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 ${\displaystyle G(A)}$ is taken to be the subspace topology in A^N, the Cartesian product of N copies of the adele ring. In this case, ${\displaystyle G(A)}$ is a topological group.

Historically the idèles (/ɪˈdɛlz/) were introduced by Chevalley (1936) under the name "élément idéal", which is "ideal element" in French, which Chevalley (1940) 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. 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. Tate (1950) defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.

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. The general construction of adelic algebraic groups by Ono (1957) 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 ${\displaystyle G=GL_{1}}$. Here the set of ideles 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

${\displaystyle \{(t,t^{-1})\},}$

the topology correctly assigned to the idele group is that induced by inclusion in A²; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A.

Inside A^N, the product K^N lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group

${\displaystyle I(K)/K^{\times }\,}$

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a compact group; the proof of this is essentially equivalent to the finiteness of the class number.

The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of the idele class group, now usually called Hecke characters or Größencharacters, give rise to the most basic class of L-functions.

For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of

G(A)/G(K).

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.

• Ring of adeles

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• Chevalley, Claude (1936), "Généralisation de la théorie du corps de classes pour les extensions infinies.", Journal de Mathématiques Pures et Appliquées (in French), 15: 359–371, JFM 62.1153.02
• Chevalley, Claude (1940), "La théorie du corps de classes", Annals of Mathematics, Second Series, 41 (2): 394–418, doi:10.2307/1969013, ISSN 0003-486X, JSTOR 1969013, MR 0002357
• Chevalley, Claude (1951), Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, Providence, R.I.: American Mathematical Society, MR 0042164
• Jaffard, Paul (1953), Anneaux d'adèles (d'après Iwasawa), Séminaire Bourbaki, Secrétariat mathématique, Paris, MR 0157859
• Ono, Takashi (1957), "Sur une propriété arithmétique des groupes algébriques commutatifs", Bulletin de la Société Mathématique de France, 85: 307–323, doi:10.24033/bsmf.1491, ISSN 0037-9484, MR 0094362
• Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026
• Weil, André (1938), "Zur algebraischen Theorie der algebraischen Funktionen.", Journal für die Reine und Angewandte Mathematik (in German), 179: 129–133, doi:10.1515/crll.1938.179.129, ISSN 0075-4102, S2CID 116472982

• Rapinchuk, A.S. (2001) [1994], "Tamagawa number", Encyclopedia of Mathematics, EMS Press