Author | Schoeneberg, Bruno. author |
---|---|
Title | Elliptic Modular Functions [electronic resource] : An Introduction / by Bruno Schoeneberg |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1974 |
Connect to | http://dx.doi.org/10.1007/978-3-642-65663-7 |
Descript | VIII, 236 p. online resource |
I. The Modular Group -- ยง 1. Inhomogeneous Linear Transformations -- ยง 2. Homogeneous Linear Transformations -- ยง 3. The Modular Group. Fixed Points -- ยง 4. Generators and Relations -- ยง 5. Fundamental Region -- II. The Modular Functions of Level One -- ยง 1. Definition and Properties of Modular Functions -- ยง 2. Extension of the Modular Group by Reflections -- ยง 3. Existence of Modular Functions. The Absolute Modular Invariant J -- ยง 4. Modular Form -- ยง 5. Entire Modular Forms -- III. Eisenstein Series -- ยง 1. The Eisenstein Series in the Case of Absolute Convergence -- ยง 2. The Eisenstein Series in the Case of Conditional Convergence -- ยง 3. The Discriminant ? -- IV. Subgroups of the Modular Group -- ยง 1. Subgroups of the Modular Group -- ยง 2. The Principal Congruence Groups -- ยง 3. Congruence Groups -- ยง 4. Fundamental Region -- ยง 5. Fundamental Regions for Special Subgroups -- ยง 6. The Quotient Space ?*/?1 -- ยง 7. Genus of the Fundamental Region -- ยง 8. The Genus of the Fundamental Region of ?0(N) -- V. Function Theory for the Subgroups of Finite Index in the Modular Group -- ยง 1. Functions for Subgroups -- ยง 2. Modular Forms for Subgroups -- ยง 3. Modular Forms of Dimension ?2 and Integrals -- ยง 4. The Riemann-Roch Theorem and Applications -- VI. Fields of Modular Functions -- ยง 1. Algebraic Field Extensions of ?(J) -- ยง 2. The Fields ?$$ \left( {\sqrt {J - 1} } \right) $$ and ?$$ \left( {{}̂3\sqrt J } \right) $$ -- ยง 3. Transformation Groups of Order n -- ยง 4. Transformation Fields of Order n -- ยง 5. The Modular Equation of Order n -- ยง 6. The Galois Group of the Modular Equation -- ยง 7. Transformations of Order n for Modular Forms -- VII. Eisenstein Series of Higher Level -- ยง 1. The Series in the Case of Absolute Convergence -- ยง 2. The Series of Dimension ?1 and ?2 -- ยง 3. Properties of the Series of Dimension ?1 and ?2. Applications -- ยง 4. Division Equation -- VIII. The Integrals of ?-Division Values -- ยง 1. The Space of ?-Division Values. Integrals -- ยง 2. An Asymptotic Formula and the Behavior of the Integrals under the Transformation T -- ยง 3. A Second Look at the Behavior of the Integrals under the Transformation T. The General Transformation Formula -- ยง 4. Consequences of the Transformation Formula -- IX. Theta Series -- ยง 1. General Theta Series. An Operator -- ยง 2. Special Theta Series -- ยง 3. Behavior of the Theta Series under Modular Transformations -- ยง 4. Behavior of the Theta Series under Congruence Groups. Gaussian Sums -- ยง 5. Examples and Applications -- Literature -- Index of Definitions -- Index of Notations