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

TL;DR

This paper studies the variation of Iwasawa invariants of fine Selmer groups of elliptic curves over function fields, extending ideas from number fields and exploring analogues of Conjecture A in this context.

Contribution

It extends the analysis of Iwasawa invariants of Selmer groups to global function fields and connects this to conjectural frameworks like Conjecture A.

Findings

Established relations between Iwasawa invariants across various $Z_p$-extensions.

Connected Selmer group invariants to an analogue of Conjecture A.

Applied Kleine's techniques to the function field setting.

Abstract

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 $K$.