Genus of abstract modular curves with level \(\ell\) structure

To any bounded family of $\mathbb F_\ell$-linear representations of the etale fundamental of a curve $X$ one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level $\ell$ structure ($Y_0(\ell), Y_1(\ell), Y(\ell)$ etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to $\infty$ with $\ell$. I will sketch a purely algebraic proof of the growth of the genus - working in particular in positive characteristic.