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This paper can be seen as an update to part of the author's dissertation. Westudy 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 thespectral 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 explicitcalculations 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 notethat the multiplicative spectral sequence $E_{1}^{s,t} =H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$ collapses on the first page by noticingthat all maps on each page are necessarily trivial, and this allows us todescribe the above group cohomology groups and all cup products. Finally wenote some connections to cohomology of central division algebras over$\mathbb{Q}_{p}$ and point out some future research directions.

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