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Generalized Quillen–Lichtenbaum Conjecture for Artin L-functions and equivariant K-theory

Establish, for every finite Galois extension F′/F of number fields with group G and every E-linear representation ρ: G → GL_N(O_E) for a number field E, the equality up to powers of 2 that for all k ≥ 1, the product over τ ∈ Gal(E/Q) of the leading coefficients L^*(O_F, τ ∘ ρ, 1 − k) equals ± the ratio of # π^G_{2k−2}(K(O_{F′}) ⊗ M(ρ)) to the size of the torsion subgroup of π^G_{2k−1}(K(O_{F′}) ⊗ M(ρ)) multiplied by the product of regulators ∏_{τ ∈ Gal(E/Q)} R_{k, τ ∘ ρ}, where M(ρ) is an integral equivariant Moore spectrum for ρ.

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Background

The classical Quillen–Lichtenbaum Conjecture relates special values of Dedekind zeta functions of number fields to sizes of algebraic K-groups and is largely settled via the work of Rost–Voevodsky and results in Iwasawa theory. This paper proposes an equivariant refinement that connects Artin L-functions attached to Galois representations and equivariant algebraic K-theory with coefficients in integral equivariant Moore spectra.

The conjecture seeks to extend this bridge to general number fields and general Galois representations, incorporating regulators and torsion in equivariant K-groups, and asserting a normed product formula across embeddings of E. The authors prove many cases (finite fields, function fields, and certain totally real abelian cases) and formulate the general statement as an explicit conjecture.

References

Conjecture (Generalized QLC) Let F′/F be a G-Galois extension of number fields for a finite group G and ρ: G→ GL_N(O_E) be an E-linear Galois representation for a number field E. Then, up to a powers of 2, we have an equality \prod_{\tau\in Gal(E/\Q)} L*(O_F,\tau\circ \rho,1-k)= \pm \frac{# \piG_{2k-2}(K(O_{F'})\otimes M (\rho))}{#\piG_{2k-1}(K(O_{F'})\otimes M (\rho))\mathrm{torsion}\cdot \prod{\tau\in Gal(E/\Q)} R_{k,\tau\circ \rho} \qquad k \geq 1, where L*(O_F,ρ,1-k) is the leading coefficient of the Taylor series of the L-function at s=1-k and R_{k,ρ} is the k-th regulator of ρ.

— Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions (2405.03578 - Elmanto et al., 6 May 2024) in Conjecture (Generalized QLC), Introduction, Section “Future work”