Moduli stacks of Galois representations and the p-adic local Langlands correspondence for GL₂ (Qₚ)

We give a categorical formulation of the p-adic local Langlands correspondence for GL₂ (Qₚ), as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of Gal (Qₚ/Qₚ). Moreover, we relate our version of the p-adic local Langlands correspondence for GL₂ (Qₚ) to the cohomology of modular curves through a local-global compatibility formula.