Adelic quotient groups on arithmetic surfaces

On an arithmetic surface there is group of Parshin-Beilinson adeles and natural subgroups of this group similar to the group of usual adeles on a number field and a subgroup which is the additive group of this field. But the group of Parshin-Beilinson adeles does not take into account the fibre of the arithmetic surface over the infinite point of the base, and therefore the Parshin-Beilinson adelic quotient group is not compact as it is in the case of a number field. I will talk on extension of Parshin-Beilinson adelig group on arithmetic surface and corresponding natural subgroups when the fibre over the infinite point of the base is taken into account, and such that the correpsonding quotient group is compact. I will discuss also an analogy with a projective algebraic surface fibered over a projective curve and defined over a finite field.