The singular elliptic curve y^2 = x^3

Consider the singular elliptic curve

E/Q: y² = x³

which is singular at the point S = (0,0).

Even though this curve is singular, we can still use the usual rule for adding points to get a group for all of the non-singular points: E_ns(Q) = E(Q) \ S. When we do this we find something interesting: the group of non-singular points on this curve is isomorphic to the rationals under addition, or that (E_ns, +) is isomorphic to (Q,+). And because this is true, we can see that E_ns(Q) isn't finitely generated, which is always the case with non-singular curves (the Mordell-Weil theorem).

To see why (E_ns, +) is isomorphic to (Q,+), we use the function

φ: E_ns(Q) → Q

defined by

φ(P) = φ(x,y) = x / y if P ≠O and

φ(O) = 0

This has an inverse

φ^-1: Q → E_ns(Q)

defined by

φ ^-1 (t) = (1 / t²,1 / t³) if t ≠0 and

φ ^-1 (0) = O

It's easy to see that φ is one-to-one and onto. Seeing why φ is a homomorphism is a bit more complicated.

Suppose that P_i = (x_i,y_i) are elements of E_ns with φ(P_i) = t_i.

What we want is that if P₁ + P₂ = P₃, then φ(P₁) + φ(P₂) = φ(P₃), or that t₁ + t₂ = t₃.

If we have that P₁ + P₂ = P₃ then P₁, P₂ and -P₃ are collinear. From the point-slope form of a line we have that the line through P₁ and P₂ is given by

y - y₁ = m (x - x₁)

where

m = (y₂ - y₁) / (x₂ - x₁)

or that

(x₂ - x₁) (y - y₁) = (y₂ - y₁) (x - x₁)

This line also passes through -P₃ = (x₃, -y₃) so we have that

(x₂ - x₁) (-y₃ - y₁) = (y₂ - y₁) (x₃ - x₁)

We also have that P₁ = (1 / t₁²,1 / t₁³), P₂ = (1 / t₂²,1 / t₂³) and P₃ = (1 / t₃²,1/ t ₃³). Substituting x₁ = 1/t₁², etc, we find that we have that

-(t₁ - t₂) (t₁ + t₃)( t₂ + t₃) (t₁ + t₂ - t₃) / (t₁³ t₂³ t₃³) = 0

If t₁, t₂ and t₃ are all different and non-zero, this gives us that t₁ + t₂ - t₃ = 0 or that t₁ + t₂ = t₃, so φ is a homomorphism like we want. The other cases can be handled similarly.