Schertz R. Complex Multiplication

Cambridge University Press, UK, 2010. — 555 p. — (New Mathematical Monographs 15). — ISBN13: 9780511774676The 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.Elliptic functions
Modular functions
Basic facts from number theory
Factorisation of singular values
The Reciprocity Law
Generation of ring class fields and ray class fields
Integral basis in ray class fields
Galois module structure
Berwick’s congruences
Cryptographically relevant elliptic curves
The class number formulae of Curt Meyer
Arithmetic interpretation of class number formulae