Some Fitting ideal computations in Iwasawa Theory over $\Q$ and in the Theory of Drinfeld Modules

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.