Sec. 3] geometric interpretation 23

3.10. A -rational point on the projective space is actually a morphism

x: Spec → Pn of schemes. The valuative criterion for properness implies that

there is a unique extension to a morphism from Spec ,

Spec

x

∩

Pn

Spec .

x

♥

♥

♥

♥

♥

♥

♥

3.11. Observation (Arakelov). Let x ∈ Pn( ), and let

x: Spec →

Pn

be its extension to Spec . Then

hnaive(x) = deg

(

x∗(O(1),

.

min

)

)

.

Proof. Write x in coordinates as (x0 : . . . : xn) for x0, . . . , xn ∈ . Then the

pullback of Xi ∈

Γ(Pn

, O(1)) is xi ∈ Γ(Spec ,

x∗(O(1))).

Since O(1) is generated

by the global sections Xi, we have

x∗(O(1))

= x0, . . . , xn = gcd(x0, . . . , xn) · .

We choose an index i0 such that xi0 = 0. Using this section, we obtain

deg

(x∗(O(1)),

. ) = log #

(

x∗(O(1))/xi0 x∗(O(1))

)

− log xi0

= log #

(

gcd(x0, . . . , xn) /xi0

)

− log min

i=0,...,n

xi0

xi

= log

xi0

gcd(x0, . . . , xn)

+ log max

i=0,...,n

xi

xi0

= log

max

i=0,...,n

|xi|

gcd(x0, . . . , xn)

,

which is exactly the logarithmic height of x.

3.12. This motivates the following general definition.

Definition. Let X be an arithmetic variety, and let L be a hermitian line bundle

on X .

Then the height with respect to L of the -rational point x ∈ X ( ) is given by

hL (x) = deg

(

x∗L

)

.

Here, x: Spec → X is the extension of x to Spec .

3.13. Examples. i) Let X = Pn , and let L be the invertible sheaf O(1)

equipped with the minimum metric. Then, as shown in Observation 3.11, the height

with respect to L is the naive height.