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Apparently for the subgroups $\Gamma(N) \subset SL_2(\mathbb{Z})$ there exists an exact sequence $$ 1 \to \Gamma(N) \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/N\mathbb{Z})\to0 $$ I understand that $\Gamma(N)$ can be embedded in $SL_2(\mathbb{Z})$. Also, since the sequence is exact we have that $$ SL_2(\mathbb{Z}/N\mathbb{Z}) = SL_2(\mathbb{Z})/\Gamma(N) $$ But I fail to understand exactly the latter relation in terms of these groups.

a) Is there any specific example that will illustrate this latter injective morphism of the exact sequence in a clear way?

b) And also, what is the importance, if any,of this quotient group $ SL_2(\mathbb{Z}/N\mathbb{Z})$? For modular forms only the congruence group is important, in the sense that this is the family of groups one studies.

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    $\begingroup$ I am not sure I understand the question. It is a subgroup, so the injective map is just the inclusion as a subgroup. $\endgroup$ Commented Nov 7, 2016 at 9:47

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$ \DeclareMathOperator{\SL}{SL} \newcommand{\Z}{\mathbb{Z}} $ a) The map $\SL_2(\Z) \to \SL_2(\Z/N\Z)$ acts on a matrix by reducing all its entries mod $N$. Since $\Gamma(N)$ is the kernel of this map, this means an element of $\Gamma(N)$ is congruent to the identity matrix mod $N$. More explicitly, its elements are matrices in $\SL_2(\Z)$ that are of the form $$ \begin{pmatrix} a & b\\ c & d \end{pmatrix} \qquad a \equiv d \equiv 1 \pmod{N}, \quad b \equiv c \equiv 0 \pmod{N} \, . $$

b) I don't think $\SL_2(\Z/N\Z)$ is of any particular importance, except for its connection to congruence subgroups. Since $\Gamma(1)/\Gamma(N) \cong \SL_2(\Z/N\Z)$, then the index of $\Gamma(N)$ in $\Gamma(1)$ is the order of the group $\SL_2(\Z/N\Z)$, which happens to be $n^3 \prod_{p \mid n} \left(1 - \frac{1}{p}\right)$. Since the modular curve $X(1) = \Gamma(1) \backslash \mathcal{H}$ parametrizes elliptic curves over $\mathbb{C}$, and $X(N) = \Gamma(N) \backslash \mathcal{H}$ parametrizes elliptic curves over $\mathbb{C}$ with given $N$-torsion, the size of the index probably has some connection with elliptic curves.

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