Fine Selmer groups of modular forms

TL;DR

This paper compares Iwasawa invariants of fine Selmer groups of p-adic Galois representations and CM modular forms with those of ideal class groups over p-adic Lie extensions of a number field.

Contribution

It introduces new comparisons between the Iwasawa invariants of fine Selmer groups of Galois representations and ideal class groups in p-adic Lie extensions.

Findings

Established relationships between the corank and μ-invariants of fine Selmer groups and ideal class groups.

Compared Iwasawa μ- and l0-invariants of CM modular forms with ideal class groups.

Provided insights into the structure of Selmer groups over trivialising multiple Z_p-extensions.

Abstract

We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let $K$ be a number field, let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$, and choose a $G_K$-invariant lattice ${T \subseteq V}$. We study the fine Selmer groups of ${A = V/T}$ over suitable $p$-adic Lie extensions $K_\infty/K$, comparing their corank and $\mu$-invariant to the corank and the $\mu$-invariant of the Iwasawa module of ideal class groups in $K_\infty/K$. In the second part of the article, we compare the Iwasawa $\mu$- and $l_0$-invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $\mathbb{Z}_p$-extensions of $K$.