Model Companions and Geometry of Globally Valued Fields

Definitions and mathematical statement

Write $T_e=\mathrm{GVF}_e$ for the theory of globally valued fields referred to in the supplied source. The record does not specify its language, axioms, the meaning or range of $e$, or its logical semantics. Consequently, the questions below are relative to the intended source definitions; the record alone does not determine a complete formalization of globally valued fields.

For clarity, in a classical first-order presentation, a model $M\models T_e$ is existentially closed among models of $T_e$ if, for every extension $M\subseteq N\models T_e$, every finite tuple $\bar a$ from $M$, and every quantifier-free formula $\varphi(\bar x,\bar y)$, \[ N\models\exists\bar x\,\varphi(\bar x,\bar a) \quad\Longrightarrow\quad M\models\exists\bar x\,\varphi(\bar x,\bar a). \] A class is elementary if it is the class of models of a first-order theory in the specified language. A theory $T_e^*$ is a model companion of $T_e$ if it is model complete, every model of $T_e$ embeds into a model of $T_e^*$, and every model of $T_e^*$ embeds into a model of $T_e$. Model completeness means that every embedding between models of $T_e^*$ is elementary. These definitions explain the classical terminology only; if the intended framework is continuous logic, its corresponding semantic definitions are required instead.

1. Do the existentially closed globally valued fields form an elementary class in the intended language and semantics? The source presents this as equivalent to asking whether $\mathrm{GVF}_e$ has a model companion. That equivalence is retained as a source formulation; its applicability cannot be checked without the missing axioms and logical framework. The record also refers to the theories $\mathrm{GVF}_e$ in the plural, without specifying the intended parameter range.
2. Is there a Berkovich-type geometry for globally valued fields in which closed points represent global heights on residue rings of an algebra? The source does not specify the permitted algebras, their base, the meaning of residue ring, the height axioms or normalization, the topology, or the required correspondence between closed points and heights. Thus this is a geometric existence question whose precise requirements remain unspecified.
3. What is the asymptotic first-order theory of number fields regarded as globally valued fields? The cleaned statement additionally asks for the asymptotic first-order theory of globally valued fields satisfying Northcott's principle. Neither the asymptotic regime nor the precise Northcott condition is specified. In particular, the record supplies no choice of height normalization, bounding parameters, or degree convention. The Northcott extension occurs in the cleaned statement but not in the supplied original excerpt, so its exact source formulation requires review.

The supplied registry labels these questions open. Its dated assessment says that the existence of model companions remains conjectural in the cited foundations text and that existential-closure results for selected fields do not settle the full theory or the other questions. These are supplied curation claims, not independently verified mathematical conclusions.

Short English statement

The source asks whether existentially closed globally valued fields admit a first-order axiomatization or model companion, whether global heights have a Berkovich-type geometric interpretation, and what asymptotic theories arise from number fields and fields satisfying Northcott's principle. The logical framework, height conventions, geometric requirements, and asymptotic regime need clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006580$, title: Model Companions and Geometry of Globally Valued Fields; source problem code catalogue entry. Collection: Oberwolfach Reports Open Problems; category: Logic; proposed year: 2026. No individual proposer is identified.
• Model Theory and Arithmetic (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/3, pp. 157--216. Supplied report reference and literature-source label: Oberwolfach Report 3/2026. https://doi.org/10.4171/owr/2026/3. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Ben Yaacov--Destic--Hrushovski--Szachniewicz, Globally Valued Fields. Supplied literature-source link: https://ora.ox.ac.uk/objects/uuid:0526cc37-8e0d-4c41-bc2d-a62178aae2e1/files/d736665338.
• Literature assessment dated exactly 2026-08-22: supplied status open. The supplied assessment states that model companions for the theories $\mathrm{GVF}_e$ remain conjectural in the foundations text and that known existential-closure results for selected fields do not establish the full theory or settle the other questions. This is supplied dated curation derived from the OWR material, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and reports that the official report was checked directly and that the cleaned statement repairs OCR, notation, and source-dependent wording while preserving the complete target. These are supplied verification claims; no source visit or independent verification is asserted here. The record was updated on 2026-09-16T12:06:53Z.