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.