On the local equivariant Tamagawa number conjecture for Tate motives

The local equivariant Tamagawa number conjecture (local ETNC) for a motive predicts a precise relationship between the local arithmetic complex and the root numbers which appear in the (conjectural) functional equations of the $L$-functions. In this paper, we prove the local ETNC for the Tate motives under a certain unramified condition at $p$. Our result gives a generalization of the previous works by Burns–Flach and Burns–Sano. Our strategy basically follows those works and builds upon the classical theory of Coleman maps and its generalization by Perrin-Riou.

💡 Research Summary

This paper presents a proof of the local equivariant Tamagawa number conjecture (local ETNC) for Tate motives ℤ_p(j) under specific unramified conditions at an odd prime p. The work generalizes previous results by Burns-Flach and Burns-Sano, employing strategies from those studies and building fundamentally on the classical theory of Coleman maps and its generalization by Perrin-Riou.

The central problem addressed is the formulation and verification of the local ETNC. While the global ETNC predicts a relationship between the determinant of a global arithmetic cohomology complex and the special values of L-functions, its local counterpart predicts a relationship between the determinant of local arithmetic complexes and the root numbers appearing in the (conjectural) functional equations of these L-functions. For a finite abelian extension of number fields K/k with Galois group G, the authors define a graded invertible module Ξ_loc_{K/k,S}(j) by tensoring the determinant modules of local Galois cohomology complexes RΓ(K_v, ℤ_p(j)) for finite places v and a module X_K(j) related to archimedean places. They then construct a canonical isomorphism ϑ_loc^{j}{K/k,S} from ℂ_p ⊗ Ξ_loc{K/k,S}(j) to the group algebra ℂ_p