On the Mod p Cohomology of Pro-p Iwahori Subgroups.

다국어 초록 (Multilingual Abstract)

Let G be a split and connected reductive Zp-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 Fp-coefficients of the unipotent pro-p group N = N(Zp) and the Lie algebra n = Lie(NFp). We proceed by arguing that N is a p-valued group using ideas of Schneider and Zabradi, which by a result of Sorensen gives us a spectral sequence E1s,t = Hs,t(g, Fp) ⇒ Hs+t(N, Fp), where g = Fp ⊗Fp[π] gr N is the graded Fp-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 Grose-Klonne, we show that the dimensions of the Fp-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 SLn and GLn over Qp for n = 2, 3, 4 and over a quadratic extension F/Qp for n = 2. Here we again use the spectral sequence E1s,t = Hs,t(g,Fp) ⇒ Hs+t(I,Fp) due to Sorensen, but in this chapter we do explicit calculations with an ordered basis of I, which gives us a basis of g = (Fp ⊗Fp[π] I) that we use to calculate Hs,t(g,Fp). We note that the spectral sequence E1s,t = Hs,t(g,Fp) 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 Qp and point out some future research directions.