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Elliptic modular functions; an introduction

Author: Bruno Schoeneberg
Publisher: Berlin, New York, Springer, 1974.
Series: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 203.
Edition/Format:   Print book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Schoeneberg, Bruno.
Elliptic modular functions.
Berlin, New York, Springer, 1974
(OCoLC)622383981
Document Type: Book
All Authors / Contributors: Bruno Schoeneberg
ISBN: 354006382X 9783540063827 038706382X 9780387063829
OCLC Number: 1009036
Description: viii, 232 pages illustrations 24 cm.
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 ?$$ \left( {\sqrt {J - 1} } \right) $$ and ?$$ \left( {{}^3\sqrt J } \right) $$.- 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.
Series Title: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 203.
Responsibility: [by] B. Schoeneberg. Translated from the German by J.R. Smart and E.A. Schwandt.
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