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

Benjamin Bakker

New York University

May 2, 2014

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.