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) . The Montréal functor appears as the‘Whittaker coefficient’ for the universal Galois representation, in the sense of the geometric Langlands program. 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.