Gonality of modular curves in characteristic~$p$

Let $k$ be an algebraically closed field of characteristic $p$. Let $X(p^e;N)$ be the curve parameterizing elliptic curves with full level $N$ structure (where $p \nmid N$) and full level $p^e$ Igusa structure. By modular curve, we mean a quotient of any $X(p^e;N)$ by any subgroup of $\left( (\Z/p^e\Z)^\times \times \SL_2(\Z/N\Z) \right)/\{\pm 1\}$. 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 $X_0(N)$ for $p \nmid N$. As an application, we prove the function field analogue of a uniform boundedness conjecture for the image of Galois on torsion of elliptic curves.

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Published 1 January 2007