Sec. 4] the adelic picard group 33

4.26. Remarks. a) The concept of an adelically metrized invertible sheaf

as described here is essentially that introduced by V. V. Batyrev and Yu. I. Manin

in [B/M]. It was used, for example, by E. Peyre in [Pe02].

A small difference is that Batyrev and Manin fix norms on the spaces L (x) only

for x ∈ X(

ν

), not for x ∈ X(

ν

). Correspondingly, their concept of adelic metric

leads to a height for points rational over the base field and not to an absolute height.

b) There is a somewhat different concept of an adelic Picard group, which is due

to S. Zhang [Zh95b]. His definition leads to by far smaller groups. In fact, the

adelically metrized invertible sheaves that are globally induced by a model are

topologically dense in PicZh(X).

This has the following consequence, which is highly interesting for many applica-

tions. There is a continuous -multilinear map

PicZh(X) × . . . × PicZh(X)

dim X+1 times

−→ ,

which is called the adelic intersection product. It is uniquely determined by the

condition that it agrees with the arithmetic intersection product of H. Gillet and

C. Soulé [G/S90, Theorem 4.2.3] when restricted to adelically metrized invertible

sheaves induced by models. More details are given in [Zh95b].