Partial weight one modularity for Galois representations associated to mod $p$ Hilbert modular forms

Let $p$ be an odd prime. Let $\rho: G_F \to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a Galois representation of a totally real field $F$. For a small partial weight one weight $(k,0)$, we prove that modularity of $\rho$ can be characterised using $p$-adic Hodge theory, as conjectured by Diamond and Sasaki. We show that if $\rho$ is modular with respect to a partial weight one mod $p$ Hilbert modular form, then each of its local representations has a crystalline lift with prescribed Hodge--Tate weights. Conversely, if for each $v|p$ the restriction $\rho|_{G_{F_v}}$ has a crystalline lift with certain irregular weights, we show that $\rho$ arises from a partial weight one Hilbert modular form. Our method consists of translating results from regular to irregular weights. We do this globally, relating modularity of regular weights to modularity of irregular weights and vice versa, and also use the local, $p$-adic Hodge theory analogue of this, which is recent work of the author.