This dissertation consists of two main topics. In the first one, we talk about an equivariant reformulation of the plus part of the Main Conjecture of Iwasawa theory in terms of the abstract $p$-adic Tate module constructed by Greither and Popescu(\cite{GP}) and the cyclotomic units of $\Q.$ Then we discuss how the Selmer module, introduced by Burns, Kurihara and Sano(\cite{BKS}) could be an unconditional replacement for the $p$-adic Tate module. In the second part, we talk about how the Euler factors in equivariant $L$-functions of Drinfeld modules relate to Fitting ideals of certain modules. This discussion takes us through the theory of $t$-motives and helps in defining the first `` etale" and ``crystalline" cohomology groups of a Drinfeld module defined over a finite field.

For an abelian extension K/F of algebraic number fields, the classical Coates-Sinnott Conjecture and its refinements predict the subtle and deep relationship between the special values of L-functions and the structure of the étale cohomology groups attached to this extension. This is a beautiful conjecture relating the arithmetic side and the Galois side of number fields. In this thesis, we aim to delve deeper along this direction to propose “higher Coates-Sinnott” Conjectures which reveal more information about these two important arithmetic objects. Our main tool is the theory of Fitting ideals and the Equivariant Iwasawa Theory developed by Greither-Popescu. In this thesis, after presenting the relevant history and preliminaries in Chapters 1-3, we first propose our “higher Coates-Sinnott Conjecture” with 2 different formulations in Chapter 4, and show that one of the formulations implies the other. Both formulations involve the language of Fitting ideals. Next, in Chapter 5 we develop the key technical tools regarding Fitting ideals, which are crucial for our theorems. Some of the results already in the literature, and based on these result we prove a technical tool about higher Fitting ideals. In Chapter 6, we are able to show that our (stronger) conjecture is true away from the component associated to the Teichmüller character. Concerning the mysterious Teichmüller component, we are able to prove a weaker “character-by-character” result rather than the desired fully equivariant one. We also discuss the related Iwasawa modules using the information we have obtained. Lastly, in the appendix, we describe how to make use of the work of Gambheera-Popescu to remove the assumption that Iwasawa’s µ = 0 Conjecture is true, which was necessary in the work of Greither-Popescu.

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.