Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$. Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$, under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$, generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$.For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

中文翻译：

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代数 λ 不变量在可解扩展上的变化

修复奇数素数p. 让$\mathcal{D}_n$表示数域的非阿贝尔扩展ķ这样$K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $并且其伽罗瓦群具有形式$ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g }\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $在哪里G> 0 和$0 \lt n'\leq n$. 给定一个模块化的伽罗瓦表示$\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$这是p- 普通的也p-尊敬的，我们将写$\mathcal{H}(\overline{\rho})$为相关的飞騨家族。使用格林伯格的塞尔默原子概念，我们证明了代数 λ-不变量的精确公式\begin{方程} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{} 的零个数 \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big )^{\wedge}\big) \end{方程}根本$f\in\mathcal{H}(\overline{\rho})$, 在假设下$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$至少一种形式F0. 然后我们可以很容易地推断出$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$沿分支是恒定的$\mathcal{H}(\overline{\rho})$, 将 Emerton、Pollack 和 Weston 的一个定理推广为$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$.例如，如果$\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$有一个结构p-adic Lie 扩展，那么我们的公式包括以下情况： (i)$\mathcal{D}_{\infty}/K$是一个G-折叠假泰特塔，或（ii）$\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$尺寸 ≤ 3 并且是亲p-团体。