On the local Tamagawa number conjecture for Tate motives over tamely ramified fields

TL;DR

This paper proves the local Tamagawa number conjecture for Tate motives over certain tamely ramified fields, extending previous results to new ramified cases using advanced $(, )$-module theory.

Contribution

It provides a new proof of the conjecture for unramified extensions and establishes it for $Q_p(2)$ over specific tamely ramified fields, expanding the scope of known cases.

Findings

Proved the conjecture for unramified extensions using $(, )$-modules.

Extended the conjecture's validity to $Q_p(2)$ over tamely ramified fields.

Introduced a new approach leveraging Cherbonnier and Colmez's reciprocity law.

Abstract

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(\phi, \Gamma_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.