Algebraic Geometry Seminar
The Geometry of the Frey-Mazur Conjecture
Speaker: Benjamin Bakker , Courant Institute of Mathematical Sciences
Location: Warren Weaver Hall 317
Date: Tuesday, December 10, 2013, 3:30 p.m.
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.