Higgs Bundles Arising from p-Adic Galois Representations

Definitions and mathematical statement

The supplied question refers to an equivalence $S$ in Theorem 4 and an embedding in Example 2, neither of which is included in the record. The formulation below is therefore conditional on those missing definitions and hypotheses.

In the intended geometric setting, a Higgs bundle consists of a finite locally free $\mathcal O_X$-module $E$ and a Higgs field \[ \theta\in H^0\!\left(X,\operatorname{End}_{\mathcal O_X}(E)\otimes_{\mathcal O_X}\Omega_X^1(-1)\right), \qquad \theta\wedge\theta=0. \] The supplied excerpt describes this condition by requiring the field to induce a morphism of $\mathcal O_X$-algebras \[ \operatorname{Sym}_{\mathcal O_X}\!\left((\Omega_X^1)^\vee(1)\right) \longrightarrow \operatorname{End}_{\mathcal O_X}(E). \] Here $(1)$ and $(-1)$ denote Tate twists. The base, the hypotheses on $X$, and the precise differential sheaf must be recovered from the omitted theorem.

Let $\mathcal R$ denote exactly the representation category in Example 2, let $\iota:\mathcal R\hookrightarrow\mathcal C$ denote its designated embedding, and let $\mathcal H$ denote the Higgs-bundle category in Theorem 4. Write $T_S:\mathcal C\xrightarrow{\sim}\mathcal H$ for transport toward Higgs bundles under the stated equivalence $S$, using its inverse if necessary. These symbols are placeholders for the omitted source definitions.

Requested characterization. Find necessary and sufficient conditions, expressed in terms of a Higgs bundle $(E,\theta)\in\mathcal H$, for \[ (E,\theta)\in\operatorname{EssIm}(T_S\circ\iota), \] that is, for \[ \exists\,\rho\in\operatorname{Ob}(\mathcal R) \quad\text{such that}\quad (E,\theta)\cong T_S(\iota(\rho)). \] The task is to characterize this essential image in the general setting of the omitted theorem, extending the cases of line bundles and abeloid varieties. The supplied question describes the general problem as very difficult and reports that the answers in those cases are already subtle and are expressed using moduli spaces of $v$-line bundles. These are supplied source qualifications, not independently verified conclusions. The later assumption that $X$ is a smooth projective curve belongs to the following discussion of moduli spaces and cannot be imposed on this question from the supplied text alone.

Short English statement

Characterize exactly which Higgs bundles arise from representations through the embedding and the $p$-adic Simpson equivalence cited in the source. The source reports known descriptions for line bundles and abeloid varieties using moduli spaces of $v$-line bundles. A fully specified statement requires the omitted Theorem 4 and Example 2, including their geometric hypotheses and representation category.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006005$, source problem code catalogue entry, titled "Higgs Bundles Arising from p-Adic Galois Representations." The record supplies no proposer attribution.
• Supplied report citation: Arithmetic Geometry (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/33, pp. 1855--1912. https://doi.org/10.4171/owr/2024/33. The excerpt identifies the target as Question 5 and refers to Theorem 4, Example 2, and bibliography entries [6] and [8], whose complete contents are not supplied.
• Supplied literature reference labeled "Heuer, rank-one image": https://arxiv.org/abs/2207.13657.
• Supplied literature reference labeled "Heuer--Mann--Werner, abeloids": https://arxiv.org/abs/2107.09403.
• Supplied literature reference labeled "Heuer, p-adic Simpson correspondence": https://link.springer.com/article/10.1007/s00222-025-01321-4.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general characterization open and reports the essential image understood in rank one and for abeloids. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. No linked source has been visited or independently checked for this draft.