Appears in collection : 2019 - T2 - WS2 - Rational points on irrational varieties

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 Q (equivalently, classification of all rational points on certain modular curves XH). 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.