Session S21 - Galois representations and automorphic forms

No date set.

On the mod $p$ cohomology of unipotent groups

Daniel Kongsgaard

University of California, San Diego, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\mathcal{G}$ be a split and connected reductive $\mathbb{Z}_p$-group and let $\mathcal{N}$ be the unipotent radical of a Borel subgroup. In this poster we study the cohomology with trivial $\mathbb{F}_p$-coefficients of the nilpotent pro-$p$-group $N = \mathcal{N}(\mathbb{Z}_p)$ and the Lie algebra $\mathfrak{n} = \operatorname{Lie}(\mathcal{N}_{\mathbb{F}_p})$. We proceed by arguing that $N$ is a $p$-valued group using ideas of Schneider (unpublished) and Zábrádi, which by a result of Sørensen gives us a spectral sequence $E_1^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_p) \Longrightarrow H^{s+t}(N,\mathbb{F}_p)$, where $\mathfrak{g} = \mathbb{F}_p \otimes_{\mathbb{F}_p[\pi]} \operatorname{gr} N$ is the graded Lie $\mathbb{F}_p$-algebra attached to $N$ as in Lazards work. We then argue that $\mathfrak{g} \cong \mathfrak{n}$ by looking at the Chevalley constants, and, using results of Polo and Tilouine and ideas from Große-Klönne, we show that the dimensions of the $\mathbb{F}_p$-cohomology of $\mathfrak{n}$ and $N$ agree, which allows us to conclude that the spectral sequence collapses on the first page.

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