Editing Complex multiplication (section)

==Example of the imaginary quadratic field extension==

[[Image:Lattice torsion points.svg|right|thumb|300px|An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ''ω''<sub>1</sub> and ''ω''<sub>2</sub>. The four-torsion is also shown, corresponding to the lattice 1/4 Λ containing Λ.

The example of an elliptic curve corresponding to the Gaussian integers occurs when ω<sub>2</sub> = ''i'' ''ω''<sub>1</sub>.]]

Consider an imaginary quadratic field <math display="inline">K = \Q\left(\sqrt{-d}\right) , \, d \in \Z, d > 0</math>.
An elliptic function <math>f</math> is said to have '''complex multiplication''' if there is an algebraic relation between <math>f(z)</math> and <math>f(\lambda z)</math> for all <math>\lambda</math> in <math>K</math>.

Conversely, Kronecker conjectured – in what became known as the ''[[Kronecker Jugendtraum]]'' – that every abelian extension of <math>K</math> '''could be obtained''' by the (roots of the) equation of a suitable elliptic curve with complex multiplication.  To this day this remains one of the few cases of [[Hilbert's twelfth problem]] which has actually been solved.

An example of an elliptic curve with complex multiplication is

:<math>\mathbb{C}/ (\theta \mathbb{Z}[i])</math>

where '''Z'''[''i''] is the [[Gaussian integer]] ring, and ''θ'' is any non-zero complex number. Any such complex [[torus]] has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as

:<math>Y^2 = 4X^3 - aX</math>

for some <math>a \in \mathbb{C} </math>, which demonstrably has two conjugate order-4 [[automorphism]]s sending

:<math>Y \to \pm iY,\quad X \to -X</math>

in line with the action of ''i'' on the [[Weierstrass elliptic function]]s.

More generally, consider the lattice Λ, an additive group in the complex plane, generated by <math>\omega_1,\omega_2</math>. Then we define the Weierstrass function of the variable <math>z</math> in <math>\mathbb{C}</math> as follows:
:<math>\wp(z;\Lambda) = \wp(z;\omega_1,\omega_2) = \frac{1}{z^2} + \sum_{(m,n)\ne (0,0)} \left\{\frac{1}{(z+m\omega_1+n\omega_2)^2} - \frac{1}{\left(m\omega_1+n\omega_2\right)^2}\right\},</math>
and
:<math>g_2 = 60\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-4}</math>
:<math>g_3 =140\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-6}.</math>
Let <math>\wp'</math> be the derivative of <math>\wp</math>. Then we obtain an isomorphism of complex Lie groups:
:<math>w\mapsto(\wp(w):\wp'(w):1) \in \mathbb{P}^2(\mathbb{C}) </math>
from the complex torus group <math>\mathbb{C}/\Lambda</math> to the projective elliptic curve defined in homogeneous coordinates by
:<math>E = \left\{ (x:y:z) \in \mathbb{C}^3 \mid y^2z = 4x^3 - g_2xz^2 - g_3 z^3 \right\} </math>
and where the point at infinity, the zero element of the group law of the elliptic curve, is by convention taken to be <math>(0:1:0)</math>.  
If the lattice defining the elliptic curve is actually preserved under multiplication by (possibly a proper subring of) the ring of integers <math>\mathfrak{o}_K</math> of <math>K</math>, then the ring of analytic automorphisms of <math> E = \mathbb{C}/\Lambda</math> turns out to be isomorphic to this (sub)ring. 

If we rewrite <math>\tau = \omega_1/\omega_2</math> where <math>\operatorname{Im}\tau > 0</math> and <math>\Delta(\Lambda) = g_2(\Lambda)^3 - 27g_3(\Lambda)^2</math>, then
:<math> j(\tau)=j(E)=j(\Lambda)=2^63^3g_2(\Lambda)^3/\Delta(\Lambda)\ .</math>
This means that the [[j-invariant]] of <math>E</math> is an [[algebraic number]] – lying in <math>K</math> – if <math>E</math> has complex multiplication.