The plectic conjecture over local fields

Abstract: The étale cohomology of varieties over $\mathbf{Q}$ enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. They conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analogue of this conjecture for local Shimura varieties. This implies that, for p-adically uniformized global Shimura varieties, we obtain an action of the local plectic group on the level of complexes. The proof crucially uses Fargues–Scholze's results on the cohomology of moduli spaces of local shtukas.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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