The equivariant Tamagawa Number Conjecture for abelian extensions of imaginary quadratic fields

TL;DR

This paper proves an Iwasawa-theoretic version of a conjecture related to elliptic units, leading to results on the Tamagawa number conjecture for abelian extensions of imaginary quadratic fields.

Contribution

It establishes the conjecture in the semi-simple case and, under a $$-vanishing hypothesis, in the general case for these fields.

Findings

Proves the Iwasawa-theoretic conjecture for elliptic units.

Derives the $p$-part of the Tamagawa number conjecture at $s=0$.

Extends results under a standard $$-vanishing hypothesis.

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 $s = 0$ for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard $\mu$-vanishing hypothesis is satisfied, also in the general case.