The Generalized Quillen–Lichtenbaum Conjecture of Elmanto–Zhang: integral equivariant $K$-theoretic formulas for special values of Artin $L$-functions

Elmanto and Zhang (arXiv:2405.03578, 2024) formulated a Generalized Quillen–Lichtenbaum Conjecture (their Conjecture 1.9): for every finite-group Galois extension of number fields and every integral Galois representation, the norm of the leading value of the associated Artin $L$-function at $1-k$ should equal, up to a power of $2$, an alternating order of equivariant algebraic $K$-groups with coefficients in an equivariant Moore spectrum $\mathrm{M}(\rho)$, corrected by regulators. They proved it only for finite fields, function fields of curves over finite fields, and totally real number fields with parity constraints, and only for representations conjugate to sums of inductions of abelian characters. The conjecture remains open in its stated generality.

Background

For a number field $F$ with ring of integers $\mathcal{O}_F$, the Dedekind zeta function $\zeta_F(s)$ encodes the arithmetic of $F$. If $F'/F$ is a Galois extension with finite group $G$ and $\rho\colon G\to\mathrm{GL}_N(\mathbb{C})$ is a complex representation, the Artin $L$-function $L(\mathcal{O}_F,\rho,s)$ is an Euler product built from the action of decomposition groups on $\rho$; the factorization $\zeta_{F'}(s)=\prod_{\rho}L(\mathcal{O}_F,\rho,s)^{\dim\rho}$, where the product ranges over the irreducible complex representations $\rho$ of $G$, generalizes $\zeta(s)=\prod_{\chi}L(s,\chi)$. Its special values at nonpositive integers connect to algebraic $K$-theory, the theory of the groups $\mathrm{K}_i(R)$ of a ring $R$ introduced by Quillen. The classical Quillen–Lichtenbaum Conjecture (QLC) predicts, for $k\geq2$:

$$\zeta^{*}_F(1-k)=\pm\ \frac{\#\mathrm{K}_{2k-2}(\mathcal{O}_F)}{\#\mathrm{K}_{2k-1}(\mathcal{O}_F)_{\mathrm{torsion}}}\cdot R^{B}_k(F),$$

where $\zeta^{*}_F(1-k)$ is the leading Taylor coefficient at $s=1-k$ and $R^B_k(F)$ is the Borel regulator, a real number measuring the free part of the odd $K$-group. This was proved using the Bloch–Kato conjecture established by Rost and Voevodsky (up to the Iwasawa main conjecture at the prime $2$).

Elmanto and Zhang (arXiv:2405.03578) proposed an equivariant, integral refinement for Artin $L$-functions. A Moore spectrum $\mathrm{M}(A)$ for an abelian group $A$ is a connective spectrum (a spectrum with homotopy concentrated in nonnegative degrees) whose only nonzero integral homology is $A$ in degree $0$; its formation is not functorial in $A$. Given an $E$-linear representation $\rho\colon G\to\mathrm{GL}_N(\mathcal{O}_E)$ for a number field $E$, an equivariant Moore spectrum $\mathrm{M}(\rho)$ is, roughly, a $G$-equivariant spectrum built from equivariant cells ($G$-CW spectrum) whose integral homology is $\mathcal{O}_E^{\oplus N}$ in degree $0$ with $G$-action $\rho$. The conjecture's right-hand side involves the equivariant $K$-groups $\pi^G_{*}(\mathrm{K}(\mathcal{O}_{F'})\otimes\mathrm{M}(\rho))$: the homotopy groups witnessing the $G$-action on the $K$-theory spectrum of $\mathcal{O}_{F'}$, with coefficients in $\mathrm{M}(\rho)$; the subscript 'torsion' denotes the torsion subgroup, and $R_{k,\rho}$ denotes the $k$-th regulator attached to $\rho$, generalizing the Borel regulator.

Two facts shape the problem. First, by the Brauer induction theorem (Serre, Graduate Texts in Mathematics 42), every complex $G$-representation is a virtual combination of characters induced from abelian characters of subgroups — but not necessarily an actual direct sum of such. Elmanto and Zhang construct $\mathrm{M}(\rho)$ as a connective $G$-CW spectrum whenever $\rho$ is conjugate to a direct sum of inductions of abelian characters on subgroups (their Theorem 1.2, building on the second author's work in Advances of Mathematics 399 (2022)), and this is the class for which their main identities are proved. Second, existence of equivariant Moore spectra is genuinely subtle: Steenrod asked whether a $G$-action on an abelian group $A$ can always be promoted to the Moore spectrum $\mathrm{M}A$, and Carlsson (Inventiones Mathematicae 64, 1981) answered negatively for $G=C_2\times C_2$; even when a lift exists it need not be unique. Whether $G$-CW models exist for general $\rho$ — a strictly weaker demand than promoting the action on the fixed spectrum $\mathrm{M}A$ — is not known and is itself part of the open problem.

Elmanto and Zhang prove their integral identity (Theorem 1.3): for finite fields, where they even determine the full group structure of the equivariant $K$-groups; for function fields of curves over finite fields; and for totally real number fields with parity constraints — in the last case only up to an undetermined power of $2$ when the extension is non-abelian. Their method lifts the Möbius inversion formula for the factorization of zeta into $L$-functions to the $E_1$-page of an equivariant spectral sequence, using the spectral Mackey functor structure (the system of restriction and transfer maps) on equivariant $K$-theory. They close their introduction with Conjecture 1.9 (Generalized QLC), the target of this problem, and state in Section 1.4 that removing the monomial-type representation hypothesis and treating general (not necessarily totally real) number fields together with regulators and free parts of odd $K$-groups are their central open issues.

Problem Statement

Prove or refute the Generalized Quillen–Lichtenbaum Conjecture (Conjecture 1.9 of Elmanto–Zhang, arXiv:2405.03578) in its full stated generality. Let $F'/F$ be a $G$-Galois extension of number fields for a finite group $G$, let $\rho\colon G\to\mathrm{GL}_N(\mathcal{O}_E)$ be an $E$-linear Galois representation for a number field $E$, and let $k\geq1$. The conjecture asserts that, up to powers of $2$, one has

$$\prod_{\tau\in\mathrm{Gal}(E/\mathbb{Q})}L^{*}(\mathcal{O}_{F},\tau\circ\rho,1-k)=\pm\frac{\#\pi^{G}_{2k-2}(\mathrm{K}(\mathcal{O}_{F^\prime})\otimes\mathrm{M}(\rho))}{\#\pi^{G}_{2k-1}(\mathrm{K}(\mathcal{O}_{F^\prime})\otimes\mathrm{M}(\rho))_{\mathrm{torsion}}}\cdot\prod_{\tau\in\mathrm{Gal}(E/\mathbb{Q})}R_{k,\tau\circ\rho},$$

where $L^{*}(\mathcal{O}_F,\tau\circ\rho,1-k)$ is the leading coefficient of the Taylor expansion at $s=1-k$ and $R_{k,\tau\circ\rho}$ is the $k$-th regulator. The product over $\tau\in\mathrm{Gal}(E/\mathbb{Q})$ reproduces the source's notation for an arbitrary number field $E$ and expresses the norm from $E$ to $\mathbb{Q}$ of the corresponding leading values (in the finite-field case the source writes the analogous product as $\mathrm{Norm}_{E/\mathbb{Q}}$). The conjecture is posed for every finite group $G$, every $E$-linear Galois representation $\rho$, every $G$-Galois extension of number fields $F'/F$, and every $k\geq1$. A refutation would consist either of a specific $G$ and $\rho$ for which no such $G$-CW integral Moore spectrum exists, or of a specific instance in which the displayed identity fails by more than a power of $2$.

References

1. Elden Elmanto and Ningchuan Zhang. Equivariant algebraic K-theory and Artin L-functions. arXiv:2405.03578 (2024), DOI 10.48550/arXiv.2405.03578. https://arxiv.org/abs/2405.03578
2. Gunnar Carlsson. A counterexample to a conjecture of Steenrod. Inventiones Mathematicae 64 (1981), 171–174, DOI 10.1007/BF01393939. https://doi.org/10.1007/BF01393939
3. Ningchuan Zhang. Analogs of Dirichlet L-functions in chromatic homotopy theory. Advances in Mathematics 399 (2022), article 108267, DOI 10.1016/j.aim.2022.108267. https://doi.org/10.1016/j.aim.2022.108267
4. Ningchuan Zhang. Borel's rank theorem for Artin L-functions. Proceedings of the American Mathematical Society 151 (2023), 4621–4632, DOI 10.1090/proc/16493. https://doi.org/10.1090/proc/16493
5. Jean-Pierre Serre. Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer-Verlag, 1977. ISBN 978-0-387-90190-9, DOI 10.1007/978-1-4684-9458-7. https://doi.org/10.1007/978-1-4684-9458-7
6. Qingyuan Bai, Shachar Carmeli, Branko Juran, and Florian Riedel. On the K-theory of algebraic tori. arXiv:2507.12954 (2025). https://arxiv.org/abs/2507.12954
7. Chunhui Wei. Noncommutative Quillen-Lichtenbaum Conjecture. arXiv:2604.27373 (2026). https://arxiv.org/abs/2604.27373
8. Ryo Komiya. On the higher algebraic K-groups of arithmetically equivalent number fields. arXiv:2607.26685 (2026). https://arxiv.org/abs/2607.26685
9. Saad El Boukhari. A Coates-Sinnott-type Theorem for First Derivatives of Artin L-Functions. arXiv:2608.01486 (2026). https://arxiv.org/abs/2608.01486
10. Shay Ben-Moshe. The Redshift Bound from Quillen-Lichtenbaum. arXiv:2608.02180 (2026). https://arxiv.org/abs/2608.02180