Newton Strata Meeting Weakly Admissible Loci

Definitions and mathematical statement

Write $\mathcal F=\mathcal F(G,b,\mu)$ for the period-domain flag space appearing in the supplied record, and let $\mathcal F^{\mathrm{wa}}\subseteq\mathcal F$ denote its weakly admissible locus. For each $[b']\in B(G)$, write $\mathcal F^{[b']}=\mathcal F(G,b,\mu)^{[b']}$ for the Newton stratum with that index.

In the usual local-field interpretation, if $F$ is the underlying nonarchimedean local field, $\breve F$ is the completion of its maximal unramified extension, and $\sigma$ is its Frobenius automorphism, then \[ B(G)=G(\breve F)/{\sim_\sigma},\qquad b_1\sim_\sigma b_2\ \Longleftrightarrow\ b_2=g^{-1}b_1\sigma(g)\text{ for some }g\in G(\breve F). \] This explains the customary notation conditionally; the supplied record does not specify the underlying field or the hypotheses on $G$, $b$, and $\mu$.

Classification problem. For fixed period-domain data $(G,b,\mu)$ in the intended source setting, determine the subset \[ \mathcal S(G,b,\mu) :=\left\{[b']\in B(G): \mathcal F(G,b,\mu)^{[b']}\cap \mathcal F(G,b,\mu)^{\mathrm{wa}} \neq\varnothing\right\}. \] Thus the requested characterization must give necessary and sufficient conditions on $[b']$ for this intersection to be nonempty.

Unresolved formulation qualifications. The record does not provide the precise class of permitted period-domain data, the construction and indexing convention for its Newton strata, the definition of weak admissibility in this setting, or the field or geometric category in which nonemptiness is intended. These must be recovered from the source before a fully specified mathematical formulation can be accepted; no restriction to basic or Hodge--Newton-decomposable cases is imposed here.

Short English statement

For the intended fixed data $(G,b,\mu)$, characterize exactly which Newton strata contain a weakly admissible point. The supplied record omits hypotheses and conventions needed to make the period-domain setting fully precise. Its dated curation labels the problem partially solved and describes results in important basic and Hodge--Newton-decomposable cases; these claims require independent review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30004060$, title Newton Strata Meeting Weakly Admissible Loci, source problem code catalogue entry. The record assigns proposed year $2019$, names no proposer, and belongs to the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Arithmetic of Shimura Varieties (2020), Oberwolfach Reports, pp. 65--131, DOI 10.4171/owr/2019/2. Supplied source URL: https://doi.org/10.4171/owr/2019/2. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Chen--Fargues--Shen, p-adic period domains, https://arxiv.org/abs/1710.06935.
• Supplied literature reference: Oberwolfach report on arithmetic of Shimura varieties, https://doi.org/10.4171/OWR/2019/2.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem partially solved, reports answers for important basic and Hodge--Newton-decomposable cases, and states that other Newton strata remain unresolved and that later structure theorems do not supply the requested complete classification. The background expressly requests independent verification of time-sensitive status and cited claims.