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

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.