Characterization of Integral Models of Shimura Varieties

Definitions and mathematical statement

For an integral domain $R$ with fraction field $F$, and a Shimura variety $V$ defined over $F$, the generic-fibre condition for an integral model consists of an $R$-scheme $\mathcal V$ together with an isomorphism \[ \mathcal V\times_{\operatorname{Spec}R}\operatorname{Spec}F\;\cong\;V. \] This supplies terminology only: the source does not specify the allowable rings, Shimura varieties, levels, or additional conditions on $\mathcal V$, such as flatness, finiteness, or an extension property.

Problem. Find a good general characterization of integral models of Shimura varieties.

The intended characterization must go beyond merely restating the generic-fibre condition, but the supplied record does not specify its desired form or the class of models to be characterized. Consequently, a uniquely determined quantified mathematical statement cannot be extracted without additional hypotheses. In particular, the record does not specify whether the intended bases are global or local, which reduction conditions are allowed, or which properties distinguish the intended models.

Short English statement

The problem asks for a general characterization of integral models of Shimura varieties. The source leaves the base, the class of models, and the required characterizing properties unspecified, so the precise scope requires mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004471$, source problem code catalogue entry, title Characterization of Integral Models of Shimura Varieties. No individual proposer is supplied; the recorded proposed year is 2020.
• Supplied report citation: Arithmetic Geometry (2021), Oberwolfach Reports, pp. 1023--1082, DOI 10.4171/owr/2020/20. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2020/20. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated 2026-08-22: supplied dated curation, not independently verified. It labels the problem open and states that a good general characterization remains open. It further states that Fargues' related characterization of integral local-Shimura models does not settle this extracted question. No separate reference for that characterization is supplied. The background expressly requests independent verification of time-sensitive status and cited claims.