**Theorem**. Let $\ell$ be prime, and $Q, R \ge 1$ such that $(\ell, Q, R)$ are pairwise coprime. Let $N = QR$ and for simplicity assume $N \ge 4$. Then $W_Q$ preserves $M_k(\Gamma_1(N), \mathbf{Z}[1/N, \zeta_Q])$.

*Proof*. Let $M_k^{\mathrm{wk}}(\Gamma_1(N), A)$ denote the space of weakly modular forms (possibly meromorphic at the cusps) with $q$-expansions in the ring $A$. If $A$ is a subring of $\mathbf{C}$ then
$$M_k(\Gamma_1(N), A) = M_k^{\mathrm{wk}}(\Gamma_1(N), A) \cap M_k(\Gamma_1(N), \mathbf{C})$$
so it suffces to show that $M_k^{\mathrm{wk}}(\Gamma_1(N), \mathbf{Z}[1/N, \zeta_Q])$ is stable under $W_Q$.

If $A$ is a $\mathbf{Z}[1/N]$-algebra, then elements of $M_k^{\mathrm{wk}}(\Gamma_1(N), A)$ can be interpreted as rules sending "test objects" $(E, P_Q, P_R, \omega) / B$ to elements of $B$. Here $B$ is a $A$-algebra, $E$ is an elliptic curve over $B$, $P_Q$ and $P_R$ are points of exact order $Q$, $R$ respectively, and $\omega$ is a global differential on $E$. These have to satisfy various conditions (the main ones are compatibility with base change and homogeneity in $\omega$ of weight $k$).

So it suffices to show that $W_Q$ makes sense on test objects if $A = \mathbf{Z}[1/N, \zeta_Q]$. The map will send $(E, P_Q, P_R, \omega) / B$ to $(E', P_Q', P_R', \omega')/B$, where all but one of these objects are simple to define:

- $E' = E/\langle P_Q \rangle$
- $P_R' = \pi(P_R)$, where $\pi : E \to E / \langle P_Q\rangle$ is the quotient map
- $\omega' = \pi_*(\omega)$

The hard one is $P_Q'$: one checks that $E[Q] / \langle P_Q\rangle$ has a unique generator $P_Q'$ characterised by the Weil pairing $\langle P_Q, P_Q'\rangle = \zeta_Q$ (and since it is unique, its formation is compatible with base-change). $\square$

(Caveat: this construction gives an operator whose square is something like $\langle Q\rangle_R Q^k$ where $\langle -\rangle_R$ denotes the diamond operator for something that is 1 (mod Q) and Q (mod R). Some people prefer to normalise away the $Q^k$, but this may not be possible if $Q$ is odd without introducing an extraneous $\sqrt{Q}$ into your ring.)

The Manin constant and the modular degreemath.u-psud.fr/~cesnavicius/Manin-degree.pdf the authors have proved bounds on the denominators of modular forms at the cusps, using adelic techniques (see Section 4). Maybe you could email them and ask whether their results are sufficient to prove what you want. $\endgroup$ – François Brunault Feb 28 '20 at 8:49