Let $j$ be the modular invariant and let $\tau$ be a point in the upper half-plane. Let $\mathfrak o_\tau$ consist of all $f\in \mathbf C(j)$ which are defined at $\tau$. Let $\mathfrak O_\tau$ consist of all $f\in F_N$ defined at $\tau$. Here $F_N$ is a the field of modlar functions of level $N$ rational over $\mathbf Q(\zeta_N)$, see this question for a description. Is it true that $\mathfrak O_\tau$ is the integral closure of $\mathfrak o_\tau$ in $F_N$?

Let $j$ be the modular invariant and let $\tau$ be a point in the upper half-plane. Let $\mathfrak o_\tau$ consist of all $f\in \mathbf Q(j)$ which are defined at $\tau$. Let $\mathfrak O_\tau$ consist of all $f\in F_N$ defined at $\tau$. Here $F_N$ is a the field of modlar functions of level $N$ rational over $\mathbf Q(\zeta_N)$, see this question for a description. Is it true that $\mathfrak O_\tau$ is the integral closure of $\mathfrak o_\tau$ in $F_N$? Alternatively, we may ask the question for the extension $\mathbf C(j)\subset \mathbf C \cdot F_N$.