Matrix Factorization over Curve Patching Fields

Definitions and mathematical statement

Let $F_U$ and $F_P$ be subfields of a common field $F_{\wp}$, with compatible inclusions of $F$, such that \[ F=F_U\cap F_P\qquad\text{inside }F_{\wp}. \] The supplied source requires these to be patching fields associated with a curve. It does not specify the curve, its base, the construction of the fields, or the meanings of the indices $U,P,\wp$. The question below is therefore conditional on a choice of such patching data; the displayed intersection identity alone does not specify the intended geometric setting.

For a linear algebraic group $G$ over $F$ and a field extension $K/F$, write $G(K)$ for its group of $K$-rational points. Regard $G(F_U)$ and $G(F_P)$ as subgroups of $G(F_{\wp})$ via the given field inclusions. Say that $G$ has the factorization property for these data if the multiplication map of sets \[ \mu_G:G(F_U)\times G(F_P)\longrightarrow G(F_{\wp}),\qquad (g_U,g_P)\longmapsto g_Ug_P, \] is surjective. Equivalently, this means \[ \forall g\in G(F_{\wp})\ \exists g_U\in G(F_U)\ \exists g_P\in G(F_P):\quad g=g_Ug_P. \] No uniqueness of the factors is required.

Question. For which linear algebraic groups $G/F$ does this factorization property hold for the curve patching fields in the source? The extracted formulation imposes no connectedness or rationality hypothesis on $G$. It does not clarify whether the desired characterization is relative to fixed patching data or must hold uniformly over all admissible curve patching data; this quantifier remains unresolved.

Short English statement

Determine which linear algebraic groups over $F$ allow every point over $F_{\wp}$ to be written, in the stated order, as a point over $F_U$ times a point over $F_P$. The geometric construction of the patching fields and the intended scope of quantification over those fields require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001253$, source problem code catalogue entry, titled Matrix Factorization over Curve Patching Fields; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2009 and supplies no individual proposer attribution.
• Quadratic Forms and Linear Algebraic Groups (2009), Oberwolfach Reports, pp. 1375--1426, DOI 10.4171/owr/2009/25. Supplied report link: https://doi.org/10.4171/owr/2009/25.
• Harbater--Hartmann--Krashen, Applications of patching. Supplied literature link: https://arxiv.org/abs/0809.4481.
• Supplied literature assessment dated 2026-08-21: the record labels the problem partially_solved and reports that patching gives factorization for rational and retract-rational connected groups, while the extension to arbitrary other linear algebraic groups remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied record labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification description is supplied metadata, not a claim of source consultation or verification for this draft.