Adelic algebraic group

In number theory and arithmetic geometry, the adelic points of an algebraic group ${\displaystyle G}$ over a global field ${\displaystyle K}$ form a topological group denoted ${\displaystyle G(\mathbb {A} _{K})}$, where ${\displaystyle \mathbb {A} _{K}}$ is the adele ring of ${\displaystyle K}$. For a linear algebraic group, ${\displaystyle G(\mathbb {A} _{K})}$ may be described as the restricted product of the local groups ${\displaystyle G(K_{v})}$ over all places ${\displaystyle v}$ of ${\displaystyle K}$, with respect to compact open subgroups ${\displaystyle G({\mathcal {O}}_{v})}$ at almost all non-archimedean places.

Adelic groups provide the natural setting for automorphic forms and automorphic representations. Their basic quotients, such as ${\displaystyle G(K)\backslash G(\mathbb {A} _{K})}$, encode arithmetic information from all completions of ${\displaystyle K}$ at once. Important examples include the idele group ${\displaystyle \mathbb {A} _{K}^{\times }=\mathbb {G} _{m}(\mathbb {A} _{K})}$, adelic general linear groups ${\displaystyle \operatorname {GL} _{n}(\mathbb {A} _{K})}$, adelic tori, and adelic points of reductive groups. Tamagawa measures and Tamagawa numbers are defined using Haar measures on such groups.