Recovering elliptic curves from their \(p\)-torsion

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.