Ph.D. Thesis, UC San Diego, 2022.

Paper abstract bibtex

Paper abstract bibtex

Let G be a split and connected reductive Z_p-group and let N be the unipotent radical of a Borel subgroup. In the first chapter of this dissertation we study the cohomology with trivial F_p-coefficients of the unipotent pro-p group N = N(Z_p) and the Lie algebra n = Lie(N_\F_p\). We proceed by arguing that N is a p-valued group using ideas of Schneider and Zábrádi, which by a result of Sørensen gives us a spectral sequence E_\1\\textasciicircum\s,t\ = H\textasciicircum\s,t\(g, F_p) ⇒ H\textasciicircum\s+t\(N, F_p), where g = (F_p ⊗_\F_p[π]\ gr N) is the graded F_p-Lie algebra attached to N as in Lazards work. We then argue that g ≈ 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 F_p-cohomology of n and N agree, which allows us to conclude that the spectral sequence collapses on the first page.In the second chapter we study the mod p cohomology of the pro-p Iwahori subgroups I of SL_n and GL_n over Q_p for n = 2, 3, 4 and over a quadratic extension F/Q_p for n = 2. Here we again use the spectral sequence E_\1\\textasciicircum\s,t\ = H\textasciicircum\s,t\(g,F_p) ⇒ H\textasciicircum\s+t\(I,F_p) due to Sørensen, but in this chapter we do explicit calculations with an ordered basis of I, which gives us a basis of g = (F_p ⊗_\F_p[π]\ I) that we use to calculate H\textasciicircum\s,t\(g,F_p). We note that the spectral sequence E_\1\\textasciicircum\s,t\ = H\textasciicircum\s,t\(g,F_p) collapses on the first page by noticing that all maps on each page are necessarily trivial. Finally we note some connections to cohomology of quaternion algebras over Q_p and point out some future research directions.

@phdthesis{kongsgaard_mod_2022, title = {On the mod p cohomology of pro-p {Iwahori} subgroups}, url = {https://escholarship.org/uc/item/6rg894gp}, abstract = {Let G be a split and connected reductive Z\_p-group and let N be the unipotent radical of a Borel subgroup. In the first chapter of this dissertation we study the cohomology with trivial F\_p-coefficients of the unipotent pro-p group N = N(Z\_p) and the Lie algebra n = Lie(N\_\{F\_p\}). We proceed by arguing that N is a p-valued group using ideas of Schneider and Zábrádi, which by a result of Sørensen gives us a spectral sequence E\_\{1\}{\textasciicircum}\{s,t\} = H{\textasciicircum}\{s,t\}(g, F\_p) ⇒ H{\textasciicircum}\{s+t\}(N, F\_p), where g = (F\_p ⊗\_\{F\_p[π]\} gr N) is the graded F\_p-Lie algebra attached to N as in Lazards work. We then argue that g ≈ 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 F\_p-cohomology of n and N agree, which allows us to conclude that the spectral sequence collapses on the first page.In the second chapter we study the mod p cohomology of the pro-p Iwahori subgroups I of SL\_n and GL\_n over Q\_p for n = 2, 3, 4 and over a quadratic extension F/Q\_p for n = 2. Here we again use the spectral sequence E\_\{1\}{\textasciicircum}\{s,t\} = H{\textasciicircum}\{s,t\}(g,F\_p) ⇒ H{\textasciicircum}\{s+t\}(I,F\_p) due to Sørensen, but in this chapter we do explicit calculations with an ordered basis of I, which gives us a basis of g = (F\_p ⊗\_\{F\_p[π]\} I) that we use to calculate H{\textasciicircum}\{s,t\}(g,F\_p). We note that the spectral sequence E\_\{1\}{\textasciicircum}\{s,t\} = H{\textasciicircum}\{s,t\}(g,F\_p) collapses on the first page by noticing that all maps on each page are necessarily trivial. Finally we note some connections to cohomology of quaternion algebras over Q\_p and point out some future research directions.}, language = {en}, urldate = {2022-06-21}, school = {UC San Diego}, author = {Kongsgaard, Daniel}, year = {2022}, keywords = {algebraic geometry, algebraic topology, mathematics, mentions sympy}, }

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