Class field theory classifies the abelian extension of a number (or more generally, global) field using data intrinsic to the number field – line bundles over it. This is a very powerful theory and can be used to answer a great many classical questions in algebraic number theory.
Nevertheless, the general theory offers little to no clues about generating the abelian extensions directly. In the simplest case of , we can in fact explicitly parametrize the abelian extensions in terms of a simple analytic function, the exponential. In short, every abelian extension of is contained in a cyclotomic field for some $n$ determined by the ramification in the abelian extension. This is a classical result predating class field theory known as the Kronecker-Weber theorem.
The problem for more general number fields looks hopelessly hard except in a few special cases. One of these is the case of quadratic imaginary number fields (that is, fields of the form for and square-free. In this case, the abelian extensions are generated by torsion points on a a certain Elliptic curve. This is in analogy to the case of where the abelian extensions were generated by torsion points on the one dimensional group scheme . This is known as complex multiplication.
Despite the parallels, the theory of complex multiplication is a lot harder than Kornecker-Weber. In fact, I believe there is no known proof of complex multiplication that does not already assume the main theorems of class field theory (in the case of quadratic imaginary number fields). For example, Silverman takes around 100 pages to develop the theory completely.
In spite of the difficult nature of the proofs, the results are very elegant and justify the effort that goes into the proofs. Hilbert has a famous quote towards this:
The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science.
I believe that there is a way to prove the main theorems of complex multiplication, undercutting much of the hard work done in Silverman’s book and I would like to explain that in this post.
The first few sections set up the general theory and follow the standard sources fairly closely. Skip directly to section 3 for the novel part.
Warning: I have not seen anyone else prove complex multiplication along these lines and it is possible that I am missing some subtlety. Proceed with caution!
Prerequisites: I will assume familiarity with the basic theory of Elliptic Curves (as in Silverman’s first book) and the main theorems of global Class Field Theory via ideals (but not their proofs!). I will gloss over many easy to prove results, especially in the early stages. A complete reference is, as mentioned above, Silverman’s second book on elliptic curves.