$K(\pi,1)$ Theorems without Cyclotomic Class-Number Conditions

Definitions and mathematical statement

Let $k$ be a number field, let $p$ be a rational prime, and let $\mathcal O_k$ be the ring of integers. Write $S_p$ for the set of nonzero prime ideals of $\mathcal O_k$ above $p$. Let $S$ be a finite set of nonzero prime ideals with $S\cap S_p=\varnothing$, and let $T$ be a set of nonzero prime ideals of Dirichlet density one; explicitly, \[ \lim_{s\downarrow1}\frac{\sum_{\mathfrak q\in T}(\mathrm N\mathfrak q)^{-s}}{\log(1/(s-1))}=1. \] The restored question in the supplied record is whether, for every such $k,p,S,T$, there exists a finite subset $T_1\subseteq T$ such that \[ X=\operatorname{Spec}(\mathcal O_k)\setminus(S\cup T_1) \quad\text{is an $\acute{e}tale$ $K(\pi,1)$ for $p$}. \] Here the removed prime ideals are viewed as closed points. In the usual pro-$p$ interpretation, if $G=\pi_1^{\acute{e}t}(X,\bar x)(p)$ is the maximal pro-$p$ quotient of the $\acute{e}tale$ fundamental group, the required property is that the natural comparison maps \[ H^i_{\mathrm{cont}}(G,{\mathbb{F}}_p)\longrightarrow H^i_{\acute{e}t}(X,{\mathbb{F}}_p) \] are isomorphisms in every degree $i\geq0$. The precise $K(\pi,1)$-for-$p$ and real-place conventions of the cited theorem should be confirmed before treating this interpretation as definitive.

Crucially, the question imposes neither $\zeta_p\notin k$, where $\zeta_p$ is a primitive $p$-th root of unity, nor $p\nmid|\operatorname{Cl}(k)|$, where $\operatorname{Cl}(k)$ is the ideal class group. Both auxiliary restrictions are to be removed. The original extraction calls this removal of condition $(*)$ from the theorem; the complete theorem and the text of $(*)$ are not reproduced.

Short English statement

Can finitely many primes from any Dirichlet-density-one set be added to $S$, making the resulting arithmetic curve an $\acute{e}tale$ $K(\pi,1)$ for $p$, without either the cyclotomic or class-number restriction?

Sources

• Local UnsolvedMath registry, permanent ID $30000779$, source code catalogue entry, "$K(\pi,1)$ Theorems without Cyclotomic Class-Number Conditions." No individual proposer is supplied.
• Algebraische Zahlentheorie (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/30, pp. 1741--1792: https://doi.org/10.4171/owr/2007/30. This is the sole supplied external reference.
• The literature triage checked 2026-08-21 labels the conjecture open and says that the cited report does not establish a later proof or counterexample. This does not independently establish its current status.