A Study of Fine Selmer Groups Over Function Fields via Greenberg Neighbourhoods

Greenberg examined the local behavior of Iwasawa invariants as functions on the the set of all $\mathbb{Z}_p$-extensions of a number field $F$. Kleine later extended these ideas to explore the variation of Iwasawa invariants in the context of Selmer groups of elliptic curves across different $\mathbb{Z}_p$-extensions of $F$. Let $K$ be a global function field of characteristic $p$. In this article, we investigate the relation between Iwasawa invariants of fine Selmer groups of an elliptic curve over $K$ across various $\mathbb{Z}_p$-extensions of $K$, utilizing Kleine's techniques. Furthermore, we connect this analysis to an analogue of Conjecture A by Coates and Sujatha for different $\mathbb{Z}_p$-extensions of the function field $K$.