|The Plectic Conjecture over Local Fields|
The étale cohomology of varieties over 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 includes (the generic fibers of) Lubin–Tate spaces, Drinfeld upper half spaces, and more generally Rapoport–Zink spaces. The proof crucially uses Scholze's theory of diamonds.