Idele

In number theory and arithmetic geometry, the adelic points of an algebraic group G {\displaystyle G} over a global field K {\displaystyle K} form a topological group denoted G ( A K ) {\displaystyle G(\mathbb {A} _{K})} , where A K {\displaystyle \mathbb {A} _{K}} is the adele ring of K {\displaystyle K} . For a linear algebraic group, G ( A K ) {\displaystyle G(\mathbb {A} _{K})} may be described as the restricted product of the local groups G ( K v ) {\displaystyle G(K_{v})} over all places v {\displaystyle v} of K {\displaystyle K} , with respect to compact open subgroups G ( O v ) {\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 G ( K ) ∖ G ( A K ) {\displaystyle G(K)\backslash G(\mathbb {A} _{K})} , encode arithmetic information from all completions of K {\displaystyle K} at once. Important examples include the idele group A K × = G m ( A K ) {\displaystyle \mathbb {A} _{K}^{\times }=\mathbb {G} _{m}(\mathbb {A} _{K})} , adelic general linear groups GL n ⁡ ( A K ) {\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.