Adeles and Restricted Products

Let $F$ be a number field with completions $F_v$.

Given compact open subrings $\mathcal O_v\subset F_v$ at almost all finite $v$, the restricted product $\prod_v' F_v=\{(x_v): x_v\in \mathcal O_v \text{ for almost all finite }v\}.$

The adele ring is $\mathbb{A}_F:=\prod_v' F_v$ (with $\mathcal O_v$ the local integer rings at finite $v$).

Key property: $F$ embeds diagonally into $\mathbb{A}_F$, and automorphic forms live on $G(F)\backslash G(\mathbb{A}_F)$ (see automorphic forms ).