Cohomology vanishing for Ekedahl--Oort strata on Hilbert--Blumenthal Shimura varieties

We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl–Oort strata of the special fiber of Hilbert–Blumenthal Shimura varieties. This work vastly extends to all Ekedahl–Oort strata previous results of Diamond–Kassaei and Goldring–Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\le 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring–Koskivirta on the set of possible weights of automorphic forms.