(Complex multiplication and class field theory)
Let \(E/\CC\) have complex multiplication by the order \(R\) of discriminant \(D\) in the imaginary quadratic field \(K\), and let \(h(R) \da \abs{\Pic(R)}\) be the class number of that order. Then:
\(j(E)\) is an algebraic integer;
\([K(j(E)) : K] = h(R)\), and \(K(j(E))/K\) is abelian, with \[ \Gal\big( K(j(E)) / K \big) \iso \Pic(R) ; \]
\(j(E) \in \ZZ\) if and only if \(h(R) = 1\).
There are exactly thirteen orders with \(h(R) = 1\), so exactly thirteen values of \(j\) arise this way, namely those of discriminant \[ D \in \ts{ -3,\ -4,\ -7,\ -8,\ -11,\ -12,\ -16,\ -19,\ -27,\ -28,\ -43,\ -67,\ -163 } . \]
(Thirteen orders, nine fields)
The two counts refer to different objects: fields versus orders.
Nine is the number of imaginary quadratic fields of class number one: \(d_K \in \ts{-3,-4,-7,-8,-11,-19,-43,-67,-163}\). That is the Baker–Heegner–Stark theorem, and it is a statement about maximal orders only.
Thirteen is the number of imaginary quadratic orders of class number one, and an order need not be maximal. Beyond the nine maximal ones there are four more, each a non-maximal order of conductor \(f\) inside a field already on the list: \[ D = -12 \ (f=2 \text{ in } \QQ(\sqrt{-3})), \quad -16 \ (f=2 \text{ in } \QQ(i)), \quad -27 \ (f=3 \text{ in } \QQ(\sqrt{-3})), \quad -28 \ (f=2 \text{ in } \QQ(\sqrt{-7})) . \] Endomorphism rings of elliptic curves range over all orders, not just maximal ones, so the relevant count for CM \(j\)-invariants is thirteen. The \(\tau = 2i\) curve is exactly the \(D = -16\) entry, and its \(j\)-invariant is the rational integer \(66^3 = 287496\). For example, \(D=-4\) gives \(j=1728\), \(D=-3\) gives \(j=0\), and \(D=-163\) gives \(j=-640320^3\), which explains the near-integrality of \(e^{\pi\sqrt{163}}\).
What the theorem buys is a bridge in the direction that was blocked. CM is immediate from an imaginary-quadratic lattice parameter \(\tau\), while recovering such a parameter from an algebraic equation is transcendental. The class field theory statement replaces that passage by an arithmetic test on \(j\) itself: a transcendental or non-integral \(j\) rules out CM by a class-number-one order immediately. For \(y^2=x(x-1)(x-2)\), the value \(j=\tfrac{2^6\cdot7^3}{3^2}\) is not an algebraic integer, hence cannot be a CM \(j\)-invariant.
The Galois statement is the reason the subject is called complex multiplication and not just “curves with extra endomorphisms”: \(K(j)\) is the ring class field of \(R\), and for \(R = \OO_K\) it is the Hilbert class field, the maximal unramified abelian extension of \(K\). So the \(j\)-invariants of CM curves generate the abelian extensions of an imaginary quadratic field, which is Kronecker’s Jugendtraum for that case: the values of a transcendental function at special points play the role that roots of unity play for \(\QQ\). The isomorphism \(\Gal \iso \Pic(R)\) is induced by the action of the class group on lattices — a class \([\mathfrak{a}]\) sends \(\CC/\Lambda\) to \(\CC/\mathfrak{a}^{-1}\Lambda\), which is another curve with the same \(R\), and this action is simply transitive on the isomorphism classes. That is also why \(j\) has exactly \(h(R)\) conjugates: they are the \(j\)-invariants of the \(h(R)\) curves in one orbit.
Dependencies
T-CRVCMComplex multiplication, and $\operatorname{End}(E,p_0)$ as an order in an imaginary quadratic fielduses
Backlinks
FE-CRVSSPRIMESFixing the curve and varying $p$, and the density of supersingular primesrelated-to