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m (moved Congruence problem to Congruence subgroup problem: usual description)
(tex,msc.mr,zbl,correction of various errors)
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Is every subgroup of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249202.png" /> is the ring of integers in an algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249203.png" /> (cf. [[Algebraic number theory|Algebraic number theory]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249204.png" /> is a connected [[Linear algebraic group|linear algebraic group]] defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249205.png" />, a [[Congruence subgroup|congruence subgroup]]? This is the classical statement of the congruence problem. A contemporary version of it is based on the concept of the congruence kernel, which is a measure of the deviation from a positive solution. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249207.png" /> be the completions of the groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249208.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c0249209.png" /> in the topologies defined by all subgroups of finite index and all congruence subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492010.png" />, respectively. Then there is a surjective continuous homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492011.png" />. The kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492012.png" /> is called the congruence kernel and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492013.png" />. The positive solution of the classical congruence problem is equivalent to proving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492014.png" />. In its modern form, the congruence problem is that of computing the congruence kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492015.png" />.
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{{TEX|done}}
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{{MSC|}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492017.png" /> is the ring of integers, it was known already in the 19th century that the congruence problem has a negative solution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492018.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492019.png" />, it was proved in 1965 that every subgroup of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492020.png" /> is a congruence subgroup (see [[#References|[1]]]). After this, the congruence problem was solved [[#References|[1]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492025.png" /> denotes the symplectic group. The results are as follows for these groups; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492026.png" /> only for totally imaginary fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492027.png" />, in which case the congruence kernel is isomorphic to the (cyclic) group of roots of unity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492028.png" />. It turned out that the same result holds for simply-connected Chevalley groups other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492029.png" /> (see [[#References|[3]]]). The condition of being simply connected is essential, because it follows from the strong approximation theorem (cf. [[Linear algebraic group|Linear algebraic group]]) that the congruence kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492030.png" /> of a non-simply-connected semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492031.png" /> is infinite. For every non-semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492034.png" /> is a maximal semi-simple subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492035.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492036.png" /> for a solvable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492037.png" />.
 
  
A more general form of the congruence problem is obtained by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492038.png" /> by the ring
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The ''congruence subgroup problem'' deals with the following question:
  
<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/c/c024/c024920/c02492039.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492040.png" /> is any finite set of inequivalent norms of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492041.png" /> containing all Archimedean norms. In this situation, the congruence kernel, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492042.png" />, depends in an essential way on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492043.png" /> (see [[#References|[4]]], [[#References|[5]]]).
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Is every subgroup of finite index in $\def\O{\mathcal{O}}G_\O$, where $\O$ is the ring of integers in an algebraic number field $k$ (cf.
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[[Algebraic number theory|Algebraic number theory]]) and $G$ is a connected
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[[Linear algebraic group|linear algebraic group]] defined over $k$, a
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[[Congruence subgroup|congruence subgroup]]?
  
====References====
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This is the classical statement of the congruence subgroup problem. A contemporary version of it is based on the concept of the ''congruence subgroup kernel'', which is a measure of the deviation from a positive solution. Let $\def\G{\Gamma}\G$ denote the group $G_\O$, and let $\hat\G$ and $\bar\G$ be the completions of the group $\G$ in the topologies defined by all its subgroups of finite index and all congruence subgroups of $\G$, respectively. Then there is a surjective continuous homomorphism $\pi:\hat\G \to \bar\G$. The kernel of $\pi$ is called the congruence subgroup kernel and is denoted by $c(G)$. The positive solution of the classical congruence subgroup problem is equivalent to proving $c(G) = 1$. In its modern form, the congruence subgroup problem is that of computing the congruence subgroup kernel $c(G)$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bass,   J. Milnor,   J.-P. Serre,  "Solution of the congruence subgroup problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492045.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492046.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492047.png" />)"  ''Publ. Math. IHES'' , '''33'''  (1967) pp. 421–499</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Le problème des groupes de congruence pour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024920/c02492048.png" />"  ''Ann. of Math.'' , '''92'''  (1970) pp. 489–527</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Matsumoto,   "Sur les sous-groupes arithmétiques des groupes semi-simples dépolyés"  ''Ann. Sci. Ecole Norm. Sup. (4)'' , '''2'''  (1969) pp. 1–62</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. i Tekhn. Algebra. Topol. Geom.'' , '''11'''  (1974)  pp. 5–37</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Raghunathan,  "On the congruence subgroup problem"  ''Publ. Math. IHES'' , '''46'''  (1946) pp. 107–161</TD></TR></table>
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If $\def\SL{\textrm{SL}}\G=\SL(n,\Z)$, where $\Z$ is the ring of integers, it was known already in the 19th century that the congruence subgroup problem has a negative solution for $n=2$. For $n\ge 3$, it was proved in 1965 that every subgroup of finite index in $\SL(n,\Z)$ is a congruence subgroup (see
 +
{{Cite|BaMiSe}}). After this, the congruence subgroup problem was solved
 +
{{Cite|BaMiSe}} for $\G = \SL(n,\O)$, $n\ge 3$, and $\def\Sp{\textrm{Sp}}\Sp(2n,\O)$, $n \ge 2$, where $\Sp$ denotes the symplectic group. The results are as follows for these groups; $c(G)\ne 1$ only for totally imaginary fields $k$, in which case the congruence subgroup kernel is isomorphic to the (cyclic) group of roots of unity in $k$. It turned out that the same result holds for simply-connected Chevalley groups other than $\SL(2,k)$ (see
 +
{{Cite|Ma}}). The condition of being simply connected is essential, because it follows from the strong approximation theorem (cf.
 +
[[Linear algebraic group|Linear algebraic group]]) that the congruence subgroup kernel $c(G)$ of a non-simply-connected semi-simple group $G$ is infinite. For every non-semi-simple group $G$ one has $c(G)=c(H)$, where $H$ is a maximal semi-simple subgroup of $G$; in particular, $c(G) = 1$ for a solvable group $G$.
  
 +
A more general form of the congruence subgroup problem is obtained by replacing $\O$ by the ring
  
 +
$$\O_S = \{x \in k:v(x) \le 1 \textrm{ for all } v \notin S\}$$
 +
where $S$ is any finite set of inequivalent valuations of the field $k$ containing all Archimedean valuations. In this situation, the congruence subgroup kernel, denoted by $c(G,S)$, depends in an essential way on $S$ (see
 +
{{Cite|Pl}},
 +
{{Cite|Ra}}).
  
====Comments====
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====References====
The congruence problem is usually called the congruence subgroup problem.
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{|
 +
|-
 +
|valign="top"|{{Ref|BaMiSe}}||valign="top"|  H. Bass,  J. Milnor,  J-P. Serre,  "Solution of the congruence subgroup problem for $\SL_n$ ($n\ge 3$) and $\Sp_{2n}$ ($n\ge 2$)"  ''Publ. Math. IHES'', '''33'''  (1967)  pp. 421–499    {{MR|0244257}}  {{ZBL|0174.05203}}
 +
|-
 +
|valign="top"|{{Ref|Pl}}||valign="top"|  V.P. Platonov,  "Algebraic groups"  ''J. Soviet Math.'', '''4''' :  5  (1975)  pp. 463–482  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'', '''11'''  (1974)  pp. 5–37  {{MR|0466334}}  {{ZBL|0386.20019}} {{ZBL|0305.20023}}   
 +
|-
 +
|valign="top"|{{Ref|Ra}}||valign="top"|  M. Raghunathan,  "On the congruence subgroup problem"  ''Publ. Math. IHES'', '''46'''  (1976)  pp. 107–161    {{MR|0507030}}  {{ZBL|0347.20027}}
 +
|-
 +
|valign="top"|{{Ref|Se}}||valign="top"|  J.-P. Serre,  "Le problème des groupes de congruence pour $\SL_2$"  ''Ann. of Math.'', '''92'''  (1970)  pp. 489–527  {{MR|0272790}}  {{ZBL|0239.20063}} 
 +
|-
 +
|}

Revision as of 10:36, 2 March 2012



The congruence subgroup problem deals with the following question:


Is every subgroup of finite index in $\def\O{\mathcal{O}}G_\O$, where $\O$ is the ring of integers in an algebraic number field $k$ (cf. Algebraic number theory) and $G$ is a connected linear algebraic group defined over $k$, a congruence subgroup?

This is the classical statement of the congruence subgroup problem. A contemporary version of it is based on the concept of the congruence subgroup kernel, which is a measure of the deviation from a positive solution. Let $\def\G{\Gamma}\G$ denote the group $G_\O$, and let $\hat\G$ and $\bar\G$ be the completions of the group $\G$ in the topologies defined by all its subgroups of finite index and all congruence subgroups of $\G$, respectively. Then there is a surjective continuous homomorphism $\pi:\hat\G \to \bar\G$. The kernel of $\pi$ is called the congruence subgroup kernel and is denoted by $c(G)$. The positive solution of the classical congruence subgroup problem is equivalent to proving $c(G) = 1$. In its modern form, the congruence subgroup problem is that of computing the congruence subgroup kernel $c(G)$.

If $\def\SL{\textrm{SL}}\G=\SL(n,\Z)$, where $\Z$ is the ring of integers, it was known already in the 19th century that the congruence subgroup problem has a negative solution for $n=2$. For $n\ge 3$, it was proved in 1965 that every subgroup of finite index in $\SL(n,\Z)$ is a congruence subgroup (see [BaMiSe]). After this, the congruence subgroup problem was solved [BaMiSe] for $\G = \SL(n,\O)$, $n\ge 3$, and $\def\Sp{\textrm{Sp}}\Sp(2n,\O)$, $n \ge 2$, where $\Sp$ denotes the symplectic group. The results are as follows for these groups; $c(G)\ne 1$ only for totally imaginary fields $k$, in which case the congruence subgroup kernel is isomorphic to the (cyclic) group of roots of unity in $k$. It turned out that the same result holds for simply-connected Chevalley groups other than $\SL(2,k)$ (see [Ma]). The condition of being simply connected is essential, because it follows from the strong approximation theorem (cf. Linear algebraic group) that the congruence subgroup kernel $c(G)$ of a non-simply-connected semi-simple group $G$ is infinite. For every non-semi-simple group $G$ one has $c(G)=c(H)$, where $H$ is a maximal semi-simple subgroup of $G$; in particular, $c(G) = 1$ for a solvable group $G$.

A more general form of the congruence subgroup problem is obtained by replacing $\O$ by the ring

$$\O_S = \{x \in k:v(x) \le 1 \textrm{ for all } v \notin S\}$$ where $S$ is any finite set of inequivalent valuations of the field $k$ containing all Archimedean valuations. In this situation, the congruence subgroup kernel, denoted by $c(G,S)$, depends in an essential way on $S$ (see [Pl], [Ra]).

References

[BaMiSe] H. Bass, J. Milnor, J-P. Serre, "Solution of the congruence subgroup problem for $\SL_n$ ($n\ge 3$) and $\Sp_{2n}$ ($n\ge 2$)" Publ. Math. IHES, 33 (1967) pp. 421–499 MR0244257 Zbl 0174.05203
[Pl] V.P. Platonov, "Algebraic groups" J. Soviet Math., 4 : 5 (1975) pp. 463–482 Itogi Nauk. i Tekhn. Algebra. Topol. Geom., 11 (1974) pp. 5–37 MR0466334 Zbl 0386.20019 Zbl 0305.20023
[Ra] M. Raghunathan, "On the congruence subgroup problem" Publ. Math. IHES, 46 (1976) pp. 107–161 MR0507030 Zbl 0347.20027
[Se] J.-P. Serre, "Le problème des groupes de congruence pour $\SL_2$" Ann. of Math., 92 (1970) pp. 489–527 MR0272790 Zbl 0239.20063
How to Cite This Entry:
Congruence subgroup problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_subgroup_problem&oldid=21407
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article