Gonality of abstract modular curves in positive characteristic

Abstract

Let$C$be a smooth, separated and geometrically connected curve over a finitely generated field$k$of characteristic$p\geqslant 0$,$\unicode[STIX]{x1D702}$the generic point of$C$and$\unicode[STIX]{x1D70B}_{1}(C)$its étale fundamental group. Let$f:X\rightarrow C$be a smooth proper morphism, and$i\geqslant 0$,$j$integers. To the family of continuous$\mathbb{F}_{\ell }$-linear representations$\unicode[STIX]{x1D70B}_{1}(C)\rightarrow \text{GL}(R^{i}f_{\ast }\mathbb{F}_{\ell }(j)_{\overline{\unicode[STIX]{x1D702}}})$(where$\ell$runs over primes$\neq p$), one can attach families of abstract modular curves$C_{0}(\ell )$and$C_{1}(\ell )$, which, in this setting, are the analogues of the usual modular curves$Y_{0}(\ell )$and$Y_{1}(\ell )$. If$i\not =2j$, it is conjectured that the geometric and arithmetic gonalities of these abstract modular curves go to infinity with$\ell$(for the geometric gonality, under a certain necessary condition). We prove the conjecture for the arithmetic gonality of the abstract modular curves$C_{1}(\ell )$. We also obtain partial results for the growth of the geometric gonality of$C_{0}(\ell )$and$C_{1}(\ell )$. The common strategy underlying these results consists in reducing by specialization theory to the case where the base field$k$is finite in order to apply techniques of counting rational points.