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.


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