Elliptic Modular Functions - B. Schoeneberg

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Sommaire:
I. The Modular Group.- ? 1. Inhomogeneous Linear Transformations.- ? 2. Homogeneous Linear Transformations.- ? 3. The Modular Group. Fixed Points.- ? 4. Generators and Relations.- ? 5. Fundamental Region.- II. The Modular Functions of Level One.- ? 1. Definition and Properties of Modular Functions.- ? 2. Extension of the Modular Group by Reflections.- ? 3. Existence of Modular Functions. The Absolute Modular Invariant J.- ? 4. Modular Form.- ? 5. Entire Modular Forms.- III. Eisenstein Series.- ? 1. The Eisenstein Series in the Case of Absolute Convergence.- ? 2. The Eisenstein Series in the Case of Conditional Convergence.- ? 3. The Discriminant ?.- IV. Subgroups of the Modular Group.- ? 1. Subgroups of the Modular Group.- ? 2. The Principal Congruence Groups.- ? 3. Congruence Groups.- ? 4. Fundamental Region.- ? 5. Fundamental Regions for Special Subgroups.- ? 6. The Quotient Space ?*/?1.- ? 7. Genus of the Fundamental Region.- ? 8. The Genus of the Fundamental Region of ?0(N).- V. Function Theory for the Subgroups of Finite Index in the Modular Group.- ? 1. Functions for Subgroups.- ? 2. Modular Forms for Subgroups.- ? 3. Modular Forms of Dimension ?2 and Integrals.- ? 4. The Riemann-Roch Theorem and Applications.- VI. Fields of Modular Functions.- ? 1. Algebraic Field Extensions of ?(J).- ? 2. The Fields ? dollars dollars \left( {\sqrt {J - 1} } \right) dollars dollars and ? dollars dollars \left( {{}^3\sqrt J } \right) dollars dollars.- ? 3. Transformation Groups of Order n.- ? 4. Transformation Fields of Order n.- ? 5. The Modular Equation of Order n.- ? 6. The Galois Group of the Modular Equation.- ? 7. Transformations of Order n for Modular Forms.- VII. Eisenstein Series of Higher Level.- ? 1. The Series in the Case of Absolute Convergence.- ? 2. The Series of Dimension?1 and ?2.- ? 3. Properties of the Series of Dimension ?1 and ?2. Applications.- ? 4. Division Equation.- VIII. The Integrals of ?-Division Values.- ? 1. The Space of ?-Division Values. Integrals.- ? 2. An Asymptotic Formula and the Behavior of the Integrals under the Transformation T.- ? 3. A Second Look at the Behavior of the Integrals under the Transformation T. The General Transformation Formula.- ? 4. Consequences of the Transformation Formula.- IX. Theta Series.- ? 1. General Theta Series. An Operator.- ? 2. Special Theta Series.- ? 3. Behavior of the Theta Series under Modular Transformations.- ? 4. Behavior of the Theta Series under Congruence Groups. Gaussian Sums.- ? 5. Examples and Applications.- Literature.- Index of Definitions.- Index of Notations.