Finiteness of Tate–Shafarevich Sets for Algebraic Groups

Definitions and mathematical statement

A field $K$ is finitely generated if it is finitely generated as a field extension of its prime field. An absolutely almost simple algebraic $K$-group $G$ is a nontrivial connected semisimple algebraic group whose base change to an algebraic closure of $K$ has no proper nontrivial connected normal algebraic subgroup.

The supplied statement requires $K$ to have "good characteristic," but does not define that requirement or specify its dependence on $G$. It also does not specify the ambient family of places. Write $V$ for that intended family, let $S\subseteq V$ be finite, and write $K_v$ for the completion at $v$, whenever the intended notion of place supplies such a completion. These are placeholders for missing source specifications, not additional choices resolving them.

For a field extension $L/K$, let $H^1(L,G)$ denote the nonabelian Galois cohomology pointed set of $G_L$, equivalently the set of isomorphism classes of $G_L$-torsors, with distinguished point the trivial torsor. Extension of scalars gives the localization map \[ \operatorname{loc}_{V\setminus S}\colon H^1(K,G)\longrightarrow\prod_{v\in V\setminus S}H^1(K_v,G). \] Define its distinguished fiber by \[ \mathcal{X}(G)_S :=\operatorname{loc}_{V\setminus S}^{-1}\left((1_v)_{v\in V\setminus S}\right) =\left\{\xi\in H^1(K,G):\xi|_{K_v}=1_v\text{ for every }v\in V\setminus S\right\}, \] where $1_v$ is the class of the trivial $G_{K_v}$-torsor. This is a pointed set; no group structure is assumed.

The supplied finiteness question asks whether, for every finitely generated field $K$ and every absolutely almost simple $K$-group $G$ satisfying the intended good-characteristic hypothesis, and every finite $S$ in the intended family $V$, one has \[ \#\mathcal{X}(G)_S<\infty. \] The hypothesis on characteristic and the family $V$ must be recovered before this becomes a fully specified conjecture. In particular, this draft does not assert the claim for arbitrary families of places. The original statement additionally asks whether "Question 2" has a positive answer, with the displayed finiteness question as a particular consequence. The content of Question 2 is absent from the supplied record, so that broader target cannot be formulated here.

Short English statement

The question asks whether there are only finitely many isomorphism classes of $G$-torsors over $K$ that become trivial over every specified completion outside a finite exceptional set $S$. The intended places, the meaning of good characteristic, and the broader referenced Question 2 remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003797$, source problem code catalogue entry, title: Finiteness of Tate--Shafarevich Sets for Algebraic Groups. Collection: Oberwolfach Reports Open Problems. 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, DOI 10.4171/owr/2018/21, pp. 1235--1291. https://doi.org/10.4171/owr/2018/21. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• 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 states that the Tate--Shafarevich finiteness conjecture follows in known type-A, spinor, and related cases, while the general finitely generated good-characteristic conjecture remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report, preserving its hypotheses, notation, and open target. This is a supplied verification claim; no source consultation or mathematical verification is asserted by this draft.