$L^p$ Cohomology at the Semisimple Rank Threshold

Definitions and mathematical statement

Let $F$ be a local field, Archimedean or non-Archimedean, and let $\mathbf G$ be a semisimple algebraic group defined over $F$. Put $G=\mathbf G(F)$, with its local-field topology, and let \[ r=\operatorname{rk}_F(\mathbf G), \] the dimension of a maximal $F$-split torus of $\mathbf G$. No positive-rank assumption is stated, so $r=0$ is included.

For $1<p<\infty$, let $L^p(G)$ denote the space of Haar-measurable functions modulo equality almost everywhere whose absolute value has integrable $p$-th power, with respect to a left Haar measure on $G$. To make the notation explicit, use the conventional left regular action \[ (g\cdot f)(x)=f(g^{-1}x). \] Here the scalar field may be $\mathbb R$ or $\mathbb C$; the supplied source does not specify it. Interpret $H^k_{\mathrm{ct}}(G,V)$ as ordinary continuous group cohomology: the cohomology of the inhomogeneous complex $C^k_{\mathrm{ct}}(G,V)=C(G^k,V)$, with $C^0_{\mathrm{ct}}(G,V)=V$, using the differential \[ \begin{aligned} (dc)(g_1,\ldots,g_{k+1}) ={}&g_1\cdot c(g_2,\ldots,g_{k+1})\\ &+\sum_{i=1}^{k}(-1)^i c(g_1,\ldots,g_i g_{i+1},\ldots,g_{k+1})\\ &+(-1)^{k+1}c(g_1,\ldots,g_k). \end{aligned} \] Thus $H^k_{\mathrm{ct}}=\ker d/\operatorname{im}d$, without taking the closure of the coboundaries, and $H^0_{\mathrm{ct}}(G,V)=V^G$.

For every such $F$ and $\mathbf G$, the supplied problem asks both:

1. Is it true that \[ \forall p\in(1,\infty)\;\forall k\in\mathbb Z_{\geq 0},\qquad k<r\ \Longrightarrow\ H^k_{\mathrm{ct}}(G,L^p(G))=\{0\}? \]
2. Is it true that \[ \exists p\in(1,\infty)\qquad H^r_{\mathrm{ct}}(G,L^p(G))\neq\{0\}? \] The exponent in the second question may depend on $F$ and $\mathbf G$.

When $r=0$, the first question has no eligible cohomological degrees, and the second concerns invariant vectors in degree zero.

The coefficient scalar field, regular-action convention, and distinction between ordinary and reduced cohomology are not explicitly defined in the supplied passage; the conventions above make this draft readable and require source confirmation. The source also does not spell out whether connectedness is part of its convention for a semisimple algebraic group. No additional assumptions such as simplicity, isotropy, or simple connectedness have been inserted.

Short English statement

For the group of points of a semisimple algebraic group over any local field, ask whether continuous cohomology with regular $L^p$ coefficients vanishes in every nonnegative degree below the $F$-rank for every $1<p<\infty$, and whether it is nonzero in the rank degree for at least one such $p$. The supplied record labels the problem partially solved; coefficient and cohomology conventions require confirmation.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30006193$, title $L^p$ Cohomology at the Semisimple Rank Threshold, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied original excerpt begins with the marker 253]. No individual proposer is identified; the supplied proposed year is 2025.
• Source report: Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6. https://doi.org/10.4171/owr/2025/6. The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. This is supplied verification metadata, not an independent verification performed for this draft.
• Supplied literature reference: Bourdon and Remy, nonvanishing $L^p$-cohomology. https://doi.org/10.5802/jep.232.
• Supplied literature reference: Lopez Neumann, vanishing in degree two. https://arxiv.org/abs/2302.09307.
• Supplied literature reference: OWR 2025/6 geometric group theory report. https://oa.tib.eu/renate/server/api/core/bitstreams/6f61d540-8e50-41f3-97a8-6e8214672460/content.
• Supplied literature assessment, checked 2026-08-22: status partially solved. The supplied assessment states that, for real and non-Archimedean semisimple groups, substantial vanishing below the rank and nonvanishing in the rank degree are proved in important ranges, including large $p$ and many group types, while the all-local-field, all-$p$ formulation remains open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.