For every prime p≥ 11 and every nonsplit très ramifiée representation bar rhocong omega&0&1 , let M be a primitive projective-augmented module of the corresponding Steinberg type, and let G_M denote its ghost series. At every point of the relevant component of p-adic weight space, the Newton polygon of the characteristic power series of the operator U_p on the corresponding space agrees with the Newton polygon of G_M.

Local ghost conjecture

The conjecture compares actual Up spectra with conjectural ghost series. The latest result concerns nonsplit très ramifiée representations and establishes the local statement under explicit hypotheses.

Known results

A 2022 precursor formulated the local conjecture and developed its ghost-series and Newton-polygon framework.
A later related paper describes the generic local version as solved by Liu–Truong–Xiao–Zhao.
A preprint states the conjecture for $p\ge 11$ and $2\le a\le p-5$ , with boundary cases and smaller primes excluded.

September 2026 local theorem

Liyan Wang's preprint The Local Ghost Theorem in the Très Ramifié Case claims the conjecture for $p\ge 11$ and the specified nonsplit très ramifiée representations, including ghost duality and finite-minor estimates. The claim is unverified; global très ramifié applications and analogous peu ramifié cases remain ongoing.

Current status (as of September 2026): The specified local très ramifiée case is claimed solved for $p\ge 11$ , but this preprint result is unverified and the global and peu ramifiée cases remain open.

Local ghost conjecture

Primary source

Additional references

Progress summary

Solutions 0