Admissibility and Finite Length on the Drinfeld Tower

Definitions and mathematical statement

Let $p$ be a prime, let $F$ be a $p$-adic field, understood in the usual sense of a finite extension of ${\mathbb{Q}}_p$, and put $G=\operatorname{GL}_2(F)$. For each integer $n\geq 0$, let $\Sigma_n$ denote the level-$n$ space of the Drinfeld tower referred to in the supplied record. Write $\mathcal O(\Sigma_n)$ for its space of global analytic functions and retain the source notation \[ V_n=\mathcal O(\Sigma_n)^*. \] The intended action on a dual of a function space is the contragredient action: if the action on functions is $(g\cdot f)(x)=f(g^{-1}x)$, then $(g\cdot\lambda)(f)=\lambda(g^{-1}\cdot f)$.

Question. For every $p$-adic field $F$ and every integer $n\geq 0$, is $V_n$ an admissible locally analytic representation of $\operatorname{GL}_2(F)$, and does $V_n$ have finite length in the intended category of locally analytic representations? Finite length means the existence of a finite composition series \[ 0=V_{n,0}\subset V_{n,1}\subset\cdots\subset V_{n,r}=V_n \] in that category, with each successive quotient nonzero and simple.

Unresolved conventions. The supplied record does not specify the geometric base field or component convention for $\Sigma_n$, the coefficient field of the representations, the topology on $\mathcal O(\Sigma_n)$, or whether ${}^*$ denotes a continuous dual with the strong topology or another dual. It also does not specify whether local analyticity is over $F$ or over ${\mathbb{Q}}_p$, the corresponding admissibility convention, or the category and exactness convention used for finite length. These choices must be recovered from the cited source before the question determines a fully specified representation-theoretic statement; no particular choice is asserted here.

Short English statement

For every $p$-adic field $F$ and every level $n\geq 0$ of the Drinfeld tower, ask whether the dual of its analytic functions is an admissible locally analytic $\operatorname{GL}_2(F)$-representation of finite length. The supplied record leaves several geometric, coefficient, topology, and category conventions unresolved.

Sources

• Supplied local UnsolvedMath registry, permanent ID $30004089$, title Admissibility and Finite Length on the Drinfeld Tower, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record gives proposed year $2019$ and names no individual proposer.
• 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 supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Dospinescu--Le Bras. https://annals.math.princeton.edu/2017/186-2/p01.
• Supplied literature reference: Patel--Schmidt--Strauch. https://arxiv.org/abs/1201.3646.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2019/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports admissibility for all levels when $F={\mathbb{Q}}_p$, admissibility at the first level for arbitrary $p$-adic fields, and finite length at the first level; it describes the full all-field, all-level conjecture as remaining open. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked directly and the clean statement restored definitions while preserving the target. This is a supplied verification claim; no source visit or independent verification is asserted in this draft.