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 algebra, number theory and topology and a deeper knowledge of the theory of functions. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto­ morphic forms of integral dimensions belonging to SL(2,Z). The theory is developed first via the Riemann mapping theorem and then again with the help of Eisenstein series. An investigation of the subgroups of SL(2, Z) and the introduction of automorphic functions and forms belonging to these groups folIows. Special attention is given to the subgroups of finite index in SL (2, Z) and, among these, to the so-called congruence groups. The decisive role in this setting is assumed by the Riemann-Roch theorem. Since its proof may be found in the literature, only the pertinent basic concepts are outlined. For the extension of the theory, special fields of modular functions­ in particular the transformation fields of order n-are studied. Eisen­ stein series of higher level are introduced which, in case of the dimension - 2, allow the construction of integrals of the 3 rd kind. The properties of these integrals are discussed at length.

Inhalt

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.