Today we’ll state the Main Theorem of Complex Multiplication. We have our standing assumptions. Let be a quadratic imaginary field and the ring of integers. For a prime of define to be the completion at and the ring of integers. For a fractional ideal of define to be the fractional ideal .
We will need a few facts about modules over . We know is a Dedekind domain, so these will easily follow from the sequence of posts about the structure of modules over Dedekind domains. We will denote the -primary part of an -module, , to be . This is just the set of elements of the module annihilated by some power of .
Back in our posts on this topic we only talked about finitely generated modules, but some things are true in more generality. For example, if is a torsion -module, then the natural map is an isomorphism. This is essentially the part of the theorem that says if we attempt to decompose a module into a projective and torsion part, the torsion part has a nice primary decomposition. We also will need that for any fractional the natural map is an isomorphism. This comes from the inclusion .
Putting the above to facts together gives us that . Now let be an idele. The fractional ideal generated by this is . Given any fractional ideal of we define . Since we can define multiplication by on all of piece by piece on the -primary parts .
Let be an elliptic curve with CM by . Fix some . We were calling our surjection from idele class field theory . Choose some with the property that . Let be a fractional ideal and fix a complex analytic isomorphism .
The Main Theorem of Complex Multiplication tells us that with this setup there is a unique isomorphism which makes the following diagram commute:
This gives us the following simple way to translate between the analytic action of multiplying by and the algebraic action of by . The proof involves several reductions to a manageable case, and then a lot of work. As usual, the statements are all we are really interested in right now. The next two posts will be some applications of this, so we might prove a few more things there.