Timeline for Cosets representatives of congruence subgroups
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28, 2022 at 14:31 | comment | added | François Brunault | For reference, what was wrong is that in Shimura's description, the condition $(c,d)=1$ should be replaced by $(c,d,N/d)=1$, see the answers by Max Horn and me. | |
| Apr 7, 2016 at 0:45 | history | edited | Michael Albanese | CC BY-SA 3.0 |
added 2 characters in body
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| Dec 20, 2011 at 23:08 | answer | added | François Brunault | timeline score: 3 | |
| Dec 20, 2011 at 22:21 | comment | added | François Brunault | The output by Sage is correct, I think it gives you representatives for the left quotient $\Gamma_0(N) \backslash \Gamma$. | |
| Dec 20, 2011 at 14:27 | answer | added | George McNinch | timeline score: 1 | |
| Dec 20, 2011 at 11:43 | answer | added | Nick | timeline score: 0 | |
| Dec 19, 2011 at 17:30 | comment | added | Max Horn | Yes, for prime $N$, we have that $R=Z/NZ$ is a finite field, in particular all elements are units, and it is well known that the projective line has the coset representatives $(1,1),\ldots,(1,N),(N,1)=(0,1)$. | |
| Dec 19, 2011 at 15:57 | answer | added | Igor Rivin | timeline score: 1 | |
| Dec 19, 2011 at 15:48 | comment | added | Igor Rivin | Does everything work for prime $N?$ | |
| Dec 19, 2011 at 15:17 | answer | added | Max Horn | timeline score: 3 | |
| Dec 19, 2011 at 13:10 | comment | added | Max Horn |
Actually, my comment (and my now deleted answer) is incorrect. Assuming you mean the group of matrices $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$ for which $c$ is divisible by $N$, then of course $1\in\Gamma$ is in the coset represented by $(N,1)$ (and (0,1) is just another rep for that). Moreover, $(1,0)$ and $(1,N)$ also represent the same coset.
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| Dec 19, 2011 at 12:37 | history | edited | Max Horn | CC BY-SA 3.0 |
Rewrote the matrices to be clearer (I did not understand them at all beforehand)
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| Dec 19, 2011 at 12:32 | comment | added | Max Horn | The missing representatives are (0,1) for the coset of 1, and (1,0). | |
| Dec 19, 2011 at 11:20 | history | asked | Nick | CC BY-SA 3.0 |