Sec. 4] the adelic picard group 25
Since X ( ) is compact, there is a positive constant D such that
for every x ∈ X ( ). This implies the claim.
b) Clearly, for every x ∈ X ( ), one has
h(L1⊗L2)(x) = deg
= 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
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
(i(x)). It, therefore, suﬃces to show
fundamental finiteness for the height function hO(1) on
Part a) together with Observation 3.11 shows that h
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 =
for a prime number p and K =