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?

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

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