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 ${\rho }_{\mathcal{E}}[p]:{\pi }_1(B)\to {\mathrm{GL}}_2({\mathbb{F}}_p)$. We prove that if ${\rho }_{\mathcal{E}}[p]\cong {\rho }_{{\mathcal{E}}^{\prime }}[p]$ then $\mathcal{E}$ and ${\mathcal{E}}^{\prime }$ 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.

Comments Welcome! v2: Many improvements to the exposition and some proofs, based on suggestions of the referee. To appear in Ann. of Math

References in corpus (2)

• On families of n-congruent elliptic curves
• On the Frey-Mazur conjecture over low genus curves