## Return to Question

2 added 22 characters in body

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

1

# 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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 36, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8358002305030823, "perplexity_flag": "head"}