Given an irreducible smooth complex-projective curve $X$, I will say that a subgroup $\Gamma< SL(2, {\mathbb R})$ weakly uniformizes $X$ if [corrected] there exists a nonconstant holomorphic map from ${\mathbb H}^2/\Gamma$ to $X$.
One formulation of the Shumura-Taniyama conjecture is that if $X$ is (the set of complex points of) an elliptic curve defined over ${\mathbb Q}$, then there exists a congruence-subgroup $\Gamma< SL(2, {\mathbb Z})$ which weakly uniformizes $X$.
(I learned this from the paper by Barry Mazur "Number theory as a gadfly, but misremembered the statement.)
Question: What if we remove the assumption that $X$ is elliptic: Does this result still hold?
Remark. 1. One reformulation of Belyi's theorem is that if $X$ is a complex-projective curve over $\bar{\mathbb Q}$ then $X$ is uniformized by a finite index subgroup $\Gamma$ of $SL(2, {\mathbb Z})$, in the sense that ${\mathbb H}^2/\Gamma$ is biholomorphic to an open (and dense) subset of $X$.
- I am a topologist, not a number-theorist, so my interest in this question is purely casual.
Update. (Thanks to a comment by François Brunault). The paper
Baker, Matthew H.; González-Jiménez, Enrique; Gonzáles, Josep; Poonen, Bjorn, Finiteness results for modular curves of genus at least 2, Am. J. Math. 127, No. 6, 1325-1387 (2005). ZBL1127.11041.
contains some results suggesting that the answer is strongly negative, i.e. that for each genus $\ge 2$ only finitely many curves over ${\mathbb Q}$ are weakly uniformized by congruence subgroups of $SL(2, {\mathbb Z})$. They conjecture this finiteness property under the extra assumption that the holomorphic map is a morphism defined over ${\mathbb Q}$. I have no feel for how strong this restriction is. They prove this conjecture under certain extra assumptions.
