Let $\Gamma\subset \mathrm{SL}_2(\mathbb Z)$ be a congruence subgroup and $h$ the fan width of $\Gamma$ (i.e; the minimum $h>0$ such that $\left(% \begin{array}{cc} 1 & h \\ 0 & 1 \\ \end{array}% \right)\in\Gamma.)$ and let $f$ be a weakly modular function with respect to $\Gamma.$ Let
$$q_h:\mathcal{H}\longrightarrow D^*=\{z\in\mathbb{C}\;:\; 0<|z|<1\}\;\;\; z \longmapsto e^{\frac{2\pi iz}{h}}$$ In this note (page 29) I find this statement :"Define $g$ by $g = f\circ q^{-1}_{h}$. That is, $g(q_h) = f(z).$ Although $q_h$ is not invertible, the above definition makes sense, and $g$ has a Laurent expansion"
Can someone explain to me why the definition $g = f\circ q^{-1}_{h}$ makes sense although $q_h$ not invertible ?
Can we take $g(q)=f\left(\frac{h\log(q)}{2\pi i }\right)$ ?
Thanks.
The formula $$g(q) = f\left(\frac{h \log(q)}{2 \pi i}\right)$$ makes sense despite the multi-valuedness of the $\log$ function, because the values of $\log$ are determined up to an integer multiple of $2 \pi i$. Choosing a different value of $\log(q)$ will therefore change the argument of $f$ by an integer multiple of $h$. But $f(z+h) = f(z)$ by the assumption that $h$ is the fan width. So there is no ambiguity in the definition of $g$.