Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $\rho^\mathrm{ nr} : G_K \to \mathrm{ Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of an unramified representation $\rho^\mathrm{ nr}_{ \mathbb Q_p} : G_{ \mathbb Q_p} \to \mathrm{ Gl}_r(\mathbb Z_p)$. In this paper we consider the $\mathrm{ Gal}(N/K)$-equivariant local $\epsilon$-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.
Categories: math.NT