Adeles and restricted products

Let $F$ be a global field. For each nonarchimedean place $v$ , let $\mathcal O_v\subset F_v$ be the valuation ring. The finite adele ring is the restricted product

\mathbb A_{F,f} = \prod_{v\nmid\infty}'F_v = \left\{ (x_v)_v:x_v\in\mathcal O_v \text{ for all but finitely many }v \right\}.

For a number field,

\mathbb A_F = \left(\prod_{v\mid\infty}F_v\right) \times\mathbb A_{F,f}.

A global function field has no archimedean places, so its full adele ring is the corresponding restricted product over all closed points of its curve.

Restricted-product topology

A basic open set is a product $\prod_v U_v$ , where $U_v\subset F_v$ is open and $U_v=\mathcal O_v$ for all but finitely many $v$ . This topology makes $\mathbb A_F$ a locally compact topological ring. It is not the subspace topology inherited from the unrestricted direct product.

Global diagonal

The diagonal embedding $F\hookrightarrow\mathbb A_F$ has discrete image, and the additive quotient $\mathbb A_F/F$ is compact. These facts underlie adelic Fourier analysis and the product formula.

For an algebraic group $G$ , its adelic points are themselves a restricted product, using integral models or hyperspecial subgroups at almost all places. Automorphic forms then live on $G(F)\backslash G(\mathbb A_F)$ .

Measures

A restricted product of local Haar measures requires almost-all normalizations, often $\operatorname{vol}(\mathcal O_v)=1$ . Tamagawa measures incorporate convergence factors and are additional global data; they are not automatic from the set-theoretic restricted product.

References

John Tate, “Fourier analysis in number fields and Hecke's zeta functions,” in Algebraic Number Theory, 1967.
André Weil, Basic Number Theory, Springer, 1967.