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 which 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.