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/Qp by Bloch and Kato. We use the theory of (φ,ΓK)
Nearby in the stack
-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