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.

The classical Tannaka reconstruction theorem \cite{tannaka1939dualitatssatz} allows one to recover a compact group $G$, up to isomorphism, from the monoidal category of finite dimensional representations of $G$ over $\mathbb{C}$, $\text{Rep}_{\mathbb{C}}(G)$, as the tensor preserving automorphisms of the forgetful functor $\text{Rep}_{\mathbb{C}}(G) \longrightarrow \text{Vec}_{\mathbb{C}}$. Now let $G$ be a profinite group, $K$ a finite extension of $\mathbb{Q}_p$ and $\text{Ban}_G(K)$ the category of $K$-Banach space representations (of $G$). $\text{Ban}_G(K)$ can be equipped with a tensor product bi-functor $\hat\otimes_K$ and forgetful functor $\omega : \text{Ban}_G(K) \longrightarrow \text{Ban}(K)$. Using an anti-equivalence of categories (\cite{schneider2002banach}[Thm 2.3]) between $\text{Ban}_G(K)$ and the category of Iwasawa $G$-modules, we prove that a profinite group $G$ can be recovered from $\text{Ban}_G(K)$ as the group of tensor preserving automorphisms of $\omega$, in particular $G \cong \text{Aut}^\otimes(\omega)$.

Part I: Starting with a rational odd prime p > 2 and a cyclic extension F=Q whose Galois group has order coprime to p, Kurihara's conjecture [24] gives an explicit description of all higher Fitting ideals of large p-power quotients of the classical Iwasawa module X, over correspondingly large p-power quotients of the classical Iwasawa algebra Λ = Zp[[T]]. The generators of these higher Fitting ideals are, essentially, special values of equivariant L-functions. A complete proof of Kurihara's conjecture was recently given by Popescu-Stone in full generality [41]. This dissertation conjectures a generalization of Kurihara's conjecture to so-called "semi-nice" extensions F=k where F is CM and k is totally real. In particular, this generalized conjecture specializes to Kurihara's original setting with k = Q and F a CM field given by the fixed field of F by the kernel of an odd Dirichlet character χ of order coprime to p, such that χ is not the Teichmüller character ω. Under certain hypotheses a proof of the generalized conjecture is given, away from the Teichmüller component. The methods of proof employed for the generalized conjecture are similar to those used by Popescu and Stone in their proof of Kurihara's conjecture.

Part II: From a potentially semistable representation of the absolute Galois group of a p-adic eld L=Qp, Breuil and Schneider [4] construct a locally algebraic representation BS(). The Breuil-Schneider conjecture asserts the equivalence between BS(ρ) carrying a GLn-invariant norm, and the existence of a certain (φ;N)-module with admissible ltration. In the indecomposable case, an unconditional proof of BS(ρ) was given by Sorensen [40]. Assuming Sorensen's result for subrepresentations and quotient representations of ρ, we prove BS(ρ) is true under some additional hypotheses on ρ.