Complex Multiplication 1

Today we’ll really get into the class field theory of CM elliptic curves. Let {K/\mathbb{Q}} be a quadratic imaginary field. Let {E/\mathbb{C}} be an elliptic curve with the property that {End(E)\simeq R_K}, e.g. take {E=\mathbb{C}/R_K}. At the end of the last post we constructed a group homomorphism {F:Gal(\overline{K}/K)\rightarrow Cl(R_K)} which was defined by the property that {E^\sigma=F(\sigma)\cdot E}.

Now we know that {Cl(R_K)} is an abelian group, so in fact {F} factors through (a now relabelled) {F: Gal(K^{ab}/K)\rightarrow Cl(R_K)} where {K^{ab}} is the maximal abelian extension of {K}. For the rest of the post we’ll sketch why a few things are true:

1) It turns out that {H=K(j(E))} is the Hilbert class field of {K}.

2) Last time we stated that {[\mathbb{Q}(j(E)): \mathbb{Q}]=[K(j(E)): K]=h_K}, but given the previous statement, the proof is now easy from class field theory.

3) If {E_1, \ldots, E_h} are a complete set of representatives of {Ell(R_K)}, then {j(E_1), \ldots, j(E_h)} form a complete set of Galois conjugates for {j(E)}.

4) For any prime {\frak{p}} of {K}, we have {j(E)^{Frob_\frak{p}}=j(F(Frob_\frak{p})\cdot E)}. Extending to all fractional ideals we get {j(E)^{\psi_{H/K}(\frak{a})}=j(\frak{a}\cdot E)}.

Let {L=\overline{K}^{\ker F}}. One can check directly that {L=K(j(E))}. This shows that {L/K} is abelian. Let {G=Gal(L/K)}. By class field theory there is some associated conductor {\frak{m}}. Considering the composition with the Artin map {I_\frak{m}\stackrel{\Psi_{L/K}}{\rightarrow} G\stackrel{F}{\rightarrow} CL(R_K)} one can see this is just the natural projection map. Now using the injectivity of {F} one can extrapolate that {\frak{m}=(1)} showing {L/K} is unramified and hence contained in the Hilbert class field.

But class field theory tells us that {I_\frak{m}} surjects onto {Cl(R_K)} and hence {F} is an isomorphism which shows {H=L} and hence {K(j(E))} is the Hilbert class field of {K}. We’ve already discussed the second item. The third just follows from identifying {Ell(R_K)} with {\{j(E_1), \ldots, j(E_h)\}} and using the fact that {Ell(R_K)} is a {Cl(R_K)}-torsor. The fourth just comes from the fact that {I_\frak{m}} consists of all fractional ideals because {\frak{m}=(1)}.

I know, this was pretty skimpy on details, but the point of this series should be to see some of the ideas and results from basic CM theory for elliptic curves and not full blown explanations. The results in this post are really cool in my mind because it takes these very classical purely field theory questions and converts them to geometric questions about elliptic curves and vice-versa. Given a quadratic imaginary {K}, it is easy to cook up an elliptic curve {E} with CM by {R_K}. If you want to know the maximal unramified abelian extension of {K}, we now know we need only figure out {j(E)} because {H=K(j(E))}.


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