The Buzzard–Diamond–Jarvis conjecture for unitary groups

Fix an imaginary CM field $F$ in which $p$ is unramified, let $F^+$ be its maximal totally real subfield, and set

$$S:=\coprod_{w\mid p}\operatorname{Hom}(k_w,\overline{\mathbb F}_p).$$

Let $({\mathbb Z}^2_+)^S_0$ consist of tuples $a$ satisfying $a_{\sigma,1}+a_{\sigma^c,2}=0$ for every $w\mid p$ and every $\sigma\in\operatorname{Hom}(k_w,\overline{\mathbb F}_p)$. A Serre weight is an element $a\in({\mathbb Z}^2_+)^S_0$ such that $a_{\sigma,1}-a_{\sigma,2}\leq p-1$ for every such $w$ and $\sigma$. Let $\bar r\colon G_F\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ be a continuous irreducible modular representation, and let $W^{\operatorname{BDJ}}(\bar r)$ be the set of Serre weights whose local component at every place $w\mid p$ lies in $W^{\operatorname{BDJ}}(\bar r|_{G_{F_w}})$. Buzzard–Diamond–Jarvis conjecture. If $a\in({\mathbb Z}^2_+)^S_0$ is a Serre weight, then $\bar r$ is modular of weight $a$ if and only if

$$a\in W^{\operatorname{BDJ}}(\bar r).$$

This conjecture gives the expected global description of the Serre weights in which a mod-$p$ Galois representation is modular, in terms of the corresponding local Buzzard–Diamond–Jarvis weight sets. The statement is presented here without resolution evidence, so its status remains open.