Valuative Form of Grothendieck's Section Conjecture

Definitions and mathematical statement

Let $k$ be a number field, that is, a finite extension of ${\mathbb{Q}}$, and fix an algebraic closure $\bar k$. Write $G_k=\operatorname{Gal}(\bar k/k)$. Use the standard convention that a hyperbolic curve $X/k$ is a smooth, geometrically connected curve whose smooth projective compactification $\overline X$ has genus $g$ and whose geometric boundary $(\overline X\setminus X)(\bar k)$ has cardinality $r$, with $2g-2+r>0$. This convention allows both proper and nonproper curves; the supplied statement imposes no further restriction.

Choose a geometric base point $\bar x$ of $X$. The arithmetic étale fundamental group fits into the exact sequence of profinite groups \[ 1\longrightarrow \pi_1^{\mathrm{\acute et}}(X_{\bar k},\bar x) \longrightarrow \pi_1^{\mathrm{\acute et}}(X,\bar x) \xrightarrow{p}G_k\longrightarrow 1. \] A Galois section is a continuous group homomorphism \[ s:G_k\longrightarrow\pi_1^{\mathrm{\acute et}}(X,\bar x), \qquad p\circ s=\operatorname{id}_{G_k}. \] Sections are considered up to conjugation by $\pi_1^{\mathrm{\acute et}}(X_{\bar k},\bar x)$.

Write $K=k(X)$ for the function field. A valuation of $K$ is a map $w:K\to\Gamma\cup\{\infty\}$, where $\Gamma$ is a totally ordered abelian group, such that $w(0)=\infty$, $w(a)\in\Gamma$ for $a\ne0$, $w(ab)=w(a)+w(b)$ for nonzero $a,b$, and $w(a+b)\ge\min\{w(a),w(b)\}$.

Source question (valuative formulation attributed to Grothendieck's section conjecture). For every number field $k$, every hyperbolic curve $X/k$, and every continuous Galois section $s$ of the displayed exact sequence, does there exist a suitable $k$-valuation $w$ of $k(X)$ that induces the conjugacy class $[s]$?

Unresolved qualification. The supplied record does not define "suitable $k$-valuation" or the relation by which such a valuation induces a section of this étale fundamental group sequence. In particular, it does not specify the restriction of $w$ to $k$, its rank, residue-field conditions, allowed centers, or the required relation with decomposition groups. The general valuation definition above supplies notation only; it does not determine the admissible valuations. Consequently, the question retains these source terms explicitly and is not yet a fully specified valuative assertion. No equivalence with a rational-point formulation, and no replacement by a birational or local statement, is asserted here.

Short English statement

The supplied question asks whether every continuous Galois section of a hyperbolic curve over a number field comes, up to geometric conjugacy, from a suitable valuation of its function field. The source leaves both suitability and the meaning of "comes from" unspecified, so the valuative target requires clarification. Its open status is a supplied curation label.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002353$, title Valuative Form of Grothendieck's Section Conjecture, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record attributes the conjecture to Grothendieck, lists proposed year $2013$, and names no individual proposer.
• Supplied report citation: The Arithmetic of Fields (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/30, pp. 1771--1818. https://doi.org/10.4171/owr/2013/30. The supplied title year is retained alongside the DOI's $2013$ component; their relationship has not been independently checked. No problem-specific page is supplied.
• Stix, On Galois sections for hyperbolic p-adic curves. https://www.math.uni-frankfurt.de/~stix/research/preprints/owr-STIX-Jun13.pdf.
• Pop--Stix, Arithmetic in the fundamental group of a p-adic curve. https://arxiv.org/abs/1111.1354.
• Saidi, Section conjecture over function fields. https://arxiv.org/abs/1512.01207.
• Supplied literature assessment, checked at the exact recorded date 2026-08-21: the number-field Grothendieck section conjecture remains unresolved; valuative and p-adic analogues are proved; conditional reductions over finitely generated fields do not establish the number-field assertion. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record itself requests independent verification of time-sensitive status and cited claims; the scope and hypotheses of the claimed analogues are not specified here.
• Supplied verification metadata labels the statement corrected_verified and says it was checked against the official Oberwolfach report. This is a source-provided verification claim, not verification performed for this draft. The supplied problem status is open.