Sec. 4] the adelic picard group 25

Since X ( ) is compact, there is a positive constant D such that

1

D

. 2(x)

.

1

(x)

D

for every x ∈ X ( ). This implies the claim.

b) Clearly, for every x ∈ X ( ), one has

h(L1⊗L2)(x) = deg

(

x∗(L1

⊗L2)

)

= deg

(

(x∗L1)⊗(x∗L2)

)

= deg

(x∗L1)

+ deg

(x∗L2)

= hL1 (x) + hL2 (x) .

c) There is some n ∈ such that L ⊗n is very ample. Part b) shows that it suﬃces

to verify the assertion for L

⊗n.

Thus, we may assume that L is very ample.

Let i: X → Pn be the closed embedding induced by L . Then L = i∗O(1).

It follows from Tietze’s Theorem that there is a hermitian metric on O(1) such

that L = i∗O(1). Then hL (x) = h

O(1)

(i(x)). It, therefore, suﬃces to show

fundamental finiteness for the height function hO(1) on

Pn(

).

Part a) together with Observation 3.11 shows that h

O(1)

differs from hnaive by a

bounded summand. Fact 1.3 yields the assertion.

3.16. Remark. It should be noted that there is a strong formal analogy

of the concept of a height on an arithmetic variety to the concept of a degree in

algebraic geometry over a ground field. The only obvious difference is that the role

of the sections of an invertible sheaf is now played by small sections, say, of norm

less than one. Nevertheless, it seems that the height of a point is actually some

sort of arithmetic intersection number.

This is an idea that has been formalized first by S. Yu. Arakelov [Ara] for two-

dimensional arithmetic varieties and later by H. Gillet and C. Soulé [G/S90] for

arithmetic varieties of arbitrary dimension.

We will not give any details on arithmetic intersection theory here as this is not

formally necessary for an understanding of the next chapters. To get an impres-

sion, the reader is advised to consult the articles [G/S90, G/S92] of H. Gillet

and C. Soulé, the textbook [S/A/B/K], and the references therein. The arti-

cle [B/G/S] is a good starting point, as well. It explains, in particular, how to

construct a height not only for points but for algebraic cycles.

The particular case of the arithmetic intersection theory on a curve over a number

field had been developed earlier. The articles of S. Yu. Arakelov [Ara] and G. Falt-

ings [Fa84] present the point of view taken before around 1990, which is a bit

different from today’s.

4. The adelic Picard group

i. The local case. Metrics induced by a model.

4.1. Let K be an algebraically closed valuation field. The cases we have in

mind are K =

p

for a prime number p and K =

∞

= .