Recovering elliptic curves from their ptorsion
Recovering elliptic curves from their ptorsion

Benjamin Bakker, NYU
IAS Room S101
Please note special day (Friday) and time (11:00 a.m.) Given an elliptic curve E over a field k, its ptorsion E[p] gives a 2dimensional representation of the Galois group G_k over F_p. The FreyMazur conjecture asserts that for k=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 FreyMazur conjecture over geometric function fields: for a complex curve C with function field k(C), any two elliptic curves over k(C) with isomorphic ptorsion 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.