2.10. Homogeneous spaces 61

or, equivalently, one can subtract both equations to get

C(δ1,δ2) :

e1 − e2 = δ2Y

2

−

δ1X2,

e3 − e2 = δ2Y 2 − δ1δ2Z2.

The space C(δ1,δ2) is the intersection of two conics, and it may have

rational points or not. If (δ1,δ2,δ3) is in the image of δ, however,

then the space C(δ1,δ2) must have a rational point; i.e., there are

X, Y, Z ∈ Q that satisfy the equations of C(δ1,δ2). Moreover, if

X0,Y0,Z0 ∈ Q are the coordinates of a point in C(δ1,δ2), then

P = (e1 + δ1X0

2,

δ1δ2X0Y0Z0) (2.12)

is a rational point on E(Q) such that δ(P ) = (δ1,δ2,δ3). The spaces

C(δ1,δ2) are called homogeneous spaces and are extremely helpful

when we try to calculate the Mordell-Weil group of an elliptic curve.

We record our findings in the form of a proposition, for later use:

Proposition 2.10.3. Let E/Q be an elliptic curve with Weierstrass

equation

y2

= (x−e1)(x−e2)(x−e3), with ei ∈ Z and e1 +e2 +e3 = 0.

Let δ : E(Q)/2E(Q) → ΓΔ be the injection given by Corollary 2.9.7,

and let δ(E) := δ(E(Q)/2E(Q)) be the image of δ in ΓΔ. Then:

(1) If (δ1,δ2,δ3) ∈ δ(E), then the space C(δ1,δ2) has a point

(X0,Y0,Z0) with rational coordinates, X0,Y0,Z0 ∈ Q.

(2) Conversely, if C(δ1,δ2) has a rational point (X0,Y0,Z0),

then E(Q) has a rational point

P = (e1 + δ1X0

2,

δ1δ2X0Y0Z0).

(3) Since δ is a homomorphism and δ(E) is the image of δ, it

follows that δ(E) is a subgroup of ΓΔ. In particular:

• If (δ1,δ2,δ3) and (δ1,δ2,δ3) are elements of the image,

then their product (δ1 · δ1, δ2 · δ2, δ3 · δ3) is also in the

image;

• If (δ1,δ2,δ3) ∈ δ(E) but (δ1,δ2,δ3) ∈ ΓΔ is not in the

image, then their product (δ1 · δ1, δ2 · δ2, δ3 · δ3) is not

in the image δ(E);

• If C(δ1,δ2) and C(δ1,δ2) have rational points, then

C(δ1 · δ1, δ2 · δ2) also has a rational point;