UC San Diego

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.