# Computing the index $[\text{SL}(2,\mathbb{Z}) : \Gamma(N)]$

Hi I have a question about congruence subgroups $Γ(N)$ of $SL(2,\mathbb{Z})$.

How to compute the index $[SL(2,\mathbb{Z}): Γ(N)]$?

BTW, can anyone tell me double cosets decomposition $\sigma^{-1}_{a}\Gamma\sigma_{b}$, where $a$ and $b$ are two cusps and $\sigma$ is an element in $SL(2,\mathbb{R})$.

Thanks

As far as I remember: A First Course in Modular Forms by Diamond and Shurman explains this in the first chapter. Miyake and Shimura probaly also address this as well in their books on modular forms.

The following strategy works for GL(N) and SL(N) as well.

First of all, $\Gamma(N)$ is normal in $\Gamma(1)=SL_2(\mathbb{Z})$ with the quotient being isomorphic to $SL_2(\mathbb{Z}/N)$.

Next, the isomorphism $$\mathbb{Z}/N \cong \bigoplus_{p^k || N} \mathbb{Z}/p^k$$ induces an isomorphism $$SL_2(\mathbb{Z}/N) \cong \bigoplus\limits_{p^k || N} SL_2(\mathbb{Z}/p^k).$$

Then if $k=1$, we have by the Bruhat decomposition of $SL_2(\mathbb{Z}/p)$ the following coset decomposition $$SL_2(\mathbb{Z}/p) = B(\mathbb{Z}/p) \amalg N(\mathbb{Z}/p) w_0 B(\mathbb{Z}/p)$$ for $B$ being the upper diagonal, and $N$ the strict upper diagonal matrices. So the cardinality is here $p(p-1) + p^2(p-1)$. The element $w_0$ is the Weyl element, i.e., zeros on the diagonal and $\pm 1$ on the antidiagonal (the order doesn't matter).

For $k \geq 2$, you should also apply the Iwahori decomposition of $\Gamma_0(p)$ together with $$SL_2(\mathbb{Z}) = \Gamma_0(p) \amalg \Gamma_1(p) w_0 \Gamma_0(p).$$

I leave the computation for $k \geq 2$ as an exercise to you.

(Caution, this is for $GL(2)$) You can also look at lemma 1 in http://link.springer.com/content/pdf/10.1007%2FBF01355984.pdf