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

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 $(\varphi ,{\Gamma }_K)$-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.

45 pages, LaTeX; extensive revisions and clarifications based on feedback; to appear in Algebra & Number Theory

References in corpus (1)

• On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules

Cited by in corpus (1)

• $\epsilon$-isomorphisms for rank one $(\phi ,\Gamma )$-modules over Lubin-Tate Robba rings