Fine Selmer groups of modular forms

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 $K$.

References in corpus (3)

• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• Fine Selmer groups of congruent $p$-adic Galois representations
• Fine Selmer groups and ideal class groups