Class Fields of Complex Multiplication: translation of "Die Klassenk\"orper der komplexen Multiplikation" by Max Deuring

TL;DR

This paper provides an English translation of Max Deuring's 1958 monograph on class fields of complex multiplication, detailing the analytic approach linking elliptic functions, modular functions, and algebraic numbers.

Contribution

It offers a faithful translation of a foundational work on complex multiplication, preserving original notation and exposition for historical and educational purposes.

Findings

Systematic exposition of complex multiplication theory

Interrelations between elliptic and modular functions

Historical preservation of original mathematical notation

Abstract

This is an English translation of the monograph "Die Klassenk\"orper der komplexen Multiplikation" by Max Deuring, published in 1958 as Enzyklop\"adie der Mathematischen Wissenschaften, Band I-2, Heft 10, Teil II by B.G. Teuber Verlagsgesellschaft, Stuttgart. It gives a systematic exposition of the analytic method in the theory of complex multiplication -- the interrelations between elliptic functions, modular functions and algebraic numbers. This translation was done as an exercise for learning the German language. The mathematical notation and terminology are kept as closely as possible to those given in the German original work. No attempt has been made to modernize the exposition, other than rectifying a (very) few obviously-correctable typographical errors.