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| 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 [[topological space|topology]] is straightforward only in case ''G'' is a [[linear algebraic group]]. In the case of ''G'' 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 representation]]s, and the [[arithmetic of quadratic form]]s.
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| 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 <math>G(A)</math> is taken to be the [[subspace topology]] in ''A''<sup>''N''</sup>, the [[Cartesian product]] of ''N'' copies of the adele ring.
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| ==Ideles==
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| An important example, the '''idele group''' ''I''(''K''), is the case of <math>G = GL_1</math>. Here the set of '''ideles''' (also ''idèles''; {{IPAc-en|ɪ|ˈ|d|ɛ|l|s}}) 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 <math>GL_1</math> lies in two-dimensional [[affine space]] as the '[[hyperbola]]' defined parametrically by
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| :{(''t'', ''t''<sup>−1</sup>)},
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| the topology correctly assigned to the idele group is that induced by inclusion in ''A''<sup>2</sup>; composing with a projection, it follows that the ideles carry a [[finer topology]] than the subspace topology from ''A''. | |
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| Inside ''A''<sup>''N''</sup>, the product ''K''<sup>''N''</sup> 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]]
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| :''I''(''K'')/''K''<sup>×</sup>
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| 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. | |
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| The study of the [[Galois cohomology]] of idele class groups is a central matter in [[class field theory]]. [[Character (group theory)|Characters]] of the idele class group, now usually called [[Hecke character]]s, give rise to the most basic class of [[L-function]]s.
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| ==Tamagawa numbers==
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| {{see also|Weil conjecture on Tamagawa numbers}}
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| For more general ''G'', the '''Tamagawa number''' is defined (or indirectly computed) as the measure of
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| :''G''(''A'')/''G''(''K'').
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| [[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 ''c''ω with ''c'' a non-zero element of ''K'', the [[product formula]] for [[valuation (algebra)|valuation]]s 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 group]]s contains important parts of classical [[quadratic form]] theory.
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| ==History of the terminology==
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| Historically the ''idèles'' were introduced by {{harvs|txt|last=Chevalley|authorlink=Claude Chevalley|year=1936}} under the name "élément idéal", which is "ideales Element" in French, which {{harvtxt|Chevalley|1940}} then abbreviated to "idèle". (In these papers he also gave the ideles a rather non-[[Hausdorff topology]].) This was to formulate [[class field theory]] for infinite extensions in terms of topological groups. {{harvtxt|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. {{harvtxt|Tate|1950}} defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.
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| {{harvtxt|Chevalley|1951}} defined the ring of adeles in the function field case, under the name "repartitions". The term ''adèle'' (short for additive idèles, and also a French girls' name) was in use shortly afterwards {{harv|Jaffard|1953}} and may have been introduced by [[André Weil]]. The general construction of adelic algebraic groups by {{harvtxt|Ono|1957}} followed the algebraic group theory founded by [[Armand Borel]] and [[Harish-Chandra]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Chevalley | first1=Claude | title=Généralisation de la théorie du corps de classes pour les extensions infinies. | language=French | jfm=62.1153.02 | year=1936 | journal=Journal de Mathématiques Pures et Appliquées | volume=15 | pages=359–371}}
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| *{{Citation | last1=Chevalley | first1=Claude | title=La théorie du corps de classes | jstor=1969013 | mr=0002357 | year=1940 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=41 | pages=394–418}}
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| *{{Citation | last1=Chevalley | first1=Claude | title=Introduction to the Theory of Algebraic Functions of One Variable | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys, No. VI | mr=0042164 | year=1951}}
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| *{{Citation | last1=Jaffard | first1=Paul | title=Anneaux d'adèles (d'après Iwasawa) | url=http://www.numdam.org/item?id=SB_1954-1956__3__23_0 | publisher=Secrétariat mathématique, Paris | series=Séminaire Bourbaki, | mr=0157859 | year=1953}}
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| *{{Citation | last1=Ono | first1=Takashi | title=Sur une propriété arithmétique des groupes algébriques commutatifs | url=http://www.numdam.org/item?id=BSMF_1957__85__307_0 | mr=0094362 | year=1957 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=85 | pages=307–323}}
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| *{{Citation | last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305–347}}
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| *{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Zur algebraischen Theorie der algebraischen Funktionen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002174502 | language=German | doi=10.1515/crll.1938.179.129 | year=1938 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=179 | pages=129–133}}
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| ==External links==
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| *{{springer|first=A.S. |last=Rapinchuk|id=T/t092060|title=Tamagawa number}}
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| [[Category:Topological groups]]
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| [[Category:Algebraic number theory]]
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| [[Category:Algebraic groups]]
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