Title: Iwasawa Invariants, Massey Products and the Quotient of Pseudo-null Module

Abstract: Gold's criterion and the results of McCallum and Sharifi establish a connection between the Iwasawa invariant $\lambda \geq 2$ and the vanishing of cup products in cyclotomic $\mathbb{Z}_p$-extensions. We propose two directions for generalizing these results. First, by replacing cup products with Massey products, we establish a relationship between $\lambda \geq n$ and the vanishing of $n$-fold Massey products. Second, by replacing the cyclotomic $\mathbb{Z}_p$-extension with more general $\mathbb{Z}_p$-extensions, we construct $S$-ramified $\mathbb{Z}_p$-extensions and demonstrate their connection to Greenberg's pseudo-null conjecture.