# Elliptic Modular Functions

## An Introduction

• Bruno Schoeneberg
Book

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 203)

1. Front Matter
Pages N1-VIII
2. Bruno Schoeneberg
Pages 1-25
3. Bruno Schoeneberg
Pages 26-49
4. Bruno Schoeneberg
Pages 50-70
5. Bruno Schoeneberg
Pages 71-103
6. Bruno Schoeneberg
Pages 104-126
7. Bruno Schoeneberg
Pages 127-153
8. Bruno Schoeneberg
Pages 154-183
9. Bruno Schoeneberg
Pages 184-202
10. Bruno Schoeneberg
Pages 203-226
11. Back Matter
Pages 227-236

### Introduction

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.

### Keywords

Elliptische Modulfunktion Finite Functions Modular form congruence construction convergence development group integral knowledge number theory operator proof transformation

#### Authors and affiliations

• Bruno Schoeneberg
• 1
1. 1.Universität HamburgGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-65663-7
• Copyright Information Springer-Verlag Berlin Heidelberg 1974
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-65665-1
• Online ISBN 978-3-642-65663-7
• Series Print ISSN 0072-7830