The epsilon constant conjecture for higher dimensional unramified twists of ${\mathbb{Z}}_p^r(1)$

Abstract

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 ${\rho }_{{\mathbb{Q}}_p}^{\mathrm{nr}}:G_{{\mathbb{Q}}_p}\to {\mathrm{Gl}}_r({\mathbb{Z}}_p)$. 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.

43 pages

References in corpus (3)

• Equivariant epsilon constant conjectures for weakly ramified extensions
• The equivariant local $\epsilon$-constant conjecture for unramified twists of ${\mathbb{Z}}_p(1)$
• An equivariant Iwasawa main conjecture for local fields