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

Daigle, Jay; Flach, Matthias

doi:10.2140/ant.2016.10.1221

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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

DOI: 10.2140/ant.2016.10.1221

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 $(φ, Γ)$ -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 $ℚ_{p} (2)$ over certain tamely ramified extensions.

Keywords

Tamagawa number conjecture

Mathematical Subject Classification 2010

Primary: 14F20

Secondary: 11G40, 18F10, 22A99

Milestones

Received: 25 August 2015

Revised: 9 March 2016

Accepted: 18 May 2016

Published: 30 August 2016

Authors

Jay Daigle
Department of Mathematics Occidental College 1600 Campus Road Los Angeles, CA 90041 United States

Matthias Flach
Department of Mathematics Caltech 253/37 Pasadena, CA 91125 United States

Vol. 10, No. 6, 2016