# When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?

What is special about $$\mathbb{Z}_p$$-extensions which are motivic to ensure that their $$\mu$$ invariant is zero? Is there a simple conceptual reason.

Here are some examples.

1. Let $$F$$ be a totally real field and $$F^{cyc}$$ the cyclotomic $$\mathbb{Z}_p$$ extension. The $$\mu$$ invariant of the $$p$$-Class group tower is conjectured to be zero. This is known when $$F$$ is a abelian.
2. There is an analogue for quadratic imaginary fields, let $$K$$ be a quadratic imaginary field in which a prime $$p$$ splits into $$\mathfrak{p}\mathfrak{p}^*$$. Let $$K_{\mathfrak{p}}^{\infty}/K$$ be unique $$\mathbb{Z}_p$$-extension which is unramified outside $$\mathfrak{p}$$ and $$K_{\mathfrak{p}^*}^{\infty}/K$$ be unique $$\mathbb{Z}_p$$-extension which is unramified outside $$\mathfrak{p}^*$$. The $$\mu$$ invariants of these $$\mathbb{Z}_p$$ extensions are known to be zero (except for $$p=2,3$$). On the other hand, there are infinitely many other $$\mathbb{Z}_p$$-extensions. These $$\mu$$ invariants in general are not expected to vanish. These two special $$\mathbb{Z}_p$$-extensions come from division points on elliptic curves with complex multiplication.

There are many non-abelian analogues. Can one simply expect that a version of $$\mu=0$$ should hold whenever there is a motive involved?

• arxiv.org/abs/1703.06550 seems relevant. I understand the original example of Iwasawa of a positive mu-invariant is reviewed there. – Chris Wuthrich Nov 13 '18 at 0:00