Abstract

I will survey how to prove the modularity of elliptic curves defined over the rational numbers, as pioneered by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond and Taylor. I will also mention the case of elliptic curves defined over real quadratic fields, more recently completed by Freitas, Le Hung and Siksek. I will then explain why the case of imaginary quadratic fields is qualitatively different from the previous ones. Finally, I will discuss joint work in progress with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. This has an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen, Khare and Thorne.