Adele group

2020 Mathematics Subject Classification: Primary: 20G35 [MSN][ZBL]

An adele group (also adèle group) is the restricted topological direct product $$\prod_{\nu\in V}\; G_{k_\nu}(G_{O_\nu})$$ of the group $G_{k_\nu}$ with distinguished invariant open subgroups $G_{O_\nu}$. (See #Comment below for the definition of the restricted topological product.) Here $G_k$ is a linear algebraic group, defined over a global field $k$, $V$ is the set of valuations (cf. Valuation) of $k$, $k_\nu$ is the completion of $k$ with respect to $\nu\in V$, and $O_\nu$ is the ring of integer elements in $k_\nu$. The adèle group of an algebraic group $G$ is denoted by $G_A$. Since all groups $G_{k_\nu}$ are locally compact and since $G_{O_\nu}$ is compact, $G_A$ is a locally compact group.

Examples. 1) If $G_k$ is the additive group $k^+$ of the field $k$, then $G_A$ has a natural ring structure, and is called the adèle ring of $k$; it is denoted by $A_k$. 2) If $G_k$ is the multiplicative group $k^*$ of the field $k$, then $G_A$ is called the idèle group of $k$ (the idèle group is the group of units in the adèle ring $A_k$). 3) If $G_k={\rm GL}(n,k)$ is the general linear group over $k$, then $G_A$ consists of the elements $g=(g_\nu)\in\prod_{\nu\in V} G_\nu$ for which $g_\nu\in {\rm GL}(n,O_\nu)$ for almost all valuations $\nu$.

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 [We]. 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 $G_k$ in $G_A$ is a discrete subgroup in $G_A$, called the subgroup of principal adèles. If $\infty$ is the set of all Archimedean valuations of $k$, then $$G_{A(\infty)} = \prod_{\nu\in\infty} G_{k_\nu} \times \prod_{\nu\in\infty} G_{O_\nu}$$ is known as the subgroup of integer adèles. If $G_k = k^*$, then the number of different double cosets of the type $G_k x G_{A(\infty)}$ of the adèle group $G_A$ is finite and equal to the number of ideal classes of $k$. 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 $G_A/G_k$. It has been shown[Bo] that $G_A/G_k$ is compact if and only if the group $G$ is $k$-anisotropic (cf. Anisotropic group). Another problem that has been solved are the circumstances under which the quotient space $G_A/G_k$ over an algebraic number field has finite volume in the Haar measure. Since $G_A$ is locally compact, such a measure always exists, and the volume of $G_A/G_k$ in the Haar measure is finite if and only if the group $G$ has no rational $k$-characters (cf. Character of a group). The number $\tau(G)$ — the volume of $G_A/G_k$ — is an important arithmetical invariant of the algebraic group G (cf. Tamagawa number). It was shown on the strength of these results [Bo] that the decomposition $$G_A = \bigcup_{i=1}^m G_k x_i G_{A(\infty)}$$ is valid for an arbitrary algebraic group $G$. If $k$ 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 [Ha]. For various arithmetical applications of adèle groups see [Pl],[Pl2].

Comment

Let $I$ be an index set. For each $\nu\in I$ let $G_\nu$ be a locally compact group and $O_\nu$ an open compact subgroup. The (topological) restricted direct product of the $G_\nu$ with respect to the $O_\nu$, above denoted by $$G = \prod_{\nu\in I} G_\nu(O_\nu),$$ consists (as a set) of all $(x_\nu)\in\prod_{\nu\in I} G_\nu$ such that $x_\nu$ in $O_\nu$ for all but finitely many $\nu$. The topology on $G$ is defined by taking as a basis at the identity the open subgroups $\prod_{\nu\in I} U_\nu$ with $U_\nu$ an open neighbourhood of $G_\nu$ for all $\nu$ and $U_\nu = O_\nu$ for all but finitely many $\nu$. This makes $G$ a locally compact topological group.

References

[Bo]	A. Borel, "Some finiteness properties of adèle groups over number fields" Publ. Math. IHES : 16 (1963) pp. 5–30 MR0202718 Zbl 0135.08902
[CaFr]	J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) MR0215665 Zbl 0153.07403
[Ha]	G. Harder, "Minkowskische Reduktionstheorie über Funktionenkörpern" Invent. Math., 7 (1969) pp. 33–54 MR0284441 Zbl 0242.20046
[Kn]	M. Kneser, "Strong approximation", Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc. (1966) pp. 187–198 MR0213361 Zbl 0201.37904
[Pl]	V.P. Platonov, "Algebraic groups" J. Soviet Math., 4 : 5 (1975) pp. 463–482 Itogi Nauk. Algebra Topol. Geom., 11 (1973) pp. 5–37 MR0466334 Zbl 0386.20019 Zbl 0305.20023
[Pl2]	V.P. Platonov, "The arithmetic theory of linear algebraic groups and number theory" Trudy Mat. Inst. Steklov., 132 (1973) pp. 162–168 (In Russian) Zbl 0305.20023
[Ta]	T. Tamagawa, "Adéles", Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc. (1966) pp. 113–121 MR0212025 Zbl 0178.23801
[We]	A. Weil, "Adèles and algebraic groups", Princeton Univ. Press (1961) MR1603471 MR0670072 Zbl 0493.14028 Zbl 0118.15801
[We2]	A. Weil, "Basic number theory", Springer (1974) MR0427267 Zbl 0326.12001