Genus Finiteness for Algebraic Groups

Definitions and mathematical statement

Let $K$ be a field, let $\overline K$ be an algebraic closure, and let $G$ be an absolutely almost simple simply connected algebraic group over $K$. A $K$-form of $G$ is an algebraic group $G'$ over $K$ such that $G'_{\overline K}\simeq G_{\overline K}$. A maximal $K$-torus of an algebraic group $H$ is a torus defined over $K$ whose base change to $\overline K$ is a maximal torus of $H_{\overline K}$. Write \[ \mathcal T_K(H)=\{[T]_K:T\text{ is a maximal }K\text{-torus of }H\}, \] where $[T]_K$ denotes the $K$-isomorphism class of the torus $T$. Interpreting "the same maximal $K$-tori" as equality of these sets, define \[ \operatorname{gen}_K(G)=\{[G']_K:G'\text{ is a }K\text{-form of }G,\ \mathcal T_K(G')=\mathcal T_K(G)\}. \] Thus each maximal $K$-torus of either group must be $K$-isomorphic to a maximal $K$-torus of the other.

Question. Under what hypotheses on $K$ and $G$ is \[ \#\operatorname{gen}_K(G)<\infty? \] In particular, does this finiteness hold for every such $G$ over every field $K$ finitely generated over its prime field and having "good characteristic"?

Unresolved qualifications. The supplied record does not define "good characteristic" or state the characteristic restrictions intended for each group type, so the particular question remains conditional on that missing specification. The interpretation of equal maximal tori above uses their abstract $K$-isomorphism classes; the record does not specify additional compatibility of embeddings. The displayed genus follows the explicit statement by allowing all $K$-forms. The background separately mentions inner forms and spinor groups without specifying whether a narrower genus or additional hypotheses were intended.

Short English statement

Determine when only finitely many $K$-isomorphism classes of forms of $G$ have exactly the same maximal tori up to $K$-isomorphism. The supplied question particularly asks about finitely generated fields of "good characteristic", but leaves that characteristic condition unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003799$, title "Genus Finiteness for Algebraic Groups", source code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2018.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, pp. 1235--1291, DOI 10.4171/owr/2018/21. Source URL and supplied literature reference, labeled "Oberwolfach Report 21": https://doi.org/10.4171/owr/2018/21. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No precise problem location within the report is supplied.
• Supplied literature assessment dated 2026-08-21: the record labels the problem open, describes the extraction as a contextual fragment rather than a resolved proposition, and states that the associated genus problem for inner forms and spinor groups remains unresolved. This is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report for preservation of hypotheses, notation, and target. This verification claim is supplied metadata, not an independent verification performed for this draft.