The equivariant Tamagawa number conjecture for abelian extensions of imaginary quadratic fields

Abstract

We prove the Iwasawa-theoretic version of a conjecture of Mazur-Rubin and Sano in the case of elliptic units. This allows us to derive the p-part of the equivariant Tamagawa number conjecture at s D 0 for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard -vanishing hypothesis is satisfied, also in the general case.

On the rank of Leopoldt’s and Gross’s regulator maps

“…In Proposition 3.1 we illustrate Theorem 1.8 with a classical application to the semisimplicity of Iwasawa modules attached to Kk 1 =K. Similar results (especially in the case where p is non-split in k) have also been obtained by Bullach and Hofer [4], with applications to the p-part of the eTNC for abelian extensions of imaginary quadratic fields.…”