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.

中文翻译：

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虚二次域上的分割质数$ \ mathbb {Z} _p $-扩展的$ \ mu $不变式的消失

假设$ \ mathbb {K} $是一个虚数二次方，$ p $是一个有理素数，它将$ \ mathbb {K} $分成两个不同的素数$ \ mathfrak {p} $和$ \ mathfrak {\ overline {p }} $和$ \ mathbb {K} _ \ infty $在$ \ mathfrak {p} $之外的$ \ mathbb {K} $唯一的$ \ mathbb {Z} _p $扩展名。对于$ \ mathbb {K} $的有限阿贝尔扩展名$ \ mathbb {L} $，我们定义$ \ mathbb {L} _ \ infty = \ mathbb {LK} _ \ infty $，然后让$ X（\ mathbb {L} _ \ infty）$是$ \ mathbb {L} _ \ infty $的最大阿贝尔文$ p $扩展名的Galois群，在$ \ mathbb {L} _ \ infty $的素数之外，这是无分支的$ \ mathfrak {p} $以上。我们使用椭圆单元的欧拉系统和Sinnott方法的适当概括，给出了一个相当基本且完全独立的证明，即$ X（\ mathbb {L} _ \ infty）$始终是有限生成的$ \ mathbb {Z } _p $-模块。这是费托罗-华盛顿定理的分裂质数$ \ mathbb {Z} _p $-扩展的类似物，用于圈式$ \ mathbb {Z} _p $-扩展。我们的证明简化并阐明了Schneps，Gillard和Oukhaba-Viguié的早期工作。