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A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence subgroup $\Gamma_n(m)$ of level $m$: $$ A\, \Gamma_n(m)\, A^{-1}=\Gamma_n(m). $$ What can be said about the matrix $A?$

A modification of Emerton's answer to the quoted question seems to imply (if I am not mistaken) that $A$ is of the form $1/d \cdot B,$ where $d$ is a natural number and all coefficients of a nonsingular matrix $B$ are of the form $p/m,$ where $p \in \mathbf Z.$ Any thoughts, and especially references, would be very much appreciated.

A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence subgroup $\Gamma_n(m)$ of level $m$: $$ A\, \Gamma_n(m)\, A^{-1}=\Gamma_n(m). $$ What can be said about the matrix $A?$

A modification of Emerton's answer to the quoted question seems to imply (if I am not mistaken) that $A$ is of the form $1/d \cdot B,$ where $d$ is a natural number and all coefficients of a nonsingular matrix $B$ are of the form $p/m,$ where $p \in \mathbf Z.$ Any thoughts, and especially references, would be very much appreciated.

A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence subgroup $\Gamma_n(m)$ of level $m$: $$ A\, \Gamma_n(m)\, A^{-1}=\Gamma_n(m). $$ What can be said about the matrix $A?$ A modification of Emerton's answer to the quoted question seems to imply (if I am not mistaken) that $A$ is of the form $1/d \cdot B,$ where $d$ is a natural number and all coefficients of a nonsingular matrix $B$ are of the form $p/m,$ where $p \in \mathbf Z.$ Any thoughts, and especially references, would be very much appreciated.

A question quite similar to this question. Let $n \geqslant 3$ and $m \geqslant 2$ be natural numbers and suppose that a matrix $A \in \mathrm{GL}(n,\mathbf Q)$ normalizes the principal congruence subgroup $\Gamma_n(m)$ of level $m$: $$ A\, \Gamma_n(m)\, A^{-1}=\Gamma_n(m). $$ What can be said about the matrix $A?$ 

A modification of Emerton's answer to the quoted question seems to imply (if I am not mistaken) that $A$ is of the form $1/d \cdot B,$ where $d$ is a natural number and all coefficients of a nonsingular matrix $B$ are of the form $p/m,$ where $p \in \mathbf Z.$ Any thoughts, and especially references, would be very much appreciated.