On the Mod p Cohomology of Pro-p Iwahori Subgroups of SL_n(\mathbb Q_p)

2022/11/13 by Daniel Kongsgaard, Kongsgaard, Daniel
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2211.06819

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 SL_n(\mathbb Q_p) (and GL_n(ℚ_p)) for n=2 and n=3. Here we use the spectral sequence E₁^s,t = H^s,t(\mathfrakg,\mathbbF_p) \Longrightarrow H^s+t(I,\mathbbF_p) due to Sorensen, and we do explicit calculations with an ordered basis of I, which gives us a basis of \mathfrakg = \mathbbF_p ⊗_\mathbbF_p[π] gr I that we use to calculate H^s,t(\mathfrakg,\mathbbF_p). We note that the multiplicative spectral sequence E₁^s,t = H^s,t(\mathfrakg,\mathbbF_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 ℚ_p and point out some future research directions.

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