For positive integers $n$ and $L$, denote by $SL\_n(Z,L)$ the level $L$ congruence subgroup of $SL\_n(Z)$, i.e. the kernel of the homomorphism $SL\_n(Z)\rightarrow SL\_n(Z/LZ)$.


For $n$ at least $3$, it is known that $SL\_n(Z,L)$ is normally generated (as a subgroup of $SL\_n(Z)$) by Lth powers of elementary matrices. Indeed, this is essentially equivalent to the congruence subgroup problem for $SL\_n(Z)$.


However, this fails for $SL\_2(Z,L)$ since $SL\_2(Z)$ does not have the congruence subgroup property.


Question : Is there a nice generating set for $SL\_2(Z,L)\ ?$ I'm sure this is in the literature somewhere, but I have not been able to find it.



Generators for congruence subgroups of SL_2
Hi Andy,


I don't know if you are still interested in this, but I just found the reference:


MR0049937 (14,250d)
Grosswald, Emil
On the parabolic generators of the principal congruence subgroups of the modular group.
Amer. J. Math. 74, (1952). 435--443.


It is based on the previous work of H.Frasch (1933) who gave an explicit set of free generators for principal congruence subgroups Gamma(p) in PSL(2,Z), for prime p's.


-Ignat


