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In Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1985), 76–91, Roland Gillard shows the following result (I mainly follow the MR review here):

Let $K$ be an imaginary quadratic field and let $p > 3$ be a prime number which splits in K into $(p)=\mathfrak{p}\mathfrak{p'}$. Let $K_{\infty}$ be the unique $\mathbb{Z}_{p}$-extension of $K$ unramified outside $\mathfrak{p}$ (thus noncyclotomic). Let $F$ be a finite abelian extension of $K$ and let $M$ be the maximal abelian $p$-extension of $F$ unramified outside $\mathfrak{p}$. Then Theorem 3.4 states that $\mathrm{Gal}(M/FK_{\infty})$ is $\mathbb{Z}_{p}$-torsion-free; in particular its $\mu$-invariant is 0.

Question: does anyone know if the vanishing of this $\mu$-invariant is also proven somewhere when $p=3$ (even special cases would be of interest)?

Note: with some work, you can get a PDF of the article in question without a subscription by following the link from here: http://www.ams.org/dmr/JournalListJ.html

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I guess your extensions are unramified outside of $\mathfrak{p}$ rather than $p$, right? –  Filippo Alberto Edoardo Sep 4 '12 at 3:13
    
You might try to look at Hida's paper in Compositio 2011, journals.cambridge.org/… –  Filippo Alberto Edoardo Sep 4 '12 at 3:54
    
Thanks for pointing out the typos - now corrected. I'll look at the Compositio paper later today. –  Henri Johnston Sep 4 '12 at 10:39
    
There has been a lot of activity in that field recently. You can also check the works of Ming-Lun Hsieh. –  Olivier Sep 4 '12 at 11:52
    
I guess Hida's paper was not much of a help, right? –  Filippo Alberto Edoardo Sep 24 '12 at 1:42

2 Answers 2

If you look at Theorem 1 in Hida's paper quoted in my comment, I think that you'll get what you are looking for.

Since $F=\mathbb{Q}$ (I stick to Hida's notation) and you work with the maximal unramified-outside-of-$\mathfrak{p}$-extension, the prime-to -$p$ part of the conductor to be considered in condition (S) is trivial and $p$ is certainly split in $F$. So the theorem applies. Now, split the maximal unramified-outside-of-$\mathfrak{p}$ extension $K(\mathfrak{p}^\infty)/K$ as the compositum $K(\mathfrak{p}^\infty)=K'K_\infty$, where $[K':K]$ is prime to $p$. The theorem tells you - if you are willing to believe it - that the projection of the $p$-adic $L$-function $\varphi$ (seen as a measure on the big Galois group $\mathrm{Gal}(K(\mathfrak{p}^\infty)/K)=\Gamma\times\mathrm{Gal}(K'/K)$ where $\Gamma$ is the Galois group of your extension) to a measure on $\Gamma$ has trivial $\mu$ invariant: this projection, indeed, corresponds to the branch character $\psi_0=\mathrm{id}$.

A last word of warning: some techniques resorting from the study of Hilbert-Blumenthal Abelian Varieties require that the totally real base field be different from the rationals. I rapidly skim the paper, without seeing any sort of hypothesis $F\neq\mathbb{Q}$, but if you intend to apply it in a research paper, I'd advise you to double-check this assumption.

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As an old-comment to a question who has been deleted: the problem with the primes $2,3$ often comes from the fact that the moduli space of Hilbert-Blumenthal Abelian Varieties is a nice space only if $6$ is invertible in the base, and this techniques are used by Katz to construct his CM $p$-adic function, which is the one used in Gillard's paper. One can try to use the full strength of Deligne-Rapoport theory of moduli stack to make everything works for (at least) also $p=3$. –  Filippo Alberto Edoardo Sep 28 '12 at 3:04
    
I emailed both Hida and Ming-Lun Hsieh to ask them about this, and they both said that actually the result I am looking for does not follow from the work of either of them. I am sorry about this - I should have contacted them before placing the bounty on this question. –  Henri Johnston Oct 5 '12 at 11:20
    
Ah, do you then have any idea on where what I wrote is wrong? –  Filippo Alberto Edoardo Oct 6 '12 at 2:45
    
Ming-Lun Hsieh said (with minor edits) "In the works of Hida and I, we consider the $\mu$-invariant of anticyclotomic $\mathbb{Z}_{p}$-extension or the $\mu$-invariant of the $\mathbb{Z}_{p}^{2}$-extension of an imaginary quadratic fields. The latter is shown to be always zero in the compositio paper of Hida and my IMRN paper with Burungale. However, what you want to know is the $\mu$-invariant of the $\mathbb{Z}_{p}$-extension unramified outside $\mathfrak{p}$, $p=\mathfrak{p}\mathfrak{p}^c$. [ctd] –  Henri Johnston Oct 6 '12 at 10:54
    
"I do not believe these $\mu$-invariants are relevant in a trivial way, so the "answer" in the link was not really correct." –  Henri Johnston Oct 6 '12 at 10:55
up vote 1 down vote accepted

The question has now been answered by this preprint: http://arxiv.org/abs/1311.3565

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