Elliptic Modular Functions: An Introduction

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Product Details

  • ISBN-13: 9783642656651
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 7/31/2012
  • Series: Grundlehren der mathematischen Wissenschaften Series , #203
  • Edition description: Softcover reprint of the original 1st ed. 1974
  • Edition number: 1
  • Pages: 252
  • Product dimensions: 6.00 (w) x 9.00 (h) x 0.53 (d)

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—$$
\left( {\sqrt {J - 1} } \right)
$$ and—$$
\left( {{}sub3\sqrt J } \right)
$$.- § 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.

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