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The epsilon constant conjecture for higher dimensional unramified twists of 
${\mathbb Z}_p^r$(1)
Published online by Cambridge University Press: 29 June 2021
Abstract
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.
Keywords
MSC classification
Secondary:
11S40: Zeta functions and $L$-functions
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- © Canadian Mathematical Society 2021
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