Cohomology of the Drinfeld Tower

Definitions and mathematical statement

Let $p$ be a prime, let $K$ be the field described in the supplied record as a $p$-adic field, and let $\mathcal M_\infty$ denote the Drinfeld tower over $K$ referred to there. A tower is a system of spaces with transition maps; the notation $\mathcal M_\infty$ does not, by itself, specify whether cohomology is taken on an infinite-level space or constructed from the cohomology of finite levels. The record supplies neither the defining moduli problem nor the dimension, levels, transition maps, or geometric base change for this tower.

For a space $X$, a specified étale coefficient sheaf $\mathscr L$, and an integer $i\geq 0$, write $H^i_{\mathrm{\acute et}}(X,\mathscr L)$ for its étale cohomology, with the site and coefficient conventions fixed. For a specified coherent $\mathcal O_X$-module $\mathscr F$, write $H^i(X,\mathscr F)$ for its sheaf cohomology on a specified geometric site. These notations describe the kinds of objects requested; they do not select coefficient sheaves, a geometric category, or a construction at infinite level.

Source research question. Describe the $p$-adic étale cohomology and the coherent cohomology associated with $\mathcal M_\infty$, extending the understanding that the supplied record attributes to compactly supported $\ell$-adic étale cohomology. Schematically, the requested objects have the forms \[ H^i_{\mathrm{\acute et}}(\mathcal M_\infty,\mathscr L) \qquad\text{and}\qquad H^i(\mathcal M_\infty,\mathscr F), \] where the admissible degrees, coefficient objects, and meaning of cohomology of the tower remain to be specified. The record does not determine whether the coherent question concerns $\mathcal O$ alone or a larger class of coherent sheaves, or whether support conditions are intended for either requested theory.

Unresolved qualifications. This is a request for a cohomological description, not a fully specified conjectural formula. The supplied literature assessment additionally mentions unresolved admissibility and finite-length assertions, but supplies no acting group, representation category, coefficient field, topology, or actual assertions to formalize. Those phrases therefore remain unformulated qualifications rather than claims that every cohomology group has either property. A precise theorem statement requires these missing choices and hypotheses.

Short English statement

The problem asks for a description of the $p$-adic étale and coherent cohomology of the Drinfeld tower, beyond the compactly supported $\ell$-adic theory described as known in the source. The tower, coefficients, support conditions, and representation-theoretic assertions need further specification. The supplied open status is dated curation, and this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30004088$, exact source problem code catalogue entry, title Cohomology of the Drinfeld Tower, collection Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2019.
• Supplied report citation: Non-Archimedean Geometry and Applications (2020), Oberwolfach Reports, pp. 513--575, DOI 10.4171/owr/2019/8. https://doi.org/10.4171/owr/2019/8. The background attributes the problem to workshop open-problem discussions collected in these reports.
• Supplied literature source, labeled "Drinfeld-tower etale cohomology": https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/factorisation-de-la-cohomologie-etale-padique-de-la-tour-de-drinfeld/79733067FC7CB7574B408744E4387F91. The record supplies no author list or further bibliographic details for this link.
• Supplied literature source, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2019/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that important cases of Drinfeld-tower étale and coherent cohomology are known, while general descriptions, admissibility, and finite-length assertions remain unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and that the clean statement restored definitions while preserving the mathematical target. These are supplied verification claims; no source visit or independent verification is asserted here.