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



Vlad Crisąn (Mathematisches Institut, Georg-August-Universität Göttingen, Germany)

Katharina Müller (Mathematisches Institut, Georg-August-Universität Göttingen, Germany)


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.


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