Iwasawa theory of fine Selmer groups over global fields

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 $\ell \ne p.$ We relate our study with a conjecture of Jannsen.

Accepted for publication in Mathematische Zeitschrift

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