A result due to B. Conrad (<http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf>, Theorem A.1) states that the Atkin-Lehner operator $w\_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M\_k(\Gamma\_0(N))$. In other words, if $f\in M\_k(\Gamma\_0(N))$ has coefficients in $\mathbb{Z}[1/Q]$ then $w\_{Q,k}(f)$ has coefficients in $\mathbb{Z}[1/Q]$.


Does anyone know of a corresponding result over $\Gamma\_1(N)$? My guess is that $w\_{k,Q}$ is $\mathbb{Z}[1/Q][\zeta\_Q]$-integral on $M\_k(\Gamma\_1(N))$ but I can't find a reference for this.


I am however, aware of a weaker result (Theorem 5.4 in <https://arxiv.org/abs/1807.00391>), which implies that $w\_{k,Q}$ is $\mathbb{Q}(\zeta\_Q)$-integral on $M\_k(\Gamma\_1(N))$.



Integrality of Atkin-Lehner operator for $\Gamma_1(N)$
**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.)


