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Asian Journal of Mathematics
Volume 24 (2020)
Number 2
The vanishing of the $\mu$-invariant for split prime $\mathbb{Z}_p$-extensions over imaginary quadratic fields
Pages: 267 – 302
DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n2.a5
Authors
Abstract
Let $\mathbb{K}$ be an imaginary quadratic field, $p$ a rational prime which splits in $\mathbb{K}$ into two distinct primes $\mathfrak{p}$ and $\mathfrak{\overline{p}}$, and $\mathbb{K}_\infty$ the unique $\mathbb{Z}_p$-extension of $\mathbb{K}$ unramified outside of $\mathfrak{p}$. For a finite abelian extension $\mathbb{L}$ of $\mathbb{K}$, we define $\mathbb{L}_\infty = \mathbb{LK}_\infty$, and let $X (\mathbb{L}_\infty)$ be the Galois group of the maximal abelian $p$-extension of $\mathbb{L}_\infty$ which is unramified outside the primes of $\mathbb{L}_\infty$ lying above $\mathfrak{p}$. We use the Euler system of elliptic units and a suitable generalisation of Sinnott’s method to give a rather elementary and completely self-contained proof that $X (\mathbb{L}_\infty)$ is always a finitely generated $\mathbb{Z}_p$-module. This is the analogue for this split prime $\mathbb{Z}_p$-extension of the Ferrero-Washington theorem for the cyclotomic $\mathbb{Z}_p$-extension. Our proof simplifies and clarifies earlier work by Schneps, Gillard, and Oukhaba–Viguié.
Keywords
Iwasawa theory, $p$-adic $\mathbb{L}$-functions, split prime $\mu$-conjecture
2010 Mathematics Subject Classification
11G05, 11R23
Received 23 April 2019
Accepted 24 May 2019
Published 8 September 2020