Complex multiplication

In

There is also the higher-dimensional complex multiplication theory of abelian varieties A having enough endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules.

Example of the imaginary quadratic field extension

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

Conversely, Kronecker conjectured – in what became known as the

${\displaystyle K}$

An example of an elliptic curve with complex multiplication is

${\displaystyle \mathbb {C} /(\theta \mathbb {Z} [i])}$

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

${\displaystyle Y^{2}=4X^{3}-aX}$

for some ${\displaystyle a\in \mathbb {C} }$, which demonstrably has two conjugate order-4 automorphisms sending

${\displaystyle Y\to \pm iY,\quad X\to -X}$

in line with the action of i on the Weierstrass elliptic functions.

More generally, consider the lattice Λ, an additive group in the complex plane, generated by ${\displaystyle \omega _{1},\omega _{2}}$. Then we define the Weierstrass function of the variable ${\displaystyle z}$ in ${\displaystyle \mathbb {C} }$ as follows:

${\displaystyle \wp (z;\Lambda )=\wp (z;\omega _{1},\omega _{2})={\frac {1}{z^{2}}}+\sum _{(m,n)\neq (0,0)}\left\{{\frac {1}{(z+m\omega _{1}+n\omega _{2})^{2}}}-{\frac {1}{\left(m\omega _{1}+n\omega _{2}\right)^{2}}}\right\},}$

and

${\displaystyle g_{2}=60\sum _{(m,n)\neq (0,0)}(m\omega _{1}+n\omega _{2})^{-4}}$

${\displaystyle g_{3}=140\sum _{(m,n)\neq (0,0)}(m\omega _{1}+n\omega _{2})^{-6}.}$

Let ${\displaystyle \wp '}$ be the derivative of ${\displaystyle \wp }$. Then we obtain an isomorphism of complex Lie groups:

${\displaystyle w\mapsto (\wp (w):\wp '(w):1)\in \mathbb {P} ^{2}(\mathbb {C} )}$

from the complex torus group ${\displaystyle \mathbb {C} /\Lambda }$ to the projective elliptic curve defined in homogeneous coordinates by

${\displaystyle E=\left\{(x:y:z)\in \mathbb {C} ^{3}\mid y^{2}z=4x^{3}-g_{2}xz^{2}-g_{3}z^{3}\right\}}$

and where the point at infinity, the zero element of the group law of the elliptic curve, is by convention taken to be ${\displaystyle (0:1:0)}$. If the lattice defining the elliptic curve is actually preserved under multiplication by (possibly a proper subring of) the ring of integers ${\displaystyle {\mathfrak {o}}_{K}}$ of ${\displaystyle K}$, then the ring of analytic automorphisms of ${\displaystyle E=\mathbb {C} /\Lambda }$ turns out to be isomorphic to this (sub)ring.

If we rewrite ${\displaystyle \tau =\omega _{1}/\omega _{2}}$ where ${\displaystyle \operatorname {Im} \tau >0}$ and ${\displaystyle \Delta (\Lambda )=g_{2}(\Lambda )^{3}-27g_{3}(\Lambda )^{2}}$, then

${\displaystyle j(\tau )=j(E)=j(\Lambda )=2^{6}3^{3}g_{2}(\Lambda )^{3}/\Delta (\Lambda )\ .}$

This means that the j-invariant of ${\displaystyle E}$ is an algebraic number – lying in ${\displaystyle K}$ – if ${\displaystyle E}$ has complex multiplication.

Abstract theory of endomorphisms

The ring of endomorphisms of an elliptic curve can be of one of three forms: the integers Z; an

is a unique factorization domain.

Here ${\displaystyle (1+{\sqrt {-163}})/2}$ satisfies α² = α − 41. In general, S[α] denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.

Alternatively,

${\displaystyle e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+743.99999999999925007\dots \,}$

an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.

Singular moduli

The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.

If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in the ring of integers O_K of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus. The values j(a) are then real algebraic integers, and generate the Hilbert class field H of K: the field extension degree [H:K] = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic to the ideal class group of K. The class group acts on the values j(a) by [b] : j(a) → j(ab).

In particular, if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z[i]) = j(i) = 1728.

See also

• Algebraic Hecke character
• Heegner point
• Hilbert's twelfth problem
• Lubin–Tate formal group, local fields
• global function field
  case
• Wiles's proof of Fermat's Last Theorem

Citations