Elliptic Modular Functions: An Introduction
Elliptic Modular Functions: An Introduction
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椭圆模函数：简介
DOI：
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发表时间：
1974
期刊：
影响因子：
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通讯作者：
E. Schwandt
中科院分区：
文献类型：
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作者：
Bruno Schoeneberg;J. R. Smart;E. Schwandt
I.模群。-1。非齐次线性变换。-2。齐次线性变换。-3。模群。不动点。-4。生成元和关系。-5。基本区域。-ii。第一级的模函数。-1。模函数的定义和性质。-2。模群的反射扩张。-3。模函数的存在性。绝对模不变J.-4.模形式.-5.整模形式.-III.Eisenstein级数.-1.绝对收敛情况下的Eisenstein级数.-2.条件收敛情况下的Eisenstein级数.-3.判别式？.-IV.模群的子群.-1.模群的子群.-2.主同余群.-3.同余群.-4.基本区域.-5.特殊子群的基本区域.-6.商空间？*/？1.-7.基本域的亏格。-8.？0(N)的基本域的亏格。-V.模群中有限指标子群的函数论。-1.子群的函数。-2.子群的模形式。-3.维？2和积分的模形式。-4.Riemann-Roch定理及其应用。-VI.模函数的域。-1.？的代数场扩张(J)。-2域？$$Left({Sqrt{J-1}})
Ight)$$和？$$Left({{}^3sqrt J}
-3.n阶变换群。-4.n阶变换域。-5.n阶模方程。-6.模方程的伽罗瓦群。-7.模形式的n阶变换。-vii.更高级别的Eisenstein级数。-1.绝对收敛的级数。-2.维度？1和？2的级数。-3.维度？1和？2的级数的性质。应用。-4.除法方程的积分。-viii.？除法的积分值。-1。？-除值的空间。积分。-2。渐近公式和积分在变换下的行为T。-3。再看积分在变换下的行为。T。一般变换公式。-4。变换公式的结果。-IX。Theta系列。-1.一般Theta系列。算子。-2。特殊的Theta级数。-3。Theta级数在模变换下的行为。-4。Theta级数在同余群下的行为。高斯和。-5。例子和应用。-文献。-定义索引。-符号索引。
I. The Modular Group.- 1. Inhomogeneous Linear Transformations.- 2. Homogeneous Linear Transformations.- 3. The Modular Group. Fixed Points.- 4. Generators and Relations.- 5. Fundamental Region.- II. The Modular Functions of Level One.- 1. Definition and Properties of Modular Functions.- 2. Extension of the Modular Group by Reflections.- 3. Existence of Modular Functions. The Absolute Modular Invariant J.- 4. Modular Form.- 5. Entire Modular Forms.- III. Eisenstein Series.- 1. The Eisenstein Series in the Case of Absolute Convergence.- 2. The Eisenstein Series in the Case of Conditional Convergence.- 3. The Discriminant ?.- IV. Subgroups of the Modular Group.- 1. Subgroups of the Modular Group.- 2. The Principal Congruence Groups.- 3. Congruence Groups.- 4. Fundamental Region.- 5. Fundamental Regions for Special Subgroups.- 6. The Quotient Space ?*/?1.- 7. Genus of the Fundamental Region.- 8. The Genus of the Fundamental Region of ?0(N).- V. Function Theory for the Subgroups of Finite Index in the Modular Group.- 1. Functions for Subgroups.- 2. Modular Forms for Subgroups.- 3. Modular Forms of Dimension ?2 and Integrals.- 4. The Riemann-Roch Theorem and Applications.- VI. Fields of Modular Functions.- 1. Algebraic Field Extensions of ?(J).- 2. The Fields ?$$ left( {sqrt {J - 1} }
ight) $$ and ?$$ left( {{}^3sqrt J }
ight) $$.- 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.