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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.

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.