On the existence of special elements in odd $K$-theory groups

TL;DR

This paper explores the relationship between conjectural special elements in odd K-theory groups and the Equivariant Tamagawa Number Conjecture (ETNC) for abelian extensions of imaginary quadratic fields, providing partial proofs under specific conditions.

Contribution

It establishes a connection between special elements in odd K-theory and ETNC, offering partial validation of the conjecture in the semi-simple case for certain abelian extensions.

Findings

Partial proof of the conjecture for specific abelian extensions

Clarification of the relationship between special elements and ETNC

Advancement in understanding K-theory in number fields

Abstract

Let $k$ be an imaginary quadratic number field, and $F/k$ a finite abelian extension of Galois group $G$. We investigate the relationship between the conjectural special elements introduced in \cite{Burns-DeJeu-Gangl} and ETNC in the semi-simple case. This provides a partial proof of the conjecture for $F/k$ under certain conditions.