Arithmetic Kei Theory

Ariel Davis, Tomer M. Schlank

Abstract

A min\CJKtilde\CJKnospace 圭 (kei), or 2-quandle, is an algebraic structure one can use to produce a numerical invariant of links, known as coloring invariants. Motivated by Mazur’s analogy between prime numbers and knots, we define for every finite kei $\mathscr{K}$ an analogous coloring invariant $\textnormal{col}_{\mathscr{K}}(n)$ of square-free integers. This is achieved by defining a fundamental kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ for every such $n$ . We conjecture that the asymptotic average order of $\textnormal{col}_{\mathscr{K}}$ can be predicted to some extent by the colorings of random braid closures. This conjecture is fleshed out in general, building on previous work, and then proven for several cases.

M. C. Escher, Knot, 1966

1 Introduction

There is an analogy that likens prime numbers $p$ to knots in the sphere $S^{3}$ . This analogy is the observation of Barry Mazur regarding the embedding

\textnormal{Spec}(\mathbb{F}_{p})\hookrightarrow\textnormal{Spec}(\mathbb{Z})

(1)

of schemes, as seen through the eyes of étale topology and homotopy. Étale topology is a tool in algebraic geometry developed by Grothendieck et al in [BD06]. It enriches the theory of schemes with topological notions such as well-behaved cohomology groups for finite locally-constant sheaves. While the concept was originally conceived with smooth varieties over fields in mind, one can attach an étale topology (or more precisely, an étale site) to any scheme. In the case of spectra of rings of integers in number fields, this provides the framework for Artin-Verdier duality. Through Artin-Verdier duality we are justified in thinking about the geometric objects attached to rings of integers as smooth threefolds - see [AV64] or [Maz73] for details. The étale homotopy type defined by M. Artin and Mazur in [AM06] takes this idea even further, in granting access to a wider array of hompotopical tools by producing for any scheme $X$ a profinite approximation $\acute{E}t(X)$ of a homotopy type. The homotopy type $\acute{E}t(\textnormal{Spec}(\mathbb{Z}))$ is simply-connected, and $\acute{E}t(\textnormal{Spec}(\mathbb{F}_{p}))$ is a profinite circle. Therefore étale homotopy sees (1) as the embedding of a circle into a simply-connected threefold - that is, a knot - and for square-free $n\in\mathbb{N}$ , the embedding

\textnormal{Spec}(\mathbb{Z}/n\mathbb{Z})\simeq\coprod_{p|n}\textnormal{Spec}(\mathbb{F}_{p})\hookrightarrow\textnormal{Spec}(\mathbb{Z})

as a disjoint union of knots - a link. One is motivated to understand how far we can take this analogy. Various notions from knot theory have been interpreted in number theory based on this analogy, many of which can be found in Morishita’s book [Mor12]. One salient example is the linking number $link(K_{1},K_{2})$ of two oriented knots $K_{1},K_{2}$ , which corresponds to the homology class

[K_{1}]\in H_{1}(S^{3}\setminus K_{2};\mathbb{Z})\simeq\mathbb{Z}.

A theorem by Gauss famously states that the linking number is symmetric in $K_{1},K_{2}$ . The Jacobi symbol $\left(\frac{p}{q}\right)$ of odd primes $p\neq q$ is seen as analogous to

(-1)^{link(K_{1},K_{2})},

and quadratic reciprocity as a manifestation the linking number’s symmetry. A more sophisticated example is the Alexander module of a knot $K$ , closely related to the Alexander polynomial of $K$ . This has an arithmetic interpretation as the Iwasawa module of a prime $p$ . The analogy between knots and primes does not produce de facto links in $S^{3}$ for individual square-free integers $n\in\mathbb{N}$ . That is, while certain knot-theoretical notions are emulated in number theory, the values that they take are not recovered knot-theoretically from any knot or link. Indeed if that were the case, we would expect actual symmetry in quadratic reciprocity.

There is however a way in which one can say that random square-free integers resemble random links. Alexander’s theorem states in [Ale23] that every link is obtained as the closure $\overline{\sigma}$ of a braid $\sigma$ . For $k\in\mathbb{N}$ , the set of all braids on $k$ strands, together with concatenation, form the Artin braid group $B_{k}$ - see [Art47]. The naive notion of a random braid on $k$ strands is ill-defined because $B_{k}$ is infinite discrete for $k\geq 2$ . That said, if a function $f\colon B_{k}\to\mathbb{R}$ factors through a finite quotient of $B_{k}$ , then we consider the locally-constant extension $\widehat{f}$ of $f$ to $\widehat{B}_{k}$ , the profinite completion of $B_{k}$ . In this case we interpret “the distribution of $f$ ” to mean the distribution of $\widehat{f}$ , where $\widehat{B}_{k}$ has normalized Haar measure. Such is the case with the number of connected components in the closure $\overline{\sigma}$ . For $\sigma\in B_{k}$ , the number of connected components of $\overline{\sigma}$ equals

|\pi_{0}(\overline{\sigma})|=|\{1,\dots,k\}/\sigma|.

This is determined by how $\sigma$ permutes the endpoints of the $k$ strands. Hence

|\pi_{0}(\overline{\sigma})|\colon B_{k}\to\mathbb{N}

factors through the symmetric group $S_{k}$ . As per the above, the closure of a random braid $\sigma\in B_{k}$ is a knot with probability $\frac{1}{k}$ . For square-free $n\in\mathbb{N}$ , the connected components of $\acute{E}t(\textnormal{Spec}(\mathbb{Z}/n\mathbb{Z}))$ correspond to the prime factors of $n$ . A square-free number $n\in\mathbb{N}$ of magnitude $X$ is prime with probability roughly $\frac{1}{\log X}$ . From here one draws the analogy that

\begin{matrix}\textnormal{Random square-free integers $n\in\mathbb{N}$ of magnitude $X$ resemble}\\ \textnormal{closures of random braids on approximately $\log X$ strands.}\end{matrix}

(2)

Another numerical invariant of links, introduced by Fox in [CF12]¹¹1See [Prz06] for a more comprehensive treatment of the subject., is the number of $3$ -colorings of a link $L$ . A $3$ -coloring of a link diagram $D$ constitutes a choice of color $x\in\mathbb{Z}/3\mathbb{Z}$ for each arc in $D$ such that at each crossing, the colors $x,y,z$ as in fig. 1 must satisfy $y+z=2x$ . This condition ensures that the number of $3$ -colorings is invariant under Reidemeister moves, and by [Rei48] is therefore an isotopy invariant of the link.

Figure 1:

The fact that $z$ in fig. 1 is determined by $x,y$ is necessary in order to transport colorings across a Reidemeister $2$ move, as depicted in fig. 2b. The concept of $3$ -colorings thus generalizes to coloring links using a set of colors $\mathscr{K}$ equipped with a binary operator

(x,y)\mapsto{{}^{x}{y}}.

Here a $\mathscr{K}$ -coloring of a crossing as in fig. 1 is valid if $z={{}^{x}{y}}$ . For $\mathscr{K}$ -colorings to be transported bijectively across all Reidemeister moves,

(a) R1

(b) R2

Figure 2:

this operator must satisfy the following corresponding axioms:

Definition 1.1 ([Tak43]²²2The terminology and Kanji notation min\CJKtilde\CJKnospace 圭 are taken from [Tak43], where the notion was first defined. See [Joy82] for keis in relation to knot theory, where they are called $2$ -quandles.).

A min\CJKtilde\CJKnospace 圭 , or kei, is a set $\mathscr{K}$ with binary operator

\mathscr{K}\times\mathscr{K}\to\mathscr{K}\;\;,\;\;\;\;x,y\mapsto{{}^{x}{y}}

satisfying

1.

${{}^{x}{x}}=x$ for all $x\in\mathscr{K}$ .
2.

${{}^{x}{({{}^{x}{y}})}}=y$ for all $x,y\in\mathscr{K}$ .
3.

${{}^{x}{({{}^{y}{z}})}}={{}^{({{}^{x}{y}})}{({{}^{x}{z}})}}$ for all $x,y,z\in\mathscr{K}$ .

Keis are therefore an abstaction of Fox’s three colors, and kei-colorings a generalization of $3$ -colorings. For finite kei $\mathscr{K}$ , the number $\textnormal{col}_{\mathscr{K}}(L)$ of $\mathscr{K}$ -colorings of a link $L$ is therefore an isotopy invariant of $L$ .

The first goal of this paper is to define a number-theoretical analogue of $\mathscr{K}$ -colorings. As is turns out, the kei-colorings of a link $L$ are goverened by a universal kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ associated to $L$ . That is, the colorings of $L$ are in natural bijection with representations of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ :

\textnormal{Col}_{\mathscr{K}}(L)\simeq\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L},\mathscr{K}),

where ${\mathscr{K}\textnormal{ei}}$ denotes the category of keis and suitable morphisms. The kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ is constructed explicitly with generators and relations extracted from a diagram $D$ of $L$ : to every arc in $D$ there is a corresponding generator, and to every crossing a corresponding relation. It is not difficult to see that $\mathscr{K}$ -colorings of a diagram $D$ correspond to morphisms $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}\to\mathscr{K}$ . The road to $\mathscr{K}$ -coloring squarefree integers $n$ rests on the construction of an analogous $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ for squarefree $n\in\mathbb{N}$ . For this, a presentation extracted from a diagram is unsuitable. While an instrinsically topological description of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ is already given in [Joy82], it is the following presentation, based on the work of Winker ([Win84]), that we make the most of (for details see section 3.1). We take $M_{L}$ to be the unique branched double cover of $S^{3}$ with ramification locus equal to $L$ , and $\widetilde{M}_{L}$ to be the universal cover of $M_{L}$ . Then as a set,

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}=\pi_{0}(\widetilde{M}_{L}\times_{S^{3}}L).

The arithmetic analogue of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ follows immediately. For simplicity assume $n$ is odd:

Definition 1.2 (definition 3.4).

Let $n\in\mathbb{N}$ be odd squarefree. By $\mathfrak{L}_{n}$ we denote the field

\mathfrak{L}_{n}=\mathbb{Q}(\sqrt{n^{*}})\;\;,\;\;\;\;n^{*}:=(-1)^{\frac{n-1}{2}}n.

The field $\mathfrak{L}_{n}$ is the unique³³3This assuming $n$ is odd. For even $n$ there are three such fields. In the body of the paper we extend this definition for even $n$ . Of the three quadratic fields that ramify precisely at $n$ we choose $\mathbb{Q}(\sqrt{n^{*}})$ with $n^{*}=\pm n$ s.t. $n^{*}=2\mod 8$ . While one can consider the other options, this is the one we find most suitable to our needs. quadratic number field ramified at precisely $n$ . As such, the morphism

\textnormal{Spec}(\mathcal{O}_{\mathfrak{L}_{n}})\to\textnormal{Spec}(\mathbb{Z})

is analogous to the branched double cover $M_{L}\to S^{3}$ .

Definition 1.3 (definition 3.7).

Let $n\in\mathbb{N}$ be squarefree. By $\mathfrak{F}_{n}$ we denote the maximal unramified extension of $\mathfrak{L}_{n}$ . We then define $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ to be the profinite set

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}:=\textnormal{Spec}(\mathcal{O}_{\mathfrak{F}_{n}})\times_{\textnormal{Spec}(\mathbb{Z})}\textnormal{Spec}(\mathbb{Z}/n\mathbb{Z})\simeq\{\mathfrak{p}\textnormal{ prime in }\mathcal{O}_{\mathfrak{F}_{n}}\textnormal{ s.t. }\mathfrak{p}|n\}.

Theorem 1.4 (proposition 3.15 + proposition 3.12 + definition 3.13).

1.

The set $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ has a natural structure of a (profinite) kei.

For finite $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ , the set of continuous morphisms

\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{K})

is finite.

This allows us to define $\mathscr{K}$ -colorings of squarefree integers $n\in\mathbb{N}$ using $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ by mimicking the coloring of links:

Definition 1.5 (definition 3.17).

Let $n\in\mathbb{N}$ be squarefree and let $\mathscr{K}$ be a finite kei. We define

\textnormal{col}_{\mathscr{K}}(n):=\left|\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{K})\right|\in\mathbb{N}.

Next, it is our goal to understand the behavior of this function. In particular, we wish to understand the average asymptotic order of $\textnormal{col}_{\mathscr{K}}(n)$ .

Definition 1.6 (definition 4.5).

Let $\mathscr{K}$ be a finite kei and let $X\geq 1$ . We denote by

\textnormal{Avg}_{\mathscr{K}}(X)=\frac{\sum\limits_{\begin{subarray}{c}1\leq n\leq X\\ n\;\textrm{sqr-free}\end{subarray}}\textnormal{col}_{\mathscr{K}}(n)}{\sum\limits_{\begin{subarray}{c}1\leq n\leq X\\ n\;\textrm{sqr-free}\end{subarray}}1}\in\mathbb{Q}

the average of $\textnormal{col}_{\mathscr{K}}(n)$ over the set of squarefree integers $1\leq n\leq X$ .

Recall the idea in (2), as pertaining to certain random variables on $\widehat{B}_{k}$ . In similar fashion we may interpret the distribution of $\mathscr{K}$ -colorings of closures of random braids as above. In [DS23] we compute the average number of $\mathscr{K}$ -colorings of braid closures $\overline{\sigma}$ of braids $\sigma$ in $B_{k}$ , and prove the following theorem:

Theorem 1.7 ([DS23, Theorem 1.1]).

Let $\mathscr{K}$ be a finite kei. There exists an integer-valued polynomial $P_{\mathscr{K}}\in\mathbb{Q}[x]$ such that for all $k\gg 0$ ,

\int\limits_{\widehat{B}_{k}}{\textnormal{c}\widehat{\textnormal{o}}\textnormal{l}}_{\mathscr{K}}(\sigma)d\mu=P_{\mathscr{K}}(k).

This, and (2) motivate us to form a conjecture about the growth of $\textnormal{Avg}_{\mathscr{K}}(X)$ as $X$ tends to infinity. In simplified form:

Conjecture 1.8 (definition 4.6 + 4.8).

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ be finite. Then there exists $a_{\mathscr{K}}>0$ such that

\textnormal{Avg}_{\mathscr{K}}(X)=a_{\mathscr{K}}\log^{\deg P_{\mathscr{K}}}(X)\cdot(1+o(1))

as $X\to\infty$ .

The second goal of this paper is to verify the conjecture for several examples of finite keis. In particular we verify the conjecture for all keis $\mathscr{K}$ of size $|\mathscr{K}|=3$ . One can easily show that every such kei is isomorphic to one of three keis, encoded in the following table by

\varphi\colon\{1,2,3\}\to S_{3}\;\;,\;\;\;\;\varphi_{i}(j)={{}^{i}{j}}.

$\mathscr{K}$	$\varphi_{1}$	$\varphi_{2}$	$\varphi_{3}$
$\mathscr{T}_{3}$	Id	Id	Id
$\mathscr{J}$	Id	Id	$(1,2)$
$\mathscr{R}_{3}$	$(2,3)$	$(1,3)$	$(1,2)$

We note that one recovers Fox’s $3$ -colorings mentioned above as $\mathscr{R}_{3}$ -colorings of links. We begin by identifying the function $\textnormal{col}_{\mathscr{K}}$ for each $\mathscr{K}\in\{\mathscr{T}_{3},\mathscr{J},\mathscr{R}_{3}\}$ :

Proposition 1.9.

For $1<n\in\mathbb{N}$ squarefree we have:

1.

Proposition 6.2:

$\textnormal{col}_{\mathscr{T}_{3}}(n)=3^{\omega(n)},$

where $\omega(n)$ is the number of prime divisors of $n$ .
2.

Proposition 7.3:

$\textnormal{col}_{\mathscr{J}}(n)=\sum_{abc=n}\left(\frac{a}{b}\right).$
3.

Proposition 8.8:

$\textnormal{col}_{\mathscr{R}_{3}}(n)=3\left|Cl\left({\mathfrak{L}_{n}}\right)\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right|.$

The polynomials

P_{\mathscr{T}_{3}}(x)=\frac{x^{2}+3x+2}{2}\;\;,\;\;\;\;P_{\mathscr{J}}(x)=2x+1\;\;,\;\;\;\;P_{\mathscr{R}_{3}}(x)=6

are computed in [DS23, §7]. Using methods of analytic number theory, we prove 1.8 for these three cases, and for additional related keis - see propositions 6.4, 7.7 and 8.14.

1.1 Structure of the Paper

In section 2 we lay the groundwork for defining $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . Here you’ll find definitions and statements for the algebraic theory of keis, some old, some new. Then in section 3 we define $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ itself, and $\mathscr{K}$ -colorings of $n$ for finite kei $\mathscr{K}$ . In section 4 we formulate the main conjecture of this paper. Subsequent sections are devoted to proving the conjecture in for several examples of keis: section 5 contains computations of some preliminary estimations we’ll need for the actual results. Sections 6 through 8 contain concrete examples:

•

section 6 - for trivial keis $\mathscr{T}_{\mathcal{a}}$
•

section 7 - for a kei $\mathscr{J}$ introduced in [Joy82] and variants.
•

section 8 - for the dihedral kei $\mathscr{R}_{3}$ .

1.2 Relation to Other Works

Our work is grounded in arithmetic statistics, an active field of research with a long, rich history history. For example, our proof of the main conjecture in the case of $\mathscr{K}=\mathscr{R}_{3}$ relies heavily on the Davenport-Heilbronn theorem [DH71] and further refinements [BST13] on counting cubic fields. The Cohen-Lenstra-Martinet heuristics [CL84] on the distribution of class groups is very much an ongoing endeavor [SW23]. In the case where one replaces $\textnormal{Spec}(\mathbb{Z})$ with $\textnormal{Spec}(\mathbb{F}_{p}[t])$ , the relation with étale topology has been studied in works such as [EVW16] and [FW18]. Others have considered keis and related structures in a number-theoretical setting, with different motivation ([Tak19]). The notions obtained there are different, and unrelated to arithmetic statistics as far as we know.

1.3 Acknowledgements

We thank Ohad Feldheim, Ofir Gorodetski, Aaron Landesman, Lior Yanovski, David Yetter and Tamar Ziegler for useful discussion. TMS was supported by the US-Israel Binational Science Foundation under grant 2018389. TMS was supported by ISF1588/18 and the ERC under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 101125896).

2 Keis and Profinite Keis

2.1 Keis and Augmented Keis

Definition 2.1.

A kei is a set $\mathscr{K}$ with a binary operator

\mathscr{K}\times\mathscr{K}\to\mathscr{K}\;\;,\;\;\;\;(x,y)\mapsto{{}^{x}{y}}

satisfying the following three axioms:

(K1)

$\forall x\in\mathscr{K}$ , ${{}^{x}{x}}=x$ .
(K2)

$\forall x,y\in\mathscr{K}$ , ${{}^{x}{({{}^{x}{y}})}}=y$ .
(K3)

$\forall x,y,z\in\mathscr{K}$ , ${{}^{x}{({{}^{y}{z}})}}={{}^{({{}^{x}{y}})}{({{}^{x}{z}})}}$ .

A morphism $f\colon\mathscr{K}\to{\mathscr{K}^{\prime}}$ of keis is a map satisfying

\forall x,y\in\mathscr{K}\;\;,\;\;\;\;f({{}^{x}{y}})={{}^{f(x)}{f(y)}}.

Together with morphisms, keis form a category, denoted by ${\mathscr{K}\textnormal{ei}}$ .

Example 2.2.

Let $G\in{\mathscr{G}\textnormal{rp}}$ be a group. Then the set $S=\{g\in G\;|\;g^{2}=1\}\subseteq G$ of involutions in $G$ is a kei with structure given by conjugation:

{{}^{g}{h}}=ghg^{-1}.

Definition 2.3.

A kei $\mathscr{T}\in{\mathscr{K}\textnormal{ei}}$ is trivial if ${{}^{x}{y}}=y$ for all $x,y\in\mathscr{T}$ . For ${\mathcal{a}}\in\mathbb{N}$ , we denote by $\mathscr{T}_{\mathcal{a}}$ the trivial kei with $|\mathscr{T}_{\mathcal{a}}|={\mathcal{a}}$ . This is well-defined up to isomorphism.

Definition 2.4.

An augmented kei $(\mathscr{K},G,\alpha)$ consists of a set $\mathscr{K}\in{\mathscr{S}\textnormal{et}}$ , a group $G\in{\mathscr{G}\textnormal{rp}}$ acting on $\mathscr{K}$ from the left, and an augmentation map - a function

\alpha\colon\mathscr{K}\to G\;,\;\;x\mapsto\alpha_{x}

satisfying:

1.

For all $x\in\mathscr{K}$ , $\alpha_{x}(x)=x$ .
2.

For all $x\in\mathscr{K}$ , $\alpha_{x}^{2}=1_{G}.$
3.

For all $g\in G$ and $x\in\mathscr{K}$ , $\alpha_{g(x)}=g\alpha_{x}g^{-1}\in G$ .

A morphism of augmented keis $(\mathscr{K},G,\alpha)\to(\mathscr{K}^{\prime},G^{\prime},\alpha^{\prime})$ consists of a function $f\colon\mathscr{K}\to\mathscr{K}^{\prime}$ and group homomorphism $\widetilde{f}\colon G\to G^{\prime}$ satisfying

1.

For every $x\in\mathscr{K}$ , $\alpha^{\prime}_{f(x)}=\widetilde{f}(\alpha_{x})\in G^{\prime}$ .
2.

For every $x\in\mathscr{K}$ and $g\in G$ , $f(g(x))=\widetilde{f}(g)(f(x))\in\mathscr{K}^{\prime}$ .

In other words, the following diagram commutes:

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(3)

Together with morphisms, augmented keis form a category ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ .

Example 2.5.

Let $G\in{\mathscr{G}\textnormal{rp}}$ group and let $X\subseteq G$ be a union of conjugacy classes in $G$ s.t. $x^{2}=1_{G}$ for all $x\in X$ . Let $\iota\colon X\hookrightarrow G$ denote the inclusion. Then $(X,G,\iota)$ is an augmented kei, where for all $x\in X$ and $g\in G$ ,

g(x)=gxg^{-1}.

As the name suggests, for $(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ , the set $\mathscr{K}$ has a canonical kei structure. We omit the proof of the following easily verifiable proposition:

Proposition 2.6.

For every $(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ , the operator

\mathscr{K}\times\mathscr{K}\to\mathscr{K}\;\;\;,\;\;\;\;\;\;(x,y)\mapsto\alpha_{x}(y)

defines a kei structure on $\mathscr{K}$ . This extends to a functor

{\textnormal{Kei}_{\bullet}}\colon{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}\to{\mathscr{K}\textnormal{ei}}\;\;\;,\;\;\;\;\;\;\mathscr{A}=(\mathscr{K},G,\alpha)\mapsto\mathscr{K}.

Definition 2.7.

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ and let $x\in\mathscr{K}$ . We denote by

\varphi_{x}=\varphi_{\mathscr{K},x}\colon\mathscr{K}\to\mathscr{K}\;\;:\;\;\;\;\forall y\in\mathscr{K}\;,\;\varphi_{x}(y)={{}^{x}{y}}

By (K2), $\varphi_{x}$ is a bijection and by (K3), $\varphi_{x}\in\textnormal{Aut}_{\mathscr{K}\textnormal{ei}}(\mathscr{K})$ . By $\textnormal{Inn}(\mathscr{K})$ we denote the subgroup

\textnormal{Inn}(\mathscr{K})=\langle\{\varphi_{x}\;|\;x\in\mathscr{K}\}\rangle\leq\textnormal{Aut}_{\mathscr{K}\textnormal{ei}}(\mathscr{K}),

and by $\varphi_{\mathscr{K}}\colon\mathscr{K}\to\textnormal{Inn}(\mathscr{K})$ the map

\varphi_{\mathscr{K}}\colon x\longmapsto\varphi_{\mathscr{K},x}.

The following notion of conciseness is not new - see [Joy82, Thm 10.2] for one of several examples of prior usage. We rely heavily on this property, and therefore see fit to give it a name:

Definition 2.8.

An augmented kei $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ is concise if the image of $\alpha$ generates $G$ , that is,

\langle\{\alpha_{x}\;|\;x\in\mathscr{K}\}\rangle=G.

Proposition 2.9.

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ . Then $(\mathscr{K},\textnormal{Inn}(\mathscr{K}),\varphi_{\mathscr{K}})$ is a concise augmented kei.

Proof.

Let $x\in\mathscr{K}$ . Then

\varphi_{x}(x)={{}^{x}{x}}=x,

and for all $y\in\mathscr{K}$ ,

\varphi_{x}^{2}(y)={{}^{x}{({{}^{x}{y}})}}=y.

Therefore $\varphi_{x}^{2}=\textnormal{Id}_{\mathscr{K}}$ . For $g=\varphi_{y}\in\textnormal{Inn}(\mathscr{K})$ we have for all $z\in\mathscr{K}$ ,

(\varphi_{g(x)}\circ g)(z)=(\varphi_{\varphi_{y}(x)}\circ\varphi_{y})(z)={{}^{({{}^{y}{x}})}{({{}^{y}{z}})}}={{}^{y}{({{}^{x}{z}})}}=(g\circ\varphi_{x})(z).

That is,

\varphi_{g(x)}=g\circ\varphi_{x}\circ g^{-1}.

This equality holds for all $g\in\langle\{\varphi_{y}\;|\;y\in\mathscr{K}\}\rangle=\textnormal{Inn}(\mathscr{K})$ . Hence $(\mathscr{K},\textnormal{Inn}(\mathscr{K}),\varphi_{\mathscr{K}})$ is a concise augmented kei. ∎

Definition 2.10.

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ . By $\mathscr{A}_{\mathscr{K}}\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ we denote the augmented kei

\mathscr{A}_{\mathscr{K}}:=(\mathscr{K},\textnormal{Inn}(\mathscr{K}),\varphi_{\mathscr{K}}).

Proposition 2.11.

Let $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ be concise, and let $f\colon\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime}$ be a surjective morphism in ${\mathscr{K}\textnormal{ei}}$ . Then there is a unique $\rho\colon G\to\textnormal{Inn}(\mathscr{K}^{\prime})$ s.t.

(f,\rho)\colon(\mathscr{K},G,\alpha)\to\mathscr{A}_{\mathscr{K}^{\prime}}

is a morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . Furthermore, $\rho$ is surjective.

Proof.

The desired $\rho\colon G\to\textnormal{Inn}(\mathscr{K}^{\prime})$ must satisfy, for all $x\in\mathscr{K}$ and $g\in G$ :

(i):\;\rho(\alpha_{x})=\varphi_{f(x)}\;\;\;\;\textrm{and}\;\;\;\;(ii):\;\rho(g)(f(x))=f(g(x)).

Let $F_{\mathscr{K}}\in{\mathscr{G}\textnormal{rp}}$ denote the free group on the set $\mathscr{K}$ . The maps $\alpha\colon\mathscr{K}\to G$ and

\varphi_{f}\colon\mathscr{K}\to\textnormal{Inn}(\mathscr{K}^{\prime})\;\;,\;\;\;\;x\mapsto\varphi_{f(x)}.

extend to respective group homomorphisms

\widetilde{\alpha}\colon F_{\mathscr{K}}\to G\;\;,\;\;\;\;\widetilde{\varphi_{f}}\colon F_{\mathscr{K}}\to\textnormal{Inn}(\mathscr{K}^{\prime}).

Condition $(i)$ is equivalent to the commuting of the diagram

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Since $\mathscr{A}$ is concise, $\widetilde{\alpha}$ is surjective, then $\rho$ - should it exist - is unique. Let $x_{1},\dots,x_{n}\in\mathscr{K}$ and let $\varepsilon_{1}\dots,\varepsilon_{n}\in\{\pm 1\}$ . Then for all $y\in\mathscr{K}$ ,

\widetilde{\varphi_{f}}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)(f(y))=(\varphi_{f(x_{1})}\circ\cdots\circ\varphi_{f(x_{n})})(f(y))=

={{}^{f(x_{1})}{(\dots{{}^{f(x_{n})}{f(y)}}\dots)}}=f\left({{}^{x_{1}}{\dots({{}^{x_{n}}{y}})\dots}}\right)=f(\alpha_{x_{1}}^{\varepsilon_{1}}\cdots\alpha_{x_{n}}^{\varepsilon_{n}}(y)).

If $x_{1},\dots,x_{n}\in\mathscr{K}$ are s.t. $\widetilde{\alpha}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)=\alpha_{x_{1}}^{\varepsilon_{1}}\cdots\alpha_{x_{n}}^{\varepsilon_{n}}=1$ , then for all $y\in\mathscr{K}$ ,

\widetilde{\varphi_{f}}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)(f(y))=f(\alpha_{x_{1}}^{\varepsilon_{1}}\cdots\alpha_{x_{n}}^{\varepsilon_{n}}(y))=f(y).

Since $f$ is surjective, we have

\widetilde{\alpha}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)=1\;\;\Longrightarrow\;\;\widetilde{\varphi_{f}}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)=\textnormal{Id}_{\mathscr{K}^{\prime}}.

Hence there exists $\rho\colon G\to\textnormal{Inn}(\mathscr{K}^{\prime})$ satisfying condition $(i)$ :

\rho\circ\widetilde{\alpha}=\widetilde{\varphi_{f}}.

Furthermore, since $\mathscr{A}$ is concise, for all $g=\alpha_{x_{1}}^{\varepsilon_{1}}\cdots\alpha_{x_{n}}^{\varepsilon_{n}}\in G$ we have

\rho(g)\circ f=\widetilde{\varphi_{f}}\left(x_{1}^{\varepsilon_{1}}\cdots x_{n}^{\varepsilon_{n}}\right)\circ f=f\circ\alpha_{x_{1}}^{\varepsilon_{1}}\cdots\alpha_{x_{n}}^{\varepsilon_{n}}=f\circ g.

Therefore $\rho$ satsifies condition $(ii)$ as well, hence

(f,\rho)\colon(\mathscr{K},G,\alpha)\to(\mathscr{K}^{\prime},\textnormal{Inn}(\mathscr{K}^{\prime}),\varphi_{\mathscr{K}^{\prime}})

is a morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . Also, $\rho$ is surjective:

\rho(G)=\rho(\langle\{\alpha_{x}\;|\;x\in\mathscr{K}\}\rangle)=\langle\{\varphi_{x^{\prime}}\;|\;x^{\prime}\in\mathscr{K}^{\prime}\}\rangle=\textnormal{Inn}(\mathscr{K}^{\prime}).

∎

Definition 2.12.

Let $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ be concise and let $f\colon\mathscr{K}\to\mathscr{K}^{\prime}$ be a morphism in ${\mathscr{K}\textnormal{ei}}$ . For $\mathscr{K}^{\prime\prime}:=f(\mathscr{K})\leq\mathscr{K}^{\prime}$ We denote by

\rho_{\mathscr{A},f}\colon G\twoheadrightarrow\textnormal{Inn}(\mathscr{K}^{\prime\prime})

the homomorphism $\rho$ from proposition 2.11. If $\mathscr{A}={\mathscr{A}_{\mathscr{K}}}$ or if $f=\textnormal{Id}_{\mathscr{K}}$ , we respectively denote by:

\rho_{f}:=\rho_{{\mathscr{A}_{\mathscr{K}}},f}\colon\textnormal{Inn}(\mathscr{K})\twoheadrightarrow\textnormal{Inn}(\mathscr{K}^{\prime\prime})\;\;,\;\;\;\;\rho_{\mathscr{A}}:=\rho_{\mathscr{A},\textnormal{Id}_{\mathscr{K}}}\colon G\twoheadrightarrow\textnormal{Inn}(\mathscr{K}).

The following proposition is a straightforward consequence of proposition 2.11. The proof, a routine check, is omitted.

Proposition 2.13.

The map

{\mathscr{A}_{\bullet}}\colon\mathscr{K}\mapsto{\mathscr{A}_{\mathscr{K}}}=(\mathscr{K},\textnormal{Inn}(\mathscr{K}),\varphi_{\mathscr{K}})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}

is functorial in surjective morphisms in ${\mathscr{K}\textnormal{ei}}$ , mapping $f\colon\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime}$ to

(f,\rho_{f})\in\textnormal{Hom}_{{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}}(\mathscr{A}_{\mathscr{K}},\mathscr{A}_{\mathscr{K}^{\prime}}).

Accordingly, the map $\mathscr{K}\mapsto\textnormal{Inn}(\mathscr{K})$ that sends $f\colon\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime}$ to

\rho_{f}\in\textnormal{Hom}_{\mathscr{G}\textnormal{rp}}(\textnormal{Inn}(\mathscr{K}),\textnormal{Inn}(\mathscr{K}^{\prime}))

is functorial in surjective morphisms.

Proposition 2.14.

(f(\mathscr{K}),\widetilde{f}(G),\alpha^{\prime}_{|f(\mathscr{K})})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}

is a sub-augmented kei of $\mathscr{A}^{\prime}$ .

Proof.

Let $y=f(x)\in f(\mathscr{K})$ and let $h=\widetilde{f}(g)\in\widetilde{f}(G)$ . Then

\alpha^{\prime}_{y}=\alpha^{\prime}_{f(x)}=\widetilde{f}(\alpha_{x})\in\widetilde{f}(G)

and

h(y)=\widetilde{f}(g)(f(x))=f(g(x))\in f(\mathscr{K}).

Therefore $(f(\mathscr{K}),\widetilde{f}(G),\alpha^{\prime}_{|f(\mathscr{K})})$ is a sub-augmented kei of $\mathscr{A}^{\prime}$ . ∎

Definition 2.15.

Let $\mathscr{A}=(\mathscr{K},G,\alpha),\mathscr{A}^{\prime}=(\mathscr{K}^{\prime},G^{\prime},\alpha^{\prime})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ and let $(f,\widetilde{f})\colon\mathscr{A}\to\mathscr{A}^{\prime}$ be a morphism. We define the image of $(f,\widetilde{f})$ to be the augmented kei

\operatorname{im}(f,\widetilde{f}):=(f(\mathscr{K}),\widetilde{f}(G),\alpha^{\prime}_{|f(\mathscr{K})})\leq\mathscr{A}^{\prime}.

We say $(f,\widetilde{f})$ is surjective if

\operatorname{im}(f,\widetilde{f})=\mathscr{A}^{\prime},

that is, if $f\colon\mathscr{K}\to\mathscr{K}^{\prime}$ and $\widetilde{f}\colon G\to G^{\prime}$ are both surjective.

Lemma 2.16.

Let $(f,\widetilde{f})\colon\mathscr{A}\twoheadrightarrow\mathscr{A}^{\prime}$ be a surjective morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . Assume $\mathscr{A}$ is concise. Then $\mathscr{A}^{\prime}$ is also concise.

Proof.

Write $\mathscr{A}=(\mathscr{K},G,\alpha)$ and $\mathscr{A}^{\prime}=(\mathscr{K}^{\prime},G^{\prime},\alpha^{\prime})$ . We have the following equality of subgroups of $G^{\prime}$ :

\langle\{\alpha^{\prime}_{y}\}_{y\in\mathscr{K}^{\prime}}\rangle=\langle\{\alpha^{\prime}_{f(x)}\}_{x\in\mathscr{K}}\rangle=\langle\{\widetilde{f}(\alpha_{x})\}_{x\in\mathscr{K}}\rangle=\widetilde{f}\left(\langle\{\alpha_{x}\}_{x\in\mathscr{K}}\rangle\right)=\widetilde{f}(G)=G^{\prime}.

∎

Corollary 2.17.

Let $\mathscr{A}=(\mathscr{K},G,\alpha),\mathscr{A}^{\prime}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ be concise and let $(f,\widetilde{f})\colon\mathscr{A}\to\mathscr{A}^{\prime}$ be a surjective morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . Then

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\rho_{f}}$} }}\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{-23.00768pt}{-24.26387pt}\pgfsys@lineto{0.19234pt}{-24.26387pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.39232pt}{-24.26387pt}\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}{-14.95723pt}{-29.63052pt}\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{$\scriptstyle{\rho_{\mathscr{A}^{\prime}}}$} }}\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}}

Proof.

The statements of proposition 2.11 and proposition 2.13 are interpreted in terms of an adjunction

{\mathscr{A}_{\bullet}}\dashv{\textnormal{Kei}_{\bullet}}\;\;,\;\;\;\;\leavevmode\hbox to136.86pt{\vbox to20.48pt{\pgfpicture\makeatletter\hbox{\hskip 68.42783pt\lower-11.49452pt\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}{-68.42783pt}{-8.98358pt}\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 22.95668pt\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}{-18.65114pt}{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{${{\mathscr{A}_{\bullet}}\colon{\mathscr{K}\textnormal{ei}}^{\textnormal{surj}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 22.95668pt\hfil&\hfil\hskip 57.47113pt\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}{-29.16562pt}{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{${{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}_{\textnormal{cs}}^{\textnormal{surj}}\colon{\textnormal{Kei}_{\bullet}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 33.47116pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-22.31447pt}{-4.07248pt}\pgfsys@lineto{0.88554pt}{-4.07248pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.08553pt}{-4.07248pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{1.2855pt}{-8.89468pt}\pgfsys@lineto{-21.9145pt}{-8.89468pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-22.11449pt}{-8.89468pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\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}},

where ${\mathscr{K}\textnormal{ei}}^{\textnormal{surj}}\subseteq{\mathscr{K}\textnormal{ei}}$ is the subcategory of all keis and surjective morphisms, ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}_{\textnormal{cs}}^{\textnormal{surj}}\subseteq{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ is the subcategory of concise augmented keis and surjective morphisms. The desired square is obtained from the unit of the adjunction:

∎

Proposition 2.18.

Let $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ and let $H\unlhd G$ . Let

\alpha_{H}\colon H\backslash\mathscr{K}\to H\backslash G\;\;,\;\;\;\;\alpha_{H}\colon Hx\mapsto H\alpha_{x},

where $H\backslash\mathscr{K}$ is the quotient in ${\mathscr{S}\textnormal{et}}$ . Then $(H\backslash\mathscr{K},H\backslash G,\alpha_{H})$ is an augmented kei and the quotient map $(\mathscr{K},G,\alpha)\to(H\backslash\mathscr{K},H\backslash G,\alpha_{H})$ is a morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . In particular $H\backslash\mathscr{K}$ inherits a well-defined kei structure from $\mathscr{K}$ , and the quotient map $\mathscr{K}\to H\backslash\mathscr{K}$ is a morphism in ${\mathscr{K}\textnormal{ei}}$ .

Proof.

It suffices to show that $\alpha_{H}$ is a well-defined function and that the $H\backslash G$ -action on $H\backslash\mathscr{K}$ is well-defined. Let $x\in\mathscr{K}$ , let $g\in G$ and let $h\in H$ . Then

\alpha_{h(x)}=h\alpha_{x}h^{-1}=h(\alpha_{x}h\alpha_{x}^{-1})\alpha_{x}\in H\alpha_{x},

(hg)(x)=h(g(x))\in Hg(x)\;\;,\;\;\;\;g(h(x))=(ghg^{-1})(g(x))\in Hg(x).

The aumgented kei axioms on $(H\backslash\mathscr{K},H\backslash G,\alpha_{H})$ are seen to hold by routine lifting to $\mathscr{A}$ . Consequently $H\backslash\mathscr{K}$ is a kei with structure induced from $\mathscr{K}$ :

{{}^{\overline{x}}{\overline{y}}}=\overline{{{}^{x}{y}}},

that is, the quotient map $\mathscr{K}\to H\backslash\mathscr{K}$ is a morphism in ${\mathscr{K}\textnormal{ei}}$ . ∎

Definition 2.19.

Let $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ and let $H\unlhd G$ . We denote by $H\backslash\mathscr{A}$ we denote the augmented kei

H\backslash\mathscr{A}:=(H\backslash\mathscr{K},H\backslash G,\alpha_{H})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}},

Proposition 2.20.

Let $\mathscr{A}=(\mathscr{K},G,\alpha),\mathscr{A}^{\prime}=(\mathscr{K}^{\prime},G^{\prime},\alpha^{\prime})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ and let $(f,\widetilde{f})\colon\mathscr{A}\to\mathscr{A}^{\prime}$ . Denote by $H_{\widetilde{f}}:=\ker(\widetilde{f}\colon G\to G^{\prime})\unlhd G$ . Then $(f,\widetilde{f})$ factors through the quotient

H_{\widetilde{f}}\backslash\mathscr{A}\to\mathscr{A}^{\prime}

Proof.

By proposition 2.18, $H_{\widetilde{f}}\backslash\mathscr{A}\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ and $\mathscr{A}\to H_{\widetilde{f}}\backslash\mathscr{A}$ is a morphism in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . Let $x\in\mathscr{K}$ , let $g\in G$ and let $h\in H_{\widetilde{f}}$ . Then $\widetilde{f}(h)=1$ , hence

\widetilde{f}(hg)=\widetilde{f}(h)\widetilde{f}(g)=\widetilde{f}(g),

f(h(x))=\widetilde{f}(h)(f(x))=f(x).

This suffices to show that $(f,\widetilde{f})$ factors through $H_{\widetilde{f}}\backslash\mathscr{A}$ . ∎

Proposition 2.21.

Let $\mathscr{A}=(\mathscr{K},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ be concise, let $\mathscr{K}^{\prime}\in{\mathscr{K}\textnormal{ei}}$ , and let $f\colon\mathscr{K}\to\mathscr{K}^{\prime}$ be a morphism in ${\mathscr{K}\textnormal{ei}}$ . Let $\mathscr{K}^{\prime\prime}\leq\mathscr{K}^{\prime}$ denote the image $\mathscr{K}^{\prime\prime}:=f(\mathscr{K})$ of $f$ . Then $f$ factors through

H_{f}\backslash\mathscr{K}\to\mathscr{K}^{\prime},

where

H_{f}:=\ker(\rho_{\mathscr{A},f}\colon G\twoheadrightarrow\textnormal{Inn}(\mathscr{K}^{\prime\prime}))\unlhd G.

Proof.

By proposition 2.11, $f\colon\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime\prime}$ lifts to a surjective morphism

(f,\rho_{\mathscr{A},f})\colon\mathscr{A}\twoheadrightarrow\mathscr{A}_{\mathscr{K}^{\prime\prime}}

in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . By proposition 2.20, $(f,\rho_{\mathscr{A},f})$ factors through

H_{f}\backslash\mathscr{A}\to\mathscr{A}_{\mathscr{K}^{\prime\prime}}.

Forgetting to ${\mathscr{K}\textnormal{ei}}$ , we find that $f\colon\mathscr{K}\to\mathscr{K}^{\prime}$ factors through

H_{f}\backslash\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime\prime}\hookrightarrow\mathscr{K}^{\prime}.

∎

2.2 Profinite Keis and Augmented Keis

In this section we discuss profinite keis, which are formal cofiltered limits of finite keis. Closely related notions were extensively developed in [BCJ⁺24]. Stone duality is the statement that profinite sets are equivalent to Stone spaces - compact, Hausdorff totally-disconnected topological spaces. More precisely, the fully-faithful embedding

\delta\colon{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}\hookrightarrow{\mathscr{T}\textnormal{op}}

of finite sets as discrete topological spaces lifts to a fully faithful functor

\widehat{\delta}\colon\textnormal{Pro-}{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}\hookrightarrow{\mathscr{T}\textnormal{op}}.

Similarly, the embedding of finite groups into topological groups extends to a fully-faithful functor

\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}\hookrightarrow{\mathscr{G}\textnormal{rp}}^{\mathscr{T}\textnormal{op}}.

Hence for $\mathfrak{G},\mathfrak{G}^{\prime}\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ we denote by

\textnormal{Hom}_{\mathscr{G}\textnormal{rp}}^{\textnormal{cont}}(\mathfrak{G},\mathfrak{G}^{\prime}):=\textnormal{Hom}_{\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}}(\mathfrak{G},\mathfrak{G}^{\prime}).

Like groups, the algberaic theory of keis consists of finitely-many operations of finite arity. An analogous statement for keis holds as well: the induced functor

\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\hookrightarrow{\mathscr{K}\textnormal{ei}}^{\mathscr{T}\textnormal{op}}

is fully-faithful ([Joh82, §VI.2]). In considering topological augmented keis, it is not difficult to see that a similar argument works: the enduced functor

\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}\hookrightarrow{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\mathscr{T}\textnormal{op}}

is fully-faithful. We therefore make no distinction between profinite (augmented) keis and their corresponding topological (augmented) keis. For $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}^{\prime}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ and $\mathfrak{A},\mathfrak{A}^{\prime}\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ we denote by

\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}^{\prime}):=\textnormal{Hom}_{\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}^{\prime}),

\textnormal{Hom}_{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\mathfrak{A},\mathfrak{A}^{\prime}):=\textnormal{Hom}_{\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}}(\mathfrak{A},\mathfrak{A}^{\prime}).

It is perhaps worth mentioning that the forgetful functors

are defined once as naturally extending the finite counterparts

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and again as restricted from the topological counterparts

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These two definitions in fact coincide. Also worth mentioning is that because ${\mathscr{K}\textnormal{ei}}$ and ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ have a well-behaved notion of images (see proposition 2.14 for ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ ), profinite keis and augmented keis can be presented as limits of cofiltered diagrams with surjective maps. This can be done canonically: For $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , Let ${\mathscr{D}}_{\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}$ denote the category whose objects are

\{(\mathscr{K},f)\;\;|\;\;\;\;\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\;,\;\;f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}\},

and

\textnormal{Hom}((\mathscr{K},f),(\mathscr{K}^{\prime},f^{\prime}))=\{h\colon\mathscr{K}\twoheadrightarrow\mathscr{K}^{\prime}\;|\;h\circ f=f^{\prime}\}.

Then ${\mathscr{D}}_{\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}$ is cofiltered, and

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\simeq\lim_{\begin{subarray}{c}\longleftarrow\\ {\mathscr{D}}_{\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}\end{subarray}}\mathscr{K}.

We set a convention: When we write

\displaystyle{\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}=\lim_{\begin{subarray}{c}\longleftarrow\\ {\mathscr{D}}\end{subarray}}\mathscr{K}_{d}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}},

(4)

it is to be understood that ${\mathscr{D}}$ is a cofiltered diagram in ${\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , and that the morphisms $\mathscr{K}_{d}\to\mathscr{K}_{d^{\prime}}$ in the diagram are all surjections. Likewise for

\displaystyle{\mathfrak{A}=\lim_{\begin{subarray}{c}\longleftarrow\\ {\mathscr{D}}\end{subarray}}\mathscr{A}_{d}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}}.

Definition 2.22.

\overline{\langle\{\alpha_{x}\;|\;x\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\}\rangle}=\mathfrak{G}.

For finite $\mathfrak{A}$ , this notion of concisesness coincides with definition 2.8.

Lemma 2.23.

Let $\mathfrak{A}\twoheadrightarrow\mathfrak{A}^{\prime}$ be a surjective morphism in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ . Assume $\mathfrak{A}$ is concise. Then so is $\mathfrak{A}^{\prime}$ .

Proof.

Denote $\mathfrak{A}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathfrak{G},\alpha)$ and $\mathfrak{A}^{\prime}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}^{\prime},\mathfrak{G}^{\prime},\alpha^{\prime})$ . Because $\mathfrak{A}$ is concise, $\overline{\langle\{\alpha_{x}\}_{x\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}\rangle}=\mathfrak{G}$ . Therefore

\overline{\langle\{\alpha^{\prime}_{y}\}_{y\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}^{\prime}}\rangle}=\overline{\langle\{\alpha^{\prime}_{f(x)}\}_{x\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}\rangle}=\overline{\langle\{\widetilde{f}(\alpha_{x})\}_{x\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}\rangle}=

=\widetilde{f}\left(\overline{\langle\{(\alpha_{x})\}_{x\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}}\rangle}\right)=\widetilde{f}(\mathfrak{G})=\mathfrak{G}^{\prime}.

Hence $\mathfrak{A}^{\prime}$ is concise. ∎

Suppose $\displaystyle{\mathfrak{A}=\lim_{\begin{subarray}{c}\longleftarrow\\ {\mathscr{D}}\end{subarray}}\mathscr{A}_{d}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}}$ is concise. Then the projections $\mathfrak{A}\to\mathscr{A}_{d}$ are all surjective. By lemma 2.23, each $\mathscr{A}_{d}\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ is also concise.

Proposition 2.24.

Let $\mathfrak{A}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathfrak{G},\alpha)\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ be concise. Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ be finite and let $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}$ be a surjective morphism in $\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . Then there is a unique morphism $\rho\colon\mathfrak{G}\to\textnormal{Inn}(\mathscr{K})$ s.t.

(f,\rho)\colon\mathfrak{A}\to\mathscr{A}_{\mathscr{K}}=(\mathscr{K},\textnormal{Inn}(\mathscr{K}),\varphi_{\mathscr{K}})

is a morphism in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ .

Proof.

Write $\mathfrak{A}$ as the limit

\mathfrak{A}=\lim\limits_{\begin{subarray}{c}\longleftarrow\\ d\in{\mathscr{D}}\end{subarray}}\mathscr{A}_{d}\;\;,\;\;\;\;\mathscr{A}_{d}=(\mathscr{K}_{d},G_{d},\alpha_{d})\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}

of a cofiltered diagram in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ with surjections, so that

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}=\lim\limits_{\begin{subarray}{c}\longleftarrow\\ d\in{\mathscr{D}}\end{subarray}}\mathscr{K}_{d}.

Therefore $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}$ factors through $f_{d}\colon\mathscr{K}_{d}\twoheadrightarrow\mathscr{K}$ for some $d\in{\mathscr{D}}$ . The morphism $\mathfrak{A}\to\mathscr{A}_{d}$ is surjective, therefore $\mathscr{A}_{d}$ is concise by lemma 2.23. Proposition 2.11 implies the existence of a canonical morphism

(f_{d},\rho_{d})\colon\mathscr{A}_{d}\to\mathscr{A}_{\mathscr{K}}

in ${\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ . The composition

(f,\rho)\colon\mathfrak{A}\to\mathscr{A}_{d}\xrightarrow{(f_{d},\rho_{d})}\mathscr{A}_{\mathscr{K}}

is a morphism in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ lifting $f$ . For all $d\in{\mathscr{D}}$ there is at most one such map $(f_{d},\rho_{d})$ . Moreover the diagram ${\mathscr{D}}$ is cofiltered. Therefore $(f,\rho)$ , independent of $d$ , is uniquely defined by $f$ . ∎

The following proposition lists versions of propositions from the previous section for profinite augmented keis. The proofs are fundamentally the same, and are therefore omitted.

Proposition 2.25.

1. (see 2.18):

N\backslash\mathfrak{A}:=(N\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},N\backslash\mathfrak{G},\alpha_{N})\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}},

such that the quotient map $\mathfrak{A}\to N\backslash\mathfrak{A}$ is a morphism in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ .

2. (see 2.20):

Let $(f,\widetilde{f})\colon\mathfrak{A}\to\mathfrak{A}^{\prime}$ be a morphism in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ . Denote by $N_{\widetilde{f}}:=\ker(\widetilde{f})$ . Then $(f,\widetilde{f})$ factors through the quotient

$N_{\widetilde{f}}\backslash\mathfrak{A}\to\mathfrak{A}^{\prime}.$

3. (see 2.21):

Let $\mathfrak{A}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathfrak{G},\alpha)\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ be concise, let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ be finite, and let $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\to\mathscr{K}$ be a morphism in $\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . Then $f$ factors through $N_{f}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}$ , where

N_{f}:=\ker\left(\rho_{\mathfrak{A},f}\colon\mathfrak{G}\to\textnormal{Inn}(f(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}))\right)\unlhd\mathfrak{G}.

Definition 2.26.

A profinite group $\mathfrak{G}\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ is small if for all $d\in\mathbb{N}$ , $\mathfrak{G}$ has finitely-many open subgroups of index $d$ .

Proposition 2.27.

Let $\mathfrak{G}\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ . Then $\mathfrak{G}$ is small iff for all $G\in{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ ,

\left|\textnormal{Hom}_{{\mathscr{G}\textnormal{rp}}}^{\textnormal{cont}}(\mathfrak{G},G)\right|<\infty.

Proof.

Let $H\leq\mathfrak{G}$ be an open subgroup of index $d$ . Then

N_{H}:=\bigcap_{gH\in\mathfrak{G}/H}gHg^{-1}\leq\mathfrak{G}

is open and normal, satisfying

\mathfrak{G}/N_{H}\hookrightarrow\prod_{\mathfrak{G}/H}\mathfrak{G}/gHg^{-1}.

Therefore every open subgroup $H$ of index $d$ contains an open normal subgroup $N_{H}$ of index $[\mathfrak{G}:N_{H}]\leq d^{d}$ . It follows that $\mathfrak{G}$ is small if and only if for all $d\in\mathbb{N}$ there are finitely-many normal open subgroups $N\unlhd\mathfrak{G}$ of index $[\mathfrak{G}:N]\leq d$ . This in turn occurs if and only if for all $G\in{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ ,

\left|\textnormal{Hom}_{{\mathscr{G}\textnormal{rp}}}^{\textnormal{cont}}(\mathfrak{G},G)\right|<\infty.

∎

We therefore give the following analogous definition:

Definition 2.28.

Let $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . We say min\CJKtilde\CJKnospace 圭 is small if for every $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ ,

\left|\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathscr{K})\right|<\infty.

Proposition 2.29.

Let $\mathfrak{A}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathfrak{G},\alpha)\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ be concise. If $\mathfrak{G}$ is a small profinite group and if $\left|\mathfrak{G}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\right|<\infty$ , then $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ is small.

Proof.

A finite kei $\mathscr{K}$ has finitely-many sub-keis, therefore min\CJKtilde\CJKnospace 圭 is small iff for all $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ there are finitely-many surjective morphisms $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}$ . Since $\mathfrak{A}$ is concise, by proposition 2.24 any surjective morphism $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}$ extends uniquely to a morphism

(f,\rho_{\mathfrak{A},f})\colon\mathfrak{A}\twoheadrightarrow\mathscr{A}_{\mathscr{K}}

in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ . Denote by

H_{f}:=\ker(\rho_{\mathfrak{A},f}\colon\mathfrak{G}\twoheadrightarrow\textnormal{Inn}(\mathscr{K}))\unlhd\mathfrak{G},

H_{\mathscr{K}}:=\bigcap_{f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}}H_{f}\unlhd\mathfrak{G}.

Since $\textnormal{Inn}(\mathscr{K})$ is finite, and $\rho_{\mathfrak{A},f}$ continuous, then $H_{f}\unlhd\mathfrak{G}$ is open. We also have

\left|\textnormal{Hom}_{{\mathscr{G}\textnormal{rp}}}^{\textnormal{cont}}(\mathfrak{G},\textnormal{Inn}(\mathscr{K}))\right|<\infty

because $\mathfrak{G}\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ is small. Therefore $H_{\mathscr{K}}\unlhd\mathfrak{G}$ is open. By proposition 2.25, every $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathscr{K}$ factors through $H_{f}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}$ , and therefore through

\overline{f}\colon H_{\mathscr{K}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\to\mathscr{K}.

The quotient $H_{\mathscr{K}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}$ is finite because $\mathfrak{G}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}$ is finite, and the fibers of the map

H_{\mathscr{K}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}\twoheadrightarrow\mathfrak{G}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}

are transitive $H_{\mathscr{K}}\backslash\mathfrak{G}$ -sets - therefore finite. We conclude that for all $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ ,

\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathscr{K})^{\textnormal{surj}}\xleftarrow{\sim}\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}}(H_{\mathscr{K}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}},\mathscr{K})^{\textnormal{surj}}\in{{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}}.

Hence min\CJKtilde\CJKnospace 圭 is small. ∎

2.3 Disjoint Unions in ${\mathscr{K}\textnormal{ei}}$

In this section we shall discuss disjoint unions of keis. Much like the tensor product of non-commutative rings, this is not the coproduct in ${\mathscr{K}\textnormal{ei}}$ . It is however a symmetric monoidal structure on ${\mathscr{K}\textnormal{ei}}$ , with the empty kei playing the role of the unit. This discussion can be extended to profinite keis as well. The statements and their proofs are similar enough that we omit them in the profinite case.

Proposition 2.30.

Let $\mathscr{K}_{1},\mathscr{K}_{2}\in{\mathscr{K}\textnormal{ei}}$ . Then the disjoint union $\mathscr{K}_{1}\sqcup\mathscr{K}_{2}$ is a kei with structure

y∈𝒦1φ𝒦1,x(y)yy∈𝒦2yφ𝒦2,x(y).{{}^{x}{y}}=\;\;\;\;\;\;\begin{tabular}[]{|c||c|c|}\hline\cr&$x\in\mathscr{K}_{1}$&$x\in\mathscr{K}_{2}$\\ \hline\cr\hline\cr$y\in\mathscr{K}_{1}$&$\varphi_{\mathscr{K}_{1},x}(y)$&$y$\\ \hline\cr$y\in\mathscr{K}_{2}$&$y$&$\varphi_{\mathscr{K}_{2},x}(y)$\\ \hline\cr\end{tabular}\;\;.

	${{}^{x}{y}}=\;\;\;\;\;\;\begin{tabular}[]{\|c\|\|c\|c\|}\hline\cr&$x\in\mathscr{K}_{1}$&$x\in\mathscr{K}_{2}$\\ \hline\cr\hline\cr$y\in\mathscr{K}_{1}$&$\varphi_{\mathscr{K}_{1},x}(y)$&$y$\\ \hline\cr$y\in\mathscr{K}_{2}$&$y$&$\varphi_{\mathscr{K}_{2},x}(y)$\\ \hline\cr\end{tabular}\;\;.$		∈xK1	∈xK2

This puts a symmetric monoidal structure $({\mathscr{K}\textnormal{ei}},\sqcup,\emptyset)$ on ${\mathscr{K}\textnormal{ei}}$ .

Proof.

The first two kei axioms are easily verified. As for the third, let $x,y\in\mathscr{K}_{1}\sqcup\mathscr{K}_{2}$ . If $x,y\in\mathscr{K}_{i}$ for some $i$ , w.l.o.g. $i=1$ , then

\varphi_{x}\circ\varphi_{y}=(\varphi_{\mathscr{K}_{1},x}\circ\varphi_{\mathscr{K}_{1},y})\sqcup\textnormal{Id}_{\mathscr{K}_{2}}=

=(\varphi_{\mathscr{K}_{1},{{}^{x}{y}}}\circ\varphi_{\mathscr{K}_{1},x})\sqcup\textnormal{Id}_{\mathscr{K}_{2}}=\varphi_{{{}^{x}{y}}}\circ\varphi_{x}.

otherwise, w.l.o.g. $x\in\mathscr{K}_{1}$ and $y\in\mathscr{K}_{2}$ , then

\varphi_{x}\circ\varphi_{y}=\varphi_{\mathscr{K}_{1},x}\sqcup\varphi_{\mathscr{K}_{2},y}=\varphi_{y}\circ\varphi_{x}=\varphi_{{{}^{x}{y}}}\circ\varphi_{x}.

∎

Let $T\in{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}$ . The disjoint union of $T$ copies of the terminal object $*\in{\mathscr{K}\textnormal{ei}}$ recovers the trivial kei $\mathscr{T}\in{\mathscr{K}\textnormal{ei}}$ with underlying set $T$ . Hence there is a functor

\iota_{T}\colon{\mathscr{K}\textnormal{ei}}^{T}\to{\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}\;\;\;\;,\;\;\;\;\;\;\;\;(\mathscr{K}_{t})_{t\in T}\longmapsto(\;\;\bigsqcup_{t\in T}\mathscr{K}_{t}\to\mathscr{T}\;\;).

The functor $\iota_{T}$ preserves small limits and filtered colimits, and therefore has a left adjoint

\lambda_{T}\colon{\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}\to{\mathscr{K}\textnormal{ei}}^{T}.

The functor $\iota_{T}$ is also fully faithful, therefore $\lambda_{T}$ can be computed in terms of the unit

\mathscr{L}\to(\iota_{T}\circ\lambda_{T})(\mathscr{L}).

We explicitly construct $\lambda_{T}(\mathscr{L})$ for $\mathscr{L}\in{\mathscr{K}\textnormal{ei}}$ , making use of an augmentation on $\mathscr{L}$ :

Proposition 2.31.

Let $\mathscr{T}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ be trivial, let $(\mathscr{L},G,\alpha)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ be concise, and let $\eta\colon\mathscr{L}\to\mathscr{T}$ be a morphism in ${\mathscr{K}\textnormal{ei}}$ . For $t\in\mathscr{T}$ , we denote by $\mathscr{L}_{t}:=\eta^{-1}(t)\subseteq\mathscr{L}$ and by $H_{t}\leq G$ the subgroup $H_{t}:=\langle\{\alpha_{x}\;|\;x\in\mathscr{L}\setminus\mathscr{L}_{t}\}\rangle\leq G$ . Then

1.

The set $\bigsqcup_{T}H_{t}\backslash\mathscr{L}_{t}$ is well-defined, and a quotient of $\mathscr{L}$ in ${\mathscr{K}\textnormal{ei}}$ .

For all $(\mathscr{K}_{t})_{T}\in{\mathscr{K}\textnormal{ei}}^{T}$ , let $\mathscr{K}_{\mathscr{T}}\in{\mathscr{K}\textnormal{ei}}$ denote the disjoint union

\mathscr{K}_{\mathscr{T}}:=\bigsqcup_{T}\mathscr{K}_{t}.

Then there is a natural bijection

\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}}(\mathscr{L},\mathscr{K}_{\mathscr{T}})\simeq\prod_{t\in T}\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}(H_{t}\backslash\mathscr{L}_{t},\mathscr{K}_{t}).

Proof.

Let $t\in\mathscr{T}$ . Then for all $y\in\mathscr{L}_{t}$ and all $x\in\mathscr{L}$ ,

\eta(\alpha_{x}(y))=\eta({{}^{x}{y}})={{}^{\eta(x)}{\eta(y)}}=\eta(y)=t.

(5)

Since $(\mathscr{L},G,\alpha)$ is concise, it follows that the $\mathscr{L}_{t}$ are $G$ -invariant. For all $g\in G$ and $x\in\mathscr{L}$ ,

g\alpha_{x}g^{-1}=\alpha_{g(x)},

where $g(x)\in\mathscr{L}_{\eta(x)}$ . Hence the subgroups

H_{t}=\langle\{\alpha_{x}\;|\;x\in\mathscr{L}\setminus\mathscr{L}_{t}\}\rangle\;\;,\;\;\;\;H^{\prime}_{t}:=\langle\{\alpha_{x}\;|\;x\in\mathscr{L}_{t}\}\rangle\leq G

are normal. Next, let $x\in\mathscr{L}$ and let $h\in H_{\eta(x)}$ . Then

\alpha_{h(x)}\alpha_{x}^{-1}=h\alpha_{x}h^{-1}\alpha_{x}^{-1}\in[H_{\eta(x)},H^{\prime}_{\eta(x)}]\leq H_{\eta(x)}\cap H^{\prime}_{\eta(x)}.

Let $t\in\mathscr{T}$ . If $t\neq\eta(x)$ , then $H^{\prime}_{\eta(x)}\leq H_{t}$ . Hence for all $t\in\mathscr{T}$ ,

\alpha_{h(x)}\alpha_{x}^{-1}\in H_{t}.

Let $x,y\in\mathscr{L}$ . Then for all $h\in H_{\eta(y)}$ ,

{{}^{x}{h(y)}}=\alpha_{x}h(y)=\alpha_{x}h\alpha_{x}^{-1}(\alpha_{x}(y))\in H_{\eta(y)}({{}^{x}{y}})=H_{\eta({{}^{x}{y}})}({{}^{x}{y}}),

and for all $h\in H_{\eta(x)}$ ,

{{}^{h(x)}{y}}=\alpha_{h(x)}(y)=\alpha_{h(x)}\alpha_{x}^{-1}(\alpha_{x}(y))\in H_{\eta({{}^{x}{y}})}({{}^{x}{y}}).

Therefore the set $\lambda_{\eta}:=\hskip 2.0pt\cdot\bigcup_{T}H_{t}\backslash\mathscr{L}_{t}$ is a kei with structure inherited from $\mathscr{L}$ . For all $\overline{x},\overline{y}\in\lambda_{\eta}$ with $\eta(x)\neq\eta(y)$ , we have $\alpha_{x}\in H_{\eta(y)}$ , therefore

{{}^{\overline{x}}{\overline{y}}}=\overline{{{}^{x}{y}}}=\overline{\alpha_{x}y}=\overline{y}.

Hence $\lambda_{\eta}$ is the disjoint union

\lambda_{\eta}=\bigsqcup_{T}H_{t}\backslash\mathscr{L}_{t}\in{\mathscr{K}\textnormal{ei}}.

Finally, let $(\mathscr{K}_{t})_{t\in T}\in{\mathscr{K}\textnormal{ei}}^{T}$ and let $f\colon\mathscr{L}\to\mathscr{K}_{\mathscr{T}}$ be a morphism in ${\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}$ . Then for all $t\in\mathscr{T}$ , for all $y\in\mathscr{L}_{t}$ and all $x\in\mathscr{L}\setminus\mathscr{L}_{t}$ ,

f(\alpha_{x}(y))=f({{}^{x}{y}})={{}^{f(x)}{f(y)}}=f(y).

Therefore $f_{|\mathscr{L}_{t}}$ is well-defined on the quotient $H_{t}\backslash\mathscr{L}_{t}$ , hence $f$ factors through

\lambda_{\eta}=\bigsqcup_{T}H_{t}\backslash\mathscr{L}_{t}\xrightarrow{\overline{f}}\mathscr{K}_{\mathscr{T}}.

We conclude that

\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}}(\mathscr{L},\mathscr{K}_{\mathscr{T}})\simeq\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}_{/\mathscr{T}}}\big{(}\bigsqcup_{T}N_{t}\backslash\mathscr{L}_{t},\mathscr{K}_{\mathscr{T}}\big{)}\simeq\prod_{T}\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}(N_{t}\backslash\mathscr{L}_{t},\mathscr{K}_{t}).

∎

3 Fundamental Arithemtic Keis

In this section we define the fundamental kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ of square-free integers $n\in\mathbb{N}$ by emulating the construction of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ , the fundamental kei of a link $L$ in $S^{3}$ . In the case of traditional links, $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ is intimately tied to the automorphism group of some infinite branched cover of the sphere. The construction is similar here, and since infinite Galois groups are profinite, $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ will in fact be a profinite kei. We then analogously define $\mathscr{K}$ -colorings of $n$ in terms of said $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ .

3.1 The Fundamental Kei of a link

Our definition of the fundamental arithmetic kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is motivated by a presentation of the fundamental kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ found in the introduction. The purpose of this section is to justify this presentation. Neither object $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L},\mathscr{Q}_{L}$ discussed here - nor indeed the general notion of a quandle - are needed anywhere else in the paper. If you are comfortable with $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ as presented in the introduction, feel free to skip this part.

Proposition 3.1.

Let $L\subseteq S^{3}$ be a link. Let $M_{L}$ denote the branched double cover of $S^{3}$ with ramification locus $L$ and let $\widetilde{M}_{L}$ denote the universal cover of $M_{L}$ . Then there is an augmented kei

\left(\pi_{0}(\widetilde{M}_{L}\times_{S^{3}}L)\;,\;\textnormal{Aut}_{\mathscr{T}\textnormal{op}}(\widetilde{M}_{L}/S^{3})\;,\;m\right)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}

that recovers the fundamental kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}$ that corepresents colorings of $L$ :

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}\simeq\pi_{0}(\widetilde{M}_{L}\times_{S^{3}}L).

An augmented quandle $(Q,G,\alpha)$ , much like an augmented kei, consists of a set $Q$ , a group $G$ acting on $Q$ , and an augmentation map $\alpha\colon Q\to G$ , satisfying two of the three augmented kei axioms:

\forall x\in Q\;,\;\;g\in G\;:\;\;\alpha_{x}(x)=x\;\;\textnormal{and}\;\;\alpha_{g(x)}=g\alpha_{x}g^{-1}.\;

The set $Q$ , together with the binary operator $(x,y)\mapsto\alpha_{x}(y)$ , is an algebraic structure known as a quandle. To be more specific, quandles are a more general notion than keis, obtained by relaxing the second axiom, requiring that each element act merely as a permutation (rather than an involution). Quandles to oriented links are what keis are to unoriented links: much like keis, one can use quandles to color an oriented link $L$ , and these colorings are similarly controled by a fundamental quandle $\mathscr{Q}_{L}$ attached to $L$ . A topological definition of $\mathscr{Q}_{L}$ is found in [Joy82, §14], where structure is given by an augmentation with $\pi_{1}(S_{L})$ . This boils down to the following presentation⁵⁵5See [Nos17, p.17]:

We take $U_{L}\subseteq S^{3}$ to denote a tubular neighborhood of $L$ , $S_{L}$ to denote the complement $S_{L}=S^{3}\setminus U$ , and $\partial U_{L}\subseteq S_{L}$ the boundary of $U_{L}$ . We fix a base point $*\in S_{L}$ . Then $\mathscr{Q}_{L}$ is defined as the set of paths $*\rightsquigarrow\partial U_{L}$ in $S_{L}$ - up to homotopy. The structure on $\mathscr{Q}_{L}$ is defined by an augmentation with $\mathscr{G}_{L}=\pi_{1}(S_{L})$ . Here $\mathscr{G}_{L}$ acts on $\mathscr{Q}_{L}$ via concatenation, and the augmentation map $m\colon\mathscr{Q}_{L}\to\mathscr{G}_{L}$ maps a path $x$ to a meridian $m_{x}\in\mathscr{G}_{L}$ , looping once around a component of $L$ in a manner consistent with the orientation of $L$ .

Proof.

(proposition 3.1) Let $\widetilde{S}_{L}$ denote the universal covering of $S_{L}$ , and let

\mathscr{G}_{L}:=\pi_{1}(S_{L})\simeq\textnormal{Aut}(\widetilde{S}_{L}/S_{L}).

The pullback $\widetilde{S}_{L}\times_{S_{L}}\partial U_{L}$ in ${\mathscr{T}\textnormal{op}}$ consists of paths $*\rightsquigarrow\partial U$ in $S_{L}$ , up to homotopy that fixes both endpoints. Passing to connected components we have an isomorphism

\pi_{0}(\widetilde{S}_{L}\times_{S_{L}}\partial U){\xrightarrow{\sim}}\mathscr{Q}_{L}

of $\mathscr{G}_{L}$ -sets. We denote by $H\unlhd\mathscr{G}_{L}$ the subgroup

H=\langle m_{x}^{2}\mid x\in\mathscr{Q}_{L}\rangle\unlhd\mathscr{G}_{L},

and by

Y=H\backslash\widetilde{S}_{L}

the covering of $S_{L}$ corresponding to $H$ . By [Joy82, Thm. 10.2],

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}\simeq H\backslash\mathscr{Q}_{L}\simeq\pi_{0}(Y\times_{S_{L}}\partial U),

with structure on $H\backslash\mathscr{Q}_{L}$ given by augmentation with the group

H\backslash\mathscr{G}_{L}\simeq\textnormal{Aut}(Y/{S_{L}}).

For every branched cover $Z\to S^{3}$ that is unramified away from $L$ , we denote by $Z^{\circ}=Z\times_{S^{3}}S_{L}$ . In [Win84, §5], the group $H\backslash\mathscr{G}_{L}$ sits in an exact sequence

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;.

From the proof in [Win84], it is not hard to see that

H\backslash\mathscr{G}_{L}\simeq\textnormal{Aut}_{\mathscr{T}\textnormal{op}}((\widetilde{M}_{L})^{\circ}/S_{L})\simeq\textnormal{Aut}_{\mathscr{T}\textnormal{op}}(\widetilde{M}_{L}/S^{3}).

The pullback $M_{L}\times_{S^{3}}U_{L}\subseteq M_{L}$ is a disjoint union of solid tori that homotopy retracts onto $M_{L}\times_{S^{3}}L$ , and has boundary

\partial(M_{L}\times_{S^{3}}U_{L})=M_{L}^{\circ}\times_{S_{L}}\partial U_{L}\subseteq M_{L}.

The same holds for any covering space of $M_{L}$ . Therefore

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{L}=\pi_{0}((\widetilde{M}_{L})^{\circ}\times_{S_{L}}\partial U_{L})\simeq\pi_{0}(\widetilde{M}_{L}\times_{S^{3}}U_{L})\simeq\pi_{0}(\widetilde{M}_{L}\times_{S^{3}}L).

∎

3.2 The Fundamental Kei of an Arithmetic Link

Definition 3.2.

We denote by ${\mathbb{N}^{*}}$ the set of square-free positive integers.

Definition 3.3.

For $n=2^{k}m\in\mathbb{N}$ with $m$ odd, we define

n^{*}=(-1)^{\frac{m-1}{2}}n.

It is easily verified that $(n_{1}n_{2})^{*}=n_{1}^{*}n_{2}^{*}$ . Thus for $n\in{\mathbb{N}^{*}}$ we have

n^{*}=\prod_{p|n}p^{*}.

Definition 3.4.

Let $1\neq n\in{\mathbb{N}^{*}}$ . By $\mathfrak{L}_{n}$ we denote the quadratic number field

\mathfrak{L}_{n}=\mathbb{Q}(\sqrt{n^{*}}).

The field $\mathfrak{L}_{n}$ is ramified at precisely $n$ . We define the field $\mathfrak{F}_{n}$ to be the maximal unramified extension of $\mathfrak{L}_{n}$ :

\mathfrak{F}_{n}=\mathfrak{L}_{n}^{un}.

The extension $\mathfrak{F}_{n}/\mathbb{Q}$ is Galois. By $\mathfrak{G}_{n}$ we denote the Galois group

\mathfrak{G}_{n}=\textnormal{Gal}(\mathfrak{F}_{n}/\mathbb{Q})\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}.

Proposition 3.5.

Let $1\neq m,n\in{\mathbb{N}^{*}}$ s.t. $m|n$ . Then

\mathfrak{F}_{m}\subseteq\mathfrak{F}_{n},

and $\mathfrak{F}_{m}$ is the maximal subfield of $\mathfrak{F}_{n}$ that is unramified over $\mathbb{Q}$ away from $m$ .

Proof.

All primes $p|n$ are doubly-ramified in $\mathfrak{L}_{n}$ and in $\mathfrak{L}_{m}\mathfrak{L}_{n}=\mathfrak{L}_{m}\mathfrak{L}_{n/m}$ . A short computation of ramification indices shows that the extension

\mathfrak{L}_{m}\mathfrak{L}_{n}/\mathfrak{L}_{n}

is unramified. Since $\mathfrak{F}_{m}/\mathfrak{L}_{m}$ is unramified, so is the tower $\mathfrak{F}_{m}\mathfrak{L}_{n}/\mathfrak{L}_{m}\mathfrak{L}_{n}/\mathfrak{L}_{n}$ , whereby

\mathfrak{F}_{m}\subseteq\mathfrak{F}_{m}\mathfrak{L}_{n}\subseteq\mathfrak{L}_{n}^{un}=\mathfrak{F}_{n}.

Finally, the ramification index of any prime $\mathfrak{p}$ in $\mathfrak{F}_{n}$ is at most $2$ . Therefore any field $F\subseteq\mathfrak{F}_{n}$ that contains $\mathfrak{L}_{m}$ and is unramified over $\mathbb{Q}$ away from $m$ is necessarily unramified over $\mathfrak{L}_{m}$ . Therefore $\mathfrak{F}_{m}$ is the maximal subfield of $\mathfrak{F}_{n}$ that is unramified over $\mathbb{Q}$ away from $m$ . ∎

Definition 3.6.

For $p$ prime, we denote by

\tau_{p}\in\textnormal{Gal}(\mathfrak{L}_{p}/\mathbb{Q})

the only non-trivial element in $\textnormal{Gal}(\mathfrak{L}_{p}/\mathbb{Q})\simeq\mathbb{Z}/2\mathbb{Z}$ .

Definition 3.7.

Let $n\in{\mathbb{N}^{*}}$ . We define $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\in{\mathscr{T}\textnormal{op}}$ to be

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}:=\left|\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{F}_{n}}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\right)\right|\in{\mathscr{T}\textnormal{op}}.

As a set, $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ comprises all primes $\mathfrak{p}$ in $\mathfrak{F}_{n}$ dividing $n$ . For every $F\subseteq\mathfrak{F}_{n}$ finite, Galois over $\mathbb{Q}$ , $|\textnormal{Spec}\left(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\right)|$ is finite discrete. The forgetful functor

\textrm{Schemes}^{\textnormal{op}}\to{\mathscr{T}\textnormal{op}}

preserves cofiltered limits, therefore

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}=|\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{F}_{n}}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\right)|\simeq\lim_{\begin{subarray}{c}\longleftarrow\\ F\end{subarray}}|\textnormal{Spec}\left(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\right)|\in\textnormal{Pro-}{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}.

The profinite group $\mathfrak{G}_{n}$ acts continuously on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ via $\mathfrak{p}\mapsto g(\mathfrak{p})$ .

The goal now is to describe a topological kei structure on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ .

Definition 3.8.

Let $n\in{\mathbb{N}^{*}}$ and let $m|n$ be prime. By $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}$ we denote

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}=|\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{F}_{n}}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\right)|\subseteq\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}.

For $p|n$ prime, the $\mathfrak{G}_{n}$ -action on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ restricts to a transitive action on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}$ . For $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}$ , the stabilizer of $\mathfrak{p}$ in $\mathfrak{G}_{n}$ is the decomposition group of $\mathfrak{p}$ , denoted by

D_{\mathfrak{p}}:=\textnormal{Stab}_{\mathfrak{G}_{n}}(\mathfrak{p}).

Definition 3.9.

Let $n\in{\mathbb{N}^{*}}$ and let $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . By $\I_{\mathfrak{p}}\leq\mathfrak{G}_{n}$ we denote the inertia group

\I_{\mathfrak{p}}:=\{g\in\mathfrak{G}_{n}\;|\;\forall\alpha\in\mathcal{O}_{\mathfrak{F}_{n}}\;,\;g(\alpha)-\alpha\in\mathfrak{p}\}\leq\mathfrak{G}_{n}

over $p$ , the rational prime under $\mathfrak{p}$ . The extension $\mathfrak{F}_{n}/\mathfrak{L}_{p}$ is unramified at $p$ and $\mathfrak{L}_{p}/\mathbb{Q}$ is totally ramified at $p$ , therefore

\I_{\mathfrak{p}}\simeq\textnormal{Gal}(\mathfrak{L}_{p}/\mathbb{Q})\simeq\mathbb{Z}/2\mathbb{Z}.

We denote the unique nontrivial element in $\I_{\mathfrak{p}}$ by

\mathfrak{m}_{n,\mathfrak{p}}\in\I_{\mathfrak{p}}\leq\mathfrak{G}_{n}.

Proposition 3.10.

Let $n\in{\mathbb{N}^{*}}$ , let $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ and let $g\in\mathfrak{G}_{n}$ . Then

1.

$\mathfrak{m}_{n,\mathfrak{p}}(\mathfrak{p})=\mathfrak{p}$ .
2.

$\mathfrak{m}_{n,\mathfrak{p}}^{2}=1_{\mathfrak{G}_{n}}$ .
3.

$g\mathfrak{m}_{n,\mathfrak{p}}g^{-1}=\mathfrak{m}_{n,g(\mathfrak{p})}$ .

Proof.

Claim $(1)$ holds because the inertia group $\I_{\mathfrak{p}}$ is a subgroup of the decomposition group $D_{\mathfrak{p}}$ . In particular $\mathfrak{m}_{n,\mathfrak{p}}\in\I_{\mathfrak{p}}$ stabilizes $\mathfrak{p}$ :

\mathfrak{m}_{n,\mathfrak{p}}(\mathfrak{p})=\mathfrak{p}.

Claim $(2)$ holds because $\I_{\mathfrak{p}}\simeq\mathbb{Z}/2\mathbb{Z}$ - therefore $\mathfrak{m}_{n,\mathfrak{p}}\in\I_{\mathfrak{p}}$ is an involution. Claim $(3)$ holds because conjugation by $g$ maps $\I_{\mathfrak{p}}$ isomorphically onto $\I_{g(\mathfrak{p})}$ . As the sole non-trivial elements in their respective groups, $\mathfrak{m}_{n,\mathfrak{p}}$ is mapped to $\mathfrak{m}_{n,g(\mathfrak{p})}$ :

g\mathfrak{m}_{n,\mathfrak{p}}g^{-1}=\mathfrak{m}_{n,g(\mathfrak{p})}.

∎

Proposition 3.11.

Let $n\in{\mathbb{N}^{*}}$ . Then the function

\mathfrak{m}_{n}\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathfrak{G}_{n}\;\;,\;\;\;\;\mathfrak{p}\mapsto\mathfrak{m}_{n,\mathfrak{p}}

is continuous.

Proof.

Let $p|n$ be prime. The $\mathfrak{G}_{n}$ -action on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}$ is continuous and transitive. Fix some $\mathfrak{p}_{0}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}$ . The stabilizer $D_{\mathfrak{p}_{0}}\leq\mathfrak{G}_{n}$ is closed, the quotient map $\mathfrak{G}_{n}\to\mathfrak{G}_{n}/D_{\mathfrak{p}_{0}}$ is open, and the map

\mathfrak{G}_{n}/D_{\mathfrak{p}_{0}}\xrightarrow{gD_{\mathfrak{p}_{0}}\longmapsto g(\mathfrak{p}_{0})}\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}

is a homeomorphism. The composition

\mathfrak{G}_{n}\xrightarrow{g\mapsto g(\mathfrak{p}_{0})}\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}\xrightarrow{\mathfrak{m}_{n}}\mathfrak{G}_{n}\;\;,\;\;\;\;g\mapsto g\mathfrak{m}_{n,\mathfrak{p}_{0}}g^{-1}

is also continuous. It follows that $(\mathfrak{m}_{n})_{|\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}}\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}\to\mathfrak{G}_{n}$ is continuous. Since $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\coprod_{p|n}\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}\in{\mathscr{T}\textnormal{op}}$ , we conclude that $\mathfrak{m}_{n}\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathfrak{G}_{n}$ is continuous. ∎

Proposition 3.12.

Let $n\in{\mathbb{N}^{*}}$ . Then $(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathfrak{G}_{n},\mathfrak{m}_{n})$ is a topological augmented kei.

Proof.

Let $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ and let $g\in\mathfrak{G}_{n}$ . From proposition 3.10,

\mathfrak{m}_{n,\mathfrak{p}}(\mathfrak{p})=\mathfrak{p}\;\;\;\;\textrm{,}\;\;\;\;\;\;\;\;\mathfrak{m}_{n,\mathfrak{p}}^{2}=1_{\mathfrak{G}_{n}}\;\;\;\;\;\;\;\;\textrm{and}\;\;\;\;\;\;\;\;g\mathfrak{m}_{n,\mathfrak{p}}g^{-1}=\mathfrak{m}_{n,g(\mathfrak{p})}.

It follows that $(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathfrak{G}_{n},\mathfrak{m}_{n})$ is an augmented kei. The $\mathfrak{G}_{n}$ -action on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is continuous, and by proposition 3.11 the augmentation map $\mathfrak{m}_{n}\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathfrak{G}_{n}$ is continuous. We conclude that $(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathfrak{G}_{n},\mathfrak{m}_{n})$ is a topological augmented kei. ∎

Definition 3.13.

Let $n\in{\mathbb{N}^{*}}$ . The augmented arithmetic kei of $n$ is defined as

\mathfrak{A}_{n}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathfrak{G}_{n},\mathfrak{m}_{n})\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}.

The fundamental arithmetic kei of $n$ is defined to be the kei $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , with structure induced from the augmentation with $\mathfrak{G}_{n}$ :

{{}^{\mathfrak{p}}{\mathfrak{q}}}:=\mathfrak{m}_{n,\mathfrak{p}}(\mathfrak{q}).

Proposition 3.14.

Let $n\in{\mathbb{N}^{*}}$ . Then $\mathfrak{A}_{n}\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ is concise.

Proof.

Let $N\leq\mathfrak{G}_{n}$ denote the closed subgroup

N=\overline{\langle\{\mathfrak{m}_{n,\mathfrak{p}}\;|\;\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\}\rangle}\leq\mathfrak{G}_{n}.

From proposition 3.10, $g\mathfrak{m}_{n,\mathfrak{p}}g^{-1}=\mathfrak{m}_{n,g(\mathfrak{p})}$ for all $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ and $g\in\mathfrak{G}_{n}$ , hence $N\unlhd\mathfrak{G}_{n}$ is normal. Since $\mathfrak{F}_{n}/\mathbb{Q}$ is unramified away from $n$ and $\I_{\mathfrak{p}}\leq N$ for all $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ , the quotient $\mathfrak{G}_{n}/N$ corresponds to a nowhere-ramified Galois extension of $\mathbb{Q}$ . By the Hermite-Minkowski theorem this is $\mathbb{Q}$ itself, hence

\overline{\langle\{\mathfrak{m}_{n,\mathfrak{p}}\;|\;\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\}\rangle}=N=\mathfrak{G}_{n}.

Hence $\mathfrak{A}_{n}\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ is concise. ∎

Proposition 3.15.

Let $n\in{\mathbb{N}^{*}}$ . Then $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ is small.

Proof.

Let $d\in\mathbb{N}$ . The open subgroups of $\mathfrak{G}_{n}$ of index $d$ correspond to number fields $L\subseteq\mathfrak{F}_{n}$ of degree $[L:\mathbb{Q}]=d$ . Such $L$ are unramified away from $n$ , therefore satisfy

\textnormal{Disc}(L)\leq n^{d}.

By the Hermite-Minkowsk theorem, there are finitely many such fields $L$ . Hence $\mathfrak{G}_{n}\in\textnormal{Pro-}{\mathscr{G}\textnormal{rp}}^{\textnormal{fin}}$ is small. By proposition 3.14, $\mathfrak{A}_{n}\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ is concise. For each $p|n$ , the group $\mathfrak{G}_{n}$ acts transitively on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,p}$ . Therefore

\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\{p|n\;\textrm{prime}\}\in{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}.

By proposition 2.29, $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is small. ∎

Example 3.16.

Following [Yam97], for $n=3,7,11,19,43,67,143$ , we have $\mathfrak{F}_{n}=\mathfrak{L}_{n}$ . In terms of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ , these $n$ are indistinguishable from the unknot, since

\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq*.

3.3 $\mathscr{K}$ -Coloring Arithmetic Links

Definition 3.17.

Let $n\in{\mathbb{N}^{*}}$ and let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . We define sets

\textnormal{Col}_{\mathscr{K}}(n):=\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{K})

and

\textnormal{Col}_{\mathscr{K}}^{\textnormal{dom}}(n):=\{f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\twoheadrightarrow\mathscr{K}\}\subseteq\textnormal{Col}_{\mathscr{K}}(n).

By proposition 3.15, $\textnormal{Col}_{\mathscr{K}}(n)$ is finite. We define $\textnormal{col}_{\mathscr{K}}(n),\textnormal{col}_{\mathscr{K}}^{\textnormal{dom}}(n)\in\mathbb{N}$ via

\textnormal{col}_{\mathscr{K}}(n):=\left|\textnormal{Col}_{\mathscr{K}}(n)\right|\;\;,\;\;\;\;\textnormal{col}^{\textnormal{dom}}_{\mathscr{K}}(n):=\left|\textnormal{Col}^{\textnormal{dom}}_{\mathscr{K}}(n)\right|.

Definition 3.18.

Let $n\in{\mathbb{N}^{*}}$ , let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , and let $F\subseteq\mathfrak{F}_{n}$ be finite and Galois over $\mathbb{Q}$ . A $\mathscr{K}$ -coloring $f\in\textnormal{Col}_{\mathscr{K}}(n)$ of $n$ is said to be defined on $F$ if $f$ is well-defined on the quotient $\textnormal{Gal}(\mathfrak{F}_{n}/F)\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . We say that $\textnormal{Col}_{\mathscr{K}}(n)$ is defined on $F$ if every $f\in\textnormal{Col}_{\mathscr{K}}(n)$ is defined on $F$ .

Lemma 3.19.

Let $n\in{\mathbb{N}^{*}}$ , let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ and let $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{K}$ be a kei morphism. Denote by $\mathscr{K}^{\prime}=f(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\leq\mathscr{K}$ , and by

H_{f}:=\ker\left(\rho_{\mathfrak{A}_{n},f}\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{K}^{\prime})\right)\unlhd\mathfrak{G}_{n}.

Let $F_{f}=(\mathfrak{F}_{n})^{H_{f}}\subseteq\mathfrak{F}_{n}$ . Then $f$ factors through

H_{f}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\textnormal{Spec}\left(\mathcal{O}_{F_{f}}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\right)\to\mathscr{K},

which is to say that $f$ is defined on $F_{f}$ .

Proof.

By proposition 3.14, $\mathfrak{A}_{n}\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ is concise. The claim therefore follows from proposition 2.25 (proposition 2.21) ∎

Lemma 3.20.

Let $n\in{\mathbb{N}^{*}}$ . Then the construction $\textnormal{Col}_{\mathscr{K}}(n)$ assembles to a functor

\textnormal{Col}_{\bullet}(n)\colon{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\to{\mathscr{S}\textnormal{et}}^{\textnormal{fin}}

that preserves finite limits.

Proof.

As is the case for all corepresentable functors, the functor

\textnormal{Hom}_{\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},-)\colon\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\to{\mathscr{S}\textnormal{et}}

preserves all limits. The canonical embedding

{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\hookrightarrow\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}

preserves finite limits. The composition

\textnormal{Col}_{\bullet}(n)\colon{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\hookrightarrow\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}\to{\mathscr{S}\textnormal{et}}

therefore preserves finite limits. By proposition 3.15, $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is small, so these finite limits in ${\mathscr{S}\textnormal{et}}$ are in fact computed in ${\mathscr{S}\textnormal{et}}^{\textnormal{fin}}$ . ∎

Proposition 3.21.

Let $m,n\in{\mathbb{N}^{*}}$ be s.t. $m|n$ . Let $H\leq\mathfrak{G}_{n}$ denote the closed subgroup

H:=\overline{\langle\mathfrak{m}_{n,\mathfrak{p}}\;|\;\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\setminus\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}\rangle}\leq\mathfrak{G}_{n}.

Then

\mathfrak{F}_{n}^{H}=\mathfrak{F}_{m}

and

H\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}\simeq\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{m}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}.

Proof.

The elements $\{\mathfrak{m}_{\mathfrak{p}}\}_{\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\setminus\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}}$ are precisely the nontrivial inertia elements for primes in $\mathfrak{F}_{n}$ over all $p|\frac{n}{m}$ . Therefore $\mathfrak{F}_{n}^{H}\subseteq\mathfrak{F}_{n}$ is the maximal subfield ramified over $\mathbb{Q}$ at most at $m$ . By proposition 3.5,

\mathfrak{F}_{n}^{H}=\mathfrak{F}_{m}

- and $H=\ker(\mathfrak{G}_{n}\twoheadrightarrow\mathfrak{G}_{m})\unlhd\mathfrak{G}_{n}$ accordingly. Let

\mathfrak{A}_{n,m}=(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m},\mathfrak{G}_{n},(\mathfrak{m}_{n})_{|\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}})\in\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}.

By proposition 2.25 (proposition 2.20), the canonical morphism

(f,\widetilde{f})\colon\mathfrak{A}_{n,m}\to\mathfrak{A}_{m}\;\;:\;\;\;\;f(\mathfrak{p})=\mathfrak{p}\cap\mathcal{O}_{\mathfrak{F}_{m}}\;\;,\;\;\;\;\widetilde{f}(g)=g_{|\mathfrak{F}_{m}}

factors through the quotient $H\backslash\mathfrak{A}_{n,m}$ . The map $H\backslash\mathfrak{A}_{n,m}\to\mathfrak{A}_{m}$ is an isomorphism because it induces isomorphisms $H\backslash\mathfrak{G}_{n}{\xrightarrow{\sim}}\mathfrak{G}_{m}$ and

H\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}=H\backslash\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{F}_{n}}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\right){\xrightarrow{\sim}}\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{F}_{n}^{H}}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\right)=\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{m}.

By proposition 2.18, the kei structure on $H\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}$ is induced by that of $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}\in{\mathscr{K}\textnormal{ei}}$ , therefore we obtain an isomorphism

H\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,m}{\xrightarrow{\sim}}\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{m}\in\textnormal{Pro-}{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}.

∎

Lemma 3.22.

Let $n\in{\mathbb{N}^{*}}$ . Then $\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is the trivial kei $\{p|n\;\textrm{prime}\}$ :

\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\mathscr{T}_{\omega(n)}\in{\mathscr{K}\textnormal{ei}}.

Proof.

As a set,

\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}=\textnormal{Gal}(\mathfrak{F}_{n}/\mathbb{Q})\backslash\left|\textnormal{Spec}(\mathcal{O}_{\mathfrak{F}_{n}}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z})\right|\simeq\left|\textnormal{Spec}(\mathbb{Z}/n\mathbb{Z})\right|=\{p|n\;\textrm{prime}\}.

The kei structure on $\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is induced by Galois action on $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . These act trivially on sets of rational primes. Therefore $\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\in{\mathscr{K}\textnormal{ei}}$ is trivial:

\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\mathscr{T}_{\omega(n)}.

∎

Proposition 3.23.

Let $\mathscr{T}\in{\mathscr{K}\textnormal{ei}}$ be trivial. There is a natural isomorphism

\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{T})\simeq\mathscr{T}^{\{p|n\;\textnormal{prime}\}}.

Proof.

Let $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{T}$ be a kei morphism. Denote by $\mathscr{T}^{\prime}:=f(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\leq\mathscr{T}$ , also trivial. Therefore $\textnormal{Inn}(\mathscr{T}^{\prime})=\{\textnormal{Id}\}$ , and

H_{f}:=\ker\Big{(}\rho_{\mathfrak{A}_{n},f}\colon\mathfrak{G}_{n}\to\textnormal{Inn}(\mathscr{T}^{\prime})\Big{)}=\mathfrak{G}_{n}.

By lemma 3.19, $f$ factors through $\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . By lemma 3.22, $\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ is a trivial kei with underlying set $\{p|n\;\textnormal{prime}\}$ . Therefore

\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{T})\simeq\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}(\mathfrak{G}_{n}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{T})\simeq\mathscr{T}^{\{p|n\;\textnormal{prime}\}}.

∎

Proposition 3.24.

Let $n\in{\mathbb{N}^{*}}$ be square-free and let $\mathscr{K}_{1},\mathscr{K}_{2}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . Then

\textnormal{col}_{\mathscr{K}_{1}\sqcup\mathscr{K}_{2}}(n)=\sum_{n=n_{1}n_{2}}\textnormal{col}_{\mathscr{K}_{1}}(n_{1})\cdot\textnormal{col}_{\mathscr{K}_{2}}(n_{2}).

Proof.

Let $\mathscr{T}_{2}=\{*_{1},*_{2}\}\in{\mathscr{K}\textnormal{ei}}$ . By proposition 3.23, there is a natural bijection

\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{T}_{2})\simeq\mathscr{T}_{2}^{\{p|n\;\textnormal{prime}\}}\simeq\{(n_{1},n_{2})\;|\;n=n_{1}n_{2}\}.

for $\eta\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{T}_{2}$ corresponding to $(n_{1},n_{2})$ , we denote by

\textnormal{Col}_{k_{1}\sqcup\mathscr{K}_{2}}(n)_{(n_{1},n_{2})}:=\textnormal{Hom}_{/\eta}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{K}_{1}\sqcup\mathscr{K}_{2}).

By proposition 2.31 and proposition 3.21, there is a bijection

\textnormal{Col}_{\mathscr{K}_{1}\sqcup\mathscr{K}_{2}}(n)_{(n_{1},n_{2})}\simeq

\simeq\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(H_{n,n_{1}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,n_{1}},\mathscr{K}_{1})\times\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(H_{n,n_{2}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n,n_{2}},\mathscr{K}_{2})\simeq

\simeq\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n_{1}},\mathscr{K}_{1})\times\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n_{2}},\mathscr{K}_{2}).

where $H_{n,m}:=\textnormal{Gal}(\mathfrak{F}_{n}/\mathfrak{F}_{m})$ for all $m|n$ as in proposition 3.21. Therefore

\textnormal{Col}_{\mathscr{K}_{1}\sqcup\mathscr{K}_{2}}(n)_{(n_{1},n_{2})}\simeq\textnormal{Col}_{\mathscr{K}_{1}}(n_{1})\times\textnormal{Col}_{\mathscr{K}_{2}}(n_{2}),

and

\textnormal{col}_{\mathscr{K}_{1}\sqcup\mathscr{K}_{2}}(n)=\sum_{n=n_{1}n_{2}}\left|\textnormal{Col}_{k_{1}\sqcup\mathscr{K}_{2}}(n)_{(n_{1},n_{2})}\right|=\sum_{n=n_{1}n_{2}}\textnormal{col}_{\mathscr{K}_{1}}(n_{1})\cdot\textnormal{col}_{\mathscr{K}_{2}}(n_{2}).

∎

4 Main Conjecture

While the analogy of Barry Mazur likens positive square-free integers to links, the fibration of the affine line $\mathbb{A}^{1}_{\mathbb{F}_{p}}$ over $\mathbb{F}_{p}$ , paints a sharper picture, likening a degree- $k$ separable polynomial $f(t)\in\mathbb{F}_{p}[t]$ to the closure of a braid on $k=\log_{p}\left|\mathbb{F}_{p}[t]/f(t)\right|$ strands. By coarse analogy, $n\in{\mathbb{N}^{*}}$ should be a braid on $\log n$ strands. The scope of this claim is limited by the lack of a Frobenius map, more so by the fact that $\log n\notin\mathbb{N}$ . Nevertheless, the claim is bolstered by statistical phenomena about random numbers: A random number of magnitude $X$ is prime with probability $\approx\left(\log X\right)^{-1}$ , mirroring the fact that the closure of a random braids $\sigma\in B_{k}$ is connected with probability $1/k$ . The aim of this paper is to use asymptotic statistical phenomena of kei-coloring invariants to lend further credence to this notion.

For a finite kei $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , basic statistical invariants about $\mathscr{K}$ -coloring of random links admit particularly straightforward description when viewed as closures of braids: Let $k\in\mathbb{N}$ and let $B_{k}$ be the Artin braid group on $k$ strands. The braid closure $\overline{\sigma}$ of a braid $\sigma\in B_{k}$ is a link in $S^{3}$ . The function mapping $\sigma\in B_{k}$ to $\textnormal{col}_{\mathscr{K}}(\overline{\sigma})\in\mathbb{N}$ extends to a locally constant function on the profinite completion $\widehat{B}_{k}$ . By [DS23] there exists for every $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ an integer-valued polynomial $P_{\mathscr{K}}(t)\in\mathbb{Q}[t]$ s.t. for all $k\gg 0$ ,

P_{\mathscr{K}}(k)=\int\limits_{\widehat{B}_{k}}\textnormal{col}_{\mathscr{K}}(\overline{\sigma})d\mu,

where $\mu$ is the Haar measure on $\widehat{B}_{k}$ . We use this to further bolster the notion that a random $n\in{\mathbb{N}^{*}}$ should be thought of as the closure of a random braid on $\approx\log X$ strands.

4.1 The Hilbert Polynomial of a Finite Kei

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . Recall in [DS23] that there exists an integer-valued polynomial $P_{\mathscr{K}}(x)\in\mathbb{Q}[x]$ s.t. for all $k\gg 0$ ,

P_{\mathscr{K}}(k)=\int\limits_{\widehat{B}_{k}}{\textnormal{c}\widehat{\textnormal{o}}\textnormal{l}}_{\mathscr{K}}d\mu,

where $\mu$ is the Haar measure on the profinite braid group $\widehat{B}_{k}$ . The degree of $P_{\mathscr{K}}$ is explicitly calculated in terms of $\mathscr{K}$ :

\deg P_{\mathscr{K}}=-1+\max_{\mathscr{K}^{\prime}\leq\mathscr{K}}\left|\textnormal{Inn}(\mathscr{K}^{\prime})\backslash\mathscr{K}^{\prime}\right|.

4.2 Statistics of $\mathscr{K}$ -Coloring Random Arithmetic Links

Definition 4.1.

Let $s\in{\mathbb{N}^{*}}$ be square-free. By ${\mathbb{N}^{*}_{s}}$ we denote the set of square-free elements coprime to $s$ :

{\mathbb{N}^{*}_{s}}:=\{n\in{\mathbb{N}^{*}}\;|\;(n,s)=1\}.

Definition 4.2.

Let $s\in{\mathbb{N}^{*}}$ , let $f\colon{\mathbb{N}^{*}_{s}}\to\mathbb{C}$ , and let $L\in\mathbb{C}$ . We say $f$ converges to $L$ if for all $\varepsilon>0$ there exists $u_{\varepsilon}\in{\mathbb{N}^{*}_{s}}$ s.t. for all $u\in{\mathbb{N}^{*}_{s}}$ ,

u_{\varepsilon}|u\;\;\Longrightarrow\;\;\left|f(u)-L\right|<\varepsilon.

We shall use either of the following notations:

\lim_{u\in{\mathbb{N}^{*}_{s}}}f(u)=L\;\;\;\;\;\;\;\;,\textnormal{}\;\;\;\;\;\;\;\;f(u)\xrightarrow{u\in{\mathbb{N}^{*}}}L.

Definition 4.3.

Let $s\in{\mathbb{N}^{*}}$ be squarefree. By $\gamma_{s}\in\mathbb{R}$ we denote

\gamma_{s}:=\prod_{p\nmid s}(1-p^{-2})\cdot\prod_{p|s}(1-p^{-1})=\zeta(2)^{-1}\cdot\prod_{p|s}(1+p^{-1})^{-1}.

Remark 4.4.

For square-free $s$ , the number of $s$ -coprime square-free integers up to $X$ is

\left|{\mathbb{N}^{*}_{s}}\cap[1,X]\right|=\gamma_{s}X(1+o(1)).

Therefore the probability of $n\in{\mathbb{N}^{*}_{s}}\cap[1,X]$ being prime is

\frac{\pi(X)+O_{s}(1)}{\left|{\mathbb{N}^{*}_{s}}\cap[1,X]\right|}=\frac{\pi(X)+O_{s}(1)}{X}\frac{X}{\left|{\mathbb{N}^{*}_{s}}\cap[1,X]\right|}=\frac{1+o(1)}{\gamma_{s}\log X}.

In likening a squarefree integer to the closure of a braid, the correct number of strands is purported to be inverse to the probability of primality. In light of this computation, for fixed $s\in{\mathbb{N}^{*}}$ we should think of $n\in{\mathbb{N}^{*}_{s}}\cap[1,X]$ as the closure of a braid with $\approx\gamma_{s}\log X$ strands.

Definition 4.5.

Let $f\colon{\mathbb{N}^{*}}\to\mathbb{C}$ and let $s\in{\mathbb{N}^{*}}$ . For $1\leq X\in\mathbb{R}$ we denote by

\mathcal{N}_{f,s}(X):=\sum_{n\in{\mathbb{N}^{*}_{s}}\cap[1,X]}f(n)

and

\mathcal{E}_{f,s}(X)=\frac{\mathcal{N}_{f,s}(X)}{|{\mathbb{N}^{*}_{s}}\cap[1,X]|}.

For $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , we denote by

\mathcal{N}_{\mathscr{K},s}(X):=\mathcal{N}_{\textnormal{col}_{\mathscr{K}},s}(X)\;\;\;\;\textrm{and}\;\;\;\;\;\;\;\;\mathcal{E}_{\mathscr{K},s}(X):=\mathcal{E}_{\textnormal{col}_{\mathscr{K}},s}(X).

Definition 4.6.

Let $f\colon{\mathbb{N}^{*}}\to\mathbb{C}$ , let $0\leq\beta\in\mathbb{N}$ and let $0\neq c\in\mathbb{C}$ . We say $f$ has generic summatory type $(\beta,c)$ if for every $s\in{\mathbb{N}^{*}}$ there exists a limit

c_{s}(f):=\lim_{X\to\infty}\frac{\mathcal{E}_{f,s}(X)}{(\gamma_{s}\log X)^{\beta}}\in\mathbb{C}\setminus\{0\},

and if

c(f):=\lim_{s\in{\mathbb{N}^{*}}}c_{s}(f)=c.

We denote by ${\mathscr{W}(\beta,c)}$ the set of all such functions:

{\mathscr{W}(\beta,c)}:=\{f\colon{\mathbb{N}^{*}}\to\mathbb{C}\;|\;f\;\textrm{has generic summatory type}\;(\beta,c)\}.

In the case of $f=\textnormal{col}_{\mathscr{K}}$ for some $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ , we denote by

c_{s}(\mathscr{K}):=c_{s}(\textnormal{col}_{\mathscr{K}})\;\;\;\;\textnormal{and}\;\;\;\;\;\;\;\;c(\mathscr{K}):=c(\textnormal{col}_{\mathscr{K}}).

Remark 4.7.

In light of remark 4.4, for $f\in{\mathscr{W}(\beta,c)}$ and $s\in{\mathbb{N}^{*}}$ ,

\gamma_{s}^{1+\beta}c_{s}(f)=\lim_{X\to\infty}\frac{\mathcal{N}_{f,s}(X)}{X\log^{\beta}(X)}.

In practice, it is the RHS here that we will be computing for every $s\in{\mathbb{N}^{*}}$ .

We are now ready to state the main conjecture of the paper:

Conjecture 4.8.

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . Let $P_{\mathscr{K}}(t)\in\mathbb{Q}[t]$ be the Hilbert polynomial of $\mathscr{K}$ , as defined in [DS23]. Then there exists $0<c\in\mathbb{Q}$ s.t.

\textnormal{col}_{\mathscr{K}}\in{\mathscr{W}(\deg P_{\mathscr{K}},c)}.

5 Preliminary Numerical Computations

A few preliminary definitions will be useful. In the following discussion of arithmetic functions, we follow standard notation: For arithmetic functions $f,g\colon\mathbb{N}\to\mathbb{C}$ , the convolution $f*g\colon\mathbb{N}\to\mathbb{C}$ is defined via

(f*g)(n):=\sum_{n_{1}n_{2}=n}f(n_{1})g(n_{2}).

By $\delta_{1}\colon\mathbb{N}\to\mathbb{C}$ we denote the unit with respect to convolution:

\delta_{1}(n)=\delta_{1,n}=\begin{cases}1&,\;n=1\\ 0&,\;n\neq 1\end{cases}\;\;.

By $\mathds{1},\mu\colon\mathbb{N}\to\mathbb{C}$ we respectively denote the constant function $\mathds{1}(n)=1$ and the Möbius function $\mu$ , its inverse with respect to convolution. This is the multiplicative arithmetic function satisfying for all prime powers $p^{\nu}\neq 1$ ,

\mu(p^{\nu})=\begin{cases}-1&,\;\nu=1\\ 0&,\;\nu>1\end{cases}.

The function $\omega\colon\mathbb{N}\to\mathbb{C}$ counts distinct prime divisors:

\omega(n):=|\{p|n\;\textnormal{prime}\}|.

Definition 5.1.

Let $s\in{\mathbb{N}^{*}}$ . We denote by $\mathds{1}_{(s)},\mathds{1}_{s}\;\colon\mathbb{N}\to\mathbb{C}$ the functions

\mathds{1}_{s}(n)=\begin{cases}1&,\;(n,s)=1\\ 0&,\;(n,s)>1\end{cases}\;\;\;\;,\;\;\;\;\;\;\;\;\mathds{1}_{(s)}(n)=\begin{cases}1&,\;n|s^{k}\;\textnormal{ for some }k\gg 0\\ 0&,\;\textnormal{otherwise}\end{cases}\;\;.

We denote by $\mu_{s},\mu_{(s)}\colon\mathbb{N}\to\mathbb{C}$ the functions

\mu_{s}(n)=\mathds{1}_{s}(n)\mu(n)\;\;,\;\;\;\;\mu_{(s)}(n)=\mathds{1}_{(s)}(n)\mu(n).

These are all multiplicative arithmetic functions, and they satisfy

\mathds{1}_{s}*\mu_{s}=\mathds{1}_{(s)}*\mu_{(s)}=\delta_{1}\;\;,\;\;\;\;\mathds{1}_{s}*\mathds{1}_{(s)}=\mathds{1}.

5.1 Divisor Counting

Let ${\mathcal{a}}\in\mathbb{N}$ . Then for all $0<n\in\mathbb{N}$

\mathds{1}^{*{\mathcal{a}}}(n)=\underbrace{\mathds{1}*\cdots*\mathds{1}}_{{\mathcal{a}}}(n)=d_{\mathcal{a}}(n),

where $d_{\mathcal{a}}$ is the ${\mathcal{a}}$ -fold divisor counting function. This is the multiplicative function satisfying

\mathds{1}^{*{\mathcal{a}}}(p^{\nu})={{\mathcal{a}}+\nu-1\choose{\mathcal{a}}-1}

For all prime powers $p^{\nu}$ . We begin by stating some standard results regarding the counting of divisors:

Lemma 5.2.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ . Then for all $m,n\in\mathbb{N}$ ,

\mathds{1}^{*{\mathcal{a}}}(mn)\leq\mathds{1}^{*{\mathcal{a}}}(m)\mathds{1}^{*{\mathcal{a}}}(n).

Proof.

There is an equality

{{\mathcal{a}}+\nu-1\choose{\mathcal{a}}-1}=\frac{({\mathcal{a}}+\nu-1)\cdots(\nu+1)}{({\mathcal{a}}-1)!}=\prod_{k=1}^{{\mathcal{a}}-1}(1+\frac{\nu}{k}).

The desired inequality immediately follows from the fact that

(1+\frac{\nu}{k})(1+\frac{\nu^{\prime}}{k})\geq 1+\frac{\nu+\nu^{\prime}}{k}.

∎

Lemma 5.3.

Let ${\mathcal{b}}\in\mathbb{N}$ and let $n\in{\mathbb{N}^{*}}$ . Then

\sum_{m|n}{\mathcal{b}}^{\omega(m)}=({\mathcal{b}}+1)^{\omega(n)}.

Proof.

Since $n$ is square-free, we have

\mathds{1}^{*{\mathcal{b}}}(n)=\prod_{p|n}\mathds{1}^{*{\mathcal{b}}}(p)={\mathcal{b}}^{\omega(n)}.

Hence

({\mathcal{b}}+1)^{\omega(n)}=\mathds{1}^{*({\mathcal{b}}+1)}(n)=\sum_{mm^{\prime}=n}\mathds{1}^{*{\mathcal{b}}}(m)\mathds{1}(m^{\prime})=\sum_{m|n}{\mathcal{b}}^{\omega(m)}.

∎

Lemma 5.4.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ . Then there exists $C=C_{\mathcal{a}}>0$ s.t. the term

\theta_{\mathcal{a}}(X):=\sum_{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)-\frac{X\log^{{\mathcal{a}}-1}(X)}{({\mathcal{a}}-1)!}

is bounded for all $X\geq 1$ by

|\theta_{\mathcal{a}}(X)|\leq\begin{cases}CX(\log^{{\mathcal{a}}-2}(X)+1)&,\;{\mathcal{a}}\geq 2\\ C&,\;{\mathcal{a}}=1\end{cases}\;\;.

In particular, as $X\to\infty$ we have

\sum_{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)=\frac{X\log^{{\mathcal{a}}-1}(X)}{({\mathcal{a}}-1)!}(1+o(1)).

Proof.

If ${\mathcal{a}}=1$ , then for all $X\geq 1$ ,

|\theta_{1}(X)|=|{\lfloor{X}\rfloor}-X|\leq 1.

If ${\mathcal{a}}\geq 2$ , then there exists $D_{\mathcal{a}}(t)\in\mathbb{R}[t]$ with leading term $\frac{t^{\mathcal{a}}}{({\mathcal{a}}-1)!}$ s.t.

\sum_{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)=XD_{\mathcal{a}}(\log X)+O(X^{1-{\frac{1}{{\mathcal{a}}-1}}}(\log^{{\mathcal{a}}-2}(X)).\;

Hence

\frac{\theta_{\mathcal{a}}(X)}{X}=\frac{1}{X}\sum_{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)-\frac{\log^{{\mathcal{a}}-1}(X)}{({\mathcal{a}}-1)!}=O(\log^{{\mathcal{a}}-2}(X)).

The function

\frac{\theta_{\mathcal{a}}(X)}{X(\log^{{\mathcal{a}}-2}(X)+1)}\colon[1,\infty)\to\mathbb{R}

is then bounded not only for $X\gg 0$ , but for all $X\in[1,\infty)$ because

X\geq 1\;\Longrightarrow\;\log^{{\mathcal{a}}-2}(X)+1\geq 1.

Hence there is some $C_{\mathcal{a}}>0$ s.t. for all $X\geq 1$ ,

|\theta_{\mathcal{a}}(X)|\leq C_{\mathcal{a}}\cdot X(\log^{{\mathcal{a}}-2}(X)+1).

∎

We have the following immediate corollary in the same spirit of stating a bound for all $X\geq 1$ :

Lemma 5.5.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ . Then there is a constant $C=C_{\mathcal{a}}>0$ s.t. for all $X\geq 1$ ,

\sum_{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)\leq CX(\log^{{\mathcal{a}}-1}(X)+1).

Lemma 5.6.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ and let $\epsilon>0$ . Then for every prime $p$ ,

\sum_{\nu=0}^{\infty}\frac{\mathds{1}^{*{\mathcal{a}}}(p^{\nu})}{p^{\epsilon\nu}}=(1-p^{-\epsilon})^{-{\mathcal{a}}}.

More generally, for every $s\in{\mathbb{N}^{*}}$ ,

\sum_{n=1}^{\infty}\frac{\mathds{1}_{(s)}^{*{\mathcal{a}}}(n)}{n^{\epsilon}}=\prod_{p|s}(1-p^{-\epsilon})^{-{\mathcal{a}}}.

Proof.

From the formal equality of power series

(1-t)^{-{\mathcal{a}}}=\sum_{\nu=0}^{\infty}{{\mathcal{a}}+\nu-1\choose{\mathcal{a}}-1}t^{\nu},

we have

\sum_{\nu=0}^{\infty}\frac{\mathds{1}^{*{\mathcal{a}}}(p^{\nu})}{p^{\epsilon\nu}}=\sum_{\nu}{{\mathcal{a}}+\nu-1\choose{\mathcal{a}}-1}p^{-\epsilon\nu}=(1-p^{-\epsilon})^{-{\mathcal{a}}}.

The function $\mathds{1}_{(s)}^{*a}$ is multiplicative, satisfying

\mathds{1}_{(s)}^{*{\mathcal{a}}}(p^{\nu})=\begin{cases}\mathds{1}^{*{\mathcal{a}}}(p^{\nu})&,\;p|s\\ 0&,\;p\nmid s,\;\nu\geq 1\end{cases}\;\;.

Hence

\sum_{n=1}^{\infty}\frac{\mathds{1}_{(s)}^{*{\mathcal{a}}}(n)}{n^{\epsilon}}=\prod_{p|s}\sum_{\nu}\frac{\mathds{1}^{*{\mathcal{a}}}(p^{\nu})}{p^{\epsilon\nu}}=\prod_{p|s}(1-p^{-\epsilon})^{-{\mathcal{a}}}.

∎

Definition 5.7.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ and let $f\colon\mathbb{N}\to\mathbb{R}$ be multiplicative, s.t.

0\leq f(n)\leq\mathds{1}^{*{\mathcal{a}}}(n)

for all $n\in\mathbb{N}$ . For every prime $p$ we denote by

M_{p}^{({\mathcal{a}})}(f):=(1-p^{-1})^{\mathcal{a}}\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu}}\in(0,1],

and we denote by $M^{({\mathcal{a}})}(f)$ the infinite product

M^{({\mathcal{a}})}(f):=\prod_{p}M_{p}^{({\mathcal{a}})}(f)\in[0,1].

Lemma 5.8.

Let ${\mathcal{a}}\in\mathbb{N}$ and let $f\colon\mathbb{N}\to\mathbb{R}$ be a function s.t. $0\leq f\leq\mathds{1}^{*{\mathcal{a}}}$ . If the product $M^{({\mathcal{a}})}(f)$ converges (to nonzero limit), then so does the infinite sum

\sum_{p}\frac{{\mathcal{a}}-f(p)}{p}.

Proof.

For all prime $p$ ,

1-M_{p}^{({\mathcal{a}})}(f)=M_{p}^{({\mathcal{a}})}(\mathds{1}^{*{\mathcal{a}}})-M_{p}^{({\mathcal{a}})}(f)\geq(1-p^{-1})^{\mathcal{a}}\cdot\frac{\mathds{1}^{*{\mathcal{a}}}(p)-f(p)}{p}\geq 2^{-{\mathcal{a}}}\frac{{\mathcal{a}}-f(p)}{p}.

Hence

0\leq M_{p}^{({\mathcal{a}})}(f)\leq 1-2^{-{\mathcal{a}}}\frac{{\mathcal{a}}-f(p)}{p}\leq 1.

The convergence of $M^{({\mathcal{a}})}(f)$ implies the convergence of

\prod_{p}\left(1-2^{-{\mathcal{a}}}\frac{{\mathcal{a}}-f(p)}{p}\right).

The convergence of the desired sum then follows, as

\sum_{p}\frac{{\mathcal{a}}-f(p)}{p}=2^{\mathcal{a}}\cdot\sum_{p}2^{-{\mathcal{a}}}\frac{{\mathcal{a}}-f(p)}{p}.

∎

Lemma 5.9.

Let $f\colon\mathbb{N}\to\mathbb{R}$ be a function s.t.

0\leq f\leq\mathds{1}_{(s)}^{*{\mathcal{b}}}

for some $b\in\mathbb{N}$ and $s\in{\mathbb{N}^{*}}$ . Then for all $i\in\mathbb{N}$ , the series

\sum_{n=1}^{\infty}\frac{f(n)}{n}\log^{i}(n)

converges.

Proof.

Let $i\in\mathbb{N}$ . Then $\log^{i}(n)<\sqrt{n}$ for $n\gg 0$ . Convergence is therefore dominated by

\sum_{n=1}^{\infty}\frac{f(n)}{\sqrt{n}}\leq\sum_{n=1}^{\infty}\frac{\mathds{1}_{(s)}^{*{\mathcal{b}}}(n)}{\sqrt{n}}=\prod_{p|s}(1-p^{-\frac{1}{2}})^{-{\mathcal{b}}}.

∎

Lemma 5.10.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ and let $f\colon\mathbb{N}\to\mathbb{R}$ be multiplicative s.t.

0\leq f\leq\mathds{1}_{(s)}^{*{\mathcal{b}}}

for some ${\mathcal{b}}\in\mathbb{N}$ and $s\in{\mathbb{N}^{*}}$ . For $g\colon\mathbb{N}\to\mathbb{C}$ and $X\geq 1$ we denote by

\mathcal{M}_{g}(X):=\sum_{1\leq n\leq X}g(X).

Then as $X\to\infty$ ,

\lim_{X\to\infty}\frac{\mathcal{M}_{f*\mathds{1}^{*{\mathcal{a}}}}(X)}{X(\log X)^{{\mathcal{a}}-1}}=\frac{1}{({\mathcal{a}}-1)!}\cdot\sum_{m=1}^{\infty}\frac{f(m)}{m}.

Proof.

Let $X\geq 0$ . Then

\mathcal{M}_{f*\mathds{1}^{*{\mathcal{a}}}}(X)=\sum_{m\leq X}f(m)\sum_{a\leq X/m}\mathds{1}^{*{\mathcal{a}}}(a)=\sum_{m\leq X}f(m)\cdot\left(\frac{X}{m}\frac{\log^{{\mathcal{a}}-1}(X/m)}{({\mathcal{a}}-1)!}+\theta_{\mathcal{a}}(X/m)\right).

Therefore

\frac{\mathcal{M}_{f*\mathds{1}^{*{\mathcal{a}}}}(X)}{X\log^{{\mathcal{a}}-1}(X)}=\frac{1}{({\mathcal{a}}-1)!}\sum_{m\leq X}\frac{f(m)}{m}\left(1-\frac{\log m}{\log X}\right)^{{\mathcal{a}}-1}+\frac{1}{\log^{{\mathcal{a}}-1}(X)}\sum_{m\leq X}\frac{f(m)}{m}\cdot\frac{m}{X}\theta_{\mathcal{a}}(X/m).

Considering the first summand, it follows from lemma 5.9 that for all $i>0$ ,

0\leq\sum_{m\leq X}\frac{f(m)}{m}\left(\frac{\log m}{\log X}\right)^{i}\leq\frac{1}{\log^{i}(X)}\sum_{m=1}^{\infty}\frac{f(m)}{m}\log^{i}(m)\xrightarrow{X\to\infty}0.

Therefore

\sum_{m\leq X}\frac{f(m)}{m}\left(1-\frac{\log m}{\log X}\right)^{{\mathcal{a}}-1}=\sum_{m\leq X}\frac{f(m)}{m}+o(1)=\sum_{m=1}^{\infty}\frac{f(m)}{m}+o(1).

Turning to the remaining summand, we first address the case ${\mathcal{a}}\geq 2$ . By lemma 5.4 there is a constant $C>0$ s.t.

\left|\frac{m}{X}\theta_{\mathcal{a}}(X/m)\right|\leq C\left(\log^{{\mathcal{a}}-2}(X/m)+1\right)\leq C\left(\log^{{\mathcal{a}}-2}(X)+1\right)

for all $1\leq m\leq X$ . Therefore

\left|\sum_{m\leq X}\frac{f(m)}{m}\cdot\frac{m}{X}\theta_{\mathcal{a}}(X/m)\right|\leq\sum_{m\leq X}\frac{f(m)}{m}\cdot\left|\frac{m}{X}\theta_{\mathcal{a}}(X/m)\right|\leq

\leq C\left(\log^{{\mathcal{a}}-2}(X)+1\right)\sum_{m=1}^{\infty}\frac{f(m)}{m}=o(\log^{{\mathcal{a}}-1}(X)).

In the case ${\mathcal{a}}=1$ , this summand is bounded, for some constant $C>0$ , by

\left|\sum_{m\leq X}\frac{f(m)}{X}\theta_{1}(X/m)\right|\leq\frac{C}{X}\sum_{m\leq X}f(m)\leq\frac{C}{X}\sum_{m\leq X}\mathds{1}_{(s)}^{*{\mathcal{a}}}(m)=O\left(X^{-1}\log^{\omega(s)}(X)\right)=o(1).

Thus for any ${\mathcal{a}}\geq 1$ ,

\frac{\mathcal{M}_{f*\mathds{1}^{*{\mathcal{a}}}}(X)}{X\log^{{\mathcal{a}}-1}(X)}=\frac{1}{({\mathcal{a}}-1)!}\left(\sum_{m=1}^{\infty}\frac{f(m)}{m}+o(1)\right)+\frac{o(\log^{{\mathcal{a}}-1}(X))}{\log^{{\mathcal{a}}-1}(X)}=

=\frac{1}{({\mathcal{a}}-1)!}\sum_{m=1}^{\infty}\frac{f(m)}{m}+o(1).

∎

Lemma 5.11.

Let $k\in\mathbb{N}$ and let $0\leq b_{i}\leq a_{i}$ for $1\leq i\leq k$ . Then

\prod_{i}a_{i}-\prod_{i}b_{i}\leq\sum_{i}(a_{i}-b_{i})\prod_{j\neq i}a_{j}.

Proof.

If $a_{i}=0$ for some $i$ then both sides of the inequality equal $0$ . We assume therefore that $a_{i}>0$ for all $i$ . Then $0\leq\frac{b_{i}}{a_{i}}\leq 1$ for all $i$ , therefore

1-\prod_{i}\frac{b_{i}}{a_{i}}\leq\sum_{i}\left(1-\frac{b_{i}}{a_{i}}\right).

Thus

\prod_{i}a_{i}-\prod_{i}b_{i}=\prod_{j}a_{j}\left(1-\prod_{i}\frac{b_{i}}{a_{i}}\right)\leq\prod_{j}a_{j}\sum_{i}\left(1-\frac{b_{i}}{a_{i}}\right)=\sum_{i}(a_{i}-b_{i})\prod_{j\neq i}a_{j}.

∎

Proposition 5.12.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ , and let $f\colon\mathbb{N}\to\mathbb{R}$ be multiplicative s.t.

0\leq f(n)\leq\mathds{1}^{*{\mathcal{a}}}(n)

for all $n\in\mathbb{N}$ . For $X\geq 1$ we denote by

\mathcal{M}_{f}(X):=\sum_{1\leq n\leq X}f(X).

Then

\lim_{X\to\infty}\frac{\mathcal{M}_{f}(X)}{X\log^{{\mathcal{a}}-1}(X)}=\frac{1}{({\mathcal{a}}-1)!}M^{({\mathcal{a}})}(f).

Proof.

For all $s\in{\mathbb{N}^{*}}$ we define the function $f_{s}^{({\mathcal{a}})}\colon\mathbb{N}\to\mathbb{R}$ via

f_{s}^{({\mathcal{a}})}=(f\cdot\mathds{1}_{(s)})*\mathds{1}_{s}^{*{\mathcal{a}}}.

Since $f,\mathds{1}_{(s)},\mathds{1}_{s}$ are multiplicative, then so is $f_{s}^{({\mathcal{a}})}$ , satisfying for all prime powers $p^{\nu}\neq 1$ ,

f_{s}^{({\mathcal{a}})}(p^{\nu})=\sum_{i=0}^{\nu}f(p^{i})\mathds{1}_{(s)}(p^{i})\mathds{1}_{s}^{*{\mathcal{a}}}(p^{\nu-i})=\begin{cases}f(p^{\nu})&,\;p|s\\ \mathds{1}^{*{\mathcal{a}}}(p^{\nu})&,\;p\nmid s\end{cases}\;\;.

Therefore for all $s\in{\mathbb{N}^{*}}$ ,

0\leq f\leq f_{s}^{({\mathcal{a}})}\leq\mathds{1}^{*{\mathcal{a}}}.

The function $f_{s}^{({\mathcal{a}})}$ can be written

f_{s}^{({\mathcal{a}})}=(f\cdot\mathds{1}_{(s)})*\mathds{1}_{s}^{*{\mathcal{a}}}=\left((f\cdot\mathds{1}_{(s)})*\mathds{1}^{*{\mathcal{a}}}\right)*\mu_{(s)}^{*{\mathcal{a}}}.

Note that $\mu_{(s)}^{*{\mathcal{a}}}(a)=0$ for $a>s^{{\mathcal{a}}}$ . Since $0\leq f\mathds{1}_{(s)}\leq\mathds{1}^{*{\mathcal{a}}}\mathds{1}_{(s)}=\mathds{1}_{(s)}^{*{\mathcal{a}}}$ , by lemma 5.10 for all $X\geq 1$ , we have

\frac{\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)}{X}=\frac{1}{X}\sum_{ab\leq X}\mu_{(s)}^{*{\mathcal{a}}}(a)\left((f\cdot\mathds{1}_{(s)})*\mathds{1}^{*{\mathcal{a}}}\right)(b)=

=\sum_{a\leq s^{\mathcal{a}}}\frac{\mu_{(s)}^{*{\mathcal{a}}}(a)}{a}\cdot\frac{a}{X}\sum_{b\leq\frac{X}{a}}\left((f\cdot\mathds{1}_{(s)})*\mathds{1}^{*{\mathcal{a}}}\right)(b)=

=\sum_{a\leq s^{\mathcal{a}}}\frac{\mu_{(s)}^{*{\mathcal{a}}}(a)}{a}\cdot\frac{\log^{{\mathcal{a}}-1}(X/a)}{({\mathcal{a}}-1)!}\cdot\sum_{m=1}^{\infty}\frac{(f\cdot\mathds{1}_{(s)})(m)}{m}\cdot(1+o(1))=

=\prod_{p|s}(1-p^{-1})^{\mathcal{a}}\cdot\frac{\log^{{\mathcal{a}}-1}(X)}{({\mathcal{a}}-1)!}\cdot\prod_{p|s}\sum_{\nu}\frac{f(p^{\nu})}{p^{\nu}}\cdot(1+o(1))=

=\frac{\log^{{\mathcal{a}}-1}(X)}{({\mathcal{a}}-1)!}\cdot\prod_{p|s}M_{p}^{({\mathcal{a}})}(f)\cdot(1+o(1)).

For all $s\in{\mathbb{N}^{*}}$ and $X\geq 1$ , $\mathcal{M}_{f}(X)\leq\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)$ . Therefore

0\leq\limsup_{X\to\infty}\frac{\mathcal{M}_{f}(X)}{X\log^{{\mathcal{a}}-1}(X)}\leq\inf_{s\in{\mathbb{N}^{*}}}\limsup_{X\to\infty}\frac{\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)}{X\log^{{\mathcal{a}}-1}(X)}=

=\frac{1}{({\mathcal{a}}-1)!}\prod_{p}M_{p}^{({\mathcal{a}})}(f)=\frac{1}{({\mathcal{a}}-1)!}M^{({\mathcal{a}})}(f).

Should the product $M^{({\mathcal{a}})}(f)$ diverge, it necessarily diverges to $0$ , whereby

\lim_{X\to\infty}\frac{\mathcal{M}_{f}(X)}{X(\log X)^{{\mathcal{a}}-1}}=0=\frac{1}{({\mathcal{a}}-1)!}M^{({\mathcal{a}})}(f),

as desired. We assume therefore that $M^{({\mathcal{a}})}(f)$ converges. By lemma 5.11, for all $s\in{\mathbb{N}^{*}}$ and $n\in\mathbb{N}$ ,

0\leq f_{s}^{({\mathcal{a}})}(n)-f(n)\leq\sum_{p^{\nu}||n}f_{s}^{({\mathcal{a}})}(n/p^{\nu})(f_{s}^{({\mathcal{a}})}(p^{\nu})-f(p^{\nu}))=

=\sum_{\begin{subarray}{c}p^{\nu}||n\\ p\nmid s\end{subarray}}f_{s}^{({\mathcal{a}})}(n/p^{\nu})(\mathds{1}^{*{\mathcal{a}}}(p^{\nu})-f(p^{\nu}))\leq\sum_{\begin{subarray}{c}p^{\nu}||n\\ p\nmid s\end{subarray}}\mathds{1}^{*{\mathcal{a}}}(n/p^{\nu})(\mathds{1}^{*{\mathcal{a}}}(p^{\nu})-f(p^{\nu})).

Summing over $1\leq n\leq X$ we have

0\leq\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)-\mathcal{M}_{f}(X)\leq\sum_{n\leq X}\sum_{\begin{subarray}{c}p^{\nu}\|n\\ p\nmid s\end{subarray}}\mathds{1}^{*{\mathcal{a}}}(n/p^{\nu})\left(\mathds{1}^{*{\mathcal{a}}}(p^{\nu})-f(p^{\nu})\right)\leq

\leq\sum_{\begin{subarray}{c}p\nmid s\end{subarray}}\left[\sum_{p^{2}|n}^{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)+\sum_{p||n}^{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n/p)\left(\mathds{1}^{*{\mathcal{a}}}(p)-f(p)\right)\right].

By lemma 5.2, the first summand is bounded as follows:

0\leq\sum_{p\nmid s}\sum_{p^{2}|n}^{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)=\sum_{\begin{subarray}{c}p\nmid s\end{subarray}}\sum_{m\leq\frac{X}{p^{2}}}\mathds{1}^{*{\mathcal{a}}}(mp^{2})\leq\sum_{\begin{subarray}{c}\\ p\nmid s\end{subarray}}\sum_{m\leq\frac{X}{p^{2}}}\mathds{1}^{*{\mathcal{a}}}(m)\mathds{1}^{*{\mathcal{a}}}(p^{2})=

={{\mathcal{a}}+1\choose 2}\sum_{p\nmid s}^{p\leq\sqrt{X}}\sum_{m\leq X/p^{2}}\mathds{1}^{*{\mathcal{a}}}(m).

Therefore there exists $C=C_{\mathcal{a}}>0$ s.t. for all $X\geq 1$ ,

\sum_{p\nmid s}\sum_{\begin{subarray}{c}p^{2}|n\end{subarray}}^{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n)\leq\sum_{\begin{subarray}{c}p\nmid s\end{subarray}}^{p\leq\sqrt{X}}C\frac{X}{p^{2}}\left(\log^{{\mathcal{a}}-1}(X/p^{2})+1\right)\leq CX\left(\log^{{\mathcal{a}}-1}(X)+1\right)\sum_{p\nmid s}\frac{1}{p^{2}}.

The second summand is likewise bounded for all $X\geq 1$ by

\sum_{p\nmid s}\sum_{\begin{subarray}{c}p||n\end{subarray}}^{n\leq X}\mathds{1}^{*{\mathcal{a}}}(n/p)(\mathds{1}^{*{\mathcal{a}}}(p)-f(p))\leq\sum_{\begin{subarray}{c}p\nmid s\end{subarray}}({\mathcal{a}}-f(p))\sum_{m\leq\frac{X}{p}}\mathds{1}^{*{\mathcal{a}}}(m)\leq

\leq\sum_{\begin{subarray}{c}p\nmid s\end{subarray}}^{p\leq X}({\mathcal{a}}-f(p))\cdot C\frac{X}{p}\left(\log^{{\mathcal{a}}-1}(X/p)+1\right)\leq

\leq CX\left(\log^{{\mathcal{a}}-1}(X)+1\right)\sum_{p\nmid s}\frac{{\mathcal{a}}-f(p)}{p}.

Hence there is a constant $C>0$ such that for all $s\in{\mathbb{N}^{*}}$ and $X\geq 1$ ,

0\leq\frac{\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)-\mathcal{M}_{f}(X)}{X(\log^{{\mathcal{a}}-1}(X)+1)}\leq C\sum_{p\nmid s}\frac{1}{p^{2}}+C\sum_{p\nmid s}\frac{{\mathcal{a}}-f(p)}{p}.

The series $\sum_{p}\frac{{\mathcal{a}}-f(p)}{p}$ converges following lemma 5.8 and the assumed convergence of $M^{({\mathcal{a}})}(f)$ . The series $\sum_{p}p^{-2}$ converges as well, therefore

\sum_{p\nmid s}\frac{1}{p^{2}}+\sum_{p\nmid s}\frac{{\mathcal{a}}-f(p)}{p}\xrightarrow{s\in{\mathbb{N}^{*}}}0.

It follows that there is uniform convergence for $X\in[2,\infty[$ :

\frac{\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)}{X(\log X)^{{\mathcal{a}}-1}}\xrightarrow{s\in{\mathbb{N}^{*}}}\frac{\mathcal{M}_{f}(X)}{X(\log X)^{{\mathcal{a}}-1}}.

Thus we obtain the desired limit

\lim_{X\to\infty}\frac{\mathcal{M}_{f}(X)}{X\log^{{\mathcal{a}}-1}(X)}=\lim_{s\in{\mathbb{N}^{*}}}\lim_{X\to\infty}\frac{\mathcal{M}_{f_{s}^{({\mathcal{a}})}}(X)}{X\log^{{\mathcal{a}}-1}(X)}=

=\lim_{s\in{\mathbb{N}^{*}}}\frac{1}{({\mathcal{a}}-1)!}\prod_{p|s}M_{p}^{({\mathcal{a}})}(f)=\frac{1}{({\mathcal{a}}-1)!}M^{({\mathcal{a}})}(f).

∎

5.2 Sums of Kronecker Symbols

Recall that the Kronecker symbol $\left(\frac{a}{b}\right)\in\{0,\pm 1\}$ , defined for $a,b\in\mathbb{Z}$ with $b>0$ , satisfies for primes $p\neq 2$ :

\left(\frac{p}{2}\right)=\left(\frac{2^{*}}{p}\right)=\left(\frac{2}{p}\right)=(-1)^{\frac{p^{2}-1}{8}}.

Proposition 5.13.

Let $\chi\colon\mathbb{Z}\to\mathbb{C}$ be a non-principal character of modulus $d$ and let $s\in{\mathbb{N}^{*}}$ . Then there exists $C>0$ , independent of $\chi$ and $s$ , s.t.

\left|\sum_{A\leq n\leq B}(\chi\cdot\mathds{1}_{s})(n)\right|\leq C\cdot 2^{\omega(s)}\sqrt{d}\log d

for all $A,B\in\mathbb{R}$ .

Proof.

Let $A,B\in\mathbb{R}$ . A run-through of $s$ -coprime integers is obtained by Möbius inversion:

\sum_{A\leq n\leq B}\chi(n)\mathds{1}_{s}(n)=\sum_{u|s}\mu(u)\sum_{\begin{subarray}{c}A\leq n\leq B\end{subarray}}^{u|n}\chi(n)=\sum_{u|s}\mu(u)\chi(u)\sum_{\frac{A}{u}\leq n\leq\frac{B}{u}}\chi(n).

By the theorem of Pólya-Vinogradov⁷⁷7see [Pól18] or [Hil88], there is a constanct $C>0$ s.t.

\left|\sum_{A\leq n\leq B}\chi(n)\mathds{1}_{s}(n)\right|\leq\sum_{u|s}\left|\sum_{\frac{A}{u}\leq n\leq\frac{B}{u}}\chi(n)\right|\leq C\cdot 2^{\omega(s)}\sqrt{d}\log d.

∎

Proposition 5.14.

For $\chi\colon\mathbb{Z}\to\mathbb{C}$ a non-principal Dirichlet character with modulus $d$ , for $s\in{\mathbb{N}^{*}}$ and $1\leq Y\leq X$ , denote by

S_{\chi,s}(X)=\sum\limits_{1\leq n\leq X}\mu_{s}^{2}(n)\chi(n)\;\;,\;\;\;\;S_{\chi,s}(Y,X)=\sum\limits_{Y<n\leq X}\mu_{s}^{2}(n)\chi(n).

Then there is a constant $0<C\in\mathbb{R}$ s.t. for all such $\chi$ , $s\in{\mathbb{N}^{*}}$ and $X\geq Y\geq 1$ ,

|S_{\chi,s}(X)|\;,\;\;|S_{\chi,s}(Y,X)|\leq C\cdot X^{\frac{1}{2}}\cdot 2^{\frac{1}{2}\omega(s)}d^{\frac{1}{4}}\log^{\frac{1}{2}}(d).

Proof.

S_{\chi,s}(X)=\sum_{1\leq n\leq X}\mu^{2}(n)\mathds{1}_{s}(n)\chi(n)=\sum_{a=1}^{\infty}\mu(a)\sum_{\begin{subarray}{c}a^{2}|n\end{subarray}}^{1\leq n\leq X}\mathds{1}_{s}(n)\chi(n)=

=\sum_{a}\mu(a)\sum_{1\leq m\leq\frac{X}{a^{2}}}\mathds{1}_{s}(a^{2}m)\chi(a^{2}m)=\sum_{1\leq a\leq\sqrt{X}}\mu_{s}(a)\chi(a)^{2}\sum_{1\leq m\leq\frac{X}{a^{2}}}\mathds{1}_{s}(m)\chi(m).

Let $1\leq Z\leq\sqrt{X}$ be some parameter to be determined later. Then

S_{\chi,s}(X)=\left(\sum_{1\leq a\leq Z}+\sum_{Z<a\leq\sqrt{X}}\right)\mu_{s}(a)\chi(a)^{2}\sum_{1\leq m\leq\frac{X}{a^{2}}}\chi(m)\mathds{1}_{s}(m).

(6)

By proposition 5.13 there is $C^{\prime}>0$ s.t. the first summand in (6) is bounded by

\left|\sum_{1\leq a\leq Z}\mu_{s}(a)\chi(a)^{2}\sum_{1\leq m\leq\frac{X}{a^{2}}}\chi(m)\mathds{1}_{s}(m)\right|\leq\sum_{a\leq Z}\left|\sum_{m\leq\frac{X}{a^{2}}}\chi(m)\cdot\mathds{1}_{s}(m)\right|\leq

\leq Z\cdot C^{\prime}2^{\omega(s)}\sqrt{d}\log d.

The second summand in (6) is bounded, for some constant $C^{\prime\prime}>0$ , by

\left|\sum_{Z<a\leq\sqrt{X}}\mu_{s}(a)\chi(a)^{2}\sum_{1\leq m\leq\frac{X}{a^{2}}}\chi(m)\mathds{1}_{s}(m)\right|\leq\sum_{a>Z}\sum_{m\leq\frac{X}{a^{2}}}1\leq\sum_{a>Z}\frac{X}{a^{2}}\leq C^{\prime\prime}\cdot\frac{X}{Z}.

Therefore taking $C=\max\{C^{\prime},C^{\prime\prime},1\}$ ,

\left|S_{\chi,s}(X)\right|\leq Z\cdot C\cdot 2^{\omega(s)}\sqrt{d}\log d+X\cdot\frac{C}{Z}.

(7)

If $X<2^{\omega(s)}d^{\frac{1}{2}}\log d$ , then

\left|S_{\chi,s}(X)\right|\leq\sum_{1\leq n\leq X}|\mu_{s}^{2}(n)\chi(n)|\leq X<2C\cdot X^{\frac{1}{2}}\left(2^{\omega(s)}d^{\frac{1}{2}}\log d\right)^{\frac{1}{2}}=

=2C\cdot X^{\frac{1}{2}}\cdot 2^{\frac{1}{2}\omega(s)}d^{\frac{1}{4}}\log^{\frac{1}{2}}(d).

Otherwise, take $Z=X^{\frac{1}{2}}\big{(}2^{\omega(s)}d^{\frac{1}{2}}\log d\big{)}^{-\frac{1}{2}}$ . The modulus $d$ of a non-principal Dirichlet character $\chi$ is at least $3$ , whereby $\sqrt{d}\log(d)>1$ . Hence

1\leq Z\leq X^{\frac{1}{2}}2^{-\frac{1}{2}\omega(s)}\leq X^{\frac{1}{2}},

and (7) then reads:

\left|S_{\chi,s}(X)\right|\leq 2C\cdot X^{\frac{1}{2}}\cdot 2^{\frac{1}{2}\omega(s)}d^{\frac{1}{4}}\log^{\frac{1}{2}}(d).

For all $1\leq Y\leq X$ , we then have

\left|S_{\chi,s}(Y,X)\right|\leq\left|S_{\chi,s}(X)\right|+\left|S_{\chi,s}(Y)\right|\leq 4C\cdot X^{\frac{1}{2}}\cdot 2^{\frac{1}{2}\omega(s)}d^{\frac{1}{4}}\log^{\frac{1}{2}}(d).

∎

Definition 5.15.

A function $B\colon\mathbb{N}^{2}\to\mathbb{C}$ is a bi-character with slopes $1\leq c_{1},c_{2}\in\mathbb{N}$ if

•

for all $a\in\mathbb{N}$ , there exist Dirichlet character $\chi_{a}$ of modulus $c_{1}a$ and $M_{a}\in\mathbb{C}$ with $|M_{a}|\leq 1$ s.t.

$B(a,-)=M_{a}\cdot(\chi_{a})_{|\mathbb{N}}\colon\mathbb{N}\to\mathbb{C},$
•

for all $b\in\mathbb{N}$ , there exist Dirichlet character $\psi_{b}$ of modulus $c_{2}b$ and $N_{b}\in\mathbb{C}$ with $|N_{b}|\leq 1$ s.t.

$B(-,b)=N_{b}\cdot(\psi_{b})_{|\mathbb{N}}\colon\mathbb{N}\to\mathbb{C}.$

We say $B$ is non-principal at $a=a_{0}$ (resp. at $b=b_{0}$ ) if either $M_{a_{0}}=0$ (resp. $N_{b_{0}}=0$ ) or $\chi_{a_{0}}$ (resp. $\psi_{b_{0}}$ ) is non-principal.

Proposition 5.16.

Let $s\in{\mathbb{N}^{*}}$ and let $B\colon\mathbb{N}^{2}\to\mathbb{C}$ be a bi-character with slopes $c_{1},c_{2}$ . Assume $a_{0},b_{0}\in\mathbb{N}$ are such that $B$ is non-principal at all $a\geq a_{0}$ and at all $b\geq b_{0}$ . Then there exists a constant $C>0$ , independent of $s,a_{0},b_{0},B$ s.t. for all $X\geq 1$ ,

\left|\sum_{\begin{subarray}{c}a\geq a_{0}\\ b\geq b_{0}\end{subarray}}^{ab\leq X}\mu_{s}^{2}(ab)B(a,b)\right|\leq C\cdot 2^{\frac{1}{2}\omega(s)}c_{1}^{\frac{1}{8}}c_{2}^{\frac{1}{8}}\left(1+\log\max\{c_{1},c_{2}\}\right)X^{\frac{7}{8}}\left(1+\log X\right)^{\frac{1}{2}}.

Proof.

The function $B$ satisfies $B(a,b)=0$ whenever $(a,b)\neq 1$ , therefore

\mu_{s}^{2}(ab)B(a,b)=\mu_{s}^{2}(a)\mu_{s}^{2}(b)B(a,b).

Let $X_{1},X_{2}>0$ , to be determined later, s.t. $X_{1}X_{2}=X$ . Then

\sum_{\begin{subarray}{c}a\geq a_{0}\\ b\geq b_{0}\end{subarray}}^{ab\leq X}\mu_{s}^{2}(a)\mu_{s}^{2}(b)B(a,b)=\Bigg{(}\underbrace{\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}}_{(I)}+\underbrace{\sum_{b=b_{0}}^{{\lfloor{X_{2}}\rfloor}}\sum_{a=a_{0}}^{{\lfloor{\frac{X}{b}}\rfloor}}}_{(II)}-\underbrace{\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\sum_{b=b_{0}}^{{\lfloor{X_{2}}\rfloor}}}_{(III)}\Bigg{)}\mu_{s}^{2}(a)\mu_{s}^{2}(b)B(a,b).

The first summand $(I)$ is bounded by

|(I)|=\left|\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\mu_{s}^{2}(a)\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)B(a,b)\right|\leq\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\left|\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)B(a,b)\right|=

=\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}|M_{a}|\left|\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)\chi_{a}(b)\right|\leq\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\left|\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)\chi_{a}(b)\right|.

For all $a\geq a_{0}$ , $\chi_{a}$ is non-principal of modulus $c_{1}a$ . By proposition 5.14 there exists $C^{\prime}>0$ , independent of $s,c_{1},a$ , s.t.

\left|\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)\chi_{a}(b)\right|=\left|S_{\chi_{a},s}(b_{0}-1,X/a)\right|\leq C^{\prime}\cdot(X/a)^{\frac{1}{2}}2^{\frac{1}{2}\omega(s)}(c_{1}a)^{\frac{1}{4}}\log^{\frac{1}{2}}(c_{1}a)

for all $X\geq 1$ . Since $a,c_{1}\geq 1$ , we have $\log c_{1},\log a\geq 0$ , therefore

\log(c_{1}a)=\log c_{1}+\log a\leq(1+\log c_{1})(1+\log a).

Therefore

|(I)|\leq\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\left|\sum_{b=b_{0}}^{{\lfloor{\frac{X}{a}}\rfloor}}\mu_{s}^{2}(b)\chi_{a}(b)\right|\leq C^{\prime}\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{1}{2}}c_{1}^{\frac{1}{4}}\left(1+\log c_{1}\right)^{\frac{1}{2}}\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}a^{-\frac{1}{4}}(1+\log a)^{\frac{1}{2}}.

There is a constant $C^{\prime\prime}>0$ s.t. for all $Y\geq 1$

\sum_{1\leq a\leq Y}a^{-\frac{1}{4}}(1+\log a)^{\frac{1}{2}}\leq C^{\prime\prime}Y^{\frac{3}{4}}(1+\log Y)^{\frac{1}{2}}.

Hence

|(I)|\leq C^{\prime}C^{\prime\prime}\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{1}{2}}c_{1}^{\frac{1}{4}}\left(1+\log_{2}c_{1}\right)^{\frac{1}{2}}X_{1}^{\frac{3}{4}}(1+\log X_{1})^{\frac{1}{2}}.

Likewise,

|(II)|\leq C^{\prime}C^{\prime\prime}\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{1}{2}}c_{2}^{\frac{1}{4}}\left(1+\log_{2}c_{2}\right)^{\frac{1}{2}}X_{2}^{\frac{3}{4}}(1+\log X_{2})^{\frac{1}{2}}.

For the summand $(III)$ , again by proposition 5.14,

\left|(III)\right|=\left|\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\sum_{b=b_{0}}^{{\lfloor{X_{2}}\rfloor}}\mu_{s}^{2}(a)\mu_{s}^{2}(b)B(a,b)\right|\leq\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}\left|\sum_{b=b_{0}}^{{\lfloor{X_{2}}\rfloor}}\mu_{s}^{2}(b)\chi_{a}(b)\right|\leq

\leq\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}C^{\prime}2^{\frac{1}{2}\omega(s)}X_{2}^{\frac{1}{2}}(c_{1}a)^{\frac{1}{4}}\log(c_{1}a)^{\frac{1}{2}}\leq

\leq C^{\prime}2^{\frac{1}{2}\omega(s)}X_{2}^{\frac{1}{2}}c_{1}^{\frac{1}{4}}\left(1+\log_{2}c_{2}\right)^{\frac{1}{2}}\sum_{a=a_{0}}^{{\lfloor{X_{1}}\rfloor}}a^{\frac{1}{4}}(1+\log a)^{\frac{1}{2}}\leq

\leq C^{\prime}C^{\prime\prime\prime}2^{\frac{1}{2}\omega(s)}X_{2}^{\frac{1}{2}}c_{1}^{\frac{1}{4}}\left(1+\log_{2}c_{2}\right)^{\frac{1}{2}}X_{1}^{\frac{5}{4}}(1+\log X_{1})^{\frac{1}{2}}.

We now take $X_{1}=c_{1}^{-\frac{1}{6}}c_{2}^{\frac{1}{6}}X^{\frac{1}{2}}$ and $X_{2}=c_{1}^{\frac{1}{6}}c_{2}^{-\frac{1}{6}}X^{\frac{1}{2}}$ , so that

X^{\frac{1}{2}}c_{1}^{\frac{1}{4}}X_{1}^{\frac{3}{4}}=X^{\frac{1}{2}}c_{2}^{\frac{1}{4}}X_{2}^{\frac{3}{4}}=X_{2}^{\frac{1}{2}}c_{1}^{\frac{1}{4}}X_{1}^{\frac{5}{4}}=c_{1}^{\frac{1}{8}}c_{2}^{\frac{1}{8}}X^{\frac{7}{8}}.

For $X\geq c_{1}^{-\frac{1}{3}}c_{2}^{\frac{1}{3}}$ we have $X_{1}\leq X$ , and for $X\geq c_{1}^{\frac{1}{3}}c_{2}^{-\frac{1}{3}}$ we have $X_{2}\leq X$ . Then for such $X$ we find, taking the above into account, that

\left|\sum_{\begin{subarray}{c}a\geq a_{0}\\ b\geq b_{0}\end{subarray}}^{ab\leq X}\mu_{s}^{2}(ab)B(a,b)\right|\leq|(I)|+|(II)|+|(III)|\leq

\leq C^{\prime}(2C^{\prime\prime}+C^{\prime\prime\prime})\cdot 2^{\frac{1}{2}\omega(s)}c_{1}^{\frac{1}{8}}c_{2}^{\frac{1}{8}}X^{\frac{7}{8}}\left(1+\log\max\{c_{1},c_{2}\}\right)^{\frac{1}{2}}\left(1+\log X\right)^{\frac{1}{2}}.

However if w.l.o.g. $c_{2}\geq c_{1}$ and $1\leq X\leq c_{1}^{-\frac{1}{3}}c_{2}^{\frac{1}{3}}$ , then the bounded is obtained in simpler fashion:

X=X^{\frac{1}{8}}X^{\frac{7}{8}}\leq c_{1}^{-\frac{1}{24}}c_{2}^{\frac{1}{24}}X^{\frac{7}{8}}\leq c_{1}^{\frac{1}{8}}c_{2}^{\frac{1}{8}}X^{\frac{7}{8}}

and

\log X\leq\frac{\log c_{2}}{3}\leq\log\max\{c_{1},c_{2}\}.

By lemma 5.5 there exists $\widetilde{C}>0$ s.t.

\left|\sum_{\begin{subarray}{c}a\geq a_{0}\\ b\geq b_{0}\end{subarray}}^{ab\leq X}\mu_{s}^{2}(ab)B(a,b)\right|\leq\sum_{\begin{subarray}{c}a\geq a_{0}\\ b\geq b_{0}\end{subarray}}^{ab\leq X}1\leq\sum_{\begin{subarray}{c}n\leq X\end{subarray}}(\mathds{1}*\mathds{1})(n)\leq\widetilde{C}X(1+\log X)\leq

\leq\widetilde{C}\cdot 2^{\frac{1}{2}\omega(s)}c_{1}^{\frac{1}{8}}c_{2}^{\frac{1}{8}}X^{\frac{7}{8}}\left(1+\log\max\{c_{1},c_{2}\}\right)^{\frac{1}{2}}\left(1+\log X\right)^{\frac{1}{2}}.

The claim therefore holds for all $X\geq 1$ with $C:=\max\{C^{\prime}(2C^{\prime\prime}+C^{\prime\prime\prime}),\widetilde{C}\}$ . ∎

Proposition 5.17.

The functions $B,B^{\prime},B^{\prime\prime}\colon\mathbb{N}^{2}\to\mathbb{C}$ ,

B(a,b)=\mathds{1}_{2}(ab)\left(\frac{b}{a}\right)\;,\;\;B^{\prime}(a,b)=\mathds{1}_{2}(ab)\left(\frac{2b}{a}\right)=\left(\frac{2}{a}\right)B(a,b)

B^{\prime\prime}(a,b)=\mathds{1}_{2}(ab)\left(\frac{b}{2a}\right)=\left(\frac{b}{2}\right)B(a,b)

are bi-characters with slopes $c_{1},c_{2}\leq 8$ . Moreover, $B$ (resp. $B^{\prime}$ ; $B^{\prime\prime}$ ) is non-principal at $a\geq 2$ and at $b\geq 2$ ( $a\geq 2$ and $b\geq 1$ ; $a\geq 1$ and $b\geq 2$ ).

Proof.

Let $h\colon\mathbb{Z}\to\mathbb{C}$ denote the function

h(n)=\left(\frac{2}{n}\right)=\left(\frac{n}{2}\right)=\mathds{1}_{2}(n)(-1)^{\frac{n^{2}-1}{8}},

a Dirichlet character of modulus $8$ . For odd $a,b\geq 1$ , let $\chi_{a},\psi_{b}\colon\mathbb{Z}\to\mathbb{C}$ be

\chi_{a}(n)=\mathds{1}_{2}(n)\left(\frac{n}{a}\right)\;,\;\;\psi_{b}(n)=\mathds{1}_{2}(n)(-1)^{\frac{b-1}{2}\frac{n-1}{2}}\left(\frac{n}{b}\right).

Then $\chi_{a}$ (resp. $\psi_{b}$ ) is a Dirichlet character of modulus $2a$ (resp. $4b$ ). For $n\geq 1$ ,

\psi_{b}(n)=\mathds{1}_{2}(n)\left(\frac{b}{n}\right)

- trivially for $n$ even, by quadratic reciprocity for $n$ odd. For all $a\geq 1$ we have

B(a,-)=\mathds{1}_{2}(a)\cdot\chi_{a}\;,\;\;B^{\prime}(a,-)=h(a)\cdot\chi_{a}\;,\;\;B^{\prime\prime}(a,-)=\mathds{1}_{2}(a)\cdot h\chi_{a},

to be interpreted as zero for $a$ even. Likewise, for all $b\geq 1$ we have

B(-,b)=\mathds{1}_{2}(b)\cdot\psi_{b}\;,\;\;B^{\prime}(-,b)=\mathds{1}_{2}(b)\cdot h\psi_{b}\;,\;\;B^{\prime\prime}(-,b)=h(b)\cdot\psi_{b}.

The function $h\chi_{a}$ (resp. $h\psi_{b}$ ) is a Dirichlet character of modulus $8a$ (resp. $8b$ ). Moreover, $|\mathds{1}_{2}(n)|,|h(n)|\leq 1$ for all $n$ . Therefore $B$ (resp. $B^{\prime}$ ; $B^{\prime\prime}$ ) is a bi-character with slopes $(c_{1},c_{2})=(2,4)$ (resp. $(2,8)$ ; $(8,4)$ ). For odd $m\geq 2$ , the function $n\mapsto\left(\frac{n}{m}\right)$ is clearly non-principal of modulus $m$ . Since

\mathds{1}_{2}(n)\;\;\;\;(\;\textnormal{resp. }\mathds{1}_{2}(n)(-1)^{\frac{m-1}{2}\frac{n-1}{2}}\;\;;\;\;h(n)\;\;;\;\;\mathds{1}_{2}(n)(-1)^{\frac{m-1}{2}}h(n)\;)

has modulus $2$ (resp. $4\;;\;8\;;\;8$ ), coprime to $m$ , it follows for all odd $m\geq 2$ that $\chi_{m}$ (resp. $\psi_{m}$ ; $h\chi_{m}$ ; $h\psi_{m}$ ) is non-principal. Moreover, for $m=1$ we have $h\chi_{1}=h\psi_{1}=h$ , non-principal. ∎

Lemma 5.18.

Let ${\mathcal{a}}>-1$ and let $1\leq{\mathcal{b}}\in\mathbb{N}$ . Then for all $X\geq 1$ ,

\sum_{1\leq n\leq X}\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot{\mathcal{b}}^{\omega(n)}\leq\frac{1}{1+{\mathcal{a}}}X^{1+{\mathcal{a}}}(1+\log X)^{{\mathcal{b}}-1}.

Proof.

The proof is by induction on ${\mathcal{b}}$ . The claim holds for ${\mathcal{b}}=1$ : since ${\mathcal{a}}>-1$ , for all $X\geq 1$ we have

\sum_{n\leq X}\mu(n)^{2}n^{{\mathcal{a}}}\leq\int\limits_{0}^{X}t^{{\mathcal{a}}}dt=\frac{1}{1+{\mathcal{a}}}X^{1+{\mathcal{a}}}.

Assume next that the claim holds for some ${\mathcal{b}}\in\mathbb{N}$ . By lemma 5.3,

\sum_{n\leq X}\mu^{2}(n)n^{{\mathcal{a}}}({\mathcal{b}}+1)^{\omega(n)}=\sum_{\begin{subarray}{c}m,d\geq 1\\ md\leq X\end{subarray}}\mu^{2}(md)(md)^{{\mathcal{a}}}{\mathcal{b}}^{\omega(m)}\leq

\leq\sum_{1\leq d\leq X}d^{{\mathcal{a}}}\sum_{1\leq m\leq\frac{X}{d}}\mu^{2}(m)m^{{\mathcal{a}}}{\mathcal{b}}^{\omega(m)}.

By the induction assumption, this is bounded by

\frac{1}{1+{\mathcal{a}}}\sum_{d\leq X}d^{{\mathcal{a}}}(X/d)^{1+{\mathcal{a}}}(1+\log(X/d))^{{\mathcal{b}}-1}\leq\frac{1}{1+{\mathcal{a}}}X^{1+{\mathcal{a}}}\sum_{d\leq X}d^{-1}(1+\log X)^{{\mathcal{b}}-1}\leq

\leq\frac{1}{1+{\mathcal{a}}}X^{1+{\mathcal{a}}}(1+\log X)^{{\mathcal{b}}}.

Hence the statement holds for all $1\leq{\mathcal{b}}\in\mathbb{N}$ , ${\mathcal{a}}>-1$ and $X\geq 1$ . ∎

Lemma 5.19.

Let ${\mathcal{a}}>-1$ . Then there exists $0<C\in\mathbb{R}$ s.t. for all $X\geq 1$ ,

\left|\sum_{1\leq n\leq X}\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot 2^{\frac{1}{2}\omega(n)}\right|\leq C\cdot X^{1+{\mathcal{a}}}(1+\log X)^{\frac{1}{2}}.

Proof.

For all $n$ we have

\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot 2^{\frac{1}{2}\omega(n)}=\left(n^{{\mathcal{a}}}\right)^{\frac{1}{2}}\left(\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot 2^{\omega(n)}\right)^{\frac{1}{2}}.

By the Cauchy-Schwartz inequality,

\sum_{n=1}^{{\lfloor{X}\rfloor}}\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot 2^{\frac{1}{2}\omega(n)}\leq\sqrt{\sum_{n=1}^{{\lfloor{X}\rfloor}}n^{{\mathcal{a}}}}\cdot\sqrt{\sum_{n=1}^{{\lfloor{X}\rfloor}}\mu^{2}(n)\cdot n^{{\mathcal{a}}}\cdot 2^{\omega(n)}}.

Since ${\mathcal{a}}>-1$ , and following lemma 5.18, this is bounded by

C\cdot\sqrt{X^{1+{\mathcal{a}}}}\cdot\sqrt{X^{1+{\mathcal{a}}}(1+\log X)}=C\cdot X^{1+{\mathcal{a}}}\cdot(1+\log X)^{\frac{1}{2}}.

∎

Proposition 5.20.

Let $s\in{\mathbb{N}^{*}}$ . For all $1\leq X\in\mathbb{R}$ , denote by

T_{s}(X)=\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X\end{subarray}}\mu_{s}^{2}(ab)\left(\frac{b}{a}\right).

Then there exists a constant $C>0$ s.t. for all $1\leq X\in\mathbb{R}$ ,

\left|T_{s}(X)\right|\leq C\cdot 2^{\frac{1}{2}\omega(s)}\cdot X^{\frac{7}{8}}(1+\log X)^{\frac{1}{2}}.

Proof.

We write

T_{s}(X)=\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X\end{subarray}}\mu_{s}^{2}(ab)\left(\frac{b}{a}\right)=

=\left(\overbrace{\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X\end{subarray}}^{2\nmid a,\;2\nmid b}}^{(I)}+\overbrace{\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X\end{subarray}}^{2\nmid a,\;2||b}}^{(II)}+\overbrace{\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X\end{subarray}}^{2||a,\;2\nmid b}}^{(III)}\right)\mu_{s}^{2}(ab)\left(\frac{b}{a}\right).

(8)

We have

(I)=\sum_{\begin{subarray}{c}a,b\geq 2\end{subarray}}^{ab\leq X}\mu_{s}^{2}(ab)\mathds{1}_{2}(ab)\left(\frac{b}{a}\right).

(II)=\sum_{\begin{subarray}{c}2b,a\geq 2\end{subarray}}^{2b\cdot a\leq X}\mu_{s}^{2}(2b\cdot a)\mathds{1}_{2}(ab)\left(\frac{2b}{a}\right)=\sum_{\begin{subarray}{c}b\geq 1,\;a\geq 2\end{subarray}}^{ab\leq X/2}\mu_{s}^{2}(ab)\mathds{1}_{2}(ab)\left(\frac{2b}{a}\right).

(III)=\sum_{\begin{subarray}{c}b,2a\geq 2\end{subarray}}^{b\cdot 2a\leq X}\mu_{s}^{2}(b\cdot 2a)\mathds{1}_{2}(b)\left(\frac{b}{2a}\right)=\sum_{\begin{subarray}{c}b\geq 2,\;a\geq 1\end{subarray}}^{ab\leq X/2}\mu_{s}^{2}(ab)\mathds{1}_{2}(ab)\left(\frac{b}{2a}\right).

By propositions 5.17 and 5.16 there exists $C>0$ s.t.

|(I)|\leq C2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(1+\log X)^{\frac{1}{2}},

|(II)|\leq C2^{\frac{1}{2}\omega(s)}(X/2)^{\frac{7}{8}}(1+\log(X/2))^{\frac{1}{2}}\leq C2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(1+\log X)^{\frac{1}{2}},

and

|(III)|\leq C2^{\frac{1}{2}\omega(s)}(X/2)^{\frac{7}{8}}(\log(1+X/2))^{\frac{1}{2}}\leq C2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(1+\log X)^{\frac{1}{2}}.

Therefore

\left|T_{s}(X)\right|\leq\left|(I)\right|+\left|(II)\right|+\left|(III)\right|\leq 3C2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(1+\log X)^{\frac{1}{2}}.

∎

6 Trivial Keis $\mathscr{T}_{a}$

Recall that a kei $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}$ is trivial if ${{}^{x}{y}}=y$ for all $x,y\in\mathscr{K}$ . The following assertions about trivial keis are easily shown:

Lemma 6.1.

1.

Let $\mathscr{T}\in{\mathscr{K}\textnormal{ei}}$ be trivial. Then every sub-kei $\mathscr{T}^{\prime}\leq\mathscr{T}$ is trivial.
2.

Let $\mathscr{T},\mathscr{T}^{\prime}\in{\mathscr{K}\textnormal{ei}}$ be trivial. Then any function $f\colon\mathscr{T}^{\prime}\to\mathscr{T}$ is a kei morphism.

Recall in [DS23] the computation of the Hilbert polynomial of $\mathscr{T}_{a}$ :

P_{\mathscr{T}_{\mathcal{a}}}(k)={{\mathcal{a}}+k-1\choose{\mathcal{a}}-1}=\frac{k^{{\mathcal{a}}-1}}{({\mathcal{a}}-1)!}+O(k^{{\mathcal{a}}-2}).

6.1 Interpretation of $\mathscr{T}_{\mathcal{a}}$ -colorings

Proposition 6.2.

Let $0\leq{\mathcal{a}}\in\mathbb{N}$ and let $n\in{\mathbb{N}^{*}}$ . Then

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}(n)={\mathcal{a}}^{\omega(n)}.

Proof.

This follows directly from proposition 3.23:

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}(n)=\left|\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{T}_{\mathcal{a}})\right|=\left|\mathscr{T}_{\mathcal{a}}^{\{p|n\;\textnormal{prime}\}}\right|={\mathcal{a}}^{\ omega(n)}.

∎

6.2 Proof of Main Conjecture for $\mathscr{K}=\mathscr{T}_{\mathcal{a}}$ .

Lemma 6.3.

Let ${\mathcal{a}}\in\mathbb{N}$ . The infinite product

\prod_{p}\frac{(1+p^{-1})^{\mathcal{a}}}{(1+{\mathcal{a}}p^{-1})}

converges.

Proof.

For ${\mathcal{a}}=0,1$ , the claim is obvious. Assume that ${\mathcal{a}}\geq 2$ . For all prime $p$ ,

1+{\mathcal{a}}p^{-1}\leq(1+p^{-1})^{\mathcal{a}}=1+{\mathcal{a}}p^{-1}+\sum_{i=2}^{\mathcal{a}}{{\mathcal{a}}\choose i}p^{-i}\leq 1+{\mathcal{a}}p^{-1}+2^{\mathcal{a}}p^{-2}\leq(1+{\mathcal{a}}p^{-1})(1+2^{\mathcal{a}}p^{-2}).

For $p\geq 4^{\mathcal{a}}$ , all but finitely many, we have

1\leq\frac{(1+p^{-1})^{\mathcal{a}}}{(1+{\mathcal{a}}p^{-1})}\leq 1+2^{\mathcal{a}}p^{-2}\leq 1+p^{-3/2}.

The convergence of

\prod_{p}(1+p^{-3/2})=\zeta(3)^{-1}\zeta(3/2)

guarantees the convergence of

\prod_{p}\frac{(1+p^{-1})^{\mathcal{a}}}{(1+{\mathcal{a}}p^{-1})}.

∎

Proposition 6.4.

Let $1\leq{\mathcal{a}}\in\mathbb{N}$ . Then $\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}$ has generic summatory type

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}\in{\mathscr{W}({\mathcal{a}}-1,\textstyle{\frac{1}{({\mathcal{a}}-1)!}})}.

Proof.

In proposition 6.2 we saw that

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}(n)={\mathcal{a}}^{\omega(n)}

for all $n\in{\mathbb{N}^{*}}$ . Let $s\in{\mathbb{N}^{*}}$ . The function $\mu_{s}^{2}(n){\mathcal{a}}^{\omega(n)}\colon\mathbb{N}\to\mathbb{R}$ is multiplicative, and satisfies

0\leq\mu_{s}^{2}(n){\mathcal{a}}^{\omega(n)}=\mu_{s}^{2}(n)\mathds{1}^{*{\mathcal{a}}}(n)\leq\mathds{1}^{*{\mathcal{a}}}(n)

for all $n\in\mathbb{N}$ . By proposition 5.12,

\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{T}_{\mathcal{a}},s}(X)}{X(\log X)^{{\mathcal{a}}-1}}=\frac{1}{({\mathcal{a}}-1)!}\prod_{p}(1-p^{-1})^{\mathcal{a}}{\mathcal{a}}\sum_{\nu}\frac{\mu_{s}^{2}(p^{\nu}){\mathcal{a}}^{\omega(p^{\nu})}}{p^{\nu}}=

=\frac{1}{({\mathcal{a}}-1)!}\prod_{p|s}(1-p^{-1})^{\mathcal{a}}\cdot\prod_{p\nmid s}(1-p^{-1})^{\mathcal{a}}(1+{\mathcal{a}}p^{-1}).

Therefore

c_{s}(\mathscr{T}_{\mathcal{a}})=\gamma_{s}^{1-{\mathcal{a}}}\lim_{X\to\infty}\frac{\mathcal{E}_{\mathscr{T}_{\mathcal{a}},s}(X)}{\log^{{\mathcal{a}}-1}(X)}=\gamma_{s}^{-{\mathcal{a}}}\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{T}_{\mathcal{a}},s}(X)}{X\log^{{\mathcal{a}}-1}(X)}=

=\frac{1}{({\mathcal{a}}-1)!}\cdot\prod_{p|s}\frac{(1-p^{-1})^{\mathcal{a}}}{(1-p^{-1})^{\mathcal{a}}}\cdot\prod_{p\nmid s}\frac{(1-p^{-1})^{\mathcal{a}}(1+{\mathcal{a}}p^{-1})}{(1-p^{-2})^{{\mathcal{a}}}}=

=\frac{1}{({\mathcal{a}}-1)!}\prod_{p\nmid s}\frac{1+{\mathcal{a}}p^{-1}}{(1+p^{-1})^{\mathcal{a}}}.

By lemma 6.3, this product converges. Moreover, it follows that

\lim_{s\in{\mathbb{N}^{*}}}c_{s}(\mathscr{T}_{\mathcal{a}})=\frac{1}{({\mathcal{a}}-1)!}\lim_{s\in{\mathbb{N}^{*}}}\prod_{p\nmid s}\frac{1+{\mathcal{a}}p^{-1}}{(1+p^{-1})^{\mathcal{a}}}=\frac{1}{({\mathcal{a}}-1)!}.

Therefore $\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}\colon{\mathbb{N}^{*}}\to\mathbb{R}$ has summatory type $({\mathcal{a}}-1,\frac{1}{({\mathcal{a}}-1)!})$ . ∎

7 The Joyce kei $\mathscr{J}$

In this section we recall the kei $\mathscr{J}$ that was introduced in [Joy82, §6]. We will address here not only colorings of $n\in{\mathbb{N}^{*}}$ by $\mathscr{J}$ , but by $\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}$ where $\mathscr{T}_{\mathcal{a}}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ is the trivial kei on ${\mathcal{a}}\in\mathbb{N}$ elements.

Definition 7.1.

The kei $\mathscr{J}$ has underlying set and structure

wx+x+x+x−x−x−x+yyy.\mathscr{J}=\{x_{+},x_{-},y\}\;\;,\;\;\;\;\begin{tabular}[]{|cc||c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup${{}^{z}{w}}$}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil$z$\hfil\lx@intercol\vrule\lx@intercol\\ \cline{3-5}\cr&&$x_{+}$&$x_{-}$&$y$\\ \hline\cr\hline\cr\hbox{\multirowsetup$w$}&\vrule\lx@intercol\hfil$x_{+}$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$x_{+}$&$x_{+}$&$x_{-}$\\ \cline{2-5}\cr&\vrule\lx@intercol\hfil$x_{-}$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$x_{-}$&$x_{-}$&$x_{+}$\\ \cline{2-5}\cr&\vrule\lx@intercol\hfil$y$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$y$&$y$&$y$\\ \hline\cr\end{tabular}\;\;.

	$\mathscr{J}=\{x_{+},x_{-},y\}\;\;,\;\;\;\;\begin{tabular}[]{\|cc\|\|c\|c\|c\|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup${{}^{z}{w}}$}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil$z$\hfil\lx@intercol\vrule\lx@intercol\\ \cline{3-5}\cr&&$x_{+}$&$x_{-}$&$y$\\ \hline\cr\hline\cr\hbox{\multirowsetup$w$}&\vrule\lx@intercol\hfil$x_{+}$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$x_{+}$&$x_{+}$&$x_{-}$\\ \cline{2-5}\cr&\vrule\lx@intercol\hfil$x_{-}$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$x_{-}$&$x_{-}$&$x_{+}$\\ \cline{2-5}\cr&\vrule\lx@intercol\hfil$y$\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&$y$&$y$&$y$\\ \hline\cr\end{tabular}\;\;.$	wz		z			x+	x-	y

The kei $\mathscr{J}$ satisfies

\textnormal{Inn}(\mathscr{J})=\{\textnormal{Id}_{\mathscr{J}},\varphi_{y}\}\simeq\mathbb{Z}/2\mathbb{Z}.

For all $z\in\mathscr{J}$ , $\varphi_{z}\neq\textnormal{Id}$ iff $z=y$ . The proper sub-quandles of $\mathscr{J}$ are

\emptyset,\{x_{+}\},\{x_{-}\},\{y\},\{x_{+},x_{-}\},

all trivial.

Lemma 7.2.

Let $\{x_{0},y_{0}\}=\mathscr{T}_{2}$ be a trivial kei. Then the map

\eta_{0}\colon\mathscr{J}\to\{x_{0},y_{0}\}\;\;,\;\;\;\;x_{\pm}\mapsto x_{0}\;,\;\;y\mapsto y_{0}

is a kei morphism.

Recall in [DS23] the computation of the Hilbert polynomial of $\mathscr{J}$ :

P_{\mathscr{J}}(k)=2k+1,

and more generally, for $0\leq{\mathcal{a}}\in\mathbb{N}$ ,

P_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}(k)=2{k+{\mathcal{a}}+1\choose{\mathcal{a}}+1}-{k+{\mathcal{a}}\choose{\mathcal{a}}}=\frac{2}{({\mathcal{a}}+1)!}k^{{\mathcal{a}}+1}+O(k^{\mathcal{a}}).

7.1 Interpretation of $\mathscr{J}$ -colorings

Proposition 7.3.

Let $n\in{\mathbb{N}^{*}}$ . Then

\textnormal{col}_{\mathscr{J}}(n)=\sum_{ab|n}\left(\frac{b}{a}\right),

where $\left(\frac{b}{a}\right)$ is the Kronecker symbol.

Proof.

Consider the kei morphism $\eta_{0}\colon\mathscr{J}\to\{x_{0},y_{0}\}\simeq\mathscr{T}_{2}$ from lemma 7.2. By lemma 3.20 and proposition 3.23 there is a map

\textnormal{Col}_{\mathscr{J}}(n)\xrightarrow{\eta_{0}\circ-}\textnormal{Col}_{\mathscr{T}_{2}}(n)\simeq\mathscr{T}_{2}^{\{p|n\;\textnormal{prime}\}}\simeq\{(n_{x},n_{y})\in\mathbb{N}^{2}\;|\;n_{x}n_{y}=n\}.

For $n_{1},n_{2}\in\mathbb{N}$ s.t. $n_{1}n_{2}=n$ , we denote by $\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}\subseteq\textnormal{Col}_{\mathscr{J}}(n)$ the fiber

\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=\left\{f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{J}\;|\;\forall\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\;,\;\;f(\mathfrak{p})\in\begin{cases}\{x_{\pm}\}&\mathfrak{p}|n_{1}\\ \{y\}&\mathfrak{p}|n_{2}\end{cases}\;\;\;\;\right\}

over $(n_{1},n_{2})$ , and by

\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}:=\left|\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}\right|\in\mathbb{N},

s.t.

\textnormal{col}_{\mathscr{J}}(n)=\sum_{n_{1}n_{2}=n}\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}.

We show that for all $n_{1},n_{2}\in\mathbb{N}$ s.t. $n_{1}n_{2}=n$ ,

\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=\prod_{p|n_{1}}\left(1+\left(\frac{p}{n_{2}}\right)\right)=\begin{cases}0&,\;\exists\;p|n_{1}\;\left(\frac{p}{n_{2}}\right)=-1\\ 2^{\omega(n_{1})}&,\;\forall\;p|n_{1}\;\left(\frac{p}{n_{2}}\right)=1\end{cases}\;\;.

If $n_{1}=n$ , then for all $p|n$ we have $\left(\frac{p}{1}\right)=1$ , and indeed

\textnormal{col}_{\mathscr{J}}(n)_{(n,1)}=\left|\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\{x_{\pm}\})\right|=\textnormal{col}_{\mathscr{T}_{2}}(n)=2^{\omega(n)}.

If $n_{1}=1$ , then $\left(\frac{p}{n}\right)=1$ vacuously for all $p|1$ , and

\textnormal{col}_{\mathscr{J}}(n)_{(1,n)}=\left|\textnormal{Hom}_{{\mathscr{K}\textnormal{ei}}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\{y\})\right|=1=2^{\omega(1)}.

Let $n_{1},n_{2}\neq 1$ with $n_{1}n_{2}=n$ . We denote by $\rho_{n,n_{2}}$ the map

\rho_{n,n_{2}}\colon\mathfrak{G}_{n}=\textnormal{Gal}(\mathfrak{F}_{n}/\mathbb{Q})\twoheadrightarrow\textnormal{Gal}(\mathfrak{L}_{n_{2}}/\mathbb{Q})\simeq\mathbb{Z}/2\mathbb{Z}\simeq\textnormal{Inn}(\mathscr{J}).

where $\mathfrak{L}_{n_{2}}=\mathbb{Q}(\sqrt{n_{2}^{*}})$ is the unique quadratic field in $\mathfrak{F}_{n}$ that ramifies at precisely $n_{2}$ . Any $f\in\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}$ is necessarily surjective, thus by proposition 2.24, lifts uniquely to

(f,\rho_{\mathfrak{A}_{n},f})\colon(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathfrak{G}_{n},\mathfrak{m}_{n})\to(\mathscr{J},\textnormal{Inn}(\mathscr{J}),\varphi)

in $\textnormal{Pro-}{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}^{\textnormal{fin}}$ . For all $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ ,

\rho_{\mathfrak{A}_{n},f}(\mathfrak{m}_{\mathfrak{p}})=\varphi_{f(\mathfrak{p})}\neq 1\;\iff\;\rho_{\mathfrak{A}_{n},f}(\mathfrak{m}_{\mathfrak{p}})=\varphi_{y}\;\iff\;\mathfrak{p}\in f^{-1}(y)\;\iff\;\mathfrak{p}|n_{2}.

Therefore $\rho_{\mathfrak{A}_{n},f}\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{J})$ corresponds to $\mathfrak{L}_{n_{2}}\subseteq\mathfrak{F}_{n}$ :

\rho_{\mathfrak{A}_{n},f}=\rho_{n,n_{2}}\;\;,\;\;\;\;\ker\rho_{\mathfrak{A}_{n},f}=\textnormal{Gal}(\mathfrak{F}_{n}/\mathfrak{L}_{n_{2}}).

It also follows from proposition 2.25 that every $f\in\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}$ factors through the quotient

\mathscr{K}:=\ker\rho_{n,n_{2}}\backslash\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\simeq\textnormal{Spec}\left(\mathcal{O}_{\mathfrak{K}_{n_{2}}}\otimes\mathbb{Z}/n\mathbb{Z}\right)\to\mathscr{J}

in ${\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ . The kei structure on $\mathscr{K}$ is defined by the induced augmentation with $G:=\textnormal{Gal}(\mathfrak{K}_{n_{2}}/\mathbb{Q})$ , satisfying ${{}^{\mathfrak{q}}{\mathfrak{p}}}=\mathfrak{m}_{\mathfrak{q}}(\mathfrak{p})$ for all $\mathfrak{p},\mathfrak{q}\in\mathscr{K}$ , where $\mathfrak{m}_{\mathfrak{q}}=1_{G}$ iff $\mathfrak{q}|n_{1}$ . Then for all $\mathfrak{p},\mathfrak{q}\in\mathscr{K}$ over respective rational primes $p,q$ ,

{{}^{\mathfrak{q}}{\mathfrak{p}}}\neq\mathfrak{p}\;\iff\;q|n_{2},\;p|n_{1},\;\textnormal{ and }\;\left(\frac{n_{2}^{*}}{p}\right)=\left(\frac{p}{n_{2}}\right)=1.

Suppose there exists a prime $p|n_{1}$ s.t. $\left(\frac{p}{n_{2}}\right)=-1$ . Let $\mathfrak{p},\mathfrak{q}\in\mathscr{K}$ be s.t. $\mathfrak{p}|p$ and $\mathfrak{q}|n_{2}$ ( $n_{2}\neq 1$ ). Then on one hand, ${{}^{\mathfrak{q}}{\mathfrak{p}}}=\mathfrak{p}$ . On the other hand, for any $f\in\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}$ ,

f(\mathfrak{p})\in\{x_{\pm}\}\;\;\Longrightarrow\;\;{{}^{f(\mathfrak{q})}{f(\mathfrak{p})}}={{}^{y}{f(\mathfrak{p})}}\neq f(\mathfrak{p}).

This is a contradiction, therefore $\textnormal{Col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=\emptyset$ in these conditions:

\exists\;p|n_{1}\;\left(\frac{p}{n_{2}}\right)=-1\;\Longrightarrow\;\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=0.

If on the other hand $\left(\frac{p}{n_{2}}\right)=1$ for all $p|n_{1}$ , then $f$ is determined by bijections

\textnormal{Spec}(\mathcal{O}_{\mathfrak{K}_{n_{2}}}\otimes\mathbb{F}_{p}){\xrightarrow{\sim}}\{x_{\pm}\}

for all $p|n_{1}$ , each independent of the other. Therefore

\forall\;p|n_{1}\;\left(\frac{p}{n_{2}}\right)=1\;\Longrightarrow\;\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=2^{\omega(n_{1})}.

It follows that for any $n_{1},n_{2}$ s.t. $n_{1}n_{2}=n$ ,

\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=\prod_{p|n_{1}}\left(1+\left(\frac{p}{n_{2}}\right)\right)=\sum_{b|n_{1}}\left(\frac{b}{n_{2}}\right).

In total,

\textnormal{col}_{\mathscr{J}}(n)=\sum_{n_{1}n_{2}=n}\textnormal{col}_{\mathscr{J}}(n)_{(n_{1},n_{2})}=\sum_{n_{1}n_{2}=n}\sum_{b|n_{1}}\left(\frac{b}{n_{2}}\right)=\sum_{ab|n}\left(\frac{b}{a}\right).

∎

Proposition 7.4.

Let $0\leq{\mathcal{a}}\in\mathbb{N}$ . Then for all $n\in{\mathbb{N}^{*}}$ ,

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}\sqcup\mathscr{J}}(n)=\sum_{abc=n}({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right).

Proof.

Let $n\in{\mathbb{N}^{*}}$ . By proposition 6.2,

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}={\mathcal{a}}^{\omega(n)}.

By proposition 3.24 and lemma 5.3,

\textnormal{col}_{\mathscr{T}_{\mathcal{a}}\sqcup\mathscr{J}}(n)=\sum_{n_{1}n_{2}=n}\textnormal{col}_{\mathscr{T}_{\mathcal{a}}}(n_{1})\textnormal{col}_{\mathscr{J}}(n_{2})=\sum_{n_{1}n_{2}=n}{\mathcal{a}}^{\omega(n_{1})}\sum_{ab|n_{2}}\left(\frac{b}{a}\right)=

=\sum_{abc=n}\left(\frac{b}{a}\right)\sum_{n_{1}|c}{\mathcal{a}}^{\omega(n_{1})}=\sum_{abc=n}({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right).

∎

7.2 Proof of the Main Conjecture for $\mathscr{K}=\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}$ .

Proposition 7.5.

Let $0\leq{\mathcal{a}}\in\mathbb{N}$ , let $s\in{\mathbb{N}^{*}}$ and let $X\geq 1$ . Then

\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)=2\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)-\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)+\sum_{n\leq X}\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}T_{sn}(X/n),

where for $v\in{\mathbb{N}^{*}}$ and $1\leq Y\in\mathbb{R}$ ,

T_{v}(Y)=\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq Y\end{subarray}}\mu_{v}^{2}(ab)\left(\frac{b}{a}\right).

Proof.

Let $n\in{\mathbb{N}^{*}}$ . By lemma 5.3, we write

\textnormal{col}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}(n)=\sum_{abc=n}({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right)=

=\left(\sum_{\begin{subarray}{c}abc=n\\ a=1\end{subarray}}+\sum_{\begin{subarray}{c}abc=n\\ b=1\end{subarray}}-\sum_{\begin{subarray}{c}abc=n\\ a=b=1\end{subarray}}+\sum_{\begin{subarray}{c}abc=n\\ a,b\geq 2\end{subarray}}\right)({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right)=

=2\sum_{c|n}({\mathcal{a}}+2)^{\omega(c)}-({\mathcal{a}}+1)^{\omega(n)}+\sum_{\begin{subarray}{c}abc=n\\ a,b\geq 2\end{subarray}}{\mathcal{a}}^{\omega(c)}\left(\frac{b}{a}\right).

Therefore by proposition 6.2,

\textnormal{col}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}(n)=2\textnormal{col}_{\mathscr{T}_{{\mathcal{a}}+2}}(n)-\textnormal{col}_{\mathscr{T}_{{\mathcal{a}}+1}}(n)+\sum_{\begin{subarray}{c}abc=n\\ a,b\geq 2\end{subarray}}({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right).

Thus for $X\geq 1$ , summing over $1\leq n\leq X$ we have

\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)-2\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)+\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)=\sum_{1\leq n\leq X}\mu_{s}^{2}(n)\sum_{\begin{subarray}{c}abc=n\\ a,b\geq 2\end{subarray}}({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right)=

\sum_{\begin{subarray}{c}abc\leq X\\ a,b\geq 2\end{subarray}}\mu_{s}^{2}(abc)({\mathcal{a}}+1)^{\omega(c)}\left(\frac{b}{a}\right)=\sum_{1\leq c\leq X}\mu_{s}^{2}(c)({\mathcal{a}}+1)^{\omega(c)}\sum_{\begin{subarray}{c}ab\leq X/c\\ a,b\geq 2\end{subarray}}\mu_{sc}^{2}(ab)\left(\frac{b}{a}\right)=

=\sum_{1\leq c\leq X}\mu_{s}^{2}(c)({\mathcal{a}}+1)^{\omega(c)}T_{sc}(X/c).

∎

Proposition 7.6.

Let $0\leq{\mathcal{a}}\in\mathbb{N}$ and let $s\in{\mathbb{N}^{*}}$ . Then

\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)-2\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)+\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)=o(X\log^{{\mathcal{a}}+1}(X)).

Proof.

By proposition 7.5,

\mathcal{N}_{J\sqcup T_{a},s}(X)-2\mathcal{N}_{T_{a+2},s}(X)+\mathcal{N}_{T_{a+1},s}(X)=\sum_{n\leq X}\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}T_{sn}(X/n),

which equals

\left(\overbrace{\sum_{1\leq n\leq\frac{X}{Y}}}^{(I)}+\overbrace{\sum_{\frac{X}{Y}<n\leq X}}^{(II)}\right)\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}T_{sn}(X/n)

(9)

for some $1<Y<X$ to be determined later. By proposition 5.20, there exists $C>0$ s.t. the first summand in (9) is bounded by

|(I)|\leq\sum_{1\leq n\leq\frac{X}{Y}}\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}|T_{sn}(X/n)|\leq

\leq C\cdot\sum_{n\leq\frac{X}{Y}}\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}\cdot 2^{\frac{1}{2}\omega(sn)}(X/n)^{\frac{7}{8}}(\log(X/n)+1)^{\frac{1}{2}}\leq

\leq C\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(\log X+1)^{\frac{1}{2}}\sum_{n\leq\frac{X}{Y}}\mu_{s}^{2}(n)({\mathcal{a}}+1)^{\omega(n)}\cdot 2^{\frac{1}{2}\omega(n)}n^{-\frac{7}{8}}.

Let ${\mathcal{b}}=\lceil({\mathcal{a}}+1)\sqrt{2}\rceil$ . Then by lemma 5.18,

|(I)|\leq C\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(\log X+1)^{\frac{1}{2}}\sum_{n\leq\frac{X}{Y}}\mu_{s}^{2}(n){\mathcal{b}}^{\omega(n)}n^{-\frac{7}{8}}\leq

\leq C\cdot 2^{\frac{1}{2}\omega(s)}X^{\frac{7}{8}}(\log X+1)^{\frac{1}{2}}\cdot 8(X/Y)^{\frac{1}{8}}(1+\log(X/Y))^{{\mathcal{b}}-1}\leq

\leq 8C\cdot 2^{\frac{1}{2}\omega(s)}X\cdot Y^{-\frac{1}{8}}(\log X+1)^{{\mathcal{b}}}.

For $X\gg 0$ we may take

Y=(\log X+1)^{8{\mathcal{b}}}\leq X,

for which we then have

|(I)|\leq C\cdot 2^{\frac{1}{2}\omega(s)}X.

We turn to $(II)$ , the second summand in (9):

(II)=\sum_{\frac{X}{Y}<n\leq X}\mu_{s}(n)^{2}({\mathcal{a}}+1)^{\omega(n)}T_{sn}(X/n)=

=\sum_{\frac{X}{Y}<n\leq X}\mu_{s}^{2}(n)\mathds{1}^{*({\mathcal{a}}+1)}(n)\sum_{\begin{subarray}{c}a,b\geq 2\\ ab\leq X/n\end{subarray}}\mu_{sn}^{2}(ab)\left(\frac{b}{a}\right),

which is bounded by

|(II)|\leq\sum_{\frac{X}{Y}<n\leq X}\mathds{1}^{*({\mathcal{a}}+1)}(n)\sum_{\begin{subarray}{c}a,b\geq 1\\ ab\leq X/n\end{subarray}}1\leq\sum_{\begin{subarray}{c}a,b\geq 1\\ ab\leq Y\end{subarray}}\sum_{n\leq\frac{X}{ab}}\mathds{1}^{*({\mathcal{a}}+1)}(n).

By lemma 5.5 there exists $C^{\prime}>0$ s.t.

|(II)|\leq C^{\prime}\sum_{\begin{subarray}{c}ab\leq Y\end{subarray}}\frac{X}{ab}\left(\log^{\mathcal{a}}(X/ab)+1\right)\leq

\leq C^{\prime}X\left(\log^{\mathcal{a}}(X)+1\right)\sum_{\begin{subarray}{c}ab\leq Y\end{subarray}}\frac{1}{ab}\leq C^{\prime}X\left(\log^{\mathcal{a}}(X)+1\right)\cdot C^{\prime\prime}\log^{2}(Y)=

=C^{\prime}C^{\prime\prime}X\left(\log^{\mathcal{a}}(X)+1\right)(8{\mathcal{b}}\log(\log X+1))^{2}.

Hence

\left|\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)-2\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)+\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)\right|=|(\ref{Eqn: J-col split T_su(X/u) sum J_+ case})|\leq|(I)|+|(II)|=

=O_{s}(X)+O_{\mathcal{a}}\left(X\log^{\mathcal{a}}(X)(\log\log X)^{2}\right)=o(X\log^{{\mathcal{a}}+1}(X)).

∎

Proposition 7.7.

Let $0\leq{\mathcal{a}}\in\mathbb{N}$ . Then $\textnormal{col}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}$ is of generic summatory type $({\mathcal{a}}+1,\frac{2}{({\mathcal{a}}+1)!})$ :

\textnormal{col}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}\in{\mathscr{W}({\mathcal{a}}+1,\textstyle{\frac{2}{({\mathcal{a}}+1)!}})}

Proof.

Let $s\in{\mathbb{N}^{*}}$ . Following proposition 7.6,

\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)-2\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)+\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)}{X\log^{{\mathcal{a}}+1}(X)}=0.

By proposition 6.4, $\textnormal{col}_{\mathscr{T}_{{\mathcal{a}}+1}}\in{\mathscr{W}({\mathcal{a}},\frac{1}{{\mathcal{a}}!})}$ and $\textnormal{col}_{\mathscr{T}_{{\mathcal{a}}+2}}\in{\mathscr{W}({\mathcal{a}}+1,\frac{1}{({\mathcal{a}}+1)!})}$ . For all $s\in{\mathbb{N}^{*}}$ we therefore have $\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+1},s}(X)=o\left(X\log^{{\mathcal{a}}+1}(X)\right)$ and

\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)}{X\log^{{\mathcal{a}}+1}(X)}=2\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)}{X\log^{{\mathcal{a}}+1}(X)}.

Therefore

c_{s}(\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}):=\gamma_{s}^{-{\mathcal{a}}-1}\lim_{X\to\infty}\frac{\mathcal{E}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}},s}(X)}{\log^{{\mathcal{a}}+1}(X)}=

=2\gamma_{s}^{-{\mathcal{a}}-1}\lim_{X\to\infty}\frac{\mathcal{E}_{\mathscr{T}_{{\mathcal{a}}+2},s}(X)}{\log^{{\mathcal{a}}+1}(X)}=2c_{s}(\mathscr{T}_{{\mathcal{a}}+2})>0,

and

\lim_{s\in{\mathbb{N}^{*}}}c_{s}(\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}})=2\lim_{s\in{\mathbb{N}^{*}}}c_{s}(\mathscr{T}_{{\mathcal{a}}+2})=\frac{2}{({\mathcal{a}}+1)!}.

Thus

\textnormal{col}_{\mathscr{J}\sqcup\mathscr{T}_{\mathcal{a}}}\in{\mathscr{W}({\mathcal{a}}+1,\textstyle{\frac{2}{({\mathcal{a}}+1)!}})}.

∎

8 The Dihedral kei $\mathscr{R}_{3}$

Definition 8.1.

Let $D_{3}\in{\mathscr{G}\textnormal{rp}}$ denote the dihedral group of a triangle. and let $\sigma\colon D_{3}\to\{\pm 1\}$ denote the homomorphism

\sigma\colon D_{3}\simeq(\mathbb{Z}/3\mathbb{Z})\rtimes\{\pm 1\}\to\{\pm 1\}.

By $\mathscr{R}_{3}\subseteq D_{3}$ we denote the set of reflections in $D_{3}$ :

\mathscr{R}_{3}:=D_{3}\setminus\ker\sigma=\sigma^{-1}(-1).

The set $\mathscr{R}_{3}$ is comprised of involutions and is closed under conjugation. Therefore $\mathscr{R}_{3}$ is a kei with structure given by conjugation: ${{}^{g}{h}}:=ghg^{-1}$ .

Recall in [DS23] the computation of the Hilbert polynomial of $\mathscr{R}_{3}$ :

P_{\mathscr{R}_{3}}(k)=6.

8.1 Interpretation of $\mathscr{R}_{3}$ -colorings

Lemma 8.2.

Let $\mathscr{K}\in{\mathscr{K}\textnormal{ei}}^{\textnormal{fin}}$ be such that

1.

the map $\varphi\colon\mathscr{K}\to\textnormal{Inn}(\mathscr{K})$ is an injection, and
2.

for every proper sub-kei $\mathscr{K}^{\prime}<\mathscr{K}$ , the set $\{\varphi_{x}\;|\;x\in\mathscr{K}^{\prime}\}\subseteq\textnormal{Inn}(\mathscr{K})$ generates a proper subgroup of $\textnormal{Inn}(\mathscr{K})$ :

$\langle\{\varphi_{\mathscr{K},x}\;|\;x\in\mathscr{K}^{\prime}\}\rangle<\textnormal{Inn}(\mathscr{K}).$

Then for all $n\in{\mathbb{N}^{*}}$ there is a bijection

\textnormal{Col}_{\mathscr{K}}^{\textnormal{dom}}(n)\simeq\{\rho\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{K})\;|\;(\rho\circ\mathfrak{m}_{n})(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\subseteq\varphi(\mathscr{K})\subseteq\textnormal{Inn}(\mathscr{K})\}.

Proof.

Let $n\in{\mathbb{N}^{*}}$ and let $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\twoheadrightarrow\mathscr{K}$ . Then by proposition 2.24, $f$ extends uniquely to a morphism $(f,\rho)\colon\mathfrak{A}_{n}\twoheadrightarrow\mathscr{A}_{\mathscr{K}}$ . The map $\rho\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{K})$ satisfies

\rho(\mathfrak{m}_{\mathfrak{p}})=\varphi_{f(\mathfrak{p})}\in\varphi(\mathscr{K}).

We therefore have a map

\textnormal{Col}_{\mathscr{K}}^{\textnormal{dom}}(n)\to\{\rho\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{K})\;|\;\rho\circ\mathfrak{m}_{n}=\varphi\circ f\}\subseteq\textnormal{Hom}_{\mathscr{G}\textnormal{rp}}^{\textnormal{cont}}(\mathfrak{G}_{n},\textnormal{Inn}(\mathscr{K}))^{\textnormal{surj}}.

Conversely, let $\rho\colon\mathfrak{G}_{n}\twoheadrightarrow\textnormal{Inn}(\mathscr{K})$ be such that $(\rho\circ\mathfrak{m}_{n})(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\subseteq\varphi(\mathscr{K})\subseteq\textnormal{Inn}(\mathscr{K})$ . Since $\varphi$ is an embedding, there is a unique continuous map $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{K}$ s.t.

\rho\circ\mathfrak{m}_{n}=\varphi\circ f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\textnormal{Inn}(\mathscr{K}).

This $f$ is a kei morphism: for all $\mathfrak{p},\mathfrak{q}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ ,

\varphi_{f({{}^{\mathfrak{p}}{\mathfrak{q}}})}=\rho(\mathfrak{m}_{{{}^{\mathfrak{p}}{\mathfrak{q}}}})=\rho(\mathfrak{m}_{\mathfrak{p}}\mathfrak{m}_{\mathfrak{q}}\mathfrak{m}_{\mathfrak{p}}^{-1})=\rho(\mathfrak{m}_{\mathfrak{p}})\rho(\mathfrak{m}_{\mathfrak{q}})\rho(\mathfrak{m}_{\mathfrak{p}})^{-1}=

=\varphi_{f(\mathfrak{p})}\varphi_{f(\mathfrak{q})}\varphi_{f(\mathfrak{p})}^{-1}=\varphi_{{{}^{f(\mathfrak{p})}{f(\mathfrak{q})}}}.

It follows that $f({{}^{\mathfrak{p}}{\mathfrak{q}}})={{}^{f(\mathfrak{p})}{f(\mathfrak{q})}}$ because $\varphi$ is an embedding. This $f$ is also surjective: Let $\mathscr{K}^{\prime}=f(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\leq\mathscr{K}$ . Then

\textnormal{Inn}(\mathscr{K})=\rho(\mathfrak{G}_{n})=\langle\{\rho(\mathfrak{m}_{\mathfrak{p}})\;|\;\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\}\rangle=\langle\{\varphi_{f(\mathfrak{p})}\;|\;\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\}\rangle=\langle\{\varphi_{\mathscr{K},x}\;|\;x\in\mathscr{K}^{\prime}\}\rangle.

It follows from the assumption on $\mathscr{K}$ that $f(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})=\mathscr{K}$ , that is, $f$ is surjective. Hence we obtain a map

\{\textnormal{Hom}_{\mathscr{G}\textnormal{rp}}^{\textnormal{cont}}(\mathfrak{G}_{n},\textnormal{Inn}(\mathscr{K}))^{\textnormal{surj}}\;|\;(\rho\circ\mathfrak{m}_{n})(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\subseteq\varphi(\mathscr{K})\}\to\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}^{\textnormal{cont}}(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n},\mathscr{K})^{\textnormal{surj}}=

=\textnormal{Col}_{\mathscr{K}}^{\textnormal{dom}}(n).

These constructions are inverse to one another. ∎

Proposition 8.3.

Let $\iota\colon\mathscr{R}_{3}\hookrightarrow D_{3}$ denote the inclusion. Then $(\mathscr{R}_{3},D_{3},\iota)$ is an augmented kei, and

(\mathscr{R}_{3},D_{3},\iota)\simeq\mathscr{A}_{\mathscr{R}_{3}}\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}.

Proof.

The set $\mathscr{R}_{3}$ is a conjugacy class of involutions in $D_{3}$ , hence $\mathscr{A}:=(\mathscr{R}_{3},D_{3},\iota)\in{\mathscr{A}\textnormal{ug}{\mathscr{K}\textnormal{ei}}}$ . This $\mathscr{A}$ is concise because $\langle\mathscr{R}_{3}\rangle=D_{3}$ . By proposition 2.11, $\textnormal{Id}_{\mathscr{R}_{3}}$ lifts to a morphism

(\textnormal{Id}_{\mathscr{R}_{3}},\rho)\colon\mathscr{A}\twoheadrightarrow\mathscr{A}_{\mathscr{R}_{3}}.

Suppose $g\in\ker\rho$ . Then $gxg^{-1}=x$ for all $x\in\mathscr{R}_{3}$ . Since $\mathscr{R}_{3}$ generates $D_{3}$ , it follows that $g\in Z(D_{3})=\{1\}$ . Hence $\rho$ is a group isomorphism, and

(\textnormal{Id}_{\mathscr{R}_{3}},\rho)\colon(\mathscr{R}_{3},D_{3},\iota){\xrightarrow{\sim}}\mathscr{A}_{\mathscr{R}_{3}}.

∎

Corollary 8.4.

The kei $\mathscr{R}_{3}$ satisfies the conditions of lemma 8.2

Proof.

Following proposition 8.3, the map $\varphi\colon\mathscr{R}_{3}\to\textnormal{Inn}(\mathscr{R}_{3})$ is an embedding. Moreover, the non-empty proper sub-keis of $\mathscr{R}_{3}$ are the singletons, and for all $x\in R_{3}$ ,

|\langle\varphi_{\mathscr{R}_{3},x}\rangle|=2<6=|\textnormal{Inn}(\mathscr{R}_{3})|.

∎

An order is an integral domain $R$ that is free as a $\mathbb{Z}$ -module. For order $R$ , the ring $R_{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}R$ is a number field. As such $R\subseteq\mathcal{O}_{R_{\mathbb{Q}}}$ , where $\mathcal{O}_{L}$ , the ring of integers in a number field $L$ , is the maximal order in $L$ . The discriminant $\textnormal{Disc}(R)$ of an order $R$ is the volume of $R$ as a lattice with the trace form:

\textnormal{Disc}(R)=\det([\textnormal{Tr}(x_{i}x_{j})]_{i,j})\in\mathbb{Z},

where $\{x_{i}\}$ is a basis for $R$ as a $\mathbb{Z}$ -module, and $\textnormal{Tr}(x)\in\mathbb{Z}$ is the trace of multiplication by $x$ . The discriminant of a number field $L$ is defined to be

\textnormal{Disc}(L):=\textnormal{Disc}(\mathcal{O}_{L}).

A fundamental discriminant is the discriminant of a quadratic number field. A quadratic number field $L$ can be written uniquely as $L=\mathbb{Q}(\sqrt{m})$ with $m\in\mathbb{Z}$ square-free. For such $L=\mathbb{Q}(\sqrt{m})$ ,

\textnormal{Disc}(L)=\begin{cases}m&,\;m=1\mod 3\\ 4m&,\;m=2,3\mod 4\end{cases}\;\;.

Every square-free integer $m\in\mathbb{Z}$ can be written uniquely as $m=u^{\prime}n=un^{*}$ with $n\in{\mathbb{N}^{*}_{2}}$ and $u,u^{\prime}\in\{\pm 1,\pm 2\}$ . Hence every fundamental discriminant is of the form

\Delta=u\cdot n^{*}\;\;,\;\;\;\;n\in{\mathbb{N}^{*}_{2}}\;,\;u\in\{1,-4,8,-8\}.

Definition 8.5.

Let $0\neq D\in\mathbb{Z}$ . By $H_{3}(D)$ we denote the set of isomorphism classes of cubic orders $R$ with discriminant $\textnormal{Disc}(R)=D$ :

H_{3}(D)=\left\{R/\mathbb{Z}\;|\;\begin{subarray}{c}rk_{\mathbb{Z}}(R)=3\;\textnormal{ and }\\ \;\textnormal{Disc}(R)=D\end{subarray}\right\}/_{\sim},

and by $h_{3}(D):=|H_{3}(D)|\in\mathbb{N}$ its cardinality.

The following is a well-known fact about cubic number fields, due to [Has30]:

Lemma 8.6.

Let $L$ be a cubic number field. Then $\textnormal{Disc}(L)=df^{2}$ , where $d\in\mathbb{Z}$ is a fundamental discriminant, and $f\in\mathbb{Z}$ is such that for every prime $p$ , $p|f$ iff $p$ is totally ramified in $L$ .

Proposition 8.7.

Let $\Delta\in\mathbb{Z}$ be a fundamental discriminant. Then

H_{3}(\Delta)=\left\{\mathcal{O}_{L}/\mathbb{Z}\;|\;\begin{subarray}{c}[L:\mathbb{Q}]=3\;\textnormal{ and }\\ \;\textnormal{Disc}(L)=\Delta\end{subarray}\right\}/_{\sim}.

Proof.

Let $R\in H_{3}(\Delta)$ . Denote by $L:=\mathbb{Q}\otimes_{\mathbb{Z}}R$ , a cubic number field. Let $\mathcal{O}_{L}$ be the maximal order in $L$ . Then

\Delta=\textnormal{Disc}(R)=[\mathcal{O}_{L}:R]^{2}\textnormal{Disc}(\mathcal{O}_{L})=[\mathcal{O}_{L}:R]^{2}\textnormal{Disc}(L).

There exists $f\in\mathbb{Z}$ s.t.

\textnormal{Disc}(L)=df^{2},

where $d$ is a fundamental discriminant. Thus

\Delta=[\mathcal{O}_{L}:R]^{2}f^{2}d.

No two fundamental discriminants differ by a perfect square, hence $R=\mathcal{O}_{L}$ is the maximal order in $L$ , and $\textnormal{Disc}(L)=\Delta$ . Hence

H_{3}(\Delta)=\left\{\mathcal{O}_{L}/\mathbb{Z}\;|\;\begin{subarray}{c}[L:\mathbb{Q}]=3\;\textnormal{ and }\\ \;\textnormal{Disc}(L)=\Delta\end{subarray}\right\}/_{\sim}.

∎

Proposition 8.8.

Let $1<n\in{\mathbb{N}^{*}}$ . Then

\textnormal{col}_{\mathscr{R}_{3}}(n)=3\left|Cl(\mathfrak{L}_{n})\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right|=\begin{cases}3+6h_{3}(n^{*})&,\;2\nmid n\\ 3+6h_{3}(4n^{*})&,\;2|n\end{cases}

=\begin{cases}3+6h_{3}(n)&,\;n=1\mod 4\\ 3+6h_{3}(-n)&,\;n=3\mod 4\\ 3+6h_{3}(4n)&,\;n=2\mod 8\\ 3+6h_{3}(-4n)&,\;n=6\mod 8\end{cases}

where $Cl(\mathfrak{L}_{n})$ is the class group of $\mathfrak{L}_{n}=\mathbb{Q}(\sqrt{n^{*}})$ .

Proof.

Since $n>1$ , $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\neq\emptyset$ . The image of a coloring $f\colon\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}\to\mathscr{R}_{3}$ is therefore a nonempty sub-kei of $\mathscr{R}_{3}$ . The proper nonempty sub-keis of $\mathscr{R}_{3}$ are the three singletons in $\mathscr{R}_{3}$ . Therefore

\textnormal{col}_{\mathscr{R}_{3}}(n)=3\textnormal{col}_{\mathscr{T}_{1}}(n)+\textnormal{col}_{\mathscr{R}_{3}}^{\textnormal{dom}}(n)=3+\textnormal{col}_{\mathscr{R}_{3}}^{\textnormal{dom}}(n).

Let $\eta\colon\mathfrak{G}_{n}\to\{\pm 1\}$ denote the homomorphism

\eta\colon\mathfrak{G}_{n}=\textnormal{Gal}(\mathfrak{F}_{n}/\mathbb{Q})\twoheadrightarrow\textnormal{Gal}(\mathfrak{L}_{n}/\mathbb{Q}){\xrightarrow{\sim}}\{\pm 1\}.

This is the unique morphism that maps $\mathfrak{m}_{\mathfrak{p}}\mapsto-1$ for all $\mathfrak{p}\in\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}$ . Since $\mathscr{R}_{3}=\sigma^{-1}(-1)\subseteq D_{3}$ , then by lemma 8.2 and corollary 8.4, we have

\textnormal{Col}_{\mathscr{R}_{3}}^{\textnormal{dom}}(n)\simeq\{\rho\colon\mathfrak{G}_{n}\twoheadrightarrow D_{3}\;|\;(\rho\circ\mathfrak{m}_{n})(\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n})\subseteq\mathscr{R}_{3}\}=\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}

where $\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}$ is the set of surjective lifts

The $D_{3}$ -action on $\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}$ via conjugation,

g(\rho)(\gamma):=g\rho(\gamma)g^{-1},

is free because $D_{3}$ has trivial center. Therefore

\textnormal{col}_{R_{3}}(n)=3+6\cdot|D_{3}\backslash\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}|.

The quotient $D_{3}\backslash\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}$ classifies $D_{3}$ -Galois number fields $\widetilde{L}$ containing $\mathfrak{L}_{n}$ s.t. $\widetilde{L}/\mathfrak{L}_{n}$ is unramified. Since $\textnormal{Gal}(\mathfrak{L}_{n}/\mathbb{Q})$ acts via sign on the class group $Cl(\mathfrak{L}_{n})$ in its entirety, every unramified $\mathbb{Z}/3\mathbb{Z}$ -extension $N/\mathfrak{L}_{n}$ is Galois over $\mathbb{Q}$ with

\textnormal{Gal}(N/\mathbb{Q})\simeq\mathbb{Z}/3\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}\simeq D_{3}.

Hence these $\widetilde{L}$ are precisely the unramified $\mathbb{Z}/3\mathbb{Z}$ -Galois extensions of $\mathfrak{L}_{n}$ , of which there are

|D_{3}\backslash\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}|=\left|\mathbb{P}_{\mathbb{F}_{3}}\left(Cl(\mathfrak{L}_{n})\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right)\right|=\frac{\left|Cl(\mathfrak{L}_{n})\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right|-1}{2}.

Such extensions are classified by cubic fields $L$ with discriminant

\textnormal{Disc}(L)=\textnormal{Disc}(\mathfrak{L}_{n})=\begin{cases}n^{*}&,\;2\nmid n\\ 4n^{*}&,\;2|n\end{cases}\;\;.

It follows by proposition 8.7 that

D_{3}\backslash\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}\simeq H_{3}(\textnormal{Disc}(\mathfrak{L}_{n})).

All in all,

\textnormal{col}_{R_{3}}(n)=3+6\cdot|D_{3}\backslash\textnormal{Hom}_{/\eta}(\mathfrak{G}_{n},D_{3})^{\textnormal{surj}}|=

=3+6\cdot\frac{\left|Cl(\mathfrak{L}_{n})\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right|-1}{2}=3\left|Cl(\mathfrak{L}_{n})\otimes_{\mathbb{Z}}\mathbb{F}_{3}\right|=

=3+6h_{3}(\textnormal{Disc}(\mathfrak{L}_{n}))=\begin{cases}3+6h_{3}(n^{*})&,\;2\nmid n\\ 3+6h_{3}(4n^{*})&,\;2|n\end{cases}\;\;.

∎

8.2 Proof of the Main Conjecture for $\mathscr{K}=\mathscr{R}_{3}$

Lemma 8.9.

Let $\Delta$ be a fundamental discriminant, let $p|\Delta$ be prime, and let $R/\mathbb{Z}$ be a cubic order with discriminant $\Delta$ . Then

\mathbb{Z}_{p}\otimes_{\mathbb{Z}}R\simeq\mathbb{Z}_{p}\times\mathbb{Z}_{p}[{\sqrt{\Delta}}/2].

Proof.

Since $\Delta$ is a fundamental discriminant, then $R$ is the maximal order $R=\mathcal{O}_{L}$ in the non-Galois cubic number field $L=\mathbb{Q}\otimes_{\mathbb{Z}}R$ . Since $p|\Delta=\textnormal{Disc}(L)$ , then $p$ is partially ramified in $L$ , hence there is some ramified quadratic $M/\mathbb{Q}_{p}$ s.t.

\mathbb{Z}_{p}\otimes_{\mathbb{Z}}R\simeq\mathbb{Z}_{p}\times\mathcal{O}_{M}.

The normal closure $\widetilde{L}$ of $L$ contains $\mathbb{Q}(\sqrt{\Delta})$ . Which is to say that $\mathbb{Q}(\sqrt{\Delta})$ is a sub-quotient of $L\otimes_{\mathbb{Q}}L$ . Therefore $\mathbb{Q}_{p}(\sqrt{\Delta}):=\mathbb{Q}_{p}\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{\Delta})$ is a sub-quotient of

(L\otimes_{\mathbb{Q}}L)\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\simeq(L\otimes_{\mathbb{Q}}\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}(L\otimes_{\mathbb{Q}}\mathbb{Q}_{p})\simeq

\simeq(\mathbb{Q}_{p}\times M)\otimes_{\mathbb{Q}_{p}}(\mathbb{Q}_{p}\times M)\simeq\mathbb{Q}_{p}\times M^{4}.

Hence $M=\mathbb{Q}_{p}(\sqrt{\Delta})$ . For all $p|\Delta$ ,

\mathcal{O}_{M}=\mathcal{O}_{\mathbb{Q}_{p}(\sqrt{\Delta})}=\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{Z}[\textstyle{\frac{\Delta+\sqrt{\Delta}}{2}}]=\mathbb{Z}_{p}[\sqrt{\Delta}/2],

and

\mathbb{Z}_{p}\otimes_{\mathbb{Z}}R\simeq\mathbb{Z}_{p}\times\mathcal{O}_{M}\simeq\mathbb{Z}\times\mathbb{Z}_{p}[\sqrt{\Delta}/2].

∎

Corollary 8.10.

Let $R$ be a cubic order s.t. $\Delta:=\textnormal{Disc}(R)$ is a fundamental discriminant. Then there exists even $n\in{\mathbb{N}^{*}}$ s.t. $\Delta=4n^{*}$ iff

\mathbb{Z}_{2}\otimes_{\mathbb{Z}}R\in\{\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{2}],\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-6}]\}.

Proof.

The set of fundamental discriminants equals

\{c\cdot m^{*}\;|\;m\in\mathbb{N}^{*}_{2},\;c=1,-4,-8,8\}.

The proof of the corollary is straightforward under the assumption that $2\nmid\Delta$ since neither $\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{2}]$ nor $\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-6}]$ is étale over $\mathbb{Z}_{2}$ . Assuming therefore that $2|\Delta$ , we have

\mathbb{Z}_{2}\otimes_{\mathbb{Z}}R\simeq\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{\Delta}/2]

For $m\in\mathbb{N}^{*}_{2}$ we have

-m^{*}\equiv 3,-1\mod 8\;,\;\;-2m^{*}\equiv 6,-2\mod 16\;,\;\;2m^{*}\equiv 2,-6\mod 16,

Hence

$\Delta$	$\mathbb{Z}_{2}\otimes_{\mathbb{Z}}R$	$\mathbb{Z}_{2}[\sqrt{\Delta}/2]=$
$-4m^{*}$	$\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-m^{*}}]$	$\mathbb{Z}_{2}[\sqrt{3}],\mathbb{Z}_{2}[\sqrt{-1}]$
$-8m^{*}$	$\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-2m^{*}}]$	$\mathbb{Z}_{2}[\sqrt{6}],\mathbb{Z}_{2}[\sqrt{-2}]$
$8m^{*}$	$\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{2m^{*}}]$	$\mathbb{Z}_{2}[\sqrt{2}],\mathbb{Z}_{2}[\sqrt{-6}]$

Having done a comprehensive sweep of all fundamental discriminants, we can verify that there exists even $n=2m\in{\mathbb{N}^{*}}$ s.t. $\textnormal{Disc}(R)=4n^{*}=8m^{*}$ iff

\mathbb{Z}_{2}\otimes_{\mathbb{Z}}R\in\{\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{2}],\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-6}]\}.

∎

Definition 8.11.

For prime $p$ we define the following sets $S_{p}$ and $S_{p}^{\circ}$ of cubic $\mathbb{Z}_{p}$ -algerbas up to isomorphism:

S_{p}=\{W(\mathbb{F}_{p^{3}})\}\cup\{\mathbb{Z}_{p}\times\mathcal{O}_{\mathbb{Q}_{p}[x]/(x^{2}-\alpha)}\;|\;\alpha\in\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}\},

- which for $p\neq 2$ equals

S_{p}=\{W(\mathbb{F}_{p^{3}})\;,\;\mathbb{Z}_{p}^{3}\;,\;\mathbb{Z}_{p}\times\mathbb{Z}_{p}[\sqrt{\alpha_{p}}]\;,\;\mathbb{Z}_{p}\times\mathbb{Z}_{p}[\sqrt{p}]\;,\;\mathbb{Z}_{p}\times\mathbb{Z}_{p}[\sqrt{\alpha_{p}p}]\},

where $\alpha_{p}$ is a quadratic non-residue modulo $p$ and $W$ denotes the ring of Witt vectors. The set $S_{p}^{\circ}$ consists of all étale cubic $\mathbb{Z}_{p}$ -algebras:

S_{p}^{\circ}=\{W(\mathbb{F}_{p^{3}}),\mathbb{Z}_{p}^{3},\mathbb{Z}_{p}\times W(\mathbb{F}_{p^{2}})\}.

For $p=2$ we also define the set $S_{2}^{\prime}$ of $\mathbb{Z}_{2}$ -algebras:

S_{2}^{\prime}=\{\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{2}],\mathbb{Z}_{2}\times\mathbb{Z}_{2}[\sqrt{-6}]\}.

Let $s\in{\mathbb{N}^{*}}$ and let $p$ be prime. We define the sets $\Sigma_{s,p}$ and $\Sigma_{s,p}^{\prime}$ :

\Sigma_{s,p}=\begin{cases}S_{p}^{\circ}&,\;p|2s\\ S_{p}&,\;p\nmid 2s\end{cases}\;\;\;\;\;\;\;\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Sigma_{s,p}^{\prime}=\begin{cases}S_{p}^{\circ}&,\;2\nmid s,\;p|s\\ S_{p}&,\;2\nmid s,\;2\neq p\nmid s\\ S_{2}^{\prime}&,\;2\nmid s,\;p=2\\ \emptyset&,\;2|s,\;p=2\\ S_{p}^{\circ}&,\;2|s,\;2\neq p|s\\ S_{p}&,\;2|s,\;p\nmid s\end{cases}

We also define for $p=\infty$ , the following sets of $\mathbb{R}$ -algebras:

\Sigma_{s,\infty}=\Sigma_{s,\infty}^{\prime}=\{\mathbb{R}^{3},\mathbb{R}\times\mathbb{C}\}.

Proposition 8.12.

Let $s\in{\mathbb{N}^{*}}$ and let $R/\mathbb{Z}$ be a cubic order. Then

R\in\bigcup_{\begin{subarray}{c}n\in{\mathbb{N}^{*}}\\ n\textnormal{ odd}\end{subarray}}H_{3}(n^{*})\;\;\;\;\;\;\textnormal{ - resp.}\;\;R\in\bigcup_{\begin{subarray}{c}n\in{\mathbb{N}^{*}}\\ n\textnormal{ even}\end{subarray}}H_{3}(4n^{*})

iff for all prime $p$ ,

\mathbb{Z}_{p}\otimes_{\mathbb{Z}}R\in\Sigma_{s,p}\;\;\;\;\;\;\textnormal{ - resp. }\mathbb{Z}_{p}\otimes_{\mathbb{Z}}R\in\Sigma_{s,p}^{\prime}.

Proof.

Following lemma 8.6 and proposition 8.7, the property that $\textnormal{Disc}(R)$ is a fundamental discriminant is completely determined locally: For prime $p$ , let

R_{p}:=\mathbb{Z}_{p}\otimes R.

Then $R\in H_{3}(\Delta)$ for some fundamental discriminant $\Delta$ iff for all prime $p$ , $R_{p}$ is a regular ring either isomorphic to the Witt vectors $W(\mathbb{F}_{p^{3}})$ , or of the form $R_{p}\simeq\mathbb{Z}_{p}\times\mathcal{O}_{M}$ for some degree- $2$ étale $\mathbb{Q}_{p}$ -algebra $M$ :

R_{p}\in\{W(\mathbb{F}_{p^{3}})\}\cup\{\mathbb{Z}_{p}\times\mathcal{O}_{\mathbb{Q}_{p}[x]/(x^{2}-\alpha)}\;|\;\alpha\in\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}\}=S_{p}.

Furthermore, for every prime $p$ , $p\nmid\Delta$ iff $R_{p}/\mathbb{Z}_{p}$ is étale, which is to say that

p\nmid\Delta\;\iff\;R_{p}\in S_{p}^{\circ}.

(10)

The set $\{n^{*}\;|\;\textnormal{odd }n\in{\mathbb{N}^{*}}\}$ equals the set of odd fundamental discriminants, hence $R\in H_{3}(n^{*})$ for some odd $n\in{\mathbb{N}^{*}}$ iff

R_{p}\in\begin{cases}S_{p}&,\;p\neq 2\\ S_{p}^{\circ}&,\;p=2\end{cases}\;\;.

By corollary 8.10, the set $\{4n^{*}\;|\;\textnormal{even }n\in{\mathbb{N}^{*}}\}=\{8m^{*}\;|\;m\in\mathbb{N}^{*}_{2}\}$ equals the set of fundamental discriminants $\Delta$ for which

\mathbb{Q}_{2}(\Delta)=\mathbb{Q}_{2}(\sqrt{2}),\mathbb{Q}_{2}(\sqrt{-6}).

Following lemma 8.9, $R\in H_{3}(4n^{*})$ for some even $n\in{\mathbb{N}^{*}}$ iff

R_{p}\in\begin{cases}S_{p}&,\;p\neq 2\\ S_{2}^{\prime}&,\;p=2\end{cases}\;\;.

By (10), placing further $s$ -coprimality conditions on $\textnormal{Disc}(R)$ is tantamount to intersecting any prior local conditions on $R$ with $S_{p}^{\circ}$ for all $p|s$ : The cubic order $R$ is in $H_{3}(n^{*})$ for some odd $n\in{\mathbb{N}^{*}_{s}}$ iff for every prime $p$ ,

R_{p}\in\begin{cases}S_{p}&,\;2\neq p\nmid s\\ S_{2}^{\circ}&,\;2=p\nmid s\\ S_{p}\cap S_{p}^{\circ}&,\;2\neq p|s\\ S_{2}^{\circ}\cap S_{p}^{\circ}&,\;2=p|s\end{cases}\;\;\;\;=\begin{cases}S_{p}&,\;p\nmid 2s\\ S_{p}^{\circ}&,\;p|2s\end{cases}\;\;\;\;=\Sigma_{s,p}.

Similarly, $R$ is in $H_{3}(4n^{*})$ for some even $n\in{\mathbb{N}^{*}}$ iff for every prime $p$ ,

R_{p}\in\begin{cases}S_{p}&,\;2\neq p\nmid s\\ S_{2}^{\prime}&,\;2=p\nmid s\\ S_{p}\cap S_{p}^{\circ}&,\;2\neq p|s\\ S_{2}^{\prime}\cap S_{p}^{\circ}&,\;2=p|s\end{cases}\;\;\;\;=\begin{cases}S_{p}&,\;2\neq p\nmid s\\ S_{2}^{\prime}&,\;2=p\nmid s\\ S_{p}^{\circ}&,\;2\neq p|s\\ \emptyset&,\;2=p|s\end{cases}\;\;\;\;=\Sigma_{s,p}^{\prime}.

∎

Proposition 8.13.

Let $s\in{\mathbb{N}^{*}}$ . Then

\lim_{X\to\infty}\frac{1}{X}\left(\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})\right)=\frac{\gamma_{s}}{3},

where

\gamma_{s}=\prod_{p|s}(1-p^{-1})\prod_{p\nmid s}(1-p^{-2}).

Proof.

Let $X\geq 1$ . By proposition 8.12, the sum

\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})

counts cubic orders $R/\mathbb{Z}$ for which $\mathbb{Z}_{p}\otimes R\in\Sigma_{s,p}$ for all primes $p$ , and

|\textnormal{Disc}(R)|=|n^{*}|=n\leq X.

Likewise, the sum

\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})

counts cubic orders $R/\mathbb{Z}$ for which $\mathbb{Z}_{p}\otimes R\in\Sigma_{s,p}^{\prime}$ for all primes $p$ , and

|\textnormal{Disc}(R)|=|4n^{*}|=4n\leq 4X.

By Theorem 8 in [BST13],

\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})=C(\Sigma_{s,\infty})\prod_{p}C(\Sigma_{s,p}),

...\lim_{X\to\infty}\frac{1}{4X}\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})=C(\Sigma_{s,\infty}^{\prime})\prod_{p}C(\Sigma_{s,p}^{\prime}),

where for $\widetilde{S}_{p}$ a set of isomorphism classes of cubic $\mathbb{Z}_{p}$ -algebras, $C(\widetilde{S}_{p})$ is a local density factor defined as

C(\widetilde{S}_{p})=(1-p^{-1})\sum_{R\in\widetilde{S}_{p}}\frac{1}{\textnormal{Disc}_{p}(R)}\cdot\frac{1}{|\textnormal{Aut}(R)|},

and for $\widetilde{S}_{\infty}$ a set of cubic $\mathbb{R}$ -algebras,

C(\widetilde{S}_{\infty})=\frac{1}{2}\sum_{R\in\widetilde{S}_{\infty}}\frac{1}{|\textnormal{Aut}(R)|}.

Hence

\lim_{X\to\infty}\frac{1}{X}\left(\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})\right)=

=C(\Sigma_{s,\infty})\prod_{p}C(\Sigma_{s,p})+4C(\Sigma_{s,\infty}^{\prime})\prod_{p}C(\Sigma_{s,p}^{\prime}).

For all every prime $p\neq 2$ , $\Sigma_{s,p}=\Sigma_{s,p}^{\prime}$ and

\sum_{R\in\Sigma_{s,p}}\frac{1}{\textnormal{Disc}_{p}(R)}\cdot\frac{1}{|\textnormal{Aut}(R)|}=\begin{cases}\frac{1}{6}+\frac{1}{2}+\frac{1}{3}=1&,\;p|s\\ \frac{1}{6}+\frac{1}{2}+\frac{1}{3}+\frac{1}{p}\cdot\frac{1}{2}+\frac{1}{p}\cdot\frac{1}{2}=1+p^{-1}&,\;p\nmid s\end{cases}\;\;.

Hence

C(\Sigma_{s,p})=C(\Sigma_{s,p}^{\prime})=(1-p^{-1})\sum_{R\in\Sigma_{s,p}}\frac{1}{\textnormal{Disc}_{p}(R)}\cdot\frac{1}{|\textnormal{Aut}(R)|}=\begin{cases}1-p^{-1}&,\;p|s\\ 1-p^{-2}&,\;p\nmid s\end{cases}\;\;.

For the case at $\infty$ ,

C(\Sigma_{s,\infty})=C(\Sigma_{s,\infty}^{\prime})=\frac{1}{2}(\frac{1}{6}+\frac{1}{2})=\frac{1}{3}.

For $p=2$ , we compute $C(\Sigma_{s,2})+4C(\Sigma_{s,2}^{\prime})$ . If $s$ is even, then

C(\Sigma_{s,2})+4C(\Sigma_{s,2}^{\prime})=C(\Sigma_{s,2})=\frac{1}{2}=1-2^{-1}

if $s$ is odd, then

C(\Sigma_{s,2})+4C(\Sigma_{s,2}^{\prime})=\frac{1}{2}+4\cdot\frac{1}{2}\left(\frac{1}{8}\cdot\frac{1}{2}+\frac{1}{8}\cdot\frac{1}{2}\right)=\frac{3}{4}=1-2^{-2}.

Hence regardless of parity of $s$ , we have

\lim_{X\to\infty}\frac{1}{X}\left(\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})\right)=

=C(\Sigma_{s,\infty})\prod_{p}C(\Sigma_{s,p})+4C(\Sigma_{s,\infty}^{\prime})\prod_{p}C(\Sigma_{s,p}^{\prime})=

=\frac{1}{3}\prod_{p|s}(1-p^{-1})\prod_{p\nmid s}(1-p^{-2})=\frac{\gamma_{s}}{3}.

∎

Proposition 8.14.

The function $\textnormal{col}_{\mathscr{R}_{3}}$ has generic summatory type

\textnormal{col}_{\mathscr{R}_{3}}\in{\mathscr{W}(0,5)}.

Proof.

Let $s\in{\mathbb{N}^{*}}$ . For $n=1$ we have $\text{\begin{CJK}{UTF8}{} \!\!\!\!\!\! min\CJKtilde\CJKnospace 圭 \!\!\!\! \end{CJK}}_{n}=\emptyset$ , therefore

\textnormal{col}_{\mathscr{R}_{3}}(n)=|\textnormal{Hom}_{\mathscr{K}\textnormal{ei}}(\emptyset,\mathscr{R}_{3})|=1.

For all $X\geq 1$ we have

\mathcal{N}_{\mathscr{R}_{3},s}(X)=1+\sum_{(n,s)=1}^{2\leq n\leq X}\textnormal{col}_{\mathscr{R}_{3}}(n)=1+\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}(3+6h_{3}(n^{*}))+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}(3+6h_{3}(4n^{*}))=

-2+3\sum_{1\leq n\leq X}\mu_{s}^{2}(n)+6\left(\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})\right).

Therefore

\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{R}_{3},s}(X)}{X}=

=3\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{T}_{1},s}(X)}{X}+6\lim_{X\to\infty}\left(\sum_{\begin{subarray}{c}1\neq n\;\textnormal{odd}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(n^{*})+\sum_{\begin{subarray}{c}n\;\textnormal{even}\\ (n,s)=1\end{subarray}}^{n\leq X}h_{3}(4n^{*})\right)=

=3\gamma_{s}+6\cdot\frac{\gamma_{s}}{3}=5\gamma_{s}.

We find that

c_{s}(\mathscr{R}_{3})=\lim_{X\to\infty}\frac{\mathcal{E}_{\mathscr{R}_{3},s}(X)}{\log^{0}(X)}=\lim_{X\to\infty}\frac{\mathcal{N}_{\mathscr{R}_{3},s}(X)}{\gamma_{s}X}=5

for all $s\in{\mathbb{N}^{*}}$ , and subseuqently

\lim_{s\in{\mathbb{N}^{*}}}c_{s}(\mathscr{R}_{3})=5.

Hence

\textnormal{col}_{\mathscr{R}_{3}}\in{\mathscr{W}(0,5)}.

∎

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Arithmetic Kei Theory

Abstract

1 Introduction

Definition 1.1 ([Tak43]222The terminology and Kanji notation ​​​​​​ min\CJKtilde\CJKnospace 圭 ​​​​ are taken from [Tak43], where the notion was first defined. See [Joy82] for keis in relation to knot theory, where they are called 22-quandles.).

Definition 1.2 (definition 3.4).

Definition 1.3 (definition 3.7).

Theorem 1.4 (proposition 3.15 + proposition 3.12 + definition 3.13).

Definition 1.5 (definition 3.17).

Definition 1.6 (definition 4.5).

Theorem 1.7 ([DS23, Theorem 1.1]).

Conjecture 1.8 (definition 4.6 + 4.8).

Proposition 1.9.

1.1 Structure of the Paper

1.2 Relation to Other Works

1.3 Acknowledgements

2 Keis and Profinite Keis

2.1 Keis and Augmented Keis

Definition 2.1.

Example 2.2.

Definition 2.3.

Definition 2.4.

Example 2.5.

Proposition 2.6.

Definition 2.7.

Definition 2.8.

Proposition 2.9.

Proof.

Definition 2.10.

Proposition 2.11.

Proof.

Definition 2.12.

Proposition 2.13.

Proposition 2.14.

Proof.

Definition 2.15.

Lemma 2.16.

Proof.

Corollary 2.17.

Proof.

Proposition 2.18.

Proof.

Definition 2.19.

Proposition 2.20.

Proof.

Proposition 2.21.

Proof.

2.2 Profinite Keis and Augmented Keis

Definition 2.22.

Lemma 2.23.

Proof.

Proposition 2.24.

Proof.

Proposition 2.25.

Definition 2.26.

Proposition 2.27.

Proof.

Definition 2.28.

Proposition 2.29.

Proof.

2.3 Disjoint Unions in 𝒦​ei{\mathscr{K}\textnormal{ei}}

Proposition 2.30.

Proof.

Proposition 2.31.

Proof.

3 Fundamental Arithemtic Keis

3.1 The Fundamental Kei of a link

Proposition 3.1.

Proof.

3.2 The Fundamental Kei of an Arithmetic Link

Definition 3.2.

Definition 3.3.

Definition 3.4.

Proposition 3.5.

Proof.

Definition 3.6.

Definition 3.7.

Definition 3.8.

Definition 3.9.

Proposition 3.10.

Proof.

Definition 1.1 ([Tak43]²²2The terminology and Kanji notation min\CJKtilde\CJKnospace 圭 are taken from [Tak43], where the notion was first defined. See [Joy82] for keis in relation to knot theory, where they are called $2$ -quandles.).

2.3 Disjoint Unions in ${\mathscr{K}\textnormal{ei}}$

3.3 $\mathscr{K}$ -Coloring Arithmetic Links

4.2 Statistics of $\mathscr{K}$ -Coloring Random Arithmetic Links