On the plectic conjecture

Abstract

The plectic conjecture of Nekovář–Scholl predicts extra symmetries in the cohomology of Hilbert modular varieties. More precisely, they conjecture that the complex computing its ℓ-adic intersection cohomology has a natural action of the plectic Galois group, which extends the usual Galois action. Nekovář–Scholl also predict such an extension for more general Shimura varieties. After describing a conjectural application to the construction of higher-rank Kolyvagin systems, we prove analogs of the plectic conjecture over global function fields and local fields. Namely, we prove a version for moduli spaces of global shtukas, which are analogs of Shimura varieties over function fields, as well as a version for local Shimura varieties, which are analogs of Shimura varieties over p-adic fields. The proofs rely on fusion, which in the p-adic case was recently developed by Fargues–Scholze. Using p-adic uniformization theorems, we deduce the plectic conjecture for certain (global) Shimura varieties after restricting to a decomposition group.