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

Abstract

The local Tamagawa number conjecture, which was 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 $\left(\phi ,\Gamma \right)$-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 ${\mathbb{Q}}_p\left(2\right)$ over certain tamely ramified extensions.