#  $v_{1}$-periodic motivic homotopy over prime fields

Hana Jia Kong  Institute for Advanced Study<EMAIL_ADDRESS>and  J.D.
Quigley  Max Planck Institute for Mathematics<EMAIL_ADDRESS>
###### Abstract.

We compute the motivic stable homotopy groups of a variant of the connective
image-of-$J$ spectrum over prime fields of characteristic not two. Together
with the analogous computation over algebraically closed fields, this yields
information about $v_{1}$-periodic motivic stable homotopy groups over
arbitrary base fields of characteristic not two.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Coefficient rings of $H\mathbb{Z}/2^{n}$
  4. 4 Coefficient rings of $kq$
  5. 5 Coefficient rings of $L$ over prime fields
  6. A Figures

##  1\. Introduction

Motivic stable homotopy theory, or the stable homotopy theory of algebraic
varieties, was developed by Morel and Voevodsky in [MV99] to apply powerful
techniques from stable homotopy theory to problems in algebraic geometry and
number theory. Since motivic stable homotopy groups are the universal stable
$\mathbb{A}^{1}$-invariant, a great deal of effort has gone into understanding
the _motivic stable stems_ , i.e., the motivic stable homotopy groups of the
unit in the motivic stable homotopy category. This paper studies certain
patterns in an approximation to the motivic stable stems over prime fields of
characteristic not two. Using analogous results over algebraically closed
fields, this establishes the existence of similar patterns over arbitrary base
fields of characteristic not two.

To situate our results, we briefly survey the existing analyses of the motivic
stable stems. Over arbitrary base fields, the low-dimensional Milnor–Witt
stems111The $n$-th Milnor–Witt stem is the sum over $i\in\mathbb{Z}$ of
$\pi_{n+i,i}^{F}(\mathbb{S})$, where $\mathbb{S}$ is the unit. See Section 1.4
for our indexing conventions on motivic stable homotopy groups. are connected
to Milnor–Witt K-theory [Mor12] and Milnor K-theory, Hermitian K-theory, and
motivic cohomology [RSØ19, RSØ21]. Over certain base fields, various
completions of the motivic stable stems have been computed in larger ranges
[BCQ21, BI22, DI10, DI17b, Isa19, IWX20, OØ14, Wil16, WØ17].

The most well-studied pattern in the motivic stable stems is
$\eta$-periodicity, where $\eta$ is the first motivic Hopf map. The
$\eta$-periodic motivic stable stems have been computed over various base
fields [AM17, CQ21, GI15, GI16, OR20, Wil18]. Over general base fields and
Dedekind domains, the $\eta$-periodic sphere spectrum sits in a fiber sequence
with connective Witt theory [BH20, Bac22].

The pattern we study in this paper, $v_{1}$-periodicity, is much less well-
understood. The $v_{1}$-periodic motivic stable stems have been computed over
the algebraically closed fields of characteristic zero [CQ21], and a small
number of $v_{1}$-periodic families have been produced over general base
fields [Qui21a, Qui21b]. In [BIK22], the coefficients of a motivic spectrum
$L$ which captures the $v_{1}$-periodic phenomena in the motivic stable stems
(cf. 1.1) are computed over the complex numbers and the real numbers.

Our main result (A) is a computation of the coefficients of $L$ over all prime
fields of characteristic not two and all algebraically closed fields. Since
any field sits between a prime field and an algebraically closed field in a
sequence of field extensions, our computations identify the $v_{1}$-periodic
phenomena which is present over every base field of characteristic not two.

###  1.1. Summary of results

From now on, we work in the $2$-complete setting and only work over base
fields $F$ of characteristic not two. In [BH20], Bachmann and Hopkins defined
Adams operations on the very effective cover of Hermitian K-theory [ARØ20]

$\psi^{3}:kq\to kq.$

Following [BIK22], we define a motivic spectrum $L$ by

$L:=\operatorname{fib}(\psi^{3}-1:kq\to kq).$

######  Remark 1.1.

In classical stable homotopy theory, the $v_{1}$-periodic stable stems are
captured by the connective cover of the $K(1)$-local sphere, $\tau_{\geq
0}L_{K(1)}\mathbb{S}$, where $L_{K(1)}$ denotes Bousfield localization with
respect to the first Morava K-theory. It is believed that $L$ is a motivic
analogue of a slight variant of $\tau_{\geq 0}L_{K(1)}\mathbb{S}$, and thus
the coefficients $\pi_{**}^{F}(L)$ can be thought of as the “$v_{1}$-periodic
$F$-motivic stable stems.” In fact, over $F=\mathbb{C}$, the Betti realization
of $L$ is equivalent to $\tau_{\geq 0}L_{K(1)}\mathbb{S}$ above degree
$2$.222The motivic image-of-$J$ spectrum $j$ (see 2.1) is equivalent in all
degrees, but for some technical reasons explained in [BIK22, Introduction], we
will work with $L$ instead of $j$. Moreover, over algebraically closed fields
$F=\bar{F}$ of characteristic zero, the groups $\pi_{**}^{F}(L)$ (computed in
[BIK22]) coincide with the $v_{1}$-periodic $F$-motivic stable stems (computed
in [CQ21]) above topological degree $2$.

Our main result is the following:

######  Theorem A.

The groups $\pi_{**}^{F}(L)$ are described in Section 5 for the following base
fields $F$:

  1. (1)

For $F=\bar{F}$ algebraically closed in Section 5.1.

  2. (2)

For $F=\mathbb{F}_{q}$ a finite field of characteristic not two in Section
5.2.

  3. (3)

For $F=\mathbb{Q}_{q}$ in Section 5.3 ($q$ odd) and Section 5.4 ($q=2$).

  4. (4)

For $F=\mathbb{R}$ in Section 5.5.

  5. (5)

For $F=\mathbb{Q}$ in Section 5.6.

######  Remark 1.2.

The groups $\pi_{**}^{F}(L)$ were already computed for $F=\mathbb{C}$ and
$F=\mathbb{R}$ by Belmont, Isaksen, and the first author in [BIK22], so our
new contribution is the computation over prime fields and the $q$-adic
rationals.

One simple consequence of our computations is that a famous pattern of
$2$-torsion in the classical stable stems sits in $\pi_{**}^{F}(L)$ for all
fields $F$.

######  Corollary 1.3.

Let $F$ be any field of characteristic not two. Then $\pi_{4k-1,2k}^{F}(L)$
contains a summand of order at least the $2$-component of
$\operatorname{denom}\left(\frac{B_{2k}}{4k}\right)$, where $B_{2k}$ is the
$2k$-th Bernoulli number.

###### Proof.

Any field $F$ sits in a sequence of field extensions

$k\to F\to\bar{F}$

between a prime field $k$ and its algebraic closure $\bar{F}$. Inspecting the
results of Section 5, we see that the corollary holds for $k$ and $\bar{F}$,
and moreover, the relevant summands in $k$ base change to the corresponding
summands in $\bar{F}$. It follows by naturality that they must be nonzero in
$F$. ∎

######  Remark 1.4.

More generally, the groups $\pi_{*}(\tau_{\geq 0}L_{K(1)}\mathbb{S})$ appear
as subgroups of $\pi_{**}^{F}(L)$ in bidegrees approximately of the form
$(2n,n)$ for all primes fields and algebraically closed fields. Therefore the
classical $v_{1}$-periodic stable stems sit interestingly inside
$\pi_{**}^{F}(L)$ for all fields of characteristic not two. In particular, we
recover the $v_{1}$-periodic elements constructed in [Qui21a, Qui21b].

1.3 and 1.4 highlight some features of $v_{1}$-periodic motivic stable
homotopy theory which are independent of the base field. We include the
following remark to illustrate an aspect of the computation which does depend
on the base field.

######  Remark 1.5.

As we will explain below, we compute $\pi_{**}^{F}(L)$ using the effective
slice spectral sequence. When $F=\mathbb{C}$, there is an important family of
$d_{1}$-differentials

$d_{1}(\iota v_{1}^{2}\tau^{n})=\iota\tau^{n+1}h_{1}^{3}$

which occur for all $n\geq 0$. However, when $F=\mathbb{F}_{q}$, the similar
differential $d_{1}(\iota x_{q}v_{1}^{2}\tau^{n})$ occurs if and only if

$\begin{cases}\nu(q-1)+\nu(n+1)>3\quad&\text{ if }q\equiv 1\mod 4,\\\
\text{unconditionally}\quad&\text{ if }q\equiv 3\mod 4\text{ and }n\equiv
0\mod 2,\\\ \nu(q^{2}-1)+\nu(n+1)>4\quad&\text{ if }q\equiv 3\mod 4\text{ and
}n\equiv 1\mod 2,\end{cases}$

where $\nu(-)$ denotes dyadic valuation. Further discussion appears in Section
5.2.

We compute $\pi_{**}^{F}(L)$ over prime fields using the effective slice
spectral sequence (ESSS), a powerful tool which has been used to great effect
in recent years (e.g., [KRØ20, RØ16, RSØ19, RSØ21]). The $E_{1}$-term of the
ESSS for $L$ is comprised of shifted copies of
$\pi_{**}^{F}(H\mathbb{Z}/2^{n})$ for varying $n$. The coefficient rings of
$H\mathbb{Z}/2$ and $H\mathbb{Z}$ have been recorded over prime fields [Hil11,
Orm11, OØ13, Kyl15] (see Section 2 for a summary), but as far as we are aware,
the coefficient rings of $H\mathbb{Z}/2^{n}$ have not appeared in previous
literature.

######  Theorem B.

The groups $\pi_{**}^{F}(H\mathbb{Z}/2^{n})$ are described for all $n\geq 1$
and all
$F\in\\{\bar{F},\mathbb{F}_{q},\mathbb{Q}_{q},\mathbb{R},\mathbb{Q}\\}$ in
Section 3.

Even with these groups in hand, describing the $E_{1}$-term of the ESSS for
$L$ in a manner suitable for computations is a difficult problem. One novel
aspect of our work is the concise graphical calculus developed in Section 4
and Section 5. Previous computations over $\mathbb{F}_{q}$ (e.g., [Kyl15,
WØ17]) or $\mathbb{Q}_{q}$ (e.g., [OØ13]) often use different charts to
display information for different congruence classes of $q$, while
computations over $\mathbb{R}$ (e.g., [BI22, DI17a, BIK22]) usually use
different charts for different Milnor–Witt stems. In our work, however, we are
able to display complicated data, like the $E_{1}$-page of the ESSS for $L$
over $\mathbb{F}_{q}$, $q$ odd, in a single chart (Figure 13).333Roughly
speaking, the tradeoff is that instead of using multiple charts containing a
small variety of symbols, we use a single chart containing a large variety of
symbols.

Once we have described the $E_{1}$-term of the ESSS for $L$, we are tasked
with computing differentials. In all of the cases we consider, the
differentials are forced by comparison with the ESSS for $L$ over
algebraically closed fields, the ESSS for $kq$ over the given prime field, or
in the case $F=\mathbb{Q}$, comparison with the ESSS for $L$ over $\mathbb{R}$
and $\mathbb{Q}_{q}$.

######  Theorem C.

The effective slice spectral sequence for $kq$ is explicitly described in
Section 4 for the following base fields:

  1. (1)

For $F=\bar{F}$ algebraically closed in Section 4.1.

  2. (2)

For $F=\mathbb{F}_{q}$ a finite field in Section 4.2.

  3. (3)

For $F=\mathbb{Q}_{q}$ in Section 4.3 ($q$ odd) and Section 4.4 ($q=2$).

  4. (4)

For $F=\mathbb{R}$ in Section 4.5.

  5. (5)

For $F=\mathbb{Q}$ in Section 4.6.

######  Remark 1.6.

The effective slice spectral sequences for $kq$ over $F=\bar{F}$ and
$F=\mathbb{R}$ were already described in [BIK22]. In fact, [RØ16], Röndigs and
Østvær expressed the differentials in the ESSS for $kq$ in terms of Steenrod
operations over any field of characteristic not two. C provides the more
explicit analysis in terms of generators and relations needed to analyze the
ESSS for $L$.

While describing _how_ the differentials in the ESSS for $L$ are produced is
easy, actually determining _which_ differentials occur and describing the
pattern intelligibly is difficult. As we alluded to in 1.5, the characteristic
of the base field (or its residue field) determines whether or not certain
$d_{1}$-differentials occur. Our approach is to find arithmetic conditions
under which differentials occur, and then describe the groups in the next page
of the ESSS in terms of these conditions. We refer the reader to Section 5.2
for examples and further discussion.

######  Remark 1.7.

One consequence of our analysis is that the Hasse map

$\pi_{**}^{\mathbb{Q}}(E)\to\prod_{\nu}\pi_{**}^{\mathbb{Q}_{\nu}}(E),$

where $\nu$ ranges over all places, is injective for $E=kq$ and $E=L$. In the
terminology of Ormsby–Østvær [OØ13], $kq$ and $L$ satisfy the motivic Hasse
principle. In [BCQ21], Balderrama, Culver, and the second author showed that
the $E_{2}$-term of the motivic Adams spectral sequence converging to the
motivic stable stems satisfies the motivic Hasse principle. Since $kq$ and $L$
serve as first approximations to the motivic sphere spectrum, it is
interesting to wonder if the motivic sphere spectrum itself might satisfy the
motivic Hasse principle.

######  Remark 1.8.

In [OØ13, Rmk. 5.14], Ormsby–Østvær explain how their computation of
$\pi_{**}^{\mathbb{Q}}BPGL\langle 1\rangle$ and [OØ13, Lem. 2.9] can be used
to recover the Rognes–Weibel computation of the $2$-complete algebraic
K-theory of $\mathbb{Q}$ from [RW00]. Similarly, our computations can be used
to recover the $2$-complete Hermitian K-theory of $\bar{F}$, $\mathbb{F}_{q}$,
$\mathbb{Q}_{q}$, $\mathbb{R}$, and $\mathbb{Q}$ using [Kyl15, Lem. 1.5.12]
which relates $\pi_{**}kq$ to $\pi_{**}KQ$. For example, using C, we can
recover some of the Hermitian K-theory computations of Friedlander [Fri76] and
Berrick–Karoubi [BK05].

###  1.2. Forthcoming work

In a forthcoming companion paper [KQ22], we analyze the coefficients of $L$
over rings of $S$-integers in number fields, where $S$ contains the
archimedean and dyadic places. In contrast with the explicit computations
carried out in this paper, which mirror the generator-and-relation approach
employed in [BIK22], our approach in [KQ22] takes a more abstract approach,
identifying $\pi_{**}^{\mathcal{O}_{F,S}}L$ in terms of arithmetic invariants
like motivic cohomology and Steenrod operations. This approach builds on the
ideas introduced by Röndigs–Spitzweck–Østvær in their computations of the
first and second motivic stable stems [RSØ19, RSØ21] and
Kylling–Röndigs–Østvær in their analysis of the Hermitian K-theory of Dedekind
domains [KRØ20]. Based on the computations in _loc. cit._ , we expect that the
special values of Dedekind zeta functions can naturally be related to the
coefficients of $L$ using this approach.

###  1.3. Outline

In Section 2, we recall the spectrum $L$ and its essential properties. We also
recall the coefficients of $H\mathbb{Z}/2$ and $H\mathbb{Z}$ over our fields
of interest.

In Section 3, we compute the coefficient rings of $H\mathbb{Z}/2^{n}$ for all
$n\geq 1$ over the relevant fields. Our approach using the Adams spectral
sequence may be of independent interest.

In Section 4, we recall the ESSS for $kq$ over algebraically closed fields and
the reals from [BIK22], and explicitly describe the ESSS for $kq$ over prime
fields and the $q$-adic rationals. The results in this section are used in
Section 5 to produce some differentials in the next section.

In Section 5, we recall the ESSS for $L$ over algebraically closed fields and
the reals from [BIK22], and analyze the ESSS for $L$ over prime fields and the
$q$-adic rationals.

###  1.4. Conventions

  1. (1)

We implicitly work in the $2$-complete setting.

  2. (2)

We work over fields of characteristic not two. In particular, $\mathbb{F}_{q}$
always refers to a finite field of odd characteristic.

  3. (3)

Motivic stable homotopy groups are bigraded in the form $(s,w)$, where $s$
denotes the stem and $w$ denotes the motivic weight.

  4. (4)

We use the abbreviation ‘ESSS’ for the effective slice spectral sequence.

  5. (5)

We write $s_{*}(X)$ for the slices of a motivic spectrum $X$.

  6. (6)

The horizontal axis in ESSS charts is always the stem $s$ and the vertical
axis is always the “Adams–Novikov filtration” $f$, which is twice the
effective slice filtration minus the stem.

  7. (7)

$E_{r}^{F}(X)$ denotes the $E_{r}$-page of the ESSS for an $F$-motivic
spectrum $X$. If the field is understood, we will sometimes suppress it from
the notation.

  8. (8)

We take the elements $\mathsf{h}$ and $\rho$ to be as defined in the
conventions of [BIK22].

###  1.5. Acknowledgments

The authors thank Oliver Röndigs for discussions related to [KRØ20] and a
reminder about the convergence of the ESSS. The first author was supported by
National Science Foundation grant DMS-1926686. The second author is grateful
to the Max Planck Institute for Mathematics in Bonn for its hospitality and
financial support.

##  2\. Background

In this section, we provide the background necessary for our computations. The
motivic spectrum $L$ and its essential properties are recalled in Section 2.1,
and the coefficients of the Eilenberg–MacLane spectra $H\mathbb{Z}/2$ and
$H\mathbb{Z}$ over our fields of interest are recalled in Section 2.2.

###  2.1. The motivic spectrum $L$ and its slices

In [BH20], Bachmann and Hopkins constructed a unital ring map

$\psi^{3}:kq\left[\frac{1}{3}\right]\to kq\left[\frac{1}{3}\right]$

whose Betti realization is the classical Adams operation $\psi^{3}$. We define

$L:=\operatorname{fib}\left(\psi^{3}-1:kq\left[\frac{1}{3}\right]\to
kq\left[\frac{1}{3}\right]\right).$

We will use the effective slice spectral sequence (ESSS) [Lev08, RSØ19]

$E_{1}^{s,f,w}(X)=\pi_{s,w}(s_{\frac{s+f}{2}}(X))\Rightarrow\pi_{s,w}(X)$

to study $\pi_{**}^{F}L$ for various base fields $F$. The differentials in the
ESSS have the form

$d_{r}:E_{r}^{s,f,w}\to E_{r}^{s-1,f+2r-1,w},$

cf. [BIK22, Thm. 2.7].

The $E_{1}$-page of the ESSS for $L$ can be computed using the fiber sequence
of slices

$s_{*}L\to s_{*}kq\xrightarrow{\psi^{3}-1}s_{*}kq,$

which implies

$\pi_{**}s_{*}L\cong K\oplus\Sigma^{-1}C$

where

$K:=\operatorname{ker}(\pi_{**}(\psi^{3}-1))\quad\text{ and }\quad
C:=\operatorname{coker}(\pi_{**}(\psi^{3}-1)).$

As in [BIK22], we will write $\iota x$ for the image of $x\in\Sigma^{-1}C$
under the inclusion into $\pi_{**}s_{*}L$.

We use the grading convention $(s,f_{AN},w)$ as in [BIK22]. Here $s$ is the
stem, $w$ is the weight, and $f_{AN}$ is the “Adams–Novikov filtration”, which
equals twice the stem minus the slice filtration $2f-s$.

By [ARØ20, Thm. 17], the slices of $kq$ can be expressed by the formula

(1) $s_{*}kq=H\mathbb{Z}[h_{1},v_{1}^{2}]/(2h_{1}),$

where $|v_{1}^{2}|=(4,0,2)$ and $|h_{1}|=(1,1,1)$. We explain the meaning of
this formula. In $H\mathbb{Z}[h_{1},v_{1}^{2}]/(2h_{1})$, a monomial of tri-
degree $(s,f_{AN},w)$ contributes a summand of the motivic Eilenberg–MacLane
spectrum $\Sigma^{s,w}HA$ to the $w$th slice $s_{w}kq$.

######  Remark 2.1.

In [BH20], Bachmann and Hopkins showed that the map $\psi^{3}-1$ factors

(2) $kq\xrightarrow{\psi^{3}-1}\Sigma^{4,2}ksp$

where the target is the very effective cover of $\Sigma^{4,2}kq$. After
localizing at $2$, the fiber of (2) defines a motivic spectrum $j$ that is
analogous to the classical image-of-$J$ spectrum.

It can be shown that the slices of $j$ can be expressed by

$s_{*}j=H\mathbb{Z}\otimes\\{\alpha\text{-family classes}\\}$

where “$\alpha\text{-family}$ classes” refers to the classical Adams–Novikov
classes (at the prime $2$) in filtration $1$ and their $\alpha_{1}$-power
multiples.

The slices of $L$ and $j$ differs by a tower of suspensions of
$H\mathbb{F}_{2}$, and in slice filtrations $-1$ and $2$. More precisely, the
very effective cover functor gives $\Sigma^{4,2}ksp\to kq$, which induces a
map $j\to L.$ The difference between $s_{*}L$ and $s_{*}j$ is given by

$s_{*}\operatorname{cofib}(j\to
L)=H\mathbb{Z}\otimes\Big{(}\mathbb{Z}[h_{1}]/(2h_{1})\\{\iota\\}\oplus\mathbb{F}_{2}\\{\bar{\alpha}_{2}\\}\Big{)}$

where $|\iota|=(-1,1,0)$ and $|\bar{\alpha}_{2}|=(3,1,2).$

One can compute the coefficients of $j$ over prime fields using the techniques
in this manuscript, but the computation is slightly messier in low degrees.

######  Theorem 2.2.

Over the base fields
$F=\bar{F},\mathbb{F}_{q},\mathbb{Q}_{q},\mathbb{R},\mathbb{Q}$, the slice
spectral sequence for $L$ and the slice spectral sequence for $kq$ converge
strongly to $\pi_{**}^{F}(L_{2}^{\wedge})$ and
$\pi_{**}^{F}(kq_{2}^{\wedge})$.

###### Proof.

The case for $\mathbb{R}$ is proved in [BIK22]. By [BIK22], the limit of the
$2$-completed slice tower of $L$ is $(2,\eta)$-completed $L$. Over the
specified base fields, the $(2,\eta)$-completions of $L$ and $kq$ coincide
with the $2$-completions of $L$ and $kq$ by [HKO11] (in characteristic zero)
and [WØ17, Prop. 5.10] (in positive characteristic). The result then follows
from [Boa99, Thm. 7.1] and the calculations in §4 for $kq$ and §5 for $L$. ∎

###  2.2. Coefficient rings of $H\mathbb{Z}/2$ and $H\mathbb{Z}$

For future reference, we record the coefficient rings of $H\mathbb{Z}/2$ and
$H\mathbb{Z}$ over various base fields in this section. We refer the reader to
[IØ20, Sec. 2.1] for a summary of how $\pi_{**}^{F}(H\mathbb{Z}/2)$ can be
computed using Milnor K-theory [Mil70] and the Bloch–Kato Conjecture [Voe03,
Voe11]. All of the computations of $\pi_{**}^{F}(H\mathbb{Z}/2)$ appear in
[IØ20], except the case $F=\mathbb{Q}$ which we pull from [OØ13]. Once
$\pi_{**}^{F}(H\mathbb{Z}/2)$ is known, the groups $\pi_{**}^{F}(H\mathbb{Z})$
can be computed using the $\rho$-Bockstein spectral sequence [Hil11] and
motivic Adams spectral sequence [DI10], or by consulting previous motivic
cohomology computations.

####  2.2.1. Algebraically closed fields

These results follow, for instance, from [DI10]. Let $F=\bar{F}$ be an
algebraically closed field. Then we have

$\pi_{**}^{F}(H\mathbb{Z}/2)\cong\mathbb{Z}/2[\tau],\quad\pi_{**}^{F}(H\mathbb{Z})\cong\mathbb{Z}_{2}[\tau],$

where $|\tau|=(0,-1)$.

####  2.2.2. Finite fields

These results appear, for instance, in [Kyl15]. We have

$\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z}/2)\cong\begin{cases}\mathbb{Z}/2[\tau,u]/(u^{2})\quad&\text{
if }q\equiv 1\mod 4,\\\ \mathbb{Z}/2[\tau,\rho]/(\rho^{2})\quad&\text{ if
}q\equiv 3\mod 4,\end{cases}$

and

$\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\cong\begin{cases}\mathbb{Z}\\{1\\}\oplus\bigoplus_{i\geq
0}\mathbb{Z}/2^{\nu(q-1)+\nu(i+1)}\\{u\tau^{i}\\}\quad&\text{ if }q\equiv
1\mod 4,\\\ \mathbb{Z}\\{1\\}\oplus\bigoplus_{i\geq 0,\
even}\mathbb{Z}/2\\{\rho\tau^{i}\\}\\\ \oplus\bigoplus_{i\geq 0,\
odd}\mathbb{Z}/2^{\nu(q^{2}-1)+\nu(i+1)-1}\\{\rho\tau^{i}\\}\quad&\text{ if
}q\equiv 3\mod 4,\end{cases}$

where $|\tau|=(0,-1)$ and $|u|=|\rho|=(-1,-1)$.

In order to make our analysis of finite fields independent of the congruence
class of $q$, we make the following definition.

######  Definition 2.3.

For all $i\geq 0$ and odd prime powers $q$, we define

$s_{q}(i):=\begin{cases}\nu(q-1)+\nu(i+1),\quad&\text{ if }q\equiv 1\mod 4,\\\
1\quad&\text{ if }q\equiv 3\mod 4\text{ and }i\equiv 0\mod 2,\\\
\nu(q^{2}-1)+\nu(i+1)-1\quad&\text{ if }q\equiv 3\mod 4\text{ and }i\equiv
1\mod 2.\end{cases}$

With this notation, we have for all odd prime powers $q$ that

$\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\cong\mathbb{Z}\\{1\\}\oplus\bigoplus_{i\geq
0}\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}\tau^{i}\\},$

where $x_{q}=u$ if $q\equiv 1\mod 4$ and $x_{q}=\rho$ if $q\equiv 3\mod 4$.

####  2.2.3. $\mathbb{Q}_{q}$ with $q$ odd

These results appear in [Orm11]. We have

$\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z}/2)\cong\begin{cases}\mathbb{Z}/2[\tau,\pi,u]/(\pi^{2},u^{2})\quad&\text{
if }q\equiv 1\mod 4,\\\
\mathbb{Z}/2[\tau,\pi,\rho]/(\rho^{2},\rho\pi+\pi^{2})\quad&\text{ if }q\equiv
3\mod 4\end{cases}$

where $|\tau|=(0,-1)$, $|\pi|=|u|=|\rho|=(-1,-1)$. The homotopy groups of
$H\mathbb{Z}$ over $\mathbb{Q}_{q}$, $q$ odd, appear in [Orm11, Thm. 5.8]:

$\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z})\cong\begin{cases}\mathbb{Z}\\{1\\}\quad&\text{
if }(*,*)=(0,0),\\\
\mathbb{Z}\\{\pi\\}\oplus\mathbb{Z}/2^{s_{q}(0)}\\{x_{q}\\}\quad&\text{ if
}(*,*)=(-1,-1),\\\ \mathbb{Z}/2^{s_{q}(0)}\\{x_{q}\pi\\}\quad&\text{ if
}(*,*)=(-2,-2),\\\
\mathbb{Z}/2^{s_{q}(i-1)}\\{\pi^{\epsilon-1}x_{q}\tau^{i-1}\\}\quad&\text{ if
}(*,*)=(-\epsilon,-i+1-\epsilon),\ i\geq 2,\ \epsilon=1\text{ or
}2,\end{cases}$

where $x_{q}=u$ if $q\equiv 1\mod 4$ and $x_{q}=\rho$ if $q\equiv 3\mod 4$.
Equivalently, there is an additive isomorphism

(3)
$\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z})\cong\left(\mathbb{Z}\\{1\\}\oplus\bigoplus_{i\geq
0}\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}\tau^{i}\\}\right)\otimes\mathbb{Z}[\pi]/(\pi^{2})\cong\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\otimes\mathbb{Z}[\pi]/(\pi^{2});$

if $q\equiv 1\mod 4$, this is an isomorphism of graded rings.

####  2.2.4. $\mathbb{Q}_{2}$

These results appear in [OØ13]. We have

$\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z}/2)\cong\mathbb{Z}/2[\tau,\pi,u,\rho]/(\rho^{3},u^{2},\pi^{2},\rho
u,\rho\pi,\rho^{2}+u\pi),$

where $|\tau|=(0,-1)$, $|\pi|=|u|=|\rho|=(-1,-1)$. The coefficients of
$H\mathbb{Z}$ over $\mathbb{Q}_{2}$ appear in the proof of [OØ13, Thm. 3.19]
as

$\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z})\cong\begin{cases}\mathbb{Z}\\{1\\}\quad&\text{
if }(*,*)=(0,0),\\\
\mathbb{Z}\\{u\\}\oplus\mathbb{Z}\\{\pi\\}\oplus\mathbb{Z}/2\\{\rho\\}\quad&\text{
if }(*,*)=(-1,-1),\\\
\mathbb{Z}\\{y_{m}\tau^{m}\\}\oplus\mathbb{Z}/2^{s_{3}(m)}\\{z_{m}\tau^{m}\\}\quad&\text{
if }(*,*)=(-1,-m-1),\ m>0,\\\
\mathbb{Z}/2^{s_{3}(m)}\\{\rho^{2}\tau^{m}\\}\quad&\text{ if
}(*,*)=(-2,-m-2),\ m\geq 0,\\\ 0\quad&\text{ otherwise.}\end{cases}$

where $y_{m}$ is $u$ for $m$ odd and $\pi$ for $m$ even, and $z_{m}$ is $\pi$
for $m$ odd and $\rho$ for $m$ even.

####  2.2.5. $\mathbb{R}$

These results appear, for instance, in [Hil11]. We have

$\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2)\cong\mathbb{Z}/2[\tau,\rho],\quad\pi_{**}^{\mathbb{R}}(H\mathbb{Z})\cong\mathbb{Z}[\tau^{2},\rho]/(2\rho).$

####  2.2.6. $\mathbb{Q}$

These results appear in [OØ13, Sec. 5]. The groups
$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)$ can be obtained from the mod two Milnor
K-theory of $\mathbb{Q}$ [OØ13, Prop. 5.3],

$k_{n}^{M}(\mathbb{Q})\cong\begin{cases}\mathbb{Z}/2\\{1\\}\quad&\text{ if
}n=0,\\\ \mathbb{Z}/2\\{\rho\\}\oplus\bigoplus_{p\geq 2,\
prime}\mathbb{Z}/2\\{[p]\\},\quad&\text{ if }n=1,\\\
\mathbb{Z}/2\\{\rho^{2}\\}\oplus\bigoplus_{p\geq 3,\
prime}\mathbb{Z}/2\\{a_{p}\\},\quad&\text{ if }n=2,\\\
\mathbb{Z}/2\\{\rho^{n}\\}\quad&\text{ if }n\geq 3,\end{cases}$

by tensoring with $\mathbb{Z}/2[\tau]$. The multiplicative structure and
$\rho$-module structure is described further in [OØ13, Props. 5.3-5.4]. The
Hasse map

$k^{M}_{*}(\mathbb{Q})\to\prod_{v}k_{*}^{M}(\mathbb{Q}_{v})$

sends pure symbols to their obvious images in $\prod
k_{1}^{M}(\mathbb{Q}_{v})$, $a_{p}$ to the unique nonzero class in
$k_{2}^{M}(\mathbb{Q}_{p})$ and to $0$ in $k_{2}^{M}(\mathbb{Q}_{\ell})$,
$\ell\neq p$.

######  Remark 2.4.

To simplify our comparison with $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})$ and
$\pi_{**}^{\mathbb{Q}_{\nu}}(H\mathbb{Z}/2)$ in the sequel, we note that

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)\cong\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2)\oplus\mathbb{Z}/2[\tau]\\{[2]\\}\oplus\bigoplus_{q\text{
odd}}\mathbb{Z}/2[\tau]\\{[q],a_{q}\\}.$

The summand $\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2)$ maps isomorphically onto
$\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2)$ under the map

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)\to\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2),$

and for each prime $q$, the elements $[q]$ and $a_{q}$ map to $\pi$ and $\pi
x_{q}$, respectively, under

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)\to\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z}/2).$

The groups $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})$ are more complicated.
Specializing [OØ13, Thm. 5.13] to the case $n=0$, we have

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})\cong A\oplus B\oplus C,$

where $A=\bigoplus_{p\equiv 3\mod 4,\ prime}A_{p}$ with $A_{p}$ the bigraded
abelian group defined by

$(A_{p})_{**}=\begin{cases}\mathbb{Z}\quad&\text{ if }(*,*)=(-1,-1),\\\
\mathbb{Z}/2\quad&\text{ if }(*,*)=(-2,-2r-2),\ r\geq 0,\\\
\mathbb{Z}/2^{s_{p}(2r+1)}\quad&\text{ if }(*,*)=(-2,-2r-3),\ r\geq 0,\\\
0\quad&\text{ otherwise,}\end{cases}$

$B=\bigoplus_{p\equiv 1\mod 4,\ prime}B_{p}$, with

$(B_{p})_{**}=\begin{cases}\mathbb{Z}\quad&\text{ if }(*,*)=(-1,-1),\\\
\mathbb{Z}/2^{s_{p}(r)}\quad&\text{ if }(*,*)=(-2,-2-r),\ r\geq 0,\\\
0\quad&\text{ otherwise,}\end{cases}$

and $C=C^{\prime}(0)\oplus C^{\prime\prime\prime}(0)$, with

$C^{\prime}(0)=\mathbb{Z}_{2}[\rho,\tau^{2}]/(2\rho),$
$C^{\prime\prime\prime}(0)_{**}=\begin{cases}\mathbb{Z}\quad&\text{ if
}(*,*)=(-1,-2r-1),\ r\geq 0,\\\ \mathbb{Z}/2^{3+\nu(r+1)}\quad&\text{ if
}(*,*)=(-1,-2r-2),\ r\geq 0,\\\ 0\quad&\text{ otherwise.}\end{cases}$

######  Remark 2.5.

The groups $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})$ have an alternative
description which will be useful in the sequel. Let

$D_{q}:=\begin{cases}A_{q}\quad&\text{ if }q\equiv 3\mod 4,\\\
B_{q}\quad&\text{ if }q\equiv 1\mod 4,\end{cases}$

so

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})\cong C\oplus\bigoplus_{q\text{
odd}}D_{q}.$

We have

$(D_{q})_{**}\cong\begin{cases}\pi_{*+1,*+1}^{\mathbb{F}_{q}}(H\mathbb{Z})\quad&\text{
if }(*,*)\neq(0,0),\\\ 0\quad&\text{ if }(*,*)=(0,0).\end{cases}$

Here, $\pi_{*+1,*+1}^{\mathbb{F}_{q}}(H\mathbb{Z})$ is shorthand for the
$\pi$-divisible part of $\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z})$, cf. (3).
Moreover, we observe that the summands $C^{\prime}(0)$ and
$C^{\prime\prime\prime}(0)$ of $C$ can be identified with familiar objects:

$C^{\prime}(0)\cong\pi_{**}^{\mathbb{R}}(H\mathbb{Z}),$

and there is an obvious inclusion

$C^{\prime\prime\prime}(0)\hookrightarrow\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z})$

with image those subgroups generated by classes of the form $\pi\tau^{m}$,
$m\geq 0$.

[OØ13, Sec. 5] implies that all of these identifications are realized via the
maps

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})\to\pi_{**}^{\mathbb{Q}_{\nu}}(H\mathbb{Z}),$

where $\mathbb{Q}_{\nu}$ ranges over all places.

##  3\. Coefficient rings of $H\mathbb{Z}/2^{n}$

In Section 5, we will see that the slices of $L$ are comprised of suspensions
of the Eilenberg–MacLane spectra $H\mathbb{Z}/2^{n}$ for various $1\leq
n\leq\infty$. In this section, we record the coefficients of these spectra
over our fields of interest.

The motivic Adams spectral sequence [DI10] for $H\mathbb{Z}$ has the following
form.

$E_{1}^{s,w,t}=\pi_{t-s,w}(H\mathbb{Z}/2)[h_{0}]\implies\pi_{t-s,w}H\mathbb{Z}.$

Here we use the Adams grading; the element $h_{0}$ has degree
$(s,w,t)=(1,0,1).$ The abutment is known, and this information determines the
differentials (see [Kyl15] for $F=\mathbb{F}_{q}$ and [OØ13] for
$F=\mathbb{Q}_{q},\mathbb{Q}$).

Similarly, we have the motivic Adams spectral sequence for
$H\mathbb{Z}/2^{n}$:

(4)
$E_{1}^{s,w,t}=\pi_{t-s,w}(H\mathbb{Z}/2)[h_{0}]/h_{0}^{n}\implies\pi_{t-s,w}(H\mathbb{Z}/2^{n}).$

The differentials can be recovered from the motivic Adams spectral sequence
for $H\mathbb{Z}$.

######  Remark 3.1.

Over algebraically closed fields and over the real numbers, we can compute
$\pi_{**}^{F}(H\mathbb{Z}/2^{n})$ directly using the long exact sequence in
homotopy associated to the cofiber sequence

$H\mathbb{Z}\xrightarrow{\cdot 2^{n}}H\mathbb{Z}\to H\mathbb{Z}/2^{n}.$

We have

$\pi_{**}^{\bar{F}}(H\mathbb{Z}/2^{n})\cong\mathbb{Z}/2^{n}[\tau]\text{\quad
and
\quad}\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2^{n})\cong\mathbb{Z}/2^{n}[\tau^{2},\rho]/(2\rho).$

######  Proposition 3.2.

The differentials in the $F$-motivic Adams spectral sequence for
$H\mathbb{Z}/2^{n}$ are determined via the Leibniz rule by the following
($i\geq 1$):

  1. (1)

When $F=\mathbb{F}_{q}$:

$d_{s_{q}(i-1)}\tau^{i}=x_{q}\tau^{i-1}h_{0}^{s_{q}(i-1)},~{}s_{q}(i-1)<n,$

where $x_{q}=u$ if $q\equiv 1\mod 4$ and $x_{q}=\rho$ if $q\equiv 3\mod 4$.

  2. (2)

When $F=\mathbb{Q}_{q}$:

$d_{s_{q}(i-1)}\tau^{i}=x_{q}\tau^{i-1}h_{0}^{s_{q}(i-1)},~{}s_{q}(i-1)<n,$

where $x_{q}=u$ if $q\equiv 1\mod 4$ and $x_{q}=\rho$ if $q\equiv 3\mod 4$.

  3. (3)

When $F=\mathbb{Q}_{2}$:

$d_{1}\tau=\rho h_{0},\quad
d_{3+\nu(i)}\tau^{2i}=\pi\tau^{2i-1}h_{0}^{3+\nu(i)},\ \nu(i)<n-3.$

  4. (4)

When $F=\mathbb{Q}$:

$d_{1}\tau=\rho h_{0},\quad d_{1}[p]\tau=(\rho^{2}+a_{p})h_{0}\quad p\equiv
3\mod 4,$ $d_{s_{p}(i-1)}[p]\tau^{i}=a_{p}\tau^{i-1}h_{0}^{s_{p}(i-1)},\quad
1\leq s_{p}(i-1)<n\text{~{}and~{}}i\text{~{}even for}~{}p\equiv 3\mod 4,$
$d_{3+\nu(i)}\tau^{2i}h_{0}=[2]\tau^{2i-1}h_{0}^{4+\nu(i)},\quad\text{ with
}\nu(i)<n-3.$

###### Proof.

  1. (1)

This follows from [Kyl15, Lem. 4.2.1, Lem. 4.2.2].444Note that for $q\equiv 3$
mod $4$, the exponent of $h_{0}$ in the target is off by one in _loc. cit._

  2. (2)

This follows from [OØ13, Thm. 3.16, Thm. 3.17].

  3. (3)

This follows from [OØ13, Thm. 3.18, Thm. 3.19].

  4. (4)

This follows from [OØ13, Thm. 5.5, Thm. 5.8].

∎

######  Proposition 3.3.

The coefficient rings of $H\mathbb{Z}/2^{n}$, $1\leq n<\infty$, are given by
the following formulas. In cases (1) and (2), we write $i(w)$ for
$\text{min}(s_{q}(w-1),n).$

  1. (1)

When $F=\mathbb{F}_{q}$ with $q\equiv 1$ mod $4$:

$\pi_{s,w}^{\mathbb{F}_{q}}H\mathbb{Z}/2^{n}=\begin{cases}\mathbb{Z}/2^{n}\\{1\\}\quad&\text{
if }s=0,w=0,\\\ \mathbb{Z}/2^{i(-w)}\\{2^{n-i(-w)}\tau^{-w}\\}\quad&\text{ if
}s=0,w\leq-1,\\\ \mathbb{Z}/2^{i(-w)}\\{\tau^{-1-w}x_{q}\\}\quad&\text{ if
}s=-1,w\leq-1,\\\ 0\quad&\text{ otherwise.}\end{cases}$

Here $x_{q}=u$ if $q\equiv 1\mod 4$ and $x_{q}=\rho$ if $q\equiv 3\mod 4$.

  2. (2)

When $F=\mathbb{Q}_{q}$ with $q\equiv 1$ mod $4$:

$\pi_{s,w}^{\mathbb{Q}_{q}}H\mathbb{Z}/2^{n}=\begin{cases}\mathbb{Z}/2^{i(-w)}\\{2^{n-i(-w)}\tau^{-w}\\}\quad&\text{
if }s=0,w\leq 0,\\\
\mathbb{Z}/2^{i(-w)}\\{\tau^{-1-w}x_{q}\\}\oplus\mathbb{Z}/2^{i(-w-1)}\\{2^{n-i(-w-1)}\tau^{-1-w}\pi\\}\quad&\text{
if }s=-1,w\leq-1\\\ \mathbb{Z}/2^{i(-w-1)}\\{\tau^{-2-w}\pi
x_{q}\\}\quad&\text{ if }s=-2,w\leq-2\\\ 0\quad&\text{ otherwise,}\end{cases}$

  3. (3)

When $F=\mathbb{Q}_{2}$:

$\pi_{s,w}^{\mathbb{Q}_{2}}(H\mathbb{Z}/2^{n})\cong\begin{cases}\mathbb{Z}/2^{i(-w)}\\{2^{n-i(-w)}\tau^{-w}\\}\quad&\text{
if }s=0,w\leq 0,\\\
\mathbb{Z}/2^{n}\\{\tau^{-1-w}u\\}\oplus\mathbb{Z}/2^{i(-w)}\\{2^{n-i(-w)}\tau^{-1-w}\pi\\}\\\
\oplus\mathbb{Z}/2\\{2^{n-1}\tau^{-1-w}\rho\\}\quad&\text{ if
}s=-1,w\leq-2\text{ even},\\\
\mathbb{Z}/2^{i(-1-w)}\\{2^{n-i(-1-w)}\tau^{-1-w}u\\}\oplus\mathbb{Z}/2^{n}\\{\tau^{-1-w}\pi\\}\\\
\oplus\mathbb{Z}/2\\{\tau^{-1-w}\rho\\}\quad&\text{ if }s=-1,w\leq-1\text{
odd},\\\
\mathbb{Z}/2^{i(-w-1)}\\{2^{n-i(-w-1)}\tau^{-2-w}\rho^{2}\\}\quad&\text{ if
}s=-2,w\leq-1,\\\ 0\quad&\text{ otherwise,}\end{cases}$

where $i(w)=1$ when $w$ is odd and $\text{min}(2+\nu(w),n)$ when $w$ is even.

  4. (4)

When $F=\mathbb{Q}$:

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2^{n})\cong
C{{}^{\prime\prime\prime}}_{**}(0)\oplus
C{{}^{\prime}}_{**}(0)\oplus\bigoplus_{q\text{ odd}}D_{q}(n),$

where

$(D_{q}(n))_{**}\cong\begin{cases}\pi_{*+1,*+1}^{\mathbb{F}_{q}}(H\mathbb{Z}/2^{n})\quad&\text{
if }(*,*)\neq(0,0),\\\ 0\quad&\text{ if }(*,*)=(0,0),\end{cases}$
$C^{\prime}_{**}(0)\cong\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2^{n}),$

and

$C^{\prime\prime\prime}(0)\hookrightarrow\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z}/2^{n})$

can be identified with the subgroups generated by classes of the form
$\pi\tau^{m}$, $m\geq 0$. Here, $\pi_{*+1,*+1}^{\mathbb{F}_{q}}$ is shorthand
for the $\pi$-divisible part of $\pi_{**}^{\mathbb{Q}_{q}}$.

###### Proof.

The results follow from the motivic Adams spectral sequence in (4) and
differentials in 3.2. ∎

##  4\. Coefficient rings of $kq$

Recall the slices for $kq$ in Equation 1. In this section, we describe the
effective slice spectral sequence for $kq$ over various base fields.

###  4.1. Algebraically closed fields

Let $F=\bar{F}$ be an algebraically closed field. The effective slice spectral
sequence for $kq$ over $F$ was computed in [BIK22, Sec. 3]. The $E_{1}$-term
of the ESSS is depicted in Figure 2.

The differential $d_{1}(v_{1}^{2})=\tau h_{1}^{3}$ follows by Betti
realization [BIK22, Prop. 3.2], with all remaining differentials following by
the Leibniz rule. The spectral sequence collapses at $E_{2}$; the resulting
$E_{\infty}$-page is depicted in Figure 3.

###  4.2. Finite fields

The $E_{1}$-page of the ESSS over $\mathbb{F}_{q}$ can be obtained from the
$E_{1}$-page of the ESSS over algebraically closed fields (Figure 2) by
letting a rectangle denote $\mathbb{Z}$ instead of $\mathbb{Z}[\tau]$, and
then juxtaposing a copy of the $E_{1}$-page, shifted by $(-1,1)$, with each
rectangle replaced by a diamond denoting the $u$\- or $\rho$-divisible part
$\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})$. The resulting $E_{1}$-page is
depicted in Figure 4.

The differential $d_{1}(v_{1}^{2})=\tau h_{1}^{3}$ follows from base change to
the algebraic closure, and all remaining differentials follow by the Leibniz
rule. The resulting pattern of $d_{1}$-differentials is depicted in Figure 4.

The ESSS collapses for degree reasons at $E_{2}$. The $E_{\infty}$-page is
shown in Figure 5. The hidden extensions $\mathsf{h}\cdot
2v_{1}^{2+4k}u=\tau^{2}h_{1}^{3}v_{1}^{4k}$, $k\geq 0$, follow by comparison
to Kylling’s computation of $\pi_{**}^{\mathbb{F}_{q}}(kq)$ via the motivic
Adams spectral sequence [Kyl15, Sec. 4].

###  4.3. $\mathbb{Q}_{q}$ with $q$ odd

The $E_{1}$-page of the ESSS over $\mathbb{Q}_{q}$ can be obtained from the
$E_{1}$-page of the ESSS over $\mathbb{F}_{q}$ using the additive isomorphism
(3), which implies that there is an additive isomorphism

$E_{1}^{\mathbb{Q}_{q}}(kq)\cong
E_{1}^{\mathbb{F}_{q}}(kq)\otimes\mathbb{Z}[\pi]/(\pi^{2}),$

where $|\pi|=(-1,0,-1)$. Graphically, this means the $E_{1}$-term over
$\mathbb{Q}_{q}$ consists of two copies of the $E_{1}$-page over
$\mathbb{F}_{q}$: one copy as it appears in Figure 4, and one copy shifted by
$(-1,1)$. The result is depicted in Figure 6.

As in the case of finite fields, all $d_{1}$-differentials follow from base
change to the algebraic closure and
$\pi^{\delta}x_{q}^{\epsilon}\tau^{n}$-linearity for the appropriate choices
of $\delta,\epsilon\in\\{0,1\\}$. The resulting $E_{\infty}$-page is identical
the the $E_{\infty}$-page over $\mathbb{F}_{q}$, tensored with
$\mathbb{Z}[\pi]/(\pi^{2})$. Hidden extensions follow from similar arguments.
The resulting groups are depicted in Figure 7.

###  4.4. $\mathbb{Q}_{2}$

Since $\rho^{2}$ is non-zero over $\mathbb{Q}_{2}$, the ESSS over
$\mathbb{Q}_{2}$ has a different $d_{1}$-differential pattern. The
$d_{1}$-differentials in the ESSS for $KQ$ (and consequently, $kq$) were
computed in terms of Steenrod operations in [RØ16, Thm. 5.5]. The $E_{1}$-term
is depicted in Figure 8.

In Figure 8, the red differentials are given by taking $Sq^{2}$, and the brown
differentials are given by multiplying by $\tau$.

######  Example 4.1.

We explain the $d_{1}$ calculation by way of example.

  1. (1)

There is a red differential between the class $\mathbb{Z}/2[\tau]\\{h_{1}\\}$
at $(1,1)$ and the class $\mathbb{Z}/2[\tau]\\{h_{1}^{2}\rho^{2}\\}$ at
$(0,4)$. The formula for the differential is

$d_{1}(\tau^{4n+2}h_{1})=\tau^{4n+1}\rho^{2}h_{1}^{2}\text{~{}and~{}}d_{1}(\tau^{4n+3}h_{1})=\tau^{4n+2}\rho^{2}h_{1}^{2},n\geq
0.$

Therefore, on the $E_{2}$-page, we have a
$\mathbb{Z}/2\\{1,\tau^{3}\\}[\tau^{4}]$ in the degree of the target and a
$\mathbb{Z}/2\\{1,\tau\\}[\tau^{4}]$ in the degree of the source.

  2. (2)

There is a brown differential between the class
$\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})\\{v_{1}^{2}\\}$ at $(2,2)$ and the
class $\mathbb{Z}/2[\tau]\\{\rho^{2}h_{1}^{3}\\}$ at $(1,5)$. The formula for
the differential is

$d_{1}(\tau^{n}\rho^{2}v_{1}^{2})=\tau^{n+1}\rho^{2}h_{1}^{3},n\geq 0.$

Therefore, on the $E_{2}$-page, we have a $\mathbb{Z}/2$ in the degree of the
target. The surviving group in the degree of the source is obtained from
$\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$ by decreasing the order of
$2$-torsion by exactly $1$ in each weight (thus is trivial in even weights).

  3. (3)

In the case of (2), the classes in $\mathbb{Z}/2[\tau]\\{h_{1}^{2}\\}$ at
$(2,2)$ also support red differentials to the classes in
$\mathbb{Z}/2[\tau]\\{\rho^{2}h_{1}^{3}\\}$ at $(1,5)$. Using the differential
formulas, we see that the following linear combinations of the source classes
survive:

$\tau^{4n+2}h_{1}^{2}+\tau^{4n}\rho^{2}v_{1}^{2}\quad\text{~{}and~{}}\quad\tau^{4n+3}h_{1}^{2}+\tau^{4n+1}\rho^{2}v_{1}^{2},\quad
n\geq 0.$

The $E_{2}$-page is depicted in Figure 9. There are no room for
$d_{2}$-differentials, therefore we have $E_{2}=E_{\infty}.$

###  4.5. $\mathbb{R}$

The ESSS for $kq$ over $\mathbb{R}$ is the subject of [BIK22, Sec. 4]. There
are two interesting differentials over $\mathbb{R}$ [BIK22, Prop. 5.2]:

$d_{1}(v_{1}^{2})=\tau h_{1}^{3},\quad d_{1}(\tau^{2})=\rho^{2}\tau h_{1}.$

The $E_{1}$-page is depicted in [BIK22, Fig. 5], and the $E_{\infty}$-page is
depicted in [BIK22, Figs. 6-8].

###  4.6. $\mathbb{Q}$

We will now analyze the ESSS for $kq$ over $\mathbb{Q}$. To begin, recall the
presentations of $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)$ and
$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})$ from 2.4 and 2.5. Since
$E_{1}^{\mathbb{Q}}(kq)$ is comprised of shifts of these groups, it naturally
decomposes as

$E_{1}^{\mathbb{Q}}(kq)\cong E_{1}^{+}(kq)\oplus E_{1}^{-}(kq),$

where $E_{1}^{+}(kq)\subseteq E_{1}^{\mathbb{Q}}(kq)$ is obtained by making
the replacements

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)\rightsquigarrow\bigoplus_{q\text{
odd}}\mathbb{Z}/2[\tau]\\{[q],a_{q}\\},\quad\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})\rightsquigarrow\bigoplus_{q\text{
odd}}D_{q},$

and $E_{1}^{-}(kq)\subseteq E_{1}^{\mathbb{Q}}(kq)$ is obtained by making the
replacements

$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)\rightsquigarrow\pi_{**}^{\mathbb{R}}(H\mathbb{Z}/2)\oplus\mathbb{Z}/2[\tau]\\{[2]\\},\quad\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})\rightsquigarrow
C\cong\pi_{**}^{\mathbb{R}}(H\mathbb{Z})\oplus C^{\prime\prime\prime}(0).$

Graphically, we can obtain $E_{1}^{\mathbb{Q}}(kq)$ from $E_{1}^{\bar{F}}(kq)$
(Figure 2) by replacing each bullet by a copy of
$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)$ and replacing each square by a copy of
$\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)$; the replacements are shown in Figure
10.

The purpose of this decomposition is to understand the map

$E_{1}^{\mathbb{Q}}(kq)\to E_{1}^{\mathbb{Q}_{\nu}}(kq)$

for each place $\nu$ of $\mathbb{Q}$. In particular, we see that the Hasse map

$E_{1}^{\mathbb{Q}}(kq)\to\prod_{\nu}E_{1}^{\mathbb{Q}_{\nu}}(kq)$

is injective. It follows that the nontrivial $d_{1}$-differentials over
$\mathbb{Q}$ are generated by

$d_{1}([q]\tau^{i}v_{1}^{2k})=[q]\tau^{i+1}h_{1}^{3}v_{1}^{2k-2},\quad
d_{1}(a_{q^{\prime}}\tau^{i}v_{1}^{2k})=a_{q^{\prime}}\tau^{i+1}h_{1}^{3}v_{1}^{2k-2},\quad
d_{1}(\tau^{2})=\rho^{2}\tau h_{1},$

where $q$ ranges over all primes, $q^{\prime}$ ranges over all odd primes, and
$i$ and $k$ range over all nonnegative integers. Note that none of the more
exotic differentials (red ones in Figure 8) from the $\mathbb{Q}_{2}$-case
lift to $\mathbb{Q}$ since the sources and targets do not lift to
$\mathbb{Q}$.

There is no room for longer differentials, so $E_{2}^{\mathbb{Q}}(kq)\cong
E_{\infty}^{\mathbb{Q}}(kq)$. All hidden extensions can be handled by
comparison with the local places.

######  Remark 4.2.

Our description of the $E_{1}$-term over $\mathbb{Q}$ makes clear that the map

$E_{1}^{\mathbb{Q}}(kq)\to\prod_{\nu}E_{1}^{\mathbb{Q}_{\nu}}(kq)$

is injective. Since all of the differentials over $\mathbb{Q}$ are lifted from
differentials over the local places, we conclude that the Hasse map

$\pi_{**}^{\mathbb{Q}}(kq)\to\prod_{\nu}\pi_{**}^{\mathbb{Q}_{\nu}}(kq)$

is injective, i.e., $kq$ satisfies the motivic Hasse principle in the sense of
[OØ13, Sec. 4],

##  5\. Coefficient rings of $L$ over prime fields

In this section, we describe the effective slice spectral sequence for $L$
over various base fields.

###  5.1. Algebraically closed fields

The ESSS for $L$ over algebraically closed fields was computed in [BIK22, Sec.
3]. The multiplicative generators of the $E_{1}$-term appear in [BIK22, Table
3] and the $E_{1}$-term is shown in [BIK22, Fig. 3]. The $d_{1}$-differentials
appear in [BIK22, Table 4] and the resulting $E_{2}=E_{\infty}$-page appears
in [BIK22, Fig. 4]. Hidden extensions are handled in [BIK22, Prop. 3.16].

For ease of reference, we have reproduced [BIK22, Fig. 3] and [BIK22, Fig. 4]
in Figure 11 and Figure 12, respectively.

###  5.2. Finite fields

To compute the $E_{1}$-term of the ESSS for $L$ over finite fields, we must
analyze the effect of $\psi^{3}-1$ on the slices of $kq$. Comparison with the
algebraically closed case implies that the only possible nontrivial action of
$\psi^{3}-1$ is on the integer slices, i.e.,

$\psi^{3}-1:\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\\{v_{1}^{2k}\\}\to\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\\{v_{1}^{2k}\\},$

which may be identified with

$\cdot 2^{\nu(3^{2k}-1)}=\cdot
2^{\nu(k)+3}:\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\\{v_{1}^{2k}\\}\to\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})\\{v_{1}^{2k}\\}.$

Recalling the functions $s_{q}(i)$ and notation $x_{q}$ from 2.3, we may write
this more explictily as

$\cdot 2^{\nu(k)+3}:\mathbb{Z}\\{v_{1}^{2k}\\}\oplus\bigoplus_{i\geq
0}\mathbb{Z}/2^{s_{q}(i)}\\{v_{1}^{2k}x_{q}\tau^{i}\\}.$

The kernel and cokernel of

$\cdot 2^{\nu(k)+3}:\mathbb{Z}\\{v_{1}^{2k}\\}\to\mathbb{Z}\\{v_{1}^{2k}\\}$

are $0$ and $\mathbb{Z}/2^{\nu(k)+3}$, respectively. On the components indexed
by $x_{q}v_{1}^{2k}\tau^{i}$,

$\cdot
2^{\nu(k)+3}:\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\to\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\},$

we have

$\operatorname{ker}\cong\begin{cases}\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\quad&\text{
if }s_{q}(i)\leq\nu(k)+3,\\\
\mathbb{Z}/2^{\nu(k)+3}\\{2^{s_{q}(i)-\nu(k)-3}x_{q}v_{1}^{2k}\tau^{i}\\}\quad&\text{
if }s_{q}(i)>\nu(k)+3,\end{cases}$

and

$\operatorname{coker}\cong\begin{cases}\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\quad&\text{
if }s_{q}(i)\leq\nu(k)+3,\\\
\mathbb{Z}/2^{s_{q}(i)-\nu(k)-3}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\quad&\text{ if
}s_{q}(i)>\nu(k)+3.\end{cases}$

######  Definition 5.1.

For each $k\geq 0$, we define graded groups $K_{q}(k)$ and $C_{q}(k)$ as
follows. Let

$K_{q}(0)=C_{q}(0):=\pi_{-1,*}^{\mathbb{F}_{q}}(H\mathbb{Z})$

denote the $x_{q}$-divisible part of $\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})$.
For all $k\geq 1$, let

$K_{q}(k):=\bigoplus_{i\geq 0}\operatorname{ker}(\cdot
2^{\nu(k)+3}:\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\to\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}),$
$C_{q}(k):=\bigoplus_{i\geq 0}\operatorname{coker}(\cdot
2^{\nu(k)+3}:\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\}\to\mathbb{Z}/2^{s_{q}(i)}\\{x_{q}v_{1}^{2k}\tau^{i}\\})$

be the sums of the kernels and cokernels, respectively, described above.

Let $\tilde{K}_{q}(k)$ denote the groups obtained from $K_{q}(k)$ as follows.
If $k$ is even, then $\tilde{K}_{q}(k):=K_{q}(k)$. If $k$ is odd, then:

  1. (1)

If $s_{q}(i)\leq\nu(k)+3$, then the $i$-th summand of $\tilde{K}_{q}(k)$ is
the group obtained from the $i$-th summand of $K_{q}(k)$ by decreasing the
order of $2$-torsion by exactly $1$ in each summand.

  2. (2)

If $s_{q}(i)>\nu(k)+3$, the $i$-th summand of $\tilde{K}_{q}(k)$ is the same
as the $i$-th summand in $K_{q}(k)$.

Let $\tilde{C}_{q}(k)$ denote the groups obtained from $C_{q}(k)$ as follows.
If $k$ is even, then $\tilde{C}_{q}(k):=C_{q}(k)$. If $k$ is odd, then
$\tilde{C}_{q}(k)$ is obtained by decreasing the order of $2$-torsion by
exactly $1$ in each summand of $C_{q}(k)$.

5.1 allows us to concisely describe the $E_{1}$\- and $E_{\infty}$-terms of
the ESSS for $L$ in chart form. The $E_{1}$-term is depicted in Figure 13.

All of the differentials follow from base change to the algebraic closure and
$x_{q}$-linearity of the $d_{1}$-differentials. Since $s_{q}(i)\geq 1$ for all
$i\geq 0$, the effect of each _nontrivial_ differential is easy to describe:
simply reduce the order of $2$-torsion in the source and target by $2$.
However, describing which differentials are nontrivial is somewhat subtle.

The red differentials out of the dark green boxes (i.e., out of
$\mathbb{Z}/2^{?}\\{\iota\tau^{i}v_{1}^{2k}\\}$) and dark green left-pointing
triangles (i.e., out of $C_{q}(k)$) are nontrivial for all $i\geq 0$, but the
green differentials out of the right-pointing triangles (i.e., out of
$K_{q}(k)$) are trivial on some summands. Recall that the generator of the
$i$-th summand in $K_{q}(k)$ for $k$ odd is

$\begin{cases}x_{q}v_{1}^{2k}\tau^{i}\quad&\text{ if }s_{q}(i)\leq 3,\\\
2^{s_{q}(i)-\nu(k)-3}x_{q}v_{1}^{2k}\tau^{i}\quad&\text{ if
}s_{q}(i)>3.\end{cases}$

Here, we have used that $\nu(k)=0$ if $k$ is odd to rewrite $\nu(k)+3$ as $3$.

If $s_{q}(i)\leq 3$, then there is a differential

$d_{1}(x_{q}v_{1}^{2k}\tau^{i})=x_{q}h_{1}^{3}v_{1}^{2k-2}\tau^{i+1},$

but if $s_{q}(i)>\nu(k)+3$, then we must have

$d_{1}(2^{s_{q}(i)-\nu(k)-3}x_{q}v_{1}^{2k}\tau^{i})=0$

since

$d_{1}(2^{s_{q}(i)-\nu(k)-3}v_{1}^{2k})=0$

by base change to the algebraic closure.

######  Remark 5.2.

In practice, it is straightforward to determine on which summands the green
differentials are nontrivial for a given $q$. For instance, if $q=3$, then:

  * •

If $i\not\equiv 3\mod 4$, then $s_{3}(i)\leq 3$. Indeed, if $i$ is even, then
$s_{3}(i)=1\leq 3$, and if $i\equiv 1\mod 4$, then
$s_{3}(i)=\nu(3^{2}-1)+\nu(i+1)-1=2+\nu(i+1)=3\leq 3$.

  * •

If $i\equiv 3\mod 4$, then $s_{3}(i)=\nu(3^{2}-1)+\nu(i+1)-1=2+\nu(i+1)>3$.

This leads us to the $E_{2}$-term. There is no room for higher differentials,
so $E_{2}=E_{\infty}$. All hidden extensions follow from comparison to the
algebraic closure or $\tau^{n}x_{q}$-linearity. The resulting groups are
depicted in Figure 14.

###  5.3. $\mathbb{Q}_{q}$ with $q$ odd

As with $kq$ (Section 4.3), the computation of $\pi_{**}^{\mathbb{Q}_{q}}(L)$
for $q$ odd follows from the analogous computation over $\mathbb{F}_{q}$
(Section 5.2). The additive isomorphism (3) implies that

$E_{1}^{\mathbb{Q}_{q}}(L)\cong
E_{1}^{\mathbb{F}_{q}}(L)\otimes\mathbb{Z}[\pi]/(\pi^{2}).$

Thus the $E_{1}$-term over $\mathbb{Q}_{q}$ can be obtained from Figure 13 by
superimposing an identical copy shifted by $(-1,1)$. Moreover, the
$d_{1}$-differentials occur in both copies; this follows as usual from base
change to the algebraic closure and
$\pi^{\delta}x_{q}^{\epsilon}\tau^{n}$-linearity for the appropriate choices
of $n\geq 0$ and $\delta,\epsilon\in\\{0,1\\}$.

There is no room for further differentials, so $E_{2}=E_{\infty}$. By the
above discussion, the $E_{\infty}$-term satisfies

$E_{\infty}^{\mathbb{Q}_{q}}(L)\cong
E_{\infty}^{\mathbb{F}_{q}}(L)\otimes\mathbb{Z}[\pi]/(\pi^{2}),$

so it can be obtained from Figure 14 by superimposing the same picture shifted
by $(-1,1)$.

###  5.4. $\mathbb{Q}_{2}$

As in §5.2, we calculate the kernels and cokernels of the reduced Adams
operation.

######  Definition 5.3.

For all $k\geq 1$, let

$\displaystyle K_{2}(k)\\{x\\}:=$ $\displaystyle\bigoplus_{i\geq
0}\operatorname{ker}(\cdot
2^{\nu(k)+3}:\mathbb{Z}/2^{s_{3}(i)}\\{xv_{1}^{2k}\tau^{i}\\}\to\mathbb{Z}/2^{s_{3}(i)}\\{xv_{1}^{2k}\tau^{i}\\})$
$\displaystyle C_{2}(k)\\{x\\}:=$ $\displaystyle\bigoplus_{i\geq
0}\operatorname{coker}(\cdot
2^{\nu(k)+3}:\mathbb{Z}/2^{s_{3}(i)}\\{xv_{1}^{2k}\tau^{i}\\}\to\mathbb{Z}/2^{s_{3}(i)}\\{xv_{1}^{2k}\tau^{i}\\})$

be the sums of kernels and cokernels, respectively, for $x$ being either
$z_{i}$, $y_{i}$, or $\rho^{2}$.

Define $K_{2}(0)=C_{2}(0)$ to be $\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$,
and define $\underline{K}_{2}(0)$ to be
$\pi_{-1,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$.

We define $\underline{C}_{2}(k),k\geq 0$ to be the sum

$\underline{C}_{2}(k):=C_{2}(k)\\{z_{i}\\}\oplus\bigoplus_{i\geq
0}\mathbb{Z}/2^{\nu(k)+3}\\{y_{i}\tau^{i}\\}\oplus\mathbb{Z}/2^{\nu(k)+3}\\{u\\}.$

Let $\tilde{K}_{2}(k)$ denote the groups obtained from $K_{2}(k)$ by
decreasing the order of $2$-torsion by exactly $1$ in each summand except the
$(4i-1)$-th summand for all positive integer $i$.

Let $\tilde{C}_{2}(k)$ denote the groups obtained from $C_{2}(k)$ by
decreasing the order of $2$-torsion by exactly $1$ in each summand; similar
for ${\tilde{\underline{C}}_{2}(k)}$

We use 5.3 to desribe the $E_{1}$-term and $E_{\infty}$-term of the ESSS for
$L$. The $E_{1}$-term is depicted in Figure 15.

In Figure 15, the red differentials are given by taking $Sq^{2}$, and the
brown differentials are given by multiplying by $\tau$. The calculation is
similar to the case for $kq$ in §4.4. The $E_{2}$-page is depicted in Figure
16.

Every potential higher differential hits an $\eta$-periodic class, or is from
the unit $1$. By comparing with the $\eta$-inverted result in [Wil18, Cor.
11], there is no possibility for nontrivial higher differentials. Therefore,
over $\mathbb{Q}_{2}$, the ESSS for $L$ collapses at $E_{2}$.

###  5.5. $\mathbb{R}$

The ESSS for $L$ over $\mathbb{R}$ was computed in [BIK22, Sec. 5]. The
multiplicative generators of the $E_{1}$-term appear in [BIK22, Table 8] and
the $E_{1}$-term is depicted in [BIK22, Fig. 10]. The values of the
$d_{1}$-differentials on the multiplicative generators appear in [BIK22, Table
10], and these values determine all of the $d_{1}$-differentials using the
Leibniz rule in conjunction with the relations recorded in [BIK22, Table 9].
The higher differentials are described in [BIK22, Prop. 5.11]. The resulting
$E_{\infty}$-page is depicted in [BIK22, Figs. 13-19].

###  5.6. $\mathbb{Q}$

To compute $E_{1}^{\mathbb{Q}}(L)$, we break it into four manageable pieces.
First, there is a decomposition

$E_{1}^{\mathbb{Q}}(L)\cong
K^{\mathbb{Q}}(L)\oplus\Sigma^{-1}C^{\mathbb{Q}}(L),$

where $K^{\mathbb{Q}}(L)$ is the kernel of $\psi^{3}-1$ and
$C^{\mathbb{Q}}(L)$ is the cokernel of $\psi^{3}-1$. There are further
decompositions

$K^{\mathbb{Q}}(L)\cong K^{+}(L)\oplus K^{-}(L),$ $C^{\mathbb{Q}}(L)\cong
C^{+}(L)\oplus C^{-}(L),$

where

$K^{+}(L):=\operatorname{ker}(\psi^{3}-1:E_{1}^{+}(kq)\to E_{1}^{+}(kq)),$
$K^{-}(L):=\operatorname{ker}(\psi^{3}-1:E_{1}^{-}(kq)\to E_{1}^{-}(kq)),$
$C^{+}(L):=\operatorname{coker}(\psi^{3}-1:E_{1}^{+}(kq)\to E_{1}^{+}(kq)),$
$C^{-}(L):=\operatorname{coker}(\psi^{3}-1:E_{1}^{-}(kq)\to E_{1}^{-}(kq)).$

Here, $E_{1}^{+}(kq)$ and $E_{1}^{-}(kq)$ were defined in Section 4.6. We then
have

$E_{1}^{\mathbb{Q}}(L)\cong K^{+}(L)\oplus
K^{-}(L)\oplus\Sigma^{-1}C^{+}(L)\oplus\Sigma^{-1}C^{-}(L).$

Since $E_{1}^{+}(kq)$ can be expressed entirely in terms of
$E_{1}^{\mathbb{F}_{q}}(kq)$, $q$ odd, the groups $K^{+}(L)$ and $C^{+}(L)$
can be described using the computations from Section 5.2. More precisely,
$K^{+}(L)\oplus\Sigma^{-1}C^{+}(L)$ is obtained graphically by taking the sum
over all odd primes of Figure 13, then shifting the entire picture by
$(-1,1)$.

The computation of $K^{-}(L)$ and $C^{-}(L)$ also largely follows from
previous computations. Since $E_{1}^{\mathbb{R}}(kq)$ is a summand of
$E_{1}^{-}(kq)$, the kernel and cokernel of $\psi^{3}-1$ over $\mathbb{R}$ are
summands in $K^{-}(L)\oplus\Sigma^{-1}C^{-}(L)$; the relevant groups are
depicted graphically in [BIK22, Fig. 10]. The complementary summand of
$E_{1}^{-}(kq)$ is obtained by replacing $\pi_{**}(H\mathbb{Z})$ by
$C^{\prime\prime\prime}(0)$ and $\pi_{**}(H\mathbb{Z}/2)$ by
$\mathbb{Z}/2[\tau]\\{[2]\\}$. Since $\psi^{3}-1$ is necessarily trivial on
$\mathbb{Z}/2[\tau]\\{[2]\\}$, we only need to analyze its effect on each copy
of $C^{\prime\prime\prime}(0)$, but this was already done in Section 5.4 since
$C^{\prime\prime\prime}(0)$ may be identified with the summand in
$\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z})$ generated by classes of the form
$\pi\tau^{m}$, $m\geq 0$, or equivalently, by the classes of the form
$y_{m}\tau^{m}$ for $m$ even and $z_{m}\tau^{m}$ for $m$ odd. Note that any
nontrivial $d_{1}$-differential over $\mathbb{Q}_{2}$ involving $\pi\tau^{m}$
must include it in both the source and target, so all such
$d_{1}$-differentials lift to $\mathbb{Q}$.

The preceding discussion implies that the map

$E_{1}^{\mathbb{Q}}(L)\to\prod_{\nu}E_{1}^{\mathbb{Q}_{\nu}}(L)$

is injective. The $d_{1}$-differentials on $E_{1}^{+}(L)$ are precisely the
lifts of the $d_{1}$-differentials from Section 5.2 (which occur on the
$\pi$-divisible part of the ESSS over $\mathbb{Q}_{q}$, $q$ odd). The
$d_{1}$-differentials on $E_{1}^{-}(L)$ are the lifts of the
$d_{1}$-differentials over $\mathbb{R}$ ([BIK22, Table 10]) and the lifts of
the $d_{1}$-differentials over $\mathbb{Q}_{2}$ for which the source
(equivalently, the target) lift to $\mathbb{Q}$.

Since all of the $d_{1}$-differentials over $\mathbb{Q}$ are lifted from
$d_{1}$-differentials over the local places, the map

$E_{2}^{\mathbb{Q}}(L)\to\prod_{\nu}E_{2}^{\mathbb{Q}_{\nu}}(L)$

is also injective. There cannot be any higher differentials involving
$E_{2}^{+}(L)$, since these would imply higher differentials in
$E_{2}^{\mathbb{Q}_{q}}(L)$ for $q$ odd. On $E_{2}^{-}(L)$, or more generally
on $E_{r}^{-}(L)$ for $r\geq 2$, the nontrivial differentials are precisely
the lifts of the nontrivial $d_{r}$-differentials in $E_{r}^{\mathbb{R}}(L)$
described in [BIK22, Prop. 5.11].

The $E_{\infty}$-term can be described as follows. We have

$E_{\infty}^{\mathbb{Q}}(L)\cong E_{\infty}^{+}(L)\oplus E_{\infty}^{-}(L).$

The piece $E_{\infty}^{+}(L)$ is the sum over all odd primes $q$ of
$E_{\infty}^{\mathbb{F}_{q}}(L)$ (see Figure 14), shifted by $(-1,1)$. The
piece $E_{\infty}^{-}(L)$ is the sum of $E_{\infty}^{\mathbb{R}}(L)$ ([BIK22,
Figs. 13-19]) and the portion of $E_{\infty}^{\mathbb{Q}_{2}}(L)$ (Figure 16)
generated by $\pi\tau^{m}$ and $\iota\pi\tau^{m}$, $m\geq 0$. All hidden
extensions follow from comparison with the local places.

######  Remark 5.4.

Since the Hasse map for $L$ is injective on the $E_{1}$-term of the ESSS and
every differential over $\mathbb{Q}$ is lifted from a differential over some
$\mathbb{Q}_{\nu}$, the Hasse map

$\pi_{**}^{\mathbb{Q}}(L)\to\prod_{\nu}\pi_{**}^{\mathbb{Q}_{\nu}}(L)$

is injective, i.e., $L$ satisfies the motivic Hasse principle of [OØ13, Sec.
4].

##  Appendix A Figures

Figure 1. A graphical depiction of $\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z}/2)$,
$\pi_{**}^{\mathbb{F}_{5}}(H\mathbb{Z})$,
$\pi_{**}^{\mathbb{F}_{3}}(H\mathbb{Z})$,
$\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z}/2)$,
$\pi_{**}^{\mathbb{Q}_{2}}(H\mathbb{Z})$,
$\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z}/2)$,
$\pi_{**}^{\mathbb{Q}_{5}}(H\mathbb{Z})$, and
$\pi_{**}^{\mathbb{Q}_{3}}(H\mathbb{Z})$. A bullet $\bullet$ represents
$\mathbb{Z}/2$, a black square $\blacksquare$ represents $\mathbb{Z}$, a black
square with a number $\blacksquare_{n}$ represents $\mathbb{Z}/{2^{n}}$. The
$x$ axis is $-s$ and the $y$ axis is $-w.$ Figure 2. The $E_{1}$-page of the
ESSS for $kq$ over an algebraically closed field. A bullet $\bullet$
represents $\mathbb{Z}/2[\tau]$ and a square $\blacksquare$ represents
$\mathbb{Z}[\tau]$. This figure appears with generators labeled as [BIK22,
Fig. 1]. Figure 3. The $E_{\infty}$-page of the ESSS for $kq$ over an
algebraically closed field. A black bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a red bullet $\bullet$ represents $\mathbb{Z}/2$, and a
square $\blacksquare$ represents $\mathbb{Z}[\tau]$. This figure appears with
generators labeled as [BIK22, Fig. 2]. Figure 4. The $E_{1}$-page of the ESSS
for $kq$ over $\mathbb{F}_{q}$. A bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a square $\blacksquare$ represents $\mathbb{Z}$, and a
diamond $\blackdiamond$ represents the $u$-divisible (if $q\equiv 1\mod 4$) or
$\rho$-divisible (if $q\equiv 3\mod 4$) part of
$\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})$. Figure 5. The $E_{\infty}$-page of
the ESSS for $kq$ over $\mathbb{F}_{q}$. A black bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a red bullet $\bullet$ represents $\mathbb{Z}/2$, a
square $\blacksquare$ represents $\mathbb{Z}$, and a diamond $\blackdiamond$
represents $\bigoplus_{i\geq
0}\mathbb{Z}/2^{\nu(q-1)+\nu(i+1)-1}\\{2\tau^{i}\\}$. The orange lines
indicate hidden $\mathsf{h}$-extensions. Figure 6. The $E_{1}$-page of the
ESSS for $kq$ over $\mathbb{Q}_{q}$. A bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a square $\blacksquare$ represents $\mathbb{Z}$, and a
diamond $\blackdiamond$ represents the $u$-divisible (if $q\equiv 1\mod 4$) or
$\rho$-divisible (if $q\equiv 3\mod 4$) part of
$\pi_{**}^{\mathbb{Q}_{q}}(H\mathbb{Z})$. Classes in olive are
$\pi$-divisible. Figure 7. The $E_{\infty}$-page of the ESSS for $kq$ over
$\mathbb{Q}_{q}$. A black bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a
red bullet $\bullet$ represents $\mathbb{Z}/2$, a square $\blacksquare$
represents $\mathbb{Z}$, and a diamond $\blackdiamond$ represents the group
obtained from the $u$-divisible (if $q\equiv 1\mod 4$) or $\rho$-divisible (if
$q\equiv 3\mod 4$) part of $\pi_{**}^{\mathbb{F}_{q}}(H\mathbb{Z})$ by
dividing the order by two. Classes in olive are $\pi$-divisible, and orange
lines indicate hidden $\mathsf{h}$-extensions. Figure 8. The $E_{1}$-page of
the ESSS for $kq$ over $\mathbb{Q}_{2}$. A bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a circle $\circ$ represents
$\mathbb{Z}/2[\tau^{2}]\\{\rho\\}$, a black square $\blacksquare$ represents
$\mathbb{Z}$, a rectangle $\square$ represents $\mathbb{Z}[\tau^{2}]\\{\tau
u\\}\oplus\mathbb{Z}\\{u\\}$, a diamond
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>represents $\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$, and an olive square
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>represents the $\pi$-divisible part of
$\pi_{-1,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$. Classes in olive are
$\pi$-divisible. The differentials are $h_{1}$-periodic; we only draw the
first occurrences. Figure 9. The $E_{\infty}$-page of the ESSS for $kq$ over
$\mathbb{Q}_{2}$. A bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a red
bullet $\bullet$ represents $\mathbb{Z}/2$, a bullet with subscript
$\bullet_{n}$ represents $\mathbb{Z}/2[\tau^{4}]\\{1,\tau^{n}\\}$, a bullet
with label $-1$ represents
$\mathbb{Z}/2[\tau]\\{\tau^{2}\\}\oplus\mathbb{Z}/2$, a circle $\circ$
represents $\mathbb{Z}/2[\tau^{2}]\\{\rho\\}$, a square $\blacksquare$
represents $\mathbb{Z}$, a rectangle $\square$ represents
$\mathbb{Z}[\tau^{2}]\\{\tau u\\}\oplus\mathbb{Z}\\{u\\}$, a diamond
${\blackdiamond}$ represents $\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$, and a
dark green diamond
${\color[rgb]{0.1796875,0.546875,0.33984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1796875,0.546875,0.33984375}\blackdiamond}$
represents $2\pi_{-2,*}^{\mathbb{Q}_{2}}(H\mathbb{Z})$. A curve represents
$(\mathbb{Z}/2[\tau^{4}])^{2}$ which is some linear combination of the classes
it connects (a faded bullet represents an $E_{1}$-page class that does not
exist on the $E_{2}$-page).

Figure 10. A graphical depiction of $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z}/2)$
(left) and $\pi_{**}^{\mathbb{Q}}(H\mathbb{Z})$ (right). In the left-hand
picture, a black bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a blue
bullet $\bullet$ represents $\bigoplus_{q\text{
prime}}\mathbb{Z}/2[\tau]\\{[q]\\}$, and a red bullet $\bullet$ represents
$\bigoplus_{q\text{ odd}}\mathbb{Z}/2[\tau]\\{a_{q}\\}$. In the right-hand
picture, a black square $\blacksquare$ represents $\mathbb{Z}$, a black bullet
$\bullet$ represents $\mathbb{Z}/2[\tau^{2}]$, a blue square $\blacksquare$
represents $\bigoplus_{q\text{ odd}}\mathbb{Z}\\{[q]\\}$, a red square
$\blacksquare$ represents $C^{\prime\prime\prime}(0)$, and a blue bullet
$\bullet$ represents $\bigoplus_{q\text{ odd}}\bigoplus_{i\geq
0}\mathbb{Z}/2^{s_{q}(i)}\\{[q]x_{q}\tau^{i}\\}$. Figure 11. The $E_{1}$-page
of the ESSS for $L$ over algebraically closed fields. A bullet $\bullet$
represents $\mathbb{Z}/2[\tau]$, a square $\blacksquare$ represents
$\mathbb{Z}[\tau]$, and a square with the positive integer $n$ as its right
subscript $\blacksquare_{n}$ represents $\mathbb{Z}/2^{n}[\tau]$. Elements of
the form $\iota x$ appear in dark green. This figure appears with generators
labeled as [BIK22, Fig. 3]. Figure 12. The $E_{\infty}$-page of the ESSS for
$L$ over algebraically closed fields. A bullet $\bullet$ represents
$\mathbb{Z}/2[\tau]$, a red bullet $\bullet$ represents $\mathbb{Z}/2$, a
square $\blacksquare$ represents $\mathbb{Z}[\tau]$, and a square with the
positive integer $n$ as its right subscript $\blacksquare_{n}$ represents
$\mathbb{Z}/2^{n}[\tau]$. Hidden extensions are depicted with orange lines.
This figure appears with generators labeled as [BIK22, Fig. 3]. Figure 13. The
$E_{1}$-page of the ESSS for $L$ over $\mathbb{F}_{q}$. A bullet $\bullet$
represents $\mathbb{Z}/2[\tau]$, a square $\blacksquare$ represents
$\mathbb{Z}$, a square with the positive integer $n$ as its left subscript
${}_{n}\blacksquare$ represents $\mathbb{Z}/2^{n}$, an isosceles triangle with
apex pointing right and nonnegative integer $n$ as its right subscript
$\blacktriangleright_{n}$ represents $K_{q}(n)$, and an isosceles triangle
with apex pointing left and nonnegative integer $n$ as its right subscript
$\blacktriangleleft_{n}$ represents $C_{q}(n)$. Elements of the form $\iota x$
appear in dark green. Figure 14. The $E_{\infty}$-page of the ESSS for $L$
over $\mathbb{F}_{q}$. A bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a
red bullet $\bullet$ represents $\mathbb{Z}/2$, a blue bullet $\bullet$
represents $\mathbb{Z}/2\\{1\\}\oplus\mathbb{Z}/2[\tau]\\{\tau^{2}\\}$, a
green bullet $\bullet$ represents the cokernel of the corresponding green
$d_{1}$-differential, a square $\square$ represents $\mathbb{Z}$, a square
with the positive integer $n$ as its left subscript represents
$\mathbb{Z}/2^{n}$, an isosceles triangle with apex pointing right and
nonnegative integer $n$ as its right subscript $\blacktriangleright_{n}$
represents $\tilde{K}_{q}(n)$, and an isosceles triangle with apex pointing
left and nonnegative integer $n$ as its right subscript
$\blacktriangleleft_{n}$ represents $\tilde{C}_{q}(n)$. Hidden extensions are
depicted with orange lines. Figure 15.  The $E_{1}$-page of the ESSS for $L$
over $\mathbb{Q}_{2}$. A bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a
triangle $\blacktriangle$ represents $(\mathbb{Z}/2[\tau])^{3}$, a square
$\blacksquare$ represents $\mathbb{Z}$, a square with the positive integer $n$
as its left subscript ${}_{n}\blacksquare$ represents $\mathbb{Z}/2^{n}$, an
isosceles triangle with apex pointing left and nonnegative integer $n$ as its
right subscript $\blacktriangleleft_{n}$
($\underline{\blacktriangleleft}_{n}$) represents $C_{2}(n)$
($\underline{C}_{2}(n)$), and an isosceles triangle with apex pointing right
and nonnegative integer $n$ as its right subscript $\blacktriangleright_{n}$
($\underline{\blacktriangleright}_{n}$) represents ${K}_{2}(n)$
($\underline{K}_{2}(n)$). Elements of the form $\iota x$ appear in dark green.
The differentials are $h_{1}$-periodic; we only draw the first occurences.
Figure 16. The $E_{\infty}$-page of the ESSS for $L$ over $\mathbb{Q}_{2}$. A
bullet $\bullet$ represents $\mathbb{Z}/2[\tau]$, a red bullet $\bullet$
represents $\mathbb{Z}/2$, a green bullet $\bullet$ represents
$\mathbb{Z}/2[\tau^{4}]$, a bullet with subscript $\bullet_{n}$ represents
$\mathbb{Z}/2[\tau^{4}]\\{1,\tau^{n}\\}$, a triangle $\blacktriangle$
represents $(\mathbb{Z}/2[\tau])^{3}$, a red triangle
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\blacktriangle$
represents $(\mathbb{Z}/2)^{3}$, a overlined triangle
$\overline{\blacktriangle}$ represents
$\mathbb{Z}/2[\tau^{2}]\oplus\mathbb{Z}/2[\tau^{2}]\\{\tau^{3}\\}$, a green
triangle
$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\blacktriangle$
represents
$\mathbb{Z}/2[\tau^{2}]\oplus\mathbb{Z}/2[\tau^{2}]\\{\tau^{3}\\}\oplus\mathbb{Z}/2[\tau^{4}]\\{1,\tau,\tau^{3}\\}$,
a square $\blacksquare$ represents $\mathbb{Z}$, a square with the positive
integer $n$ as its left subscript ${}_{n}\blacksquare$ represents
$\mathbb{Z}/2^{n}$, an isosceles triangle with apex pointing left and
nonnegative integer $n$ as its right subscript
$\blacktriangleleft_{n}({\color[rgb]{0.1796875,0.546875,0.33984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1796875,0.546875,0.33984375}\blacktriangleleft_{n}},\underline{\blacktriangleleft_{n}},{\color[rgb]{0.1796875,0.546875,0.33984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1796875,0.546875,0.33984375}\underline{\blacktriangleleft}_{n}})$
represents $C_{2}(n)$ ($\tilde{C}_{2}(n)$, $\underline{C}_{2}(n)$,
$\tilde{\underline{C}_{2}}(n)$), and an isosceles triangle with apex pointing
right and nonnegative integer $n$ as its right subscript
$\blacktriangleright_{n}({\underline{\blacktriangleright}_{n}},{\color[rgb]{0.1796875,0.546875,0.33984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1796875,0.546875,0.33984375}\blacktriangleright_{n}})$
represents ${K}_{2}(n)$ (${\underline{K}_{2}}(n)$, $\tilde{K_{2}}(n)$). A
curve represents $(\mathbb{Z}/2[\tau^{4}])^{2}$ which is some linear
combination of the classes it connects (a faded bullet represents a
$E_{1}$-page class that does not exist on the $E_{2}$-page).

## References

  * [AM17] Michael Andrews and Haynes Miller.  Inverting the Hopf map.  J. Topol., 10(4):1145–1168, 2017.
  * [ARØ20] Alexey Ananyevskiy, Oliver Röndigs, and Paul Arne Østvær.  On very effective hermitian $K$-theory.  Math. Z., 294(3-4):1021–1034, 2020.
  * [Bac22] Tom Bachmann.  $\eta$-periodic motivic stable homotopy theory over Dedekind domains.  J. Topol., 15(2):950–971, 2022.
  * [BCQ21] William Balderrama, Dominic Leon Culver, and J.D. Quigley.  The motivic lambda algebra and motivic Hopf invariant one problem.  arXiv preprint arXiv:2112.07479, 2021.
  * [BH20] Tom Bachmann and Michael J Hopkins.  $\eta$-periodic motivic stable homotopy theory over fields.  arXiv preprint arXiv:2005.06778, 2020.
  * [BI22] Eva Belmont and Daniel C. Isaksen.  $\mathbb{R}$-motivic stable stems.  J. Topol., 15(4):1755–1793, 2022.
  * [BIK22] Eva Belmont, Daniel C Isaksen, and Hana Jia Kong.  $\mathbb{R}$-motivic $v_{1}$-periodic homotopy.  arXiv preprint arXiv:2204.05937, 2022.
  * [BK05] A. J. Berrick and M. Karoubi.  Hermitian $K$-theory of the integers.  Amer. J. Math., 127(4):785–823, 2005.
  * [Boa99] J. Michael Boardman.  Conditionally convergent spectral sequences.  In Homotopy invariant algebraic structures (Baltimore, MD, 1998), volume 239 of Contemp. Math., pages 49–84. Amer. Math. Soc., Providence, RI, 1999.
  * [CQ21] Dominic Leon Culver and J.D. Quigley.  $kq$-resolutions I.  Trans. Amer. Math. Soc., 374(7):4655–4710, 2021.
  * [DI10] Daniel Dugger and Daniel C. Isaksen.  The motivic Adams spectral sequence.  Geom. Topol., 14(2):967–1014, 2010.
  * [DI17a] Daniel Dugger and Daniel Isaksen.  Low-dimensional Milnor-Witt stems over $\mathbb{R}$.  Ann. $K$-Theory, 2(2):175–210, 2017.
  * [DI17b] Daniel Dugger and Daniel C. Isaksen.  $\mathbb{Z}/2$-equivariant and $\mathbb{R}$-motivic stable stems.  Proc. Am. Math. Soc., 145(8):3617–3627, 2017.
  * [Fri76] Eric M. Friedlander.  Computations of K-theories of finite fields.  Topology, 15:87–109, 1976.
  * [GI15] Bertrand J. Guillou and Daniel C. Isaksen.  The $\eta$-local motivic sphere.  J. Pure Appl. Algebra, 219(10):4728–4756, 2015.
  * [GI16] Bertrand J. Guillou and Daniel C. Isaksen.  The $\eta$-inverted $\mathbb{R}$-motivic sphere.  Algebr. Geom. Topol., 16(5):3005–3027, 2016.
  * [Hil11] Michael A. Hill.  Ext and the motivic Steenrod algebra over $\mathbb{R}$.  J. Pure Appl. Algebra, 215(5):715–727, 2011.
  * [HKO11] P. Hu, I. Kriz, and K. Ormsby.  Convergence of the motivic Adams spectral sequence.  J. $K$-Theory, 7(3):573–596, 2011.
  * [IØ20] Daniel C. Isaksen and Paul Arne Østvær.  Motivic stable homotopy groups.  In Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., pages 757–791. CRC Press, Boca Raton, FL, [2020] ©2020.
  * [Isa19] Daniel C. Isaksen.  Stable stems.  Mem. Amer. Math. Soc., 262(1269):viii+159, 2019.
  * [IWX20] Daniel C Isaksen, Guozhen Wang, and Zhouli Xu.  More stable stems.  arXiv preprint arXiv:2001.04511, 2020.
  * [KQ22] Hana Jia Kong and J.D. Quigley.  $v_{1}$-periodic motivic homotopy over rings of integers of number fields.  In preparation, 2022.
  * [KRØ20] Jonas Irgens Kylling, Oliver Röndigs, and Paul Arne Østvær.  Hermitian $K$-theory, Dedekind $\zeta$-functions, and quadratic forms over rings of integers in number fields.  Camb. J. Math., 8(3):505–607, 2020.
  * [Kyl15] Jonas Irgens Kylling.  Hermitian K-theory of finite fields via the motivic Adams spectral sequence.  Master’s thesis, University of Oslo, Norway, 2015.
  * [Lev08] Marc Levine.  The homotopy coniveau tower.  J. Topol., 1(1):217–267, 2008.
  * [Mil70] John Milnor.  Algebraic $K$-theory and quadratic forms.  Invent. Math., 9:318–344, 1969/70.
  * [Mor12] Fabien Morel.  $\mathbb{A}^{1}$-algebraic topology over a field, volume 2052 of Lecture Notes in Mathematics.  Springer, Heidelberg, 2012.
  * [MV99] Fabien Morel and Vladimir Voevodsky.  $\mathbb{A}^{1}$-homotopy theory of schemes.  Publ. Math., Inst. Hautes Étud. Sci., 90:45–143, 1999.
  * [OØ13] Kyle M. Ormsby and Paul Østvær.  Motivic Brown-Peterson invariants of the rationals.  Geom. Topol., 17(3):1671–1706, 2013.
  * [OØ14] Kyle M. Ormsby and Paul Arne Østvær.  Stable motivic $\pi_{1}$ of low-dimensional fields.  Adv. Math., 265:97–131, 2014.
  * [OR20] Kyle Ormsby and Oliver Röndigs.  The homotopy groups of the $\eta$-periodic motivic sphere spectrum.  Pacific J. Math., 306(2):679–697, 2020.
  * [Orm11] Kyle M. Ormsby.  Motivic invariants of $p$-adic fields.  J. K-Theory, 7(3):597–618, 2011.
  * [Qui21a] J.D. Quigley.  Motivic Mahowald invariants over general base fields.  Doc. Math., 26:561–582, 2021.
  * [Qui21b] J.D. Quigley.  Real motivic and $C_{2}$-equivariant Mahowald invariants.  J. Topol., 14(2):369–418, 2021.
  * [RØ16] Oliver Röndigs and Paul Østvær.  Slices of Hermitian $K$-theory and Milnor’s conjecture on quadratic forms.  Geom. Topol., 20(2):1157–1212, 2016.
  * [RSØ19] Oliver Röndigs, Markus Spitzweck, and Paul Østvær.  The first stable homotopy groups of motivic spheres.  Ann. Math. (2), 189(1):1–74, 2019.
  * [RSØ21] Oliver Röndigs, Markus Spitzweck, and Paul Arne Østvær.  The second stable homotopy groups of motivic spheres.  arXiv preprint arXiv:2103.17116, 2021.
  * [RW00] J. Rognes and C. Weibel.  Two-primary algebraic $K$-theory of rings of integers in number fields.  J. Amer. Math. Soc., 13(1):1–54, 2000.  Appendix A by Manfred Kolster.
  * [Voe03] Vladimir Voevodsky.  Reduced power operations in motivic cohomology.  Publ. Math., Inst. Hautes Étud. Sci., 98:1–57, 2003.
  * [Voe11] Vladimir Voevodsky.  On motivic cohomology with $\mathbb{Z}/l$-coefficients.  Ann. Math. (2), 174(1):401–438, 2011.
  * [Wil16] Glen M. Wilson.  Motivic stable stems over finite fields.  ProQuest LLC, Ann Arbor, MI, 2016.  Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick.
  * [Wil18] Glen Matthew Wilson.  The eta-inverted sphere over the rationals.  Algebr. Geom. Topol., 18(3):1857–1881, 2018.
  * [WØ17] Glen Matthew Wilson and Paul Østvær.  Two-complete stable motivic stems over finite fields.  Algebr. Geom. Topol., 17(2):1059–1104, 2017.