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.
Previous Page Next Page