Moduli stacks of Galois representations and the p -adic local Langlands correspondence for GL₂ ({{Q₏}})

Abstract We give a categorical formulation of the p -adic local Langlands correspondence for GL₂ ({{Q₏}}) 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 ({Q}ₚ/{{Q₏}}). 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 ({{Q₏}}) to the cohomology of modular curves through a local–global compatibility formula.