On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture

TL;DR

This paper proposes a conjecture linking special elements in higher algebraic K-theory of number fields to derivatives of Artin L-functions at negative integers, providing proofs in key cases and supporting evidence.

Contribution

It introduces a new conjecture connecting algebraic K-theory elements with L-function derivatives and proves it in significant cases, advancing understanding in number theory.

Findings

Conjecture established in certain important cases

Theoretical evidence supporting the conjecture

Numerical evidence supporting the conjecture

Abstract

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite dimensional complex representations. We prove this conjecture in certain important cases and also provide other evidence (both theoretical and numerical) in its support.