Special Zeta Values via Weil–Étale Cohomology

Definitions and mathematical statement

Let $X$ be a regular scheme that is projective and flat over $\operatorname{Spec}{\mathbb{Z}}$. Write $|X|$ for its set of closed points and $\kappa(x)$ for the residue field at $x$. Its zeta function is initially defined, in a right half-plane of absolute convergence, by \[ \zeta(X,s)=\prod_{x\in|X|}\left(1-\#\kappa(x)^{-s}\right)^{-1}. \] Whenever a meromorphic continuation exists near a specified point $s_0$, its leading term is defined by \[ \zeta^{*}(X,s_0)=\lim_{s\to s_0}(s-s_0)^{-m}\zeta(X,s), \qquad m=\operatorname{ord}_{s=s_0}\zeta(X,s). \] This definition does not assert the required continuation or identify which points are intended by "special values."

Supplied mathematical target. For every such $X$, formulate a conjectural expression for its special zeta values in terms of Weil--étale motivic, singular, and de Rham cohomology, compatible with Serre's functional equation.

Unresolved qualifications. The supplied record specifies neither the special arguments nor whether values, orders of vanishing, or leading terms are intended. It gives no cohomological coefficients, twists, comparison maps, normalizations, explicit formula, or precise version of the functional equation. Consequently, the target above is a research request, not a fully specified conjectural identity. The supplied dated literature assessment also describes the extraction as a truncated table-of-contents fragment with no identifiable mathematical statement, whereas the later statement-verification metadata describes the displayed target as self-contained and verified. These conflicting qualifications remain unresolved.

Short English statement

The supplied target asks for a cohomological description of special zeta values for every regular scheme projective and flat over $\operatorname{Spec}{\mathbb{Z}}$, compatible with Serre's functional equation. The special values and the proposed formula are unspecified, so a precise conjectural identity cannot be recovered from this record. Its open status and verification labels are supplied metadata requiring independent review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004162$, source problem code catalogue entry, title "Special Zeta Values via Weil--étale Cohomology." The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Algebraic K-theory (2020), Oberwolfach Reports, pp. 1737--1790, DOI 10.4171/owr/2019/29. Source URL and supplied literature-source link, labelled "OWR primary report": https://doi.org/10.4171/owr/2019/29. This is report-level provenance; no specific page, speaker, or conjecture number is supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open but states that the extracted record is a truncated table-of-contents fragment ending with the word "conjecture" and contains no identifiable mathematical statement to triage. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note claims manual semantic, typographic, and LaTeX review. This supplied claim does not resolve the missing mathematical specifications or the conflicting literature assessment.