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Consider the groups \begin{align*} \Gamma_0(N) \; &:= \; \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) \;:\; c \equiv 0 \mod N \biggr\} \\ \Gamma_\infty \; &:= \; \biggl\{ \begin{pmatrix} \pm 1 & n \\ 0 & \pm 1 \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) \;:\; n \in \mathbb{Z} \biggr\} . \end{align*} Via Sage or GAP (see for example https://www.gap-system.org/Manuals/pkg/congruence/doc/chap4.html#X79C44528864044C5) we can compute the generators of $\Gamma_0(N)$, where I am interested in how one can decompose any given element $\gamma \in \Gamma_\infty \backslash \Gamma_0(N)$ in terms of those generators.

Given an $\gamma = \begin{pmatrix} * & * \\ c & d \end{pmatrix} \in \Gamma_\infty \backslash \Gamma_0(N)$, we only need to care about the decomposition of $(c, d)$ by right-side multiplication.

My current method have been to

  1. Make sure that all the generators $g_1, \ldots, g_k$ are on the form $T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and possibly $J = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$, and the rest being on the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $a, c$ have the same sign, and opposite sign of $b, d$.
  2. Make the transform $\gamma \mapsto \gamma \, T^n = \gamma_1$ such that the $c(\gamma_1) = c(\gamma)$ and $|d(\gamma_1)| < |c(\gamma_1)|$ where $d(\gamma_1)$ has the opposite sign to $c(\gamma_1)$.
  3. Try to minimize $c(\gamma_1)$ by looking at the ratio $| d(\gamma_1) / c(\gamma_1)|$ and try to find a generator $g$ (or its inverse $g^{-1}$) which has a similar ratio $|a(g) / c(g)|$. When such a generator is found, map $\gamma_1 \mapsto \gamma_1 g = \gamma_2$.
  4. "Repeat" previous step until $\gamma_i$ can be expressed in terms of a generator.

Now, as an example consider $\Gamma_0(10)$, which is generated by the elements \begin{align*} T &= \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \\ U_1 &= \begin{pmatrix} 3 & -1 \\ 10 & -3 \end{pmatrix} \\ U_2 &= \begin{pmatrix} 19 & -7 \\ 30 & -11 \end{pmatrix} \\ U_3 &= \begin{pmatrix} 11 & -5 \\ 20 & -9 \end{pmatrix} \\ U_4 &= \begin{pmatrix} 7 & -5 \\ 10 & -7 \end{pmatrix} \end{align*} whose inverses are \begin{align*} T^{-1} &= \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \\ U_1^{-1} &= \begin{pmatrix} -3 & 1 \\ -10 & 3 \end{pmatrix} \\ U_2^{-1} &= \begin{pmatrix} -11 & 7 \\ -30 & 19 \end{pmatrix} \\ U_3^{-1} &= \begin{pmatrix} -9 & 5 \\ -20 & 11 \end{pmatrix} \\ U_4^{-1} &= \begin{pmatrix} -7 & 5 \\ -10 & 7 \end{pmatrix} . \end{align*}

Let's say we are given $\gamma = \begin{pmatrix} * & * \\ 10 & 9 \end{pmatrix}$, which we want to decompose. After step one, we find $\gamma_1 = \gamma \, T^{-1} = \begin{pmatrix} * & * \\ 10 & -1 \end{pmatrix}$. The closest ratio we find is given by $U_1$, and after this step we find $\gamma_2 = \gamma_1 U_1 = \begin{pmatrix} * & * \\ 20 & -7 \end{pmatrix}$. Here the closest ratio is given by $U_2^{-1}$, where $\gamma_3 = \gamma_2 U_2^{-1} = \begin{pmatrix} * & * \\ -10 & 7 \end{pmatrix} \sim U_4$. Thus \begin{equation*} \gamma = U_4 U_2 U_1^{-1}\, T. \end{equation*}

I have found this algorithm to work for many examples, but not for every example. For prime $N$ I have seem to find an extension to this technique which has worked for example I have tried.

However, one example that I am not able to find any decomposition for is \begin{equation*} \gamma = \begin{pmatrix} * & * \\ 40 & -3 \end{pmatrix} \in \Gamma_0(10). \end{equation*}

Does anyone know how to decompose this $\gamma$? How can I find a more general technique for finding decompositions?

aahlback
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