The mod $p$ Langlands lifting and automorphy conjecture for global fields

Let $F$ be a global field of characteristic different from $p$. Let $G$ be a connected reductive group, and let $G'$ be the split group over $F$ whose root datum is dual to that of $G$. Let $S$ be a finite set of primes of $F$, containing all primes above $p$ when $F$ is a number field, and let

$$\bar{\rho} \colon \Gamma_{F,S} \to G(k)$$

be a continuous representation unramified outside $S$. If $F$ is a number field, assume further that it is totally real and that $\bar{\rho}$ is odd. Mod $p$ Langlands lifting and automorphy conjecture. There is a lift

$$\rho \colon \Gamma_F \to G(\mathcal{O})$$

of $\bar{\rho}$ to the ring of integers $\mathcal{O}$ in some finite extension of $\mathbb{Q}_p$ inside $\overline{\mathbb{Q}}_p$, unramified outside a finite set of primes, with Zariski-dense image in $G$ and, in the number field case, regular Hodge--Tate cocharacters. There are also an $L$-algebraic cuspidal automorphic representation $\pi$ of $G'(\mathbb{A}_F)$ and an isomorphism $\iota \colon \mathbb{C} \to \overline{\mathbb{Q}}_p$ such that the conjectural Galois representation

$$\rho_{\pi,\iota} \colon \Gamma_F \to G(\overline{\mathbb{Q}}_p)$$

associated to $(\pi,\iota)$ is isomorphic to $\rho \colon \Gamma_F \to G(\mathcal{O}) \subset G(\overline{\mathbb{Q}}_p)$. Moreover, if $\bar{\rho}$ is absolutely irreducible, $\rho$ and $\pi$ can be taken to be unramified outside $S$. In the function field case, the automorphic Galois representation construction is known by V. Lafforgue for general $G$; in the number field case, the conjecture concerns automorphic Galois representations expected to have regular infinitesimal character, and remains open in general.