# 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.