Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

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. I will then explain an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen-Khare--Thorne.