On the equivariant Tamagawa number conjecture for Abelian extensions of a quadratic imaginary field

Abstract

Let $k$ be a quadratic imaginary field, $p$ a prime which splits in $k/\mathbb{Q}$ and does not divide the class number $h_k$ of $k$. Let $L$ denote a finite abelian extension of $k$ and let $K$ be a subextension of $L/k$. In this article we prove the $p$-part of the Equivariant Tamagawa Number Conjecture for the pair $(h^0(\mathrm{Spec}(L)),\mathbb{Z}[\mathrm{Gal}(L/K)])$.