Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions

TL;DR

This paper extends the Quillen-Lichtenbaum Conjecture to Artin L-functions, relating special values to equivariant algebraic K-groups, and proves it in many cases with detailed structural results.

Contribution

It generalizes the conjecture to a broader class of L-functions and develops a spectral sequence approach to connect zeta function factorizations with algebraic K-theory.

Findings

Proved the conjecture in many cases, except possibly up to powers of 2.

Determined the structure of equivariant algebraic K-groups over finite fields.

Developed a spectral sequence lifting M"obius inversion to algebraic K-theory.

Abstract

In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our generalization extends the scope of this conjecture to Artin $L$-functions of Galois representations of finite, function, and totally real number fields. The statement of this conjecture relates norms of the special values of these $L$-functions to sizes of equivariant algebraic $\mathrm{K}$-groups with coefficients in an equivariant Moore spectrum attached to a Galois representation. We prove this conjecture in many cases, integrally, except up to a possible factor of powers of $2$ in the non-abelian and totally real number field case. In the finite field case, we further determine the group structures of their equivariant algebraic $\mathrm{K}$-groups with coefficients in Galois representations. At heart, our method lifts the M\"obius inversion formula for factorizations of zeta functions as a product of $L$-functions, to the $E_1$-page of an equivariant spectral sequence converging to equivariant algebraic $\mathrm{K}$-groups. Additionally, the spectral Mackey functor structure on equivariant $\mathrm{K}$-theory allows us to incorporate certain ramified extensions that appear in these $L$-functions.