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 being 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 is taken to be the subspace topology in AN, the Cartesian product of N copies of the adele ring. In this case, is a topological group. Historically the idèles (ɪˈdɛlz) were introduced by under the name "élément idéal", which is "ideal element" in French, which then abbreviated to "idèle" following a suggestion of Hasse. (In these papers he also gave the ideles a non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in terms of topological groups. 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. defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles. 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 and may have been introduced by André Weil. The general construction of adelic algebraic groups by followed the algebraic group theory founded by Armand Borel and Harish-Chandra. An important example, the idele group (ideal element group) I(K), is the case of .