#### Vol. 10, No. 6, 2016

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

### Jay Daigle and Matthias Flach

Vol. 10 (2016), No. 6, 1221–1275
##### 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 $K∕{ℚ}_{p}$ 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 ${ℚ}_{p}\left(2\right)$ over certain tamely ramified extensions.

##### Keywords
Tamagawa number conjecture
##### Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 11G40, 18F10, 22A99