Class Field Theory and the Main Theorem of Complex Multiplication

Abstract

We give an exposition of the results of class field theory and the theory of elliptic curves with complex multiplication. We begin by reviewing the basic number theoretic and algebraic concepts needed to state the main theorems of local and global class field theory, including Galois theory of infinite algebraic extensions and some group cohomology. We then proceed to the statements of class field theory, both in the local and global cases, using the language of ideles and ideals. From there, we turn to elliptic curves. After outlining some basic theory, we describe how one obtains the maximal abelian extension of an imaginary quadratic field K using global class field theory and the theory of complex multiplication on elliptic curves. We finish the thesis with the main theorem of complex multiplication, which relates the action of the abelianization of the absolute Galois group of K on the torsion points of an elliptic curve E with complex multiplication as a map on the torsion points of \(\mathbb{C}\)/\(\Lambda\).