Multivariable (φ, Γ)-modules and local global compatibility for the mod p cohomology of Shimura curves

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• Stefano Morra (Université de Paris Saint-Denis)
• Tuesday 09 May 2023, 14:30-15:30
• MR15.

If you have a question about this talk, please contact Rong Zhou.

Let K be a finite unramified extension of Qp. Using techniques from perfectoid geometry we describe functors from the category of continuous finite dimensional mod p representations of Gal(Qp/K), and from the category of smooth mod p representations of GL2 , into a category of multivariable (φ, O × K)-modules. We state a conjecture towards the mod p local Langlands program relating the two functors, and prove it in several cases: if K is the completion at a p-adic place of a totally real field F, r : Gal(Q/F) → GL2 a modular Galois representation which is tame and sufficiently generic at Gal(Qp/K), and π is the smooth representation cut out by the r-eigenspace in the cohomology of a Shimura curve with infinite level at p, then the two functors produce the same (φ, O × K)-module. This is a report on a series of work joint with C. Breuil, F. Herzig, Y. Hu et B. Schraen.

This talk is part of the Number Theory Seminar series.

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