In the case of G being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers.
is taken to be the subspace topology in AN, the Cartesian product of N copies of the adele ring.
This was to formulate class field theory for infinite extensions in terms of topological groups.
Weil (1938) defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of Idealelemente was the group of invertible elements of this ring.
Chevalley (1951) defined the ring of adeles in the function field case, under the name "repartitions"; the contemporary term adèle stands for 'additive idèles', and can also be a French woman's name.
The term adèle was in use shortly afterwards (Jaffard 1953) and may have been introduced by André Weil.
For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over K, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of K, the product formula for valuations in K is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor.
The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.