 1. The Classical Modular Forms 1.1. Periodic functions 1.2. Elliptic functions 1.3. Modular functions 1.4. The Fourier expansion of Eisenstein series 1.5. The modular group 1.6. The linear space of modular forms Chapter 2. Automorphic Forms in General 2.1. The hyperbolic plane 2.2. The classification of motions 2.3. Discrete groups -- Fuchsian groups 2.4. Congruence groups 2.5. Double coset decomposition 2.6. Multiplier systems 2.7. Automorphic forms 2.8. The eta-function and the theta-function Chapter 3. The Eisenstein and the Poincare Series 3.1. General Poincare series 3.2. Fourier expansion of Poincare series 3.3. The Hilbert space of cusp forms Chapter 4. Kloosterman Sums 4.1. General Kloosterman sums 4.2. Kloosterman sums for congruence groups 4.3. The classical Kloosterman sums 4.4. Power-moments of Kloosterman sums 4.5. Sums of Kloosterman sums 4.6. The Salie sums Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 5.1. General estimates 5.2. Estimates by Kloosterman sums 5.3. Coefficients of cusp forms with theta multiplier 5.4. Linear forms in Fourier coefficients of cusp forms 5.5. Spectral analysis of the diagonal symbol Chapter 6. Hecke Operators 6.1. Introduction 6.2. Hecke operators Tn 6.3. The Hecke operators on periodic functions 6.4. The Hecke operators for the modular group 6.5. The Hecke operators with a character 6.6. An overview of newforms 6.7. Hecke eigencuspforms for a primitive character 6.8. Final remarks Chapter 7. Automorphic L-functions 7.1. Introduction 7.2. The Hecke L-functions 7.3. Twisting automorphic forms and L-functions 7.4. Converse theorems Chapter 8. Cusp Forms Associated with Elliptic Curves 8.1. The Hasse-Weil L-function 8.2. Elliptic curves Er 8.3. Computing ��(p) 8.4. A Hecke Grossencharacter 8.5. A theta series 8.6. The automorphy of f Chapter 9. Spherical Functions 9.1. Positive definite quadratic forms 9.2. Space spherical functions 9.3. The spherical functions reconsidered 9.4. Harmonic analysis on the sphere Chapter 10. Theta Functions 10.1. Introduction 10.2. An inversion formula 10.3. The congruent theta functions 10.4. The automorphy of theta functions 10.5. The standard theta function Chapter 11. Representations by Quadratic Forms 11.1. Introduction 11.2. Siegel's mass formula 11.3. Representations by Eisenstein series and cusp forms 11.4. The circle method after Kloosterman 11.5. The singular series 11.6. Equidistribution of integral points on ellipsoids Chapter 12. Automorphic Forms Associated with Number Fields 12.1. Automorphic forms attached to Dirichlet L-functions 12.2. Hecke L-functions with Grossencharacters 12.3. Automorphic forms associated with quadratic fields 12.4. Class group L-functions reconsidered 12.5. L-functions for genus characters 12.6. Automorphic forms of weight one Chapter 13. Convolution L-functions 13.1. Introduction 13.2. Rankin-Selberg integrals 13.3. Selberg's theory of Eisenstein series 13.4. Statement of general results 13.5. The scattering matrix for ��0(N) 13.6. Functional equations for the convolution L-functions 13.7. Metaplectic Eisenstein series 13.8. Symmetric power L-functions Bibliography Index