Gonality of modular curves in characteristic~$p$

By Bjorn Poonen. 2006

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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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0daa59b331b42c27c8cb20c076aa43a852b85bac

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Citations in the corpus

Expander graphs, gonality, and variation of Galois representations (2010, background): “) Poonen has shown [52] that (19) holds when U = X(1) is the moduli space of elliptic curves and (Ui) is a sequence of modular curves of increasing level.”