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

Elden Elmanto  Department of Mathematics, University of Toronto, Toronto, ON,
M5S 2E4, Canada<EMAIL_ADDRESS>and  Ningchuan Zhang  Department of
Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA  Max
Planck Institute for Mathematics, Bonn 53111, Germany<EMAIL_ADDRESS>
###### 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 Möbius 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.

###### 2020 Mathematics Subject Classification:

19F27, 19L47, 55P91

###### Contents

  1. 1 Introduction
  2. 2 Zeta and $L$-functions
  3. 3 Algebraic $\mathrm{K}$-theory and the classical Quillen–Lichtenbaum Conjectures
  4. 4 Equivariant algebraic $\mathrm{K}$-theory and Galois descent
  5. 5 Equivariant Moore spectra and its cellular filtration
  6. 6 Proof of the Main Theorem

##  1\. Introduction

The study of the Riemann $\zeta$-function has been a driving force in
mathematics since its invention. While the location of its zeros is the
subject of the celebrated Riemann Hypothesis, there is another aspect of its
study that has garnered much attention in the previous few decades: its
special values at non-positive integers, and connections with algebraic
$\mathrm{K}$-theory. The beginning of this connection is actually much older.
If $F$ is a number field and $\zeta_{F}(s)$ is the Dedekind zeta function of
$F$ (see Example 2.4), then Dirichlet’s class number formula (from early
1800’s) can be re-interpreted as

$\zeta_{F}^{*}(0)=-\frac{\\#\mathrm{K}_{0}(\mathcal{O}_{F})_{\mathrm{torsion}}}{\\#\mathrm{K}_{1}(\mathcal{O}_{F})_{\mathrm{torsion}}}\cdot
R_{F},$

where $\zeta_{F}^{*}(0)$ is the leading coefficient of the Taylor expansion of
$\zeta_{F}(s)$ around $s=0$, $\mathcal{O}_{F}$ is the ring of integers of $F$,
and $R_{F}$ is the Dirichlet regulator of $F$. On the other hand, the Birch-
Tate conjecture can be reinterpreted in $\mathrm{K}$-theoretic terms as well.
It states that for a totally real number field $F$ with $r_{1}(F)$ real
embeddings, we have an equality:

$\zeta_{F}^{*}(-1)=\zeta_{F}(-1)=(-2)^{r_{1}(F)}\cdot\frac{\\#\mathrm{K}_{2}(\mathcal{O}_{F})}{\\#\mathrm{K}_{3}(\mathcal{O}_{F})}.$

In the 70’s, Quillen defined higher algebraic $\mathrm{K}$-theory in [44] and
computed $\mathrm{K}_{*}(\mathbb{F}_{q})$ for a finite field $\mathbb{F}_{q}$
in [42]. Around the same time, Lichtenbaum studied étale cohomology groups of
number fields in [34]. These computations led to the Quillen–Lichtenbaum
Conjecture (QLC), first proposed in [35], describing $\zeta_{F}^{*}(1-k)$ in
terms of the orders of certain $\mathrm{K}$-groups, building a very compelling
bridge between the seemingly disparate fields of number theory and homotopy
theory.

To be slightly more precise, the QLC proposes that special values of the
Dedekind $\zeta$-function $\zeta_{F}$ for a number field $F$ are computed by
the higher algebraic $\mathrm{K}$-groups of its ring of integers
$\mathcal{O}_{F}$ in the sense that the following equality holds up to a power
of $2$:

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

Here, $R^{B}_{k}(F)$ is the $k$-th Borel/Beilinson regulator.111The two
regulators only differ by powers of $2$ by [12, Theorem 10.9]. We also note
that the Dirichlet’s Unit Theorem and Borel’s calculation of rational
algebraic $\mathrm{K}$-groups of number fields imply that

$\mathrm{ord}_{s=1-k}\zeta_{F}(s)=\mathrm{rank}(\mathrm{K}_{2k-1}(\mathcal{O}_{F})),\qquad
k\geq 1.$

We refer the readers to Section 3.2 for a review on the Quillen–Lichtenbaum
Conjectures for zeta functions of finite, function, and number fields.

The QLC has been settled in large part due Rost–Voevodsky’s proof of the
Bloch–Kato conjecture [55, 56]. It was a remarkable achievement. Among other
things, it led to the development of Morel and Voevodsky’s
$\mathbb{A}^{1}$-homotopy theory [39], an environment in which one can do
homotopy theory within algebraic geometry. This subject has, nowadays, taken a
life of its own.

###  1.1. What is done in this paper

In this paper, we propose and prove in some cases, a _refinement_ of the QLC.
Let us discuss the simplest possible setup — the context of finite fields. In
[42], Quillen computed algebraic $\mathrm{K}$-groups of finite fields as:

$\mathrm{K}_{t}(\mathbb{F}_{q})=\begin{cases}\mathbb{Z},&t=0;\\\
\mathbb{Z}/(q^{k}-1),&t=2k-1>0;\\\ 0,&\text{else}.\end{cases}$

This is the first complete computation of $\mathrm{K}$-groups of a class of
fields, yielding the simplest case of a QLC type equality which plainly
states:

$\zeta(\mathbb{F}_{q},-k)=\frac{1}{1-q^{k}}=-\frac{\\#\mathrm{K}_{2k}(\mathbb{F}_{q})}{\\#\mathrm{K}_{2k-1}(\mathbb{F}_{q})},\qquad
k\geq 1.$

Now, fix $m\geq 1$ and let $\mathrm{Fr}_{q}$ be the Frobenius element
(generator) of the Galois group
$\mathrm{Gal}(\mathbb{F}_{q^{m}}/\mathbb{F}_{q})\cong C_{m}$. Consider a
character
$\chi\colon\mathrm{Gal}(\mathbb{F}_{q^{m}}/\mathbb{F}_{q})\to\mathbb{C}^{\times}$.
In this context, we can also attach an $L$-function to $\chi$ given by a
simple “twist” of the zeta function of $\mathbb{F}_{q}$:

$L(\mathbb{F}_{q},\chi,s)=\frac{1}{1-\chi(\mathrm{Fr}_{q})q^{-s}}.$

######  Theorem 1.1 (Special case of Theorem 6.1 and Corollary 6.2).

Let $\sigma$ be the sign character of the group
$\mathrm{Gal}\left(\mathbb{F}_{q^{2}}/\mathbb{F}_{q}\right)=C_{2}$ and
$\mathrm{S}^{1-\sigma}$ be the representation sphere for the virtual
$C_{2}$-representation $1-\sigma$. Then

$\pi_{t}^{C_{2}}(\mathrm{K}(\mathbb{F}_{q^{2}})\otimes\mathrm{S}^{1-\sigma})=\begin{cases}\mathbb{Z}/(q^{k}+1),&t=2k-1>0;\\\
0,&\text{else.}\end{cases}$

It follows that:

$\frac{1}{1+q^{k}}=L(\mathbb{F}_{q},\sigma,-k)=\frac{\\#\pi_{2k}^{C_{2}}\left(\mathrm{K}(\mathbb{F}_{q^{2}})\otimes\mathrm{S}^{1-\sigma}\right)}{\\#\pi_{2k-1}^{C_{2}}\left(\mathrm{K}(\mathbb{F}_{q^{2}})\otimes\mathrm{S}^{1-\sigma}\right)},\quad
k\geq 1.$

We note the appearance the equivariant homotopy groups on the
$\mathrm{K}$-theory sides of this equality; this is one of the novel
contributions of the present paper. Readers unfamiliar with genuine
equivariant homotopy theory are referred to Section 4.1 for a (hopefully,
gentle) introduction to the subject.

The main theorem of the current paper produces many other cases of the
identity in Theorem 1.1 for Artin $L$-functions $L(X,\rho,s)$ (Definition
2.11) of Galois representations $\rho$ of finite, function, and totally real
number fields. Algebraic $\mathrm{K}$-groups are notoriously difficult to
compute and, even more so, equivariant algebraic $\mathrm{K}$-groups. We see
our paper as, to our knowledge, the first collection of computations of
certain equivariant algebraic $\mathrm{K}$-groups — at least the ratio of
their sizes.

To state our generalization of the QLC to Artin $L$-functions, we need to
first construct spectral liftings of Galois representations. The
representation sphere $\mathrm{S}^{1-\sigma}$ in Theorem 1.1 is an integral
equivariant Moore spectrum for the sign representation $\sigma$, in the sense
that $C_{2}$ acts on its zeroth (integral) homology group by $\sigma$. In nice
cases, we can construct an integral equivariant Moore spectrum
$\mathrm{M}(\rho)$ for a Galois representation $\rho$, following the second
author’s thesis [59, §3.3].

######  Theorem 1.2 (5.4 and Proposition 6.10).

Let $G$ be a finite group and $\rho\colon
G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be an $E$-linear representation for some
number field $E$. If $\rho$ is conjugate to a direct sum of inductions of
abelian characters on subgroups, then there exists a connective $G$-CW
spectrum $\mathrm{M}(\rho)$ whose integral homology groups are concentrated in
degree $0$ and the induced $G$-action on
$\mathrm{H}_{0}(\mathrm{M}(\rho);\mathbb{Z})=\mathcal{O}_{E}^{\oplus N}$ is
isomorphic to $\rho$.

######  Theorem 1.3 (Main Theorem, Theorems 6.1, 6.4, and 6.12).

Let the representation $\rho\colon
G=\mathrm{Aut}_{X}(Y)\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ and its equivariant
Moore spectrum $\mathrm{M}(\rho)$ be as described in Theorem 1.2. We have the
following QLC type identities as in Theorem 1.1:

  1. (1)

(Finite fields) $Y=\operatorname{Spec}\mathbb{F}_{q^{m}}$ and
$X=\operatorname{Spec}\mathbb{F}_{q}$ are finite fields with Galois group
$G=C_{m}$. Then:

$\mathrm{Norm}_{E/\mathbb{Q}}L(\mathbb{F}_{q},\rho,-k)=(-1)^{\dim{Q}(\rho\otimes\mathbb{Q})^{G}}\cdot\frac{\\#\pi_{2k}^{C_{m}}\left(\mathrm{K}(\mathbb{F}_{q^{m}})\otimes\mathrm{M}(\rho)\right)}{\\#\pi_{2k-1}^{C_{m}}\left(\mathrm{K}(\mathbb{F}_{q^{m}})\otimes\mathrm{M}(\rho)\right)},\qquad
k\geq 1.$

In particular, when $\rho=\chi$ is a $1$-dimensional representation, we have:
(Corollary 6.2)

$\pi_{*}^{C_{m}}(\mathrm{K}(\mathbb{F}_{q^{m}})\otimes\mathrm{M}(\chi))=\begin{cases}\mathbb{Z}/\mathrm{Norm}_{\mathbb{Q}[\mathrm{Im}\chi]/\mathbb{Q}}L(\mathbb{F}_{q},\chi,-k)^{-1}\cong\mathbb{Z}[\mathrm{Im}\chi]/L(\mathbb{F}_{q},\chi,-k)^{-1},&*=2k-1>0;\\\
0,&\text{else.}\end{cases}$

  2. (2)

(Function fields) $Y\to X$ is a finite morphism of integral, normal, and
affine curves over a finite field $\mathbb{F}_{q}$ such that
$\mathrm{Aut}_{X}(Y)=G$ and the induced map at the generic points (function
fields) is a $G$-Galois extension. Then:

$\mathrm{Norm}_{E/\mathbb{Q}}L(X,\rho,1-k)=(-1)^{\dim{Q}[(\mathrm{K}_{1}(Y)\oplus\mathbb{Z}_{\mathrm{triv}})\otimes\rho\otimes\mathbb{Q}]^{G}}\cdot\frac{\\#\pi_{2k-2}^{G}\left(\mathrm{K}(Y)\otimes\mathrm{M}(\rho)\right)}{\\#\pi_{2k-1}^{G}\left(\mathrm{K}(Y)\otimes\mathrm{M}(\rho)\right)},\qquad
k\geq 2.$

  3. (3)

(Totally real number fields) $Y=\operatorname{Spec}\mathcal{O}_{F^{\prime}}$
and $X=\operatorname{Spec}\mathcal{O}_{F}$ where $F^{\prime}/F$ is a
$G$-Galois extension of totally real and abelian number fields. Then:

$\mathrm{Norm}_{E/\mathbb{Q}}L(\mathcal{O}_{F},\rho,1-2n)=((-1)^{n}\cdot
2)^{r_{1}(F)\cdot(\dim{Q}\rho\otimes\mathbb{Q})}\cdot\frac{\\#\pi_{4n-2}^{G}\left(\mathrm{K}(\mathcal{O}_{F^{\prime}})\otimes\mathrm{M}(\rho)\right)}{\\#\pi_{4n-1}^{G}\left(\mathrm{K}(\mathcal{O}_{F^{\prime}})\otimes\mathrm{M}(\rho)\right)},\qquad
n\geq 1,$

where $r_{1}(F)$ is the number of real embeddings of $F$.

The equality only holds up to possible powers of $2$ if $F^{\prime}$ is only
totally real but not abelian over $\mathbb{Q}$.

We make some remarks about this result:

######  Remark 1.4 (Equivariant $\mathrm{K}$-groups).

In the case of finite fields, we see our computation as an equivariant
extension of Quillen’s celebrated computation of algebraic $\mathrm{K}$-groups
of finite fields in [42].

######  Remark 1.5 (Use of genuine equivariant homotopy theory).

Naively, one might want to express the $\mathrm{K}$-theory side in terms of
homotopy fixed points of the $G$-action on $\mathrm{K}(Y)$. However, this
formulation does not quite work because the map $Y\rightarrow X$ might have
ramification points, i.e., it is not a Galois or étale cover. To overcome
this, we use genuine equivariant homotopy theory via the language of spectral
Mackey functors as in Section 4.1. This structure helps us record the
ramification at every intermediate extension which is strictly more data than
the $G$-action on $\mathrm{K}(Y)$.

Notably, algebraic $\mathrm{K}$-theory of a Galois cover of schemes is known
to have a spectral $G$-Mackey functor/genuine $G$-spectrum structure by
Barwick [2] and Merling [37]. We extend this result to the _pseudo_ Galois
covers (in the sense of the function and number field cases of Theorem 1.3;
see also Definition 2.8) of $1$-dimensional schemes in 4.6.

######  Remark 1.6 (Integral results).

Inverting the order of the group erases the difference between equivariant
homotopy groups and homotopy groups of the homotopy fixed points. A simpler
version of Theorem 1.3, using just fixed points on $\mathrm{K}$-groups, is
easier to prove after inverting the order of the group. Hence the main
contribution of this paper is to provide an integral statement without
inverting any integer on $\mathrm{K}$-theory, except for possibly the prime
$2$ in the totally real but non-abelian number field case.

######  Remark 1.7 (Equivariant Moore spectra).

One of the key characters of the second author’s thesis [59] is the
equivariant Moore spectrum. For us, the equivariant cellular structure on this
object is crucial to the proof of Theorem 1.3. Essentially, it “animates” the
Möbius inversion formula.

Let us recall that in classical topology, the Moore spectrum associated to an
abelian group $A$ is a spectrum $\mathrm{M}A$ characterized as the unique
connective spectrum such that $\mathrm{H}_{0}(\mathrm{M}A;\mathbb{Z})\cong A$
and $\mathrm{H}_{>0}(\mathrm{M}A;\mathbb{Z})=0$. In addition, homotopy groups
of a spectrum $X$ with coefficients in an abelian group $A$ are simply
homotopy groups of $X\otimes\mathrm{M}A$. In particular, algebraic
$\mathrm{K}$-groups of a ring $R$ with coefficients in $A$ are then homotopy
groups of the spectrum $\mathrm{K}(R)\otimes\mathrm{M}A$.222This is different
from $\mathrm{K}_{*}(R)\otimes{Z}A$. The two are connected by a (non-split)
Universal Coefficient Theorem.

However, we note that the existence and uniqueness of equivariant Moore
spectra are subtle questions. If $G$ is a group acting on $A$, then Steenrod
asked if one can always promote the action to one on the Moore spectrum
$\mathrm{M}A$, so that the induced action on $\mathrm{H}_{0}$ is the
prescribed one. This has been answered in the negative by Carlsson [14]. Even
if the $G$-action can be promoted to $\mathrm{M}A$, the lifting is not unique
in general. See discussions in Remark 5.3.

Applying our method to rational equivariant algebraic $\mathrm{K}$-theory of
number fields, we recover a generalization of Borel’s rank Theorem 3.2 to
Artin $L$-functions by Gross in [23], which was reformulated by the second
author in [60].

######  Theorem 1.8 (Gross, Zhang, Theorem 6.5 and Theorem 6.14).

Let $F^{\prime}/F$ be a Galois extension of number fields and $\rho\colon
G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be an $E$-linear Galois representation
for some number field $E$. Then there exists a unique rational equivariant
Moore spectrum $M\mathbb{Q}(\rho)$ such that

$\mathrm{ord}_{s=1-n}L(\mathcal{O}_{F},\rho,s)=\dim_{E}\pi^{G}_{2n-1}(\mathrm{K}(\mathcal{O}_{F^{\prime}})\otimes
M\mathbb{Q}(\rho)).$

###  1.2. Outline of proof

We break the proof of Theorem 1.3 into two major steps: the first, more
difficult step, is a calculation of the equivariant $\mathrm{K}$-theory with
coefficients in a primitive abelian character; the second step is a reduction
to the case of primitive characters of a cyclic groups. We explain these steps
in turn.

####  1.2.1. Step I: Calculation in the case of primitive character of a
cyclic group

We begin with some purely number theoretic reductions. In the situation when
$G=C_{m}$ is cyclic, $Y/X$ is a pseudo $C_{m}$-Galois cover of schemes of
dimension $d\leq 1$, and $\rho=\chi\colon
C_{m}\hookrightarrow\mathbb{C}^{\times}$ is a primitive abelian character. We
exploit the fact that when $G$ is cyclic, conjugacy classes of characters form
a partially ordered set (by their kernels) and thus we can apply the Möbius
Inversion Formula to the factorization formula in Section 2.3:

$\displaystyle\zeta(Y,s)=\prod_{\chi\in C_{m}^{\vee}}L(X,\chi,s)$
$\displaystyle\Longrightarrow$ $\displaystyle\prod_{\chi\colon
C_{m}\hookrightarrow\mathbb{C}^{\times}}L(X,\chi,s)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}},s\right)\right)^{(-1)^{j}},$

where $\\{p_{1},\cdots,p_{\lambda}\\}$ is the set of prime factors of $m$ and
$Y/_{H}$ is the quotient scheme of $Y$ under the action of a subgroup $H\leq
C_{m}$.

In the cases of finite, function, and totally real number fields, the special
values of $L$-functions at negative integers are algebraic numbers in
$\mathbb{Q}[\zeta_{m}]$. Furthermore, the product of $L(X,\chi,d-n)$ over
injective characters is equal to the norm of this algebraic number over
$\mathbb{Q}$. The classical QLC (see Section 3.2) states that the values of
those zeta functions at negative integers are ratios between sizes of
algebraic $\mathrm{K}$-groups. In this way, we can express norms of special
values of $L$-functions as alternating products of sizes of algebraic
$\mathrm{K}$-groups:

(1.1)
$|\mathrm{Norm}_{\mathbb{Q}[\zeta_{m}]/\mathbb{Q}}L(X,\chi,d-n)|=\frac{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{2n-2d}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)\right)^{(-1)^{j}}}{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{2n-1}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)\right)^{(-1)^{j}}}.$

Let us switch gear now and set out to compute the equivariant
$\mathrm{K}$-groups with coefficients in a Moore spectrum. Let $M$ be a
dualizable $G$-spectrum. We can try to calculate the equivariant homotopy
groups of $\mathrm{K}(Y)$ with coefficients in $M$ via the equivariant Atiyah-
Hirzebruch spectral sequence (eAHSS):

(1.2)
$E_{2}^{s,t}=\mathrm{H}^{s}_{G}(M;\underline{\pi}_{t}(\mathrm{K}(Y)))\Longrightarrow\pi_{t-s}^{G}(\mathrm{K}(Y)\otimes\mathbb{D}M),$

where $\mathbb{D}M:=\underline{\mathrm{Map}}(M,\mathrm{S}^{0})$ is the
Spanier-Whitehead dual of $M$ in $G$-spectra. The $E_{2}$-page of this
spectral sequence is Bredon cohomology, the $G$-equivariant analog of singular
cohomology. These groups are difficult to compute in general, but one can try
to get a handle on them if there is a good equivariant cellular decomposition
of $M$. The latter gives an $E_{1}$-refinement to the $E_{2}$ page of the
spectral sequence (1.2). When $\mathbb{D}M=\mathrm{M}(\chi)$ is the
equivariant Moore spectrum for the character $\chi$, we have the cellular
structure discussed in 5.6, whence an equivariant filtration on the
$G$-spectrum $\mathrm{K}(Y)\otimes\mathrm{M}(\chi)$. This cellular structure
comes equipped with an explicit description of its graded pieces, described in
terms of fixed points with respect to various subgroups of a cyclic group $G$.
The $E_{1}$-page of the eAHSS (1.2) is then:

(1.3) $E_{1}^{s,t}=\bigoplus_{1\leq
k_{1}<\cdots<k_{s}\leq\lambda}\mathrm{K}_{t}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{s}}}}\right)\Longrightarrow\pi_{s+t}^{C_{m}}(\mathrm{K}(Y)\otimes\mathrm{M}(\chi)).$

To compute this spectral sequence, one then observes two further phenomena:
that the spectral sequence (1.2) collapses at the $E_{2}$-page and that the
passage from the $E_{1}$-page to the $E_{2}$-page is not too “lossy”. This
amounts to a sparsity result on the level of Bredon cohomology for these Moore
spectra; see Propositions 5.12 and 5.14. There is a $\mathrm{K}$-theoretic
input in these calculations; specifically for the former case: without
inverting the order of $G$ odd $\mathrm{K}$-groups of certain extensions
satisfy descent on the level of homotopy groups (Corollaries 4.9, 4.12 and
4.13).

Finally, the $L$-function and the equivariant homotopy theory sides of the
proof are connected by the remarkable observation that the alternating
products of sizes of algebraic $\mathrm{K}$-groups in (1.1) are precisely the
Euler numbers of the equivariant cellular chain complexes on the $E_{1}$-page
of the eAHSS in (1.3). The ratios of these Euler numbers are invariant when
passing to the $E_{2}$-page (taking cohomology), and then to the
$E_{\infty}$-page since the $E_{2}$-page has only two non-zero rows and the
eAHSS collapses there.

Everything above is summarized below in the case of primitive characters of
pseudo $C_{m}$-Galois covers of affine curves (the signs are ignored for
simplicity):

$\displaystyle\quad\leavevmode\nobreak\
|\mathrm{Norm}_{\mathbb{Q}[\zeta_{m}]/\mathbb{Q}}L(X,\chi,1-n)|$
$\displaystyle=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\left|\zeta\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}},1-n\right)\right|\right)^{(-1)^{j}}$ Möbius Inversion Formulae
Section 2.3 $\displaystyle=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\frac{\\#\mathrm{K}_{2n-2}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)}{\\#\mathrm{K}_{2n-1}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)}\right)^{(-1)^{j}}$ Quillen–Lichtenbaum Conjectures Section
3.2 $\displaystyle=\frac{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{2n-2}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)\right)^{(-1)^{j}}}{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{2n-1}\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)\right)^{(-1)^{j}}}$ $\displaystyle=\frac{\text{Euler
number}(E_{1}^{*,2n-2},d_{1})}{\text{Euler number}(E_{1}^{*,2n-1},d_{1})}$
eAHSS (1.3) $\displaystyle=\frac{\text{Euler
number}(E_{2}^{*,2n-2})}{\text{Euler number}(E_{2}^{*,2n-1})}$ Euler numbers
invariant under cohomology
$\displaystyle=\frac{\\#\mathrm{H}_{C_{m}}^{0}(\mathrm{M}(\psi_{m});\underline{\mathrm{K}}_{2n-2}(Y))/\\#\mathrm{H}_{C_{m}}^{-1}(\mathrm{M}(\psi_{m});\underline{\mathrm{K}}_{2n-2}(Y))}{\\#\mathrm{H}_{C_{m}}^{0}(\mathrm{M}(\psi_{m});\underline{\mathrm{K}}_{2n-1}(Y))}$
Propositions 5.12 and 5.14
$\displaystyle=\frac{\\#E_{2}^{0,2n-2}}{\\#E_{2}^{0,2n-1}\cdot\\#E_{2}^{-1,2n-2}}$
$\displaystyle=\frac{\\#\pi_{2n-2}^{C_{m}}\left(\mathrm{K}(Y)\otimes\mathrm{M}(\chi)\right)}{\\#\pi_{2n-1}^{C_{m}}\left(\mathrm{K}(Y)\otimes\mathrm{M}(\chi)\right)}.$
eAHSS collapses on $E_{2}$-page

####  1.2.2. Step II: Reduction to the primitive characters of cyclic groups

We notice some similarities between identities of Artin $L$-functions and
isomorphisms of equivariant homotopy groups as summarized in Table 1. We think
of these identities as a sequence of moves that one can make to reduce $G$ to
a cyclic group and the representation to an abelian character; the point is
that these moves can be both be made in the $L$-function and the
$\mathrm{K}$-theory side. In addition, $L$-functions satisfy a descent
identity ((3) in Proposition 2.12) that is similar to the descent property in
equivariant homotopy theory (see Proposition 6.8), which further reduces the
case to _primitive_ abelian characters.

Table 1. Comparison of identities and isomorphisms $L$-functions | Equivariant homotopy groups
---|---
$L(X,1_{G},s)=\zeta(X)$ | $\pi^{G}_{*}(M)\cong\pi_{*}(M^{G})$
$L(X,\rho_{1}\oplus\rho_{2},s)=L(X,\rho_{1},s)\cdot L(X,\rho_{2},s)$ | $\pi_{*}^{G}(M_{1}\oplus M_{2})=\pi_{*}^{G}(M_{1})\oplus\pi_{*}^{G}(M_{2})$
$L(X,\mathrm{Ind}_{H}^{G}\rho,s)=L(Y/_{H},\rho,s)$ | $\pi^{G}_{*}(\mathrm{Ind}_{H}^{G}M)\cong\pi_{*}^{H}(M)$

More precisely, if we have a representation of the form
$\rho=\bigoplus_{i}\mathrm{Ind}_{H_{i}}^{G}\chi_{i}$, which is conjugate to a
sum of inductions of abelian characters $\chi_{i}$ on subgroups $H_{i}\leq G$,
then we can construct a lift of it to the sphere spectrum by setting
$\mathrm{M}(\rho)=\bigoplus_{i}\mathrm{Ind}_{H_{i}}^{G}\mathrm{M}(\chi_{i})^{\oplus[E:\mathbb{Q}[\mathrm{Im}\chi_{i}]]}$;
that this is a Moore spectrum for $\rho$ is verified in Proposition 6.10. We
then prove in Theorem 6.12 that the Main Theorem 1.3 for $\rho$ reduces to the
primitive abelian character $\chi$ case by the comparison table above.

###  1.3. Related work

To our knowledge, the first instance of a $\mathrm{K}$-theoretic
interpretation of Artin $L$-functions was spelled out in the work of Kolster
[29] in which he considers the sign representation of a CM extension of number
fields. His methods proceed via étale cohomology and hence, just like we do,
use the Quillen–Lichtenbaum Conjectures. As the odd $\mathrm{K}$-groups of
number fields with complex embeddings have non-trivial free parts, the result
of this paper does not cover his. It would be interesting to cover his
situation as well by taking into account of regulators (see Section 1.4
below).

As explained in [26, Section 1.4], Bloch–Kato’s conjecture [7] concerning
cohomological interpretations of $L$-function of Artin motives implies
Lichtenbaum’s conjecture (relating étale cohomology and special values of zeta
functions) when the Artin motive is $h^{0}(F)$ for $F$ a field. Furthermore,
in [26], the Bloch–Kato conjecture was verified for Artin motives associated
to Dirichlet characters. Therefore, at least in spirit, our results are
related to the work of [26]. We leave it to interested readers to work out an
explicit relationship.

###  1.4. Future work

In future work, we hope to remove the assumptions in Theorem 1.2 and Theorem
1.3. Two major ones are:

  1. (1)

The Galois representations are assumed to be a sum of inductions of abelian
characters on subgroups in Theorem 1.2.

  2. (2)

In the number field case, we can only prove Theorem 1.3 when $n$ is even and
the number fields are totally real.

The first assumption is related to the Brauer Induction Theorem (see [48,
Theorems 10.20]), which says that the complex representation ring $R(G)$ of a
finite group $G$ is generated by abelian characters up to inductions. This
only implies any complex representation is a _virtual sum_ of inductions of
abelian characters, not necessarily a _direct_ sum. See more discussions in
Remark 6.11. In future works, we hope to construct integral equivariant Moore
spectra for general Galois representations of finite groups as $G$-CW spectra.
This will help us prove Theorem 1.3 for general higher-dimensional Galois
representations.

The second assumption is related to regulators of $\zeta$-functions and free
parts of algebraic $\mathrm{K}$-groups of number fields. On the
$\mathrm{K}$-theory side, our proof relies crucially on the finiteness of
$\mathrm{K}_{4n-1}(\mathcal{O}_{F})$ when $F$ is totally real. In this case,
the equivariant cellular chain complex to compute the Bredon cohomology
$\mathrm{H}^{*}_{G}(\mathbb{D}\mathrm{M}(\chi);\underline{\mathrm{K}}_{4n-1}(\mathcal{O}_{F}))$
is a bounded complex of finite abelian groups. Hence the size of
$\mathrm{H}^{0}_{G}$ is equal to the Euler number of the complex if it is the
only non-zero cohomology group. On the $\zeta$-function side,
$\zeta_{F}(1-k)=0$ unless $F$ is totally real and $k$ is even. If either
fails, the QLC states that

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

holds up to powers of $2$, where $\zeta^{*}_{F}(1-k)$ is the leading
coefficient in the Taylor expansion of $\zeta_{F}(s)$ at $s=1-k$, and
$R^{B}_{k}(F)$ is the $k$-th Beilinson/Borel regulator of $F$. In future
works, we plan to prove Theorem 1.3 for general number fields, using
corresponding non-equivariant Quillen–Lichtenbaum Conjecture for those number
fields (see Theorem 3.3). We thus end this introduction with what we believe
is a reasonable generalized version of QLC.

######  Conjecture 1.9 (Generalized QLC).

Let $F^{\prime}/F$ be a $G$-Galois extension of number fields for a finite
group $G$ and $\rho\colon G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be an
$E$-linear Galois representation for a number field $E$. Then, up to a powers
of $2$, we have an equality

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

where $L^{*}(\mathcal{O}_{F},\rho,1-k)$ is the leading coefficient of the
Taylor series of the $L$-function at $s=1-k$ and $R_{k,\rho}$ is the $k$-th
regulator of $\rho$.

### Notation and convention

We freely use the language and syntax of higher algebra as laid out in [36];
in particular the foundations of equivariant homotopy theory that we use is
expressed in terms of $\infty$-categories as in the work of Barwick [2].

  * •

(Euler numbers) Suppose that we have a bounded chain complex of _finite_
abelian groups

$D_{*}:=0\rightarrow D_{m}\rightarrow D_{m-1}\rightarrow\cdots\rightarrow
D_{m-k}\rightarrow 0.$

Then we set

$\text{Euler number}(D_{*}):=\prod\\#\left(D_{j}\right)^{(-1)^{j}}.$

Note that this number is invariant under taking (co)homology.

  * •

(Spanier-Whitehead duals) If $(\mathcal{C},\otimes,\mathbbm{1})$ is a
presentably symmetric monoidal $\infty$-category, then we may speak of a
dualizable object $X$ and their categorical/Spanier-Whitehead dual
$\mathbb{D}(X)$ which is calculated as the internal hom
$\underline{\mathrm{Map}}(X,\mathbbm{1})$.

  * •

(Representations) For a group $G$, we write $1_{G}$ and $\rho_{G}$ for its
complex trivial and regular representations, respectively.

  * •

(primitive/injective characters) In this paper, a primitive character
$\chi\colon G\to\mathbb{C}^{\times}$ is simply an injective group
homomorphism. So we will use the terms primitive/injective characters
interchangeably. Such representations are also called faithful in
representation theory. We remind the readers that this is different from the
notion of a _primitive Dirichlet character_ , where
$\chi\colon\left(\mathbb{Z}/N\right)^{\times}\to\mathbb{C}^{\times}$ is called
primitive if it cannot factor through $(\mathbb{Z}/N^{\prime})^{\times}$ for
any proper divisor $N^{\prime}$ of $N$ (see [41, §VII.2]).

  * •

The Euler totient function $\phi(m)$ of a positive integer $m$ is the number
of positive integers $1\leq k\leq m$ that are coprime to $m$.

  * •

We will fix the meanings of the following letters (in standard math font)
throughout the paper:

    * –

$d$ is the (Krull) dimension of a scheme $X$.

    * –

$\kappa(x)$ is the residue field of a closed point $x$ in a scheme $X$.

    * –

$k(X)$ is the function field of an integral scheme $X$.

    * –

$q$ is the size of a finite field $\mathbb{F}_{q}$.

    * –

$\ell$ is a prime number not dividing the size of the base field
$\mathbb{F}_{q}$.

    * –

$F$ and $F^{\prime}$ are number fields whose zeta functions and algebraic
$\mathrm{K}$-theory are studied.

    * –

$r_{1}(F)$ is the number of real embeddings of a number field $F$.

    * –

$E$ is a number field such that a Galois representation $\rho$ is
$\mathcal{O}_{E}$-linear.

    * –

$\mathrm{Norm}_{E/\mathbb{Q}}(x)$ is the field norm of an algebraic number $x$
in $E$ over $\mathbb{Q}$.

    * –

$N$ is the dimension of a representation $\rho$ over $E$.

    * –

$G$ is a finite group.

    * –

$C_{m}$ is the cyclic group of order $m$.

    * –

$\psi_{m}\colon C_{m}\hookrightarrow\mathbb{Z}[\zeta_{m}]^{\times}$ is a fixed
integral cyclotomic character of $C_{m}$.

    * –

$\sigma$ is the sign character/representation of $C_{2}$.

    * –

$\lambda$ is the number of distinct prime factors of an integer $m$.

### Acknowledgments

This project has benefitted from numerous conversations that the authors had
with Matthew Ando, Paul Balmer, Clark Barwick, Mark Behrens, Ted Chinburg,
Paul Goerss, Lars Hesselholt, Mike Hopkins, Annette Huber-Klawitter, Michael
Larsen, Yifeng Liu, Mona Merling, Matthew Morrow, Niranjan Ramachandran,
Catherine Ray, Charles Rezk, Nick Rozenblyum, Peter Scholze, and Zhouli Xu. In
particular, we would like to thank Tobias Barthel and Michael Mandell for a
careful reading of the draft of the paper.

EE has benefitted from an Erik Ellentuck fellowship from the Institute for
Advanced Study and expresses gratitude for perfect working conditions.

NZ was partially supported by the NSF Grant DMS-2348963/2304719 and the AMS-
Simons Travel Grant. Some of the work was done when NZ was visiting the Max
Planck Institute for Mathematics. He thanks the MPIM for their hospitality and
support.

##  2\. Zeta and $L$-functions

We begin with an exposition of some concepts in number theory and arithmetic
algebraic geometry. We hope that the exposition will be friendly to homotopy
theorists who are not necessarily well-acquainted with the apparatus of zeta
and $L$-functions. We will follow Serre’s exposition in [47].

###  2.1. Zeta functions

To begin with, the most fundamental function in the subject is arguably the
following:

######  Definition 2.1.

Let $X$ be a finite type $\mathbb{Z}$-scheme of which is relatively
equidimensional over $\mathbb{Z}$ of relative dimension $d$. We define the
Hasse–Weil zeta function attached to $X$ to be

$\zeta(X,s)=\prod_{x\in|X|}\frac{1}{1-\\#\kappa(x)^{-s}};$

where $|X|$ is the topological space (set) of closed points on the scheme $X$
and $\\#k$ denotes the cardinality of a finite field $k$. Notice that the
residue field $\kappa(x)$ of a closed point $x\in|X|$ is also finite, since
$X$ is finite type over $\mathbb{Z}$. This product is absolutely convergent
over the domain $\mathrm{Re}(s)>d$ in $\mathbb{C}$.

######  Example 2.2.

If $X=\operatorname{Spec}(\mathbb{F}_{q})$ is a finite field, then we have

$\zeta(\mathbb{F}_{q},s)=\frac{1}{1-q^{-s}}.$

######  Example 2.3.

If the structure map $X\rightarrow\operatorname{Spec}(\mathbb{Z})$ factors
through
$\operatorname{Spec}(\mathbb{F}_{q})\rightarrow\operatorname{Spec}(\mathbb{Z})$
for some finite field $\mathbb{F}_{q}$, then we can describe the Hasse-Weil
zeta function in terms of point count. Namely, we have an equality (see proof
in Example 2.15):

$\zeta(X,s)=\exp\left(\sum_{m=1}^{\infty}\frac{\\#X(\mathbb{F}_{q^{m}})}{m}\cdot
q^{-ms}\right).$

This is often called the Weil zeta function of the $\mathbb{F}_{q}$-scheme
$X$. In particular, when $X=\operatorname{Spec}\mathbb{F}_{q}$, we recover the
formula in Example 2.2.

######  Example 2.4.

Suppose that $X=\operatorname{Spec}(\mathcal{O}_{F})$ where $F$ is a number
field and $\mathcal{O}_{F}$ is its ring of integers. Then we recover the
Dedekind zeta function, in the sense we get equalities:

(2.1)
$\zeta_{F}(s):=\zeta(\mathcal{O}_{F},s)=\prod_{(0)\neq\mathfrak{p}\trianglelefteq\mathcal{O}_{F}}\left(1-(\\#\mathcal{O}_{F}/\mathfrak{p})^{-s}\right)^{-1}=\sum_{(0)\neq\mathcal{I}\trianglelefteq\mathcal{O}_{F}}\frac{1}{(\\#\mathcal{O}_{F}/\mathcal{I})^{s}},$

where the product and the summation range over all non-zero _prime_ ideals and
all non-zero ideals and of $\mathcal{O}_{F}$, respectively. The last equality
boils down to the unique factorization of ideals in the Dedekind domain
$\mathcal{O}_{F}$. When $F=\mathbb{Q}$, we recover the Riemann zeta function:

$\zeta(s):=\zeta{Q}(s)=\prod_{p\text{
prime}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$

The next theorem records analytic properties of the Hasse-Weil zeta function
in certain situations.

######  Theorem 2.5.

The $\zeta$-functions in the three examples above admit meromorphic
continuations to the complex plane.

###### Proof.

We give a sketch of proof for each case in turn.

  1. (1)

When $X=\operatorname{Spec}\mathbb{F}_{q}$, this is obvious since
$\zeta(\mathbb{F}_{q},s)=\frac{1}{1-q^{-s}}$ is a meromorphic function on
$\mathbb{C}$ with a simple pole at $s=0$.

  2. (2)

When $X$ is a smooth variety of dimension $d$ over a finite field
$\mathbb{F}_{q}$, the meromorphic continuation follows from the rationality
part of the Weil Conjectures by Dwork in [18]. This can be proved using the
Lefschetz fixed point theorem for étale cohomology. Namely, we have an
identity [38, Theorem 27.6]:

$\zeta(X,s)=\prod_{r=0}^{2d}\det\left[1-\mathrm{Fr}_{q}\cdot
q^{-s}\mid\mathrm{H}^{r}_{\acute{e}t}(\overline{X};\mathbb{Q}_{\ell})\right]^{(-1)^{r+1}},$

where
$\overline{X}=X\times_{\mathbb{F}_{q}}\operatorname{Spec}\overline{\mathbb{F}}_{q}$,
$\mathrm{Fr}_{q}$ is the $q$-th power Frobenius endomorphism on
$\overline{X}$, and $\ell$ is a prime number that does not divide $q$.

  3. (3)

When $X=\operatorname{Spec}\mathcal{O}_{F}$ for a number field $F$, the
summation (2.1) for the Dedekind zeta function $\zeta_{F}(s)$ converges
absolutely and uniformly when $\mathrm{Re}(s)\geq 1+\delta$ for every
$\delta>0$. One first extends $\zeta_{F}$ to the strip $0<\mathrm{Re}(s)\leq
1$. The analytic continuation follows from the functional equation of
$\zeta_{F}$ in this region. This is a standard result in analytic number
theory. See [41, Corollary VII.5.11.(i)]. ∎

######  Remark 2.6.

The existence of analytic/meromorphic continuation of the Hasse-Weil
$\zeta$-functions for finite type varieties over $\mathbb{Z}$ is open in
general. Some other known cases include _cellular_ schemes like
$\mathbb{A}^{d},\mathbb{P}^{d}$ over $\mathcal{O}_{F}$ (since their zeta
functions are products of translations of $\zeta_{F}(s)$) and elliptic curves
(as a consequence of the celebrated modularity theorem).

######  Proposition 2.7.

In all three examples above, $\zeta(X,s)$ is a rational number when $s$ is a
non-positive integer.

###### Proof.

Again, the case when $X$ is a smooth variety over a finite field follows from
the rationality part of the Weil Conjectures. For $X=\mathcal{O}_{F}$, one can
first show $\zeta_{F}(1-n)=0$ unless $F$ is totally real and $n$ is even. The
rationality of $\zeta_{F}(1-2n)$ when $F$ is totally real was proved in [28].
∎

###  2.2. Artin $L$-functions of Galois representations

In this subsection, we introduce $L$-functions associated to Galois
representations. We work with pseudo-Galois coverings of schemes to
incorporate ramifications.

######  Definition 2.8.

A finite surjective morphism of noetherian, normal, integral schemes
$\pi\colon Y\to X$ is called a pseudo-Galois covering if the induced map on
function fields $k(X)\to k(Y)$ is a finite Galois extension and the natural
homomorphism $\mathrm{Aut}_{X}(Y)\to\mathrm{Gal}(k(Y)/k(X))$ is an
isomorphism. A ring map $R\to R^{\prime}$ is called pseudo-Galois if the
induced map $\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R$ is a
pseudo-Galois covering. If the group $G=\mathrm{Gal}(k(Y)/k(X))$ is fixed, we
say that $\pi$ is a pseudo $G$-Galois cover.

######  Remark 2.9.

The concept in Definition 2.8 was given the name pseudo-Galois extensions by
Suslin and Voevodsky in [50, Definition 5.5].

######  Example 2.10.

  1. (1)

Any Galois covering of normal integral schemes is pseudo-Galois.

  2. (2)

Suppose that $X$ is a normal integral scheme with function field $F$ and let
$F^{\prime}$ be a finite, Galois extension of $F$. Let $Y$ be the
normalization of $X$ into $F^{\prime}$ [49, Tag 035H], then the map
$Y\rightarrow X$ is an example of a pseudo-Galois covering.

  3. (3)

As a more classical version of the above example, we can consider
$F^{\prime}/F$ is a Galois extension of number fields. Then one can check this
extension map restricts to their rings of integers
$\mathcal{O}_{F}\to\mathcal{O}_{F^{\prime}}$, which is a pseudo-Galois
extension of Dedekind domains.

  4. (4)

Geometrically, a pseudo-Galois covering of Riemann surfaces is a ramified
covering space. Given a compact Riemann surface $X$, there is a one-to-one
correspondence between ramified coverings of $X$ and Galois extensions of the
field of meromorphic functions on $X$. See [20, Theorem 8.12].

Let $\pi\colon Y\to X$ be a pseudo $G$-Galois covering, $y\in|Y|$ be a closed
point and $x=\pi(y)$. We recall the following definitions and results needed
to define Artin $L$-functions (see [49, Tag 0BSD]):

  * •

$\mathcal{D}(y)=\\{g\in G\mid g(y)=y\\}$ is the decomposition group of $y$.

  * •

One can show that there are natural surjections:

${\mathcal{D}(y)}$${\mathrm{Gal}(\kappa(y)/\kappa(x)).}$

  * •

$\mathcal{I}(y)$, the inertia subgroup of $y$, is the kernel of the surjection
above. The map $\pi$ is unramified at $x$ if $\mathcal{I}(y)$ is trivial (for
any $y$ above $x$).

  * •

$\mathrm{Fr}_{y/x}\in\mathcal{D}(y)/\mathcal{I}(y)$ corresponds to the
Frobenius element in $\mathrm{Gal}(\kappa(y)/\kappa(x))$ under the isomorphism
$\mathrm{Gal}(\kappa(y)/\kappa(x))\cong\mathcal{D}(y)/\mathcal{I}(y)$.

  * •

If $y^{\prime}$ is another point above $x$ such that $g(y)=y^{\prime}$ for
some $g\in G$. Then $g^{-1}\mathcal{D}(y^{\prime})g=\mathcal{D}(y)$,
$g^{-1}\mathcal{I}(y^{\prime})g=\mathcal{I}(y)$, and
$g^{-1}\mathrm{Fr}_{y^{\prime}/x}g=\mathrm{Fr}_{y/x}$.

  * •

We will write $\mathrm{Fr}_{x}$ and $\mathcal{I}(x)$ for the conjugacy classes
of the element $\\{\mathrm{Fr}_{y/x}\\}$ and the subgroups
$\\{\mathcal{I}(y)\\}$ for all $y$ above $x$.

######  Definition 2.11.

Suppose that $Y\rightarrow X$ is a pseudo-Galois covering of normal integral
schemes such that $G=\mathrm{Gal}(k(Y)/k(X))$. Let $\rho\colon
G\to\mathrm{GL}_{N}(\mathbb{C})$ be a representation of $G$ valued in a finite
dimensional complex vector space $V=\mathbb{C}^{N}$. The Artin $L$-function of
$\rho$ is defined to be the Euler product:

$L(X,\rho,s)=\prod_{x\in|X|}\frac{1}{\det\left[\operatorname{id}-\\#\kappa(x)^{-s}\cdot\rho(\mathrm{Fr}_{x})\left|V^{\mathcal{I}(x)}\right.\right]}.$

This product converges absolutely to a complex analytic function over the
domain $\mathrm{Re}(s)>\dim X=\dim Y$. Notice in each Euler factor:

  * •

$\mathrm{Fr}_{x}$ is represented by an element
$\mathrm{Fr}_{y/x}\in\mathcal{D}(y)/\mathcal{I}(y)\subseteq G/\mathcal{I}(y)$
for some $y$ above $x$. We are allowed to evaluate $\rho$ on
$\mathrm{Fr}_{y/x}$ only after restricting the representation to the fixed
point subspace $V^{\mathcal{I}(y)}$ by the inertia subgroup;

  * •

the determinant does not depend on the choice of $y$ over $x$.

######  Proposition 2.12 ([41, Proposition VII.10.4]).

The Artin $L$-functions satisfy the following identities with respect to
operations on representations:

  1. (1)

$L(X,1_{G},s)=\zeta(X,s).$ where $1_{G}$ is the $1$-dimensional trivial
representation of $G$.

  2. (2)

(Additivity) $L(X,\rho_{1},s)\cdot
L(X,\rho_{2},s)=L(X,\rho_{1}\oplus\rho_{2},s)$.

  3. (3)

(Descent) Notice the Galois representation $\rho$ factors uniquely as
$\rho\colon G\twoheadrightarrow
G/\ker\rho\xrightarrow{\rho^{\prime}}\mathrm{GL}_{N}(\mathbb{C})$. Then

$L(X,\rho,s)=L(X,\rho^{\prime},s),$

where the pseudo-Galois covering on the right hand side is $Y/_{\ker\rho}\to
X$. Thus it makes sense to drop the total space $Y$ from the notation of an
Artin $L$-function.

  4. (4)

(Induction) Let $H$ be a subgroup of $G$. Then for any finite-dimensional
complex representation $\rho$ of $H$, we have an induction formula:

$L(X,\mathrm{Ind}_{H}^{G}\rho,s)=L(Y/_{H},\rho,s).$

######  Corollary 2.13 (Factorization of zeta functions, [41, Corollary
VII.10.5]).

For any normal subgroup $H\trianglelefteq G$, we have

$\zeta(Y/_{H},s)=\prod_{H\subseteq\ker\rho}L(X,\rho,s)^{\dim\rho},$

where $\rho$ ranges through all irreducible complex representations of $G$
whose kernels contain $H$. If $G$ is abelian, write
$G^{\vee}=\hom(G,\mathbb{C}^{\times})$ for the group of its complex abelian
characters. Then for any subgroup $H\leq G$, we have

$\zeta(Y/_{H},s)=\prod_{\chi\in(G/H)^{\vee}}L(X,\chi,s),$

where complex abelian characters of $G/H$ are identified with those of $G$
that are trivial on $H$.

###### Proof.

This follows from Proposition 2.12 and the decomposition of the regular
representation $\rho_{G/H}$ of the quotient group $G/H$.

$\zeta(Y/_{H},s)=L(X,\mathrm{Ind}_{H}^{G}1_{H},s)=L(X,\rho_{G/H},s)=L\left(X,\bigoplus_{H\subseteq\ker\rho}\rho^{\oplus\dim\rho},s\right)=\prod_{H\subseteq\ker\rho}L(X,\rho,s)^{\dim\rho}.$

∎

######  Example 2.14.

Let
$Y/X=\operatorname{Spec}\mathbb{F}_{q^{m}}/\operatorname{Spec}\mathbb{F}_{q}$
and $\chi\colon\mathrm{Aut}(Y/X)\cong C_{m}\to\mathbb{C}^{\times}$ be a
complex character. Denote the Frobenius element by $\mathrm{Fr}$. Then we have

$L(\mathbb{F}_{q},\chi,s)=\frac{1}{1-\chi(\mathrm{Fr})q^{-s}}.$

######  Example 2.15.

When $X$ is a normal integral $\mathbb{F}_{q}$-scheme, the Artin $L$-function
of a Galois representation of $k(X)$ can be expressed as an exponential sum:
(see for example [27])

$L(X,\rho,s)=\exp\left(\sum_{m=1}^{\infty}\sum_{x\in
X(\mathbb{F}_{q^{m}})}\mathrm{tr}\left[\rho(\mathrm{Fr}_{x})|V^{\mathcal{I}(x)}\right]\cdot\frac{q^{-ms}}{m}\right)\in
1+q^{-s}E\llbracket q^{-s}\rrbracket,$

where $\rho$ is an $E$-linear representation for some subfield $E$ of
$\mathbb{C}$. It follows that the Artin $L$-function $L(X,\rho,s)$ only
depends on the character of the representation $\rho$. Note that the identity
in Example 2.3 is the special case of the one above when $\rho=1_{G}$ is the
$1$-dimensional representation.

We see that the two expressions coincide as follows. For each closed point
$x\in|X|$ with residue field $\mathbb{F}_{q^{n}}$, set $x_{m}$ to be the
closed point
$\operatorname{Spec}\mathbb{F}_{q^{nm}}\to\operatorname{Spec}\mathbb{F}_{q^{n}}\xrightarrow{x}X$.
Notice the Galois conjugates of
$x\colon\operatorname{Spec}\mathbb{F}_{q^{m}}\to X$ over $\mathbb{F}_{q}$ are
identified in $|X|$, but they are counted separately in each
$X(\mathbb{F}_{q^{m}})$. Then we have (see also [38, pages 157 – 158 and 165 –
166])

$\displaystyle\quad\exp\left(\sum_{m=1}^{\infty}\sum_{x\in
X(\mathbb{F}_{q^{m}})}\mathrm{tr}\left[\rho(\mathrm{Fr}_{x})|V^{\mathcal{I}(x)}\right]\cdot\frac{q^{-ms}}{m}\right)$
$\displaystyle=\exp\left(\sum_{m=1}^{\infty}\sum_{x\in
X(\mathbb{F}_{q^{m}})}\mathrm{tr}\left[\rho(\mathrm{Fr}_{x})|V^{\mathcal{I}(x)}\right]\cdot\frac{\\#\kappa(x)^{-s}}{[\kappa(x):\mathbb{F}_{q}]}\right)$
$\displaystyle=\exp\left(\sum_{x\in|X|}\left([\kappa(x):\mathbb{F}_{q}]\sum_{m=1}^{\infty}\mathrm{tr}\left[\rho(\mathrm{Fr}_{x_{m}})\mid
V^{\mathcal{I}(x)}\right]\cdot\frac{\\#\kappa(x_{m})^{-s}}{[\kappa(x_{m}):\mathbb{F}_{q}]}\right)\right)$
$\displaystyle=\prod_{x\in|X|}\exp\left(\left.\mathrm{tr}\left[\sum_{m=1}^{\infty}\rho(\mathrm{Fr}_{x})^{m}\frac{\\#\kappa(x)^{-ms}}{m}\right)\right|V^{\mathcal{I}(x)}\right]$
$\displaystyle=\prod_{x\in|X|}\det\left[\left.\exp\left(\sum_{m=1}^{\infty}\frac{\left(\rho(\mathrm{Fr}_{x})\cdot\\#\kappa(x)^{-s}\right)^{m}}{m}\right)\right|V^{\mathcal{I}(x)}\right]$
$\displaystyle=\prod_{x\in|X|}\det\left[\exp\left(-\log(1-\rho(\mathrm{Fr}_{x})\cdot\\#\kappa(x)^{-s})\right)\mid
V^{\mathcal{I}(x)}\right]$
$\displaystyle=\prod_{x\in|X|}\frac{1}{\det\left[1-\rho(\mathrm{Fr}_{x})\cdot\\#\kappa(x)^{-s}\mid
V^{\mathcal{I}(x)}\right]}.$

######  Example 2.16.

Let $X$ be an integral normal $\mathbb{F}_{q}$-scheme of dimension $d$, $E$ be
a number field, and
$\rho\colon\pi_{1}^{\acute{e}t}(X)\to\mathrm{GL}_{N}(\mathcal{O}_{E})\hookrightarrow\mathrm{GL}_{N}(\overline{\mathbb{Q}}_{\ell})$
be a Galois representation for some prime $\ell\nmid q$. Denote by
$\mathcal{F}_{\rho}$ the lisse sheaf on $X_{{\acute{e}t}}$ associated to
$\rho$ and $\pi\colon\overline{X}\to X$ is the base change map to
$\overline{\mathbb{F}}_{q}$. Then the Lefschetz fixed point theorem for étale
cohomology implies

(2.2) $L(X,\rho,s)=\prod_{r=0}^{2d}\det\left[1-\mathrm{Fr}\cdot
q^{-s}\mid\mathrm{H}^{r}_{\acute{e}t}(\overline{X};\pi^{\ast}\mathcal{F}_{\rho})\right]^{(-1)^{r+1}},$

It follows that $L(X,\rho,s)$ is a rational function of $q^{-s}$ whose
coefficients are algebraic numbers in $F$. In particular, $L(X,\rho,s)$ admits
a meromorphic continuation to $\mathbb{C}$.

######  Lemma 2.17.

Let $E$ be a number field and
$\rho\colon\pi_{1}^{\acute{e}t}(X)\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be an
$E$-linear Galois representation of a connected, normal, separated
$\mathbb{F}_{q}$-scheme of dimension $d$. Then

  1. (1)

For any integer $n$, if $s=n$ is not a pole for $L(X,\rho,s)$, then
$L(X,\rho,n)\in E$.

  2. (2)

Let $\sigma\in\mathrm{Aut}{Q}(E)$ be a field automorphism, then
$L(X,\sigma\circ\rho,n)=\sigma(L(X,\rho,n))$.

###### Proof.

(1) follows from the identity (2.2) and (2) is a result of the definition of
$L(X,\rho,s)$ in Example 2.15 as the exponential of a formal power series with
coefficients in $F$. ∎

######  Proposition 2.18.

Let $\pi\colon Y\to X$ be a pseudo-Galois cover of normal integral curves over
$\mathbb{F}_{q}$ with $G=\mathrm{Gal}(k(Y)/k(X))$ and $\rho\colon
G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be a Galois representation. Then

  1. (1)

The Artin $L$-function $L(X,\rho,s)$ admits a meromorphic continuation to
$\mathbb C$.

  2. (2)

For any integer $n$, if $s=n$ is not a pole for $L(X,\rho,s)$, then
$L(X,\rho,n)\in E$.

  3. (3)

Let $\sigma\in\mathrm{Aut}{Q}(E)$ be a field automorphism, then
$L(X,\sigma\circ\rho,n)=\sigma(L(X,\rho,n))$.

###### Proof.

Notice the branch locus $B$ of $\pi$ is a finite disjoint union of closed
points in the curve $X$. By Definition 2.11, we have

$L(X,\rho,s)=L(X-B,\rho,s)\cdot\prod_{x\in|B|}\frac{1}{\det[\operatorname{id}-\\#\kappa(x)^{-s}\rho(\mathrm{Fr}_{x})\mid
V^{\mathcal{I}(x)}]}.$

As $Y-\pi^{-1}(B)\to X-B$ is a $G$-Galois cover by assumption, the three
claims follow from Lemma 2.17. ∎

######  Example 2.19.

Let $F^{\prime}/F$ be a $G$-Galois extension of number fields. When
$\chi\colon G\to\mathbb{C}^{\times}$ is a $1$-dimensional complex
representation (abelian character), the Artin $L$-function of $\chi$ is
usually called the Dirichlet $L$-function. The abelian character $\chi$
necessarily factors through an abelian extension of $F$. One particular
example is when $F=\mathbb{Q}$. Then the Kronecker-Weber Theorem states any
finite abelian Galois extension of $\mathbb{Q}$ is contained in some
cyclotomic extension $\mathbb{Q}(\zeta_{N})$ of $\mathbb{Q}$. Let
$\chi\colon\left(\mathbb{Z}/N\right)^{\times}\cong\mathrm{Gal}(\mathbb{Q}(\zeta_{N})/\mathbb{Q})\to\mathbb{C}^{\times}$
be a Dirichlet character, then we have an identity:

$L(\mathbb{Z},\chi,s):=\prod_{p}\frac{1}{1-\chi(p)p^{-s}}=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},\qquad\chi(n)=0\text{
if }(n,N)\neq 1.$

Using global class field theory, there is a similar formula to express the
Artin $L$-function of an abelian Galois representation of a number field as a
summation (Hecke $L$-function). See [41, Theorem VII.10.6].

######  Proposition 2.20.

Let $F^{\prime}/F$ be a $G$-Galois extension of number fields.

  1. (1)

(Artin) The Artin $L$-function $L(\mathcal{O}_{F},\chi,s)$ of a non-trivial
complex abelian character $\chi\colon G\to\mathbb{C}^{\times}$ extends to an
_entire_ function of $s$.

  2. (2)

(Brauer) The Artin $L$-function $L(\mathcal{O}_{F},\rho,s)$ of any complex
Galois representation $\rho$ of $G$ extends to a _meromorphic_ function on the
complex plane. In particular, this meromorphic continuation is _entire_ if
$\rho$ is a direct sum of inductions of non-trivial abelian characters on
subgroups.

######  Remark 2.21.

Artin conjectured that for any non-trivial irreducible Galois representation
$\rho$ of number fields, the Artin $L$-function $L(X,\rho,s)$ extends to an
_entire_ function of $s$. He proved this conjecture when $\rho$ is
$1$-dimensional using the Artin reciprocity law (1-dimensional Langlands
correspondence). See more discussions in [40].

######  Remark 2.22.

We note the condition for holomorphic continuation in Brauer’s theorem also
appears in the assumption of our main Theorem 1.3. Indeed, both uses the
Brauer Induction Theorem. See more discussions in Remark 6.11.

Like Proposition 2.18, we also have the following algebraicity result for
special values of Artin $L$-functions of number fields.

######  Proposition 2.23 ([16, Theorem 1.2]).

Let $F^{\prime}/F$ be a $G$-Galois extension of number fields and $\rho\colon
G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$ be an $E$-linear Galois representation
for some number field $E$. Then $L(\mathcal{O}_{F},\rho,1-n)\in E$ if it is
not a pole. Moreover if $\rho^{\prime}=\tau\circ\rho$ for some
$\tau\in\mathrm{Gal}(E/\mathbb{Q})$, then
$L(\mathcal{O}_{F},\rho^{\prime},1-n)=\tau(L(\mathcal{O}_{F},\rho,1-n))$.

###  2.3. Möbius inversion formulae

In Corollary 2.13, we express zeta functions as products of $L$-functions
associated to irreducible representations of Galois groups. Ideally, we want a
procedure to extract each $L(X,\chi,s)$ for each character $\chi$ and explain
a Quillen–Lichtenbaum phenomenon for it. However, we are unable to separate
out characters which are conjugate. In fact, by [16, Theorem 1.2] and
Proposition 2.18, if the characters are conjugate then their special values
are conjugate and therefore the same from the point-of-view of
$\mathrm{K}$-theory. Noting that two abelian characters
$\chi,\tau:G\rightarrow\mathbb{C}^{\times}$ are conjugate if and only if they
have the same kernel, we prove the following key computational tool which
should be thought of as a Möbius Inversion Formula.

######  Proposition 2.24 (Möbius Inversion Formula).

Let $G$ be a finite group. Denote the group of complex characters of $G$ by
$G^{\vee}$. For a normal subgroup $H\trianglelefteq G$, we can view
$(G/H)^{\vee}$ as a subgroup of $G^{\vee}$ by

$(G/H)^{\vee}:=\hom(G/H,\mathbb{C}^{\times})\cong\\{\chi\in G^{\vee}\mid
H\subseteq\ker\chi\\}.$

Given a family of complex (meromorphic) functions $g(s,\chi)$ indexed by
$\chi\in G^{\vee}$, we define $f(s,G/H)$ for each normal subgroup
$H\trianglelefteq G$ by setting

(2.3) $f(s,G/H):=\prod_{\chi\in(G/H)^{\vee}}g(s,\chi).$

Then for any cyclic quotient $G/H\cong C_{m}$ of $G$ where
$m=p_{1}^{d_{1}}\cdots p_{\lambda}^{d_{\lambda}}$ is the prime factorization
of $m$, we have the following Möbius Inversion Formula:

$\prod_{\tau\in
G^{\vee},\ker\tau=H}g(s,\tau)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}f\left(s,(G/H)/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)\right)^{(-1)^{j}},.$

Here the $j=0$ factor is set to be $f(s,G/H)$.

###### Proof.

On the right hand side of (2.3), separating primitive (injective) characters
of $G/H$ from the non-primitive ones, we obtain:

(2.4) $\prod_{\tau\in G^{\vee},\ker\tau=H}g(s,\tau)=\prod_{\tau\colon
G/H\hookrightarrow\mathbb{C}^{\times}}g(s,\tau)=f(s,G/H)\cdot\left(\prod_{\tau\in(G/H)^{\vee}\text{,
non-primitive}}g(s,\tau)\right)^{-1}$

Notice any non-primitive character $\tau$ of $G/H\cong C_{m}$ must factor
through a further quotient $(G/H)/_{C_{p_{i}}}\cong C_{m}/C_{p_{i}}$ for some
prime factor $p_{i}$ of $m$. By (2.3), we have

(2.5)
$\prod_{i=1}^{\lambda}f(s,(G/H)/_{C_{p_{i}}})=\prod_{i=1}^{\lambda}\prod_{\tau\in\left((G/H)/_{C_{p_{i}}}\right)^{\vee}}g(s,\tau).$

Comparing this with (2.4), we observe the right hand side of (2.5) over-counts
those characters $\tau\colon G/H\to\mathbb{C}^{\times}$ whose kernels contain
$C_{p_{i}p_{j}}$ for some distinct pair $p_{i},p_{j}$ of prime factors of $m$.
The Möbius inversion formula then follows from the Inclusion and Exclusion
Principle. ∎

Next, we apply Proposition 2.24 to the factorization formulas for zeta
functions in Corollary 2.13. For $L$-functions of (pseudo-)Galois
representations of varieties over $\mathbb{F}_{q}$, we obtain:

######  Corollary 2.25.

Let $Y/X$ be a pseudo-Galois extension of normal integral curves or étale
covers of integral normal varieties over $\mathbb{F}_{q}$ for the group
$G=C_{m}$ and $\chi\colon\mathrm{Gal}(Y/X)\cong
C_{m}\hookrightarrow\mathbb{C}^{\times}$ be a primitive character. We have

$\mathrm{Norm}_{\mathbb{Q}(\zeta_{m})/\mathbb{Q}}L(X,\chi,-n)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta\left(Y/_{C_{p_{k_{1}}\cdots
p_{k_{j}}}},-n\right)\right)^{(-1)^{j}}.$

###### Proof.

By Lemma 2.17 (for Galois covers of varieties) and Proposition 2.18 (for
pseudo Galois covers of curves), we have

$\displaystyle\mathrm{Norm}_{\mathbb{Q}(\zeta_{m})/\mathbb{Q}}L(X,\chi,-n)=\prod_{\tau\in\mathrm{Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})}\tau(L(X,\chi,-n))$
$\displaystyle=\prod_{\tau\in\mathrm{Gal}(\mathbb{Q}(\zeta_{m})/\mathbb{Q})}L(X,\tau\circ\chi,-n)$
$\displaystyle=\prod_{\psi\colon\mathrm{Gal}(Y/X)\hookrightarrow\mathbb{C}^{\times}}L(X,\psi,-n).$

Setting $G=\mathrm{Gal}(Y/X)\cong C_{m}$ and $g(s,\chi)=L(X,\chi,s)$ in
Proposition 2.24, we have $f(s,G/H)=\zeta(Y/_{H},s)$ by Corollary 2.13. The
claim then follows from Proposition 2.24. ∎

For Dirichlet $L$-functions of number fields, we have

######  Proposition 2.26.

Let $F^{\prime}/F$ be a $C_{m}$-Galois extension of number fields. Suppose $m$
has exactly $\lambda$ many distinct prime factors $p_{1},\cdots,p_{\lambda}$.
Let $\chi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}$
be a primitive (injective) character.

  1. (1)

Write $\mathrm{ord}_{s=s_{0}}f(s)$ for the order of vanishing of a meromorphic
function $f(s)$ at $s=s_{0}$. Then

$\sum_{\psi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}}\mathrm{ord}_{s=s_{0}}L(\mathcal{O}_{F},\psi,s)=\sum_{j=0}^{\lambda}(-1)^{j}\left(\sum_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\mathrm{ord}_{s=s_{0}}\zeta_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(s)\right).$

In particular when $s=1-n$, we have:

$\phi(m)\cdot\mathrm{ord}_{s=1-n}L(\mathcal{O}_{F},\chi,s)=\sum_{j=0}^{\lambda}(-1)^{j}\left(\sum_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\mathrm{ord}_{s=1-n}\zeta_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(s)\right).$

  2. (2)

Write $f^{*}(s_{0})$ for the leading coefficient of the Laurent series of a
meromorphic function $f(s)$ at $s=s_{0}$. Then

$\prod_{\psi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}}L^{*}(\mathcal{O}_{F},\psi,s)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta^{*}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(s)\right)^{(-1)^{j}},$

In particular, when $F^{\prime}$ is a totally real number field (then so is
$F$) and $s=1-2n$, we have:

(2.6)
$\mathrm{Norm}_{\mathbb{Q}(\zeta_{m})/\mathbb{Q}}L(\mathcal{O}_{F},\chi,1-2n)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(1-2n)\right)^{(-1)^{j}}.$

###### Proof.

Applying the Möbius Inversion Formula in Proposition 2.24 to factorization of
Dedekind zeta function $\zeta_{F^{\prime}}(s)$ over $F$ in Corollary 2.13, we
get an identity of meromorphic functions:

$\prod_{\psi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}}L(\mathcal{O}_{F},\psi,s)=\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(s)\right)^{(-1)^{j}}.$

The first halves of (1) and (2) follow directly by taking orders of vanishing
and leading coefficients of Laurent series at $s=s_{0}$ on both sides of the
identity. Proposition 2.23 implies that:

$\displaystyle\phi(m)\cdot\mathrm{ord}_{s=1-n}L(\mathcal{O}_{F},\chi,s)$
$\displaystyle=\sum_{\psi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}}\mathrm{ord}_{s=1-n}L(\mathcal{O}_{F},\psi,s),$
$\displaystyle\mathrm{Norm}_{\mathbb{Q}(\zeta_{m})/\mathbb{Q}}L(\mathcal{O}_{F},\chi,1-2n)$
$\displaystyle=\prod_{\psi\colon\mathrm{Gal}(F^{\prime}/F)\hookrightarrow\mathbb{C}^{\times}}L(\mathcal{O}_{F},\psi,1-2n).\qed$

##  3\. Algebraic $\mathrm{K}$-theory and the classical Quillen–Lichtenbaum
Conjectures

Let $F/\mathbb{Q}$ be a number field and $\mathcal{O}{F}$ be its ring of
integers. In this section, we review the classical Quillen–Lichtenbaum
Conjecture for Dedekind $\zeta$-function attached to $F$. In particular,
having pretended to be number theorists in the previous section, we pretend to
be homotopy theorists and explain $\mathrm{K}$-theory in a (hopefully)
friendly way to number theorists. To orient us, the road from
$\zeta$-functions to algebraic $\mathrm{K}$-theory, is summarized in the
diagram below (we will elaborate more on it during some proofs in what
follows; see Section 3.2):

(3.1) ${\begin{subarray}{c}\text{Frobenius action on}\\\
\mathrm{H}_{\acute{e}t}^{1}(\mathcal{O}_{F}[1/p,\zeta_{p^{\infty}}];\mathbb{Z}_{p}(t))\end{subarray}}$${\begin{subarray}{c}\text{$p$-adic
$L$-function}\\\
\zeta_{F}(1-t)\end{subarray}}$${\mathrm{H}_{c}^{r}\left(\mathbb{Z}_{p}^{\times};\mathrm{H}_{\acute{e}t}^{s}(\mathcal{O}_{F}[1/p,\zeta_{p^{\infty}}];\mathbb{Z}_{p}(t))\right)}$${\mathrm{H}_{\acute{e}t}^{r+s}(\mathcal{O}_{F}[1/p];\mathbb{Z}_{p}(t))}$${\pi_{2t-r-s}\left(\mathrm{L}_{\mathrm{K}(1)}\mathrm{K}(\mathcal{O}_{F}[1/p])\right)}$${\mathrm{H}^{r+s}_{\mathrm{mot}}(\mathcal{O}_{F}[1/p];\mathbb{Z}_{p}(t))}$${\pi_{2t-r-s}\mathrm{K}(\mathcal{O}_{F}[1/p])^{\wedge}_{p}}$Iwasawa
Main ConjectureMazur-WilesHochschild-Lyndon-Serre SSThomason SSMotivic
SSétale-motivic comparison$\scriptstyle{\mathrm{L}_{\mathrm{K}(1)}}$

We will also discuss the case of finite fields and functions fields of curves.

###  3.1. Reminders on $\mathrm{K}$-theory

The zeroth algebraic $\mathrm{K}$-group of a ring, denoted by
$\mathrm{K}_{0}(R)$, is the Grothendieck group of finitely generated
projective modules over $R$. While this is a relatively elementary definition,
it stores rather subtle information about a ring. For example, say $D$ is a
Dedekind domain with fraction field $F$, then the ideal class group of $F$ aka
$\mathrm{Pic}(D)$ is a summand of $\mathrm{K}_{0}(D)$; more precisely we have
a canonical short exact sequence

$0\rightarrow\mathrm{Pic}(D)\longrightarrow\mathrm{K}_{0}(D)\longrightarrow\mathbb{Z}\rightarrow
0;$

where the first map is defined by observing that any ideal of $D$ is
projective. At this point, as already mentioned in the introduction, the
reader might know that $\mathrm{K}_{1}$ has something to do with units.
Indeed, if $R$ is a field or, more generally, a semilocal ring then
$\mathrm{K}_{1}(R)\cong R^{\times}$, the group of units in $R$. This could
seem unmotivated333Unless one already knows that one is looking for a higher
analog of the class number formula. but there is a check for the “correctness”
of the definition of the functor of higher $K$-groups. This comes under the
name “Bass’ fundamental theorem”. Indeed, for any ring $R$ (possibly even just
an associative, unital ring) one should have a 4-term exact sequence (where
the last map is, in fact, a split surjection)

$0\rightarrow\mathrm{K}_{n}(R)\longrightarrow\mathrm{K}_{n}(R[t])\oplus\mathrm{K}_{n}(R[t^{-1}])\longrightarrow\mathrm{K}_{n}(R[t,t^{-1}])\overset{\partial}{\longrightarrow}\mathrm{K}_{n-1}(R)\rightarrow
0.$

We refer the reader to [57, V.8] for further explanation. Hence, once one has
defined the functor of $\mathrm{K}_{n}$, one should recover the functor of
$\mathrm{K}_{n-1}$. This principle even applies for any theory of negative
$K$-groups.

Quillen solved the problem of defining higher $\mathrm{K}$-groups in [44]. He
built on an earlier definition of his which is somewhat more elementary.
Higher $\mathrm{K}$-theory was defined via Quillen’s $+$-construction [42]
which is, informally, the universal way to abelianize the fundamental group of
a space without changing its homology; we refer to [57, IV. 1] for a textbook
discussion and [25, 45, 1] for modern exposition. The positive homotopy groups
of the algebraic $\mathrm{K}$-theory of a ring is then calculated as

$\mathrm{K}_{i}(R):=\pi_{i}(\mathrm{BGL}(R)^{+})\qquad i\geq 1,$

where $\mathrm{BGL}(R):=\operatorname*{colim}\mathrm{BGL}_{n}(R)$. This is
sufficient to give meaning to the groups that appear in the
Quillen–Lichtenbaum Conjectures. From this point of view, the
Quillen–Lichtenbaum Conjectures is ultimately a relationship between the
homotopy groups of the “derived abelianization” of the classifying space of
$\mathrm{GL}_{\infty}(R)$ and zeta functions.

For the purposes of this paper, algebraic $\mathrm{K}$-theory in this guise is
not enough. For one, it is not true that $\mathrm{K}(R)$ is functorially
equivalent to $\mathrm{BGL}(R)^{+}\times\mathrm{K}_{0}(R)$.444If $R$ is not
regular, then we even have negative homotopy groups in $\mathrm{K}$-theory!
For another, we need certain extra structure of algebraic $\mathrm{K}$-theory
that is not immediately visible using the $+$-construction. The viewpoint that
we must take is the one taken by Blumberg-Gepner-Tabuada in [9]. To formulate
this perspective, we set $\mathcal{C}\mathrm{at}{}^{\mathrm{perf}}_{\infty}$
to be the $\infty$-category of small, stable idempotent $\infty$-categories.
This $\infty$-category houses the $\infty$-category of perfect complexes on
rings in the sense that there is a functor

$\mathrm{Perf}\colon\mathrm{CAlg}\longrightarrow\mathcal{C}\mathrm{at}{}^{\mathrm{perf}}_{\infty}.$

Algebraic $\mathrm{K}$-theory is then a universal functor

$\mathrm{K}\colon\mathcal{C}\mathrm{at}{}^{\mathrm{perf}}_{\infty}\longrightarrow\mathcal{S}\mathrm{pt}{},$

which preserves final objects and bifiber sequences in
$\mathcal{C}\mathrm{at}{}^{\mathrm{perf}}_{\infty}$ (Verdier sequences); we
refer to [13, Appendix A] for a modern exposition.

This viewpoint unveils clearly certain structures and properties that
algebraic $K$-theory enjoys. One that is relevant for this paper are “wrong-
way” maps for $\mathrm{K}$-groups. For example, if $f\colon R\rightarrow S$ is
a morphism of rings which is perfect [49, Tag 0685] and finite so that the
forgetful functor on derived categories preserves perfect complexes [49, Tag
0B6G]:

$f_{*}\colon\mathrm{Perf}(S)\longrightarrow\mathrm{Perf}(R),$

then we have an induced map on $\mathrm{K}$-groups
$\mathrm{K}_{*}(S)\rightarrow\mathrm{K}_{*}(R)$. Of course such a structure
could be defined using more classical methods, but we will need to keep track
of the coherences on this construction as we do via the language of spectral
Mackey functors in Section 4.2.

The last thing that the reader should keep in mind, though perhaps not
strictly needed for the purposes of this paper, is the relationship between
$\mathrm{K}$-theory and étale/motivic cohomology. The relevance of motivic
cohomology, then still a conjecture object, to values of zeta functions was
posited by Beilinson [6] and Lichtenbaum [33]. For us, these form the $E_{2}$
page of the motivic spectral sequence:

$\mathrm{H}^{i-j}_{\mathrm{mot}}(X;\mathbb{Z}(-j))\Longrightarrow\mathrm{K}_{-i-j}(X).$

Here $X$ is a smooth scheme over field, in which case this spectral sequence
is the work of many mathematicians, beginning with Bloch and Bloch–Lichtenbaum
[8], and then later Friedlander-Suslin [21], Levine [32, 31] and Voevodsky
[54].

The existence of the above spectral sequence indicates that one can access
information stored in these higher $\mathrm{K}$-groups, once we can
satisfactorily describe motivic cohomology. For the purposes of $L$-functions,
the flagship theorem in this direction is the comparison of motivic cohomology
with étale/syntomic cohomology as conjectured by Beilinson-Lichtenbaum and
proved by to Rost–Voevodsky [56, 55] away from characteristic and Geisser-
Levine [22] at the characteristic.

###  3.2. Statements of the classical Quillen–Lichtenbaum Conjectures

The most basic case of the Quillen–Lichtenbaum Conjecture comes from Quillen’s
own computation of the $\mathrm{K}$-groups of finite fields. In [42], he
computed the groups as follows:

(3.2) $\mathrm{K}_{t}(\mathbb{F}_{q})=\begin{cases}\mathbb{Z},&t=0;\\\
\mathbb{Z}/(q^{n}-1),&t=2n-1>0;\\\ 0,&\text{else}.\end{cases}$

From this, the Quillen–Lichtenbaum Conjecture reads as:

(3.3)
$\zeta(\mathbb{F}_{q},-n)=\frac{1}{1-q^{n}}=-\frac{\\#\mathrm{K}_{2n}(\mathbb{F}_{q})}{\\#\mathrm{K}_{2n-1}(\mathbb{F}_{q})},\qquad
n\geq 1.$

Notice this identity even “holds” at $n=0$ in the sense that
$\zeta(\mathbb{F}_{q},s)$ has a simple pole at $s=0$,
$\mathrm{K}_{0}(\mathbb{F}_{q})=\mathbb{Z}$ is free of rank $1$, and
$\mathrm{K}_{-1}(\mathbb{F}_{q})=0$.

For smooth affine curves over $\mathbb{F}_{q}$, the QLC states:

######  Theorem 3.1.

Let $X$ be a smooth affine curve over a finite field $\mathbb{F}_{q}$ with
smooth compactification $X\to\widetilde{X}$. Denote by $r(X)$ the number of
closed points in the complement $\widetilde{X}-X$. Then we have the following
equality for $n\geq 2$:

$\zeta(X,1-n)=\varepsilon\cdot\frac{\\#\mathrm{K}_{2n-2}(X)}{\\#\mathrm{K}_{2n-1}(X)},\qquad\varepsilon=(-1)^{r(X)}=(-1)^{\mathrm{rank}(\mathrm{K}_{1}(X))+1}.$

###### Sketch Proof.

We sketch a proof this result in steps:

  1. (1)

First, we can relate the $\zeta$ functions with étale cohomology of $X$ is via
the Weil Conjectures, which implies [4, Theorem 3.1]:

$|\zeta(X,1-n))|_{\ell}=\frac{\\#\mathrm{H}^{2}_{\acute{e}t}(X;\mathbb{Z}_{\ell}(n))}{\\#\mathrm{H}^{1}_{\acute{e}t}(X;\mathbb{Z}_{\ell}(n))},$

where $\ell$ is a prime not dividing $q$ and $|-|_{\ell}$ means $\ell$-adic
absolute value (or equivalently valuation).

  2. (2)

Thomason’s hyperdescent theorem [53] produces a spectral sequence whose
$E_{2}$-terms are étale cohomology and abuts to the homotopy groups of
$\mathrm{K}(1)$-local algebraic $\mathrm{K}$-theory of $X$:

$\mathrm{H}^{i-j}_{\acute{e}t}(X;\mathbb{Z}_{\ell}(-j))\Longrightarrow\pi_{-i-j}\left(\mathrm{L}_{\mathrm{K}(1)}\mathrm{K}(X)\right).$

In the case of curves, this spectral sequence collapses. Thomason observes in
[53, (4.6)] that:

$|\zeta(X,1-n)|_{\ell}=\frac{\\#\pi_{2n-2}\mathrm{L}_{\mathrm{K}(1)}\mathrm{K}(X)}{\\#\pi_{2n-1}\mathrm{L}_{\mathrm{K}(1)}\mathrm{K}(X)},$

  3. (3)

Next, we also have motivic spectral sequence [54, 32]

$\mathrm{H}^{i-j}_{\mathrm{mot}}(X;\mathbb{Z}(-j))\Longrightarrow\mathrm{K}_{-i-j}(X),$

which fits into a comparison diagram

${\mathrm{H}^{i-j}_{\mathrm{mot}}(X;\mathbb{Z}_{\ell}(-j))}$${\mathrm{K}_{-i-j}(X)^{\wedge}_{\ell}}$${\mathrm{H}^{i-j}_{{\acute{e}t}}(X;\mathbb{Z}_{\ell}(-j))}$${\pi_{-i-j}\left(\mathrm{L}_{\mathrm{K}(1)}\mathrm{K}(X)\right).}$Motivic
SSétale-motivic comparison$\scriptstyle{\mathrm{L}_{\mathrm{K}(1)}}$Thomason
SS

  4. (4)

There is then an isomorphism on $E_{2}$-pages thanks to the Rost–Voevodsky
theorem [55, 56] which yields an isomorphism between $\mathrm{K}(1)$-local
$\mathrm{K}$-theory and $\mathrm{K}$-theory in the above range.

  5. (5)

It remains to determine the sign. Let $Y$ be a smooth variety of dimension $d$
over $\mathbb{F}_{q}$. From Definition 2.1, it is clear that $\zeta(Y,s)>0$
when $s$ is a real number greater than $d$ (the bound is for the convergence
of the infinite product). When $Y/\mathbb{F}_{q}$ is smooth proper, the sign
of $\zeta(Y,d-n)$ is determined by the functional equation of the Weil zeta
function [38, Remark 27.13]:

$\zeta(Y,s)=\pm q^{-d\chi(\overline{Y})/2}\cdot
q^{\chi(\overline{Y})s}\cdot\zeta(Y,d-s),$

where the sign is positive if $d$ is odd and $\chi(\overline{Y})$ is the Euler
characteristic of the $\ell$-adic étale cohomology of
$\overline{Y}=Y\times_{\mathbb{F}_{q}}\operatorname{Spec}\overline{\mathbb{F}}_{q}$.
It follows that $\zeta(\widetilde{X},1-n)>0$ for the smooth compactification
$\widetilde{X}$ of $X$. Again by Definition 2.1, we have

$\zeta(\widetilde{X},1-n)=\zeta(X,1-n)\cdot\prod_{x\in|\widetilde{X}-X|}\frac{1}{1-\\#\kappa(x)^{n-1}}.$

This implies $(-1)^{r(X)}\cdot\zeta(X,1-n)>0$ for $n>1$, where
$r(X):=\\#|\widetilde{X}-X|$. Write $X=\operatorname{Spec}A$. By Dirichlet’s
Unit Theorem (see [15, Theorem 18]) and the result of Bass-Milnor-Serre [3],
we have
$r(X)=\mathrm{rank}(A^{\times})+1=\mathrm{rank}(\mathrm{K}_{1}(X))+1$.∎

Let $F$ be a number field. Denote by $\zeta^{*}_{F}(s)$ the first non-
vanishing coefficient in Taylor expansion of the Dedekind zeta function
$\zeta_{F}$ associated to $F$ at $s$, by $r_{1}(F)$ the number of real
embeddings of $F$, and by $r_{2}(F)$ the number of pairs of complex conjugate
embeddings of $F$. Let $k$ be an integer greater than $1$.

######  Theorem 3.2 (Quillen, Borel, [10, 11, 43]).

Algebraic $\mathrm{K}$-groups of $\mathcal{O}_{F}$ are all finitely generated
abelian groups. In even degrees, $\mathrm{K}_{2k}(\mathcal{O}_{F})$ are finite
when $k>0$. In odd degrees, we have

$\dim{Q}\mathrm{K}_{2k-1}(\mathcal{O}_{F})\otimes{Z}\mathbb{Q}=\mathrm{rank}(\mathrm{K}_{2k-1}(\mathcal{O}_{F}))=\mathrm{ord}_{s=1-k}\zeta_{F}(s)=\begin{cases}r_{1}(F)+r_{2}(F)-1,&k=1;\\\
r_{1}(F)+r_{2}(F),&k>1\text{ odd};\\\ r_{2}(F),&k>0\text{ even}.\end{cases}$

######  Theorem 3.3 (Quillen–Lichtenbaum Conjecture, Rost–Voevodsky).

The followings hold:

  1. (1)

Suppose that $F$ is an _abelian_ number field, then the equality

(3.4)
$\zeta^{*}_{F}(1-k)=\varepsilon\cdot\frac{\\#\mathrm{K}_{2k-2}(\mathcal{O}_{F})}{\\#\mathrm{K}_{2k-1}(\mathcal{O}_{F})_{\mathrm{tors}}}\cdot
R^{B}_{k}(F),\qquad\varepsilon=\begin{cases}1,&k\equiv 1\mod 4;\\\
(-1)^{r_{1}(F)+r_{2}(F)},&k\equiv 2\mod 4;\\\ (-1)^{r_{1}(F)},&k\equiv 3\mod
4;\\\ (-1)^{r_{2}(F)},&k\equiv 0\mod 4,\end{cases}$

holds up to powers of $2$, where $R^{B}_{k}(F)$ is the $k$-th Beilinson
regulator of $F$.

  2. (2)

If $F$ is both _totally real_ and _abelian_ , then we have the precise
equality

(3.5) $\zeta_{F}(1-2n)=((-1)^{n}\cdot
2)^{r_{1}(F)}\cdot\frac{\\#\mathrm{K}_{4n-2}(\mathcal{O}_{F})}{\\#\mathrm{K}_{4n-1}(\mathcal{O}_{F})},$

where $r_{1}(F)$ is the number of real embeddings of $F$.

  3. (3)

If $F$ is only totally real but not abelian, then (3.5) only holds up to
possible powers of $2$.

######  Remark 3.4.

Like in the affine curve case, the number $r_{1}(F)$ can also be expressed in
terms of algebraic $\mathrm{K}$-groups when $F$ is totally real, using
Dirichlet’s Unit Theorem and the Bass-Milnor-Serre Theorem:

$r_{1}(F)=\mathrm{rank}(\mathcal{O}_{F}^{\times})+1=\mathrm{rank}(\mathrm{K}_{1}(\mathcal{O}_{F}))+1.$

###### Proof sketch.

We discuss how to find the statements in the literature. Making reference to
the diagram (3.1), we note that, thanks to the Rost–Voevodsky theorem, the
only bottleneck is the Lichtenbaum Conjecture, which follows from the Iwasawa
Main Conjecture (except for the $2$-powers). It was proved by Huber-Kings in
[26, Theorem 1.4.1] for Case (1), by Wiles in [58, Theorem 1.6] at odd primes
for Cases (2) and (3), and by Kolster in [46, Theorem A.1] at the prime $2$
for Case (2), respectively. The precise powers of $2$ in Case (2) are
determined by Rognes-Weibel’s computation in [46]. The signs of special values
are determined by the functional equation of $\zeta_{F}$, see [30, page 200].
∎

##  4\. Equivariant algebraic $\mathrm{K}$-theory and Galois descent

In this section, begin by discussing the necessary equivariant homotopy theory
for the results in this paper. We claim no originality nor completeness in the
exposition. We have in mind arithmetic geometers who might not be well-
acquainted with the machinery of this theory and we follow the exposition of
[2]. We will then demonstrate the theory in practice by showing that
$K$-theory satisfies descent on the level of homotopy groups in some cases in
Section 4.3. One of the points of working with spectral Mackey functors is
that these latter statements apply for extension of rings which are not
necessarily Galois but only Galois on the fraction fields.

###  4.1. The definition of spectral Mackey functors

Let $G$ be a finite group. The basic category in genuine equivariant homotopy
theory is the $1$-category $\mathcal{O}\mathrm{rb}_{G}$ of orbits of $G$;
objects are transitive $G$-sets which means that they must be of the form
$G/H$ for some subgroup $H$, while the morphisms are $G$-equivariant
morphisms. This is an enlargement of the basic category in other equivariant
mathematics — $\mathrm{B}G$ is the category with a single object and $G$ worth
of morphisms. Thinking of the object in $\mathrm{B}G$ as the orbit $G/e$, we
have a fully faithful embedding
$\mathrm{B}G\subseteq\mathcal{O}\mathrm{rb}^{\mathrm{op}}_{G}$.

The $\infty$-category of $G$-spaces is defined to be the presheaf
category555This definition is justified by Elemendorf’s theorem [19] where he
proves that the homotopy theory of $G$-CW complexes is equivalent to this
presheaf category.

$\mathcal{S}\mathrm{pc}{}_{G}:=\mathrm{Fun}(\mathcal{O}\mathrm{rb}^{\mathrm{op}}_{G},\mathcal{S}\mathrm{pc}{}).$

Given a $G$-space $\mathcal{X}$, we will write

$\mathcal{X}(G/H)=:\mathcal{X}^{H}.$

This is suggestive notation for what follows. We will extract the following
important pieces of data out of $G$-space.

  1. (1)

the underlying $G$-space
$\mathcal{X}|_{\mathrm{B}G}\in\mathrm{Fun}(\mathrm{B}G,\mathcal{S}\mathrm{pc}{})$
obtained via restriction and the underlying space $\mathcal{X}^{e}$.

  2. (2)

the fixed points space $\mathcal{X}^{G}$.

  3. (3)

In general, we have a morphism in $\mathcal{S}\mathrm{pc}{}$

$\mathcal{X}^{G}\rightarrow\operatorname*{lim}_{\mathrm{B}G}\mathcal{X}|_{\mathrm{B}G}(=:\mathcal{X}^{hG}),$

where the target is the homotopy fixed points space.

The last point highlights a difference between the theory of $G$-spaces and
more “usual” versions of equivariant mathematics: the fixed points are _data_
and we have comparison maps to what one would otherwise guess as the correct
notion of fixed points in homotopy theory. Furthermore, for all subgroups
$H\leq G$, we do have the $H$-fixed points space $\mathcal{X}^{H}$. This
pattern persists as we go to spectra.

This definition suggests that the $\infty$-category of $G$-spectra should be

$\mathrm{Fun}(\mathcal{O}\mathrm{rb}^{\mathrm{op}}_{G},\mathcal{S}\mathrm{pt}{}).$

This is not quite the correct category because objects in it lack certain
transfer maps666These transfer maps are not just novelties for a
$G$-cohomology theory; they imply certain relations between such theories such
as the one explained in Proposition 4.7. expected from a $G$-cohomology
theories, nonetheless it is elementary and also elucidates certain important
constructions that we will encounter. We will denote the above category of
$\mathcal{S}\mathrm{pt}{}_{\mathcal{O}\mathrm{rb}_{G}}$ and refer to it as the
$\infty$-category of $\mathcal{O}\mathrm{rb}_{G}$-_spectra_. Just like in the
spaces situation, we have:

  1. (1)

the underlying $G$-_spectrum_
$\mathcal{X}|_{\mathrm{B}G}\in\mathrm{Fun}(\mathrm{B}G,\mathcal{S}\mathrm{pt}{})$
obtained via restriction and the underlying _spectrum_ $\mathcal{X}^{e}$.

  2. (2)

the fixed points _spectrum_ $\mathcal{X}^{G}$.

  3. (3)

In general, we have a morphism in $\mathcal{S}\mathrm{pt}{}$

$\mathcal{X}^{G}\rightarrow\operatorname*{lim}_{\mathrm{B}G}\mathcal{X}|_{\mathrm{B}G}(=:\mathcal{X}^{hG}),$

where the target is the homotopy fixed points _spectrum_.

Now, let $\mathrm{F}\mathrm{in}_{G}$ be the category of finite $G$-sets; this
is obtained from the category $\mathcal{O}\mathrm{rb}_{G}$ by adjoining
arbitrary finite coproducts. The $\infty$-category of genuine $G$-spectra is
built from the indexing following indexing category: we have the
$(2,1)$-category of $\mathrm{Span}_{G}$ of spans of finite $G$-sets. The
objects are finite $G$-sets $T$ and morphisms are spans of finite $G$-sets:

${S}$${T}$${U,}$$\scriptstyle{f}$$\scriptstyle{g}$

while composition is obtained by pullback. One feature of this category is
that it is semiadditive in that it is pointed admits finite products and
coproducts which coincide. There is a functor

$\mathrm{F}\mathrm{in}_{G}^{\mathrm{op}}\longrightarrow\mathrm{Span}_{G};$

defined informally by

$(T\rightarrow S)\longmapsto(S\leftarrow T\xrightarrow{=}T).$

A spectral Mackey functor (following [2]) is a an additive functor of
$\infty$-categories

$\mathcal{X}:\mathrm{Span}_{G}\rightarrow\mathcal{S}\mathrm{pt}{}.$

As always we write $\mathcal{X}(G/H)=:\mathcal{X}^{H}$ for $H$ a subgroup of
$G$.

Out of this we extract several important pieces of structure that
$\mathcal{X}$ posses:

  1. (1)

the underlying $G$-spectrum
$\mathcal{X}|_{\mathrm{B}G}\in\mathrm{Fun}(\mathrm{B}G,\mathcal{S}\mathrm{pt}{})$
obtained via restriction and the underlying spectrum $\mathcal{X}^{e}$.

  2. (2)

the fixed points spectrum $\mathcal{X}^{G}$.

  3. (3)

In general, we have a morphism in $\mathcal{S}\mathrm{pt}{}$

$\mathcal{X}^{G}\rightarrow\operatorname*{lim}_{\mathrm{B}G}\mathcal{X}|_{\mathrm{B}G}(=:\mathcal{X}^{hG})$

where the target is the homotopy fixed points spectrum.

  4. (4)

given $H^{\prime}\leq H\leq G$, then we have the restriction map

$\mathrm{Res}_{H/H^{\prime}}\colon\mathcal{X}^{H}\longrightarrow\mathcal{X}^{H^{\prime}};$

  5. (5)

and the transfer map

$\mathrm{tr}_{H^{\prime}/H}\colon\mathcal{X}^{H^{\prime}}\longrightarrow\mathcal{X}^{H}$

and $\mathrm{Res}$ and $\mathrm{tr}$ are subject to base change relations, up
to coherent homotopies, governed by the span category; see [2, Summary] for
details.

###  4.2. Spectral Mackey functors from algebraic $\mathrm{K}$-theory

Fix a pseudo $G$-Galois cover $\pi:Y\rightarrow X$; in the situation that we
are interested in, $X$ and $Y$ are both one-dimensional, integral normal
schemes. The goal of this section is to build a spectral Mackey functor in 4.6

$\underline{\mathrm{K}}_{\pi}:\mathrm{Span}_{G}\overset{\mathrm{K}}{\longrightarrow}\mathcal{S}\mathrm{pt}{};$

which takes the following values: for any $H\leq G$, we have

$\underline{\mathrm{K}}_{\pi}(G/H)=\mathrm{K}(\mathrm{Perf}(Y/_{H})).$

For any $H^{\prime}\leq H\leq G$, we have the restriction map

$p^{\ast}\colon\mathrm{K}(\mathrm{Perf}(Y/_{H}))\longrightarrow\mathrm{K}(\mathrm{Perf}(Y/_{H^{\prime}}))$

induced by the canonical map $p:Y/H^{\prime}\to Y/H$ and we have the transfer
map

$p_{\ast}\colon\mathrm{K}(\mathrm{Perf}(Y/_{H^{\prime}}))\longrightarrow\mathrm{K}(\mathrm{Perf}(Y/_{H})).$

These maps will satisfy a coherent system of compatibilities.

######  Remark 4.1.

When $\pi\colon Y\to X$ is a $G$-_Galois_ cover, Barwick and Merling
constructed $\mathrm{K}(Y)$ as a spectral $G$-Mackey functor (genuine
$G$-spectrum) in [2, 37]. The point of this subsection is to extend their
results to pseudo Galois coverings in $1$-dimensional cases.

For the rest of this subsection, we fix $X$ a one-dimensional, noetherian
integral normal scheme, $G$ a finite group and $\pi\colon Y\rightarrow X$ a
pseudo $G$-Galois cover; in particular $Y$ is also one-dimensional, noetherian
and integral. We write $k(Y)$ and $k(X)$ as the fraction fields of $Y$ and
$X$, respectively. Denote by $\widetilde{\pi}\colon k(X)\rightarrow k(Y)$ the
induced map on fraction fields; the latter map is a $G$-Galois extension. For
an intermediate extension $E$ between $k(X)$ and $k(Y)$ such that $k(Y)/E$ is
a $H$-Galois extension, the normalization $X$ into $E$ is isomorphic to the
quotient $Y/_{H}$ by the following:

######  Lemma 4.2.

Let $A\to A^{\prime}$ be a pseudo $G$-Galois extension of integral domains.
Then for any subgroup $H\leq G$, the integral closure of $A$ in
$\mathrm{Frac}(A^{\prime})^{H}$ is $(A^{\prime})^{H}$.

###### Proof.

On one hand, $(A^{\prime})^{H}$ is integral over $A$ as subring of an integral
extension $A^{\prime}$ of $A$. On the other hand, any element
$x\in\mathrm{Frac}(A^{\prime})^{H}\subseteq\mathrm{Frac}(A^{\prime})$ integral
over $A$ is necessarily in $\mathrm{Frac}(A^{\prime})^{H}\cap
A^{\prime}=(A^{\prime})^{H}$, since $A^{\prime}$ is the integral closure of
$A$ in $\mathrm{Frac}(A^{\prime})$. ∎

The usual Galois correspondence furnishes an equivalence between the orbit
category of $G$ and the category of intermediate Galois extensions between
$k(Y)$ and $k(X)$:

$\mathcal{O}\mathrm{rb}_{G}\simeq\mathrm{Gal}_{k(Y)/k(X)},\qquad
G/H\mapsto(k(Y)\times G/H)^{G}\cong k(Y)^{H}.$

The category $\mathrm{Gal}_{k(Y)/k(X)}$ is furthermore equivalent to the
category of intermediate pseudo Galois extensions.

######  Definition 4.3.

Let $\pi\colon Y\rightarrow X$ be a pseudo $G$-Galois extension. Then the
category $\mathrm{PsGal}_{Y/X}$ has as objects noetherian integral normal
schemes $Y\rightarrow W\rightarrow X$ such that $Y\rightarrow W$ is a pseudo
$H$-Galois extension for some $H$ a subgroup of $G$.

######  Lemma 4.4.

We have an equivalence of categories
$\mathrm{PsGal}_{Y/X}\simeq\mathrm{Gal}_{k(Y)/k(X)}$.

###### Proof.

The mutually inverse functors are given by:

$\displaystyle\mathrm{PsGal}_{Y/X}\longrightarrow\mathrm{Gal}_{k(Y)/k(X)}$
$\displaystyle Y^{\prime}\rightarrow Y^{\prime\prime}\longmapsto
k(Y^{\prime})\rightarrow k(Y^{\prime\prime});$
$\displaystyle\mathrm{Gal}_{k(Y)/k(X)}\longrightarrow\mathrm{PsGal}_{Y/X}$
$\displaystyle F\rightarrow
F^{\prime}\longmapsto\widetilde{X}_{F}\rightarrow\widetilde{X}_{F^{\prime}},$

where $\widetilde{X}_{F}$ is the normalization of $X$ into $F$ computed via
Lemma 4.2. ∎

On the other hand, taking finite-coproduct completion everywhere gives
equivalences

$\mathrm{Fin}_{G}\simeq\mathrm{FEt}{}_{k(Y)/k(X)}\simeq\mathrm{PsGal}^{\coprod}_{Y/X}$

Therefore taking spans everywhere gives equivalences

$\mathrm{Span}_{G}\overset{\sim}{\longrightarrow}\mathrm{Span}(\mathrm{FEt}{}_{k(Y)/k(X)})\overset{\sim}{\longrightarrow}\mathrm{Span}(\mathrm{PsGal}^{\coprod}_{Y/X}).$

To define transfer for $\mathrm{K}$-theory in our context, let us recall that
$f\colon Y\rightarrow X$ is a proper morphism of finite tor-dimension, then
the pushforward
$f_{\ast}\colon\mathrm{D}_{\mathrm{qc}}(Y)\rightarrow\mathrm{D}_{\mathrm{qc}}(X)$
preserve perfect complexes [49, Tag 0B6G]

$f_{\ast}\colon\mathrm{Perf}(Y)\longrightarrow\mathrm{Perf}(X);$

whence we have a pushforward on $\mathrm{K}$-theory.

######  Remark 4.5.

Normalization maps do not necessarily have to have finite tor-dimension hence
we are not guaranteed pushforwards on $\mathrm{K}$-theory. In our setting
$Y\rightarrow X$ will be a finite morphism of one-dimensional regular affine
schemes. In particular $Y,X$, and other intermediate schemes are all regular
and all the maps in sight are finite. Therefore, [49, Tag 068B] guarantees
that all morphisms in sight preserves perfect complexes. We use this
observation in the next construction.

######  Construction 4.6.

Fix $\pi:Y\rightarrow X$ a pseudo $G$-Galois cover of one dimensional, regular
affine schemes. We construct a functor

$\mathrm{Perf}:\mathrm{Span}(\mathrm{PsGal}^{\coprod}_{Y/X})\longrightarrow\mathcal{C}\mathrm{at}{}_{\infty}$

To do so, we recall from [2, Corollary D.18.1] that we have a Mackey functor

$\mathrm{Span}(\mathrm{DM},\mathrm{DM}_{\mathrm{FP}},\mathrm{DM})\longrightarrow\mathcal{C}\mathrm{at}{}_{\infty},$

where $\mathrm{DM}_{\mathrm{FP}}$ refers to the class of morphisms of
(spectral) Deligne-Mumford stacks which are strongly proper and morphisms have
finite Tor-amplitude. As explained above, because morphisms $\mathrm{PsGal}$
are all proper and has finite tor-dimension, we have an inclusion of
subcategories

$\mathrm{Span}(\mathrm{PsGal}^{\coprod}_{Y/X})\subset\mathrm{Span}(\mathrm{DM},\mathrm{DM}_{\mathrm{FP}},\mathrm{DM}).$

Therefore, we get a spectral Mackey functor

$\underline{\mathrm{K}}_{\pi}\colon\mathrm{Span}_{G}\overset{\sim}{\longrightarrow}\mathrm{Span}(\mathrm{PsGal}^{\coprod}_{Y/X})\xrightarrow{\mathrm{K}\circ\mathrm{Perf}}\mathcal{S}\mathrm{pt}{}.$

In particular, we can take homotopy groups and obtain abelian-group valued
Mackey functors

$\pi_{i}\underline{\mathrm{K}}_{\pi}\colon\mathrm{Span}_{G}\xrightarrow{\mathrm{K}\circ\mathrm{Perf}}\mathsf{Ab},\qquad
G/H\longmapsto\mathrm{K}_{i}(Y/_{H}).$

###  4.3. (Pseudo-)Galois descent on the level of homotopy groups

In this subsection, we study Galois descent on the level of algebraic
$\mathrm{K}$-_groups_. The case is simpler when the group order is inverted.

######  Proposition 4.7.

Let $\underline{M}$ be a $G$-Mackey functor for a finite group $G$. Then for
any subgroup $H\leq G$, the map
$\mathrm{Res}\colon\mathrm{M}(G/H)[1/|H|]\to\mathrm{M}(G/e)^{H}[1/|H|]$ is an
isomorphism.

###### Proof.

This is a standard result in the theory of (rational) $G$-Mackey functors.
Consider the restriction and the transfer maps:

${\mathrm{Res}\colon\mathrm{M}(G/H)}$${\mathrm{M}(G/e)\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}\mathrm{tr}}$

Their compositions are given by $\mathrm{tr}\circ\mathrm{Res}=|H|$ and
$\mathrm{Res}\circ\mathrm{tr}=\sum_{g\in H}g^{*}$. It is then straightforward
to check that
$\mathrm{Res}\colon\mathrm{M}(G/H)[1/|H|]\to\mathrm{M}(G/e)^{H}[1/|H|]$ is an
isomorphism with inverse $\mathrm{tr}_{*}/|H|$. ∎

Applying Proposition 4.7 to the algebraic $\mathrm{K}$-group Mackey functors,
we obtain:

######  Corollary 4.8.

Let $Y\to X$ be (pseudo-)Galois covering of schemes for a finite group $G$.
Then the map
$i^{*}\colon\mathrm{K}_{*}(X)[1/|G|]\to\mathrm{K}_{*}(Y)^{G}[1/|G|]$ is an
isomorphism.

In some special cases, this isomorphism holds without inverting the group
order.

  * •

finite Galois extensions of finite fields;

  * •

finite pseudo-Galois extensions of affine curves over finite fields or rings
of integers of number fields and $*\geq 1$ is _odd_.

For finite fields, the fixed point statement follows from Quillen’s
computation (3.3) in [42].

######  Proposition 4.9.

For any $m\geq 1$, the induced map of field extension on algebraic
$\mathrm{K}$-groups
$\mathrm{K}_{*}(\mathbb{F}_{q})\to\mathrm{K}_{*}(\mathbb{F}_{q^{m}})$
identifies the source as the Galois fixed points of the target.

###### Proof.

The $*=0$ case follows from the fact
$\mathrm{K}_{0}(\mathbb{F}_{q})=\mathrm{K}_{0}(\mathbb{F}_{q^{m}})=\mathbb{Z}$
with trivial Galois group action. The $*=2n>0$ case is trivial since
$\mathrm{K}_{2n}(\mathbb{F}_{q})=\mathrm{K}_{2n}(\mathbb{F}_{q^{m}})=0$.

When $*=2n-1$, $\mathrm{K}_{2n-1}(\mathbb{F}_{q^{m}})$ is a finite abelian
group whose orders are coprime to $q$ for all $m$. The equalizer sequences
[42]

${\mathrm{BGL}(\mathbb{F}_{q})^{+}}$${\mathrm{BU}}$${\mathrm{BU}}$$\scriptstyle{\psi^{q}}$$\scriptstyle{\operatorname{id}}$

for $\mathbb{F}_{q}$ and $\mathbb{F}_{q^{m}}$ are connected by

${\mathrm{BGL}(\mathbb{F}_{q})^{+}}$${\mathrm{BU}}$${\mathrm{BU}}$${\mathrm{BGL}(\mathbb{F}_{q^{m}})^{+}}$${\mathrm{BU}}$${\mathrm{BU}.}$$\scriptstyle{\psi^{q}-\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\psi^{q^{m-1}}+\cdots+\psi^{q}+\operatorname{id}}$$\scriptstyle{\psi^{q^{m}}-\operatorname{id}}$

The induced maps on the long exact sequences of homotopy groups break up into
maps between short exact sequences:

${0}$${\pi_{2n}(\mathrm{BU})}$${\pi_{2n}(\mathrm{BU})}$${\mathrm{K}_{2n-1}(\mathbb{F}_{q})}$${0}$${0}$${\pi_{2n}(\mathrm{BU})}$${\pi_{2n}(\mathrm{BU})}$${\mathrm{K}_{2n-1}(\mathbb{F}_{q^{m}})}$${0.}$$\scriptstyle{q^{n}-1}$$\scriptstyle{1}$$\scriptstyle{\frac{q^{nm}-1}{q^{n}-1}}$$\scriptstyle{q^{nm}-1}$

This shows that the natural map
$\mathrm{K}_{2n-1}(\mathbb{F}_{q})\to\mathrm{K}_{2n-1}(\mathbb{F}_{q^{m}})$ is
injective. We next show it is the Galois fixed point. As taking fixed points
commutes completion, we can prove the statement by completing each prime
$\ell$ not dividing $q$. Suslin’s theorem identifies
$\mathrm{K}(\overline{\mathbb{F}}_{q})^{\wedge}_{\ell}$ with
$\mathrm{KU}^{\wedge}_{\ell}$ such that Galois actions by the $q$-th power
Frobenius $\mathrm{Fr}$ corresponds to the Adams operation $\psi^{q}$. It
follows that $\mathrm{Fr}$ acts on $\mathrm{K}_{2n-1}(\mathbb{F}_{q^{m}})$ by
multiplication by $q^{nm}$, whose fixed points is then precisely
$\mathrm{K}_{2n-1}(\mathbb{F}_{q})$ from the diagram above. ∎

For affine curves and rings of integers of number fields (Dedekind domains),
the pseudo-Galois descent on $\mathrm{K}$-groups only holds in odd degrees.
For $\mathrm{K}_{1}$, this follows from the Bass-Milnor-Serre’s result in [3].

######  Proposition 4.10.

Let $F^{\prime}/F$ be a finite $G$-Galois extension of global fields,
$A=S^{-1}\mathcal{O}_{F}$ be the ring of $S$-integers in $F$ for a finite set
$S$ of primes in $F$, and $A^{\prime}$ be the integral closure of $A$ in
$F^{\prime}$. Then the induced map
$\mathrm{K}_{1}(A)\to\mathrm{K}_{1}(A^{\prime})$ exhibits the source as the
$G$-fixed points of the target.

###### Proof.

By [3, Corollary 3.4], we have natural isomorphisms $\mathrm{K}_{1}(A)\cong
A^{\times}$. The claim then follows from the facts that the functor
$(-)^{\times}$ is right adjoint to the group ring functor and
$(A^{\prime})^{G}=A$ by construction. ∎

For $\mathrm{K}_{2n-1}$ and $n\geq 2$, the fixed point statement is a result
of the following isomorphism in étale cohomology groups of Dedekind domains.

######  Proposition 4.11.

Let $A\to A^{\prime}$ be an extension of Dedekind domains that is $G$-Galois
on their fraction fields for a finite group $G$. Then for any prime $\ell$
away from the characteristic of $A$ and $A^{\prime}$ and $n\geq 2$, the
natural map

$\mathrm{H}^{1}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))\longrightarrow\mathrm{H}^{1}_{\acute{e}t}(A^{\prime};\mathbb{Z}_{\ell}(n))$

exhibits the source as the $G$-fixed point of the target.

###### Proof.

(See [30, Corollary 2.3 and Proposition 2.9]) Let $C$ (resp. $C^{\prime}$) be
the collection of closed points of $\operatorname{Spec}A$ (resp.
$\operatorname{Spec}A^{\prime}$). Localization and purity in étale cohomology
yield a long exact sequence:

$0=\bigoplus_{x\in
C}\mathrm{H}_{\acute{e}t}^{-1}(\kappa(x);\mathbb{Z}_{\ell}(n-1))\longrightarrow\mathrm{H}^{1}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))\longrightarrow\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A);\mathbb{Z}_{\ell}(n))\longrightarrow\bigoplus_{x\in
C}\mathrm{H}_{\acute{e}t}^{0}(\kappa(x);\mathbb{Z}_{\ell}(n-1)),$

where $\mathrm{Frac}(A)$ is the fraction field of $A$. Since $n\not=1$ the
Galois action on $\mathbb{Z}_{\ell}(n-1)$ on the étale site of $\kappa(x)$ for
all $x$ is nontrivial, we have that
$\mathrm{H}_{\acute{e}t}^{0}(\kappa(x);\mathbb{Z}_{\ell}(n-1))=0$, hence we
have an isomorphism
$\mathrm{H}^{1}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))\cong\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A);\mathbb{Z}_{\ell}(n))$.
As $\mathrm{Frac}(A)\to\mathrm{Frac}(A^{\prime})$ is a finite $G$-Galois
extension of fields, we have a Hochschild-Lyndon-Serre spectral sequence
connecting their étale cohomology groups:

$E_{2}^{s,t}=\mathrm{H}^{s}(G;\mathrm{H}^{t}_{\acute{e}t}(\mathrm{Frac}(A^{\prime});\mathbb{Z}_{\ell}(n)))\Longrightarrow\mathrm{H}^{s+t}_{\acute{e}t}(\mathrm{Frac}(A);\mathbb{Z}_{\ell}(n)).$

By the same reasoning as in $\kappa(x)$,
$\mathrm{H}^{0}_{\acute{e}t}(\mathrm{Frac}(A^{\prime});\mathbb{Z}_{\ell}(n))=0$
whenever $n\neq 0$, hence the edge homomorphism
$\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A);\mathbb{Z}_{\ell}(n))\to\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A^{\prime});\mathbb{Z}_{\ell}(n))^{G}$
becomes an isomorphism. The two isomorphisms assemble into the commutative
diagram below, proving the isomorphism in the statement.

$\leavevmode\hbox to207.63pt{\vbox
to57.12pt{\pgfpicture\makeatletter\hbox{\hskip
103.81563pt\lower-28.61038pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-103.81563pt}{-28.51054pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
30.31093pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.00539pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{H}^{1}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
30.31093pt\hfil&\hfil\hskip 57.2824pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.97688pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{H}^{1}_{\acute{e}t}(A^{\prime};\mathbb{Z}_{\ell}(n))^{G}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
33.28242pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
43.72763pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-39.42209pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A);\mathbb{Z}_{\ell}(n))}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
43.72763pt\hfil&\hfil\hskip 72.08798pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-43.78247pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{H}^{1}_{\acute{e}t}(\mathrm{Frac}(A^{\prime});\mathbb{Z}_{\ell}(n))^{G}.}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
48.08801pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-60.088pt}{2.36255pt}\pgfsys@lineto{-60.088pt}{-15.61311pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-60.088pt}{-15.8131pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-57.73523pt}{-8.10931pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\sim}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-29.57707pt}{11.50003pt}\pgfsys@lineto{21.84523pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.04521pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{55.72762pt}{2.36255pt}\pgfsys@lineto{55.72762pt}{-14.83754pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{55.72762pt}{-15.03752pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{58.08038pt}{-7.72153pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\sim}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-16.16037pt}{-26.01054pt}\pgfsys@lineto{7.03964pt}{-26.01054pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.23962pt}{-26.01054pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.08261pt}{-23.65778pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\sim}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qed$

######  Corollary 4.12.

Suppose that $Y\rightarrow X$ is a pseudo $G$-Galois cover of smooth affine
curves over $\mathbb{F}_{q}$ or for a finite group $G$. Then the induced map
on odd degree algebraic $\mathrm{K}$-groups
$\mathrm{K}_{2n-1}(X)\to\mathrm{K}_{2n-1}(Y)$ exhibits the source as the
$G$-fixed point of the target when $n\geq 1$.

###### Proof.

The $n=1$ case has been proved in Proposition 4.10. Harder’s theorem [24] (see
also [57, Theorem VI.6.1]) states that $\mathrm{K}_{*}(X)$ is a finite abelian
group of order coprime to $q$ when $*\geq 2$. It suffices to prove the
isomorphism after completing at each prime $\ell$ not dividing $q$. By
Rost–Voevodsky and the Thomason spectral sequence, we have natural
isomorphisms for affine curves when $n\geq 2$:

$\mathrm{K}_{2n-1}(X)^{\wedge}_{\ell}\overset{\sim}{\longrightarrow}\mathrm{K}^{\acute{e}t}_{2n-1}(X)^{\wedge}_{\ell}\overset{\sim}{\longrightarrow}\mathrm{H}^{1}_{\acute{e}t}(X;\mathbb{Z}_{\ell}(n)).$

The claim then follows from Proposition 4.11. ∎

There is a similar result in the number field case:

######  Corollary 4.13.

Let $A^{\prime}/A$ be a pseudo $G$-Galois extension of integrally closed
subdomains of number fields,777i.e. rings of $S$-integers for some finite set
of primes $S$ in the number field. then the natural map

  1. (1)

$\mathrm{K}_{2n-1}(A)\to\mathrm{K}_{2n-1}(A^{\prime})$ exhibits the source as
the $G$-fixed points of the target when $n=1$ or both $\mathrm{Frac}(A)$ and
$\mathrm{Frac}(A^{\prime})$ has no real embeddings.

  2. (2)

$\mathrm{K}_{2n-1}(A)[1/2]\to\mathrm{K}_{2n-1}(A^{\prime})[1/2]$ exhibits the
source as the $G$-fixed points of the target for general pseudo Galois
extensions $A^{\prime}/A$.

###### Proof.

Again, the $n=1$ case has been established in Proposition 4.10. By
Rost–Voevodsky and the Thomason spectral sequence, we have a natural
isomorphism when $n\geq 2$ and $\ell$ is odd or $\mathrm{Frac}(A)$ is totally
imaginary (by [46, Theorem 0.4 and Proposition 6.12]):

$\mathrm{K}_{2n-1}(A)^{\wedge}_{\ell}\cong\mathrm{H}^{1}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n)).$

The claim then follows from Proposition 4.11. ∎

The $2$-primary fixed point statement is more complicated to prove.

######  Proposition 4.14.

Suppose the fraction field $F^{\prime}$ of $A^{\prime}$ in Corollary 4.13 is a
_totally real_ number field (then so is $F=\mathrm{Frac}(A)$). Then the
natural map $\mathrm{K}_{2n-1}(A)\to\mathrm{K}_{2n-1}(A^{\prime})$ exhibits
the source as the $G$-fixed point of the target when $n\geq 1$.

###### Proof.

It remains to study the $2$-primary part of the statement when $n\geq 2$. This
was computed by Rognes-Weibel in [46, Theorem 0.6]. Notice that
$\mathrm{K}_{2n-1}(A)^{\wedge}_{2}\cong\mathrm{K}_{2n-1}(F)^{\wedge}_{2}$ by
the localization sequence in algebraic $\mathrm{K}$-theory, so we can assume
$A=\mathcal{O}_{F}[1/2]$ and $A^{\prime}=\mathcal{O}_{F^{\prime}}[1/2]$
without loss of generality. Denote the set of real embeddings of $F$ by
$\mathrm{Emb}(F,\mathbb{R})$. This is a $G$-set of size $r_{1}(F)$. There are
four cases.

  1. (1)

$n=4k>0$. The claim follows the same way as in Corollary 4.13 because of the
natural isomorphisms (the case “$n=8k+7$ of [46, Theorem 0.6]”):

$\mathrm{K}_{8k-1}(\mathcal{O}_{F}[1/2])^{\wedge}_{2}\overset{\sim}{\longrightarrow}\mathrm{H}^{1}_{\acute{e}t}(\mathcal{O}_{F}[1/2];\mathbb{Z}_{2}(4k)).$

  2. (2)

$n=4k+1>1$. We have a trivial extension on the $E_{\infty}$-page of the
motivic spectral sequence (the case “$n=8k+1$ of [46, Theorem 0.6]”):

$0\to\mathbb{Z}_{2}^{\oplus
r_{1}(F)}\longrightarrow\mathrm{K}_{8k+1}(\mathcal{O}_{F})^{\wedge}_{2}\longrightarrow\mathbb{Z}/2\to
0.$

It suffices to prove the fixed point statement on both two summands. The
$G$-action on $\mathbb{Z}/2$ is necessarily trivial and
$(\mathbb{Z}/2)^{G}=\mathbb{Z}/2$.

The $\mathbb{Z}_{2}^{\oplus r_{1}(F)}$-term is coming from
$\left[\mathrm{K}_{8k+1}(\mathbb{R})^{\wedge}_{2}\right]^{\oplus\mathrm{Emb}(F,\mathbb{R})}$.
Hence it is Galois-equivariantly isomorphic to
$\hom_{\mathsf{Set}}\left(\mathrm{Emb}(F,\mathbb{R}),\mathbb{R}\right)$. As
$F^{\prime}/F$ is a $G$-Galois extension, we have an isomorphism of $G$-sets
$\mathrm{Emb}(F^{\prime},\mathbb{R})\cong G\times\mathrm{Emb}(F,\mathbb{R})$.
This implies that $\left[\mathbb{Z}_{2}^{\oplus
r_{1}(F^{\prime})}\right]^{G}\cong\mathbb{Z}_{2}^{\oplus r_{1}(F)}$.

  3. (3)

$n=4k+2$. This is similar to the previous case, except that the extension
problem is non-trivial (the case “$n=8k+3$ of [46, Theorem 0.6]”888In the
cited reference, there is a direct decomposition of the $\mathrm{K}$-group of
interest. However, we do not know how to understand the Galois action on the
summand given by $\mathbb{Z}/2w_{4k+2}(F)$. Instead we have to understand the
Galois action on the $E_{\infty}$-page of the motivic spectral sequence which
is given in [46, Theorems 6.7]. Note that there is a shift in degrees because
this reference speaks about the $2^{\infty}$-torsion groups.). We have the
following diagram with the top and bottom rows exact by [46, Theorems 6.7 and
6.9]:

${0\to(\mathbb{Z}/2)^{\oplus
r_{1}(F)}}$${\mathrm{K}_{8k+3}(\mathcal{O}_{F}[1/2])^{\wedge}_{2}}$${\mathrm{H}^{1}_{\acute{e}t}(\mathcal{O}_{F}[1/2];\mathbb{Z}_{2}(4k+2))\to
0}$${0\to\left[(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})}\right]^{G}}$${\left[\mathrm{K}_{8k+3}(\mathcal{O}_{F^{\prime}}[1/2])^{\wedge}_{2}\right]^{G}}$${\left[\mathrm{H}^{1}_{\acute{e}t}(\mathcal{O}_{F^{\prime}}[1/2];\mathbb{Z}_{2}(4k+2))\right]^{G}}$${\mathrm{H}^{1}(G;(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})})=0}$${0\to(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})}}$${\mathrm{K}_{8k+3}(\mathcal{O}_{F^{\prime}}[1/2])^{\wedge}_{2}}$${\mathrm{H}^{1}_{\acute{e}t}(\mathcal{O}_{F^{\prime}}[1/2];\mathbb{Z}_{2}(4k+2))\to
0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The fixed point statement holds on the étale cohomology part by Proposition
4.11. For the left vertical map, the term $(\mathbb{Z}/2)^{\oplus r_{1}(F)}$
is coming from
$\left[\mathrm{K}_{8k+3}(\mathbb{R})^{\wedge}_{2}\right]^{\oplus\mathrm{Emb}(F,\mathbb{R})}$;
similarly for $F^{\prime}$. This uniform description across all fields yields
a $G$-equivariant isomorphism $(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})}\cong\mathrm{Map}_{\mathsf{Set}}(G,(\mathbb{Z}/2)^{\oplus
r_{1}(F)})$, which implies:

     * •

$(\mathbb{Z}/2)^{\oplus
r_{1}(F)}\overset{\sim}{\longrightarrow}\left[(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})}\right]^{G}$;

     * •

$\mathrm{H}^{1}\left(G;(\mathbb{Z}/2)^{\oplus
r_{1}(F^{\prime})}\right)\cong\mathrm{H}^{1}\left(e;(\mathbb{Z}/2)^{\oplus
r_{1}(F)}\right)=0$ by Shapiro’s Lemma.

The isomorphism
$\mathrm{K}_{8k+3}(\mathcal{O}_{F}[1/2])^{\wedge}_{2}\overset{\sim}{\longrightarrow}\left[\mathrm{K}_{8k+3}(\mathcal{O}_{F}[1/2])^{\wedge}_{2}\right]^{G}$
then follows from the Five Lemma.

  4. (4)

$n=4k+3$. There are natural isomorphisms
$\mathrm{K}_{8k+5}(\mathcal{O}_{F})^{\wedge}_{2}\cong\mathbb{Z}_{2}^{\oplus
r_{1}(F)}\cong\hom_{\mathsf{Set}}\left(\mathrm{Emb}(F,\mathbb{R}),\mathbb{Z}_{2}\right)$
(The case “$n=8k+5$ of [46, Theorem 0.6]”). The claim follows similar to the
$n=4k+1$ case. ∎

######  Remark 4.15.

We note that Galois descent also holds in even dimensional algebraic
$\mathrm{K}$-groups in certain cases. When $A^{\prime}/A$ is a $G$-_Galois_ of
rings of integers of number fields or affine curves over $\mathbb{F}_{q}$,
Kolster proved a “Galois codescent” in [30, Proposition 2.12] of étale
cohomology groups

$\mathrm{H}^{2}_{\acute{e}t}(A^{\prime};\mathbb{Z}_{\ell}(n))_{G}\overset{\sim}{\longrightarrow}\mathrm{H}^{2}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n)),$

where $\ell\neq 2$ in the number field case and $\ell\nmid q$ in the function
field case. As both étale cohomology groups and the Galois group are finite,
taking Pontryagin duals turns codescent into descent:

$\mathrm{H}^{2}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))\overset{\sim}{\longrightarrow}\mathrm{H}^{2}_{\acute{e}t}(A^{\prime};\mathbb{Z}_{\ell}(n))^{G}.$

By Rost–Voevodsky and the motivic spectral sequence, we have natural
isomorphisms when $n\geq 2$

$\mathrm{H}^{2}_{\acute{e}t}(A;\mathbb{Z}_{\ell}(n))\overset{\sim}{\longrightarrow}\mathrm{K}_{2n-2}(A)^{\wedge}_{\ell}.$

This yields the Galois descent on even-degree $\mathrm{K}$-groups when $n\geq
2$

$\mathrm{K}_{2n-2}(A)\overset{\sim}{\longrightarrow}\mathrm{K}_{2n-2}(A^{\prime})^{G},\qquad\text{
(invert 2 in the number field case).}$

Note that this argument fails when $A^{\prime}/A$ is only a _pseudo_ -Galois
extension or when $\ell=2$ in the number field case, because the first step in
proving [30, Proposition 2.12] uses the Tate spectral sequence for Galois
extensions.

##  5\. Equivariant Moore spectra and its cellular filtration

This section is the technical heart of the paper and uses crucially the
construction of a cellular presentation of equivariant Moore spectra due to
Rezk and the second author as in [59, §3.3]. The first key result here is
Corollary 5.10 where the norms of $L$-values are incarnated as the ratio
between Euler numbers of the $E_{1}$ page of a spectral sequence built from a
cellular filtration on equivariant Moore spectra. Turning the page once, we
compute the Bredon cohomology of these Moore spectra with coefficients in
equivariant algebraic $\mathrm{K}$-groups in Section 5.2.

###  5.1. Equivariant Moore spectra of abelian characters

Let’s recall the construction of the equivariant Moore spectra following the
second author’s thesis [59]. If $A$ is an abelian group, then the Moore
spectrum associated to $A$, denoted usually by $\mathrm{M}A$, is characterized
uniquely as a _connective_ spectrum whose homology group is given as follows:

$\pi_{*}(\mathrm{M}A\otimes\mathrm{H}\mathbb{Z})=\mathrm{H}_{*}(\mathrm{M}A;\mathbb{Z})=\begin{cases}A,&*=0;\\\
0,&\text{else}.\end{cases}$

In other words we have an equivalence

$\mathrm{M}A\otimes\mathrm{H}\mathbb{Z}\simeq\mathrm{H}A.$

Note, however, unlike the formation of Eilenberg-MacLane spectra, the
formation of Moore spectra $\mathrm{M}A$ is _not_ functorial in $A$; this
complication presents itself in the second author’s thesis.

######  Definition 5.1.

Let $A$ be an abelian group with an action by a group $G$ via the homomorphism
$\rho\colon G\to\mathrm{Aut}(A)$. An equivariant Moore spectrum of $\rho$, if
it exists, is a $G$-action on the Moore spectrum $\mathrm{M}A$ such that the
induced $G$-action on $\mathrm{H}_{0}(\mathrm{M}A;\mathbb{Z})=A$ is isomorphic
to $\rho$. In other words, $\mathrm{M}(\rho)$ is a lifting

${\mathcal{S}\mathrm{pt}{}}$${\mathrm{B}G}$${\mathsf{Ab}=\mathsf{Ab}^{\\{0\\}}}$${\mathsf{Ab}{Z}}$$\scriptstyle{\mathrm{H}_{*}(-;\mathbb{Z})}$$\scriptstyle{\rho}$$\scriptstyle{\exists\mathrm{M}(\rho)?}$

Any complex representation $\rho\colon G\to\mathrm{GL}_{N}(\mathbb{C})$ of a
finite group $G$ necessarily factors through
$\mathrm{GL}_{N}(\mathcal{O}_{E})$ for some number field $E$; we abusively
call this representation $\rho\colon G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$. An
integral equivariant Moore spectrum $\mathrm{M}(\rho)$ of $\rho$, if it
exists, is a $G$-action on the Moore spectrum of $\mathcal{O}_{E}^{\oplus N}$
such that the induced action on its zeroth homology group is given by
$\rho\colon G\to\mathrm{GL}_{N}(\mathcal{O}_{E})$.

######  Remark 5.2.

Steenrod asked the question whether $\mathrm{M}(\rho)$ always exists and
Carlsson constructed a counterexample for the group $C_{2}\times C_{2}$ in
[14].

######  Remark 5.3.

Even if $\mathrm{M}(\rho)$ exists, the lifting is not unique unless
multiplication by the group order of $G$ is an automorphism of $A$. One
example is the trivial and sign representations of $C_{2}$ on $\mathbb{Z}$.
Let $\sigma$ be the sign representation of $C_{2}$. Then depending on the
parity of $n$, any representation sphere of the form
$\mathrm{S}^{n(1-\sigma)}$ is an equivariant Moore spectrum for $\mathbb{Z}$
with the trivial or the sign $C_{2}$-action. Moreover,
Beaudry–Goerss–Hopkins–Stojanoska showed in [5, Theorem 6.22] that the set of
homotopy classes of $C_{2}$-actions on $\mathrm{S}^{0}$ can be identified as:

$[\mathrm{B}C_{2},\mathrm{B}\mathrm{GL}_{1}(\mathrm{S}^{0})]\overset{\sim}{\longleftarrow}[\mathrm{B}C_{2},\mathrm{BO}]=\widetilde{\mathrm{KO}}^{0}(\mathbb{RP}^{\infty})=\mathbb{Z}_{2}.$

The Atiyah-Segal completion theorem implies that $\mathrm{S}^{n(1-\sigma)}$
corresponds to the element $\pm n\in\mathbb{Z}\subseteq\mathbb{Z}_{2}$ on the
right hand side. Hence, the sets of homotopy classes of $C_{2}$-actions on
$\mathrm{S}^{0}$ that induce trivial and sign representations on
$\mathrm{H}_{0}(\mathrm{S}^{0};\mathbb{Z})$ can be identified with
$2\mathbb{Z}_{2}$ and $1+2\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$,
respectively.

First, we give a construction of integral equivariant Moore spectra for
abelian characters $\chi\colon G\to\mathbb{C}^{\times}$ for a finite group
$G$. This cellular filtration is essential in our identification of norms
special values of $L$-functions with sizes of equivariant algebraic
$\mathrm{K}$-groups.

######  Construction 5.4 (Equivariant Moore spectra for abelian characters,
see also [59, Construction 3.3.2]).

Let $\chi\colon G\to\mathbb{C}^{\times}$ be a complex-valued abelian character
of a finite group $G$. Then there is a unique positive integer $m$ such that
$\chi$ factors as

(5.1) ${\chi\colon
G}$${C_{m}}$${\mathbb{Z}[\zeta_{m}]^{\times}}$${\mathbb{C}^{\times},}$$\scriptstyle{\phi_{\chi}}$$\scriptstyle{\mathcal{O}_{\chi}}$$\scriptstyle{\psi_{m}}$

where $\psi_{m}$ sends a generator of $C_{m}$ to a primitive $m$-th root of
unity $\zeta_{m}$. As a result, it suffices to construct equivariant Moore
spectra for the cyclotomic characters $\psi_{m}\colon
C_{m}\to\mathbb{Z}[\zeta_{m}]^{\times}$. In turn, this is constructed as
follows:

  1. (1)

First, following a suggestion of Rezk, we construct the case of $m=p^{\nu}$ by
setting $\mathrm{M}(\psi_{p^{\nu}})$ to be the cofiber

$C_{p^{\nu}+}\rightarrow
C_{p^{\nu-1}+}\rightarrow\mathrm{M}(\psi_{p^{\nu}})[1];$

where the first map is taken in the $\infty$-category of
$\mathcal{S}\mathrm{pt}{}^{C_{p^{\nu}}}$.

  2. (2)

Second, take the (external) tensor product999For groups $H,H^{\prime}$ there
is an external tensor product functor
$\mathcal{S}\mathrm{pt}{}^{H}\times\mathcal{S}\mathrm{pt}{}^{H^{\prime}}\xrightarrow{\boxtimes}\mathcal{S}\mathrm{pt}{}^{H\times
H^{\prime}}.$

$\mathrm{M}(\psi_{m}):=\boxtimes_{p|m}\mathrm{M}\left(\psi_{p^{\nu_{p}(m)}}\right)\in\mathcal{S}\mathrm{pt}{}^{C_{m}}.$

One model for integral equivariant Moore spectrum associated to $\chi$ would
be $\phi_{\chi}^{*}\mathrm{M}(\psi_{m})$. For technical reasons explained in
the proof later, we will instead set

$\mathrm{M}(\chi):=\mathbb{D}((\phi_{\chi^{-1}})^{*}\mathrm{M}(\psi_{m})),$

where $\mathbb{D}$ is the $G$-equivariant Spanier-Whitehead dual. A priori,
$(\phi_{\chi^{-1}})^{*}\mathrm{M}(\psi_{m})$ is an equivariant Moore spectrum
attached to the character $\mathcal{O}_{\chi^{-1}}$. Taking duals inverts the
character back to $\mathcal{O}_{\chi}$. From this definition, we can see
$\ker\chi$ acts trivially on $\mathrm{M}(\chi)$. This particular choice of
equivariant Moore spectra allows us to avoid discussing transfer maps in
algebraic $\mathrm{K}$-groups in the computations (see Remarks 5.9 and 5.15).

######  Remark 5.5.

Technically, we can also set $\mathrm{M}(\chi)$ to be
$\mathbb{D}((\phi_{\chi})^{*}\mathrm{M}(\psi_{m}))$ without taking the
inverse. As $\chi^{-1}=\overline{\chi}$ is complex conjugate to $\chi$, we
have $L(X,\chi^{-1},1-n)=\overline{L(X,\chi,1-n)}$ by Proposition 2.18 and
Proposition 2.23 — their norms over $\mathbb{Q}$ are the same. So we do not
need to distinguish the integral Moore spectra of $\chi$ and $\chi^{-1}$ for
the purpose of proving Theorem 1.3.

For the purposes of this paper, we will evaluate spectra of the form
$\mathrm{Map}\left(\mathrm{M}(\psi_{m}),\mathrm{E}\right)$, where
$\mathrm{Map}$ denote $G$-equivariant maps in the genuine stable category.
Towards this end we review the skeletal filtration on $\mathrm{M}(\psi_{m})$
or, rather, on
$\Sigma^{\lambda}\mathrm{M}(\psi_{m})=\mathrm{M}(\psi_{m})[\lambda]$ which
first appeared in the second author’s thesis [59, Section 3.3].

######  Construction 5.6 (Equivariant cellular filtration on equivariant Moore
spectra).

Let $\lambda$ be the number of distinct prime factors of $m$. We have a
finite, increasing filtration

$\mathrm{sk}_{0}\mathrm{M}(\psi_{m})[\lambda]\rightarrow\mathrm{sk}_{1}\mathrm{M}(\psi_{m})[\lambda]\cdots\rightarrow\mathrm{sk}_{\lambda}\mathrm{M}(\psi_{m})[\lambda],$

whose graded pieces

$\mathrm{gr}_{j}\mathrm{M}(\psi_{m})[\lambda]:=\mathrm{cofib}\left(\mathrm{sk}_{j-1}\mathrm{M}(\psi_{m})[\lambda]\rightarrow\mathrm{sk}_{j}\mathrm{M}(\psi_{m})\right)[\lambda].$

are given by

  1. (1)

$\mathrm{gr}_{0}\mathrm{M}(\psi_{m})[\lambda]\simeq(C_{m}/C_{p_{1}\cdots
p_{\lambda}})_{+}$.

  2. (2)

For $0<j<\lambda$, $\mathrm{gr}_{j}\mathrm{M}(\psi_{m})[\lambda]$ is
equivalent to

$\left(\coprod_{1\leq
k_{1}<\cdots<k_{\lambda-j}\leq\lambda}C_{m}/C_{p_{k_{1}}\cdots
p_{k_{\lambda-j}}}\right)_{+}[j].$

  3. (3)

There is only one equivariant simplex $C_{m}$ in top dimension $\lambda$, i.e.

$\mathrm{gr}_{\lambda}\mathrm{M}(\psi_{m})[\lambda]\simeq(C_{m})_{+}[\lambda].$

The attaching maps of equivariant simplices are given by quotients of orbits;
we refer to [59, Pages 34-35] for details.

If $\mathrm{E}$ is a $C_{m}$-spectrum, then mapping the above filtration into
$\mathrm{E}$ (and shifting) we then get another finite filtered object

$\mathrm{Map}(\mathrm{sk}_{\lambda}\mathrm{M}(\psi_{m}),\mathrm{E})\rightarrow\mathrm{Map}(\mathrm{sk}_{\lambda-1}\mathrm{M}(\psi_{m}),\mathrm{E})\rightarrow\cdots\rightarrow\mathrm{Map}(\mathrm{sk}_{0}\mathrm{M}(\psi_{m}),\mathrm{E}).$

with graded pieces (in decreasing order)

$\mathrm{E}^{C_{p_{1}\cdots p_{\lambda}}}[\lambda],\cdots,\coprod_{1\leq
k_{1}<\cdots<k_{\lambda-j}\leq\lambda}\mathrm{E}^{C_{p_{k_{1}}\cdots
p_{k_{\lambda-j}}}}[\lambda-j],\cdots,\mathrm{E}.$

Taking equivariant homotopy groups, we obtain an $E_{1}$-spectral sequence
called the _equivariant Atiyah-Hirzebruch spectral sequence_ (eAHSS):

(5.2) $E_{1}^{s,t}=\bigoplus_{1\leq
k_{1}<\cdots<k_{s}\leq\lambda}\pi_{t}\mathrm{E}^{C_{p_{k_{1}}\cdots
p_{k_{s}}}}\Longrightarrow\pi_{s+t}\mathrm{Map}^{C_{m}}\left(\mathrm{M}(\psi_{m}),\mathrm{E}\right),\quad
d_{r}^{s,t}\colon E_{r}^{s,t}\to E_{r}^{s-r,t+r-1}.$

######  Remark 5.7.

The construction of the Moore spectrum $\mathrm{M}(\psi_{m})$ is an
equivariant enhancement of the Moore spectrum
$\mathrm{M}\mathbb{Z}[\zeta_{m}]$ which takes into account the $C_{m}$-action.
More precisely $\mathrm{M}(\psi_{m})^{e}\simeq M\mathbb{Z}[\zeta_{m}]$. By
construction of the skeletal filtration, we can extract a chain complex:

$\mathbb{Z}[C_{m}]\to\bigoplus_{i}\mathbb{Z}[C_{m}/C_{p_{i}}]\to\cdots\to\mathbb{Z}\left[C_{m}/C_{p_{1}\cdots
p_{\lambda}}\right].$

whose homology group concentrated in degree $0$.

######  Remark 5.8.

To our knowledge, the spectral sequence in (5.2) was first constructed by
Davis-Lück in [17]. Note that by [17, Page 236], the $E_{2}$-page of the
spectral sequence (5.2) computes Bredon cohomology with coefficients in
(abelian group valued) Mackey functors:

$E_{2}^{s,t}\cong\mathrm{H}^{s}_{C_{m}}(\mathrm{M}(\psi_{m});\underline{\pi}_{t}\mathrm{E}).$

This is analogous to the story non-equivariantly where the cellular filtration
on $X$ gives an $E_{1}$-refinement of the $E_{2}$-page of the Atiyah-
Hirzebruch spectral sequence for computing the value of an generalized
cohomology theory on $X$.

######  Remark 5.9.

If we had set $\mathrm{M}(\chi):=\phi_{\chi}^{*}\mathrm{M}(\psi_{m})$ without
taking the Spanier-Whitehead dual, we will instead obtain a _homological_
equivariant Atiyah-Hirzebruch spectral sequence with $E_{2}$-page:

$E^{2}_{s,t}\cong\mathrm{H}_{s}^{C_{m}}(\mathrm{M}(\psi_{m});\underline{\pi}_{t}\mathrm{E})\Longrightarrow\pi_{s+t}^{C_{m}}(\mathrm{E}\otimes\mathrm{M}(\psi_{m})).$

######  Corollary 5.10.

Let $\chi\colon C_{m}=\mathrm{Aut}_{X}(Y)\hookrightarrow\mathbb{C}^{\times}$
be a primitive character of the pseudo Galois cover $Y\to X$. Consider the
equivariant Atiyah-Hirzebruch spectral sequence above for the $C_{m}$-spectral
Mackey functor $\mathrm{E}=\mathrm{K}(Y)$:

$E_{1}^{s,t}\Longrightarrow\pi_{t-s}^{C_{m}}(\mathrm{K}(Y)\otimes\mathrm{M}(\chi)).$

Then in the cases of finite and function fields, we have

$\mathrm{Norm}_{\mathbb{Q}[\zeta_{m}]/\mathbb{Q}}L(X,\chi,d-n)=(-1)^{\dim{Q}[(\mathrm{K}_{1}(Y)\oplus\mathbb{Z}_{\mathrm{triv}})\otimes\chi\otimes\mathbb{Q}]^{G}}\cdot\frac{\text{Euler
number }(E^{*,2n-2d}_{1})}{\text{Euler number }(E^{*,2n-1}_{1})}.$

In the case of totally real and abelian number fields where
$X=\operatorname{Spec}\mathcal{O}_{F}$ and
$Y=\operatorname{Spec}\mathcal{O}_{F^{\prime}}$, denote the number of real
embeddings of $F$ by $r_{1}(F)$. We have a precise equality:

$\mathrm{Norm}_{\mathbb{Q}[\zeta_{m}]/\mathbb{Q}}L(X,\chi,1-2n)=((-1)^{n}\cdot
2)^{\phi(m)\cdot r_{1}(F)}\cdot\frac{\text{Euler number
}(E^{*,4n-2}_{1})}{\text{Euler number }(E^{*,4n-1}_{1})}.$

In the case of totally real number fields which are not abelian, then the
equality holds up to possible powers of $2$.

###### Proof.

We showcase the proof of the totally real and abelian field case. The other
cases are simpler.

Set $\mathrm{E}$ to be a spectral Mackey functor of the algebraic
$\mathrm{K}$-theory of the pseudo-Galois covering
$\operatorname{Spec}\mathcal{O}_{F^{\prime}}\to\operatorname{Spec}\mathcal{O}_{F}$,
constructed in 4.6. Then we obtain the $E_{1}$-page of an equivariant Atiyah-
Hirzebruch spectral sequence as in (5.2):

(5.3) $E_{1}^{s,t}=\bigoplus_{1\leq
k_{1}<\cdots<k_{s}\leq\lambda}\mathrm{K}_{t}\left(\mathcal{O}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{s}}}}}\right)\Longrightarrow\pi_{s+t}^{C_{m}}\mathrm{Map}(\mathrm{M}(\psi_{m}),\mathrm{K}(\mathcal{O}_{F^{\prime}})),\leavevmode\nobreak\
d_{r}^{s,t}\colon E_{r}^{s,t}\to E_{r}^{s-r,t+r-1}.$

Notice $F^{\prime}/F$ is $C_{m}$-Galois extension of totally real number
fields, we have $r_{1}((F^{\prime})^{H})=r_{1}(F^{\prime})/\\#H$ for any
subgroup $H\leq C_{m}$. Combining (2.6), (3.5), and (5.3), we have:
$\displaystyle\mathrm{Norm}_{\mathbb{Q}[\mathrm{Im}\chi]/\mathbb{Q}}L(\mathcal{O}_{F},\chi,1-2n)$
$\displaystyle=$ $\displaystyle\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\zeta_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}(1-2n)\right)^{(-1)^{j}}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\left(2\cdot(-1)^{n}\right)^{r_{1}\left((F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}\right)}\cdot\frac{\\#\mathrm{K}_{4n-2}\left(\mathcal{O}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}\right)}{\\#\mathrm{K}_{4n-1}\left(\mathcal{O}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}\right)}\right)^{(-1)^{j}}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\left((-1)^{n}\cdot
2\right)^{r_{1}(F^{\prime})/p_{k_{1}}\cdots
p_{k_{j}}}\right)^{(-1)^{j}}\cdot\frac{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{4n-2}\left(\mathcal{O}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}\right)\right)^{(-1)^{j}}}{\prod_{j=0}^{\lambda}\left(\prod_{1\leq
k_{1}<\cdots<k_{j}\leq\lambda}\\#\mathrm{K}_{4n-1}\left(\mathcal{O}_{(F^{\prime})^{C_{p_{k_{1}}\cdots
p_{k_{j}}}}}\right)\right)^{(-1)^{j}}}$ $\displaystyle=$
$\displaystyle\left((-1)^{n}\cdot
2\right)^{r_{1}(F)\phi(m)}\cdot\frac{\text{Euler number
}(E^{*,4n-2}_{1},d_{1})}{\text{Euler number }(E^{*,4n-1}_{1},d_{1})}.$ The
proofs for the other cases are almost the same, except for signs in the finite
and function field cases. Those will be postponed to Proposition 6.6. ∎

###  5.2. Bredon cohomology of equivariant Moore spectra of abelian characters

From Corollary 5.10, the mere existence of the cellular $E_{1}$-spectral
sequence gives an interpretation of the norm of special values of these
$L$-functions in terms of Euler numbers of the said spectral sequence. In this
subsection, we compute these Bredon cohomology groups. The key point is the
already established structure results of these Mackey functors — namely that
they are weak fixed point Mackey functors as proved in Section 4.3.

The equivariant Moore spectra admits a cell structure whose cells are
concentrated in non-positive (homological) degrees. Nonetheless, we show that
the cohomology with coefficients in an arbitrary Mackey functor is
concentrated only degrees $0$ and $-1$. We will employ the following rather
non-standard definition: a weak fixed point Mackey functor is an