Higgs Bundles Arising from p-Adic Representations

Definitions and mathematical statement

The supplied record asks for a characterization of the Higgs bundles arising from genuine representations under a $p$-adic Simpson correspondence. It does not specify the geometric setting, the version of the correspondence, or the meaning of "genuine representation." Consequently, the target can be stated conditionally, but a fully specified mathematical problem requires these missing choices.

Fix a setting in which the chosen correspondence is defined. Write $X$ for its underlying space and $\mathcal D$ for its specified sheaf of differentials, including any twist required by that version. A Higgs bundle in this setting is a pair $(E,\theta)$, where $E$ is a finite locally free $\mathcal O_X$-module and $\theta:E\to E\otimes_{\mathcal O_X}\mathcal D$ is an $\mathcal O_X$-linear map satisfying \[ (\operatorname{id}_E\otimes\wedge)\circ(\theta\otimes\operatorname{id}_{\mathcal D})\circ\theta=0 \quad\text{in}\quad \operatorname{Hom}_{\mathcal O_X}\!\left(E,E\otimes_{\mathcal O_X}\bigwedge^2\mathcal D\right). \] The actual space, base field, differential sheaf, twists, and admissible Higgs bundles must be supplied by the intended correspondence; none is fixed by this record.

Let $\mathcal H$ denote its category of Higgs bundles and let $\mathcal R_{\mathrm{gen}}$ denote the category intended by "genuine representations." If the correspondence assigns Higgs bundles by a functor \[ \mathsf S:\mathcal R_{\mathrm{gen}}\longrightarrow\mathcal H, \] the requested problem is to give an intrinsic necessary and sufficient condition $\mathcal P$ on objects of $\mathcal H$ such that \[ \forall (E,\theta)\in\operatorname{Ob}(\mathcal H),\qquad \mathcal P(E,\theta) \iff \exists V\in\operatorname{Ob}(\mathcal R_{\mathrm{gen}})\text{ with }(E,\theta)\cong\mathsf S(V). \] Thus the target is a characterization of the essential image of genuine representations. This conditional formulation specifies the requested target, not a proposed criterion or solution. The source excerpt refers to "this condition," but neither that condition nor its hypotheses is included in the supplied record.

Short English statement

Determine exactly which Higgs bundles arise from genuine representations in the intended $p$-adic Simpson correspondence. The record leaves the geometric setting, the correspondence, and "genuine" unspecified, so the essential-image question requires further source clarification. Its partially solved label and claims about known cases are supplied dated curation, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005000$, title "Higgs Bundles Arising from p-Adic Representations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2022. The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report; these are supplied verification claims, not verification performed for this draft.
• Supplied report citation: Non-Archimedean Geometry and Applications (2023), Oberwolfach Reports, pp. 231--302, DOI 10.4171/owr/2022/5. https://doi.org/10.4171/owr/2022/5. The supplied excerpt describes the question as fundamental and open and refers to preceding results and an unspecified condition checkable in interesting cases. No precise passage page or individual speaker is supplied.
• Heuer, A geometric p-adic Simpson correspondence in rank one. Supplied literature link: https://arxiv.org/abs/2207.13657.
• Heuer--Mann, The p-adic Corlette--Simpson correspondence for abeloids. Supplied literature link: https://doi.org/10.1007/s00208-022-02371-2.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved and states that the essential image of genuine representations is characterized in rank one and important geometric cases, while the general question remains open. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.