Visualizing complex multiplication

Here's an attempt to create a way to visualize elliptic curves with complex multiplication. With any luck, it will also give some insight into why the term "complex multiplication" is used.

An endomorphism of an elliptic curve is a homomorphism that maps points on the curve to other points on the curve. The multiplication-by-n maps

φ_n: P→ nP

all do this. For most elliptic curves, those are all the endomorphisms that exist, but some elliptic curves have other endomorphisms. Curves like those are said to have complex multiplication, and if we look at a few examples, it's not hard to see why it's called this.

Example 1

For the elliptic curve

y² = x³ + x

the mapping given by

φ(x,y) = (-x,iy)

is an endomorphism that’s not a multiplication-by-n map. It also has the property that

φ²(P) = φ∘φ(P) = (x,-y) = -P

Because φ² acts roughly like multiplication by -1, we can think of φ as acting roughly like multiplication by the complex number  i.

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω₁ and ω₂, we find that

ω₁ = 1.85407 – 1.85407 i

and

ω₂ = 1.85407 + 1.85407 i

Note that ω₂ = iω₂.

Here’s what the lattice L of points nω₁ + mω₂ looks like:

 

Here's what multiplication by i does to a typical point in this lattice:

 

Note that this lattice has the property that iL = L, so the same multiplication by i that maps points on the elliptic curve to other points on the curve also maps points in the lattice to other points in the lattice.

Example 2

For the elliptic curve

y² = x³ + 1

we have that the mapping given by

φ(x,y) = (ξx,y)

where

ξ = e^2πi/3, a complex cube root of 1 that's also a root of x² + x + 1 = 0

is an endomorphism that’s not a multiplication-by-n map. It also has the property that

φ³(x,y) = φ∘φ∘φ(x,y) = (x,y)

Because φ³ acts roughly like multiplication by 1, we can think of φ as acting roughly like multiplication by the complex number ξ.

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω₁ and ω₂, we find that

ω₁ = 2.10327 -1.21433 i

and

ω₂ = 2.42865 i

Note that we have that ω₂ = ξω₁.

Here’s what the lattice L of points nω₁ + mω₂ looks like:

 

Here's what multiplication by ξ does to a typical point in this lattice:

Note that this lattice has the property that ξL = L, so the same multiplication by ξ that maps points on the elliptic curve to other points on the curve also maps points in the lattice to other points in the lattice.

Example 3

Consider the elliptic curve

y² = x³ - 4320 x + 96768

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω₁ and ω₂, we find that

ω₁ = -0.296858 i

and

ω₂ = 0.419821

Note that ω₂ = √2iω_1.

Here’s what the lattice L of points nω₁ + mω₂ looks like:

 

In this lattice, if we multiply a point z by (√2i)² we get the point -2z. This looks something like this: 

This will also be reflected in the points on the elliptic curve, so this curve has complex multiplication by √2i. In this case, however, it's not at all obvious what the endomorphism is that's not a multiplication-by-n map.