# The Main Theorem of Complex Multiplication

Today we’ll state the Main Theorem of Complex Multiplication. We have our standing assumptions. Let ${K}$ be a quadratic imaginary field and ${R_K}$ the ring of integers. For a prime ${\frak{p}}$ of ${K}$ define ${K_\frak{p}}$ to be the completion at ${\frak{p}}$ and ${R_\frak{p}}$ the ring of integers. For a fractional ideal ${\frak{a}}$ of ${K}$ define ${\frak{a}_\frak{p}}$ to be the fractional ideal ${\frak{a}R_\frak{p}}$.

We will need a few facts about modules over ${R_K}$. We know ${R_K}$ 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 ${\frak{p}}$-primary part of an ${R_K}$-module, ${M}$, to be ${M[\frak{p}^\infty]}$. This is just the set of elements of the module annihilated by some power of ${\frak{p}}$.

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 ${M}$ is a torsion ${R_K}$-module, then the natural map ${\displaystyle \bigoplus_{\frak{p}}M[\frak{p}^\infty]\rightarrow M}$ 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 ${\frak{a}}$ the natural map ${(K/\frak{a})[\frak{p}^\infty]\rightarrow R_\frak{p}/\frak{a}_p}$ is an isomorphism. This comes from the inclusion ${K\hookrightarrow K_\frak{p}}$.

Putting the above to facts together gives us that ${\displaystyle K/\frak{a}\simeq \bigoplus_{\frak{p}}K_\frak{p}/\frak{a}_\frak{p}}$. Now let ${x\in \mathcal{J}_K}$ be an idele. The fractional ideal generated by this is ${\displaystyle (x)=\prod_\frak{p} \frak{p}^{ord_p(x_p)}}$. Given any fractional ideal ${\frak{a}}$ of ${K}$ we define ${x\frak{a}:=(x)\frak{a}}$. Since ${\displaystyle K/x\frak{a}\simeq \bigoplus_\frak{p} K_p/x_p\frak{a}_p}$ we can define multiplication by ${x}$ on all of ${K/\frak{a}\rightarrow K/x\frak{a}}$ piece by piece on the ${\frak{p}}$-primary parts ${(t_\frak{p})\mapsto (x_\frak{p}t_\frak{p})}$.

Let ${E/\mathbb{C}}$ be an elliptic curve with CM by ${R_K}$. Fix some ${\sigma\in Aut(\mathbb{C})}$. We were calling our surjection from idele class field theory ${\lambda: \mathcal{J}_K\rightarrow Gal(K^{ab}/K)}$. Choose some ${s\in \mathcal{J}_K}$ with the property that ${\lambda(s)=\sigma|_{K^{ab}}}$. Let ${\frak{a}}$ be a fractional ideal and fix a complex analytic isomorphism ${f:\mathbb{C}/\frak{a}\stackrel{\sim}{\rightarrow} E(\mathbb{C})}$.

The Main Theorem of Complex Multiplication tells us that with this setup there is a unique isomorphism ${f':\mathbb{C}/s^{-1}\frak{a}\stackrel{\sim}{\rightarrow} E^\sigma(\mathbb{C})}$ which makes the following diagram commute:

$\displaystyle \begin{matrix} \mathbb{C}/\frak{a} & \stackrel{s^{-1}}{\rightarrow} & \mathbb{C}/s^{-1}\frak{a} \\ \downarrow & & \downarrow \\ E(\mathbb{C}) & \stackrel{\sigma}{\rightarrow} & E^\sigma(\mathbb{C}) \end{matrix}$

This gives us the following simple way to translate between the analytic action of multiplying by ${s^{-1}}$ and the algebraic action of ${\sigma}$ by ${f(t)^{\lambda(s)}=f'(s^{-1}t)}$. 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.