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.