Adelic algebraic group
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.