The geometry of the Frey-Mazur conjecture

Benjamin Bakker, December 10th, 2013

A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial
solution would yield an elliptic curve with modular p-torsion but which was itself not
modular. The connection between an elliptic curve and its p-torsion is very deep: a
conjecture of Frey and Mazur, stating that the p-torsion group scheme actually
determines the elliptic curve up to isogeny (at least when p>13), implies an asymptotic
generalization of Fermat's last theorem. We study a geometric analogue of this conjecture,
and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces
with quaternionic multiplication---to their p-torsion group scheme is one-to-one. Our
proof involves understanding curves on a certain Shimura surface, and fundamentally
uses the interaction between its hyperbolic and algebraic properties. This is joint work
with Jacob Tsimerman.