propAthmApropA corAthmAcorA lemmathmlemma corthmcor propthmprop dfthmdf
remthmrem exthmex

# On Smirnov’s approach to the ABC conjecture

Manoel Jarra  Manoel Jarra, University of Groningen, the Netherlands, and
IMPA, Rio de Janeiro, Brazil<EMAIL_ADDRESS>
###### Abstract.

We use algebraic geometry over pointed monoids to give an intrinsic
interpretation for the compactification of the spectrum of the ring of
integers of a number field $K$, for the projective line over algebraic
extensions of ${\mathbb{F}}_{1}$ and for maps between them induced by elements
of $K$, as introduced by Alexander Smirnov in his approach to the ABC
conjecture.

###### Contents

  1. 1 Monoid schemes
  2. 2 Strong congruence spaces
  3. 3 The projective line over algebraic extensions of ${\mathbb{F}}_{1}$
  4. 4 The compactification of $\operatorname{Spec}\mathcal{O}_{K}$
  5. 5 Maps from $\overline{\operatorname{Spec}\mathcal{O}_{K}}$ to $\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$ induced by elements of $K$

## Introduction

In [Smi93], Smirnov proposes an approach to the ABC conjecture based on the
analogy between number fields and function fields of algebraic curves (see
also [Bru16]).

The main idea is to consider the “compactification”
$\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}$ of
$\operatorname{Spec}{\mathbb{Z}}$ as a curve over “the field with one element”
${\mathbb{F}}_{1}$. The curve
$\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}$ is defined as the
set of non-trivial places of the “function field” ${\mathbb{Q}}$, i.e., as the
set

$\\{[2],[3],[5],[7],[11],\dotsc\\}\cup\\{[\infty]\\},$

where $[p]$ is the class of the $p$-adic valuation

$q\mapsto v_{p}(q)=n\quad\text{if}\quad q=p^{n}\dfrac{a}{b}\enspace\text{ with
}\enspace a,b\in{\mathbb{Z}}\enspace\text{ and }\enspace p\not|ab,$

and $[\infty]$ is the class of the achimedean valuation

$q\mapsto v_{\infty}(q)=-\textup{log}(|q|).$

The degree of a point is given by

$\textup{deg}([p])=\textup{log}(p)\quad\text{and}\quad\textup{deg}([\infty])=1,$

which is the unique choice (up to common multiple) that satisfies

$\underset{[x]}{\sum}v_{x}(q)\textup{deg}([x])=0\quad\text{ for all }\quad
q\in{\mathbb{Q}}^{*}.$

The set of non-zero elements of the “field of constants” of ${\mathbb{Q}}$ is

$\\{q\in{\mathbb{Q}}^{*}\mid v_{x}(q)=0\text{ for all }[x]\\}=\\{1,-1\\}.$

The projective line ${\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}$ is
defined as $\big{\\{}[n]\;\big{|}\;n\in{\mathbb{N}}\\}\cup\\{[\infty]\\}$,
with degree map given by

$\textup{deg}([0])=1=\textup{deg}([\infty])\quad\text{and}\quad\textup{deg}([n])=\myphi(n)\quad\text{
for }\quad 0<n<\infty,$

where $\myphi$ is the Euler function.

Each “non-constant” $q=\frac{a}{b}\in{\mathbb{Q}}^{*}\backslash\\{1,-1\\}$
with $\textup{gcd}(a,b)=1$ defines a map $\myvarphi_{q}$ from
$\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}$ to
${\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}$, given by

$\begin{array}[]{ccl}[x]&\longmapsto&\smash{\left\\{\begin{array}[]{lll}[0]&\text{if
}x=p\neq\infty&\text{ and }p\mid a\\\ {}[\infty]&\text{if
}x=p\neq\infty&\text{ and }p\mid b\\\ {}[n]&\text{if }x=p\neq\infty&\text{ and
}p\nmid ab,\text{ where
}n=\textup{ord}\big{(}\overline{a}\overline{b}^{-1}\big{)}\text{ in
}{\mathbb{F}}_{p}^{*}\\\ {}[0]&\text{if }x=\infty&\text{ and }a<b\\\
{}[\infty]&\text{if }x=\infty&\text{ and }a>b.\end{array}\right.}\end{array}$

The ramification index of a point in
$\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}$ is

$\begin{array}[]{cl}e_{[p]}:=&\smash{\left\\{\begin{array}[]{ll}\textup{max}\\{k\mid
p^{k}\text{ divides }a\\}&\text{if }\myvarphi_{q}([p])=[0]\\\
{}\textup{max}\\{k\mid p^{k}\text{ divides }b\\}&\text{if
}\myvarphi_{q}([p])=[\infty]\\\ {}\textup{max}\\{k\mid p^{k}\text{ divides
}a^{n}-b^{n}\\}&\text{if
}\myvarphi_{q}([p])=[n]\end{array}\right.}\end{array}$

if $p$ is a prime number, and $e_{[\infty]}=-\textup{log}(|q|)$.

In analogy with the function field case, where the degree of a non-constant
map between curves is the degree of its divisor of zeros, the degree of
$\myvarphi_{q}$ is

$\textup{deg}(\myvarphi_{q})\;\;\;=\underset{v_{x}(f)>0}{\sum}v_{x}(f)\textup{deg}([x]).$

The arithmetic defect of a point
$[x]\in\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}$ is

$\mydelta_{[x]}:=\dfrac{(e_{[x]}-1)\deg([x])}{\textup{deg}(\myvarphi_{q})}.$

######  Conjecture ([Smi93]).

For each $\myepsilon>0$, there exists a constant $C$ satisfying

$\underset{[x]\in X(q)}{\sum}\mydelta_{[x]}\leq
2+\myepsilon+\dfrac{C}{\textup{deg}(\myvarphi_{q})},$

where $X(q)$ is the set
$\\{[x]\in\overline{\operatorname{Spec}{\mathbb{Z}}}\mid\myvarphi_{q}([x])\text{
has degree }1\\}$.

######  Theorem ([Smi93]).

If the conjecture above holds, then the ABC conjecture is true.

######  Remark.

The conjecture above can be understood as analogous to the inequality

$\underset{\mathfrak{P}\text{ prime of
}L}{\sum}\dfrac{\big{(}e(\mathfrak{P}/P)-1\big{)}\textup{deg}_{L}\mathfrak{P}}{[L:K]}\leq
2-2g_{K}+\dfrac{2g_{L}-2}{[L:K]},$

which follows from the well-known Riemann-Hurwitz formula for finite,
separable, geometric extensions of function fields $L/K$ (see [Ros02, Thm.
7.16]).

In this paper we link Smirnov’s approach with strong congruence spaces of
monoid schemes, which are topological spaces that locally characterize maps
from the corresponding coordinate monoids into fields.

The next result shows how to recover
${\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}$ by using
$\operatorname{SCong}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$, the strong
congruence space of the projective line over ${\mathbb{F}}_{1}$.

######  Theorem A (Section 3).

There is a canonical inclusion
$i:{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}\rightarrow\operatorname{SCong}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$
whose image consists of all non-generic points of
$\operatorname{SCong}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$.

### Compactification of $\operatorname{Spec}\mathcal{{\mathbb{Z}}}$

In Section 4, we construct the compactification of
$\operatorname{Spec}{\mathbb{Z}}$ as follows: by forgetting the addition the
structural sheaf of $\operatorname{Spec}{\mathbb{Z}}$, one has the monoidal
space $(\operatorname{Spec}{\mathbb{Z}})^{\bullet}$. As a topological space,
$\overline{\operatorname{Spec}{\mathbb{Z}}}$ is the disjoint union
$(\operatorname{Spec}{\mathbb{Z}})\sqcup\\{\infty\\}$, with cofinite topology
on $(\operatorname{Spec}{\mathbb{Z}}\backslash\\{0\\})\sqcup\\{\infty\\}$ and
generic point $\\{0\\}$. The sheaf
$\mathcal{O}_{\overline{\operatorname{Spec}{\mathbb{Z}}}}$ is given by

$(\overline{\operatorname{Spec}{\mathbb{Z}}})\backslash\\{\infty\\}\simeq(\operatorname{Spec}{\mathbb{Z}})^{\bullet}\quad\text{and}\quad\mathcal{O}_{\overline{\operatorname{Spec}{\mathbb{Z}}},\infty}:=[-1,1]\cap{\mathbb{Q}}.$

It comes with a canonical injection
$j:\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}\rightarrow\overline{\operatorname{Spec}{\mathbb{Z}}}$
whose image is the set of non-generic points of
$\overline{\operatorname{Spec}{\mathbb{Z}}}$.

### Maps from $\overline{\operatorname{Spec}{\mathbb{Z}}}$ to
$\operatorname{SCong}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$

Given $q\in{\mathbb{Q}}^{*}\backslash\\{1,-1\\}$, we consider the morphism of
pointed monoids

$\begin{array}[]{cccc}\mysigma_{q}:&{\mathbb{F}}_{1}[T]&\longrightarrow&({\mathbb{Z}}[q])^{\bullet}\\\
&T&\longmapsto&q,\end{array}$

where $({\mathbb{Z}}[q])^{\bullet}$ is the multiplicative pointed monoid of
${\mathbb{Z}}[q]$. It induces a continuous map
$\mysigma_{q}^{*}:\operatorname{Spec}{\mathbb{Z}}[q]\rightarrow\operatorname{SCong}{\mathbb{F}}_{1}[T]$
that sends a prime ideal $\mathfrak{p}$ to the strong prime congruence

$\mysigma_{q}^{*}(\mathfrak{p}):=\big{\\{}(a,b)\in{\mathbb{F}}_{1}[T]\times{\mathbb{F}}_{1}[T]\;\big{|}\;\overline{\mysigma(a)}=\overline{\mysigma(b)}\text{
in }{\mathbb{Z}}[q]/\mathfrak{p}\big{\\}}.$

The next result shows how the map $\myvarphi_{q}$ fits in this context (see
Theorem 5.1).

######  Theorem B.

There exists a continuous map
$\check{q}:\overline{\operatorname{Spec}{\mathbb{Z}}}\rightarrow\operatorname{SCong}{\mathbb{P}}^{1}_{{\mathbb{F}}_{1}}$
that extends $\mysigma_{q}^{*}$ and such that the diagram

${\overline{\operatorname{Spec}{\mathbb{Z}}}^{\textup{Smi}}}$${{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}}$${\overline{\operatorname{Spec}{\mathbb{Z}}}}$${\operatorname{SCong}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}}$$\scriptstyle{\myvarphi_{q}}$$\scriptstyle{j}$$\scriptstyle{i}$$\scriptstyle{\check{q}}$

commutes.

### Projective line over algebraic extensions of ${\mathbb{F}}_{1}$

In [Smi93], Smirnov also introduces the projective line over certain
“algebraic extensions of the field with one element”.

Let $\mymu_{\infty}$ be the group of complex roots of unity
$\\{z\in{\mathbb{C}}\mid z^{n}=1\text{ for some }n\geq 1\\}$ and define
${\mathbb{F}}_{1^{\infty}}:=\mymu_{\infty}\cup\\{0\\}$.

The set of “geometric points” of ${\mathbb{P}}^{1}$ is

${\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}}):={\mathbb{F}}_{1^{\infty}}\sqcup\\{\infty\\}.$

The Galois group
$G:=\textup{Gal}\big{(}{\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}}\big{)}$ acts
on ${\mathbb{F}}_{1^{\infty}}$, and hence on
${\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})$, with trivial
action on $\infty$. For each subgroup $\Gamma$ of $G$, a “schematic point” in
${\mathbb{P}}^{1,\textup{Smi}}_{\Gamma}$ is an orbit of the action

$\Gamma\times{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})\longrightarrow{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}}),$

i.e., Smirnov defines
${\mathbb{P}}^{1,\textup{Smi}}_{\Gamma}:={\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})/\Gamma$.

######  Proposition  (Section 3).

There is a canonical inclusion

$i_{\infty}:{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})\rightarrow\operatorname{SCong}_{{\mathbb{F}}_{1^{\infty}}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$

whose image consists of all non-generic points of
$\operatorname{SCong}_{{\mathbb{F}}_{1^{\infty}}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$.

Given a subgroup $\Gamma$ of $G$, we define

${\mathbb{F}}_{\Gamma}:=\\{\mylambda\in{\mathbb{F}}_{1^{\infty}}\mid
h\cdot\mylambda=\mylambda\text{ for all }h\in\Gamma\\}.$

The inclusion ${\mathbb{F}}_{\Gamma}\hookrightarrow{\mathbb{F}}_{1^{\infty}}$
induces a surjective continuous map

$\Phi_{{\mathbb{F}}_{\Gamma}}:\operatorname{SCong}_{{\mathbb{F}}_{1^{\infty}}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\rightarrow\operatorname{SCong}_{{\mathbb{F}}_{\Gamma}}{\mathbb{P}}_{{\mathbb{F}}_{\Gamma}}^{1}.$

The next result, which follows from Theorem 3.1 and Section 3, shows how we
recover Smirnov’s projective lines by using strong congruence spaces.

######  Theorem C.

If $\Gamma$ is a closed subgroup of
$\textup{Gal}\big{(}{\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}}\big{)}$, then
there exists an injective map
$j_{\Gamma}:{\mathbb{P}}^{1,\textup{Smi}}_{\Gamma}\rightarrow\operatorname{SCong}_{{\mathbb{F}}_{\Gamma}}{\mathbb{P}}_{{\mathbb{F}}_{\Gamma}}^{1}$
whose image consists of all non-generic points of
$\operatorname{SCong}_{{\mathbb{F}}_{\Gamma}}{\mathbb{P}}_{{\mathbb{F}}_{\Gamma}}^{1}$
and such that the diagram

${{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})}$${{\mathbb{P}}^{1,\textup{Smi}}_{\Gamma}}$${\operatorname{SCong}_{{\mathbb{F}}_{1^{\infty}}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}}$${\operatorname{SCong}_{{\mathbb{F}}_{\Gamma}}{\mathbb{P}}_{{\mathbb{F}}_{\Gamma}}^{1}}$$\scriptstyle{i_{\infty}}$$\scriptstyle{j_{\Gamma}}$$\scriptstyle{\Phi_{{\mathbb{F}}_{\Gamma}}}$

commutes.

### Acknowledgements

The author thanks Oliver Lorscheid for several conversations and for his help
with preparing this text. The author also thanks Eduardo Vital for useful
conversations. The present work was carried out with the support of CNPq,
National Council for Scientific and Technological Development - Brazil.

##  1\. Monoid schemes

In this section we recall the theory of monoid schemes. For more details, see
[Dei05], [CC10], [CLS12] and [CHWW15].

###  1.1. Basic definitions

A pointed monoid is a set $A$ equipped with an associative binary operation

$\displaystyle A\times A$ $\displaystyle\rightarrow A$ $\displaystyle(a,b)$
$\displaystyle\mapsto a\cdot b,$

called multiplication or product, such that $A$ has an element $0$, called
absorbing element or zero, and an element $1$, called one, satisfying $0\cdot
a=a\cdot 0=0$ and $1\cdot a=a\cdot 1=a$ for all $a$ in $A$. We sometimes use
$ab$ to denote $a\cdot b$. A pointed monoid is commutative if $a\cdot b=b\cdot
a$ for all $a,b$ in $A$. Throughout this text all monoids are commutative. A
pointed monoid is integral (or cancellative) if $ab=ac$ implies $a=0$ or
$b=c$. A pointed monoid is without zero divisors if $ab=0$ implies $a=0$ or
$b=0$. Note that if a pointed monoid is integral, then it is without zero
divisors.

An ideal of $A$ is a set $I\subseteq A$ such that $0\in I$ and $ax\in I$ for
all $a\in A$ and $x\in I$. Every ideal $I$ induces an equivalence relation
$\sim$ on $A$ generated by $\\{(0,x)\in A\times A\mid x\in I\\}$. The quotient
set $A/I:=A/\sim$ is a pointed monoid with operation given by
$[a]\cdot[b]:=[ab]$, absorbing element $[0]$ and one $[1]$. The ideal $I$ is
prime if $A/I$ is without zero divisors.

The ideal generated by a subset $E\subseteq A$ is the intersection $\langle
E\rangle$ of all ideals of $A$ that contain $E$, which is the smallest ideal
containing $E$. We use $\langle a_{i}\mid i\in I\rangle$ to denote the ideal
$\langle\\{a_{i}\\}_{i\in I}\rangle$.

A morphism of pointed monoids is a multiplicative map preserving zero and one.
We use ${\mathbb{F}}_{1}$ to denote the initial object $\\{0,1\\}$ of the
category $\mathcal{M}_{0}$ of pointed monoids.

An element $u\in A$ is invertible if there exists $y\in A$ such that $uy=1$.
We denote the set of invertible elements of $A$ by $A^{\times}$. The set
$A\backslash A^{\times}$ is a prime ideal and contains every proper ideal of
$A$. A morphism of pointed monoids $f:A\rightarrow B$ is local if
$f^{-1}(B^{\times})=A^{\times}$. A pointed group is a pointed monoid $A$ such
that $A^{\times}=A\backslash\\{0\\}$.

If $f:A\rightarrow B$ is a morphism and $I$ is an ideal of $B$, then the set

$f^{*}(I):=\\{a\in A\mid f(a)\in I\\}$

is an ideal of $A$, called pullback of $I$ along $f$. If $I$ is prime, then
$f^{*}(I)$ is also prime.

A multiplicative subset of $A$ is a multiplicatively closed subset $S\subseteq
A$ that contains $1$. We define the localization $S^{-1}A$ in the same way as
in the case of rings. The group of fractions of a pointed monoid without zero
divisors $A$ is the localization
$\textup{Frac}(A):=(A\backslash\\{0\\})^{-1}A$. It is a pointed group and the
natural map $A\rightarrow\textup{Frac}(A)$ is injective.

###  1.2. The geometry of monoids

We use $\operatorname{MSpec}A$ to denote the set of prime ideals of the
pointed monoid $A$. For $h\in A$, we denote the set
$\\{I\in\operatorname{MSpec}A\mid h\notin I\\}$ by $U_{h}$. We endow
$\operatorname{MSpec}A$ with the topology generated by the basis $\\{U_{a}\mid
a\in A\\}$, and with the sheaf of pointed monoids
$\mathcal{O}_{\operatorname{MSpec}A}$ characterized by
$\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{h})=A[h^{-1}]$ for $h\in A$,
with restriction maps

$\begin{array}[]{ccc}\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{h})&\longrightarrow&\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{gh})\\\
\dfrac{a}{h}&\longmapsto&\dfrac{ga}{gh}.\end{array}$

A monoidal space is a pair $(X,\mathcal{O}_{X})$ where $X$ is a topological
space and $\mathcal{O}_{X}$ is a sheaf of pointed monoids on $X$. A morphism
of monoidal spaces $(X,\mathcal{O}_{X})\rightarrow(Y,\mathcal{O}_{Y})$ is a
pair $(\myvarphi,\myvarphi^{\\#})$ where $\myvarphi:X\rightarrow Y$ is
continuous and
$\myvarphi^{\\#}:\mathcal{O}_{Y}\rightarrow\myvarphi_{*}\mathcal{O}_{X}$ is a
morphism of sheaves of pointed monoids such that the induced map between
stalks
$\myvarphi^{\\#}_{x}:\mathcal{O}_{Y,\myvarphi(x)}\rightarrow\mathcal{O}_{X,x}$
is local for every $x\in X$.

An affine monoid scheme is a monoidal space isomorphic to
$(\operatorname{MSpec}A,\mathcal{O}_{\operatorname{MSpec}A})$ for some pointed
monoid $A$. A monoid scheme is a monoidal space that has an open covering by
affine monoid schemes.

###  1.3. Base extension to rings

Throughout this text all rings are commutative and with unit. Let $A$ be a
pointed monoid. Define $A_{\mathbb{Z}}$ as the ring ${\mathbb{Z}}[A]/\langle
1_{\mathbb{Z}}0_{A}\rangle$. The assignment $A\mapsto A_{\mathbb{Z}}$ extends
to a functor from the category of pointed monoids to rings, which is left-
adjoint to the forgetful functor $B\mapsto B^{\bullet}$ from rings to pointed
monoids. For an affine monoid scheme $U=\operatorname{MSpec}A$, we use
$U_{\mathbb{Z}}$ to denote the affine Grothendieck scheme
$\operatorname{Spec}(A_{\mathbb{Z}})$. We also define a functor $Y\mapsto
Y^{\bullet}$ from schemes to monoidal spaces by simply forgetting the additive
structure of $\mathcal{O}_{Y}$.

Let $X$ be a monoid scheme. Let $\mathcal{U}_{X}$ be the category of all
affine open subschemes of $X$, with inclusions as morphisms. We define the
base extension of $X$ to ${\mathbb{Z}}$ (or ${\mathbb{Z}}$-realization of $X$)
as the Grothendieck scheme

$X_{\mathbb{Z}}:=\underset{U\in\mathcal{U}_{X}}{\textup{colim}}\;U_{\mathbb{Z}}.$

If $\\{U_{i}\\}_{i\in I}$ is an affine covering of a monoid scheme $X$, then
$\\{U_{i,{\mathbb{Z}}}\\}_{i\in I}$ is an affine covering of $X_{\mathbb{Z}}$.
If $U,V$ are affine open subsets of $X$, then $U_{\mathbb{Z}}\cap
V_{\mathbb{Z}}=(U\cap V)_{\mathbb{Z}}$ in $X_{\mathbb{Z}}$ (cf. [CLS12, Cor.
3.3] and [CHWW15, Section 5]).

######  Proposition .

There exists a morphism of monoidal spaces
$p:X_{\mathbb{Z}}^{\bullet}\rightarrow X$, functorial in $X$, such that if
$X=\operatorname{MSpec}A$ is affine, then

$\begin{array}[]{cccc}p:&(\operatorname{Spec}A_{\mathbb{Z}})^{\bullet}&\longrightarrow&\operatorname{MSpec}A\\\
&Q&\longmapsto&Q\cap A\end{array}$

and $p^{\\#}$ is induced by $A\hookrightarrow A_{\mathbb{Z}}$.

###### Proof.

See [Jar23, Prop. 1.1]. ∎

##  2\. Strong congruence spaces

We explain the construction of strong congruence spaces introduced by the
author in [Jar23], which is based on the congruence space of Lorscheid and Ray
(see [LR23]).

Let $A$ be a pointed monoid. A congruence on $A$ is an equivalence relation
$\mathfrak{c}\subseteq A\times A$ such that $(ab,ac)\in\mathfrak{c}$ whenever
$(b,c)\in\mathfrak{c}$ and $a\in A$. The null ideal of $\mathfrak{c}$ is the
set $I_{\mathfrak{c}}:=\\{a\in A\mid(a,0)\in\mathfrak{c}\\}$.

The congruence generated by a subset $E\subseteq A\times A$ is the
intersection $\langle E\rangle$ of all congruences on $A$ that contain $E$,
which is the smallest congruence containing $E$. The trivial congruence is
$\langle\emptyset\rangle=\\{(a,b)\in A\times A\mid a=b\\}$.

The quotient set $A/\mathfrak{c}$ is a pointed monoid with product
$[a]\cdot[b]:=[ab]$, absorbing element $[0]$ and one $[1]$. The congruence
$\mathfrak{c}$ is prime if $A/\mathfrak{c}$ is integral. If $\mathfrak{c}$ is
prime, then $I_{\mathfrak{c}}$ is a prime ideal.

Let $f:A\rightarrow B$ be a morphism of pointed monoids and $\mathfrak{d}$ a
congruence on $B$. The pullback of $\mathfrak{d}$ along $f$ is the set

$f^{*}(\mathfrak{d}):=\\{(x,y)\in A\times A\mid(f(x),f(y))\in\mathfrak{d}\\},$

which is a congruence on $A$. The congruence kernel of $f$ is
$\textup{congker}(f):=f^{*}(\mathfrak{d}_{\textup{triv}})$, where
$\mathfrak{d}_{\textup{triv}}$ is the trivial congruence on $B$.

######  Notation.

_If $\mathfrak{c}$ is a congruence, we sometimes use $a\sim_{\mathfrak{c}}b$
or $a\sim b$ instead of $(a,b)\in\mathfrak{c}$. We also use $\langle a_{i}\sim
b_{i}\mid i\in I\rangle$ to denote the congruence
$\langle\\{(a_{i},b_{i})\\}_{i\in I}\rangle$._

######  Definition .

A domain is a pointed monoid $A$ that is isomorphic to a pointed submonoid of
$K^{\bullet}$ for some field $K$. A strong prime congruence on $A$ is a
congruence $\mathfrak{c}$ such that $A/\mathfrak{c}$ is a domain. We denote
the set of strong prime congruences on A by $\operatorname{SCong}A$.

######  Proposition .

Let $A$ be a pointed monoid. Then the following are equivalent:

  1. (i)

$A$ is a domain;

  2. (ii)

$A$ is isomorphic to a pointed submonoid of $K^{\bullet}$ for some field $K$
of characteristic zero;

  3. (iii)

$A$ is integral and $\\#\\{x\in A|\,x^{n}=\myalpha\\}\leq n$ for every
$\myalpha$ in $A$ and $n\geq 1$.

###### Proof.

It follows from [Jar23, Prop. 4.5 and Cor. 4.6]. ∎

Let $k$ be a pointed monoid and $A$ a $k$-algebra. A $k$-congruence on $A$ is
a strong prime congruence $\mathfrak{c}$ such that the natural map
$k\rightarrow A/\mathfrak{c}$ is injective. The set of $k$-congruences of $A$,
endowed with the topology generated by the open sets

$U_{a,b}:=\\{\mathfrak{c}\mid(a,b)\notin\mathfrak{c}\\},$

is denoted by $\operatorname{SCong}_{k}A$. Note that
$\operatorname{SCong}A=\operatorname{SCong}_{{\mathbb{F}}_{1}}A$, and
$\operatorname{SCong}_{k}A=\emptyset$ if $k$ is not a domain or if the map
$k\rightarrow A$ is not injective.

There exists a functor $\operatorname{SCong}_{k}(-)$ from $k$-schemes to
topological spaces such that:

  1. (i)

If $X=\operatorname{MSpec}A$ is affine, then
$\operatorname{SCong}_{k}X=\operatorname{SCong}_{k}A$;

  2. (ii)

If $\myvarphi:\operatorname{MSpec}B\rightarrow\operatorname{MSpec}A$ is a
morphism of affine $k$-schemes, then
$\operatorname{SCong}_{k}\myvarphi=\big{(}\myvarphi^{\\#}(\operatorname{MSpec}A)\big{)}^{*}:\operatorname{SCong}_{k}B\rightarrow\operatorname{SCong}_{k}A$;

  3. (iii)

If $\\{U_{i}\\}_{i\in I}$ is an open covering of $X$, then
$\\{\operatorname{SCong}_{k}U_{i}\\}_{i\in I}$ is an open covering of
$\operatorname{SCong}_{k}X$.

The space $\operatorname{SCong}_{k}X$ comes with a continuous map
$\mypi_{X}:\operatorname{SCong}_{k}X\rightarrow X$, defined as follows: given
$\tilde{x}\in\operatorname{SCong}_{k}X$, the image of $\tilde{x}$ by
$\mypi_{X}$ is $\mypi_{X}(\tilde{x})=I_{\mathfrak{c}}$ if $U\subseteq X$ is an
open affine and $\mathfrak{c}$ is a $k$-congruence on $\Gamma U$ such that

$\tilde{x}=\mathfrak{c}\in\operatorname{SCong}_{k}\Gamma
U\subseteq\operatorname{SCong}_{k}X.$

We define the pointed group $\mykappa(\tilde{x}):=\textup{Frac}(\Gamma
U/\mathfrak{c})$, which does not depend on the choice of $U$.

A morphism of $k$-schemes $\myvarphi:X\rightarrow Y$ induces an injective
morphism of $k$-algebras

$\mykappa\big{(}\operatorname{SCong}_{k}(\myvarphi)(\tilde{x})\big{)}\hookrightarrow\mykappa(\tilde{x})$

for each $\tilde{x}\in\operatorname{SCong}_{k}X$.

###  2.1. The map $\mygamma:X_{\mathbb{Z}}\rightarrow\operatorname{SCong}X$

Let $A$ be a pointed monoid. The canonical inclusion $i:A\rightarrow
A_{\mathbb{Z}}^{\bullet}$ induces a continuous map
$i^{*}:\operatorname{Spec}(A_{\mathbb{Z}})\rightarrow\operatorname{SCong}A$
that sends a prime ideal $\mathfrak{p}$ to the strong prime congruence

$i^{*}(\mathfrak{p}):=\operatorname{congker}(\mypi_{\mathfrak{p}}\circ
i)=\\{(a,b)\in A\times A\mid\overline{1.a}=\overline{1.b}\text{ in
}A_{\mathbb{Z}}/\mathfrak{p}\\},$

where $\mypi_{\mathfrak{p}}:A_{\mathbb{Z}}\rightarrow
A_{\mathbb{Z}}/\mathfrak{p}$ is the natural projection.

######  Theorem 2.1.

Let $X$ be a monoid scheme. The underlying continuous map of the morphism of
monoidal spaces $p:X_{\mathbb{Z}}^{\bullet}\rightarrow X$ factorizes through a
continuous map

$\mygamma:X_{\mathbb{Z}}\rightarrow\operatorname{SCong}X$

characterized by the property that if $U$ is an open affine of $X$, then
$\mygamma(U_{\mathbb{Z}})\subseteq\operatorname{SCong}U$ and
$\mygamma|_{U_{\mathbb{Z}}}:U_{\mathbb{Z}}\rightarrow\operatorname{SCong}U$ is
induced by the inclusion $\Gamma U\rightarrow(\Gamma
U_{\mathbb{Z}})^{\bullet}$.

The map $\mygamma$ induces an injective morphism of pointed groups
$\mygamma_{x}:\mykappa(\mygamma(x))\rightarrow\mykappa(x)^{\bullet}$ for every
$x\in X_{\mathbb{Z}}$.

###### Proof.

See [Jar23, Thm. 4.1 and Prop. 4.2]. ∎

##  3\. The projective line over algebraic extensions of ${\mathbb{F}}_{1}$

Fix a pointed submonoid of ${\mathbb{F}}\subseteq{\mathbb{F}}_{1^{\infty}}$
and note that ${\mathbb{F}}$ is a pointed group. The projective line over
${\mathbb{F}}$ is the monoid scheme

${\mathbb{P}}_{\mathbb{F}}^{1}=D_{+}(Y)\cup D_{+}(X),$

where $D_{+}(Y)=\operatorname{MSpec}{\mathbb{F}}[X/Y]$ and
$D_{+}(X)=\operatorname{MSpec}{\mathbb{F}}[Y/X]$, with identifications

${\operatorname{MSpec}{\mathbb{F}}[X/Y]}$${\supset}$${U_{X/Y}}$${=}$${D_{+}(Y)\cap
D_{+}(X)}$${=}$${U_{Y/X}}$${\subset}$${\operatorname{MSpec}{\mathbb{F}}[Y/X]}$${\operatorname{MSpec}{\mathbb{F}}[(X/Y)^{\pm}].}$

$\scriptstyle\sim$

$\scriptstyle\sim$

######  Remark .

By [Jar23, Ex. 5.6], ${\mathbb{F}}$ is an algebraic extension of
${\mathbb{F}}_{1}$ in the sense of [Jar23, Def. 5.1].

######  Lemma .

Let $\mylambda\in{\mathbb{F}}^{\times}$ and $n\geq 1$ such that there is no
divisor $d>1$ of $n$ and $\mytheta\in{\mathbb{F}}$ satisfying
$\mytheta^{d}=\mylambda$. Then $\langle T^{n}\sim\mylambda\rangle$ is an
${\mathbb{F}}$-congruence of ${\mathbb{F}}[T]$.

###### Proof.

Let $\myomega\in{\mathbb{F}}_{1^{\infty}}$ such that $\myomega^{n}=\mylambda$.
The map of ${\mathbb{F}}$-algebras

$\begin{array}[]{cccc}f:&{\mathbb{F}}[T]&\rightarrow&{\mathbb{F}}_{1^{\infty}}\\\
&T&\mapsto&\myomega\end{array}$

satisfies $\langle
T^{n}\sim\mylambda\rangle\subseteq\textup{congker}(f):=\mathfrak{c}$. Let
$\myalpha\neq\mybeta$ be in ${\mathbb{F}}[T]\backslash\\{0\\}$ such that
$\myalpha\sim_{\mathfrak{c}}\mybeta$. Thus there exists two integers $0\leq
n<m$ and an element $a\in{\mathbb{F}}^{\times}$ such that
$T^{m}\sim_{\mathfrak{c}}aT^{n}$. As $\mathfrak{c}$ is prime, one has
$T^{m-n}\sim_{\mathfrak{c}}a$.

Let $\ell:=\textup{min}\big{\\{}w\geq 1\mid T^{w}\sim_{\mathfrak{c}}b\text{
for some }b\in{\mathbb{F}}^{\times}\big{\\}}$ and
$\myrho\in{\mathbb{F}}^{\times}$ such that
$T^{\ell}\sim_{\mathfrak{c}}\myrho$. Note that $\ell$ divides $n$ and
$\mathfrak{c}=\langle T^{\ell}\sim\myrho\rangle$. As
$\myrho^{n/\ell}\sim_{\mathfrak{c}}T^{n}\sim_{\mathfrak{c}}\mylambda$ and
$\mathfrak{c}$ is an ${\mathbb{F}}$-congruence, one has
$\myrho^{n/\ell}=\mylambda$, which implies $\ell=n$ and $\mathfrak{c}=\langle
T^{n}\sim\mylambda\rangle$. ∎

######  Proposition .

The set $\operatorname{SCong}_{\mathbb{F}}{\mathbb{P}}_{\mathbb{F}}^{1}$
consists of the following points:

  * •

the generic point, which corresponds to the trivial congruence;

  * •

the point zero $\langle X/Y\sim 0\rangle\in D_{+}(Y)$;

  * •

the point at infinity $\langle 0\sim Y/X\rangle\in D_{+}(X)$;

  * •

congruences $\langle(X/Y)^{n}\sim\mylambda\rangle\in D_{+}(Y)\cap D_{+}(X)$
with $\mylambda\in{\mathbb{F}}^{\times}$ and $n\geq 1$ such that there is no
divisor $d>1$ of $n$ and $\mytheta\in{\mathbb{F}}$ satisfying
$\mytheta^{d}=\mylambda$.

###### Proof.

One knows that
$\operatorname{SCong}_{\mathbb{F}}{\mathbb{P}}_{\mathbb{F}}^{1}\backslash\\{\langle
0\sim
Y/X\rangle\\}=\operatorname{SCong}_{\mathbb{F}}D_{+}(Y)\simeq\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[T]$.
Let
$\mathfrak{c}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[T]\backslash\\{=,\langle
T\sim 0\rangle\\}$. Analogous to what is in the proof of Section 3, there
exists a non-zero $\mylambda\in{\mathbb{F}}$ and $\ell=\textup{min}\\{w\geq
1\mid T^{w}\sim_{\mathfrak{c}}b\text{ for some }b\in{\mathbb{F}}^{\times}\\}$
such that $\mathfrak{c}=\langle T^{\ell}\sim\mylambda\rangle$.

Let $h:=\textup{ord}(\mylambda)$. If there exists a divisor $d>1$ of $\ell$,
and $\mytheta\in{\mathbb{F}}$ such that $\mytheta^{d}=\mylambda$, let
$\myzeta:=\mytheta^{h}$. Note that $\textup{ord}(\myzeta)=d$ and

$\\{\myzeta^{i}.\mytheta\mid 0\leq i\leq
d-1\\}\cup\\{\myzeta^{j}.\overline{T}^{\ell/d}\mid 0\leq j\leq d-1\\}$

is a subset of $\\{x\in{\mathbb{F}}[T]/\mathfrak{c}\mid x^{d}=\mylambda\\}$
with $2d$ elements. As ${\mathbb{F}}[T]/\mathfrak{c}$ is a domain, one has a
contradiction.

Therefore, the result follows from Section 3. ∎

######  Notation.

_Let
$\widetilde{{\mathbb{P}}}_{\mathbb{F}}^{1}:=\operatorname{SCong}_{\mathbb{F}}{\mathbb{P}}_{{\mathbb{F}}}^{1}$.
We introduce the following notation for the non-generic points of
$\widetilde{{\mathbb{P}}}_{\mathbb{F}}^{1}$:_

  * •

_$[n,\mylambda]:=\langle(X/Y)^{n}\sim\mylambda\rangle\in D_{+}(Y)\cap
D_{+}(X)$ for $n\geq 1$ and $\mylambda\in{\mathbb{F}}^{\times}$ such that
there is no divisor $d>1$ of $n$ and $\mytheta\in{\mathbb{F}}$ satisfying
$\mytheta^{d}=\mylambda$_;

  * •

_$[0]:=\langle X/Y\sim 0\rangle\in D_{+}(Y)$_ ;

  * •

_$[\infty]:=\langle 0\sim Y/X\rangle\in D_{+}(X)$_.

_Note that if
$[n,\mylambda]\in\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$,
then $n=1$. In this case we use $[\mylambda]$ instead of $[1,\mylambda]$. As
${\mathbb{F}}_{1}^{\times}=\\{1\\}$, if
$[n,\mylambda]\in\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$, then
$\mylambda=1$. In this case we use $[n]$ instead of $[n,1]$._

######  Corollary .

The map
$i:{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1,\textup{Smi}}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$,
given by $i([n])=[n]$, is an injection whose image is the set of non-generic
points of $\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$.

######  Corollary .

The map
$i_{\infty}:{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$,
given by $i_{\infty}(\mylambda)=[\mylambda]$, is an injection whose image is
the set of non-generic points of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$.

The inclusion ${\mathbb{F}}\hookrightarrow{\mathbb{F}}_{1^{\infty}}$ induces a
morphism
${\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\rightarrow{\mathbb{P}}_{{\mathbb{F}}}^{1}$
and, consequently, a continuous map
$\operatorname{SCong}_{\mathbb{F}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}}^{1}$.
As
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\subseteq\operatorname{SCong}_{\mathbb{F}}{\mathbb{P}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$,
by restriction one has

$\Phi_{\mathbb{F}}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}}^{1}.$

For $\myzeta$ in ${\mathbb{F}}_{1^{\infty}}$, let
$\textup{ord}_{\mathbb{F}}(\myzeta):=\textup{min}\\{w\geq
1\mid\myzeta^{w}\in{\mathbb{F}}\\}$. Note that $\Phi_{\mathbb{F}}$ is
surjective and

$\Phi^{-1}_{\mathbb{F}}([n,\mylambda])=\\{[\myzeta]\in\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\mid\textup{ord}_{\mathbb{F}}(\myzeta)=n\text{
and }\myzeta^{n}=\mylambda\\}.$

Analogously, the inclusion ${\mathbb{F}}_{1}\hookrightarrow{\mathbb{F}}$
induces a surjective continuous map

$\Psi_{\mathbb{F}}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}}^{1}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}.$

such that
$\Psi_{\mathbb{F}}^{-1}([n])=\\{[m,\mylambda]\in\widetilde{{\mathbb{P}}}_{{\mathbb{F}}}^{1}\mid\textup{ord}(\mylambda)=n/m\\}$.

######  Remark .

The map
$\Phi_{{\mathbb{F}}_{1}}=\Psi_{{\mathbb{F}}_{1^{\infty}}}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$
satisfies $\\#\Phi_{{\mathbb{F}}_{1}}^{-1}([n])=\myphi(n)$, where $\myphi$ is
the Euler function.

######  Remark .

Note that the map
$\Psi_{{\mathbb{F}}_{1^{2}}}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{2}}}^{1}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$
is a homeomorphism.

Let $G$ be the profinite group
$\textup{Gal}({\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}})$. The map
$\myvarphi\mapsto\myvarphi|_{{\mathbb{F}}_{1^{\infty}}}$ is an isomorphism
$G\overset{\sim}{\rightarrow}\textup{Aut}_{\mathcal{M}_{0}}({\mathbb{F}}_{1^{\infty}})$
(cf. [Neu99, Ex. IV.2.6]). The group $G$ acts on
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$ by

$\begin{array}[]{ccl}G\times\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}&\longrightarrow&\quad\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}\\\
\vspace{1mm}\hfil&&\\\
(g,Q)&\longmapsto&\smash{\left\\{\begin{array}[]{ll}[g\cdot\mylambda]&\text{if
}Q=[\mylambda]\text{ for some }\mylambda\in{\mathbb{F}}_{1^{\infty}};\\\
{}[\infty]&\text{if }Q=[\infty];\\\ {}Q&\text{if }Q\text{ is the generic
point.}\end{array}\right.}\end{array}$

Given a subgroup $\Gamma$ of $G$, we define the pointed subgroup of
${\mathbb{F}}_{1^{\infty}}$

${\mathbb{F}}_{\Gamma}:=\\{\mylambda\in{\mathbb{F}}_{1^{\infty}}\mid
g\cdot\mylambda=\mylambda\text{ for all }g\in\Gamma\\}.$

######  Theorem 3.1.

If $\Gamma$ is a closed subgroup of
$\textup{Gal}({\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}})$, then
$\Phi_{{\mathbb{F}}_{\Gamma}}$ induces a continuous bijection
$b_{\Gamma}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}$.

###### Proof.

Note that
$\Phi_{{\mathbb{F}}_{\Gamma}}([\myzeta])=\Phi_{{\mathbb{F}}_{\Gamma}}(g\cdot[\myzeta])$
for all $\myzeta\in({\mathbb{F}}_{1^{\infty}})^{\times}$ and $g\in\Gamma$,
thus there exists a continuous map
$b_{\Gamma}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}$
such that

${\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}}$${\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma}$${\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}}$$\scriptstyle{\Phi_{{\mathbb{F}}_{\Gamma}}}$$\scriptstyle{b_{\Gamma}}$

commutes. As $\Phi_{{\mathbb{F}}_{\Gamma}}$ is surjective, it only remains to
show that $b_{\Gamma}$ is injective.

Let $\myzeta,\myxi\in{\mathbb{F}}^{\times}$ such that
$\Phi_{{\mathbb{F}}_{\Gamma}}([\myzeta])=\Phi_{{\mathbb{F}}_{\Gamma}}([\myxi])$.
Then
$n:=\textup{ord}_{{\mathbb{F}}_{\Gamma}}(\myzeta)=\textup{ord}_{{\mathbb{F}}_{\Gamma}}(\myxi)$
and $\mylambda:=\myzeta^{n}=\myxi^{n}$. Note that
${\mathbb{Q}}({{\mathbb{F}}_{\Gamma}},\myzeta)={\mathbb{Q}}({{\mathbb{F}}_{\Gamma}},\myxi)$
and there exists
$\myvarphi\in\textup{Gal}({\mathbb{Q}}({{\mathbb{F}}_{\Gamma}},\myzeta)/{\mathbb{Q}}({{\mathbb{F}}_{\Gamma}}))$
such that $\myvarphi(\myzeta)=\myxi$. Thus there exists
$\mypsi\in\textup{Gal}({\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}}({{\mathbb{F}}_{\Gamma}}))$
such that $\mypsi|_{{\mathbb{Q}}({{\mathbb{F}}_{\Gamma}},\myzeta)}=\myvarphi$.
As $\Gamma$ is closed,
$\Gamma=\textup{Gal}({\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}}({{\mathbb{F}}_{\Gamma}}))$.
Therefore $b_{\Gamma}$ is injective. ∎

######  Remark .

For $G=\textup{Gal}({\mathbb{Q}}(\mymu_{\infty})/{\mathbb{Q}})$, the set of
non-generic points of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/G$ has the cofinite
topology, thus it is a $T_{1}$ space. Note that
${\mathbb{F}}_{G}={\mathbb{F}}_{1^{2}}$ and the set of non-generic points of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{2}}}^{1}$ is not a $T_{1}$ space,
because the point $[9,1]$ is in the closure of $[3,1]$.

We conclude that the map
$b_{\Gamma}:\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}$
is not a homeomorphism in general.

######  Remark .

Note that
$i_{\infty}:{\mathbb{P}}^{1,\textup{Smi}}({\mathbb{F}}_{1^{\infty}})\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}$
is a morphism of $G$-sets. Thus, given a subgroup $\Gamma$ of $G$, one has an
injective map
$p_{\Gamma}:{\mathbb{P}}_{\Gamma}^{1,\textup{Smi}}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma$,
whose image consists of all non-generic points of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1^{\infty}}}^{1}/\Gamma$.

If $\Gamma$ is closed, by Theorem 3.1, the map $j_{\Gamma}:=b_{\Gamma}\circ
p_{\Gamma}:{\mathbb{P}}_{\Gamma}^{1,\textup{Smi}}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}$
is an injection whose image consists of all non-generic points of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{\Gamma}}^{1}$.

##  4\. The compactification of $\operatorname{Spec}\mathcal{O}_{K}$

Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers. We use
$|-|_{0}$ to denote the trivial absolute value on $K$ and $[0]$ for the
trivial place.

As $\mathcal{O}_{K}$ is a Dedekind domain, every non-zero fractional ideal
$I\subseteq K$ has a unique decomposition

$I=\prod P^{e_{P}},$

where $P\in(\operatorname{Spec}\mathcal{O}_{K})\backslash\\{0\\}$ and
$e_{P}\in{\mathbb{Z}}$, with $e_{P}\neq 0$ for only finitely many primes $P$.
Given a constant $c>1$, each $P\in\operatorname{Spec}\mathcal{O}_{K}$ induces
a non-archimedean absolute value

$\begin{array}[]{cccl}|-|_{P}:&K&\longrightarrow&\quad{\mathbb{R}}_{\geq 0}\\\
\\\ &a&\longmapsto&\smash{\left\\{\begin{array}[]{ll}c^{-e_{P}}&\text{if
}\;a\neq 0\;\text{ and }\;a\mathcal{O}_{K}=\prod Q^{e_{Q}}\\\ {}0&\text{if
}\;a=0.\end{array}\right.}\end{array}$

We use $[P]$ to denote the equivalence class of $|-|_{P}$. Every non-trivial
non-archimedean absolute value on $K$ is equivalent to $|-|_{P}$ for some
prime $P$.

Given a (possibly archimedean) place $\mathfrak{a}$, one has the associated
pointed monoid $\mathcal{O}_{\mathfrak{a}}:=\big{\\{}a\in
K^{\bullet}\;\big{|}\;|a|\leq 1\big{\\}}$, whose maximal ideal is
$m_{\mathfrak{a}}:=\big{\\{}a\in K^{\bullet}\;\big{|}\;|a|<1\big{\\}}$, and
the pointed group
$\mykappa(\mathfrak{a}):=\mathcal{O}_{\mathfrak{a}}/m_{\mathfrak{a}}$. We
define the compactification of $\operatorname{Spec}\mathcal{O}_{K}$ as the set

$\overline{\operatorname{Spec}\mathcal{O}_{K}}:=\\{\text{places of $K$}\\}$

endowed with the topology where $[0]$ is the generic point and
$\overline{\operatorname{Spec}\mathcal{O}_{K}}\backslash\\{[0]\\}$ has the
cofinite topology. As $K$ has only a finite number of archimedean places, the
subset

$(\overline{\operatorname{Spec}\mathcal{O}_{K}})_{\textup{nA}}:=\\{\text{non-
archimedean places of }K\\}$

is open. Note that for any $f\in K$, the subset

$U_{f}:=\\{\mathfrak{a}\in\overline{\operatorname{Spec}\mathcal{O}_{K}}\mid
f\notin m_{\mathfrak{a}}\\}$

is open. We define the structure sheaf of
$\overline{\operatorname{Spec}\mathcal{O}_{K}}$ by

$\mathcal{O}_{\overline{\operatorname{Spec}\mathcal{O}_{K}}}(U):=\\{\mylambda\in
K^{\bullet}\mid\mylambda\in\mathcal{O}_{\mathfrak{a}}\text{ for all
}\mathfrak{a}\in U\\}$

for an open subset $U\subseteq\overline{\operatorname{Spec}\mathcal{O}_{K}}$.
Note that $\overline{\operatorname{Spec}\mathcal{O}_{K}}$ is a monoidal space
and the map $P\mapsto[P]$ induces an isomorphism
$(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}\simeq(\overline{\operatorname{Spec}\mathcal{O}_{K}})_{\textup{nA}}$.

##  5\. Maps from $\overline{\operatorname{Spec}\mathcal{O}_{K}}$ to
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$ induced by elements of $K$

There is a map
$K^{*}\rightarrow\textup{Hom}_{\textup{Sch}}(\operatorname{Spec}\mathcal{O}_{K},{\mathbb{P}}_{\mathbb{Z}}^{1})$
that sends a non-zero $f$ to the morphism
$\myvarphi_{f}:\operatorname{Spec}\mathcal{O}_{K}\rightarrow{\mathbb{P}}_{\mathbb{Z}}^{1}$
induced by the pair

$\begin{array}[]{ccccccccc}\myvarphi_{1}:&{\mathbb{Z}}[X/Y]&\longrightarrow&\mathcal{O}_{K}[f]&\quad\text{and}&\myvarphi_{2}:&{\mathbb{Z}}[Y/X]&\longrightarrow&\mathcal{O}_{K}[1/f]\\\
&X/Y&\longmapsto&f&&&Y/X&\longmapsto&1/f.\end{array}$

By Theorem 2.1, every $f\in K^{*}$ defines a continuous map

$\widetilde{f}=\mygamma\circ\myvarphi_{f}^{\bullet}:(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1},$

which induces an injective morphism of pointed groups
$\widetilde{f}_{P}:\mykappa(\widetilde{f}(P))\rightarrow\mykappa(P)^{\bullet}$
for every $P\in\operatorname{Spec}\mathcal{O}_{K}$.

If $P\in\operatorname{Spec}(\mathcal{O}_{K}[f])$ is non-zero, then
$\widetilde{f}(P)=\textup{congker}(\mypi_{P}^{\bullet}\circ\myvarphi_{1}^{\bullet}\circ\myiota)$,
where

${\mathbb{F}}_{1}[X/Y]\overset{\myiota}{\longrightarrow}{\mathbb{Z}}[X/Y]^{\bullet}\overset{\myvarphi_{1}^{\bullet}}{\longrightarrow}\mathcal{O}_{K}[f]^{\bullet}\overset{\mypi_{P}^{\bullet}}{\longrightarrow}(\mathcal{O}_{K}[f]/P)^{\bullet},$

i.e., in $\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$ one has

$\widetilde{f}(P)=\smash{\left\\{\begin{array}[]{ll}[0]&\text{if }f\in P\\\
{}[n]&\text{if }f\notin P\text{ and $n$ is the order of $\overline{f}$ in
$(\mathcal{O}_{K}[f]/P)^{\times}$.}\end{array}\right.}$

The map
$\widetilde{f}_{P}:\mykappa(\widetilde{f}(P))\hookrightarrow\mykappa(P)^{\bullet}$
is the inclusion ${\mathbb{F}}_{1}\hookrightarrow\mykappa(P)^{\bullet}$ if
$\widetilde{f}(P)=[0]$, and the inclusion

$\begin{array}[]{ccc}{\mathbb{F}}_{1}[T]/\langle T^{n}\sim
1\rangle&\longhookrightarrow&\mykappa(P)^{\bullet}\\\
\overline{T}&\longmapsto&\overline{f}\end{array}$

if $\widetilde{f}(P)=[n]$, where $T:=X/Y$.

Analogously, if $P$ is a non-zero prime ideal of $\mathcal{O}_{K}[1/f]$, then

$\widetilde{f}(P)=\smash{\left\\{\begin{array}[]{ll}[\infty]&\text{if }1/f\in
P\\\ {}[\textup{ord}(\overline{1/f})]&\text{if }1/f\notin
P.\end{array}\right.}$

The map
$\widetilde{f}_{P}:\mykappa(\widetilde{f}(P))\hookrightarrow\mykappa(P)^{\bullet}$
is the inclusion ${\mathbb{F}}_{1}\hookrightarrow\mykappa(P)^{\bullet}$ if
$\widetilde{f}(P)=[\infty]$, and the inclusion

$\begin{array}[]{ccc}{\mathbb{F}}_{1}[T^{-1}]/\langle T^{-m}\sim
1\rangle&\longhookrightarrow&\mykappa(P)^{\bullet}\\\
\overline{(T^{-1})}&\longmapsto&\overline{1/f}\end{array}$

if $\widetilde{f}(P)\neq[\infty]$.

If $P$ is the generic point of $\operatorname{Spec}\mathcal{O}_{K}$, then
$\widetilde{f}(P)$ is the generic point of
$\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$ and $\widetilde{f}_{P}$ is
the map

$\begin{array}[]{ccc}{\mathbb{F}}_{1}[T^{\pm}]&\longrightarrow&K^{\bullet}\\\
T&\longmapsto&f.\end{array}$

######  Definition  ([Smi93]).

An element $\mylambda\in K$ is called exceptional number if there exists an
archimedean absolute value $|-|$ such that $|\mylambda|=1$.

######  Theorem 5.1.

If $f\in K^{*}$ is not exceptional, then there exists a morphism of monoidal
spaces
$\mypsi:\overline{\operatorname{Spec}\mathcal{O}_{K}}\rightarrow{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$
that extends

$(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}\overset{\myvarphi_{f}^{\bullet}}{\longrightarrow}({\mathbb{P}}_{\mathbb{Z}}^{1})^{\bullet}\overset{p}{\longrightarrow}{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1},$

and a continuous map
$\check{f}:\overline{\operatorname{Spec}\mathcal{O}_{K}}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$
that factorizes $\mypsi$ and extends

$\widetilde{f}:(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1},$

as summarized in the following commutative diagram

${\overline{\operatorname{Spec}\mathcal{O}_{K}}}$${\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}}$${(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}}$${{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}.}$$\scriptstyle{\check{f}}$$\scriptstyle{\mypi_{{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}}}$$\scriptstyle{p\circ\myvarphi_{f}^{\bullet}}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\mypsi}$

The map $\check{f}$ induces an injective morphism
$\check{f}_{\mathfrak{a}}:\mykappa(\check{f}(\mathfrak{a}))\hookrightarrow\mykappa(\mathfrak{a})$
for each $\mathfrak{a}\in\overline{\operatorname{Spec}\mathcal{O}_{K}}$, such
that $\check{f}_{[P]}=\widetilde{f}_{P}$ for all
$P\in\operatorname{Spec}\mathcal{O}_{K}$.

###### Proof.

If $\mathfrak{a}$ is an archimedean place, as $f$ is non-exceptional, $f\in
m_{\mathfrak{a}}$ or $1/f\in m_{\mathfrak{a}}$. In this case, define

$\mypsi(\mathfrak{a}):=\smash{\left\\{\begin{array}[]{ll}\langle X/Y\rangle\in
D_{+}(Y)&\text{if }f\in m_{\mathfrak{a}}\\\ {}\langle Y/X\rangle\in
D_{+}(X)&\text{if }1/f\in m_{\mathfrak{a}}.\end{array}\right.}$

For $Q\in(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}$, define
$\mypsi([Q]):=p\circ\myvarphi_{f}^{\bullet}(Q)$. Note that
$\mypsi:\overline{\operatorname{Spec}\mathcal{O}_{K}}\rightarrow{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$
is a continuous map.

Let $T:=X/Y$ and define

$\begin{array}[]{cccc}f^{\\#}:&{\mathbb{F}}_{1}[T^{\pm}]&\longrightarrow&K^{\bullet}\\\
&T&\longmapsto&f.\end{array}$

For $U\subseteq{\mathbb{P}}^{1}_{{\mathbb{F}}_{1}}$ open, note that
$f^{\\#}\big{(}\Gamma(\mathcal{O}_{{\mathbb{P}}^{1}_{{\mathbb{F}}_{1}}},U)\big{)}\subseteq\Gamma\big{(}\mathcal{O}_{\overline{\operatorname{Spec}\mathcal{O}_{K}}},\mypsi^{-1}(U)\big{)}$
and define

$\begin{array}[]{cccc}\mypsi^{\\#}(U):&\Gamma(\mathcal{O}_{{\mathbb{P}}^{1}_{{\mathbb{F}}_{1}}},U)&\longrightarrow&\Gamma\big{(}\mathcal{O}_{\overline{\operatorname{Spec}\mathcal{O}_{K}}},\mypsi^{-1}(U)\big{)}\\\
&z&\longmapsto&f^{\\#}(z).\end{array}$

Note that $(\mypsi,\mypsi^{\\#})$ is a morphism of monoidal spaces
$\overline{\operatorname{Spec}\mathcal{O}_{K}}\rightarrow{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$
that extends
$p\circ\myvarphi_{f}^{\bullet}:(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}\rightarrow{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$.

Next we construct $\check{f}$. If $\mathfrak{a}$ is an archimedean place,
define

$\check{f}(\mathfrak{a}):=\smash{\left\\{\begin{array}[]{ll}[0]&\text{if }f\in
m_{\mathfrak{a}}\\\ {}[\infty]&\text{if }1/f\in
m_{\mathfrak{a}}.\end{array}\right.}$

For $P\in(\operatorname{Spec}\mathcal{O}_{K})^{\bullet}$, define
$\check{f}([P]):=\widetilde{f}(P)$. Note that
$\check{f}:\overline{\operatorname{Spec}\mathcal{O}_{K}}\rightarrow\widetilde{{\mathbb{P}}}_{{\mathbb{F}}_{1}}^{1}$
is continuous.

For $\mathfrak{a}\in\overline{\operatorname{Spec}\mathcal{O}_{K}}$, we define
the injective morphism
$\check{f}_{\mathfrak{a}}:\mykappa(\check{f}(\mathfrak{a}))\hookrightarrow\mykappa(\mathfrak{a})$
as follows: if $\mathfrak{a}=[P]$ for some
$P\in\operatorname{Spec}\mathcal{O}_{K}$, define
$\check{f}_{\mathfrak{a}}:=\widetilde{f}_{P}$. If $\mathfrak{a}$ is
archimedean, then $\mykappa(\check{f}(\mathfrak{a}))\simeq{\mathbb{F}}_{1}$.
In this case, define $\check{f}_{\mathfrak{a}}$ as the inclusion
${\mathbb{F}}_{1}\hookrightarrow\mykappa(\mathfrak{a})$. ∎

######  Remark .

In Theorem 5.1, if $\check{f}(\mathfrak{a})\in\operatorname{SCong}D_{+}(Y)$,
the diagram

${{\mathbb{F}}_{1}[T]}$${\mathcal{O}_{\mathfrak{a}}}$${\mykappa(\check{f}(\mathfrak{a}))}$${\mykappa(\mathfrak{a})}$$\scriptstyle{f^{\\#}}$$\scriptstyle{\check{f}_{\mathfrak{a}}}$

commutes, and if $\check{f}(\mathfrak{a})\in\operatorname{SCong}D_{+}(X)$,

${{\mathbb{F}}_{1}[T^{-1}]}$${\mathcal{O}_{\mathfrak{a}}}$${\mykappa(\check{f}(\mathfrak{a}))}$${\mykappa(\mathfrak{a}).}$$\scriptstyle{f^{\\#}}$$\scriptstyle{\check{f}_{\mathfrak{a}}}$

commutes. Thus the maps $\check{f}_{\mathfrak{a}}$ are induced by $f^{\\#}$.

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