The first chapter of this thesis serves as an introduction. The remaining two chapters each focus on different topics, all related to the theme of Iwasawa theory. These chapters are adapted from two separate papers, yet they form a coherent whole in the following sense.The second chapter studies the relation between Iwasawa $\lambda$-invariants and Massey products. For imaginary quadratic fields, Gold’s criterion provides a connection between the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension and cup products. Sands used this criterion to construct infinitely many imaginary quadratic fields with $\lambda > 1$. Massey products can be viewed as a higher-order generalization of cup products. In Chapter 2, we generalize Gold’s criterion by establishing a connection between the Iwasawa $\lambda$-invariant and Massey products. This generalization also recovers results of McCallum and Sharifi.In the third chapter, we study quotients of pseudo-null module. Let $\Lambda$ be an Iwasawa algebra and $M$ be a pseudo-null $\Lambda$-module. Take a regular element $T$ in $\Lambda$. We prove a criterion for determining when the quotient module $M/TM$ is again pseudo-null over $\Lambda/T$. This result provides a useful tool for reducing questions over $\mathbb{Z}_p^d$-extensions to those over $\mathbb{Z}_p^{d-1}$-extensions.Though the topics of two chapters seems unrelated, they all are motivated by studying the Gold's criterion. There are two main directions in which Gold’s criterion can be generalized. The first is by replacing cup products with Massey products—this is the focus of Chapter 2. The second is by considering $S$-ramified $\mathbb{Z}_p$-extensions instead of the classical cyclotomic $\mathbb{Z}_p$-extensions. By the joint papers of Matt Stokes and the author, we find that the $S$-ramified $\mathbb{Z}_p$-extensions of CM field behaves similarly to the cyclotomic $\Z_p$-extension of totally real field. For cyclotomic $\Z_p$-extension of totally real fields, Greenberg conjectured that the Iwasawa module formed by the inverse limit of class group is a pseudo-null module. Hence, we need to consider the structure of pseudo-null module to better understand $S$-ramified $\mathbb{Z}_p$-extensions. We summarize the properties of $S$-ramified $\mathbb{Z}_p$-extensions before Chapter 3 to motivate the work done in Chapter 3. The summary is a reproduction of the joint research work by Matt Stokes and the author.
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