P-torsion monodromy representations of elliptic curves over geometric function fields

Abstract: Given a complex quasiprojective curve $B$ and a non-isotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow \mathrm{GL}2(\mathbb{F}_p)$. We prove that if $\rho{\mathcal E}[p]\cong \rho_{\mathcal E'}[p]$ then $\mathcal{E}$ and $\mathcal E'$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $B$. This can be viewed as a function field analog of the Frey--Mazur conjecture, which states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p> 17$. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.