A representative of $R\Gamma (N,T)$ for higher dimensional twists of ${\mathbb{Z}}_p^r(1)$

Abstract

Let $N/K$ be a Galois extension of $p$-adic number fields and let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$. The equivariant local $\epsilon$-constant conjecture $C_{EP}^{na}(N/K,V)$ is related to the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin $L$-functions and it can be formulated as the vanishing of a certain element $R_{N/K}$ in $K_0({\mathbb{Z}}_p[G],{\mathbb{Q}}_p^c[G])$. One of the main technical difficulties in the computation of $R_{N/K}$ arises from the so-called cohomological term $C_{N/K}$, which requires the construction of a bounded complex $C_{N,T}^{\bullet }$ of cohomologically trivial modules which represents $R\Gamma (N,T)$ for a full $G_K$-stable ${\mathbb{Z}}_p$-sublattice $T$ of $V$. In this paper we generalize the construction of $C_{N,T}^{\bullet }$ in Thm. 2 of arXiv:1602.07858 to the case of a higher dimensional $T$.

22 pages

References in corpus (2)

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