Number Theory Seminar

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 B.Bakker, we prove a version of the Frey-Mazur conjecture for 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 depending only on the gonality of C. The proof method involves the hyperbolic geometry of a modular surface, and has other applications to arithmetic geometry which we will discuss.