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| In [[algebraic number theory]] and [[topological algebra]], the '''adele ring'''<ref>Also spelled: ''adèle ring'' {{IPAc-en|ə|ˈ|d|ɛ|l|_|r|ɪ|ŋ}}.</ref> (other names are the '''adelic ring''', the '''ring of adeles''') is a self-dual [[topological ring]] built on the [[Field (mathematics)|field]] of [[rational number]]s (or, more generally, any [[algebraic number field]]). It involves in a symmetric way all the completions of the field.
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| The adele ring was invented by [[Claude Chevalley]] for the purposes of simplifying and clarifying [[class field theory]]. It has also found applications outside that area.
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| The adele ring and its relation to the number field are among the most fundamental objects in number theory. The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory. It is a central principle of [[Diophantine geometry]] to study solutions of polynomials equations in number fields by looking at their solutions in the larger complete adele ring, where it is generally easier to detect solutions, and then deciding which of them come from the number field.
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| The word "adele" is short for "additive [[idele]]"<ref name=N357>Neukirch (1999) p. 357.</ref> and it was invented by [[André Weil]]. The previous name was '''the valuation vectors'''. The ring of adeles was historically preceded by '''the ring of repartitions''', a construction which avoids completions, and is today sometimes referred to as '''pre-adele'''.
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| == Definitions ==
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| The [[profinite group|profinite completion]] of the integers, <math>\widehat{\mathbb{Z}}</math>, is the [[inverse limit]] of the rings <math>\mathbb{Z} / n \mathbb{Z}</math>:
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| :<math> \widehat{\mathbb{Z}} = \varprojlim \,\mathbb{Z}/n\mathbb{Z}. </math>
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| By the [[Chinese remainder theorem]] it is isomorphic to the product of all the rings of [[p-adic integers|''p''-adic integers]]:
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| :<math> \widehat{\mathbb{Z}} = \prod_{p} \mathbb{Z}_p. </math>
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| The '''ring of integral adeles''' '''A'''<sub>'''Z'''</sub> is the product
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| :<math> \mathbb{A}_\mathbb{Z} = \mathbb{R} \times \widehat{\mathbb{Z}}.</math>
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| The '''ring of '''('''rational''')''' adeles''' '''A'''<sub>'''Q'''</sub> is the [[tensor product of algebras|tensor product]]
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| :<math> \mathbb{A}_\mathbb{Q} =\mathbb{Q}\otimes_\mathbb Z \mathbb{A}_\mathbb{Z} </math>
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| (topologized so that '''A'''<sub>'''Z'''</sub> is an open subring).
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| More generally the ring of adeles '''A'''<sub>''F''</sub> of any algebraic number field ''F'' is the tensor product
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| :<math> \mathbb{A}_F =F\otimes_\mathbb Z \mathbb{A}_\mathbb{Z}</math>
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| (topologized as the product of <math>\deg(F)</math> copies of '''A'''<sub>'''Q'''</sub>).
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| The ring of (rational) adeles can also be defined as the [[restricted product]]
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| :<math> \mathbb{A}_\mathbb{Q} = \mathbb{R} \times {\prod_{p}}' \mathbb{Q}_p </math>
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| of all the [[p-adic number|''p''-adic completions]] '''Q'''<sub>''p''</sub> and the [[real number]]s (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele (''a''<sub>∞</sup>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ''a''<sub>5</sub>, …) all but a finite number of the ''a''<sub>''p''</sub> are [[p-adic integer|''p''-adic integer]]s.<ref name=N357/>
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| The adeles of a [[function field of an algebraic variety|function field]] over a finite field can be defined in a similar way, as the restricted product of all completions.
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| ==Properties==
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| The adele ring is a [[locally compact]] complete [[Group (mathematics)|group]] with respect to its most natural topology. This group is self dual in the sense that it is topologically isomorphic to its group of characters. The adelic ring contains the number or function field as a discrete [[co-compact]] subgroup.
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| Similarly, the multiplicative group of adeles, called the group of ideles, is a locally compact group with respect to its topology defined below.
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| ==Idele group==
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| The group of invertible elements of the adele ring is the '''[[idele group]]'''.<ref name=N357/><ref>William Stein, [http://wstein.org/129/lectures/day23-24/day23.pdf "Algebraic Number Theory"], May 4, 2004, p. 5.</ref> It is ''not'' given the subset topology, as the inverse is not continuous in this topology. Instead the ideles are identified with the closed subset of all pairs (''x'',''y'') of ''A''×''A'' with ''xy''=1, with the subset topology. The idele group may be realised as the [[restricted product]] of the unit groups of the local fields with respect to the subgroup of local integral units.<ref>Neukirch (1999) pp. 357–358.</ref> The ideles form a locally compact topological group.<ref name=N361>Neukirch (1999) p. 361.</ref>
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| The '''principal ideles''' are given by the diagonal embedding of the invertible elements of the number field or field of functions and the quotient of the idele group by principal ideles is the '''idele class group'''.<ref>Neukirch (1999) pp. 358–359.</ref> This is a key object of [[class field theory]] which describes abelian extensions of the field. The product of the local reciprocity maps in [[local class field theory]] gives a homomorphism from the idele group to the Galois group of the maximal abelian extension of the number or function field. The [[Artin reciprocity law]], which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus we obtain the global reciprocity map from the idele class group to the abelian part of the absolute Galois group of the field.<ref>{{cite book | first1=Henri | last1=Cohen | author1-link=Henri Cohen (number theorist) | first2=Peter | last2=Stevenhagen | chapter=Computational class field theory | pages=497–534 | editor1-first=J.P. | editor1-last=Buhler | editor2-first=Stevenhagen | editor2-last=P. | title=Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography | series=MSRI Publications | volume=44 | publisher=[[Cambridge University Press]] | year=2008 | isbn=978-0-521-20833-8 | zbl=1177.11095 }}</ref>
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| ==Applications==
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| The self-duality of the adeles of the function field of a curve over a finite field easily implies the [[Riemann–Roch theorem]] for the curve and the duality theory for the curve.
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| As a locally compact abelian group, the adeles have a nontrivial translation invariant measure. Similarly, the group of ideles has a nontrivial translation invariant measure using which one defines a zeta integral. The latter was explicitly introduced in papers of [[Kenkichi Iwasawa]] and [[John Tate]]. The zeta integral allows one to study several key properties of the zeta function of the number field or function field in a beautiful concise way, reducing its functional equation of meromorphic continuation to a simple application of [[harmonic analysis]] and self-duality of the adeles, see [[Tate's thesis]].<ref name=N503>Neukirch (1999) p. 503</ref>
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| The ring ''A'' combined with the theory of [[algebraic group]]s leads to [[adelic algebraic group]]s.
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| For the function field of a smooth curve over a finite field the quotient of the multiplicative group (i.e. GL(1)) of its adele ring by the multiplicative group of the function field of the curve and units of integral adeles, i.e. those with integral local components, is isomorphic to the group of isomorphisms of linear bundles on the curve, and thus carries a geometric information. Replacing GL(1) by GL(''n''), the corresponding quotient is isomorphic to the set of isomorphism classes of n vector bundles on the curve, as was already observed by [[André Weil]].
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| Another key object of number theory is automorphic representations of adelic GL(''n'') which are constituents of the space of square integrable complex valued functions on the quotient by GL(''n'') of the field. They play the central role in the [[Langlands correspondence]] which studies finite-dimensional representations of the Galois group of the field and which is one of noncommutative extensions of class field theory.
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| Another development of the theory is related to the [[Tamagawa number]] for an adelic linear algebraic group. This is a volume measure relating ''G''('''Q''') with ''G''(''A''), saying how ''G''('''Q'''), which is a [[discrete group]] in ''G''(''A''), lies in the latter. A [[Weil conjecture on Tamagawa numbers|conjecture of André Weil]] was that the Tamagawa number was always 1 for a [[simply connected]] ''G''. This arose out of Weil's modern treatment of results in the theory of [[quadratic form]]s; the proof was case-by-case and took decades, the final steps were taken by [[Robert Kottwitz]] in 1988 and [[V. I. Chernousov]] in 1989. The influence of the Tamagawa number idea was felt in the theory of arithmetic of [[abelian varieties]] through its use in the statement of the [[Birch and Swinnerton-Dyer conjecture]], and through the Tamagawa number conjecture developed by [[Spencer Bloch]], [[Kazuya Kato]] and many other mathematicians.
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| ==See also==
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| *[[Schwartz–Bruhat function]]
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| ==Notes==
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| <references />
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| ==References==
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| Almost any book on modern algebraic number theory, such as:
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| *{{Citation | last1=Fröhlich | first1=A. | author1-link=Albrecht Fröhlich | last2=Cassels | first2=J. W. | author2-link=J. W. S. Cassels | title=Algebraic number theory | publisher=[[Academic Press]] | location=London and New York | isbn=978-0-12-163251-9 | year=1967 | zbl=0153.07403 }}
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| *{{Citation | last=Lang | first=Serge | author-link=Serge Lang | title=Algebraic number theory | edition=2nd | publisher=[[Springer-Verlag]] | year=1994 | series=[[Graduate Texts in Mathematics]] | volume=110 | place=New York | isbn=978-0-387-94225-4 | mr=1282723 | zbl=0811.11001 }}
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| *{{Neukirch ANT}}
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| [[Category:Algebraic number theory]]
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| [[Category:Topological algebra]]
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Kassie Mata could be the name even though it isn't the label on my birth certificate my parents provided me Selecting is how she makes cash and he or she'll be advertised soon To see comics is anything he would never give up Years ago we moved to Texas
My webpage ... omega 3