Number Theory Seminar
Benjamin Bakker
NYU
The geometry of the Frey-Mazur conjecture
Abstract: 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 analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their p-torsion is at most two-to-one, and one-to-one in special cases. 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.
Please note the special time, day, and room.
Wednesday November 20, 2013 at 2:30 PM in SEO 636