A Serre weight conjecture for geometric Hilbert modular forms in characteristic $p$

Let p be a prime and F totally real field in which is unramified. We consider mod Hilbert modular forms for F, defined as sections of automorphic line bundles on varieties level to characteristic p. For Hecke eigenform arbitrary weight (without parity hypotheses), we associate two-dimensional representation the absolute Galois group give conjectural description set weights all eigenforms from it arises. This conjecture can viewed "geometric" variant "algebraic" Serre Buzzard-Diamond-Jarvis, spirit Edixhoven's Serre's original case = Q. develop techniques studying giving rise fixed representation, prove results support conjecture, including cases partial one.