The fundamental theorem of complex multiplication

TL;DR

This paper provides a straightforward and elementary proof of the fundamental theorem of complex multiplication, clarifying its core concepts and historical development for a broad mathematical audience.

Contribution

It offers a simplified, accessible proof of the fundamental theorem of complex multiplication, enhancing understanding of its foundational role in number theory.

Findings

A clear, elementary proof of the theorem is presented.

The proof consolidates various complex multiplication results into a unified argument.

The article clarifies the theorem's significance in the context of algebraic number theory.

Abstract

The goal of this expository article is to present a proof that is as direct and elementary as possible of the fundamental theorem of complex multiplication (Shimura, Taniyama, Langlands, Tate, Deligne et al.). The article is a revision of part of my manuscript, Complex Multiplication, April 7, 2006.