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Explore the intricacies of multivariable $(\varphi, \Gamma)$-modules and their connection to local-global compatibility in this 54-minute lecture by Yongquan Hu at BIMSA. Delve into the generalization of Colmez's construction, associating admissible smooth mod $p$ representations of $\mathrm{GL}_2(K)$ with multivariable (étale) $(\varphi, \mathcal{O}_K^{\times})$-modules over a suitable ring $A$. Examine the challenges in explicitly computing these modules and discover how they can be calculated when derived from Hecke eigenspaces in mod $p$ cohomology of Shimura curves. Compare the results with multivariable $(\varphi, \mathcal{O}_K^{\times})$-modules associated with mod $p$ Galois representations. Gain insights into this collaborative research effort with Breuil, Herzig, Morra, and Schraen, advancing our understanding of local-global compatibility in number theory.
