Gonality of modular curves in characteristic p

Let k be an algebraically closed field of characteristic p. Let X(p;N) be the curve parameterizing elliptic curves with full level N structure (where p N) and full level p Igusa structure. By modular curve, we mean a quotient of any X(p;N) by any subgroup of ((Z/peZ)× × SL2(Z/NZ)) /{±1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X0(N) for p N . As an application, we prove the function field analogue of a uniform boundedness conjecture for the image of Galois on torsion of elliptic curves.