Diamond–Sasaki geometric Serre weight conjecture

Let $F$ be a totally real field in which $p$ is unramified, let $\Sigma$ be the set of embeddings $F\to L$, and let $G_F$ denote the absolute Galois group of $F$. For a weight $(k,l)\in\mathbb{Z}^{\Sigma}\times\mathbb{Z}^{\Sigma}$, write $\Xi_{\mathrm{min}}^+=\{k\in\mathbb{Z}^{\Sigma}:pk_\tau\geq k_{\operatorname{Fr}^{-1}\circ\tau}\text{ for all }\tau\in\Sigma\}\cap\mathbb{Z}_{\geq1}^{\Sigma}$. A representation is geometrically modular of weight $(k,l)$ when it arises from a mod $p$ Hilbert modular eigenform of that weight. For each $v\mid p$, a restriction $\rho|_{G_{F_v}}$ has a crystalline lift of weight $(k_\tau,l_\tau)_{\tau\in\Sigma_v}$ if it admits a crystalline characteristic-zero lift with the corresponding Hodge–Tate types. Diamond–Sasaki's geometric weight conjecture. If $\rho:G_F\to\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ is irreducible and geometrically modular of some weight, and $k\in\Xi_{\mathrm{min}}^+$, then $\rho$ is geometrically modular of weight $(k,l)$ if and only if, for every $v\mid p$, $\rho|_{G_{F_v}}$ has a crystalline lift of weight $(k_\tau,l_\tau)_{\tau\in\Sigma_v}$. The conjecture gives a geometric description of the weights of Hilbert modular forms attached to a fixed residual representation; the paper establishes supporting results but does not resolve the general assertion.