Timeline for Definition of level N congruence subgroup of an arithmetic group, useful for computations
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2013 at 10:21 | vote | accept | j0equ1nn | ||
| Dec 19, 2013 at 1:39 | comment | added | j0equ1nn | The possibility of confusion lies among readers who know less about these things than you and Alex (like possibly me). As far as I understand there is no matrix ring here because the only operation we're using is multiplication. It's a Bianchi group, not a Bianchi ring. I'm not trying to be snide, I'm trying to be precise, to avoid the possibility of my own confusion. | |
| Dec 19, 2013 at 0:08 | comment | added | Qiaochu Yuan | 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. | |
| Dec 18, 2013 at 23:20 | answer | added | j0equ1nn | timeline score: 0 | |
| Dec 18, 2013 at 22:36 | comment | added | j0equ1nn | 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. | |
| Dec 18, 2013 at 10:34 | comment | added | Alex B. | 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. | |
| Dec 18, 2013 at 7:39 | comment | added | j0equ1nn | 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. | |
| Dec 18, 2013 at 7:06 | comment | added | Qiaochu Yuan | Levels in the general case should be ideals of $\mathcal{O}$, shouldn't they? I don't see the difficulty here. | |
| Dec 18, 2013 at 6:09 | history | asked | j0equ1nn | CC BY-SA 3.0 |