3
$\begingroup$

Let $\Gamma$ be the modular group and $\mathcal M_n$ the set of all primitive matrices with determinant $n\geq1$. Recall that a primitive matrix has relatively prime entries.

The modular group $\Gamma$ acts on the set $\mathcal M _n$ from the left. The representatives of the orbits of this action can be chosen as

$$S(n)=\bigg \lbrace \begin{pmatrix}r &s\\0&t\end{pmatrix}:r>0,rt=n,(r,s,t)=1,0\leq s<t\bigg\rbrace.$$

One encounters this, for example, when studying the modular equation $\Phi_n(X,X)$. Cf. Zagier, p. 68.

Now consider a generalization of $S(n)$, namely

$$\mathcal{S}\left(n,d\right)=\bigg\lbrace \begin{pmatrix}r &s\\0&t\end{pmatrix}: r>0,rt=n,(r,t,s)=1,d \mid s, 0 \leq s < dt\bigg\rbrace.$$

These sets arise from the study of modular functions for certain subgroups of $\Gamma$ and their transformation equations.

My question is this: is there some set of matrices $\mathcal M_{n,d}$ and a subgroup $\Gamma_{n,d}\subset\Gamma$ such that $S(n,d)$ is a set of representatives for the orbits of the left action $\Gamma_{n,d}\backslash\mathcal M_n$?

$\endgroup$
1
$\begingroup$

This is a bit of a trivial example: take $$ \mathcal{M}_{n,d} = \left\{ \begin{pmatrix} r & s \\ 0 & t \end{pmatrix} ~:~ rt = n, d \mid s,~ (r,t,n)=1 \right\} $$ and the subgroup $$ \Gamma_{n,d} = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}~:~ ac = 1,~d\mid b \right\} ~\subset~ \Gamma. $$ Another idea is to write $\mathcal{S}(n,d)$ as the quotient $$\mathcal{S}(n,d) = g\Gamma g^{-1}~\backslash~ g\mathcal{M}_n g^{-1}$$ for $$g =\begin{pmatrix} d & 0 \\ 0 & 1 \end{pmatrix},$$ but $g\Gamma g^{-1}$ is not a subgroup of $\Gamma$.

$\endgroup$
1
  • $\begingroup$ This seems somewhat artificial. I am seeking a more natural group. But I admit that the question is stated rather vaguely. I will try to improve it. $\endgroup$ – Shimrod May 20 '18 at 21:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.