Progr\`es r\'ecents sur la conjecture de Zagier et le programme de Goncharov [d'apr\`es Goncharov, Rudenko, Gangl, ...]

TL;DR

This survey discusses recent advances in Zagier's conjecture on special values of Dedekind zeta functions, highlighting the proof of the n=4 case by Goncharov and Rudenko and its connections to K-theory and motives.

Contribution

It provides an overview of recent progress on Zagier's conjecture, including the proof of the n=4 case and its implications in K-theory and motive theory.

Findings

Proof of the n=4 case of Zagier's conjecture by Goncharov and Rudenko

Connections established between the conjecture, K-theory, and motives

Comprehensive survey of recent developments in the field

Abstract

This survey article is the written version of a talk given at the Bourbaki seminar in April 2021. We give an introduction to Zagier's conjecture on special values of Dedekind zeta functions, and its relation to $K$-theory of fields and the theory of motives. We survey recent progress on the conjecture and in particular the proof of the $n=4$ case of the conjecture by Goncharov and Rudenko.