Iwasawa theory of fine Selmer groups over global fields

TL;DR

This paper investigates the structure and properties of the $p^ abla$-fine Selmer groups of elliptic curves over various global fields, linking deep Iwasawa theory conjectures with new insights into Selmer group behavior.

Contribution

It extends the study of fine Selmer groups to $p$-adic Lie extensions and function fields, connecting these to Jannsen's conjecture and deepening understanding of Iwasawa theory.

Findings

Analyzed the structure of fine Selmer groups over $p$-adic Lie extensions.

Defined fine Selmer groups over function fields of characteristic $p$ and $ eq p$.

Related properties of these groups to Jannsen's conjecture.

Abstract

The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field $F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of $E$ over a $p$-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa's classical $\mu$-invariant vanishing conjecture. In this article, we study the properties of the $p^\infty$-fine Selmer group of an elliptic curve over certain $p$-adic Lie extensions of a number field. We also define and discuss $p^\infty$-fine Selmer group of an elliptic curve over function fields of characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our study with a conjecture of Jannsen.