Recovering elliptic curves from their p-torsion

Recovering elliptic curves from their p-torsion

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Benjamin Bakker, NYU
IAS Room S-101

Please note special day (Friday) and time (11:00 a.m.)  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 F_p. The Frey-Mazur 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 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.