Degeneration and Purity of Local-Field Adic Systems

Definitions and mathematical statement

Let $p$ and $\ell$ be distinct primes, let $K$ be a $p$-adic local field, and let $X$ be a smooth variety over $K$. Here a $\mathbb Q_\ell$-adic local system means a lisse sheaf of finite-dimensional $\mathbb Q_\ell$-vector spaces on the étale site of $X$. The supplied formulation concerns local systems $\mathcal L$ that are pure, but the supplied excerpt does not define purity over $K$, specify a weight, or give the additional context implicit in the original phrase "such local systems."

The two questions attributed to Moritz Kerz, in joint work with Hélène Esnault, are the following, intended for every local system in that source-specific class:

1. Does $\mathcal L$ degenerate to a limiting mixed structure at each ramification point?
2. Is the cohomology of $\mathcal L$ pure?

These questions retain unresolved qualifications. For the first, the record specifies neither a compactification, model, or degeneration of $X$, nor the meaning of a ramification point or the category and properties of the proposed limiting mixed structure. Thus the points being quantified over and the required limiting object are not determined. For the second, the record does not specify the cohomology theory, the support condition, the degrees, the arithmetic action used to measure purity, or the expected weights. In particular, it does not distinguish cohomology over $K$ from geometric cohomology over an algebraic closure of $K$. No properness hypothesis is supplied. Such hypotheses and choices cannot be inserted silently. Consequently these are source-faithful problem statements, but the supplied record does not determine fully specified mathematical assertions.

Short English statement

For pure $\mathbb Q_\ell$-adic local systems on a smooth variety over a $p$-adic local field, with $\ell\ne p$, the source asks whether ramification admits limiting mixed structures and whether cohomology is pure. The meanings of purity, the limiting structure, and the intended cohomology require further source context.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30006068$, titled Degeneration and Purity of Local-Field Adic Systems; exact source problem code catalogue entry. The supplied registry labels the problem open and the statement corrected_verified; these are supplied curation labels, not independent verification here.
• Supplied report citation: Anabelian Geometry and Representations of Fundamental Groups (2025), Oberwolfach Reports, pp. 2621--2670, DOI 10.4171/owr/2024/45. Supplied source URL: https://doi.org/10.4171/owr/2024/45. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• The original excerpt identifies the contribution Pure $\ell$-adic local systems over local fields, Moritz Kerz, joint work with Hélène Esnault, p. 2627 of the report. The excerpt calls the two questions basic open problems about "such local systems," without supplying the preceding definition of that class.
• Supplied literature source: Oberwolfach Report 45/2024, https://doi.org/10.4171/owr/2024/45. The literature assessment checked on 2026-08-22 states that the report explicitly identifies both limiting mixed structures at ramification and purity of cohomology as basic open problems. This assessment is supplied dated curation, not independently verified; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note says that the official report was checked, that the question was isolated from surrounding exposition, and that notation was repaired for publication. This records a supplied verification claim; no source visit or mathematical verification is asserted here. The proposed year and report identifier indicate 2024, whereas the supplied bibliographic citation gives 2025; that distinction is preserved without resolution.