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.
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