Mazur's program B

David Zureick-Brown

I'll discuss recent progress on Mazur's “Program B” -- the problem of classifying all possibilities for the “image of Galois” for an elliptic curve over \(\mathbf Q\) (equivalently, classification of all rational points on certain modular curves \(X_H\)).

This will including my own recent work with Jeremy Rouse which completely classifies the possibilities for the 2-adic image of Galois associated to an elliptic curve over the rationals, and work in progress for other prime powers. I will also survey other very recent results by many authors.