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arXiv · 1508.06031

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

Jay Daigle·Matthias Flach
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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.

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Jay Daigle, Matthias Flach. 2015-08-25. On the local Tamagawa number conjecture for Tate motives over tamely ramified fields. https://doi.org/10.2140/ant.2016.10.1221

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