Conjectures on the logarithmic derivatives of Artin L-functions II

TL;DR

This paper proposes a broad conjecture linking Chern classes of Gauss-Manin bundles in Arakelov geometry to the logarithmic derivatives of Artin L-functions, extending classical formulas relating periods of elliptic curves to special gamma values.

Contribution

It introduces a new conjecture generalizing the Chowla-Selberg formula and proves several cases for Dirichlet characters, connecting arithmetic geometry and L-function derivatives.

Findings

Formulated a general conjecture relating Chern classes and L-function derivatives.

Proved special cases for Dirichlet characters.

Connected geometric and number-theoretic concepts through the conjecture.

Abstract

We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of periods of elliptic curves in terms of special values of the $\Gamma$-function. We prove several special cases of this conjecture in the situation where the involved Artin characters are Dirichlet characters. This article contains the computations promised in the article {\it Conjectures sur les d\'eriv\'ees logarithmiques des fonctions L d'Artin aux entiers n\'egatifs}, where our conjecture was announced. We also give a quick introduction to the Grothendieck-Riemann-Roch theorem and to the geometric fixed point formula, which form the geometric backbone of our conjecture.