Adeles of a global function field

Let $F=K$ be a global function field. Write $F_v$ for its completion at $v$ , and $\mathcal O_v$ for the valuation ring at a nonarchimedean place. At any archimedean places put $\mathcal O_v=F_v$ .

Let $K$ be the specified global function field. the adele ring of K is the restricted product

\mathbb A_{K}=\prod_{v}{}'(F_v,\mathcal O_v),

An adele is integral at all but finitely many nonarchimedean places. No integrality condition is imposed at archimedean places.

Topology

A basic open set is $\prod_v U_v$ , with each $U_v$ open in the corresponding local field and $U_v=\mathcal O_v$ for all but finitely many nonarchimedean places. Operations are coordinatewise. This restricted-product topology is part of the object; the topology inherited from the unrestricted direct product is different.

Scope of the places

For a number field the archimedean factors are included; for a function field every place is nonarchimedean. In particular, the place at infinity of $\mathbb F_q(t)$ is still included and is nonarchimedean.

Units

An adele is a unit exactly when every coordinate is nonzero and its inverse is again an adele. The resulting idele group carries its own restricted-product group topology.

References

J. S. Milne, Class Field Theory, Chapter V, §4, Ideles, pp. 169–172, especially restricted-product topology and Notes.