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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 $p$ \mathfrak{p}$(thus noncyclotomic). Let$F$be a finite abelian extension of$K$and let$M$be the maximal abelian$p$-extension of$F$unramified outside$p$. \mathfrak{p}$. 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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# Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields

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 $p$ (thus noncyclotomic). Let $F$ be a finite abelian extension of $K$ and let $M$ be the maximal abelian $p$-extension of $F$ unramified outside $p$. 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