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An element of the adèle group, i.e. of the restricted topological direct product
+
An element of the adèle group, i.e. of 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
 +
$G_{O_\nu}$. Here $G_k$ is a [[Linear algebraic group|linear algebraic
 +
group]], defined over a [[Global field|global field]] $k$, $V$ is the
 +
set of valuations (cf.  [[Valuation|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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107401.png" /></td> </tr></table>
+
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 $\nu$.
  
of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107402.png" /> with distinguished invariant open subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107403.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107404.png" /> is a [[Linear algebraic group|linear algebraic group]], defined over a [[Global field|global field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107406.png" /> is the set of valuations (cf. [[Valuation|Valuation]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107408.png" /> is the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a0107409.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074011.png" /> is the ring of integer elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074012.png" />. The adèle group of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074013.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074014.png" />. Since all groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074015.png" /> are locally compact and since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074016.png" /> is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074017.png" /> is a locally compact group.
+
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 [[#References|[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.
  
Examples. 1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074018.png" /> is the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074019.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074021.png" /> has a natural ring structure, and is called the adèle ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074022.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074023.png" />. 2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074024.png" /> is the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074025.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074027.png" /> is called the idèle group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074028.png" /> (the idèle group is the group of units in the adèle ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074029.png" />). 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074030.png" /> is the general linear group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074032.png" /> consists of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074033.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074034.png" /> for almost all valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074035.png" />.
+
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 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 [[#References|[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 set of all Archimedean valuations of $k$, then  
 
+
$$G_{A(\infty)} = \prod_{\nu\in\infty} G_{k_\nu} \times \prod_{\nu\in\infty} G_{O_\nu}$$
The image of the diagonal imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074037.png" /> is a discrete subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074038.png" />, called the subgroup of principal adèles. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074039.png" /> is the set of all Archimedean valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074040.png" />, then
+
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
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074041.png" /></td> </tr></table>
+
equal to the number of ideal classes of $k$. The naturally arising
 
+
problem as to whether the number of such double classes for an
is known as the subgroup of integer adèles. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074042.png" />, then the number of different double cosets of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074043.png" /> of the adèle group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074044.png" /> is finite and equal to the number of ideal classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074045.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074046.png" />. It has been shown [[#References|[5]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074047.png" /> is compact if and only if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074049.png" />-anisotropic (cf. [[Anisotropic group|Anisotropic group]]). Another problem that has been solved are the circumstances under which the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074050.png" /> over an algebraic number field has finite volume in the [[Haar measure|Haar measure]]. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074051.png" /> is locally compact, such a measure always exists, and the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074052.png" /> in the Haar measure is finite if and only if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074053.png" /> has no rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074054.png" />-characters (cf. [[Character of a group|Character of a group]]). The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074055.png" /> — the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074056.png" /> — is an important arithmetical invariant of the algebraic group G (cf. [[Tamagawa number|Tamagawa number]]). It was shown on the strength of these results [[#References|[5]]] that the decomposition
+
arbitrary algebraic group is finite is connected with the reduction
 
+
theory for subgroups of principal adèles, i.e. with the construction
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074057.png" /></td> </tr></table>
+
of fundamental domains for the quotient space $G_A/G_k$. It has been shown[[#References|[5]]] that $G_A/G_k$ is compact if and only if the group $G$
 
+
is $k$-anisotropic (cf. [[Anisotropic group|Anisotropic
is valid for an arbitrary algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074059.png" /> 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 [[#References|[6]]]. For various arithmetical applications of adèle groups see [[#References|[4]]], [[#References|[7]]].
+
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|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|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|Tamagawa number]]). It was shown on the strength of these
 +
results [[#References|[5]]] 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 [[#References|[6]]]. For various
 +
arithmetical applications of adèle groups see [[#References|[4]]],[[#References|[7]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil,   "Adèles and algebraic groups" , Princeton Univ. Press (1961)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> T. Tamagawa,   "Adéles" , ''Algebraic groups and discontinuous subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966) pp. 113–121</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> M. Kneser,   "Strong approximation" , ''Algebraic groups and discontinuous subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966) pp. 187–198</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.P. Platonov,   "Algebraic groups" ''J. Soviet Math.'' , '''4''' : 5 (1975) pp. 463–482 ''Itogi Nauk. Algebra Topol. Geom.'' , '''11''' (1973) pp. 5–37</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Borel,   "Some finiteness properties of adèle groups over number fields" ''Publ. Math. IHES'' : 16 (1963) pp. 5–30</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G. Harder,   "Minkowskische Reduktionstheorie über Funktionenkörpern" ''Invent. Math.'' , '''7''' (1969) pp. 33–54</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.P. Platonov,   "The arithmetic theory of linear algebraic groups and number theory" ''Trudy Mat. Inst. Steklov.'' , '''132''' (1973) pp. 162–168 (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A. Weil,   "Basic number theory" , Springer (1974)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> A. Weil, "Adèles and algebraic groups" , Princeton
 +
Univ. Press (1961)</TD></TR><TR><TD valign="top">[2a]</TD> <TD
 +
valign="top"> T. Tamagawa, "Adéles" , ''Algebraic groups and
 +
discontinuous subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' ,
 +
Amer. Math. Soc.  (1966) pp. 113–121</TD></TR><TR><TD
 +
valign="top">[2b]</TD> <TD valign="top"> M. Kneser, "Strong
 +
approximation" , ''Algebraic groups and discontinuous subgroups'' ,
 +
''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)
 +
pp. 187–198</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 +
J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory''
 +
, Acad. Press (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD
 +
valign="top"> V.P. Platonov, "Algebraic groups" ''J. Soviet Math.'' ,
 +
'''4''' : 5 (1975) pp. 463–482 ''Itogi Nauk. Algebra Topol. Geom.'' ,
 +
'''11''' (1973) pp. 5–37</TD></TR><TR><TD valign="top">[5]</TD> <TD
 +
valign="top"> A. Borel, "Some finiteness properties of adèle groups
 +
over number fields" ''Publ. Math. IHES'' : 16 (1963)
 +
pp. 5–30</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">
 +
G. Harder, "Minkowskische Reduktionstheorie über Funktionenkörpern"
 +
''Invent. Math.'' , '''7''' (1969) pp. 33–54</TD></TR><TR><TD
 +
valign="top">[7]</TD> <TD valign="top"> V.P. Platonov, "The arithmetic
 +
theory of linear algebraic groups and number theory" ''Trudy
 +
Mat. Inst. Steklov.'' , '''132''' (1973) pp. 162–168 (In
 +
Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">
 +
A. Weil, "Basic number theory" , Springer (1974)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074060.png" /> be an index set. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074061.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074062.png" /> be a locally compact group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074063.png" /> on open compact subgroup. The restricted (topological) direct product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074064.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074065.png" />, above denoted by
+
Let $I$ be an index set. For each $\nu\in I$ let $G_\nu$ be a
 
+
locally compact group and $O_\nu$ on open compact subgroup. The restricted
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074066.png" /></td> </tr></table>
+
(topological) direct product of the $G_\nu$ with respect to the $O_\nu$, above
 
+
denoted by  
consists (as a set) of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074069.png" /> for all but finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074070.png" />. The topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074071.png" /> is defined by taking as a basis at the identity the open subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074072.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074073.png" /> an open neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074074.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074076.png" /> for all but finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074077.png" />. This makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010740/a01074078.png" /> a locally compact topological group.
+
$$G = \prod_{\nu\in I} G_\nu(O_\nu),$$
 +
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.

Revision as of 17:52, 11 September 2011

An element of the adèle group, i.e. of 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 $G_{O_\nu}$. 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 [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 $G_k$ 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[5] 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|Tamagawa number]]). It was shown on the strength of these results [5] 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 [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 $I$ be an index set. For each $\nu\in I$ let $G_\nu$ be a locally compact group and $O_\nu$ on open compact subgroup. The restricted (topological) 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.

How to Cite This Entry:
Adele group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adele_group&oldid=19563
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article