2.4. The group structure on E(Q) 29

Figure 3. Louis Mordell (1888-1972) and André Weil (1906-1998).

the three points

P = (1,0), Q = (2,0), and R = (0, −3)

are generators of E(Q). This means that any other point on E(Q)

can be expressed as a Z-linear combination of P , Q and R. In other

words,

E(Q) = {a · P + b · Q + c · R : a, b, c ∈ Z}.

For instance,

(−3, −1) = P + Q, (8, −22) = P + R, (−2, −4) = P − Q,

(−1, −4) = Q − R and (3,3) = P − R.

The proof of the theorem has three fundamental ingredients: the

so-called weak Mordell-Weil theorem E(Q)/mE(Q) is finite for any

m ≥ 2; see below); the concept of height functions on abelian groups

and the descent theorem, which establishes that an abelian group A

with a height function h, such that A/mA is finite (for some m ≥ 2),

is finitely generated.