Minimal weights of Hilbert modular forms in characteristic p

TL;DR

This paper proves that mod p Hilbert modular forms over totally real fields can be generated from forms with weights in a specific cone using partial Hasse invariants, clarifying their minimal weights.

Contribution

It establishes that all mod p Hilbert modular forms can be obtained from forms with weights in a certain cone via multiplication by partial Hasse invariants, answering a previously open question.

Findings

Every mod p Hilbert modular form arises from a form with weight in a specific cone.

The proof uses properties of the Goren-Oort stratification on Hilbert modular varieties.

The result clarifies the structure of minimal weights for these forms.

Abstract

We consider mod p Hilbert modular forms associated to a totally real field of degree d in which p is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a d-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren-Oort stratification on mod p Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.