Conjectures sur les d\'eriv\'ees logarithmiques des fonctions L d'Artin aux entiers n\'egatifs

TL;DR

This paper explores conjectures linking the arithmetic properties of hermitian bundles to the derivatives of Artin L-functions at negative integers, extending and complementing existing conjectures in number theory.

Contribution

It formulates new variants of conjectures connecting hermitian fiber bundles with derivatives of Artin L-functions, generalizing and complementing prior conjectures by Colmez, Gross-Deligne, and Beilinson.

Findings

Formulation of several conjectural variants relating hermitian bundles and L-function derivatives.

Partial results supporting the formulated conjectures.

Connections established with existing conjectures in number theory.

Abstract

We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements.