Adele group

An element of the adèle group, i.e. of the restricted topological direct product

of the group with distinguished invariant open subgroups . Here is a linear algebraic group, defined over a global field , is the set of valuations (cf. Valuation) of , is the completion of with respect to , and is the ring of integer elements in . The adèle group of an algebraic group is denoted by . Since all groups are locally compact and since is compact, is a locally compact group.

Examples. 1) If is the additive group of the field , then has a natural ring structure, and is called the adèle ring of ; it is denoted by . 2) If is the multiplicative group of the field , then is called the idèle group of (the idèle group is the group of units in the adèle ring ). 3) If is the general linear group over , then consists of the elements for which for almost all valuations .

The concept of an adèle group was first introduced by C. Chevalley (in the 1930s) for algebraic number fields, to meet certain needs of class field theory. It was generalized twenty years later to algebraic groups by M. Kneser and T. Tamagawa [1], . They noted that the principal results on the arithmetic of quadratic forms over number fields can be conveniently reformulated in terms of adèle groups.

The image of the diagonal imbedding of in is a discrete subgroup in , called the subgroup of principal adèles. If is the set of all Archimedean valuations of , then

is known as the subgroup of integer adèles. If , then the number of different double cosets of the type of the adèle group is finite and equal to the number of ideal classes of . The naturally arising problem as to whether the number of such double classes for an arbitrary algebraic group is finite is connected with the reduction theory for subgroups of principal adèles, i.e. with the construction of fundamental domains for the quotient space . It has been shown [5] that is compact if and only if the group is -anisotropic (cf. Anisotropic group). Another problem that has been solved are the circumstances under which the quotient space over an algebraic number field has finite volume in the Haar measure. Since is locally compact, such a measure always exists, and the volume of in the Haar measure is finite if and only if the group has no rational -characters (cf. Character of a group). The number — the volume of — is an important arithmetical invariant of the algebraic group G (cf. Tamagawa number). It was shown on the strength of these results [5] that the decomposition

is valid for an arbitrary algebraic group . If is a function field, it was also proved that the number of double classes of this kind for the adèle group of the algebraic group is finite, and an analogue of the reduction theory was developed [6]. For various arithmetical applications of adèle groups see [4], [7].

References

[1]	A. Weil, "Adèles and algebraic groups" , Princeton Univ. Press (1961)
[2a]	T. Tamagawa, "Adéles" , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 113–121
[2b]	M. Kneser, "Strong approximation" , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 187–198
[3]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[4]	V.P. Platonov, "Algebraic groups" J. Soviet Math. , 4 : 5 (1975) pp. 463–482 Itogi Nauk. Algebra Topol. Geom. , 11 (1973) pp. 5–37
[5]	A. Borel, "Some finiteness properties of adèle groups over number fields" Publ. Math. IHES : 16 (1963) pp. 5–30
[6]	G. Harder, "Minkowskische Reduktionstheorie über Funktionenkörpern" Invent. Math. , 7 (1969) pp. 33–54
[7]	V.P. Platonov, "The arithmetic theory of linear algebraic groups and number theory" Trudy Mat. Inst. Steklov. , 132 (1973) pp. 162–168 (In Russian)
[8]	A. Weil, "Basic number theory" , Springer (1974)

Comments

Let be an index set. For each let be a locally compact group and on open compact subgroup. The restricted (topological) direct product of the with respect to the , above denoted by

consists (as a set) of all such that in for all but finitely many . The topology on is defined by taking as a basis at the identity the open subgroups with an open neighbourhood of for all and for all but finitely many . This makes a locally compact topological group.