On the μ-invariant of fine Selmer groups

We give some examples of elliptic curves and modular forms with good ordinary reduction at a prime p such that the associated fine Selmer group defined over the cyclotomic Zp-extension of a certain number field has μ-invariant equal to zero, thus verifying a conjecture of Coates and Sujatha in [CS]. We also prove the conjecture for Galois representations associated to certain twists of CM cusp forms that arise from CM elliptic curves.