In number theory and arithmetic geometry, the adelic points of an algebraic group <math>G</math> over a global field <math>K</math> form a topological group denoted <math>G(\mathbb A_K)</math>, where <math>\mathbb A_K</math> is the adele ring of <math>K</math>. For a linear algebraic group, <math>G(\mathbb A_K)</math> may be described as the restricted product of the local groups <math>G(K_v)</math> over all places <math>v</math> of <math>K</math>, with respect to compact open subgroups <math>G(\mathcal O_v)</math> at almost all non-archimedean places.
Adelic groups provide the natural setting for automorphic forms and automorphic representations. Their basic quotients, such as <math>G(K)\backslash G(\mathbb A_K)</math>, encode arithmetic information from all completions of <math>K</math> at once. Important examples include the idele group <math>\mathbb A_K^\times=\mathbb G_m(\mathbb A_K)</math>, adelic general linear groups <math>\operatorname{GL}_n(\mathbb A_K)</math>, 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
Let <math>K</math> be a global field, and let <math>\mathbb A_K</math> be its adele ring. If <math>G</math> is an algebraic group over <math>K</math>, the notation
<math>G(\mathbb A_K)</math> denotes the group of adelic points of <math>G</math>. Informally, an element of <math>G(\mathbb A_K)</math> is a compatible collection of local points of <math>G</math>, one over each completion <math>K_v</math> of <math>K</math>.
For a linear algebraic group, the adelic group can be described as a restricted product
<math display="block">
G(\mathbb A_K)=\prod_v' G(K_v),
</math>
where <math>v</math> runs over the places of <math>K</math>. At almost all non-archimedean places <math>v</math>, the restricted product is taken with respect to the compact open subgroup <math>G(\mathcal O_v)</math>, where <math>\mathcal O_v</math> is the valuation ring of <math>K_v</math>, after choosing an integral model of <math>G</math> outside a finite set of places.
The simplest examples are the additive and multiplicative groups. For the additive group <math>\mathbb G_a</math>,
<math display="block">
\mathbb G_a(\mathbb A_K)=\mathbb A_K.
</math>
For the multiplicative group <math>\mathbb G_m</math>,
<math display="block">
\mathbb G_m(\mathbb A_K)=\mathbb A_K^\times,
</math>
the idele group of <math>K</math>. The topology on <math>\mathbb A_K^\times</math> is the restricted product topology with respect to <math>\mathcal O_v^\times</math> at almost all non-archimedean places; equivalently, it is the topology induced by the embedding
<math display="block">
x\longmapsto (x,x^{-1})
</math>
of <math>\mathbb A_K^\times</math> into <math>\mathbb A_K\times \mathbb A_K</math>. It is generally finer than the subspace topology inherited from <math>\mathbb A_K</math>.
Suppose first that <math>G</math> is smooth of dimension <math>d</math>, and let <math>\omega</math> be a nonzero left-invariant rational differential form of top degree on <math>G</math>, defined over <math>K</math>. Here "rational" means that <math>\omega</math> is a rational section of the canonical sheaf <math>\Omega^d_{G/K}</math>, or equivalently a top-degree differential form defined at the generic point of <math>G</math>, with coefficients in the function field <math>K(G)</math>. For each place <math>v</math> of <math>K</math>, the form <math>\omega</math> defines a local Haar measure <math>|\omega|_v</math> on the locally compact group <math>G(K_v)</math>. With the usual convergence normalizations at almost all places, these local measures define a measure on the restricted product <math>G(\mathbb A_K)</math>.