Let $Y$ be a Riemann surface, and $$T=\sum_{k=0}^n a_k y^k \in K(Y)[y]$$ be a polynomial in $y$ of degree $n$ over the field $K(Y)$ of meromorphic functions on $Y$. If we denote by $O$ the discrete subset of $Y$ containing all poles of the coefficients $a_k$ and the zeros of the discriminant of $T$, then $T=0$ defines an $n$-fold ramified covering $\pi: M\to Y$ together with a meromorphic function $y$ on $M$, such that for all $x_0=Y\backslash 0$ the set of roots of $T_{x_0}$ coincides with the set of values of $y$ on the preimage $\pi^{-1}(x_0)$.
Let now $Y=X(1)$ be the modular curve for $SL(2,\mathbb Z)$, which is a compactification of its fundamental domain $SL(2,\mathbb Z)\backslash \mathbb H$. In case the polynomial $T$ is such that its roots are modular functions for a congruence subgroup $\Gamma\subset SL(2,\mathbb Z)$ of index $[SL(2,\mathbb Z):\Gamma]=n$, then $T=0$ defines an $n$-fold covering over $X(1)$, which is precisely the modular curve of $\Gamma$ (a compactification of $\Gamma\backslash SL(2,\mathbb Z)$).
In case the polynomial $T$ is not of such a form, $T=0$ defines a Riemann surface $M$ that is not a modular curve, and any root $y$ of $T=0$ is not a modular function for a congruence subgroup. I am wondering if nevertheless statements can be made about $y$, for instance:
- Is there a domain $D\subset\mathbb H$ such that $y:D\to P^1(\mathbb C)$ is an isomorphism ($y$ is a Hauptmodul and $D$ is a fundamental domain for $y$)? Can $M$ be "unfolded" on the upper half-plane $\mathbb H$, with maps $\alpha_j\in SL(2,\mathbb Z)$ such that $D=\bigcup_{j=1}^n \alpha_j\mathcal F$ where $\mathcal F$ is a fundamental domain for $SL(2,\mathbb Z)$?
- Is $y$ automorphic on $M$?
Many thanks for any suggestions!