Local-global compatibility conjecture for the Galois representations

Let $F$, $S$, $T$, $Q$, $\Sigma$, the Hecke algebras $\mathbb{T}_{\Sigma,Q}$ and $\widetilde{\mathbb{T}}^S(K_{\Sigma,Q})$, and the universal deformation rings $R_{\Sigma,Q}$ be as defined in the paper. Let $\rho_{\Sigma,Q}\colon G_F\to\operatorname{GL}_2(\mathbb{T}_{\Sigma,Q})$ be the representation supplied by the Galois-representation conjecture. Local-global compatibility conjecture. For every $\Sigma$ and $Q$, $\rho_{\Sigma,Q}$ is induced from the universal representation through $R_{\Sigma,Q}\twoheadrightarrow\mathbb{T}_{\Sigma,Q}$ and satisfies the four listed trace, diamond-operator, local deformation, and compatibility-diagram conditions. The conjecture formulates the expected compatibility between local automorphic Hecke operators and the corresponding Galois representations; many cases are known up to nilpotent ideals.