Adelic algebraic group

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear algebraic group. In the case of G an abelian variety it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.

In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelic algebraic group ${\displaystyle G(A)}$ is taken to be the subspace topology in A^N, the Cartesian product of N copies of the adele ring.

An important example, the idele group I(K), is the case of ${\displaystyle G=GL_{1}}$. Here the set of ideles (also idèles /ɪˈdɛlz/) consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles. Instead, considering that ${\displaystyle GL_{1}}$ lies in two-dimensional affine space as the 'hyperbola' defined parametrically by

...
Wikipedia