I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of $E[p^{\infty}]$ is conjectured to have $\mu$ invariant zero if the Galois representation $\bar{\varrho}_{E,p}:G_{\mathbb{Q}}\rightarrow \text{Aut}(E[p])\simeq \text{GL}_2(\mathbb{F}_p)$ is irreducible.

When it is reducible, this is not true, for instance, Mazur in his Inventiones 1972 paper shows that the dual Selmer group of the modular curve $X_{0}(11)$ which is an elliptic curve prescribed by $y^2+y=x^3-x^2-10x-20$ on the other hand has $\mu=1$ at the prime $5$. The residual Galois representation is a sum of the trivial character and the mod $5$ cyclotomic character.

What does one expect over more general number fields, like imaginary quadratic fields or totally real fields for instance. In particular, if $E$ is defined over a number field $K$ such that the residual Galois representation at all primes $\mathfrak{p}|p$ is irreducible, can one expect that the dual Selmer group associated to $E[p^{\infty}]$ over $K^{cyc}$ is zero?

I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of $E[p^{\infty}]$ is conjectured to have $\mu$ invariant zero if the Galois representation $\bar{\varrho}_{E,p}:G_{\mathbb{Q}}\rightarrow \text{Aut}(E[p])\simeq \text{GL}_2(\mathbb{F}_p)$ is irreducible.

When it is reducible, this is not true, for instance, Mazur in his Inventiones 1972 paper shows that the dual Selmer group of the modular curve $X_{0}(11)$ which is an elliptic curve prescribed by $y^2+y=x^3-x^2-10x-20$ on the other hand has $\mu=1$ at the prime $5$. The residual Galois representation is a sum of the trivial character and the mod $5$ cyclotomic character.

What does one expect over more general number fields, like imaginary quadratic fields or totally real fields for instance. In particular, if $E$ is defined over a number field $K$ such that the residual Galois representation at all primes $\mathfrak{p}|p$ is irreducible, can one expect that the $\mu$ invariant of the dual Selmer group associated to $E[p^{\infty}]$ over $K^{cyc}$ is zero?