Anna Cadoret

Note on the Gonality of Abstract Modular Curves

Let S be a curve over an algebraically closed field k of characteristic \(p \geq 0\). To any family of representations \(\rho= ({\rho }_{\mathcal{l}}\, :\ {\pi }_{1}(S) \rightarrow {\mbox{ GL}}_{n}({\mathbb{F}}_{\mathcal{l}}))\) indexed by primes \(\mathcal{l} \gg0\) one can associate abstract modular curves \({S}_{\rho ,1}(\mathcal{l})\) and \({S}_{\rho }(\mathcal{l})\) which, in this setting, are the modular analogues of the classical modular curves \({Y }_{1}(\mathcal{l})\) and Y (l). The main result of this paper is that, under some technical assumptions, the gonality of \({S}_{\rho }(\mathcal{l})\) goes to \(+\infty \) with \(\mathcal{l}\). These technical assumptions are satisfied by \({\mathbb{F}}_{\mathcal{l}}\)-linear representations arising from the action of π1(S) on the etale cohomology groups with coefficients in \({\mathbb{F}}_{\mathcal{l}}\) of the geometric generic fiber of a smooth proper scheme over S. From this, we deduce a new and purely algebraic proof of the fact that the gonality of \({Y }_{1}(\mathcal{l})\), for \(p \nmid\mathcal{l}({\mathcal{l}}^{2} - 1)\), goes to \(+\infty \) with l.