Equivariant algebraic K-theory and Artin L-functions

In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic 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 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 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 E₁-page of an equivariant spectral sequence converging to equivariant algebraic K-groups. Additionally, the spectral Mackey functor structure on equivariant K-theory allows us to incorporate certain ramified extensions that appear in these L-functions.