Elliptic Modular Functions An Introduction
This book is a fully detailed introduction to the theory of modular functions of a single variable. I hope that it will fill gaps which in view ofthe lively development ofthis theory have often been an obstacle to the students' progress. The study of the book requires an elementary knowledge of...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg
Springer Berlin Heidelberg
1974, 1974
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| Edition: | 1st ed. 1974 |
| Series: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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 ?
- § 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