Characteristic Classes of Hodge–Tate Local Systems

Definitions and mathematical statement

Let $p$ be a prime, let $K/{\mathbb{Q}}_p$ be a finite extension, let $X$ be a smooth algebraic variety over $K$, and let $L$ be a Hodge--Tate local system on $X$, in the sense intended by the supplied source. The record does not specify the coefficient field, rank, or precise category of local systems.

The notation $D_{\mathrm{HT}}(L)$ denotes the Hodge--Tate object associated with $L$ in the source, and $\operatorname{gr}_mD_{\mathrm{HT}}(L)$ denotes its piece indexed by $m\in{\mathbb{Z}}$. The construction, grading convention, and base of this object are not supplied. For an object $E$ for which the Chern character is defined, $\operatorname{ch}_{i-1}(E)$ denotes its component of cohomological degree $2(i-1)$ in de Rham cohomology, as specified in the original statement.

The original transcription appears to place the characteristic class in \[ \ell_i(L)\in H^{2i-1}_{\mathrm{\acute{e}t}}(X,{\mathbb{Q}}_p). \] Its construction and normalization are not supplied. The map $\alpha_X$ is named but not defined; its target and any coefficient extensions, twists, or comparison conventions are unspecified.

Source question, with unresolved qualifications. With the source's intended meanings of these constructions, is the identity \[ \alpha_X\left(\ell_i(L)\right) = (i-1)!\sum_{m\in{\mathbb{Z}}}m\, \operatorname{ch}_{i-1}\left(\operatorname{gr}_mD_{\mathrm{HT}}(L)\right) \] valid for every such $K,X,L$ and every intended index $i$? Under the ordinary factorial convention the displayed expression requires $i\in{\mathbb{Z}}_{\geq1}$, but the source record does not explicitly state the index range. It also does not state why the sum is finite or otherwise well defined. A fully specified equality requires the missing definitions and a common target for both sides; these cannot be recovered from the supplied record alone.

Short English statement

The question asks whether a characteristic class of a Hodge--Tate local system, after applying the source's map $\alpha_X$, equals the weighted sum of Chern-character components of its Hodge--Tate graded pieces. The map, characteristic-class construction, grading conventions, and common cohomological target remain unspecified. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30006078$, title Characteristic Classes of Hodge--Tate Local Systems, source problem code catalogue entry. The supplied record is the evidence used for this draft.
• Supplied primary-source citation: Anabelian Geometry and Representations of Fundamental Groups (2025), Oberwolfach Reports, pp. 2621--2670, DOI 10.4171/owr/2024/45. https://doi.org/10.4171/owr/2024/45. The record identifies this as Oberwolfach Report 45/2024 and gives proposed year 2024; these supplied year designations are retained.
• Supplied literature-source label: "Ge, Characteristic-class question." https://math.mit.edu/~apetrov/papers/pdf/ow_2024.pdf. The record does not give a full bibliographic citation or explicitly identify a proposer.
• Literature assessment checked on 2026-08-22: the supplied curation labels the question open and states that the report poses the displayed identity as unresolved and that the cited characteristic-class literature does not establish the general $p$-adic comparison. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked and its notation repaired. These are supplied verification claims; no source was independently visited for this draft.