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§8. The Canonical Height on Projective Spaces over Global Fields 141

(1) (Archimedian case) |x|=|x| ∞ , the ordinary absolute value, or
(2) (non-Archimedian case) |a/b| p = (1/p) ord p (a)−ord p (b) ,the p-adic absolute
value where a and b are integers and p is a prime number in Z.
Let V (Q) denote the totality of these absolute values on Q. The product formula for
Q is

1 for x = 0,
|x| v =
0 for x = 0,
v∈V
or

log |x| v = 0 for x = 0.
v∈V
(8.1) Definition. A number field k is a finite extension of Q, a function field in one
variable over a field F is a finite separable extension of F(t), and a global field is
either a number field or a function field in one variable over F q , a finite field.
A global field k is a number field if and only if it has characteristic zero and is a
function field if and only if it has positive characteristic.
(8.2) Remark. Global fields have several things in common which can be used to
axiomatically characterize them. The most important feature is a family V (k) of
absolute values || v such that the product formula holds

1if x = 0,
|x| v =
0if x = 0.
v∈V (k)
It is this product formula generalizing the product formula for Q which leads to a
height function on P m (k) for a global field k.
(8.3) Notation. For y = (y 0 ,..., y m ) ∈ k m+1 −{0}, we introduce

∗
h (y 0 ,..., y m ) = max{log |y 0 | v ,..., log |y m | v }
v∈V (k)
This is a finite sum.
(8.4) Lemma. For a nonzero a in k and (y 0 ,..., y m ) ∈ k m+1 − 0, we have
∗
∗
h (ay 0 ,..., ay m ) = h (y 0 ,..., y m ).
Proof. We calculate

max{log |ay o | v ,..., log |ay m | v }
v∈V

= [log |a| v + max{log |y o | v ,..., log |y m | v }]
v∈V

∗
= max{log |y o | v ,..., log |y m | v }= h (y 0 ,..., y m ).
v∈V
∗
∗
This lemma says that h induces a function, also denoted h , on the projective
space P m (k).