Smoothness of Ekedahl–Kottwitz–Oort–Rapoport Strata

Definitions and mathematical statement

An Iwahori level and a parahoric level refer here to level subgroups $I\subseteq K$ at the prime of reduction. Write $S_I$ and $S_K$ for the corresponding reductions of Shimura varieties in the intended source setting, and write $\pi:S_I\to S_K$ for the level-forgetting morphism. This notation is conditional: the supplied record does not specify the Shimura datum, prime, integral models, residue field, or hypotheses under which these reductions and morphisms are defined.

A Kottwitz--Rapoport stratum is a member of the named stratification at Iwahori level; an Ekedahl--Kottwitz--Oort--Rapoport (EKOR) stratum is a member of the named stratification at parahoric level. The record supplies neither construction nor indexing conventions for these stratifications. Denote their index sets by $A$ and $B$, respectively, and their strata by $X_a\subseteq S_I$ and $Y_b\subseteq S_K$. Let $c(a)\in B$ denote the stratum described in the record as corresponding to $X_a$. The asserted finite morphism is denoted by \[ f_a:X_a\longrightarrow Y_{c(a)}. \] Here a finite morphism is one for which the inverse image of every affine open $\operatorname{Spec}(R)$ is affine, say $\operatorname{Spec}(T)$, with $T$ finite as an $R$-module. It is finite étale if it is additionally étale.

Question, with unresolved scope. For every parahoric-level Shimura variety within the intended but unspecified source hypotheses, are all the asserted finite morphisms $f_a$ finite étale? In symbols, is \[ \forall a\in A,\qquad f_a:X_a\longrightarrow Y_{c(a)} \quad\text{finite \'{e}tale}? \] The record also asks whether all EKOR strata are consequently smooth. If $k$ denotes the common residue field intended by the reduction setting, this asks whether \[ \forall b\in B,\qquad Y_b\longrightarrow\operatorname{Spec}(k) \quad\text{is smooth}. \] The field $k$, the relevant class of Shimura varieties, the correspondence $c$, and the precise stratum morphisms are not specified by the record. In particular, the record does not provide the smoothness of the source strata or the covering and surjectivity properties needed to justify the proposed consequence by descent. These omissions prevent an unconditional, fully specified formulation.

The supplied dated curation says that the extracted claim is truncated and that the report records a proof in the Siegel case, while classifying the general question as open. The scope and hypotheses of that reported special case are not supplied here. These are source qualifications, not independently verified mathematical or literature conclusions.

Short English statement

The question asks whether every stated finite map from an Iwahori-level Kottwitz--Rapoport stratum to its corresponding parahoric-level EKOR stratum is finite étale, and whether all EKOR strata are consequently smooth. The supplied extract leaves the general hypotheses and the justification of this consequence unresolved. Its dated curation reports a proof in the Siegel case and labels the general question open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002947$, source problem code catalogue entry, title: Smoothness of Ekedahl--Kottwitz--Oort--Rapoport Strata; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Reductions of Shimura Varieties (2016), Oberwolfach Reports, pp. 2265--2328, DOI 10.4171/owr/2015/39. Source URL and literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2015/39.
• Supplied literature reference labelled "full report text": https://ems.press/content/serial-article-files/46591.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open, describes the extracted claim as truncated, and reports a proof in the Siegel case without settlement of the general statement. This is supplied dated curation, not an independently verified assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.