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

Joint IAS/Princeton University Number Theory Seminar
Topic:Genus of abstract modular curves with level \(\ell\) structure
Speaker:Ana Cadoret
Affiliation:l'École Polytechnique; von Neumann Fellow, School of Mathematics
Date:Thursday, November 21
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/jointiasnts/2013/1121-AnaCadoret

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.