Borel's rank theorem for Artin $L$-functions

TL;DR

This paper extends Borel's rank theorem from Dedekind zeta functions to Artin L-functions by using equivariant algebraic K-theory and Moore spectra, linking algebraic K-groups with special values of L-functions.

Contribution

It introduces a novel equivariant K-theoretic framework for Artin L-functions, generalizing Borel's theorem to a broader class of number-theoretic functions.

Findings

Established a version of Borel's rank theorem for Artin L-functions.

Developed a construction involving twisting algebraic K-theory spectra with equivariant Moore spectra.

Discussed potential applications of integral equivariant Moore spectra in L-function analysis.

Abstract

Borel's rank theorem identifies the ranks of algebraic $K$-groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin $L$-functions by considering equivariant algebraic $K$-groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic $K$-theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in $L$-functions.