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In this paper, we prove Kato’s main conjecture for 
$CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.
We compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions 
$K_\infty $ of number fields K. We prove that in several natural settings the 
$\pi $-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the 
$\mu $-invariants. In the special case of a 
$\mathbb {Z}_p$-extension 
$K_\infty /K$, we also compare the Iwasawa 
$\lambda $-invariants of the fine Selmer groups, even in situations where the 
$\mu $-invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.
Let 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = 
$\frac{1}{2}$(1 − j)An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞←nAn−. Furthermore we will show that μ = 0 if and only if μ− = 0 in this new definition.
