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142 6. Proof of Mordell’s Finite Generation Theorem

(8.5) Lemma. For (y 0 ,..., y m ) ∈ Z m+1 −{0} without common factors, i.e., a Z-
reduced representative of a point in P m (Q), the height
∗
h(y 0 ,..., y m ) = h (y 0 ,..., y m ).
Proof. Since each y j is in Z, it follows that a p-adic valuation satisfies |y j | p ≤ 1or
log |y j | p ≤ 0. Since at least one y j is not divisible by p, this means that |y j | p < 1
or log |y j | p < 0, and, hence, we obtain max{log |y o | p ,..., log |y m | p }= 0. We
∗
conclude that the p-adic valuations in the sum defining h do not contribute anything,
and, therefore,

∗
h (y 0 ,..., y m ) = max{log |y o | p ,..., log |y m | p }
p∈V (Q)
= max{log |y 0 | ∞ ,..., log |y m | ∞ }.

This proves the lemma.

In view of the above two lemmas the next definition is a natural extension of
(6.3).

(8.6) Definition. For a global field k the canonical height h on P m (k) is defined by

h(P) = max{log |y o | v ,..., log |y m | v },
v∈V (k)

where P = (y 0 ,..., y m ).
Since P 1 (k) = k ∪{(0, 1)} where ∞= (0, 1) and a in k is identified with (l, a),

the canonical height on k is the function h(a) = v∈V (k) max{0, log |a| v }. The proof
that h is a height in the technical sense of (6.2) follows the lines of (6.8).