30 the concept of a height [Chap. I

Finally, there is the ambiguity caused by the choice of the section s. Two non-zero

sections s, s ∈ Γ(Spec K, OSpec

K

) differ by a factor t ∈ K∗. Thus, the correspond-

ing images in

w

differ by the summand (− log |t|w)w∈Val(K). The assertion fol-

lows.

4.19. Definition (Arithmetic degree). For an adelically metrized invertible

sheaf (L , . ) on Spec K, define its arithmetic degree by

deg (L , . ) := s(l(L , . )) .

Here, s:

w

im λ → is the summation map.

4.20. Remarks. i) The product formula implies that the summation map s

factors via

w

im λ.

ii) The arithmetic degree is a group homomorphism deg : Pic(Spec K) → .

iii) For every L ∈ Pic(Spec ), one has deg (iSpec (L )) = deg (L ). Indeed, this

is directly seen from the various definitions.

4.21. Remarks. The adelic Picard groups as defined here tend to be very

large groups. They are not designed to be particularly interesting invariants for a

purpose such as distinguishing between non-isomorphic varieties.

They just give a general framework for the determination of an individual

height function. This framework is in fact more flexible than the concept of a

height with respect to a hermitian line bundle, introduced in the definition in Sub-

section 3.12.

Further, the height function defined by an ample adelically metrized invertible

sheaf L differs only by a bounded function from that defined in a naive way by the

underlying invertible sheaf L , cf. Definition 2.10.

4.22. Definition. Let X be a regular, projective variety over , and let

L ∈ Pic(X) be an adelically metrized invertible sheaf.

Then the absolute height with respect to L of an K-valued point x ∈ X(K) for K a

number field is given by

h

L

(x) :=

1

[K : ]

deg L |x .

4.23. Example. Consider the situation that X = Pn and L = O(1),

equipped with the adelic metric, induced by the model

(Pn

, O(1), 1).

Then hL coincides with the naive height hnaive.

Proof. This is immediate from Definitions 4.22 and 4.6, combined with Defini-

tions 2.4 and 2.9.