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Difference between revisions of "Congruence subgroup"

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A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249301.png" /> of the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249302.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249303.png" /> with the following property: There exists a non-zero two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249305.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249306.png" />, where
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{{TEX|done}}
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{{MSC|20G30}}
  
<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/c024930/c0249307.png" /></td> </tr></table>
 
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249308.png" /> contains all matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c0249309.png" /> that are congruent to the unit matrix modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493010.png" />. More generally, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493011.png" /> of a linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493012.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493013.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493014.png" /> is said to be a congruence subgroup if
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A ''conguence subgroup'' is a subgroup $H$ of the general linear group $\def\GL{\textrm{GL}}\GL(n,R)$ over a ring $R$ with the following property: There exists a non-zero two-sided ideal $\def\fp{\mathfrak{p}}\fp$ of $R$ such that $H\supseteq \GL(n,R,\fp)$, where
  
<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/c024930/c02493015.png" /></td> </tr></table>
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$$\GL(n,R,\fp) = \ker(\GL(n,R)\to \GL(n,R/\fp)),$$
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that is, $H$ contains all matrices in $\GL(n,R)$ that are congruent to the unit matrix modulo $\fp$. More generally, a subgroup $H$ of a linear group $\def\G{\Gamma}\G$ of degree $n$ over $R$ is said to be a congruence subgroup if
  
for some non-zero two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493016.png" />.
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$$H\supseteq \G\cap\GL(n,R,\fp)$$
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for some non-zero two-sided ideal $\fp\subseteq$.
  
 
When
 
When
  
<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/c024930/c02493017.png" /></td> </tr></table>
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$$H = \G\cap\GL(n,R,\fp)$$
 
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$H$ is said to be the principal congruence subgroup corresponding to $\fp$. The concept of a congruence subgroup arose first for $R=\Z$. It is particularly effective and important for a Dedekind ring $R$ in the case $\G=G\cap\GL(n,R)$, where $G$ is an algebraic group defined over the field of fractions of $R$.
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493018.png" /> is said to be the principal congruence subgroup corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493019.png" />. The concept of a congruence subgroup arose first for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493020.png" />. It is particularly effective and important from the point of view of applications for a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493021.png" /> in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493023.png" /> is an algebraic group defined over the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493024.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Bass,  J. Milnor,  J.-P. Serre,  "Solutions of the congruence subgroup problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493026.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024930/c02493028.png" />)"  ''Publ. Math. IHES'' , '''33'''  (1967)  pp. 421–499</TD></TR></table>
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|valign="top"|{{Ref|BaMiSe}}||valign="top"| H. Bass,  J. Milnor,  J.-P. Serre,  "Solutions of the congruence subgroup problem for $\textrm{SL}_n$ ($n\ge 3$) and $\textrm{Sp}_{2n}$ ($n\ge 2$)"  ''Publ. Math. IHES'', '''33'''  (1967)  pp. 421–499 {{MR|0244257}}    {{ZBL|0174.05203}}
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Revision as of 11:05, 2 March 2012

2020 Mathematics Subject Classification: Primary: 20G30 [MSN][ZBL]


A conguence subgroup is a subgroup $H$ of the general linear group $\def\GL{\textrm{GL}}\GL(n,R)$ over a ring $R$ with the following property: There exists a non-zero two-sided ideal $\def\fp{\mathfrak{p}}\fp$ of $R$ such that $H\supseteq \GL(n,R,\fp)$, where

$$\GL(n,R,\fp) = \ker(\GL(n,R)\to \GL(n,R/\fp)),$$ that is, $H$ contains all matrices in $\GL(n,R)$ that are congruent to the unit matrix modulo $\fp$. More generally, a subgroup $H$ of a linear group $\def\G{\Gamma}\G$ of degree $n$ over $R$ is said to be a congruence subgroup if

$$H\supseteq \G\cap\GL(n,R,\fp)$$ for some non-zero two-sided ideal $\fp\subseteq$.

When

$$H = \G\cap\GL(n,R,\fp)$$ $H$ is said to be the principal congruence subgroup corresponding to $\fp$. The concept of a congruence subgroup arose first for $R=\Z$. It is particularly effective and important for a Dedekind ring $R$ in the case $\G=G\cap\GL(n,R)$, where $G$ is an algebraic group defined over the field of fractions of $R$.

References

[BaMiSe] H. Bass, J. Milnor, J.-P. Serre, "Solutions of the congruence subgroup problem for $\textrm{SL}_n$ ($n\ge 3$) and $\textrm{Sp}_{2n}$ ($n\ge 2$)" Publ. Math. IHES, 33 (1967) pp. 421–499 MR0244257 Zbl 0174.05203
How to Cite This Entry:
Congruence subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_subgroup&oldid=21411
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article