On the local equivariant Tamagawa number conjecture for Tate motives

Abstract

The local equivariant Tamagawa number conjecture (local ETNC) for a motive predicts a precise relationship between the local arithmetic complex and the root numbers which appear in the (conjectural) functional equations of the $L$-functions. In this paper, we prove the local ETNC for the Tate motives under a certain unramified condition at $p$. Our result gives a generalization of the previous works by Burns--Flach and Burns--Sano. Our strategy basically follows those works and builds upon the classical theory of Coleman maps and its generalization by Perrin-Riou.