16 the concept of a height [Chap. I

1.5. Remark. Let X ⊂

Pn

be a subvariety. Then every -rational point

on X is also a -rational point on Pn. We call the restriction of Hnaive to X( ) the

naive height on X. It is obvious that the naive height on X fulfills the fundamental

finiteness property.

1.6. Notation. i) For a prime number p, denote by .

p

the normalized p-

adic valuation. I.e., for x ∈ \{0}, let x =

±2v2 ·3v3

· . . . ·pkpk

v

be its decomposition

into prime factors. Then put

x

p

:=

p−vp

.

Further, 0

p

:= 0.

ii) We use .

∞

as an alternative notation for the usual absolute value,

x

∞

:=

x if x ≥ 0 ,

−x if x 0 .

1.7. Fact. For p a prime number or infinity, .

p

is indeed a valuation.

This means, for x, y ∈ ,

i) x

p

≥ 0,

ii) x

p

= 0 if and only if x = 0,

iii) xy

p

= x

p

· y

p

,

iv) x + y

p

≤ x

p

+ y

p

.

For p = ∞, one even has that x + y

p

≤ max{x

p

, y

p

}.

1.8. Fact (Product formula). For x ∈ \{0}, one has

p prime or ∞

x

p

= 1 .

1.9. Lemma. Let (x0 : . . . : xn) ∈ Pn ( ). Then

Hnaive(x0 : . . . : xn) =

p prime or ∞

max

i=0,...,n

xi

p

.

Proof. The product formula implies that the right-hand side remains unchanged

when (x0 : . . . : xn) is replaced by (λx0 : . . . : λxn) for λ = 0. Thus, we may

suppose that all xi are integers and gcd(x0, . . . , xn) = 1.

These assumptions imply that

max

i=0,...,n

xi

p

= 1

for every prime number p. Hence, the formula on the right-hand side may be

simplified to maxi=0,...,n |xi|. This is precisely the assertion.

1.10. Remark. Despite being so primitive, the naive height is actually suf-

ficient for most applications. For example, in the numerical experiments described

in Part C, we will always work with the naive height.