[Math] Elliptic Curves with CM and Class Field Theory

class-field-theoryelliptic-curvesnt.number-theory

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known that for an integral ideal $\mathfrak{m}$ of $O_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for $E/H$. (This is stated, for example, on page 135 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

My question is this: what if $E$ has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if $E$ has CM by an order of conductor $\mathfrak{f}$, then $K(j(E))$ is the ring class field of $K$ with conductor $\mathfrak{f}$, but I'm wondering if anything more can be said.

Best Answer

In fact Shimura handled the case of an abelian variety $A$ with complex multiplication by an order $O$ inside the maximal order $O_K$ of the CM field $K$. A very good modern reference is the following article by Marco Streng: An explicit version of Shimura's reciprocity law for Siegel modular functions. I also recommend his PhD thesis. Both are available here: http://pub.math.leidenuniv.nl/~strengtc/research.html

Now for the statement (see Theorem 2.2 of the paper above): let $\tau$ be the element of the Siegel space that corresponds to the abelian variety $A$ above. The Siegel modular functions of level $N$ with $q$-expansion in $\mathbb{Q}(\zeta_N)$ evaluated at $\tau$ generate an abelian extension $H(N)$ of the reflex field $K^r$ (for the type norm $\Phi$ associated to $A$).

This abelian extension is associated by class field theory to the class group $I_K(NF)/H_{\Phi,O}(N)$ where

  • $F$ is the conductor of $O$
  • $I_K(NF)$ are the fractional ideals of $K$ prime to $NF$
  • $H_{\Phi,O}(N)$ are the ideals $\mathfrak{a} \in I_K(NF)$ such that $\exists \mu \in K$ with $N_{\Phi^r,O}(\mathfrak{a})=\mu O$, $\mu \overline{\mu}=N(\mathfrak{a})\in \mathbb{Q}$ and $\mu \equiv 1 {\bmod^\times} NO$. (Here $N_{\Phi^r,O}$ is the type norm from the reflex field $K^r$ to $K$.)

Now specializing to elliptic curves, we get that

  • If $E$ has CM by $O_K$, then $K(j(E),h(E[𝔪]))$ is the ray class field associated to the class group $I_K(m)/\{ \mu O_K \mid \mu \equiv 1 {\bmod^\times} m\}$
  • If $E$ has CM by $O$ where the conductor of $O$ is $F \in \mathbb{Z}$, then $K(j(E))$ is the ring class field of $O$, meaning the extension associated to the class group $I_K(F)/\{ \text{principal ideals of $O$ primes to $F$}\}$. And finally, $K(j(E),h(E[𝔪]))$ will correspond to $I_K(mF)/\{ \mu O \mid \text{$\mu$ is prime to $F$ and}\ \mu \equiv 1 {\bmod^\times} mO\}$. (At least we have the inclusion, but I am pretty sure that in the elliptic curve case, the evaluation of Weber functions at the points of $m$-torsion over the $j$-invariant give the field generated by level-$m$ modular forms evaluated at $\tau$.) So in this case we have something intermediate between a ring class field and a ray class field. [I have not checked carefully but it should be the compositium of the ring class field of conductor $F$ and the ray class field of modulus $m$ when $m$ is prime to $F$].
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