Sec. 4] the adelic picard group 31

4.24. Lemma. Let f : X → Y be a morphism of projective varieties over ,

and let L be an adelically metrized invertible sheaf on Y .

Then, for every number field K and every x ∈ X (K),

hf

∗L

(x) = hL (f(x)) .

Proof. According to Definition 4.22, we have

hf

∗L

(x) =

1

[K : ]

deg

(

x∗(f ∗L

)

)

=

1

[K : ]

deg

(

(f◦x)∗L

)

= hL (f(x)) .

4.25. Proposition. Let X be a projective variety over .

a) Let .

1

and .

2

be adelic metrics on one and the same invertible sheaf L .

Then there is a constant C such that

|h(L

, . 1)

(x) − h(L

, . 2)

(x)| C

for every number field K and every x ∈ X(K).

b) Let L1 and L2 be two adelically metrized invertible sheaves. Then

h(L1⊗L2) = hL1 + hL2 .

c) Let L be an adelically metrized invertible sheaf such that the underlying invert-

ible sheaf is ample. Then, for every B, D ∈ , there are only finitely many points

x ∈ X(K) such that [K : ] D and

hL (x) B .

Proof. a) Over some Spec [ 1

m

], the adelic metrics .

1

and .

2

are induced

by models (X1, L1,n1) and (X2, L2,n2), respectively. We may assume without

restriction that n1 = n2 =: n, as a model (X , L , n) may always be replaced by

(X , L ⊗n , nn ) without change.

Further, there is a birational equivalence ι: X1

❴ ❴

X2 that extends the identity

map on X. It defines an isomorphism U1

❴ ❴

U2 between suitable open subschemes

completely containing the generic fibers. As X1 and X2 are proper, the complemen-

tary closed subsets are contained in finitely many special fibers. Hence, ι induces

an isomorphism over some Spec [

1

mm

] for m = 0.

As an analogous argument applies to the invertible sheaves L1 and L2, we see that

.

1

and .

2

coincide up to finitely many primes ν1, . . . , νk. Further, an adelic

metric consists of bounded metrics. Hence, there is a constant D such that

1

D

.

1,νi

≤ .

2,νi

≤ D .

1,νi

for i = 1, . . . , k.