Steinness of Local Shimura Varieties

Definitions and mathematical statement

Let $\mathcal L$ denote the class of local Shimura varieties intended in the supplied report's reference to Hansen's conjecture, reference [10, Conjecture 1.10]. The proposed assertion is \[ \forall X\in\mathcal L,\qquad X\text{ is a Stein space}. \] The quantifier is universal over the intended class, with no restrictions stated in the supplied passage.

This is a source-faithful formulation with unresolved terminology: the supplied record does not define a local Shimura variety, specify its analytic category, base field, or level, or define the intended notion of Steinness. Consequently, $\mathcal L$ and the predicate "is a Stein space" require specification from the cited conjecture before this becomes a fully precise mathematical statement. In particular, no choice between finite and infinite level, or between possible nonarchimedean analytic conventions, is imposed here.

The report says that this conjecture would make its Theorem 3 potentially applicable to every such variety. The statement and hypotheses of Theorem 3 are not supplied; this contextual remark is therefore retained only as motivation, not as an additional mathematical assertion.

Short English statement

The conjecture attributed to Hansen asks whether every local Shimura variety is Stein. The supplied passage leaves the precise class of varieties and the meaning of Steinness unspecified, so these definitions require human review against the cited conjecture.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30005574$, title Steinness of Local Shimura Varieties, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Algebraische Zahlentheorie (2023), Oberwolfach Reports, pp. 1543--1606, DOI 10.4171/owr/2023/28. Supplied source report and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2023/28. The collection describes these reports as documenting mathematical workshops and their open-problem discussions.
• Hansen, reference [10, Conjecture 1.10] as cited in the supplied report passage. Supplied literature-source link labelled "Hansen": https://www.davidrenshawhansen.net/middlev2.pdf. The record does not provide the complete bibliographic entry for reference [10] or explicitly establish that this PDF is that entry. The conjecture is attributed to Hansen; no separate proposer is identified.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified here. It labels the general conjecture open and reports established special cases including Lubin--Tate, Drinfeld, and selected ordinary or reducible settings. These reported special cases are not supplied with precise hypotheses or individual theorem citations. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was checked against the official report. This is a supplied source-verification claim, not an independent verification performed for this draft; likewise, the supplied open status is a curation label.