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 .