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

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