On the Mod $p$ Cohomology of Pro-$p$ Iwahori Subgroups of $\operatorname{SL}_{n}(\mathbb Q_{p})$

TL;DR

This paper computes the mod p cohomology of pro-p Iwahori subgroups of SL_n(Q_p) for n=2,3, using spectral sequences and explicit basis calculations, revealing structural properties and connections to division algebra cohomology.

Contribution

It provides explicit calculations of the mod p cohomology of certain p-adic subgroups using spectral sequences and basis methods, extending previous work and offering new insights.

Findings

Spectral sequence collapses at the first page.

Explicit basis allows detailed cohomology calculations.

Connections identified with cohomology of division algebras.

Abstract

This paper can be seen as an update to part of the author's dissertation. We study the mod $p$ cohomology of the pro-$p$ Iwahori subgroups $I$ of $\operatorname{SL}_{n}(\mathbb Q_{p})$ (and $\operatorname{GL}_{n}(\mathbb{Q}_{p})$) for $n=2$ and $n=3$. Here we use the spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p}) \Longrightarrow H^{s+t}(I,\mathbb{F}_{p})$ due to Sorensen, and we do explicit calculations with an ordered basis of $I$, which gives us a basis of $\mathfrak{g} = \mathbb{F}_{p} \otimes_{\mathbb{F}_{p}[\pi]} \operatorname{gr} I$ that we use to calculate $H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$. We note that the multiplicative spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$ collapses on the first page by noticing that all maps on each page are necessarily trivial, and this allows us to describe the above group cohomology groups and all cup products. Finally we note some connections to cohomology of central division algebras over $\mathbb{Q}_{p}$ and point out some future research directions.