For every number field F, prime p, Hilbert modular variety X/F, and cohomological degree i, the locally analytic p-adically completed cohomology widetildeH^i(X) should carry the canonical plectic Galois action predicted by Nekovář and Scholl. In particular, this predicts the corresponding canonical plectic Lie algebra action on widetildeH^i(X). The full conjecture over number fields remains unproved; the cited work constructs only part of the predicted action.

Nekovář–Scholl plectic conjecture

The Nekovář–Scholl conjecture predicts a canonical symmetry acting on cohomology associated with Hilbert modular varieties. Existing work proves restricted versions and provides structural evidence, but does not establish the full number-field conjecture.

Known results

Loeffler and Zerbes (2025): the comparison conjecture is proved for $[F:\mathbb{Q}]=2$ under mild assumptions, with a weaker splitting result for general $F$ .
The plectic conjecture over function fields (2021, revised 2021): an action is constructed in the global function-field setting.
The local-field version is proved in a categorical setting, but this does not settle the global conjecture.
Leonhardt (2020, revised 2022): plectic actions are extended to further Shimura varieties, without proving the conjecture.

September 2026 partial construction

Jiang and Pan construct part of the predicted plectic Lie algebra action on locally analytic $p$ -adically completed cohomology, using partial Sen operators. This strengthens the evidence for the conjecture but explicitly does not claim the full action; the advance is reported here as unverified.

Current status (as of September 2026): restricted cases and partial structures are known, but the full Nekovář–Scholl plectic conjecture for Hilbert modular varieties remains open.

Nekovář–Scholl plectic conjecture

Primary source

Additional references

Progress summary

Solutions 0