Mathematical Research Letters

Volume 14 (2007)

Number 4

Gonality of modular curves in characteristic~$p$

Pages: 691 – 701

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n4.a14

Author

Bjorn Poonen (University of California at Berkeley)

Abstract

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.

Full Text (PDF format)