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),
(x) = hL (f(x)) .
Proof. According to Definition 4.22, we have
[K : ]
[K : ]
= hL (f(x)) .
4.25. Proposition. Let X be a projective variety over .
a) Let .
be adelic metrics on one and the same invertible sheaf L .
Then there is a constant C such that
, . 1)
(x) − h(L
, . 2)
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
], the adelic metrics .
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 [
] for m = 0.
As an analogous argument applies to the invertible sheaves L1 and L2, we see that
coincide up to finitely many primes ν1, . . . , νk. Further, an adelic
metric consists of bounded metrics. Hence, there is a constant D such that
≤ D .
for i = 1, . . . , k.