Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields

2012-09-03T19:11:03Z

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

Answer by Filippo Alberto Edoardo for Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields

2012-09-28T02:56:49Z

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.

Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields - MathOverflow

Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields

Answer by Filippo Alberto Edoardo for Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields