# Recovering elliptic curves from their $p$-torsion

 Joint IAS/Princeton University Number Theory Seminar Topic: Recovering elliptic curves from their $p$-torsion Speaker: Benjamin Bakker Affiliation: New York University Date: Friday, May 2 Time/Room: 11:00am - 12:00pm/S-101 Video Link: http://video.ias.edu/jointiasnts/2014/0502-BenjaminBakker

Given an elliptic curve $E$ over a field $k$, its $p$-torsion $E[p]$ gives a 2-dimensional representation of the Galois group $G_k$ over $\mathbb F_p$. The Frey-Mazur conjecture asserts that for $k= \mathbb Q$ and $p > 13$, $E$ is in fact determined up to isogeny by the representation $E[p]$. In joint work with J. Tsimerman, we prove a version of the Frey-Mazur conjecture over geometric function fields: for a complex curve $C$ with function field $k(C)$, any two elliptic curves over $k(C)$ with isomorphic $p$-torsion representations are isogenous, provided $p$ is larger than a constant only depending on the gonality of $C$. The proof involves understanding the hyperbolic geometry of a modular surface.