The Cone of Minimal Weights for Mod p Hilbert Modular Forms

We prove that all mod p Hilbert modular forms arise via multiplication by generalized partial Hasse invariants from forms whose weight falls within a certain minimal cone. This answers a question posed by Andreatta and Goren and generalizes our previous results that treated the case where p is unramified in the totally real field. Whereas our previous work made use of deep Jacquet–Langlands type results on the Goren–Oort stratification (not yet available when p is ramified), here we instead use properties of the stratification at Iwahori level, which are more readily generalizable to other Shimura varieties.