Outer Automorphisms of p-Adic Absolute Galois Groups

Definitions and mathematical statement

Let $p$ be a prime and let $K$ be a $p$-adic field. Using the usual interpretation of this term, take $K$ to be a finite extension of ${\mathbb{Q}}_p$. Fix a separable closure $K^{\mathrm{sep}}$, and give the absolute Galois group \[ G_K=\operatorname{Gal}(K^{\mathrm{sep}}/K) \] its Krull topology. Under the conventional continuous interpretation of automorphisms of a profinite group, define \[ \operatorname{Aut}_{\mathrm{cont}}(G_K) =\{\alpha:G_K\longrightarrow G_K:\alpha\text{ is a continuous group isomorphism}\}, \qquad \operatorname{Inn}(G_K) =\{g\mapsto hgh^{-1}:h\in G_K\}, \] and \[ \operatorname{Out}_{\mathrm{cont}}(G_K) =\operatorname{Aut}_{\mathrm{cont}}(G_K)/\operatorname{Inn}(G_K). \] Problem. For each prime $p$ and each $p$-adic field $K$, determine the structure of $\operatorname{Out}_{\mathrm{cont}}(G_K)$, denoted $\operatorname{Out}(G_K)$ in the supplied source.

Unresolved formulation qualifications. The source does not explicitly define a $p$-adic field or specify whether automorphisms must be continuous. The definitions above state the conventional interpretation used for this draft. It also does not specify what constitutes a determination of the structure, or whether the intended classification includes a topology on the outer automorphism group. Thus the request remains a structural classification problem rather than a specified conjectural isomorphism.

For context, write $G_K(p)$ for the maximal pro-$p$ quotient of $G_K$, namely the quotient by the intersection of all open normal subgroups of $p$-power index. The supplied excerpt reports a family of infinite-order elements in $\operatorname{Out}(G_K(p))$ generalizing surface Dehn twists. This is attributed background, not an independently verified assertion here, and the requested classification concerns $G_K$ itself. The excerpt motivates this work through arithmetic topology, attributed there to Mazur, and an analogy with surface mapping class groups; it specifies no identification with a mapping class group. Its subsequent discussion of surface-group splittings is truncated and supplies no additional precise hypothesis for the problem.

Short English statement

Describe the outer automorphism group of the absolute Galois group of each $p$-adic field, using continuous automorphisms under the stated convention. The supplied record labels the problem open. Its reported Dehn-twist constructions concern the maximal pro-$p$ quotient and do not supply a classification of the full group.

Sources

• Local UnsolvedMath registry, permanent ID $30006074$, source code catalogue entry, in the supplied complete source record. The registry labels the statement corrected_verified and says that the official report was checked and its notation repaired; these are supplied provenance claims, not verification performed for this draft. No individual proposer is identified; the proposed year is recorded as 2024.
• Supplied report citation: Anabelian Geometry and Representations of Fundamental Groups (2025), Oberwolfach Reports, pp. 2621--2670, DOI 10.4171/owr/2024/45. Also listed as Oberwolfach Report 45/2024. https://doi.org/10.4171/owr/2024/45.
• Supplied literature reference: Gropper, Surfaces and p-adic fields I. https://arxiv.org/abs/2303.04309.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the full outer automorphism group is not classified, while arithmetic Dehn twists are studied in the maximal pro-$p$ quotient. This is supplied dated curation, not independently verified literature assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.