Descent of Rational Points from Laurent Series Fields

Definitions and mathematical statement

Let $k$ be a field and let $t$ be an indeterminate. The Laurent series field is \[ k((t))=\left\{\sum_{n\geq N}a_nt^n : N\in{\mathbb{Z}},\ a_n\in k\right\}, \] with $k$ embedded as the constant series. For a $k$-scheme $X$ and a field extension $L/k$, write \[ X(L)=\operatorname{Hom}_k(\operatorname{Spec}L,X) \] for its set of $L$-rational points.

The source considers an affine algebraic $k$-group $G$ and a homogeneous space $X$ of $G$. Geometric homogeneity refers to a $k$-action $G\times_k X\to X$ that becomes transitive over an algebraic closure of $k$. The source does not specify the precise scheme-theoretic convention for a homogeneous space or for an algebraic group; in particular, it does not state smoothness, reducedness, connectedness, or stabilizer hypotheses. These conventions remain unresolved and are not added as assumptions here.

Question. For every field $k$, every affine algebraic $k$-group $G$, and every homogeneous $G$-space $X$, with the preceding qualification concerning the intended category, does \[ X(k((t)))\neq\varnothing \quad\Longrightarrow\quad X(k)\neq\varnothing \] hold? The supplied statement imposes no restriction on the characteristic of $k$.

The supplied dated assessment distinguishes known torsor settings from the proposed extension to homogeneous spaces and mentions known characteristic-zero cases. It does not give the exact hypotheses of those results. Those qualifications are retained as supplied context, without asserting a theorem or determining the precise remaining open range.

Short English statement

Must every homogeneous space of an affine algebraic group over a field $k$ that has a point over $k((t))$ already have a point over $k$? The precise geometric conventions and the scope of the reported known cases require review.

Sources

• Supplied local registry record: UnsolvedMath permanent ID $30004320$, title Descent of Rational Points from Laurent Series Fields, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The record gives proposed year 2019 and names no proposer.
• Supplied report citation: Mini-Workshop: Rank One Groups and Exceptional Algebraic Groups (2020), Oberwolfach Reports, pp. 3243--3269, DOI 10.4171/owr/2019/52. https://doi.org/10.4171/owr/2019/52. The background attributes the problem collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Florence and Gille, Residues on affine Grassmannians. https://doi.org/10.1515/crelle-2021-0007.
• Florence, Residues manuscript. https://webusers.imj-prg.fr/~mathieu.florence/residus11nov.pdf.
• Oberwolfach Reports, torsors and homogeneous spaces. https://ems.press/content/serial-article-files/46832.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, reports that the torsor statement is established in several settings, and describes its extension to homogeneous spaces as explicitly open beyond known characteristic-zero cases. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The source itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. This supplied editorial label is not independent mathematical verification of this draft.