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138 6. Proof of Mordell’s Finite Generation Theorem

q(0) = (0, 1). Then there is a k-morphism g : P 1 (k) → P 1 (k) of degree 4 such that
the following diagram is commutative:
2
E(k) −−−−→ E(k)
 
q   q

g
P 1 (k) −−−−→ P 1 (k).

Proof. The relation between (x, y) and 2(x, y) = (x , y ) is given by considering

the tangent line y = λx + β to E at (x, y). This line goes through (x , −y ) and as
in 1(1.4) we have
f (x)

2

2x + x = λ − a and λ = .
2y
2
Using the relation y = f (x), we obtain
2
(3x + 2ax + b) 2
2
λ = ,
2
3
4(x + ax + bx + c)
and
2
2
4
x − 2bx − 8cx + (b − 4ac)
2
x = λ − a − 2x = .
3
2
4x + 4ax + 4bx + 4c
Thus g(w, x) = (g 0 (w, x), g 1 (w, x)) is given by the forms
2 2
3
3
4
g 0 (w, x) = 4wx + 4aw x + 4bw x + 4cw ,
4 2 2 3 2 4
g 1 (w, x) = x − 2bw x − 8cw x + (b − 4ac)w .
This proves the lemma.
Observe that for the map q : E(k) → P 1 (k) the inverse image q −1 (1, x) is empty
2
when y = f (x) has no solution in k and q −1 (1, x) ={(x, ±y)} when ±y are the
2
solutions of y = f (x). Now we can easily describe the height function h E on an
elliptic curve over a number field in terms of the canonical height h on P 1 (k).
(7.2) Theorem. Let E be an elliptic curve over a number field k in Weierstrass form
3
2
y = f (x) = x + bx + c. Then there is a unique function h E : E(k) → R such
that:
(1) h E (P) − (1/2)h(x(P)) is bounded, where x(P) = q(P) is the x-coordinate of
P and h is the canonical height on P 1 (k), and
(2) h E (2P) = 4h E (P) and h E (P) = h E (−P).
Moreover, h E is proper, positive, and quadratic.