On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

TL;DR

This paper proves new cases of the minus component of the equivariant Tamagawa number conjecture for the multiplicative group and CM abelian extensions, especially when a tamely ramified p-adic prime exists.

Contribution

It establishes the validity of the p-component of the eTNC under tameness conditions, extending previous results and using strategies inspired by Dasgupta and Kakde.

Findings

Proves the minus component of the eTNC for $\

$p$-component holds when a tamely ramified $p$-adic prime exists.

Extends the validity of the eTNC to new cases involving CM abelian extensions.

Abstract

The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for $\mathbb{G}_m$ and for CM abelian extensions. One of the main results states that the $p$-component of the eTNC is true when there exists at least one $p$-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture.