The aim of this book is to give an account of the state of the art in classical complex multiplication including, in particular, recent results on rings of integers and applications to cryptography using elliptic curves. All requisites needed about elliptic functions, modular functions and quadratic number fields are developed in this book and the results from class field theory are summarised in compact form. Further, most of the main results presented in the following chapters are accompanied by a plethora of numerical examples. The reader interested in the application of the various explicit results will therefore find all the necessary tools in this book.

After the early results of Abel and Kronecker at the beginning of and mid nineteenth century, Weber at the start of the twentieth century gave the first systematic account of complex multiplication in his “Lehrbuch der Algebra III”. The aim of this theory is to generate abelian extensions of quadratic imaginary number fields by values of elliptic functions and modular functions. Up until 1931 further accounts of the theory were given by Fricke (1916, 1922) and Fueter (1924, 1927). Finally, Hasse (1927) using class field theory that had developed in the meantime, presented a very short and elegant version of complex multiplication. His work contains the generation of ray class fields over a quadratic imaginary number field by singular values of the modular invariant j and Weber's τ function, using in the proof, besides class field theory, only the discriminant from the theory of elliptic functions.