Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

令 $N/K$ 为p进数域的有限伽罗瓦扩展，令 $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_ {p}})$ 是绝对伽罗瓦群 $G_K$ 的r维无分支表示，它是无分支表示 $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}的限制_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ 。在本文中，我们考虑p -adic 表示$ T = \ mathbb Z_p ^r(1)(\rho ^{ \mathrm {nr}})$ . 例如，如果A是定义在 ${{\mathbb Q}_{p}}$ 上的具有良好普通归约的维数r的阿贝尔变体，则与形式群 $ 相关联的 Tate 模块 $T = T_p\hat A $ A 的 \hat A $ 是这种形式的p进表示。我们证明了所有驯服扩展 $N/K$ 和某个弱分支扩展 $N/K$ 家族的猜想。这概括了 Izychev 和 Venjakob 在驯服案例中的先前工作以及作者在弱和广泛分支案例中的工作。