The telescope conjecture for derived Artin motives over local fields

The problem asks whether the telescope conjecture holds for derived Artin motives over local fields. The reported result addresses only a stated unequal-characteristic condition, not the entire arithmetic category.

Known results

• For a finite field $\mathbb{F}$ and arbitrary coefficient field $k$, the big derived category $\mathrm{DAM}_{\mathrm{big}}(\mathbb{F};k)$ is stratified, has weakly noetherian compact spectrum, and satisfies the telescope property.
• The big tensor-triangulated category $\mathscr{T}(\hat{\mathbb{Z}}_{p})$ is BHS-stratified and satisfies the telescope conjecture.

September 2026 claimed advance

A September 2026 arXiv report claims stratification and generic noetherianity of the Balmer spectrum for derived Artin motives over selected unequal-characteristic local fields, implying the telescope question under that condition. This claim is unverified and explicitly is not a general resolution.

Current status (as of September 2026): The telescope conjecture is claimed for selected unequal-characteristic local fields, while the general local-field case remains open and the new claim is unverified.