Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

Question

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?

Answer 1

I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous.

First of all, I'm assuming that $E$ has good ordinary reduction (otherwise, it has a $\mu$-invariant, but of a different kind and I don't think that this is what you have in mind - tell me if I'm wrong). In that situation, what do we definitely know for an elliptic curve over $\mathbb Q$? If $E[p]$ is irreducible, we know by works of M.Emerton, R.Pollack and T.Weston that the $\mu$-invariant is always zero or always non-zero in the Hida family passing through $E$ (now that I think of it, the theorem there is stated with the supplementary assumption that $E[p]$ be $p$-distinguished as a local representation of the decomposition group at $p$ and I never checked if one could prove the same result without that supplementary assumption - in which case there are issues of choices of lattices presumably). If $E[p]$ is reducible, we know by works of J.Bellaïche, R.Pollack and P.Wake that, in many situations, the $\mu$-invariant in the Hida family can be computed explicitly in terms of the $p$-adic valuations of $a_p(f)-1$ and $k(f)$ (respectively the $p$-th coefficients in the $q$-expansion of $f$ and the weight of $f$). In the example of the elliptic curve $X_0(11)$ at the prime $5$ you quote, for instance, this should amount to the following formula: $$\mu(X_0(11))=\operatorname{ord}_p(k)+1=\operatorname{ord}_5(2)+1=1.$$ (I'm not sure these results have been published yet so it goes without saying that all errors about them are solely mine.)

Now how does this bear on your question about other number fields? Well, in every instance we know something for sure about the cyclotomic $\mu$-invariant of an elliptic curve over $\mathbb Q$, this went through the consideration of the full Hida family that contains it and $p$-padic properties of this Hida family. Over $\mathbb Q$, whenever the $\mu$-invariant is definitely not trivial, this is explained by certain congruences with Eisenstein series, either because of a vanishing Bernoulli number or because of a vanishing Euler factor modulo $p$. One way to understand the (still conjectural!) vanishing of the $\mu$-invariant in the irreducible case would be to remark that everything necessary to fully account for the $\mu$-invariant in the reducible case cannot occur in the irreducible case.

Consequently, if I were to investigate the same question over a number field, my first try would be to generalize what is known about non-trivial $\mu$-invariant to that setting. If it does generalize, then you have at least a reason to formulate the generalization of Greenberg's conjecture (but full disclosure, I don't think the results I mentioned generalize, or that they do so easily, even in the case of totally real field, if only because the relation between cuspforms and Eisenstein series is very different in that case - there are several more subtle reasons).