Geometric weight-shifting operators on Hilbert modular forms in characteristic p

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author's prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial <![CDATA[ $\Theta $ ]]> -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial <![CDATA[ $\Theta $ ]]> -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.