Finiteness of Algebraic Forms with Good Reduction

Definitions and mathematical statement

Let $K$ be a field and let $G$ be an absolutely almost simple algebraic $K$-group that is either simply connected or adjoint. Here absolutely almost simple means that, over an algebraic closure of $K$, the group is nontrivial, connected, semisimple, and has no proper nontrivial connected normal algebraic subgroup. Simply connected is understood in the sense of semisimple algebraic groups; adjoint means that the center is trivial.

An inner $K$-form of $G$ is an algebraic $K$-group $H$ admitting an isomorphism $f:H_{K_s}\to G_{K_s}$, where $K_s$ is a separable closure of $K$, such that the descent cocycle $f\,{}^{\sigma}\!f^{-1}$ takes values in $\operatorname{Inn}(G)(K_s)$ for every $\sigma\in\operatorname{Gal}(K_s/K)$. The groups $H$ are counted up to $K$-isomorphism.

For a nontrivial discrete valuation $v$ of $K$, write $K_v$ for the completion and $\mathcal O_v$ for its valuation ring. Using the standard completed local convention, $H$ has good reduction at $v$ if there exists a reductive group scheme $\mathcal H$ over $\mathcal O_v$ with \[ \mathcal H\times_{\mathcal O_v}K_v\simeq H\times_K K_v. \] A reductive group scheme here is a smooth affine group scheme of finite presentation whose geometric fibers are connected reductive algebraic groups. The supplied record does not specify its local-model convention; the completed local convention is made explicit here for review against the cited source.

Problem. Determine under what hypotheses on $K$ and $G$ the field $K$ admits a natural set $V$ of discrete valuations such that \[ \text{for every finite }S\subseteq V,\qquad \#\left\{ [H]_K\;\middle|\; \begin{gathered} H\text{ is an inner }K\text{-form of }G,\\ H\text{ has good reduction at every }v\in V\setminus S \end{gathered} \right\}<\infty. \] The set $V$ is chosen before the quantification over $S$. No restriction on the characteristic of $K$, or assumption that $K$ is finitely generated, is stated in the problem. The word "natural" is not defined in the supplied record, so the admissible choices of $V$, and consequently the precise classification target, remain unresolved. In particular, this is a request to determine circumstances in which the property holds, rather than an assertion that it holds for every field and group.

Short English statement

Determine when a field $K$ and an absolutely almost simple, simply connected or adjoint $K$-group $G$ admit a natural collection of discrete valuations for which only finitely many inner forms of $G$, up to $K$-isomorphism, have good reduction outside any prescribed finite subset. The source leaves "natural" unspecified. Its supplied label is partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30003794$, source problem code catalogue entry, title Finiteness of Algebraic Forms with Good Reduction, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, pp. 1235--1291, DOI 10.4171/owr/2018/21. https://doi.org/10.4171/owr/2018/21. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The citation year 2019 and the record's 2018 year indicators are preserved without reconciliation.
• Supplied literature source, labeled "spinor groups with good reduction": https://arxiv.org/abs/1707.08062.
• Supplied literature source, labeled "good-reduction survey": https://arxiv.org/abs/2005.05484.
• Supplied literature source, labeled "workshop report": https://ems.press/content/serial-article-files/46742.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports finiteness results in important cases, including inner type $A$, spinor groups over two-dimensional global fields, and related types, while describing the arbitrary-field and general finitely-generated-field problem as open. This is supplied dated curation, not independently verified. The record expressly requests independent verification of time-sensitive status and the cited claims; the exact hypotheses of those reported results are not supplied.
• The registry marks the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied review annotations, not independent mathematical verification by this draft.