Moduli stacks of Galois representations and the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$

Abstract

We give a categorical formulation of the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$,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 $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. Moreover, we relate our version of the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ to the cohomology of modular curves through a local-global compatibility formula.

69 pages, revisions after referee report