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Let k be an algebraically closed field of characteristic p. Let X(p^e;N) bethe curve parameterizing elliptic curves with full level N structure (where pdoes not divide N) and full level p^e Igusa structure. By modular curve, wemean a quotient of any X(p^e;N) by any subgroup of ((Z/p^e Z)^* x\SL_2(Z/NZ))/{+-1}. We prove that in any sequence of distinct modular curvesover k, the k-gonality tends to infinity. This extends earlier work, in whichthe result was proved for particular sequences of modular curves, such asX_0(N) for p not dividing N. We give an application to the function fieldanalogue of a uniform boundedness statement for the image of Galois on torsionof elliptic curves.Comment: 10 pages; many improvements over previous versio

Gonality of modular curves in characteristic~$p$