# Definition of level N congruence subgroup of an arithmetic group, useful for computations

My title requests something more general than I actually require right now, so I would settle for an answer to something more specific (details below) but I would like to understand the more general concept as well.

When we mean a subgroup of $PSL_2(\mathbb{Z})$, a concise way of describing a congruence subgroup of level $N$ would be: a subgroup containing the kernel of the natural projection map $PSL_2(\mathbb{Z})\rightarrow PSL_2(\mathbb{Z}/N\mathbb{Z})$, where $N$ is minimal.

I am looking to understand how this concept generalizes if instead of subgroups of $PSL_2(\mathbb{Z})$, we are discussing subgroups of $PSL_2(\mathcal{O})$ where $\mathcal{O}$ is the ring of integers of a number field. But for now, I'm interested in the case where $\mathcal{O}$ is the ring of integers of $\mathbb{Q}(\sqrt{-d})$ for $d\in\mathbb{N}$ squarefree, and where the level is $N=2$. Is there a concise definition in this case, analogous to the one given above for the modular case?

It seems that the first thing to do here is use an integral basis to deal with the fact that the entries are no longer rational integers. How exactly escapes me, and then I wonder (departing from the $N=2$ case for a moment) would this then introduce the possibility of reducing modulo multiples of these basis elements? For instance is there such thing as a level $\frac{1+i\sqrt{3}}{2}$ congruence subgroup of $PSL_2(\mathcal{O}_3)$? (If that question is too ignorant, feel free to ignore it and address the one in the last paragraph).

• Levels in the general case should be ideals of $\mathcal{O}$, shouldn't they? I don't see the difficulty here. – Qiaochu Yuan Dec 18 '13 at 7:06
• Maybe, the thing is I can find no definition in any of my resources or in any online search. I just find mention of congruence subgroups of arithmetic groups. We shouldn't have to guess at the definition. – j0equ1nn Dec 18 '13 at 7:39
• The principal congruence subgroup consists of matrices that reduce to the identity modulo an ideal, as Qiaochu says. As for references, just google "congruence subgroups of Bianchi groups" and click on any of the first hits. – Alex B. Dec 18 '13 at 10:34
• Okay, thank you. Sorry if I wasted your time with that question; I knew they're called Bianchi groups but for some reason it didn't occur to me to call them that when searching! To be precise though, groups don't have ideals, rings do. So the ideal you speak of is an ideal of the ring the matrix group is over, and the congruence is in the entries. I'm going to type out the answer to my own question now but with credit to you guys for the help. – j0equ1nn Dec 18 '13 at 22:36
• The ideal is an ideal of the matrix ring induced from an ideal of the coefficient ring. In any case I don't think there was a serious possibility of confusion in what Alex wrote. – Qiaochu Yuan Dec 19 '13 at 0:08

Let $K$ be a number field, let $\mathcal{O}$ be its ring of integers, and let $\mathfrak{I}\vartriangleleft\mathcal{O}$. The principle congruence subgroup of level $\mathfrak{I}$ in $PSL_2(\mathcal{O})$ is the kernel of the natural projection map $PSL_2(\mathcal{O})\rightarrow PSL_2(\mathcal{O}/\mathfrak{I})$. A congruence subgroup of level $\mathfrak{I}$ in $PSL_2(\mathcal{O})$ is a subgroup containing the principle congruence subgroup of level $\mathfrak{I}$, where $\mathfrak{I}$ is maximal with respect to inclusion.
In the case where $K=\mathbb{Q}(\sqrt{-d})$ for $d\in\mathbb{N}$ squarefree, $PSL_2(\mathcal{O})$ is called a Bianchi group and denoted $\Gamma_d$. Also the principle congruence subgroup of level $\mathfrak{I}$ in $\Gamma_d$ is denoted $\Gamma_d(\mathfrak{I})$, and can be written explicitly as $P\Big\{ \begin{pmatrix} a & b\\ c & d \end{pmatrix}\in P^{-1}\Gamma_d\;\Big|\;a\equiv_{\mathfrak{I}} d\equiv_{\mathfrak{I}} 1,\; b\equiv_{\mathfrak{I}} c\equiv_{\mathfrak{I}} 0 \Big\}$, where $P$ is just the natural projection map from $SL_2$ to $PSL_2$.
• Say, I just found some notes from last October of an Alan Reid lecture I went to, where he defines this just like in the 2nd paragraph above (wish I'd remembered that quicker). He adds though the following notation for the principle congruence subgroup of level N: $\Gamma(N)$. – j0equ1nn Dec 28 '13 at 10:21