Weil-etale cohomology and special values of L-functions at zero

TL;DR

This paper develops Weil-étale cohomology for certain sheaves over imaginary number fields and links special L-function values at zero to these cohomological invariants, providing new insights into classical number theory formulas.

Contribution

It introduces a construction of Weil-étale cohomology and Euler characteristics for specific sheaves, connecting them to special L-function values at zero.

Findings

Special value of Artin L-functions at zero expressed via Weil-étale Euler characteristic.

Derived classical formulas for L-functions of tori and their class numbers.

Established a cohomological framework for understanding L-function special values.

Abstract

We construct the Weil-\'etale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary number field. Then we show that the special value of an Artin L-function of toric type at zero is given by the Weil-\'etale Euler characteristic of an appropriate $\mathbb{Z}$-constructible sheaf up to signs. As applications of our result, we will derive some classical formulas of special values of L-functions of tori and their class numbers.