Joint IAS/PU Arithmetic Geometry

I will talk about a proof of local-global compatibility at p for higher coherent cohomology mod p in weight one for Hilbert modular varieties at an unramified prime, assuming we are not in middle degree. I will discuss some key ingredients to the proof, including a geometric Jacquet-Langlands transfer map to suitable quaternionic Shimura varieties, and a doubling map in this setting.