On the local equivariant Tamagawa number conjecture for Tate motives

TL;DR

This paper proves the local equivariant Tamagawa number conjecture for Tate motives under specific unramified conditions at p, extending previous results by employing Coleman maps and Perrin-Riou theory.

Contribution

It establishes the local ETNC for Tate motives under unramified conditions, generalizing prior work by Burns--Flach and Burns--Sano.

Findings

Proves the local ETNC for Tate motives under unramified conditions at p.

Extends previous results in the field.

Utilizes Coleman maps and Perrin-Riou theory in the proof.

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.