Congruence subgroup

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