On the $\mu$-invariants of residually reducible Galois representations

The Iwasawa $\mu$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $\mu$-invariant are defined and analyzed. As an application, we show that given any reducible mod-$p$ Galois representation $\bar{\rho }$ and any choice of integer $N\ge 1$, there is a modular Galois representation lifting $\bar{\rho }$ whose associated Selmer group has $\mu$-invariant $\ge N$. This is a refinement of Serre’s conjecture in the residually reducible case.