Iwasawa's mu-invariant for noncyclotomic $\mathbf{Z}_p$ extensions of cyclotomic fields?

Question

Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.

(1) Is there any example where certain noncyclotomic $\mathbf{Z}_p$-extension $k_\infty/k$ has positive $\mu$-invariant?

(2) Let $n$ be an integer such that $m\mid n$ and $p$ does not divide $n/m$. Put $k'=\mathbf{Q}(\mu_n)$, $k'_\infty=k_\infty\cdot k'$. It is well-known that $\mu(k_\infty/k)\le \mu(k'_\infty/k')$. I wonder is it possible that the two sides differ by exactly 1. Here, for this part, I allow $\mu(k_\infty/k)=0$. I know that Iwasawa and Ozaki has constructed examples from cyclotomic fields which have positive $\mu$-invariant. But their examples all have degree divisible by $p$ over the base field. Their method does not apply to the above question.

Answer 1

I think I have a positive answer for (1). Let $p > 3$ be a prime, let $\mathbb{K}_1 = \mathbb{Q}[ \zeta_p ]$ be the $p$-cycotomic extension and let $q = 1 \bmod p$ be some rational prime. Let $\mathbb{F} \subset \mathbb{Q}[ \zeta_q ]$ be the subfield of degree $p$ over $\mathbb{Q}$. Obviously, $\mathbb{K}_1$ has $r_2+1 = (p+1)/2$ independent $\mathbb{Z}_p$-extensions. Let $\mathbb{M}$ be the composite of all $\mathbb{Z}_p$-extensions of $\mathbb{K}_1$ and $D(q) \subset X = Gal(\mathbb{M}/\mathbb{K}_1)$ be the decomposition group of some prime of $\mathbb{K}_1$ above $q$ - it is isomorphic to $\mathbb{Z}_p$. So we can choose some $\mathbb{Z}_p$ - extension of $\mathbb{K}_1$ which is fixed by $D(q)$; let this be $\mathbb{L}$. Then $q$ is totally split in $\mathbb{L}/\mathbb{Q}$. Let now $\mathbb{K}' = \mathbb{K}_1 . \mathbb{F}$ and let $\mathbb{L}' = \mathbb{L} . \mathbb{F}$. It is an exercise in the application of the ambig ideal lemma of Chevalley, to adapt Iwasawa's seminal proof to the present situation and show that $\mu(\mathbb{L}') > 0$. Of course, $\mathbb{K}'$ is a (quite simple) cyclotomic field, which answers your question in the affirmative.

For extensions $\mathbb{K}'/\mathbb{K}$ of degree not divisible by $p$, the method of Iwasawa fails. So the question (2) appears to be difficult ...

Answer 2

I believe that one can construct examples for (1). For (2) I was wondering why you state that "It is well-known that μ(k∞/k)≤μ(k′∞/k′)." - why can the mu-invariant not grow. I would expect that it is rather the divisibility of phi(n)/phi(m) by p which plays a role: the fact whether [ k' : k ] is or not divisible by p.

I think that all these questions are related to the following apparently remote question. Let l/k be a non-cyclotomic Z_p extension of some base field (e.g. abelian, but needs not be). Suppose that there exists a prime ideal Q subset k which is totally split in l/k. What can be said about such primes: (a) do they always exist, for any non - cyclotomic exgtnesion l/k? (b) if Q exists, is it necessarily unique, or, conversely, do there necessarily exist several primes which are totally split in the same extensions?