UC San Diego

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}^{s,t} = H^{s,t}(g, F_p) ⇒ H^{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}^{s,t} = H^{s,t}(g,F_p) ⇒ H^{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^{s,t}(g,F_p). We note that the spectral sequence E_{1}^{s,t} = H^{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.