Abelian varieties with complex multiplication (for pedestrians)

TL;DR

This paper provides a simplified exposition of Deligne's extension of the Main Theorem of Complex Multiplication, along with additional insights into the origins, related problems, and properties of abelian varieties with complex multiplication.

Contribution

It offers an accessible presentation of Deligne's theorem and discusses key topics in the theory of complex multiplication, including Hilbert's Twelfth Problem and motivic properties.

Findings

Extended the Main Theorem of Complex Multiplication to all automorphisms of complex numbers

Discussed the motivic nature of algebraic Hecke characters

Explored the periods of abelian varieties of CM-type

Abstract

This is the text of an article that I wrote and disseminated in September 1981, except that I've updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne's extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The addendum discusses some additional topics in the theory of complex multiplication -- the origins of the theory, Hilbert's Twelfth Problem, why algebraic Hecke characters are motivic, and the periods of abelian varieties of CM-type. (43 pages)