Ask Your Question

# Lifting matrices to $SL_2(\mathbb{Z})$

Is there a way to lift a matrix in $Sl_2(\mathbb{Z}/N\mathbb{Z})$ to $SL_2(\mathbb{Z})$ for an arbitrary N ?

I found this algorithm : here

But it seems to work only when $N$ is prime. I think that the algorithm described in Shimura (which the implementation cited above is base on) works for arbitrary N. But I don't know how to modify the one above to make it work. Does anyone know how to do it.

Ultimately my goal is to iterate through a list of representatives of $SL_2(\mathbb{Z}/24\mathbb{Z})$ in $SL_2(\mathbb{Z})$ so if anyone knows how to it without the above I would be equally (in fact probably more) grateful.

edit : Ok now I see at least one reason why the code mentionned above fails and in fact it isn't because of the arbitrary ring but because the smith normal form function doesn't return elements in $SL_2(\mathbb{Z})$ but only invertible matrices. Anyone knows how to bypass this ?

edit retag close merge delete

## 2 Answers

Sort by » oldest newest most voted

Actually, the question asks only for SL_2, while the trac ticket is more general (SL_m). The case of SL_2 is already in Sage (as long as N <= 2147483647). Try

sage: lift_to_sl2z?

more

OK I found the solution. In the code I mentioned the error was that smith_form doesn't return matrices in $SL_2(\mathbb{Z})$. It's quite easy to correct by the following snippet that you have to add just after you use smith_form :

 D, U, V = A.smith_form()
if det(U) == -1 :
U = matrix(2,2,[-1,0,0,1])* U
if det(V) == -1 :
V = V *matrix(2,2,[-1,0,0,1])
D = U*A*V

more

## Comments

Thanks for posting it, this may help resolution of the trac ticket you mentionned.

## Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

## Stats

Asked: 2013-05-31 19:29:39 +0200

Seen: 136 times

Last updated: Jun 03 '13