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.
Johansson et al. (Sun,) studied this question.