Fine Selmer groups and ideal class groups

TL;DR

This paper compares arithmetic invariants of ideal class groups and fine Selmer groups in $p$-adic Lie extensions, establishing relations between their ranks and Iwasawa invariants, and introduces new asymptotic formulas for their growth.

Contribution

It introduces new comparisons and relations between Iwasawa invariants of ideal class groups and fine Selmer groups, including novel asymptotic formulas for their growth in multiple $ ext{Z}_p$-extensions.

Findings

Relations between ranks and $ ext{mu}$-invariants of modules.

Comparison of $ ext{lambda}$-invariants for class groups and Selmer groups.

New asymptotic formulas for growth in $ ext{Z}_p$-extensions.

Abstract

Let $K_\infty/K$ be a uniform $p$-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $K_\infty$ contains sufficiently many $p$-power torsion points of $A$, then we can compare the ranks and the Iwasawa $\mu$-invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $\mathbb{Z}_p$-extensions), we can also prove relations between suitable generalisations of the Iwasawa $\lambda$-invariant of the two types of Iwasawa modules. In the literature, different kinds of Iwasawa $\lambda$-invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $\mathbb{Z}_p$-extensions.