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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 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 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 is taken to be the subspace topology in ''A''''N'', the Cartesian product of ''N'' copies of the adele ring. ==Ideles== An important example, the idele group ''I''(''K''), is the case of . Here the set of ideles (also ''idèles'' ) 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 lies in two-dimensional affine space as the 'hyperbola' defined parametrically by :, the topology correctly assigned to the idele group is that induced by inclusion in ''A''2; 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 :''I''(''K'')/''K''× 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, give rise to the most basic class of L-functions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Adelic algebraic group」の詳細全文を読む スポンサード リンク
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