The epsilon constant conjecture for higher dimensional unramified twists of $\mathbb Z_p^r(1)$

TL;DR

This paper proves the epsilon constant conjecture for higher-dimensional unramified twists of $Z_p^r(1)$ over certain ramified extensions, extending previous results to more general ramification cases.

Contribution

It generalizes the epsilon constant conjecture proof to higher-dimensional unramified twists over tame and specific wild ramified extensions.

Findings

Proved the conjecture for all tame extensions.

Extended proof to certain weakly and wildly ramified extensions.

Generalizes previous work by Izychev, Venjakob, and the authors.

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 $\rho^\mathrm{nr}_{\mathbb Q_p} : G_{\mathbb Q_p} \to \mathrm{Gl}_r(\mathbb Z_p)$. 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.