In number theory and arithmetic geometry, the adelic points of an algebraic group over a global field form a topological group is the adele ring. For a linear algebraic group may be described as the restricted product over all places, with respect to compact open subgroups at almost all non-archimedean places.

Adelic groups provide the natural setting for automorphic forms at once. Important examples include the idele group, adelic general linear groups, adelic tori, and adelic points of reductive groups. Tamagawa measures and Tamagawa numbers are defined using Haar measures on such groups.

History of the terminology

Historically the idèles () 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.

Definition

be a global field be its adele ring is an algebraic group.

For a linear algebraic group, the adelic group can be described as a restricted product outside a finite set of places..

at almost all non-archimedean places.

. For linear algebraic groups, this diagonal image is discrete under the adelic topology. is one of the basic spaces on which automorphic forms and automorphic representations are defined.

the idele group.

Tamagawa measures and Tamagawa numbers

. A Tamagawa measure is a canonically normalized Haar measure.

.

. The product formula implies that the resulting adelic measure is unchanged. This observation was one of the basic motivations for defining Tamagawa measures adelically.

is standard in the theory of automorphic forms.

For many reductive groups, Tamagawa numbers encode arithmetic information. In the case of tori they are related to class groups and Galois cohomology, while for semisimple groups they are connected with the arithmetic of quadratic forms and the volume of arithmetic quotients. This conjecture was proved over number fields by work culminating in Robert Kottwitz's proof, building on earlier work of Robert Langlands.

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