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§7. The Canonical Height and Norm on an Elliptic Curve 137

(6.8) Theorem. For the canonical height h on P m (Q) and a Q-morphism f :
P m (Q) → P m (Q) of degree d the difference
h( f (y)) − d · h(y)

is bounded on P m (Q). In particular, h is a height on P m (Q).

Proof. Using (6.6), we have an upper estimate for H( f (y)) where

d d
H( f (y)) = max | f i (y)|≤ max c( f i ) H(y) = c 2 H(y) .
i i
To obtain a lower estimate, we use (6.7)

s+d s
|b|·|y i | = max c(g ij ) H(y) | f j (y)|
i, j
j

≤ max c(g ij ) (m + 1)H(y) s max | f j (y)| .
i, j j
Since any common factor among the f j (y) divides b, by (6.7), it follows that
max j | f j (y)|=|b|H( f (y)). Using this inequality and taking the maximum over
i of the above inequality, we have

s
|b|· H(y) s+d ≤ max c(g ij ) (m + 1)H(y) |b|H( f (y)).
i, j
After cancellation, we have for some c 1 > 0 the inequality

d
c 1 · H(y) ≤ H( f (y)).
Thus for all y in P m (Q) it follows that
d
d
c 1 · H(y) ≤ H( f (y)) ≤ c 2 · H(y) ,
d
and after taking the logarithm of both sides, we see that log(H( f (y))/H(y) ) =
h( f (y)) − d · h(y) is bounded on P m (Q). This proves the theorem.

§7. The Canonical Height and Norm on an Elliptic Curve

Using the canonical height h on P m (Q) for k a number field which is defined in §8
for k = Q, we can define a height h E on E(k) the group of k-valued points on an
elliptic curve E over k. First, we need another lemma about multiplication by 2 on
an elliptic curve.
3
2
2
(7.1) Lemma. Let E be an elliptic curve defined by y = f (x) = x +ax +bx +c
over a field k. Define the function q : E(k) → P 1 (k) defined by q(x, y) = (1, x) and