######  Remark 7.

By allowing that $f$ takes value $0$, we allow “singular” metrics.

###  4 Heights on $\mathscr{P}(\mathbf{a})(F)$

In this section we will define heights on $\mathscr{P}(\mathbf{a})(F)$. An
obvious approach is to define heights to be pullbacks of heights for some
morphism to a scheme. However, such heights exhibit a drawback, they do not
satisfy the Northcott property and hence are not suitable for countings. For
the purpose of counting, we define quasi-toric heights, for which in 6 we
establish that they satisfy the Northcott property.

#### 1

In this paragraph we define heights on stacks. We give a condition that will
enable us to define heights.

######  Definition 1.

Let $k\in\mathbb{Z}$. For $v\in M_{F}$, let
$f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0}$ be an $\mathbf{a}$-homogenous function of weighted degree $d$ and let us
set

$E_{v}\mathrel{\mathop{\ordinarycolon}}=\\{\mathbf{x}\in
F_{v}^{n}-\\{0\\}|\forall j\hskip 2.84544pt|x_{j}|_{v}=1\text{ or
}x_{j}=0.\\}$

We say that a family $(f_{v})_{v}$ is generalized adelic if for almost every
$v\in M_{F}^{0}$, one has that

$f|_{E_{v}}=1.$

We use the language of $\mathbf{a}$-homogenous continuous functions to define
heights.

######  Lemma 2.

Let $(f_{v})_{v\in M_{F}}$ be a generalized adelic family of
$\mathbf{a}$-homogenous functions
$f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0}$ of weighted degree $d\in\mathbb{Z}$. For
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$, let us denote by
$\widetilde{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}(\mathbb{G}_{m})_{F_{v}}\to(\mathbb{A}^{n}-\\{0\\})_{F_{v}}$
the induced $(\mathbb{G}_{m})_{F_{v}}$-equivariant morphism defined by
$\mathbf{x}$. For every $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$, the product

$H((f_{v})_{v})(\mathbf{x})\mathrel{\mathop{\ordinarycolon}}=\prod_{v\in
M_{F}}f_{v}(\widetilde{\mathbf{x}}(1))$ (3)

is a finite product. Moreover, if $\mathbf{x}\xrightarrow{\sim}\mathbf{y}$ is
a $2$-isomorphism, then
$H((f_{v})_{v})(\mathbf{x})=H((f_{v})_{v})(\mathbf{y})$.

###### Proof.

Let $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F).$ If $\widetilde{x}_{j}(1)\neq
0$, then for almost every $v\in M_{F}^{0}$ one has
$|\widetilde{x}_{j}(1)|_{v}=1$. We conclude that for almost all $v\in
M_{F}^{0}$, one has $\widetilde{\mathbf{x}}(1)\in E_{v}$. As $(f_{v})_{v}$ is
generalized adelic, we deduce that the product (3) is indeed finite. Let now
$\mathbf{x}\xrightarrow{\sim}\mathbf{y}$ be a $2$-isomorphism, it is given by
an element $t\in\mathbb{G}_{m}(F_{v})$ such that
$t\cdot\widetilde{\mathbf{x}}(1)=\widetilde{\mathbf{y}}(1)$. By the product
formula, one has that

$\prod_{v\in M_{F}}f_{v}(\widetilde{\mathbf{y}}(1))=\prod_{v\in
M_{F}}f_{v}(t\cdot\widetilde{\mathbf{x}}(1))=\prod_{v\in
M_{F}}|t|^{k}_{v}f_{v}(\widetilde{\mathbf{x}}(1))=\prod_{v\in
M_{F}}f_{v}(\widetilde{\mathbf{x}}(1)).$

∎

######  Definition 4.

Let $(f_{v})_{v}$ be a generalized adelic family of $\mathbf{a}$-homogenous
continuous functions $F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq 0}$ of weighted
degree $k$.

  1. 1.

The function that associates to $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ the
value of the product (3), we call the resulting height defined by the family
$(f_{v})_{v}$ and we denote it by $H((f_{v})_{v})$.

  2. 2.

For $\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)],$ we define
$H((f_{v})_{v})(\mathbf{x})$ by setting it to be $H((f_{v})_{v})(\mathbf{y})$
where $\mathbf{y}\in\mathscr{P}(\mathbf{a})(F)$ is such that the isomorphism
class of $\mathbf{y}$ is $\mathbf{x}$.

When there is no confusion, we call it simply the height. When two families
consist of functions which are non-vanishing at every place and which coincide
at almost ever places, the resulting heights can be compared as follows.

######  Lemma 5.

Let $d\in\mathbb{Z}$, and let
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0})_{v}$
and
$(f_{v}^{\prime}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0})_{v}$
be two generalized adelic degree $d$ families of $\mathbf{a}$-homogenous
functions. Suppose that for almost all $v\in M_{F}$, one has that
$f_{v}=f_{v}^{\prime}$. There exist $C_{1},C_{2}>0$ such that for every
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ one has that

$C_{1}<\frac{H((f_{v})_{v})(\mathbf{x})}{H((f_{v}^{\prime})_{v})(\mathbf{x})}<C_{2}.$

###### Proof.

For $v\in M_{F}$, the function

$F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}\hskip
28.45274pt\mathbf{y}\mapsto\frac{f_{v}(\mathbf{y})}{f_{v}^{\prime}(\mathbf{y})}$

is $\mathbb{G}_{m}(F_{v})$-invariant, thus descends to a positive valued
function on $[\mathscr{P}(\mathbf{a})(F_{v})],$ which is compact by 5. We
deduce that for every $v\in M_{F}$, there exist $C_{v,1},C_{v,2}>0$ such that
for every $\mathbf{y}\in F^{n}_{v}-\\{0\\}$ one has that

$C_{v,1}<\dfrac{f_{v}(\mathbf{y})}{f_{v}^{\prime}(\mathbf{y})}<C_{v,2}.$

Let us write $H$ for $H((f_{v})_{v})$ and $H^{\prime}$ for
$H((f_{v}^{\prime})_{v})$. Let $\mathbf{y}\in F^{n}_{v}-\\{0\\}$ be a lift of
$\mathbf{x}$. Let $S$ be the finite set of places, for which $f_{v}\neq
f_{v}^{\prime}$. For every $v\not\in S$, one has that $f_{v}=f_{v}^{\prime}$.
We deduce that

$\frac{H(\mathbf{x})}{H^{\prime}(\mathbf{x})}=\prod_{v\in
S}\frac{f_{v}(\widetilde{\mathbf{x}})}{f_{v}^{\prime}(\widetilde{\mathbf{x}})}<\prod_{v\in
S}C_{v,2}$

and that

$\prod_{v\in S}C_{v,1}<\prod_{v\in
S}\frac{f_{v}(\widetilde{\mathbf{x}})}{f_{v}^{\prime}(\widetilde{\mathbf{x}})}=\frac{H(\mathbf{x})}{H^{\prime}(\mathbf{x})}.$

The statement is proven. ∎

#### 2

In this paragraph, we define stable heights on $\mathscr{P}(\mathbf{a})(F)$,
where $\mathbf{a}\in\mathbb{Z}^{n}_{\geq 1}$. These heights are pullbacks of
the heights on varieties. Such height $H$ satisfies that if
$\mathbf{x},\mathbf{y}\in\mathscr{P}(\mathbf{a})(F)$ are such that
$\mathbf{x}_{K}\cong\mathbf{y}_{K}$ for some extension $K/F$, then
$H(\mathbf{x})=H(\mathbf{y})$. The definition of the height can hence be
naturally carried to any $\overline{F}$-point of $\mathscr{P}(\mathbf{a})$.
They are called “stable” because the height of an $F$-point stays invariant
when regarding this point as a $K$-point of $[\mathscr{P}(\mathbf{a})]$. We
will see later in 6, that such heights do not always satisfy the Northcott
property.

If $v\in M_{F}$ and $|\lvert\cdot\rvert|$ is an $F_{v}$-metric on the line
bundle $\mathcal{O}(k)$ on $\mathscr{P}(\mathbf{a})_{F_{v}}$, for some
$k\in\mathbb{Z}$, we will denote by
$f_{|\lvert\cdot\rvert|}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
the $\mathbf{a}$-homogenous function of weighted degree $k$ given by Lemma 6.

If $r\geq 0$ is an integer, let us denote by $|\lvert\cdot\rvert|_{v,\max}$
the metric on $\mathcal{O}(1)$ on $\mathbb{P}^{r}$ given by the
$\mathbf{1}$-homogenous continuous function
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
of weighted degree $1$

$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}\mathbf{x}\mapsto\max_{j}(|x_{j}|_{v}).$

By a stable $F_{v}$-metric on a line bundle $L$ on locally of finite type
$F_{v}$-scheme $Z$, we mean a metric which is the restriction of a metric on
the analytic line bundle $L^{\operatorname{an}}$ on the analytic space
$Z^{\operatorname{an}}_{F_{v}}$. The metric $|\lvert\cdot\rvert|_{v,\max}$ on
the line bundle $\mathcal{O}(1)$ on the projective space is stable.

######  Definition 1.

Let $X$ be a locally of finite type $F_{v}$-algebraic stack and let $L$ be a
line bundle on $X$. An $F_{v}$-metric $|\lvert\cdot\rvert|$ on $L$ is said to
be stable, if there exist an integer $m\neq 0$, a locally of finite type
$F_{v}$-scheme $Z,$ an $F_{v}$-metrized line bundle
$(L^{\prime},|\lvert\cdot\rvert|^{\prime})$ on $Z$ with
$|\lvert\cdot\rvert|^{\prime}$ stable and a $1$-morphism of algebraic stacks
$g\mathrel{\mathop{\ordinarycolon}}X\to Z$ such that $(L^{\otimes
m},|\lvert\cdot\rvert|^{\otimes
m})=g^{*}(L^{\prime},|\lvert\cdot\rvert|^{\prime})$.

Recall that an adelic metric on the line bundle $\mathcal{O}(1)$ on
$\mathbb{P}^{n-1}$ is a collection of metrics $(|\lvert\cdot\rvert|_{v})_{v},$
where each $|\lvert\cdot\rvert|_{v}$ is a stable metric on the line bundle
$\mathcal{O}(1)$ on $\mathbb{P}^{n-1}_{F_{v}},$ and for almost all $v,$ one
has $|\lvert\cdot\rvert|_{v}=|\lvert\cdot\rvert|_{v,\max}$.

######  Definition 2.

Let $k$ be an integer. Let $(f_{v})_{v}$ be a generalized adelic family of
$\mathbf{a}$-homogenous continuous functions of weighted degree $k$. We say
that $(f_{v})_{v}$ is stable if there exists an integer $\ell\neq 0$, an
integer $r\geq 0,$ an adelic metric $(|\lvert\cdot\rvert|_{v})_{v}$ on the
line bundle $\mathcal{O}(1)$ on $\mathbb{P}^{r}$ and a $1$-morphism of
algebraic stacks
$h\mathrel{\mathop{\ordinarycolon}}\mathscr{P}(\mathbf{a})\to\mathbb{P}^{r}$
such that the line bundle $\mathcal{O}(\ell k)$ is the pullback line bundle
$h^{*}\mathcal{O}(1)$ and such that for every $v$ one has that
$f^{\ell}_{v}=f_{h^{*}|\lvert\cdot\rvert|_{v}}$. In this case, we also say
that the height $H=H((f_{v})_{v})$ is stable.

######  Lemma 3.

Let $k$ be an integer, let $(f_{v})_{v}$ be a stable generalized adelic family
of $\mathbf{a}$-homogenous continuous functions of weighted degree $k$. Let
$H=H((f_{v})_{v})$ be the corresponding height. Let
$\mathbf{x},\mathbf{y}\in\mathscr{P}(\mathbf{a})(F)$ be such that there exists
an extension $K/F$ and an isomorphism
$\mathbf{x}|_{K}\xrightarrow{\sim}\mathbf{y}_{K}$ in
$\mathscr{P}(\mathbf{a})(K)$. Then $H(\mathbf{x})=H(\mathbf{y})$.

###### Proof.

There exists an $\ell\neq 0$, an integer $r\geq 0,$ an adelic metric
$(|\lvert\cdot\rvert|_{v})_{v}$ on the line bundle $\mathcal{O}(1),$ a
$1$-morphism of algebraic stacks
$g\mathrel{\mathop{\ordinarycolon}}\mathscr{P}(\mathbf{a})\to\mathbb{P}^{r}$
such that $\mathcal{O}(\ell k)=g^{*}\mathcal{O}(1)$ and such that for each $v$
one has $f_{v}^{\ell}=f_{g^{*}|\lvert\cdot\rvert|_{v}}$. Let
$H_{\mathbb{P}^{n-1}}$ be the height on $\mathbb{P}^{n-1}(F)$ given by
$(|\lvert\cdot\rvert|_{v})_{v}$. For $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$
that

$H(\mathbf{x})^{\ell}=\prod_{v}f_{v}(\widetilde{\mathbf{x}}(1))^{\ell}=\prod_{v}f_{g^{*}|\lvert\cdot\rvert|_{v}}(\widetilde{\mathbf{x}}(1))=\prod_{v}(g^{*}||1(\mathbf{x})||_{v})^{-1}\\\
=\prod_{v}||1(g(\mathbf{x})||^{-1}_{v}=H_{\mathbb{P}^{n-1}}(g(\mathbf{x})).$

The image of $\mathbf{x}_{K}$ in $\mathbb{P}^{n-1}(K)$ is precisely
$g(\mathbf{x})_{K}$ and the image of $\mathbf{y}_{K}$ in $\mathbb{P}^{n-1}(K)$
is precisely $g(\mathbf{x})_{K}$. The existence of an isomorphism
$\mathbf{x}_{K}\xrightarrow{\sim}\mathbf{y}_{K}$ gives that
$g(\mathbf{x})_{K}=g(\mathbf{y})_{K}$. It follows that
$g(\mathbf{x})=g(\mathbf{y})$, and hence

$H(\mathbf{x})=H_{\mathbb{P}^{n-1}}(g(\mathbf{x}))=H_{\mathbb{P}^{n-1}}(g(\mathbf{y}))=H(\mathbf{y}).$

∎

We give an example of a stable family.

######  Lemma 4.

For $v\in M_{F}$, the functions
$f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
given by $\mathbf{x}\mapsto\max(|x_{j}|^{1/a_{j}}_{v})$ are continuous and
$\mathbf{a}$-homogenous of weighted degree $1$. The family $(f_{v})_{v}$ is
generalized adelic and stable. Moreover, the resulting height
$H^{\max}=H((f_{v}))$ satisfies for every
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ that $H(\mathbf{x})\geq 1$.

###### Proof.

It is evident that $f_{v}$ is continuous and that
$f_{v}(t\cdot\mathbf{x})=\max_{j}(|t^{a_{j}}x_{j}|^{1/a_{j}}_{v})=|t|_{v}\max_{j}(|x_{j}|_{v}^{1/a_{j}})=|t|_{v}f_{v}(\mathbf{x})$,
and thus for every $v\in M_{F}$ one has that $f_{v}$ is a continuous
$\mathbf{a}$-homogenous function of weighted degree $1$. Moreover, for every
$v\in M_{F}$, one has that if $\mathbf{y}\in E_{v}=\\{\mathbf{x}\in
F^{n}_{v}-\\{0\\}|\hskip 2.84544pt|x_{j}|_{v}=1\text{ or }x_{j}=0\\}$, then
$f_{v}(\mathbf{y})=1$. Thus, the family $(f_{v})_{v}$ is generalized adelic.

We will verify that $(f_{v})_{v}$ is stable. Let $v\in M_{F}$, an
$F_{v}$-metric on $\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}}$ identifies with a
continuous function
$g\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$ by
setting $g(\mathbf{x})=||1(\mathbf{x})||^{-1}$, where $\mathbf{x}\in
F^{n}_{v}-\\{0\\}$ and $1(\mathbf{x})$ is the value of
$1\in\Gamma(\mathbb{A}^{n}-\\{0\\},\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}})$ at
$\mathbf{x}$. Let us set $\ell=\operatorname{lcm}(\mathbf{a})$. Consider the
morphism
$J(\ell)\mathrel{\mathop{\ordinarycolon}}\mathbb{A}^{n}-\\{0\\}\to\mathbb{A}^{n}-\\{0\\}$
given by

$J(\ell)\mathrel{\mathop{\ordinarycolon}}\mathbf{x}\mapsto(x_{j}^{\ell/a_{j}})_{j}.$

In 1, we have established that $J(\ell)$ is $(t\mapsto t^{\ell})$-invariant
and that $J(\ell)^{*}\mathcal{O}(1)=\mathcal{O}(\ell).$ Moreover, the
following diagram is $2$-commutative:

${\mathbb{A}^{n}-\\{0\\}}$${\mathbb{A}^{n}-\\{0\\}}$${{\mathscr{P}(\mathbf{a})}}$${{\mathbb{P}^{n-1}}.}$$\scriptstyle{q^{\mathbf{a}}}$$\scriptstyle{\overline{J(\ell)}}$$\scriptstyle{J(\ell)}$$\scriptstyle{q^{\mathbf{1}}}$

By 1, it follows that the diagram

${{\widehat{\operatorname{Pic}_{v}}(\mathbb{A}^{n}-\\{0\\})}}$${{\widehat{\operatorname{Pic}_{v}}(\mathbb{A}^{n}-\\{0\\})}}$${{\widehat{\operatorname{Pic}_{v}}(\mathscr{P}(\mathbf{a}))}}$${{\widehat{\operatorname{Pic}_{v}}(\mathbb{P}^{n-1})}}$$\scriptstyle{(q^{\mathbf{a}}_{F_{v}})^{*}}$$\scriptstyle{\overline{J(\ell)}^{*}}$$\scriptstyle{J(\ell)^{*}}$$\scriptstyle{(q^{\mathbf{1}}_{F_{v}})^{*}}$

is commutative. Hence, the image of the $F_{v}$-metrized line bundle
$(\mathcal{O}(k),|\lvert\cdot\rvert|_{v,\max})$ under
$(q^{\mathbf{a}}_{F_{v}})^{*}\circ\overline{J(\ell)}^{*}$ identifies with

$\displaystyle
J(\ell)^{*}(q^{\mathbf{1}}_{F_{v}})^{*}(\mathcal{O}(k),|\lvert\cdot\rvert|_{v,\max})$
$\displaystyle=(\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}},J(\ell)^{*}(q^{\mathbf{1}}_{F_{v}})^{*}|\lvert\cdot\rvert|_{v,\max})$
$\displaystyle=(\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}},\mathbf{x}\mapsto
J(\ell)^{*}(\mathbf{x}\mapsto\max_{j}|x_{j}|_{v}))$
$\displaystyle=(\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}},\mathbf{x}\mapsto\max_{j}(|x_{j}|^{\ell/a_{j}}_{v}))$
$\displaystyle=(\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}},\mathbf{x}\mapsto
f_{v}(\mathbf{x})^{\ell}).$

The $\mathbb{G}_{m}$-invariant $F_{v}$-linearized line bundle
$(\mathcal{O}_{\mathbb{A}^{n}-\\{0\\}},\mathbf{x}\mapsto
f_{v}(\mathbf{x})^{\ell})$ is precisely the image of
$\overline{J(\ell)}^{*}(\mathcal{O}(1),|\lvert\cdot\rvert|_{v,\max})=(\mathcal{O}(k),\overline{J(\ell)}^{*}|\lvert\cdot\rvert|_{v,\max})$
under the pullback $(q^{\mathbf{a}}_{F_{v}})^{*}$, so that
$f_{v}^{\ell}=f_{J(\ell)^{*}|\lvert\cdot\rvert|_{v,\max}}.$ It follows that
the family $(f_{v})_{v}$ is stable.

Finally, let us prove the estimate $H(\mathbf{x})\geq 1$. Let
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ and let $\widetilde{\mathbf{x}}$ be
the $\mathbb{G}_{m}$-equivariant $F$-morphism
$\mathbb{G}_{m}\to\mathbb{A}^{n}-\\{0\\}$ given by $\mathbf{x}$. Let $i$ be an
index such that $\widetilde{\mathbf{x}}(1)_{i}\neq 0$. Let
$K=F\big{(}\sqrt[a_{i}]{\widetilde{x}_{i}(1)}\big{)}.$ For $v\in M_{F}$, the
absolute value $\lvert\cdot\rvert_{v}$ on $F$ admits a unique extension to
$K$. By the product formula, we get that

$\displaystyle H(\mathbf{x})$ $\displaystyle=\prod_{v\in
M_{F}}\max_{j}(|\widetilde{x}_{j}(1)|_{v}^{1/a_{j}})$
$\displaystyle=\prod_{v\in
M_{F}}|\sqrt[a_{i}]{\widetilde{x}_{i}(1)})|_{v}\max_{j}\bigg{(}\bigg{|}\frac{\widetilde{x}_{j}(1)}{(\sqrt[a_{i}]{\widetilde{x}_{j}(1)})^{a_{j}}}\bigg{|}^{1/a_{j}}_{v}\bigg{)}$
$\displaystyle=\prod_{v\in
M_{F}}\max_{j}\bigg{(}\bigg{|}\frac{\widetilde{x}_{j}(1)}{(\sqrt[a_{i}]{\widetilde{x}_{j}(1)})^{a_{j}}}\bigg{|}^{1/a_{j}}_{v}\bigg{)}$
$\displaystyle\geq 1.$

∎

#### 3

In this paragraph we present an example of a height on
$\mathscr{P}(\mathbf{a})$ which is intrinsic to stacks. We will call them
quasi-toric heights. In 6, we are going to show that they satisfy the
Northcott property. In 4, we have introduced the set
$\mathcal{D}^{\mathbf{a}}_{v}=\mathcal{O}_{v}^{n}-(\pi_{v}^{a_{1}}\mathcal{O}_{v})\times\cdots\times(\pi_{v}^{a_{n}}\mathcal{O}_{v})$.
For $v\in M_{F}^{0}$, we have defined in 4 a function
$r_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{Z}$ by

$r_{v}(\mathbf{x})=\inf\\{k\in\mathbb{Z}|\pi_{v}^{k}\cdot\mathbf{x}\in\mathcal{O}_{v}^{n}\\}$

and we have established that for every $\mathbf{x}\in F^{n}_{v}-\\{0\\}$ that

$\pi_{v}^{r_{v}(\mathbf{x})}\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}.$

######  Lemma 1.

Let $v\in M_{F}^{0}$.

  1. 1.

For every $\mathbf{x}\in F^{n}_{v}-\\{0\\}$, one has that

$r_{v}(\mathbf{x})=\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(x_{j})}{a_{j}}\bigg{\rceil}.$

  2. 2.

For every $t\in F_{v}^{\times}$ and every $\mathbf{x}\in F^{n}_{v}-\\{0\\}$,
one has that

$r_{v}(t\cdot\mathbf{x})=r_{v}(\mathbf{x})-v(t).$

  3. 3.

For every $k\in\mathbb{Z}$ one has that

$\\{\mathbf{x}\in
F^{n}_{v}-\\{0\\}|r_{v}(\mathbf{x})=k\\}=\pi_{v}^{-k}\cdot\mathcal{D}^{\mathbf{a}}_{v}.$

  4. 4.

Let $k\in\mathbb{Z}$. Suppose that for $u\in\mathbb{G}_{m}(F_{v})$ one has
$u\cdot(\pi_{v}^{k}\cdot\mathcal{D}^{\mathbf{a}}_{v})\cap(\pi_{v}^{k}\cdot\mathcal{D}^{\mathbf{a}}_{v})\neq\emptyset$,
then $v(u)=0$.

  5. 5.

The function

$f^{\\#}_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}\hskip
28.45274pt\mathbf{x}\mapsto|\pi_{v}|_{v}^{-r_{v}(\mathbf{x})}$

is $\mathbf{a}$-homogenous of weighted degree $1$ and is locally constant.

###### Proof.

  1. 1.

For every index $j$ such that $x_{j}\neq 0$, one has that
$\pi_{v}^{a_{j}k}x_{j}\in\mathcal{O}_{v}$ if and only if
$k\geq-\frac{v(x_{j})}{a_{j}}.$ We conclude

$r_{v}(\mathbf{x})=\bigg{\lceil}\sup_{\begin{subarray}{c}j=1,\dots,n\\\
x_{j}\neq
0\end{subarray}}\frac{-v(x_{j})}{a_{j}}\bigg{\rceil}=\sup_{\begin{subarray}{c}j=1,\dots,n\\\
x_{j}\neq 0\end{subarray}}\bigg{\lceil}-\frac{v(x_{j})}{a_{j}}\bigg{\rceil}.$

  2. 2.

We observe that

$r_{v}(t\cdot\mathbf{x})=\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(t^{a_{j}}x_{j})}{a_{j}}\bigg{\rceil}=\sup_{\begin{subarray}{c}j=1,\dots,n\\\
x_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(x_{j})+a_{j}v(t)}{a_{j}}\bigg{\rceil}=r_{v}(\mathbf{x})-v(t).$

  3. 3.

We verify the claim for $k=0$. Pick
$\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}.$ For every index $j$ one has
$v(x_{j})\geq 0$ and there exists an index $i$ such that
$x_{i}\not\in\pi_{v}^{a_{i}}\mathcal{O}_{v},$ i.e. such that $v(x_{i})<a_{i}$.
This implies that

$0\geq r_{v}(\mathbf{x})=\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(x_{j})}{a_{j}}\bigg{\rceil}\geq 0.$

Pick now $\mathbf{x}\in F^{n}_{v}-\\{0\\}$ such that $r_{v}(\mathbf{x})=0$.
This means that for every $j$ such that $x_{j}\neq 0$, one has that

$\frac{-v(x_{j})}{a_{j}}\leq 0,$

i.e. that $v(x_{j})\geq 0$ and that there exists an index $i$ such that

$-1<\frac{-v(x_{i})}{a_{i}}\leq 0,$

i.e. such that $0\leq v(x_{i})<a_{i}.$ We deduce that
$\mathbf{x}\in\mathcal{O}_{v}^{n}$ and that
$\mathbf{x}\not\in\pi_{v}^{a_{1}}\mathcal{O}_{v}\times\cdots\times\pi_{v}^{a_{n}}\mathcal{O}_{v}$,
i.e. one has $\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}$. Let now $k\neq 0$ be
an integer. It follows from (1) and the case $k=0$ that

$\displaystyle\\{\mathbf{x}\in F^{n}_{v}-\\{0\\}|r_{v}(\mathbf{x})=k\\}$
$\displaystyle=\\{\mathbf{x}\in
F^{n}_{v}-\\{0\\}|r_{v}(\pi_{v}^{k}\cdot\mathbf{x})=0\\}$
$\displaystyle=\\{\mathbf{x}\in
F^{n}_{v}-\\{0\\}|\pi_{v}^{k}\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}\\}$
$\displaystyle=\pi_{v}^{-k}\cdot\mathcal{D}^{\mathbf{a}}_{v}.$

  4. 4.

We have that

$\emptyset\neq\pi_{v}^{-k}\cdot(u\cdot(\pi_{v}^{k}\cdot\mathcal{D}^{\mathbf{a}}_{v})\cap(\pi_{v}^{k}\cdot\mathcal{D}^{\mathbf{a}}_{v}))=(u\cdot\mathcal{D}^{\mathbf{a}}_{v})\cap\mathcal{D}^{\mathbf{a}}_{v}.$

If $v(u)>0$, then
$(u\cdot\mathcal{D}^{\mathbf{a}}_{v})\subset\pi_{v}^{a_{1}}\mathcal{O}_{v}\times\cdots\times\pi_{v}^{a_{n}}\mathcal{O}_{v}$
and hence
$(u\cdot\mathcal{D}^{\mathbf{a}}_{v})\cap\mathcal{D}^{\mathbf{a}}_{v}=\emptyset,$
a contradiction. Suppose $v(u)<0,$ then
$(u\cdot\mathcal{D}^{\mathbf{a}}_{v})\subset\pi_{v}^{-a_{1}}\mathcal{O}_{v}\times\cdots\times\pi_{v}^{-a_{n}}\mathcal{O}_{v}-\mathcal{O}_{v}^{n},$
and hence
$u\cdot\mathcal{D}^{\mathbf{a}}_{v}\cap\mathcal{D}^{\mathbf{a}}_{v}=\emptyset$,
a contradiction. We deduce $v(u)=0$.

  5. 5.

Let $t\in F_{v}^{\times}$. One has that

$f^{\\#}_{v}(t\cdot\mathbf{x})=|\pi_{v}|_{v}^{-r_{v}(t\cdot\mathbf{x})}=|\pi_{v}|_{v}^{-r_{v}(\mathbf{x})+1}=|\pi_{v}|_{v}\cdot
f^{\\#}_{v}(\mathbf{x}),$

hence $f_{v}^{\\#}$ is $\mathbf{a}$-homogenous. We have seen in 4 that
$\mathcal{D}^{\mathbf{a}}_{v}$ is open and closed in $F_{v}^{n}$, and hence in
$F^{n}_{v}-\\{0\\}$. Hence, the sets
$\pi_{v}^{k}\cdot\mathcal{D}^{\mathbf{a}}_{v}$ are open and closed in
$F^{n}_{v}-\\{0\\}$ for every $k\in\mathbb{Z}$. It follows that $r_{v}$ is
locally constant, so is $f_{v}^{\\#}$.

∎

######  Definition 2.

Let $v\in M_{F}^{0}$. We call the $\mathbf{a}$-homogenous function
$f^{\\#}_{v}\mathrel{\mathop{\ordinarycolon}}F_{v}^{n}-\\{0\\}\to\mathbb{R}_{>0}$
of weighted degree $d\in\mathbb{Z}$ given by

$\displaystyle\quad f_{v}\mathrel{\mathop{\ordinarycolon}}\mathbf{x}\mapsto$
$\displaystyle\begin{cases}|\pi_{v}|_{v}^{-dr_{v}(\mathbf{x})}&\text{if }v\in
M_{F}^{0},\\\ (\max_{j}(|x_{j}|^{1/a_{j}}_{v}))^{d}&\text{if }v\in
M_{F}^{\infty},\end{cases}$

the toric $\mathbf{a}$-homogenous function of weighted degree $d$.

We remark that when $\mathbf{a}=\mathbf{1},$ for every $v\in M_{F}$, the toric
$\mathbf{1}$-homogenous function of weighted degree $1$ is given by
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}\mathbf{x}\mapsto\max_{j}(|x_{j}|_{v})$.

######  Definition 3.

Let
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0})_{v}$ be a family of $\mathbf{a}$-homogenous continuous functions of
weighted degree $d$. We say that the degree of the family $(f_{v})_{v}$ is
$d$.

  1. 1.

We say that $(f_{v})_{v}$ is quasi-toric if for almost all $v\in M_{F}^{0}$,
the function $f_{v}$ is toric.

  2. 2.

We say that $(f_{v})_{v}$ is toric if for every $v$ one has that
$f_{v}=f_{v}^{\\#}$, where $f_{v}^{\\#}$ is the toric $\mathbf{a}$-homogenous
function of weighted degree $d$.

For every $v\in M_{F}^{0},$ as $E_{v}\subset\mathcal{D}^{\mathbf{a}}_{v}$, one
has that $f^{\\#}_{v}|_{E_{v}}=1$. Thus every quasi-toric family $(f_{v})_{v}$
is automatically generalized adelic.

######  Definition 4.

Let $(f_{v})_{v}$ be a quasi-toric family of $\mathbf{a}$-homogenous
continuous functions of a fixed weighted degree. We say that the resulting
height $H((f_{v})_{v})$ is quasi-toric. If $(f_{v})_{v}$ is furthermore
assumed to be toric, we say that $H$ is toric and may be denoted by $H^{\\#}$.

######  Example 5.

We present a formula for the toric height in the case $F=\mathbb{Q}$. Let $d$
be a strictly positive integer. Let
$\mathbf{x}\in[\mathscr{P}(\mathbf{a})(\mathbb{Q})]$ and let
$\widetilde{\mathbf{x}}\in\mathbb{Z}^{n}$ be a lift of $\mathbf{x}$ which
satisfies $\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{p}$ for every prime $p$.
Then for every prime $p$, one has that
$f^{\\#}_{p}(\widetilde{\mathbf{x}})=1$. It follows that the toric height
defined by the degree $d$ toric family satisfies:

$H^{\\#}(\mathbf{x})=(\max(|\widetilde{x}_{j}|^{1/a_{j}}))^{d}.$

When $n=2$ and $\mathbf{a}=(4,6)$, the toric height defined by the degree $12$
toric family is sometimes called naive height (e.g. [13]).

######  Remark 6.

Quasi-toric heights are not stable. In fact, we will verify in 6, that quasi-
toric heights satisfy the Northcott heights, while the stable heights do not

######  Lemma 7.

For $v\in M_{F}$, let $f_{v}^{\\#}$ be the toric $\mathbf{a}$-homogenous
function of weighted degree $1$.

  1. 1.

For every $v\in M_{F}$ and every $\mathbf{y}\in F^{n}_{v}-\\{0\\}$, one has
that $f_{v}^{\\#}(\mathbf{y})\geq\max(|y_{j}|^{1/a_{j}}_{v})$.

  2. 2.

For every $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ one has that
$H^{\\#}(\mathbf{x})\geq H^{\max}(\mathbf{x})\geq 1,$ where $H^{\max}$ is the
height given by $(\mathbf{y}\mapsto\max(|y_{j}|^{1/a_{j}}_{v}))_{v}.$

###### Proof.

For $v\in M_{F}^{\infty}$, we recall that
$f_{v}^{\\#}(\mathbf{y})=\max(|y_{j}|^{1/a_{j}}_{v})$. Let $v\in M_{F}^{0}$.
For every $\mathbf{y}\in\mathcal{D}^{\mathbf{a}}_{v}$ one has that

$r_{v}(\mathbf{y})\geq\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\big{(}\frac{-v(y_{j})}{a_{j}}\big{)}=\mathrel{\mathop{\ordinarycolon}}r^{\max}_{v}(\mathbf{y})$

and thus

$f_{v}(\mathbf{y})=|\pi_{v}|_{v}^{-r_{v}(\mathbf{y})}\geq|\pi_{v}|_{v}^{-r^{\max}_{v}(\mathbf{y})}\geq(\max(|y_{j}|^{1/a_{j}}_{v}).$

The function $\mathbf{y}\mapsto
f_{v}^{\\#}(\mathbf{y})(\max(|y_{j}|^{1/a_{j}}_{v}))^{-1}$ is
$\mathbb{G}_{m}(F_{v})$-invariant. As by 1 any $\mathbf{z}\in
F^{n}_{v}-\\{0\\}$ writes us $t\cdot\mathbf{y}$, where
$\mathbf{y}\in\mathcal{D}^{\mathbf{a}}_{v}$ it follows that
$f^{\\#}_{v}(\mathbf{z})\geq(\max(|z_{j}|^{1/a_{j}}_{v})$ for every
$\mathbf{z}\in F^{n}_{v}-\\{0\\}.$ The first claim is proven.

Now let $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$ and let
$\widetilde{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}\mathbb{G}_{m}\to\mathbb{A}^{n}-\\{0\\}$
be the $\mathbb{G}_{m}$-equivariant morphism over $F$ defined by $\mathbf{x}$.
We have that

$H^{\\#}(\mathbf{x})=\prod_{v\in
M_{F}}f_{v}^{\\#}(\widetilde{\mathbf{x}}(1))\geq\prod_{v\in
M_{F}}\max_{j}(|\widetilde{\mathbf{x}}(1)|_{v}^{1/a_{j}})=H^{\max}(\mathbf{x})\geq
1,$

by 4. The second claim is proven. ∎

Let us end the paragraph by a proof of a lemma that will be used latter for
the proof of Proposition 2. It is motivated by [30, Theorem B.2.5].

######  Lemma 8.

Let $P_{1},\dots,P_{r+1}\in F[X_{1},\dots,X_{n}]$ be $\mathbf{a}$-homogenous
polynomials of the same weighted degree $d$. Let
$\overline{J}(P_{1},\dots,P_{r+1})\mathrel{\mathop{\ordinarycolon}}(\mathscr{P}(\mathbf{a})-\mathcal{Z}(P_{1},\dots,P_{r+1}))\to\mathbb{P}^{r}$
be the $1$-morphism given by the $t\mapsto t^{d}$-equivariant morphism
$(P_{1},\dots,P_{r+1})\mathrel{\mathop{\ordinarycolon}}(\mathbb{A}^{n}-\\{0\\})\to(\mathbb{A}^{n}-\\{0\\})$
(see 5). Let $H^{\\#}$ be the toric height on $[\mathscr{P}(\mathbf{a})(F)]$
defined by the toric degree $d$ family. There exists $C>0$ such that for all
$\mathbf{x}\in[(\mathscr{P}(\mathbf{a})-Z(P_{1},\dots,P_{r+1}))(F)]$ one has
that

$CH^{\\#}(\mathbf{x})\geq H_{\mathbb{P}^{r}}(\phi(\mathbf{x}))^{d},$

where $H_{\mathbb{P}^{r}}$ is the toric height defined by the toric degree $1$
family on $\mathbb{P}^{r}$.

###### Proof.

The strategy from the proof of [30, Theorem B.2.5] applies here. Let us denote
by $X^{\mathbf{m}}$ the monomial $X_{1}^{m_{1}},\dots,X_{n}^{m_{n}},$ where
$\mathbf{m}\in\mathbb{Z}^{n}_{\geq 0}$. We denote by $w(\mathbf{a},d)$ the
number of $\mathbf{m}\in\mathbb{Z}^{n}_{\geq 0}$ for which
$\mathbf{a}\cdot\mathbf{m}=d$, this is precisely the number of monomials which
have $\mathbf{a}$-weighted degree equal to $d$. For $i\in\\{1,\dots,r+1\\}$,
we write

$P_{i}=\sum_{\mathbf{a}\cdot\mathbf{m}=d}A_{i,\mathbf{m}}X^{\mathbf{m}},$

where the sum runs over $\mathbf{m}\in\mathbb{Z}^{n}_{\geq 0}$ for which
$\mathbf{a}\cdot\mathbf{m}=\sum_{j=1}^{n}a_{j}m_{j}=d.$ For $v\in M_{F}$, we
denote by $|P_{i}|_{v}=\max_{\mathbf{m}}|A_{i,\mathbf{m}}|_{v}$. For almost
all $v,$ one has that $|P_{i}|_{v}=1.$ For $k\in\mathbb{Z}$, we set

$\displaystyle\quad\varepsilon_{v}(k)\mathrel{\mathop{\ordinarycolon}}=$
$\displaystyle\begin{cases}k&\text{if }v\in M_{F}^{\infty},\\\ 1&\text{if
}v\in M_{F}^{0}.\end{cases}$

We note that for any $v\in M_{F}$, any $k\geq 1$ and any $z_{1},\dots,z_{k}\in
F$, one has that

$|z_{1}+\cdots+z_{k}|_{v}\leq\varepsilon_{v}(r)\max(|z_{1}|_{v},\dots,|z_{k}|_{v}).$

For $\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]$, let
$\widetilde{\mathbf{x}}\in F^{n}_{v}-\\{0\\}$ be a lift of $\mathbf{x}$. For
$i\in\\{1,\dots,r+1\\}$ and $v\in M_{F}$, we deduce that

$\displaystyle|P_{i}(\widetilde{\mathbf{x}})|_{v}$
$\displaystyle=\big{|}\sum_{\mathbf{a}\cdot\mathbf{m}}A_{i,\mathbf{m}}\widetilde{x}^{m_{1}}_{1}\cdots\widetilde{x}^{m_{n}}_{n}\big{|}_{v}$
$\displaystyle\leq\varepsilon(w(\mathbf{a},d))(\max_{\mathbf{m}}|A_{i,\mathbf{m}}|_{v})(\max_{\mathbf{m}}|\widetilde{x}_{1}^{m_{1}}\cdots\widetilde{x}_{n}^{m_{n}}|_{v})$
$\displaystyle\leq\varepsilon(w(\mathbf{a},d))|P_{i}|_{v}\max_{\mathbf{m}}\big{(}\prod_{\ell=1}^{n}\max_{j}(|\widetilde{x}_{j}|^{1/a_{j}}_{v})^{a_{\ell}m_{\ell}}\big{)}$
$\displaystyle=\varepsilon(w(\mathbf{a},d))|P_{i}|_{v}\max_{j}(|\widetilde{x}_{j}|_{v}^{1/a_{j}})^{\mathbf{a}\cdot\mathbf{m}}$
$\displaystyle=\varepsilon(w(\mathbf{a},d))|P_{i}|_{v}\max_{j}(|\widetilde{x}_{j}|_{v}^{1/a_{j}})^{d}.$

Set $C_{v}=\varepsilon(w(\mathbf{a},d))\max_{i}|P_{i}|_{v}$. For almost all
$v$, one has that $C_{v}=1$ and we set $C=\prod_{v}C_{v}$. Let us define

$r_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{Z}\hskip
28.45274pt\mathbf{y}\mapsto\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(y_{j})}{a_{j}}\bigg{\rceil}$

and

$r^{\max}_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{Q}\hskip
28.45274pt\mathbf{y}\mapsto\sup_{\begin{subarray}{c}j=1,\dots,n\\\ y_{j}\neq
0\end{subarray}}\big{(}\frac{v(y_{j})}{a_{j}}\big{)}.$

One has that

$f^{\\#}_{v}(\mathbf{y})=|\pi_{v}|_{v}^{-dr_{v}(\mathbf{y})}\geq|\pi_{v}|_{v}^{-dr^{\max}_{v}(\mathbf{y})}=\max_{j}(|y_{j}|_{v}^{1/a_{j}}|_{v})^{d}.$

We deduce that

$\max_{i}(|P_{i}(\widetilde{\mathbf{x}})|_{v})\leq
C_{v}\max_{j}(|y_{j}|_{v}^{1/a_{j}})^{d}\leq
C_{v}f^{\\#}_{v}(\widetilde{\mathbf{x}}).$

By multiplying this inequality for all $v$ we obtain that for every
$\mathbf{x}\in[(\mathscr{P}(\mathbf{a})-Z(P_{1},\dots,P_{n}))(F)]$ one has
that

$H_{\mathbb{P}^{r}}(\overline{J}(P_{1},\dots,P_{r+1})(\mathbf{x}))^{d}=\prod_{v\in
M_{F}}(\max_{i}(|P_{i}(\widetilde{\mathbf{x}})|_{v}))^{d}\leq\prod_{v}C_{v}f_{v}^{\\#}(\widetilde{\mathbf{x}})\\\
=CH^{\\#}(\mathbf{x}).$

The claim is proven. ∎

#### 4

We establish several facts on “local heights” that will be needed in 6.

Let
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0})_{v}$
be a generalized adelic family of $\mathbf{a}$-homogenous continuous function
of weighted degree $|\mathbf{a}|$. Let $H=H((f_{v})_{v})$ be the corresponding
height on $[\mathscr{P}(\mathbf{a})(F)]$. For $v\in M_{F},$ $t\in
F_{v}^{\times}$ and $\mathbf{x}\in(F_{v}^{\times})^{n}$ one has that

$f_{v}(t\cdot\mathbf{x})\prod_{j=1}^{n}|t^{a_{j}}x_{j}|_{v}^{-1}=|t|_{v}^{|\mathbf{a}|}f_{v}(\mathbf{x})\prod_{j=1}^{n}|t|_{v}^{-a_{j}}|x_{j}|_{v}^{-1}=f_{v}(\mathbf{x})\prod_{j=1}^{n}|x_{j}|_{v}^{-1}$

i.e. for $v\in M_{F}$, the continuous function

$(F_{v}^{\times})^{n}\to\mathbb{R}_{>0},\hskip 28.45274pt\mathbf{x}\mapsto
f_{v}(\mathbf{x})\prod_{j=1}^{n}|x_{j}|_{v}^{-1}$ (1)

is $(F_{v}^{\times})_{\mathbf{a}}$-invariant. Let
$H_{v}\mathrel{\mathop{\ordinarycolon}}[\mathscr{T}({\mathbf{a}})(F_{v})]\to\mathbb{R}_{>0}$
be the function induced from $(F_{v})_{\mathbf{a}}$-invariant function (1).
For $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)],$ we write
$H_{v}(\mathbf{x})$ for what is technically
$H_{v}([\mathscr{T}({\mathbf{a}})(i_{v})](\mathbf{x})),$ where
$[\mathscr{T}({\mathbf{a}})(i_{v})]\mathrel{\mathop{\ordinarycolon}}[\mathscr{T}({\mathbf{a}})(i)(F)]\to[\mathscr{T}({\mathbf{a}})(i)(F_{v})]$
is the induced homomorphism from $(F^{\times})_{\mathbf{a}}$-invariant
homomorphism

$(F^{\times})^{n}\hookrightarrow(F_{v}^{\times})^{n}\to[\mathscr{T}({\mathbf{a}})(F_{v})].$

######  Lemma 2.

Let $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$. One has that
$H(\mathbf{x})=\prod_{v}H_{v}(\mathbf{x})$.

###### Proof.

Let $\widetilde{\mathbf{x}}\in(F^{\times})^{n}$ be a lift of $\mathbf{x}$. By
using the product formula, one gets that

$\displaystyle\prod_{v\in M_{F}}H_{v}(\mathbf{x})$ $\displaystyle=\prod_{v\in
M_{F}}\big{(}f_{v}(\widetilde{\mathbf{x}})\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}\big{)}$
$\displaystyle=\big{(}\prod_{v\in
M_{F}}f_{v}(\widetilde{\mathbf{x}})\big{)}\prod_{v\in
M_{F}}\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}$
$\displaystyle=H(\mathbf{x})$

∎

######  Lemma 3.

Let $v\in M_{F}$ and suppose that $f_{v}=f_{v}^{\\#}$ is the toric
$\mathbf{a}$-homogenous function of weighted degree $1$. Let
$\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)].$ One has that
$H^{\\#}_{v}(\mathbf{x})\geq 1$.

###### Proof.

Suppose $v\in M_{F}^{0}.$ It follows from Lemma 1 that there exists a lift
$\widetilde{\mathbf{x}}$ of $[\mathscr{T}({\mathbf{a}})(i_{v})](\mathbf{x})$
lying in $\mathcal{D}^{\mathbf{a}}_{v}$. By using that
$f^{\\#}_{v}|_{\mathcal{D}^{\mathbf{a}}_{v}}=1$ and that
$\mathcal{D}^{\mathbf{a}}_{v}\subset(\mathcal{O}_{v})^{n}$, we obtain

$H^{\\#}_{v}(\mathbf{x})=f_{v}^{\\#}(\widetilde{\mathbf{x}})\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1_{j}}=\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1_{j}}\geq
1.$

Suppose now $v\in M_{F}^{\infty}$. Let
$\widetilde{\mathbf{x}}\in(F_{v}^{\times})^{n}$ be a lift of $\mathbf{x}$. One
has that

$\displaystyle H^{\\#}_{v}(\mathbf{x})$
$\displaystyle=f^{\\#}_{v}(\widetilde{\mathbf{x}})\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}$
$\displaystyle=\big{(}\max_{k}(|\widetilde{x}_{k}|_{v}^{1/a_{k}})\big{)}^{|\mathbf{a}|}\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}$
$\displaystyle=\prod_{j=1}^{n}\big{(}\max_{k=1,\dots,n}(|\widetilde{x}_{k}|_{v}^{1/a_{k}})\big{)}^{a_{j}}\big{(}|\widetilde{x}_{j}|_{v}^{1/a_{j}}\big{)}^{-a_{j}}$
$\displaystyle\geq\prod_{j=1}^{n}\big{(}\max_{k}(|\widetilde{x}_{k}|_{v}^{1/a_{k}})\big{)}^{a_{j}}\big{(}\max_{k}(|\widetilde{x}_{k}|_{v}^{1/a_{k}})\big{)}^{-a_{j}}$
$\displaystyle=1.$

∎

###  5 Metrics induced by models

We use models with enough integral points to define metrics. We establish that
the toric metric comes from models of weighted projective stacks from 2.

#### 1

We use $\mathcal{O}_{v}$-points of $\overline{\mathscr{P}(\mathbf{a})}$ to
define $F_{v}$-metrics.

Let $v\in M_{F}^{0}$. By an $\mathcal{O}_{v}$-extension of
$\mathbf{x}\in\overline{\mathscr{P}(\mathbf{a})}(F_{v})$ we mean a pair
$(\mathbf{y},t)$ where
$\mathbf{y}\in\overline{\mathscr{P}(\mathbf{a})}(\mathcal{O}_{v})$ and
$t\mathrel{\mathop{\ordinarycolon}}\mathbf{y}_{F_{v}}\xrightarrow{\sim}\mathbf{x}$
is a $2$-isomorphism. Let $S_{\mathbf{x}}$ be the set of
$\mathcal{O}_{v}$-extensions of
$\mathbf{x}\in\overline{\mathscr{P}(\mathbf{a})}(F_{v})$. Proposition 5 gives
that the set $S_{\mathbf{x}}$ is non-empty for any
$\mathbf{x}\in\overline{\mathscr{P}(\mathbf{a})}(F_{v}).$

For $\mathbf{x}\in\overline{\mathscr{P}(\mathbf{a})}(F_{v})$ (respectively,
$\mathbf{x}\in\overline{\mathscr{P}(\mathbf{a})}(\mathcal{O}_{v})$), we will
denote by $\widetilde{\mathbf{x}}$ the canonical
$(\mathbb{G}_{m})_{F_{v}}$-equivariant morphism
$\widetilde{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}(\mathbb{G}_{m})_{F_{v}}\to\mathbb{A}^{n}_{F_{v}}$
(respectively, the canonical $(\mathbb{G}_{m})_{\mathcal{O}_{v}}$-equivariant
morphism
$\widetilde{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}(\mathbb{G}_{m})_{\mathcal{O}_{v}}\to\mathbb{A}^{n}_{\mathcal{O}_{v}}$)
induced by $\mathbf{x}.$ One has that
$q^{\mathbf{a}}\circ\widetilde{\mathbf{x}}(1)=\mathbf{x}$.

If $L$ is a line bundle on $\overline{\mathscr{P}(\mathbf{a})}$, a
$2$-isomorphism
$t\mathrel{\mathop{\ordinarycolon}}\mathbf{x}\xrightarrow{\sim}\mathbf{x}^{\prime}$,
where
$\mathbf{x},\mathbf{x}^{\prime}\in\overline{\mathscr{P}(\mathbf{a})}(F_{v}),$
induces a linear map
$L(t)\mathrel{\mathop{\ordinarycolon}}L(\mathbf{x})\xrightarrow{\sim}L(\mathbf{x}^{\prime})$.

######  Definition 1.

Let $L$ be a line bundle on $\overline{\mathscr{P}(\mathbf{a})}$. Let
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F_{v})$ and let $\ell\in L(\mathbf{x})$.
We define

$||\ell||_{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}=\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip 2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}\ell\in
a(L(t)(\mathbf{y}^{*}L))\\}\\}.$

We calculate the metrics.

######  Lemma 2.

Let $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F_{v})$. Let $k$ be an integer and
let
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
be the $\mathbf{a}$-homogenous toric function of weighted degree $k$. Let
$\ell\in\mathcal{O}(k)(\mathbf{x})$. One has that

$||\ell||_{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}=f^{\\#}_{v}(\widetilde{\mathbf{x}}(1))^{-1}|\ell|_{v}.$

###### Proof.

If
$r\mathrel{\mathop{\ordinarycolon}}\mathbf{x}^{\prime}\to\mathbf{x}^{\prime\prime}$
is a $2$-isomorphism in $\overline{\mathscr{P}(\mathbf{a})}(F_{v})$, the
induced linear map
$\mathcal{O}(k)(r)\mathrel{\mathop{\ordinarycolon}}F_{v}=\mathcal{O}(k)(\mathbf{x}^{\prime})\xrightarrow{\sim}\mathcal{O}(k)(\mathbf{x}^{\prime\prime})=F_{v}$
is the linear map $x\mapsto t^{k}x$. It follows that:

$\displaystyle||\ell||_{\mathbf{x}}$ $\displaystyle=\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip 2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}\ell\in
at^{k}(\mathbf{y}^{*}\mathcal{O}(k))\\}\\}$
$\displaystyle=\sup_{(\mathbf{y},t)\in S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip
2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}\ell\in
at^{k}((\widetilde{\mathbf{y}}(1))^{*}(q^{\mathbf{a}}_{\mathcal{O}_{v}})^{*}\mathcal{O}(k))\\}\\}$
$\displaystyle=\sup_{(\mathbf{y},t)\in S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip
2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}\ell\in
at^{k}((\widetilde{\mathbf{y}}(1))^{*}\mathcal{O}_{\mathbb{A}^{n}})\\}\\}$
$\displaystyle=\sup_{(\mathbf{y},t)\in S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip
2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}\ell\in
at^{k}\mathcal{O}_{v}\\}\\}$ $\displaystyle=\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{\inf\\{|a|_{v}\hskip 2.84544pt|\hskip 2.84544pta\in
F_{v}^{\times}\mathrel{\mathop{\ordinarycolon}}1\in
a\ell^{-1}t^{k}\mathcal{O}_{v}\\}\\}$ $\displaystyle=\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{|t|^{-k}_{v}|\ell|_{v}\\}$
$\displaystyle=|\ell|_{v}\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{|t|^{-k}_{v}\\}.$

Note that if $(\mathbf{y},t)\in S_{\mathbf{x}}$, then as
$t^{-1}\cdot\widetilde{\mathbf{x}}(1)=\widetilde{\mathbf{y}}(1)\in\mathcal{O}_{v}^{n},$
it follows from 1 that $v(t^{-1})\geq r_{v}(\widetilde{\mathbf{x}}(1))$, with
the equality if and only if
$\widetilde{\mathbf{y}}(1)\in\mathcal{D}^{\mathbf{a}}_{v}$. We deduce that

$\sup_{(\mathbf{y},t)\in
S_{\mathbf{x}}}\\{|t|^{-k}_{v}\\}=|\pi_{v}|_{v}^{-kr_{v}(\widetilde{\mathbf{x}}(1))}=f^{\\#}_{v}(\widetilde{\mathbf{x}}(1))^{-1}.$

The claim follows. ∎

We can deduce that:

######  Corollary 3.

Let $v$ be a finite place of $F$ and let $k$ be an integer. The metrics
$|\lvert\cdot\rvert|_{\mathbf{x}}$ from 1 on $\mathcal{O}(k)(\mathbf{x})$ for
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F_{v})$ define an $F_{v}$-metric
$|\lvert\cdot\rvert|$ on $\mathcal{O}(k)|_{\mathscr{P}(\mathbf{a})_{F_{v}}}.$
The $F_{v}$-metric $|\lvert\cdot\rvert|$ is the induced $F_{v}$-metric from
the function $x\mapsto f^{\\#}_{v}({\mathbf{x}})$ by 4.

###### Proof.

For $\mathbf{x}\in\mathscr{P}(\mathbf{a})(F_{v})$, let us pick
$\ell=1\in\mathcal{O}(k)(\mathbf{x})$. By 2, we obtain that
$||1||_{\mathbf{x}}=f^{\\#}_{v}(\widetilde{\mathbf{x}})^{-1}$ for every
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F_{v})$. As
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
is continuous (1) and satisfies that
$f^{\\#}_{v}(t\cdot\mathbf{x})=|t|^{k}_{v}f^{\\#}_{v}(\mathbf{x})$, we deduce
that $|\lvert\cdot\rvert|$ is the induced $F_{v}$-metric from the function
$f_{v}^{\\#}$ using Lemma 6. ∎

###  6 Finiteness property of quasi-toric heights

We say that a height is a Northcott height if for every $B>0$ there are only
finitely many points in $[\mathscr{P}(\mathbf{a})(F)]$ having the height less
than $B$. We establish that quasi-toric heights are Northcott heights. We
improve it further for heights that are degenerate if the singularities of
$f_{v}$ for $v$ infinite are logarithmic along a rational divisor. Let
$\mathbf{a}\in\mathbb{Z}_{\geq 1}^{n}.$

#### 1

We define Northcott heights.

######  Definition 1.

Suppose
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0})_{v}$ is a generalized adelic family of continuous $\mathbf{a}$-homogenous
functions. We say that the corresponding height $H((f_{v})_{v})$ is a
Northcott height if for every $B>0$, the set

$\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H((f_{v})_{v})(\mathbf{x})<B\\}$

is finite.

#### 2

We give heights which are not Northcott heights.

######  Lemma 1.

Suppose $n=1$ and $a\in\mathbb{Z}_{\geq 2}$. The set $[\mathscr{P}(a)(F)]$ is
infinite.

###### Proof.

One has that
$[\mathscr{P}(a)(F)]=[\mathscr{T}({\mathbf{a}})(F)]=F^{\times}/(F^{\times})_{a},$
where $(F^{\times})_{a}$ is the subgroup given by the non-zero $a$-th powers
in $F^{\times}$. There are infinitely many non-zero principal prime ideals in
$\mathcal{O}_{F}$ (because only finitely many prime ideals in $\mathbb{Z}$
which ramify in $\mathcal{O}_{F}$). For any of those principal prime ideals
$\mathfrak{p}$, let $b_{\mathfrak{p}}\in F^{\times}$ be a generator. Note that
for any principal prime ideals $\mathfrak{p}_{1},\mathfrak{p}_{2}$ one has
that $b_{\mathfrak{p}_{1}}b_{\mathfrak{p}_{2}}^{-1}\not\in(F^{\times})_{a},$
because $a\nmid
v_{\mathfrak{p}_{1}}(b_{\mathfrak{p}_{1}}b_{\mathfrak{p}_{2}}^{-1})=1,$ where
$v_{\mathfrak{p}_{1}}$ is the valuation corresponding to $\mathfrak{p}_{1}$.
It follows that the images in $F^{\times}/(F^{\times})_{a}$ of different
$b_{\mathfrak{p}}$ are different. It follows that
$[\mathscr{P}(a)(F)]=F^{\times}/(F^{\times})_{a}$ is infinite. ∎

######  Corollary 2.

Suppose that $n=1$ and $a\in\mathbb{Z}_{\geq 2}$. Let $(f_{v})_{v}$ be a
stable generalized adelic family of $a$-homogenous functions
$F_{v}-\\{0\\}\to\leavevmode\nobreak\ \mathbb{R}_{>0}$. For every
$x,y\in\mathscr{P}(a)(F)$ one has that
$H((f_{v})_{v})(\mathbf{x})=H((f_{v})_{v})(\mathbf{y})$. The height
$H=H((f_{v})_{v})$ is not a Northcott height.

###### Proof.

Let $x,y\in\mathscr{P}(a)(F)$ and let
$\widetilde{x},\widetilde{y}\mathrel{\mathop{\ordinarycolon}}(\mathbb{G}_{m})_{F_{v}}\to(\mathbb{A}^{1}-\\{0\\})_{F_{v}}$
be the two $(\mathbb{G}_{m})_{F_{v}}$-equivariant morphisms defined by $x$ and
$y$. Let $K=F(\sqrt[a]{\widetilde{x}(1)\widetilde{y}(1)^{-1}}).$ Note that for
$t\mathrel{\mathop{\ordinarycolon}}=\sqrt[a]{\widetilde{x}(1)\widetilde{y}(1)^{-1}}$
one has $t^{a}x=y$, thus $x_{K}\cong y_{K}$. By 3, one has that for every
$x,y\in\mathscr{P}(a)(F)$ one has $H(x)=H(y)$. Thus for every
$z,w\in[\mathscr{P}(\mathbf{a})(F)]$ one has $H(z)=H(w)$. Let $B>H(w)$, where
$w\in[\mathscr{P}(a)(F)]$. By 1, the set $\\{z|z\in[\mathscr{P}(a)(F)]\text{
and }H(z)<B\\}$ is infinite. Thus $H$ is not a Northcott height. ∎

######  Remark 3.

In fact, stable heights are not Northcott heights unless all $a_{j}$ are equal
to $1$. Indeed, every rational point of $\mathscr{P}(a_{j})$ can be seen as a
rational point of $\mathscr{P}(\mathbf{a})$ via the closed embedding
$\mathscr{P}(a_{j})\to\mathscr{P}(\mathbf{a})$ (induced from
$\mathbb{G}_{m}$-invariant morphism
$\mathbb{A}^{1}-\\{0\\}\to\mathbb{A}^{n}-\\{0\\}\xrightarrow{q^{\mathbf{a}}}\mathscr{P}(\mathbf{a})$).
Thus if for some $j$ one has $a_{j}>1$, then $\mathscr{P}(a_{j}),$ and hence
$\mathscr{P}(\mathbf{a}),$ both have infinitely many rational points. We omit
details.

#### 3

We dedicate the next paragraphs to the proof that the toric heights are
Northcott heights. By the property of the boundedness 5 of the quotients, it
is then immediate that any quasi-toric height is Northcott height.

Let $\operatorname{Div}(F)$ be the group of fractional ideals of $F$. We
define

$\operatorname{Div}(F)_{\mathbf{a}}\mathrel{\mathop{\ordinarycolon}}=\\{(x^{a_{j}})_{j}|\hskip
2.84544ptx\in\operatorname{Div}(F)\\}.$

In this paragraph we will give an estimate to the number of elements of the
abelian group $\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$
of bounded “height”. This will be useful, as in the next paragraph, we relate
the finite part of the height on $\mathscr{T}({\mathbf{a}})(F)$ with the
“height” on $\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}.$ If
$v\in M_{F}^{0},$ for a fractional ideal $x$ of $\mathcal{O}_{F}$ we define
$v(x)$ by setting it to be the exponent of the prime ideal corresponding to
$v,$ in the prime factorization of $x$.

We define set

$\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}\mathrel{\mathop{\ordinarycolon}}=\\{\mathbf{x}\in\operatorname{Div}(F)^{n}|\mathbf{x}\subset\mathcal{O}_{F}^{n}\text{
and }\forall v\in M_{F}^{0},\exists
j\mathrel{\mathop{\ordinarycolon}}v(x_{j})<a_{j}\\}.$

######  Lemma 1.

  1. 1.

Let us define

$r_{v}\mathrel{\mathop{\ordinarycolon}}\operatorname{Div}(F)^{n}\to\mathbb{Z}\hskip
28.45274pt\mathbf{x}\mapsto\sup_{\begin{subarray}{c}j=1,\dots,n\\\ x_{j}\neq
0\end{subarray}}\big{\lceil}\frac{-v(x_{j})}{a_{j}}\big{\rceil}.$

Let $\mathbf{x}\in\operatorname{Div}(F)^{n}.$ The fractional ideal

$y=y(\mathbf{x})\mathrel{\mathop{\ordinarycolon}}=\prod_{v\in
M_{F}^{0}}(\pi_{v}^{r_{v}(\mathbf{x})}\mathcal{O}_{v}\cap F)$

satisfies that

$(x_{j}y(\mathbf{x})^{a_{j}})_{j}\in\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}.$

  2. 2.

The restriction of the quotient map

$q_{\operatorname{Div}(F),\mathbf{a}}\mathrel{\mathop{\ordinarycolon}}\operatorname{Div}(F)^{n}\to\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$

to $\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}$ is a bijection
$\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}\xrightarrow{\sim}\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}.$

###### Proof.

  1. 1.

Let $v\in M_{F}^{0}$. For every $j$ one has that

$v(x_{j}y^{a_{j}})=v(x_{j})+{a_{j}r_{v}(\mathbf{x})}\geq 0.$

Let $i$ be the index such that $\frac{-v(x_{i})}{a_{i}}$ is maximal, then
$r_{v}(\mathbf{x})<1+\frac{-v(x_{i})}{a_{i}}.$ We deduce that

$v(x_{j}y^{a_{j}})=v(x_{j})+a_{j}r_{v}(\mathbf{x})<a_{j}.$

Hence, $(x_{j}y^{a_{j}})_{j}$ is an element of
$\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}.$

  2. 2.

We prove the surjectivity of $q_{\operatorname{Div}(F),\mathbf{a}}$. Let
$\mathbf{z}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$
and let $\widetilde{\mathbf{z}}\in\operatorname{Div}(F)^{n}$ be a lift of
$\mathbf{z}$. Then $(\widetilde{z}_{j}y(\mathbf{z})^{a_{j}})_{j}$ is a lift of
$\mathbf{z}$ belonging to
$\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}$. Let us prove the
injectivity. Suppose that the elements
$\mathbf{x},\mathbf{r}\in\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}$
are lying above the same element in
$\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$. Then there
exists $t\in\operatorname{Div}(F)$ such that $x_{j}=t^{a_{j}}r_{j}$ for
$j=1,\dots,n$. We need to establish that $t=(1)$. Suppose on the contrary
$t\neq(1).$ One can choose $v$ such that $v(t)\neq 0$. If $v(t)>0$, then for
every $j$ one has

$v(x_{j})=a_{j}v(t)+v(r_{j})\geq a_{j}\cdot 1+0=a_{j},$

which is a contradiction with the fact that $\mathbf{x}$ is primitive. If
$v(t)<0$, then there exists $j$ such that $a_{j}>v(r_{j})$, which gives that

$v(x_{j})=a_{j}v(t)+v(r_{j})<a_{j}-a_{j}=0,$

which is a contradiction with the fact that $\mathbf{x}$ is primitive. We
deduce that $t=(1)$, and hence $\mathbf{x}=\mathbf{r}.$ Therefore
$q_{\operatorname{Div}(F),\mathbf{a}}|_{\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}}$
is injective. The claim is proven.

∎

For every
$\mathbf{x}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$,
let us denote by $\widetilde{\mathbf{x}}$ the unique lift of $\mathbf{x}$
lying in $\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}.$ For
$\mathbf{x}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$,
we set

${{H}_{\operatorname{Ideal}}}(\mathbf{x})\mathrel{\mathop{\ordinarycolon}}=\prod_{j=1}^{n}N(\widetilde{x}_{j})$

where $N$ is the ideal norm.

######  Lemma 2.

For $m\in\mathbb{Z}_{\geq 1}$, there exists $C_{m}>0$ such that for every
$B>0$ one has that

$\\{\mathbf{y}\in\operatorname{Div}(F)^{m}_{\geq
0}|\prod_{j=1}^{n}N(y_{j})\leq B\\}\leq C_{m}B\log(1+B)^{m-1},$

where $\operatorname{Div}_{\geq
0}(F)=\operatorname{Div}(F)\cap\mathcal{O}_{F}$.

###### Proof.

We will use the following result: there exists $C_{1}>0$ such that

$|\\{y\in\operatorname{Div}(F)|y\subset\mathcal{O}_{F},N(y)\leq B\\}|\leq
C_{1}B.$

This is proven in [32, Pages 145-150]. We use the induction on $m$. Let $m\geq
2$. For $k\geq 1$, we set

$g(k)=\big{|}\big{\\{}\mathbf{x}\in\operatorname{Div}_{\geq
0}(F)^{m-1}|\prod_{j=1}^{m-1}N(x_{j})=k\big{\\}}\big{|}.$

For $B\geq 1$, using Abel’s summation formula, we get that there exists
$C_{m}^{\prime},C_{m}>0$ such that have that

$\displaystyle|\\{\mathbf{y}\in\operatorname{Div}_{\geq
0}(F)^{m}|\prod_{j=1}^{m}N(y_{j})\leq B\\}|\hskip-179.25244pt$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{y}\in\operatorname{Div}_{\geq
0}(F)^{m}\\\ N(y_{m})\leq B/\prod_{j=1}^{m-1}N(y_{j})\end{subarray}}1$
$\displaystyle\leq\sum_{\begin{subarray}{c}(y_{j})_{j=1}^{m-1}\in\operatorname{Div}_{\geq
0}(F)^{m-1}\\\ \prod_{j=1}^{m-1}N(y_{j})\leq
B\end{subarray}}\frac{C_{1}B}{\prod_{j=1}^{m-1}N(y_{i})}$
$\displaystyle=C_{1}B\sum_{k=1}^{\lfloor B\rfloor}\frac{g(k)}{k}$
$\displaystyle=C_{1}B\bigg{(}\frac{1}{\lfloor B\rfloor}\sum_{j=1}^{\lfloor
B\rfloor}g(k)+\sum_{j=1}^{\lfloor
B\rfloor-1}\big{(}\frac{1}{j}-\frac{1}{j+1}\big{)}\sum_{k=1}^{j}g(k)\bigg{)}$
$\displaystyle\leq 2C_{1}C_{m-1}\log(\lfloor
B\rfloor+1)^{m-2}+C_{1}B\sum_{j=1}^{\lfloor
B\rfloor-1}\big{(}\frac{1}{j}-\frac{1}{j+1}\big{)}C_{m-1}j\log(j+1)^{m-2}$
$\displaystyle\leq
2C_{1}C_{m-1}\log(B+1)^{m-2}+C_{1}C_{m-1}B\sum_{j=1}^{\lfloor
B\rfloor-1}\frac{\log(j+1)^{m-2}}{j}$ $\displaystyle\leq
C_{m}^{\prime}\log(B+1)^{m-2}+C_{m}^{\prime}B\log(\lfloor
B\rfloor)^{m-2}\sum_{j=1}^{\lfloor B\rfloor-1}\frac{1}{j+1}$
$\displaystyle\leq C_{m}^{\prime}\log(B+1)^{m-2}+C_{m}^{\prime}B\log(\lfloor
B\rfloor)\log(\lfloor B\rfloor+1)^{m-1}$ $\displaystyle\leq
C_{m}B\log(B+1)^{m-1}.$

The statement follows. ∎

We can estimate the number of elements of
$\operatorname{Div}(F)^{n}/\mathbf{a}\operatorname{Div}(F)$ having
$H_{\operatorname{Ideal}}$ less than $B$.

######  Corollary 3.

There exists $C>0$ such that for any $B>0$, one has

$|\\{\mathbf{x}\in\operatorname{Div}(F)^{n}/\mathbf{a}\operatorname{Div}(F)|\hskip
2.84544pt{H_{\operatorname{Ideal}}}(\mathbf{x})\leq B\\}|\leq
CB\log(1+B)^{n-1}.$

###### Proof.

By 2, there exists $C_{n}>0$ such that:

$\displaystyle|\\{\mathbf{x}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}|\hskip
2.84544pt{H}_{\operatorname{Ideal}}(\mathbf{x})<B\\}|\hskip-85.35826pt$
$\displaystyle=|\\{\mathbf{y}\in\operatorname{Div}_{\mathbf{a}-\operatorname{prim}}(F)|\prod_{j=1}^{n}N(y_{j})<B\\}|$
$\displaystyle\leq|\\{\mathbf{y}\in\operatorname{Div}(F)^{n}|{\mathbf{y}}\subset\mathcal{O}_{F}^{n},\prod_{j=1}^{n}N(y_{j})<B\\}|$
$\displaystyle\leq C_{n}B\log(1+B)^{n-1}.$

The statement is proven. ∎

#### 4

For $v\in M_{F}$, we let
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$
be the toric $\mathbf{a}$-homogenous function of weighted degree
$|\mathbf{a}|$. Let $H^{\\#}$ be the corresponding toric height on
$[\mathscr{P}(\mathbf{a})(F)]$. For $v\in M_{F}$, we let $H^{\\#}_{v}$ to be
the “local heights” from 4 i.e. the functions
$[\mathscr{T}({\mathbf{a}})(F_{v})]\to\mathbb{R}_{>0}$ induced from
$F_{v}^{\times}$-invariant maps

$(F_{v}^{\times})^{n}\to\mathbb{R}_{>0},\hskip 28.45274pt\mathbf{x}\mapsto
f^{\\#}_{v}(\mathbf{x})\prod_{j=1}^{n}|x_{j}|_{v}^{-1}.$

In this paragraph we relate $\prod_{v\in M_{F}^{0}}H^{\\#}_{v}$ and
$H_{\operatorname{Ideal}}$. For $x\in F^{\times}$, let us denote by
$\mathfrak{I}(x)\in\operatorname{Div}(F)$ the fractional ideal
$x\mathcal{O}_{F}$. We denote by $\mathfrak{I}^{n}$ the product homomorphism
$(F^{\times})^{n}\to\operatorname{Div}(F)^{n}.$ Note that if
$\mathbf{x}\in(F^{\times})_{\mathbf{a}}=\\{(x^{a_{j}})_{j}|x\in
F^{\times}\\}$, then
$\mathfrak{I}^{n}(\mathbf{x})\in\operatorname{Div}(F)_{\mathbf{a}}.$ We denote
by $\overline{\mathfrak{I}}$ the induced homomorphism

$\overline{\mathfrak{I}}\mathrel{\mathop{\ordinarycolon}}[\mathscr{T}({\mathbf{a}})(F)]=(F^{\times})^{n}/(F^{\times})_{\mathbf{a}}\to\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$
(1)

from $(F^{\times})_{\mathbf{a}}$-invariant map

$F^{\times
n}\xrightarrow{\mathbf{x}\mapsto\mathfrak{I}^{n}(\mathbf{x})}\operatorname{Div}(F)^{n}\to\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}},$

where the second homomorphism is the quotient homomorphism.

######  Lemma 2.

Let $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$. One has that

$H_{\operatorname{Ideal}}(\overline{\mathfrak{I}}(\mathbf{x}))=\prod_{v\in
M_{F}^{0}}H^{\\#}_{v}(\mathbf{x}).$

###### Proof.

Let $\widetilde{\mathbf{x}}\in(F^{\times})^{n}$ be a lift of $\mathbf{x}$. It
follows from 1 that

$\mathfrak{I}^{n}(\widetilde{\mathbf{x}})(y(\mathfrak{I}^{n}(\widetilde{\mathbf{x}}))^{a_{j}})_{j}=(\mathfrak{I}(\widetilde{x}_{j})\prod_{v\in
M_{F}^{0}}(\pi_{v}^{a_{j}r_{v}(\mathfrak{I}^{n}(\widetilde{\mathbf{x}}))}\mathcal{O}_{v}\cap\mathcal{O}_{F}))_{j}$

is the unique lift of $\overline{\mathfrak{I}}(\mathbf{x})$ lying in
$\operatorname{Div}(F)_{\mathbf{a}-\operatorname{prim}}$. We can calculate (if
$0\neq I\in\operatorname{Div}(F)$, we write $v(I)$ for the exponent of the
prime ideal corresponding to $v$ in the prime factorization of $I$):

$\displaystyle H_{\operatorname{Ideal}}(\overline{\mathfrak{I}}(\mathbf{x}))$
$\displaystyle=\prod_{j=1}^{n}N\bigg{(}\mathfrak{I}(\widetilde{x}_{j})\prod_{v\in
M_{F}^{0}}(\pi_{v}^{a_{j}r_{v}(\mathfrak{I}^{n}(\widetilde{\mathbf{x}}))}\mathcal{O}_{v}\cap\mathcal{O}_{F})\bigg{)}$
$\displaystyle=\prod_{j=1}^{n}N\bigg{(}\prod_{v\in
M_{F}^{0}}\pi_{v}^{v(\mathfrak{I}(\widetilde{x}_{j}))+a_{j}r_{v}(\mathfrak{I}^{n}(\widetilde{\mathbf{x}}))}\mathcal{O}_{v}\cap\mathcal{O}_{F}\bigg{)}$
$\displaystyle=\prod_{j=1}^{n}\prod_{v\in
M_{F}^{0}}|\pi_{v}|_{v}^{-(v(\mathfrak{I}(\widetilde{x}_{j}))+a_{j}r_{v}(\mathfrak{I}^{n}(\widetilde{\mathbf{x}})))}$
$\displaystyle=\prod_{v\in
M_{F}^{0}}\prod_{j=1}^{n}|\pi_{v}|_{v}^{-(v(\widetilde{x}_{j})+a_{j}r_{v}(\widetilde{\mathbf{x}}))}$
$\displaystyle=\prod_{v\in
M_{F}^{0}}|\pi_{v}|_{v}^{-|\mathbf{a}|r_{v}(\widetilde{\mathbf{x}})}\prod_{j=1}^{n}|\pi_{v}|_{v}^{-v(\widetilde{x}_{j})}$
$\displaystyle=\prod_{v\in
M_{F}^{0}}f_{v}(\widetilde{\mathbf{x}})\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}$
$\displaystyle=\prod_{v\in M_{F}^{0}}H^{\\#}_{v}(\mathbf{x}).$

∎

#### 5

In this paragraph we study the kernel of the homomorphism
$\overline{\mathfrak{I}}$ defined in 4. Let $U$ be the group of units of $F$
and let
$U_{\mathbf{a}}\mathrel{\mathop{\ordinarycolon}}=\\{(u^{a_{j}})_{j}|u\in
U\\}.$ As $U_{\mathbf{a}}=(F^{\times})_{\mathbf{a}}\cap U^{n}$, we have a
canonical identification of $U^{n}/U_{\mathbf{a}}$ with a subgroup of
$[\mathscr{T}({\mathbf{a}})(F)]$. Note that $U\subset\ker(\mathfrak{I})$ and
thus $U^{n}\subset\ker(\mathfrak{I}^{n}).$ It follows that

$U^{n}/U_{\mathbf{a}}\subset\ker(\overline{\mathfrak{I}}).$

We prove the following fact:

######  Lemma 1.

Let $d=\gcd(\mathbf{a})$. One has that

$(\ker({\overline{\mathfrak{I}}})\mathrel{\mathop{\ordinarycolon}}(U^{n}/U_{\mathbf{a}}))=|{\operatorname{Cl}(F)}[{d}]|,$

where ${\operatorname{Cl}(F)}[{d}]$ is the $d$-torsion subgroup of the class
group $\operatorname{Cl}(F)$.

###### Proof.

The following “snake diagram” is commutative

${U}$${{U^{n}}}$${{\ker(\overline{\mathfrak{I}})}}$${{F^{\times}}}$${{(F^{\times})^{n}}}$${{[\mathscr{T}({\mathbf{a}})(F)]}}$${1}$${0}$${{\operatorname{Div}(F)}}$${{\operatorname{Div}(F)^{n}}}$${{\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}}}$${{\operatorname{Cl}(F)}}$${{\operatorname{Cl}(F)^{n}}}$${{\operatorname{coker}(\overline{\mathfrak{I}}),}}$$\scriptstyle{t\mapsto(t^{a_{j}})_{j}}$$\scriptstyle{\mathfrak{I}}$$\scriptstyle{\mathfrak{I}^{n}}$$\scriptstyle{\overline{\mathfrak{I}}}$$\scriptstyle{t\mapsto(t^{a_{j}})_{j}}$$\scriptstyle{t\mapsto(t^{a_{j}})_{j}}$

where $\operatorname{Cl}(F)$ is the class group of $F$. All vertical and
horizontal sequences are exact. The snake lemma provides an exact sequence

$U\xrightarrow{t\mapsto(t^{a_{j}})_{j}}U^{n}\rightarrow\ker(\overline{\mathfrak{I}})\xrightarrow{\sigma}\operatorname{Cl}(F)\xrightarrow{t\mapsto(t^{a_{j}})_{j}}\operatorname{Cl}(F)^{n}\to\operatorname{coker}(\mathfrak{I}).$

It follows that
$\ker(\sigma)=\operatorname{Im}\big{(}U^{n}\to\ker(\overline{\mathfrak{I}})\big{)}=U^{n}/U_{\mathbf{a}}$.
The kernel of

$\operatorname{Cl}(F)\to\operatorname{Cl}(F)^{n}\hskip
28.45274ptt\mapsto(t^{a_{j}})_{j}$

is given by the subgroup $\operatorname{Cl}(F)[d]$. We deduce that

$|\operatorname{Cl}(F)[d]|=|\operatorname{Im}(\sigma)|=|\ker(\overline{\mathfrak{I}})/\ker(\sigma)|=(\ker({\overline{\mathfrak{I}}})\mathrel{\mathop{\ordinarycolon}}(U^{n}/U_{\mathbf{a}})).$

∎

#### 6

In this paragraph for fixed $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$, we
bound the number of elements $\mathbf{u}\in U^{n}/U_{\mathbf{a}}$ for which
$\prod_{v\in M_{F}^{\infty}}H_{v}^{\\#}(\mathbf{u}\mathbf{x})<B$.

We define an auxiliary function
$h_{\infty}\mathrel{\mathop{\ordinarycolon}}(\mathbb{R}^{M_{F}^{\infty}})^{n}\to\mathbb{R}$
by

$h_{\infty}\mathrel{\mathop{\ordinarycolon}}(\mathbf{y}_{v})_{v}\mapsto\sum_{v\in
M_{F}^{\infty}}\mathbf{a}\cdot\big{(}\max_{k=1,\dots,n}\bigg{(}\frac{y_{vk}}{a_{k}}-\frac{y_{vj}}{a_{j}}\bigg{)}\big{)}_{j}.$

We define a homomorphism

$\rho\mathrel{\mathop{\ordinarycolon}}F^{\times}\to\mathbb{R}^{M_{F}^{\infty}}\hskip
28.45274ptx\mapsto\log(|x|_{v}),$

and a homomorphism

$\rho^{n}\mathrel{\mathop{\ordinarycolon}}(F^{\times})^{n}\to(\mathbb{R}^{M_{F}^{\infty}})^{n}\hskip
28.45274pt\mathbf{x}\mapsto(\rho(x_{j}))_{j}.$

######  Lemma 1.

Let $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$ and let
$\widetilde{\mathbf{x}}\in(F^{\times})^{n}$ be its lift. For every
$\mathbf{s}\in\mathbb{R}^{n}_{>0}$, we have that

$\log\bigg{(}\prod_{v\in
M_{F}^{\infty}}H^{\\#}_{v}(\mathbf{x})\bigg{)}=h_{\infty}(\rho^{n}(\widetilde{\mathbf{x}})).$

###### Proof.

For $v\in M_{F}^{\infty}$, we have that

$\displaystyle\log(H^{\\#}_{v}(\mathbf{x}))$
$\displaystyle=\log\big{(}\max_{k}(|\widetilde{x}_{k}|^{1/a_{k}}_{v})^{|\mathbf{a}|}\prod_{j=1}^{n}|\widetilde{x}_{j}|_{v}^{-1}\big{)}$
$\displaystyle=\log\big{(}\prod_{j=1}^{n}\max_{k}(|\widetilde{x}_{k}|^{1/a_{k}}_{v})^{a_{j}}|\widetilde{x}_{j}|_{v}^{-1}\big{)}$
$\displaystyle=\log\big{(}\prod_{j=1}^{n}\max_{k}(|\widetilde{x}_{k}|^{a_{j}/a_{k}}_{v})|\widetilde{x}_{j}|^{-1}_{v}\big{)}$
$\displaystyle=\log\big{(}\prod_{j=1}^{n}\max_{k}(|\widetilde{x}_{k}|^{a_{j}/a_{k}}_{v}|\widetilde{x}_{j}|_{v}^{-1})\big{)}$
$\displaystyle=\sum_{j=1}^{n}\log\big{(}\max_{k}\bigg{(}\frac{|\widetilde{x}_{k}|^{1/a_{k}}_{v}}{|\widetilde{x}_{j}|^{1/a_{j}}_{v}}\bigg{)}^{a_{j}}\big{)}$
$\displaystyle=\sum_{j=1}^{n}a_{j}\max_{k}\bigg{(}\frac{\log(|\widetilde{x}_{k}|_{v})}{a_{k}}-\frac{\log(|\widetilde{x}_{j}|_{v})}{a_{j}}\bigg{)}$
$\displaystyle=\mathbf{a}\cdot\big{(}\max_{k}\bigg{(}\frac{\log(|\widetilde{x}_{k}|_{v})}{a_{k}}-\frac{\log(|\widetilde{x}_{j}|_{v})}{a_{j}}\bigg{)}\big{)}_{j}.$

We deduce:

$\log\big{(}\prod_{v\in
M_{F}^{\infty}}H^{\\#}_{v}(\mathbf{x})\big{)}=\sum_{v\in
M_{F}^{\infty}}\mathbf{a}\cdot\big{(}\max_{k}\bigg{(}\frac{\log(|\widetilde{x}_{k}|_{v})}{a_{k}}-\frac{\log(|\widetilde{x}_{j}|_{v})}{a_{j}}\bigg{)}\big{)}_{j}\\\
=h_{\infty}(\rho(\widetilde{\mathbf{x}})).$

∎

######  Lemma 2.

For $j=1,\dots,n$, let us define a homomorphism
$\alpha_{j}\mathrel{\mathop{\ordinarycolon}}(\mathbb{R}^{M_{F}^{\infty}})^{n}\to(\mathbb{R}^{M_{F}^{\infty}})^{n}$
by

$\alpha_{j}\mathrel{\mathop{\ordinarycolon}}(\mathbf{y}_{i})_{i}\mapsto\big{(}\frac{\mathbf{y}_{i}}{a_{i}}-\frac{\mathbf{y}_{j}}{a_{j}}\big{)}_{i}.$

Set
$\alpha=\prod_{j=1}^{n}\alpha_{j}\mathrel{\mathop{\ordinarycolon}}(\mathbb{R}^{M_{F}^{\infty}})^{n}\to(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}.$

  1. 1.

One has that

$\ker(\alpha)=\big{\\{}(a_{j}\mathbf{y})_{j}|\mathbf{y}\in\mathbb{R}^{M_{F}^{\infty}}\big{\\}}.$

  2. 2.

Let $K>0$ and let $(\mathbf{y}_{j})_{j}\in(\mathbb{R}^{M_{F}^{\infty}})^{n}$
be such that $h_{{\infty}}(\mathbf{y})<K.$ One has that

$\alpha(\mathbf{y})\in\bigg{[}\frac{-K}{a_{i}},\frac{K}{a_{j}}\bigg{]}_{v,i,j}.$

  3. 3.

Suppose $L$ is a lattice of $\mathbb{R}^{M_{F}^{\infty}}$. Then
$\alpha(L^{n})$ is contained in a lattice of
$(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}.$

###### Proof.

  1. 1.

For $k,\ell\in\\{1,\dots,n\\}$, one has that

$\ker(\alpha_{\ell})=\\{\big{(}\dfrac{a_{i}\mathbf{y}_{\ell}}{a_{\ell}}\big{)}_{i}|\mathbf{y}_{\ell}\in\mathbb{R}^{M_{F}^{\infty}}\\}=\\{(a_{i}\mathbf{y})_{i}|\mathbf{y}\in\mathbb{R}^{M_{F}^{\infty}}\\}\\\
=\\{\big{(}\dfrac{a_{i}\mathbf{y}_{k}}{a_{k}}\big{)}_{j}|\mathbf{y}_{k}\in\mathbb{R}^{M_{F}^{\infty}}\\}=\ker(\alpha_{k}).$

Therefore

$\ker(\alpha)=\bigcap_{j=1}^{n}\ker(\alpha_{j})=\ker(\alpha_{1})=\\{(a_{j}\mathbf{y})_{j}|\mathbf{y}\in\mathbb{R}^{M_{F}^{\infty}}\\}.$

  2. 2.

Let $\ell\in\\{1,\dots,n\\}.$ Note that from

$h_{\infty}(\mathbf{y})=\sum_{v\in
M_{F}^{\infty}}\mathbf{a}\cdot\big{(}\max_{k}\bigg{(}\frac{y_{vk}}{a_{k}}-\frac{y_{vj}}{a_{j}}\bigg{)}\big{)}_{j}<K,$

it follows that for every $i\in\\{1,\dots,n\\}$ and every $w\in
M_{F}^{\infty}$, one has that

$a_{\ell}\bigg{(}\frac{y_{wi}}{a_{i}}-\frac{y_{w\ell}}{a_{\ell}}\bigg{)}\leq
a_{\ell}\max_{k}\bigg{(}\frac{y_{vk}}{a_{k}}-\frac{y_{vj}}{a_{j}}\bigg{)}\leq\sum_{v\in
M_{F}^{\infty}}\mathbf{a}\cdot\big{(}\max_{k}\bigg{(}\frac{y_{vk}}{a_{k}}-\frac{y_{vj}}{a_{j}}\bigg{)}\big{)}_{j}<K,$

and hence that

$\frac{y_{wi}}{a_{i}}-\frac{y_{w\ell}}{a_{\ell}}<\frac{K}{a_{\ell}}.$

Similarly, for every $i\in\\{1,\dots,n\\}$ and every $w\in M_{F}^{\infty}$,
one has that

$\frac{y_{w\ell}}{a_{\ell}}-\frac{y_{wi}}{a_{i}}<\frac{K}{a_{i}}.$

We deduce that

$\frac{\mathbf{y}_{i}}{a_{i}}-\frac{\mathbf{y}_{j}}{a_{j}}\in\bigg{[}-\frac{K}{a_{i}},\frac{K}{a_{\ell}}\bigg{]}_{v\in
M_{F}^{\infty}}$

and hence that

$\alpha_{\ell}((\mathbf{y}_{i})_{i})=\big{(}\frac{\mathbf{y}_{i}}{a_{i}}-\frac{\mathbf{y}_{\ell}}{a_{\ell}}\big{)}_{i}\in\bigg{[}-\frac{K}{a_{i}},\frac{K}{a_{\ell}}\bigg{]}_{v,i}.$

Finally, it follows that

$\alpha((\mathbf{y}_{i})_{i})=(\alpha_{j}((\mathbf{y}_{i})_{i}))_{j}\in\bigg{[}-\frac{K}{a_{i}},\frac{K}{a_{j}}\bigg{]}_{v,i,j}.$

  3. 3.

Let $(\mathbf{y}_{j})_{j}\in L^{n}$. Then for every $i,j$ one has

$\frac{\mathbf{y}_{i}}{a_{i}}-\frac{\mathbf{y}_{j}}{a_{j}}\in\frac{L}{\prod_{j=1}^{n}a_{j}}.$

We deduce

$\alpha_{j}(\mathbf{y})=\big{(}\frac{\mathbf{y}_{i}}{a_{i}}-\frac{\mathbf{y}_{j}}{a_{j}}\big{)}_{i}\in\bigg{(}\frac{L}{\prod_{j=1}^{n}a_{j}}\bigg{)}_{i=1}^{n}.$

It follows that

$\alpha(L^{n})=(\alpha_{j}(L^{n}))_{j}\subset\bigg{(}\frac{L}{\prod_{j=1}^{n}a_{j}}\bigg{)}_{\ell=1}^{n^{2}}.$

The claim is proven.

∎

Let us set

$W\mathrel{\mathop{\ordinarycolon}}=\big{\\{}(w_{v})_{v}\in\mathbb{R}^{M_{F}^{\infty}}|\sum_{v}w_{v}=0\big{\\}}$

For $u\in U,$ we have that

$\rho(u)=(\log|u|_{v})_{v}\in W,$

and hence for $\mathbf{u}\in U^{n}$, we have that
$\rho^{n}(\mathbf{u})=(\rho(u_{j}))_{j}\in W^{n}.$

######  Lemma 3.

The following claims are valid:

  1. 1.

The homomorphism $\rho^{n}$ is of finite kernel and its image is a lattice of
$W^{n}$.

  2. 2.

One has that $W^{n}\cap\ker(\alpha)=\\{(a_{j}\mathbf{w})_{j}|\mathbf{w}\in
W\\}$.

  3. 3.

One has that $\operatorname{rk}(\ker(\alpha\circ(\rho^{n}|_{U^{n}})))\leq
r_{1}+r_{2}-1,$ where $r_{1}$ is the number of the real and $r_{2}$ is the
number of complex places of $F$.

  4. 4.

One has that $U_{\mathbf{a}}\subset\ker(\alpha\circ(\rho^{n}|_{U^{n}}))$ and
the map
$\beta\mathrel{\mathop{\ordinarycolon}}U^{n}/U_{\mathbf{a}}\to(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}},$
induced from $U_{\mathbf{a}}$-invariant map $\alpha\circ(\rho^{n}|_{U^{n}})$,
is of finite kernel.

  5. 5.

The image $\beta(U^{n}/U_{\mathbf{a}})$ is contained in a lattice of
$(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}$.

###### Proof.

  1. 1.

By Dirichlet’s unit theorem, one has that $\ker(\rho|_{U})$ is finite and that
$\rho(U)$ is a lattice of $W$. It follows that
$\ker(\rho^{n}|_{U^{n}})=\ker(\rho|_{U})^{n}$ is finite and that
$\rho^{n}(U^{n})=\rho(U)^{n}$ is a lattice of $W^{n}$.

  2. 2.

By 2 one has that
$\ker(\alpha)=\\{(a_{j}\mathbf{y})_{j}|\mathbf{y}\in\mathbb{R}^{M_{F}^{\infty}}\\}$
and thus

$W^{n}\cap\ker(\alpha)=W^{n}\cap\\{(a_{j}\mathbf{y})_{j}|\mathbf{y}\in\mathbb{R}^{M_{F}^{\infty}}\\}=\\{(a_{j}\mathbf{w})_{j}|\mathbf{w}\in
W\\}.$

  3. 3.

By (1), the kernel of
$\rho^{n}|_{\ker(\alpha\circ\rho^{n}|_{U^{n}})}\mathrel{\mathop{\ordinarycolon}}\ker(\alpha\circ\rho^{n}|_{U^{n}})\to
W^{n}$ is finite. It suffices therefore to show that the image
$\rho^{n}({\ker(\alpha\circ\rho^{n}|_{U^{n}})})$ is of rank no more than
$r_{1}+r_{2}-1$. The image $\rho^{n}({\ker(\alpha\circ\rho^{n}|_{U^{n}})})$ is
contained in $\ker(\alpha)$ and by (1), it is also contained in the lattice
$\rho^{n}(U^{n})$ of $W^{n},$ thus

$\rho^{n}({\ker(\alpha\circ\rho^{n}|_{U^{n}})})\subset\rho^{n}(U^{n})\cap\ker(\alpha)=\rho^{n}(U^{n})\cap
W^{n}\cap\ker(\alpha).$

The rank of the intersection of the lattice $\rho^{n}(U^{n})$ of $W^{n}$ and
the vector subspace $\ker(\alpha)\cap W^{n}$ of $W^{n}$ cannot be more than
$\dim(W^{n}\cap\ker(\alpha))=\dim(\\{(a_{j}\mathbf{w})_{j}|\mathbf{w}\in
W\\})=r_{1}+r_{2}-1$. The claim follows.

  4. 4.

Let $(u^{a_{j}})_{j}\in U_{\mathbf{a}}$, where $u\in U$. We have that

$\rho^{n}((u^{a_{j}})_{j})=(a_{j}\rho(u))_{j}\in\ker(\alpha).$

We deduce $U_{\mathbf{a}}\subset\ker(\alpha\circ(\rho^{n}|_{U^{n}}))$. Let us
establish that $U_{\mathbf{a}}$ is of finite index in
$\ker(\alpha\circ(\rho^{n}|_{U^{n}}))$. The map

$U\to U^{n}\hskip 28.45274ptu\mapsto(u^{a_{j}})_{j}$

is of finite kernel and thus

$\operatorname{rk}(U_{\mathbf{a}})=\operatorname{rk}(U)=r_{1}+r_{2}-1\geq\operatorname{rk}(\ker(\alpha\circ(\rho^{n}|_{U^{n}}))).$

We conclude that
$\operatorname{rk}(U_{\mathbf{a}})=r_{1}+r_{2}-1=\operatorname{rk}(\ker(\alpha\circ(\rho^{n}|_{U^{n}}))),$
and hence is $U_{\mathbf{a}}$ of finite index in
$\ker(\alpha\circ(\rho^{n}|_{U^{n}})).$ Now, it follows that the homomorphism
$\beta$, which is precisely the composite homomorphism

$\beta\mathrel{\mathop{\ordinarycolon}}U^{n}/U_{\mathbf{a}}\to
U^{n}/\ker(\alpha\circ(\rho^{n}|_{U^{n}}))\hookrightarrow(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}),$

is of finite kernel. The claim is proven.

  5. 5.

One has that $\beta(U^{n}/U_{\mathbf{a}})=\alpha(\rho(U^{n})).$ By 2 one has
that $\alpha(\rho^{n}(U^{n}))$ is contained in a lattice of
$(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}$. The statement follows.

∎

We now prove the principal result of the paragraph

######  Lemma 4.

There exists $C>0$ such that for every
$\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$ and every $B>2$, one has that

$\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|H^{\\#}_{\infty}(\mathbf{u}\mathbf{x})<B\\}\leq
C\big{(}\log(B)\big{)}^{n^{2}(r_{1}+r_{2})}.$

###### Proof.

For $\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]$, let
$\widetilde{\mathbf{x}}\in(F^{\times})^{n}$ be a lift of $\mathbf{x}$ and for
$\mathbf{u}\in U^{n}/U_{\mathbf{a}},$ let $\widetilde{\mathbf{u}}\in U^{n}$ be
a lift of $\mathbf{u}$. For $B>1$, by 1, one has

$\displaystyle\begin{split}\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|H^{\\#}_{\infty}(\mathbf{u}\mathbf{x})<B\\}&=\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|h_{\infty}(\rho^{n}(\widetilde{\mathbf{u}}\widetilde{\mathbf{x}}))<B\\}\\\
&=\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|h_{\infty}(\rho^{n}(\widetilde{\mathbf{x}})+\rho^{n}(\widetilde{\mathbf{u}}))<\log(B)\\}.\end{split}$
(5)

Further, by using 2 and the fact that $\ker(\beta)$ is finite (3), we obtain
for every $B>1$ that

$\displaystyle\begin{split}|\\{\mathbf{u}&\in
U^{n}/U_{\mathbf{a}}|h_{\infty}(\rho^{n}(\widetilde{\mathbf{x}})+\rho(\widetilde{\mathbf{u}}))<\log(B)\\}|\\\
&=\big{|}\big{\\{}\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|\alpha(\rho^{n}(\widetilde{\mathbf{x}})+\rho^{n}(\widetilde{\mathbf{u}}))\in\big{[}\frac{-\log(B)}{a_{i}},\frac{\log(B)}{a_{i}}\big{]}_{v,i,j}\big{\\}}\big{|}\\\
&=\big{|}\big{\\{}\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|\alpha(\rho^{n}(\widetilde{\mathbf{u}}))\in-\alpha(\rho^{n}(\widetilde{\mathbf{x}}))+\big{[}\frac{-\log(B)}{a_{i}},\frac{\log(B)}{a_{i}}\big{]}_{v,i,j}\big{\\}}\big{|}\\\
&=\big{|}\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|\beta(\mathbf{u})\in-\alpha(\rho^{n}(\widetilde{\mathbf{x}}))+\big{[}\frac{-\log(B)}{a_{i}},\frac{\log(B)}{a_{j}}\big{]}_{v,i,j}\\}\big{|}\\\
&=|\ker(\beta)|\cdot\big{|}\\{\beta(U^{n}/U_{\mathbf{a}})\cap\big{(}-\alpha(\rho^{n}(\widetilde{\mathbf{x}}))+\big{[}\frac{-\log(B)}{a_{i}},\frac{\log(B)}{a_{j}}\big{]}_{v,i,j}\big{)}\\}\big{|}.\end{split}$
(6)

As $\beta(U^{n}/U_{\mathbf{a}})$ is a subgroup (3) of a lattice of
$(\mathbb{R}^{M_{F}^{\infty}})^{n^{2}}$, there exists $C^{\prime}>0$ such that

$\displaystyle\begin{split}\bigg{|}\beta(U^{n}/U_{\mathbf{a}})\cap\big{(}-\alpha(\rho(\widetilde{\mathbf{x}}))+\big{[}\frac{-\log(B)}{a_{i}},\frac{\log(B)}{a_{j}}\big{]}_{v,i,j}\big{)}\\}\bigg{|}\hskip-142.26378pt&\\\
&\leq
C^{\prime}\prod_{i,j}\big{(}\frac{\log(B)}{a_{i}}+\frac{\log(B)}{a_{j}}\big{)}^{r_{1}+r_{2}}\\\
&\leq
C^{\prime}\prod_{i,j}\big{(}\log(B^{1/a_{i}+1/a_{j}})\big{)}^{r_{1}+r_{2}}\\\
&=C^{\prime}\big{(}1/a_{i}+1/a_{j}\big{)}^{n^{2}(r_{1}+r_{2})}\big{(}\log(B)\big{)}^{n^{2}(r_{1}+r_{2})}.\end{split}$
(7)

for every $B>2$. By combining $(\ref{robuu}),(\ref{robou})$ and $(\ref{robo})$
we deduce that there exists $C>0$ such that for every $B>2$ one has that

$\displaystyle\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|H^{\\#}_{\infty}(\mathbf{u}\mathbf{x})<B\\}$
$\displaystyle\leq C\big{(}\log(B)\big{)}^{n^{2}(r_{1}+r_{2})}.$

∎

#### 7

We prove there are only finitely many isomorphism classes of rational points
of bounded height on the “stacky torus” $\mathscr{T}({\mathbf{a}})$.

######  Proposition 1.

Let
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0})_{v}$
be a quasi-toric family of $\mathbf{a}$-homogenous functions of weighted
degree $|\mathbf{a}|$. There exists $C>0$ such that for every $B>0$ one has

$\big{|}\big{\\{}\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})<B\big{\\}}\big{|}<CB\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1}.$

###### Proof.

Let $(f_{v}^{\\#})_{v}$ be the family of toric $\mathbf{a}$-homogenous
continuous functions of weighted degree $|\mathbf{a}|$. We firstly prove the
statement for the family $(f_{v}=f_{v}^{\\#})_{v}$. Recall that
$\overline{\mathfrak{I}}\mathrel{\mathop{\ordinarycolon}}[\mathscr{T}({\mathbf{a}})(F)]\to\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$
is the induced homomorphism from
$\operatorname{Div}(F)_{\mathbf{a}}$-invariant homomorphism
$(F^{\times})^{n}\to\operatorname{Div}(F)^{n}\to\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$.
We set

$\displaystyle H_{0}^{\\#}(\mathbf{x})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\prod_{v\in
M_{F}^{0}}H_{v}^{\\#}(\mathbf{x})$ $\displaystyle
H_{\infty}^{\\#}(\mathbf{x})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=\prod_{v\in
M_{F}^{\infty}}H_{v}^{\\#}(\mathbf{x}).$

Recall that for every $v\in M_{F}$ we have $H^{\\#}_{v}(\mathbf{x})\geq 1$.
Using this fact and 2, for $B>0$ we deduce that

$\displaystyle\begin{split}\big{|}\big{\\{}\mathbf{x}\in[\mathscr{T}^{\mathbf{a}}(F)]|H^{\\#}(\mathbf{x})<B\big{\\}}\big{|}\hskip-56.9055pt&\\\
&\leq\big{|}\big{\\{}\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H^{\\#}_{0}(\mathbf{x})<B,H^{\\#}_{\infty}(\mathbf{x})<B\big{\\}}\big{|}\\\
&\leq\big{|}\big{\\{}\mathbf{x}\in\mathscr{T}^{\mathbf{a}}(F)|{H_{\operatorname{Ideal}}}(\overline{\mathfrak{I}}(\mathbf{x}))<B,H^{\\#}_{\infty}(\mathbf{x})<B\big{\\}}\big{|}.\end{split}$
(2)

For
$\mathbf{y}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}},$
let $\widetilde{\mathbf{y}}\in\operatorname{Div}(F)^{n}$ be a lift of
$\mathbf{y}$. We have that

$\big{|}\big{\\{}\mathbf{x}\in[\mathscr{T}^{\mathbf{a}}(F)]|{H_{\operatorname{Ideal}}}(\overline{\mathfrak{I}}(\mathbf{x}))<B,H^{\\#}_{\infty}(\mathbf{x})<B\big{\\}}\big{|}\\\
=\big{|}\\{(\mathbf{y},\mathbf{z})\in(\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}})\times\ker(\overline{\mathfrak{I}})|H_{\operatorname{Ideal}}(\mathbf{y})<B,H^{\\#}_{\infty}(\mathbf{z}\widetilde{\mathbf{y}})<B\\}\big{|}$

and thus by (2) we get that

$\big{|}\big{\\{}\mathbf{x}\in[\mathscr{T}^{\mathbf{a}}(F)]|H(\mathbf{x})<B\big{\\}}\big{|}\\\
\leq\big{|}\big{\\{}(\mathbf{y},\mathbf{z})\in(\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}})\times\ker(\overline{\mathfrak{I}})|H_{\operatorname{Ideal}}(\mathbf{y})<B,H^{\\#}_{\infty}(\mathbf{z}\widetilde{\mathbf{y}})<B\big{\\}}\big{|}.$
(3)

Recall from 5, that $U^{n}/U_{\mathbf{a}}\subset\ker(\mathfrak{I}).$ Let
$\delta$ be a set theoretical section to the quotient map
$\ker(\overline{\mathfrak{I}})\to\ker(\overline{\mathfrak{I}})/(U^{n}/U_{\mathbf{a}})$.
By 1, the group $\ker(\overline{\mathfrak{I}})/(U^{n}/U_{\mathbf{a}})$ is
finite, thus is $\delta(\ker(\overline{\mathfrak{I}})/(U^{n}/U_{\mathbf{a}}))$
finite. By using the estimate from 4, we deduce that there exists $C_{0}>0$
such that for every
$\mathbf{y}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}$
and every $B>0$ one has

$\displaystyle\begin{split}|\\{\mathbf{z}\in\ker(\overline{\mathfrak{I}})|H^{\\#}_{\infty}(\mathbf{z}\widetilde{\mathbf{y}})<B\\}|\hskip-113.81102pt&\\\
&\leq\big{|}\big{\\{}(\mathbf{u},\mathbf{d})\in(U^{n}/U_{\mathbf{a}})\times\delta(\ker(\overline{\mathfrak{I}})/(U^{n}/U_{\mathbf{a}}))\big{|}H^{\\#}_{\infty}(\mathbf{u}\mathbf{d}\widetilde{\mathbf{y}})<\log(B)\big{\\}}\big{|}\\\
&\leq\sum_{\mathbf{d}\in\delta(\ker(\overline{\mathfrak{I}})/(U^{n}/U_{\mathbf{a}}))}|\\{\mathbf{u}\in
U^{n}/U_{\mathbf{a}}|H^{\\#}_{\infty}(\mathbf{u}\mathbf{d}\widetilde{\mathbf{y}})<\log(B)\\}|\\\
&\leq C_{0}(\log(B+2))^{n^{2}(r_{1}+r_{2})}.\end{split}$ (4)

Corollary 3 gives that there exists $C_{1}>0$ such that for every $B>0$ one
has that

$\displaystyle|\\{\mathbf{y}\in\operatorname{Div}(F)^{n}/\mathbf{a}\operatorname{Div}(F)|\hskip
2.84544pt{H_{\operatorname{Ideal}}}(\mathbf{y})<B\\}|$ $\displaystyle\leq
C_{1}B\log(1+B)^{n-1}.$

We deduce that for every $B>0$ one has

$\displaystyle\big{|}\big{\\{}$
$\displaystyle\mathbf{x}\in\mathscr{T}^{\mathbf{a}}(F)|H(\mathbf{x})<B\big{\\}}\big{|}$
$\displaystyle\leq\big{|}\big{\\{}(\mathbf{y},\mathbf{z})|(\mathbf{y},\mathbf{z})\in(\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}})\times\ker(\overline{\mathfrak{I}});$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
H_{\operatorname{Ideal}}(\mathbf{y})<B,H^{\\#}_{\infty}(\mathbf{z}\widetilde{\mathbf{y}})<B\big{\\}}\big{|}$
$\displaystyle\leq\big{|}\big{\\{}\mathbf{y}\in\operatorname{Div}(F)^{n}/\operatorname{Div}(F)_{\mathbf{a}}|H_{\operatorname{Ideal}}(\mathbf{y})<B\big{\\}}\big{|}\big{(}\log(B)\big{)}^{n^{2}(r_{1}+r_{2})}$
$\displaystyle\leq C_{0}C_{1}B\log(1+B)^{n-1}\log(2+B)^{n^{2}(r_{1}+r_{2})}$
$\displaystyle\leq CB\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1},$

where $C>0$ is big enough.

Let $(f_{v})_{v}$ now be any quasi-toric family of $\mathbf{a}$-homogenous
continuous functions of weighted degree $|\mathbf{a}|$. By 5, there exists
$C^{\prime}>0$ such that

${H^{\\#}(\mathbf{x})}<C^{\prime}{H(\mathbf{x})}\hskip
28.45274pt\forall\mathbf{x}\in F^{n}_{v}-\\{0\\}.$

We deduce that

$\displaystyle|\\{\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})<B\\}|$
$\displaystyle\leq|\\{\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H^{\\#}(\mathbf{x})<C^{\prime}B\\}|$
$\displaystyle\leq C(C^{\prime}B)\log(2+C^{\prime}B)^{n^{2}(r_{1}+r_{2})+n-1}$
$\displaystyle=C^{\prime\prime}B\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1}$

for $C^{\prime\prime}\gg 0$ and every $B>0$. The statement is proven. ∎

#### 8

In this paragraph we prove that quasi-toric heights are Northcott heights. For
$j\in\\{1,\dots,n\\}$, we will denote by $\\{j\\}^{c}$, the set
$\\{1,\dots,n\\}-\\{j\\}.$ For $j\in\\{1,\dots,n\\}$, we denote by
$p^{j}\mathrel{\mathop{\ordinarycolon}}\mathbb{A}^{n}\to\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1}$
the canonical projection and by $d_{j}$ the closed immersion

$d_{j}\mathrel{\mathop{\ordinarycolon}}\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1}\to\mathbb{A}^{n}\hskip
28.45274pt(x_{i})_{i}\mapsto((x_{i})_{i\neq j},(0)_{j}).$

By abuse of notation, we will shorten $p^{j}(\mathbb{Z})$ and write simply
$p^{j}$. The morphisms $d_{j}$ are $\mathbb{G}_{m}$-equivariant when
$\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1}$ is endowed with the action of
$\mathbb{G}_{m}$ given by

$t\cdot(x_{i})_{i}=(t^{a_{i}}x_{i})_{i}.$

For every $j$, one has that
$d_{j}^{-1}(\mathbb{A}^{n}-\\{0\\})=\big{(}\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1}\big{)}-\\{0\\}$.
We deduce $\mathbb{G}_{m}$-equivariant morphisms

$d_{j}|_{(\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1})-\\{0\\}}\mathrel{\mathop{\ordinarycolon}}\big{(}\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1}\big{)}-\\{0\\}\to\mathbb{A}^{n}-\\{0\\}.$

Let $\overline{d_{j}}$ denotes the induced closed immersion of stacks
$\mathscr{P}(p^{j}(\mathbf{a}))\to\mathscr{P}(\mathbf{a})$ from the
$\mathbb{G}_{m}$-equivariant morphism
$d_{j}|_{{(\prod_{i\in\\{j\\}^{c}}}\mathbb{A}^{1})-\\{0\\}}$.

######  Lemma 1.

Let $j\in\\{1,\dots,n\\}$.

  1. 1.

Let $v\in M_{F}$ and let
$f\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0}$ be an
$\mathbf{a}$-homogenous continuous function of weighted degree
$b\in\mathbb{Z}_{\geq 0}$. The function
$f\circ(d_{j}(F_{v}))\mathrel{\mathop{\ordinarycolon}}(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\}\to\mathbb{C}$
is $p^{j}(\mathbf{a})$-homogenous, continuous and of weighted degree $b$.
Moreover, if $f$ is the toric $\mathbf{a}$-homogenous function of weighted
degree $b$, then $f\circ(d_{j}(F_{v}))$ is the toric $\mathbf{a}$-homogenous
of weighted degree $b$.

  2. 2.

Let
$(g_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0})_{v}$ be a quasi-toric family of $\mathbf{a}$-homogenous continuous
functions of weighted degree $b$ and let $H=H((g_{v})_{v})$ be the
corresponding height. The family $g_{v}\circ d_{j}(F_{v})$ is a degree $b$
quasi-toric family of $p_{I}(\mathbb{Z})(\mathbf{a})$-homogenous continuous
functions and the corresponding height $H^{j}=H((g_{v}\circ
d_{I}(F_{v}))_{v})$ satisfies that

$H^{j}=H\circ(\overline{d_{j}}(F)).$

  3. 3.

Let
$\mathfrak{i}\mathrel{\mathop{\ordinarycolon}}\mathscr{T}({\mathbf{a}})\to\mathscr{P}(\mathbf{a})$
be the inclusion induced by the $\mathbb{G}_{m}$-invariant subscheme
$\mathbb{G}_{m}^{n}\subset\mathbb{A}^{n}-\\{0\\}$. The map

$\big{(}\coprod_{j=1}^{n}[\overline{d_{j}}(F)]\coprod[\mathfrak{i}(F)]\big{)}\mathrel{\mathop{\ordinarycolon}}\coprod_{j=1}^{n}[\mathscr{P}(p^{j}(\mathbb{Z})(\mathbf{a}))(F)]\coprod[\mathscr{T}({\mathbf{a}})(F)]\to[\mathscr{P}(\mathbf{a})(F)]$

is surjective.

###### Proof.

  1. 1.

The function $f\circ d_{j}(F_{v})$ is continuous, as it coincides with the
restriction $f|_{(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\}}$ and $f$ is
continuous. Let $t\in F_{v}^{\times}$ and let
$(x_{i})_{i\in\\{j\\}^{c}}\in(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\}$. We have
that

$\displaystyle f(d_{j}(F_{v})(t\cdot(x_{i})_{i\in\\{j\\}^{c}}))$
$\displaystyle=f(d_{j}(F_{v})((t^{a_{i}}x_{i})_{i\in\\{j\\}^{c}}))$
$\displaystyle=f((t^{a_{i}}x_{i})_{i\in\\{j\\}^{c}},(0)_{j})$
$\displaystyle=|t|_{v}^{b}f((x_{i})_{i\in\\{j\\}^{c}},(0)_{j})$
$\displaystyle=|t|_{v}^{b}(f\circ d_{j}(F_{v}))((x_{i})_{i\in\\{j\\}^{c}}),$

thus $f\circ d_{j}(F_{v})$ is $p^{j}(\mathbf{a})$ homogenous of weighted
degree $b$. Suppose $f$ is the toric $\mathbf{a}$-homogenous of weighted
degree $b$, that is

$f(\mathbf{y})=|\pi_{v}|_{v}^{-b\cdot r^{\mathbf{a}}_{v}(\mathbf{y})}\text{,
where }r^{\mathbf{a}}_{v}(\mathbf{y})={\sup_{\begin{subarray}{c}j=1,\dots,n\\\
y_{j}\neq 0\end{subarray}}\bigg{\lceil}\frac{-v(y_{j})}{a_{j}}\bigg{\rceil}},$

if $v$ is finite and

$f(\mathbf{y})=({\max_{j=1,\dots,n}(|y_{j}|^{1/a_{j}}_{v})})^{-b}$

if $v$ is infinite. Suppose $v$ is finite. Let
$\mathbf{x}\in(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\}$. One has that
$d_{j}(F_{v})(\mathbf{x})=((x_{i})_{i\in\\{j\\}^{c}},(0)_{j})$ and that

$(r^{\mathbf{a}}_{v}\circ(d_{j}(F_{v})))(\mathbf{x})=\sup_{\begin{subarray}{c}i\in\\{j\\}^{c}\\\
x_{i}\neq
0\end{subarray}}\bigg{\lceil}\frac{-v(x_{i})}{a_{i}}\bigg{\rceil}=\mathrel{\mathop{\ordinarycolon}}r^{p^{j}(\mathbf{a})}(\mathbf{x}),$

because the $j$-th coordinate of $d_{j}(F_{v})(\mathbf{x})$ is zero. It
follows that the function $f\circ(d_{j}(F_{v}))$ given by

$\mathbf{x}\mapsto f(d_{j}(F_{v})(\mathbf{x}))=|\pi_{v}|_{v}^{-b\cdot
r^{\mathbf{a}}_{v}(d_{j}(F_{v})(\mathbf{x}))}=|\pi_{v}|_{v}^{-b\cdot
r^{p^{j}(\mathbf{a})}_{v}(\mathbf{x})}$

is toric.

Suppose $v$ is infinite. The toric $p^{j}(\mathbf{a})$-homogenous function of
weighted degree $b$ is the function

$f^{\prime}\mathrel{\mathop{\ordinarycolon}}(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\}\to\mathbb{R}_{>0}\hskip
28.45274pt\mathbf{x}\mapsto\max_{i\in\\{j\\}^{c}}(|x_{i}|^{1/a_{i}}_{v}).$

For $\mathbf{x}\in(\prod_{i\in\\{j\\}^{c}}F_{v})-\\{0\\},$ we have that

$f(d_{j}(F_{v})(\mathbf{x}))={\max_{i\in\\{j\\}^{c}}\big{(}|x_{i}|^{1/a_{i}}_{v}}\big{)}=f^{\prime}(\mathbf{x}),$

because the $j$-th coordinate of $d_{j}(F_{v})(\mathbf{x})$ is zero. The claim
follows.

  2. 2.

By (1), for every $v\in M_{F}$, the function $g_{v}\circ d_{I}(F_{v})$ is a
continuous $p_{I}(\mathbb{Z})(\mathbf{a})$-homogenous of weighted degree $b$
and if $g_{v}$ is toric, then $g_{v}\circ d^{j}(F_{v})$ is toric of the same
weighted degree. It follows that the family $(g_{v}\circ d^{j}(F_{v}))_{v}$ is
quasi-toric of the degree $b$. Let
$\mathbf{y}\in[\mathscr{P}(p^{j}(\mathbf{a}))(F_{v})]$ and let
$\widetilde{\mathbf{y}}\in\ (\prod_{i\in\\{j\\}^{c}}F)-\\{0\\}$ be a lift.
Then $d^{j}(F)((\widetilde{y}_{i})_{i\in\\{j\\}^{c}})$ is a lift of
$\overline{d^{j}}(F)(\mathbf{y}).$ We have that

$H^{j}(\mathbf{y})=\prod_{v\in M_{F}}(g_{v}\circ
d^{j}(F_{v}))(\widetilde{\mathbf{y}})=\prod_{v\in
M_{F}}g_{v}(d^{j}(F)(\widetilde{\mathbf{y}}))=H((g_{v})_{v})(\mathbf{y})=H(\mathbf{y}).$

The claim follows.

  3. 3.

Let $\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]$ and let
$\widetilde{\mathbf{x}}\in F^{n}-\\{0\\}$ be a lift of $\mathbf{x}$. If
$\widetilde{\mathbf{x}}\in(F^{\times})^{n}$, then
$\mathbf{x}\cong[\mathfrak{i}(F)(q^{\mathbf{a}}(F)(\widetilde{\mathbf{x}}))],$
where $q^{\mathbf{a}}$ is the quotient $1$-morphism
$\mathbb{G}_{m}^{n}\to\mathscr{T}({\mathbf{a}})$. Suppose at least one
coordinate, say the $j$-th coordinate, of $\widetilde{\mathbf{x}}$ is equal to
zero. Then,

$\widetilde{\mathbf{x}}=d_{j}(F)((\widetilde{x}_{i})_{i\in\\{j\\}^{c}})$

and so

$\mathbf{x}\cong\overline{d_{j}}(q^{p^{j}(\mathbf{a})}((\widetilde{x}_{i})_{i\in\\{j\\}^{c}})),$

where
$q^{p^{j}(\mathbf{a})}\mathrel{\mathop{\ordinarycolon}}\big{(}(\prod_{i\in\\{j\\}^{c}}\mathbb{A}^{1})-\\{0\\}\big{)}\to\mathscr{P}(p^{j}(\mathbf{a}))$,
is the quotient $1$-morphism. The claim follows.

∎

######  Theorem 2.

Let $n\geq 1$ be an integer and let $\mathbf{a}\in\mathbb{Z}^{n}_{>0}$. Let
$(f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{>0})_{v}$
be a quasi-toric family of $\mathbf{a}$-homogenous functions of weighted
degree $|\mathbf{a}|$ and let $H=H((f_{v})_{v})$.

  1. 1.

There exists $C>0$ such that for every $B>0$ one has that

$|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]-[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})\leq
B\\}|\\\
<CB^{\frac{|\mathbf{a}|-\min_{i}a_{i}}{|\mathbf{a}|}}\log(2+B^{\frac{|\mathbf{a}|-\min_{i}a_{i}}{|\mathbf{a}|}})^{n^{2}(r_{1}+r_{2})+n-1}.$

  2. 2.

There exists $C>0$ such that for every $B>0$ one has that

$|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H(\mathbf{x})\leq
B\\}|<CB\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1}.$

###### Proof.

We prove the both statements simultaneously. We apply the induction on $n$.
When $n=1,$ one has that $\mathscr{P}(a)=\mathscr{T}(a)$. Thus the first claim
is trivial and the second is proven in 1.

Suppose that the both claims are true for some $n-1\geq 0$ and let us prove
them for $n$. The map

$\big{(}\coprod_{j=1}^{n}[\overline{d_{j}}(F)]\coprod[\mathfrak{i}(F)]\big{)}\mathrel{\mathop{\ordinarycolon}}\coprod_{j=1}^{n}[\mathscr{P}(p^{j}(\mathbb{Z})(\mathbf{a}))(F)]\coprod[\mathscr{T}({\mathbf{a}})(F)]\to[\mathscr{P}(\mathbf{a})(F)]$

is surjective by 1. For $j=1,\dots,n$, by 1, one has that

$H^{j}=H((f_{v}\circ(d_{j}(F_{v})))_{v})=H\circ\overline{d}_{j},\hskip
14.22636pt\text{where
}H^{j}\mathrel{\mathop{\ordinarycolon}}=H((f_{v}\circ(d_{j}(F_{v})))_{v}).$

We deduce that

$|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H(\mathbf{x})\leq B\\}|\\\
\leq|\\{\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})\leq
B\\}|+\sum_{j=1}^{n}|\\{\mathbf{x}\in[\mathscr{P}(p^{j}(\mathbf{a}))(F)]|H^{j}(\mathbf{x})\leq
B\\}|.$ (3)

For $j=1,\dots,n$, by 1, the height $H^{j}$ is defined by degree
$|\mathbf{a}|$ quasi-toric family. The family

$(g_{v}\circ(d_{j}(F_{v}))^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}})_{v}$

is quasi-toric of degree $|p^{j}(\mathbf{a})|$ and the resulting height
$H^{\prime}=H((g_{v}\circ(d_{j}(F_{v}))^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}})_{v})$
is related to $H^{j}$ as follows

$H^{\prime}=(H^{j})^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}.$

By the induction hypothesis, we deduce that for every $j=1,\dots,n$ there
exists $C_{j}>0$ such that for every $B>1$ one has that

$\displaystyle|\\{\mathbf{x}\in[\mathscr{P}(p^{j}$
$\displaystyle(\mathbf{a}))(F)]|H^{j}(\mathbf{x})\leq B\\}|$
$\displaystyle\leq|\\{\mathbf{x}\in[\mathscr{P}(p^{j}(\mathbf{a}))(F)]|H^{j}(\mathbf{x})^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}\leq
B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}\\}|$
$\displaystyle\leq|\\{\mathbf{x}\in[\mathscr{P}(p^{j}(\mathbf{a}))(F)]|H^{\prime}(\mathbf{x})\leq
B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}\\}|$ $\displaystyle\leq
C_{j}B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}\log(2+B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}})^{n^{2}(r_{1}+r_{2})+n-1}.$

It follows that

$\displaystyle|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]-[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})\leq
B\\}|\hskip-227.62204pt$
$\displaystyle=|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H(\mathbf{x})\leq
B\\}|-|\\{\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})\leq B\\}|$
$\displaystyle\leq\sum_{j=1}^{n}|\\{\mathbf{x}\in[\mathscr{P}(p^{j}(\mathbf{a}))(F)]|H^{j}(\mathbf{x})\leq
B\\}|$
$\displaystyle\leq\sum_{j=1}^{n}C_{j}B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}}\log(2+B^{\frac{|p^{j}(\mathbf{a})|}{|\mathbf{a}|}})$
$\displaystyle\leq
C^{\prime\prime}B^{\frac{|\mathbf{a}|-\min_{i}a_{i}}{|\mathbf{a}|}}\log(2+B^{\frac{|\mathbf{a}|-\min_{i}a_{i}}{|\mathbf{a}|}})$

for $C^{\prime\prime}\gg 0$. Thus the first claim is proven for $n$. By 1,
there exists $C^{\prime}>0$ such that

$|\\{\mathbf{x}\in[\mathscr{T}({\mathbf{a}})(F)]|H(\mathbf{x})\leq
B\\}|<C^{\prime}B\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1}.$ (4)

By combining (3), (4) and the first claim, we get that there exists $C>0$ such
that for every $B>1$ one has that

$|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H(\mathbf{x})\leq B\\}|\leq
CB\log(2+B)^{n^{2}(r_{1}+r_{2})+n-1}.$

The statement is proven. ∎

######  Remark 5.

In Chapter 6, we establish that

$|\\{\mathbf{x}\in[\mathscr{P}(\mathbf{a})(F)]|H(\mathbf{x})\leq B\\}|\sim
CB,$

for some $C>0$.

###  7 Northcott property for singular heights

In this section we allow finitely many $f_{v}$ to take values in $0$ but we
require that $f_{v}^{-1}$ admits a “logarithmic singularity” over a rational
divisor. We establish that corresponding heights are Northcott heights.

#### 1

Let $K$ be a field and let $\mathbf{a}\in\mathbb{Z}_{>0}^{n}$. Let us define
weighted degrees of polynomials in $K[X_{1},\dots,X_{n}]$. For $j=1,\dots,n$,
we define $\mathbf{a}$-weighted degree (or simply weighted degree) of
polynomial $X_{j}$ by setting $\deg_{\mathbf{a}}(X_{j})=a_{j}.$ For a monomial
$cX_{1}^{d_{1}}\cdots X_{n}^{d_{n}},$ where $c\in R$, we define
$\deg_{\mathbf{a}}(cX_{1}^{d_{1}}\cdots
X_{n}^{d_{n}})=\mathbf{a}\cdot\mathbf{d}$. Finally, if $P=\sum_{i}Q_{i},$ with
$Q_{i}$ monomials, we define
$\deg_{\mathbf{a}}(P)=\max_{i}(\deg_{\mathbf{a}}(Q_{i}))$.

######  Definition 1.

We say that a polynomial $P\in K[X_{1},\dots,X_{n}]$ is
$\mathbf{a}$-homogenous if it is a sum of monomials of the same
$\mathbf{a}$-weighted degree.

######  Lemma 2.

Let $P$ be an $\mathbf{a}$-homogenous polynomial of weighted degree $d\geq 1$.

  1. 1.

For $t\in\mathbb{G}_{m}$ and $\mathbf{x}\in\mathbb{A}^{n}$, one has that
$P(t\cdot\mathbf{x})=t^{d}P(\mathbf{x})$.

  2. 2.

Let $Q$ be an $\mathbf{a}$-homogenous polynomial of weighted degree $k\geq 0$.
The polynomial $PQ$ is an $\mathbf{a}$-homogenous polynomial of weighted
degree $d+k$.

  3. 3.

Let $\\{P_{1},\dots,P_{m}\\}\in F_{v}[X_{1},\dots,X_{n}]$ be a set of non-
constant $\mathbf{a}$-homogenous polynomials. The closed subscheme
$Z(\\{P_{1},\dots,P_{m}\\})\subset\mathbb{A}^{n},$ given by the common zero
set of $P_{1},\dots,P_{m}$ is $\mathbb{G}_{m}$-invariant for the action of
$\mathbb{G}_{m}$ on $\mathbb{A}^{n}$ with weights $a_{1},\dots,a_{n}$. The
open subscheme $D(P_{1})\subset\mathbb{A}^{n},$ given by the locus where
$P_{1}$ does not vanish is $\mathbb{G}_{m}$-invariant for the same action of
$\mathbb{G}_{m}$ on $\mathbb{A}^{n}$.

###### Proof.

  1. 1.

Let $t\in\mathbb{G}_{m}$ and let $CX_{1}^{m_{1}}\cdots X_{n}^{m_{n}}$ be a
monomial. For $\mathbf{x}\in\mathbb{A}^{n}$, we have that

$CX_{1}^{m_{1}}\cdots
X_{n}^{m_{n}}(t\cdot\mathbf{x})=Ct_{1}^{a_{1}m_{1}}x_{1}^{m_{1}}\cdots
t_{n}^{a_{n}m_{n}}x_{n}^{m_{n}}=t^{\mathbf{a}\cdot\mathbf{m}}X_{1}^{m_{1}}\cdots
X_{n}^{m_{n}}(\mathbf{x}).$

Now, $P$ can be written as sum $\sum_{i}C_{i}X^{m_{1,i}}_{1}\cdots
X^{m_{n,i}}_{n}$ where for each $i$ one has $\mathbf{a}\cdot\mathbf{m}_{i}=d$
and we deduce that $P(t\cdot\mathbf{x})=t^{d}P(\mathbf{x}).$

  2. 2.

The product $PQ$ is a sum of monomials $CX^{m_{1}}_{1}\cdots
X^{m_{n}}_{n}\cdot DX^{r_{1}}_{1}\cdots X^{r_{n}}_{n},$ with
$\mathbf{a}\cdot\mathbf{m}=d$ and $\mathbf{a}\cdot\mathbf{r}=k$. It follows
that $PQ$ is $\mathbf{a}$-homogenous of weighted degree
$\mathbf{a}(\mathbf{m}+\mathbf{r})=k+d$.

  3. 3.

For every $\mathbf{x}\in\mathbb{A}^{n}$ and every $i\in\\{1,\dots,m\\}$, we
have that $P_{i}(t\cdot\mathbf{x})=t^{d}P_{i}(\mathbf{x})=0$ if and only if
$P_{i}(\mathbf{x})=0$. The claim follows.

∎

######  Definition 3.

Let $\\{P_{1},\dots,P_{m}\\}\subset F_{v}[X_{1},\dots,X_{n}]$ be a set of non-
constant $\mathbf{a}$-homogenous polynomials. We define
$\mathcal{Z}(\\{P_{1},\dots,P_{m}\\})$ to be the substack of
$\mathscr{P}(\mathbf{a})$ defined by the $\mathbb{G}_{m}$-invariant closed
subscheme $Z(P_{1},\dots,P_{m})-\\{0\\}\subset\mathbb{A}^{n}-\\{0\\}$. We
define $\mathcal{D}(P_{1})$ to be the substack of $\mathscr{P}(\mathbf{a})$
defined by the $\mathbb{G}_{m}$-invariant open subscheme
$D(P_{1})\subset\mathbb{A}^{n}-\\{0\\}$.

It follows from [57, Lemma 04YN] that $\mathcal{Z}(\\{P_{1},\dots,P_{k}\\})$
is a closed substack of $\mathscr{P}(\mathbf{a})$ and that
$\mathcal{D}(P_{1}\cdots P_{m})$ is an open substack of
$\mathscr{P}(\mathbf{a})$.

Let $Q_{1},\dots,Q_{r+1}$ be $\mathbf{a}$-homogenous polynomials of the same
weighted degree $d$. The morphism

$J(Q_{1},\dots,Q_{r+1})\mathrel{\mathop{\ordinarycolon}}(\mathbb{A}^{n}-Z(\\{Q_{1},\dots,Q_{r+1}\\}))\to\mathbb{A}^{r+1}-\\{0\\}\hskip
14.22636pt\mathbf{x}\mapsto(Q_{i}(\mathbf{x}))_{i}$

is $t\mapsto t^{d}$-equivariant (see 5 for the terminology), when the left
scheme is endowed with the action
$t\cdot_{\mathbf{a}}\mathbf{x}=(t^{a_{j}}x_{j})_{j}$ of $\mathbb{G}_{m}$ and
the right scheme is endowed with the action
$t\cdot_{\mathbf{1}}\mathbf{x}=(tx_{j})_{j}$. Let us denote by

$\overline{J(Q_{1},\dots,Q_{r+1})}\mathrel{\mathop{\ordinarycolon}}\mathscr{P}(\mathbf{a})\to\mathbb{P}^{r}$

the $1$-morphism of stacks induced by [57, Lemma 046Q].

#### 2

In this paragraph we study heights that are obtained when $v$\- adic metric is
singular. We require the singularities to be “logarithmic” over a rational
divisor. We establish that resulting heights are Northcott.

######  Definition 1.

Let $P\in F[X_{1},\dots,X_{n}]$ be a non-constant $\mathbf{a}$-homogenous
polynomial and let $S$ be a finite set of places. A collection of continuous
$\mathbb{G}_{m}(F_{v})$-invariant functions
$g_{v}\mathrel{\mathop{\ordinarycolon}}D(P)(F_{v})\to\mathbb{R}_{>0}$ for
$v\in S$, will be said to be logarithmically suitable if the following
condition is satisfied: there exists a set $\\{Q_{i}\\}_{i}$ of non-constant
$\mathbf{a}$-homogenous polynomials $Q_{i}\in F[X_{1},\dots,X_{n}]$ such that
$Q_{i}$ and $P$ have no common factors of degree at least $1,$ such that
$Z(\\{Q_{i}\\}_{i})=\\{0\\}$ and such that for every $v\in S$ and every $i$
there exist $\alpha_{v,i},\beta_{v,i}>0$ such that

$g_{v}(\mathbf{x})\leq\alpha_{v,i}\max\bigg{(}-\log\bigg{(}\frac{|P(\mathbf{x})|^{\deg_{\mathbf{a}}(Q)/\deg_{\mathbf{a}}(P)}_{v}}{|Q(\mathbf{x})|_{v}}\bigg{)}^{\beta_{v,i}},1\bigg{)}$

for every $\mathbf{x}\in D(PQ_{i})(F_{v})$.

By the quasi-compactness of the scheme $\mathbb{A}^{n}$, one can always ask
for the set $\\{Q_{i}\\}_{i}$ to be finite.

For $v\in M_{F}$, let
$f_{v}^{\\#}\mathrel{\mathop{\ordinarycolon}}F_{v}^{n}-\\{0\\}\to\mathbb{R}_{>0}$
be the $v$-adic toric $\mathbf{a}$-homogenous continuous function of weighted
degree $d\geq 1$. Let $P\in F[X_{1},\dots,X_{n}]$ be a non-constant
$\mathbf{a}$-homogenous polynomial. Let $S$ be a finite set of places and let
$(g_{v})_{v}$ be a logarithmically suitable family of continuous
$\mathbb{G}_{m}(F_{v})$-invariant functions. For $v\in S$ we define
$f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0}$ by

$\displaystyle\quad f_{v}(\mathbf{x})\mathrel{\mathop{\ordinarycolon}}=$
$\displaystyle\begin{cases}f_{v}^{\\#}g_{v}^{-1}(\mathbf{x})&\text{if
}\mathbf{x}\in D(P)(F_{v}),\\\ 0&\text{if }\mathbf{x}\in
Z(P)(F_{v}).\end{cases}$

and for $v\in M_{F}-S$ we let $f_{v}=f_{v}^{\\#}$. For every $v\in M_{F}$, the
function
$f_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0}$ is an $\mathbf{a}$-homogenous continuous function of weighted degree
$d\geq 1$. We define height $H=H((f_{v})_{v})$ and $H^{\\#}=H((f_{v}^{\\#}))$
on $\mathscr{P}(\mathbf{a})(F)$. We recall that for every
$\mathbf{x}\in\mathscr{P}(\mathbf{a})(F)$, one has that
$H^{\\#}(\mathbf{x})\geq\leavevmode\nobreak\ 1$ by 7.

Motivated by [55, Proposition 2.1], we establish that:

######  Proposition 2.

There exist $C,\beta>0$ such that for all $\mathbf{x}\in\mathcal{D}(P)(F)$ one
has

$H(\mathbf{x})\geq CH^{\\#}(\mathbf{x})\log(1+H^{\\#}(\mathbf{x}))^{-\beta}$

###### Proof.

For $\mathbf{x}\in\mathcal{D}(P)(F)$, let
$\widetilde{\mathbf{x}}\mathrel{\mathop{\ordinarycolon}}\mathbb{G}_{m}\to\mathbb{A}^{n}-\\{0\\}$
be the $\mathbb{G}_{m}$-equivariant morphism over $F$ defined by $\mathbf{x}$.
For simplicity we will write $\widetilde{\mathbf{x}}$ for
$\widetilde{\mathbf{x}}(1)$.

There exists a finite set of non-constant $\mathbf{a}$-homogenous polynomials
$\\{Q_{i}\\}_{i}$ which have no common factors with $P_{v}$ of degree at least
$1$ such that $Z(\\{Q_{i}\\}_{i})=\\{0\\},$ and such that for every $v\in S$
and every $i,$ there exist $\alpha_{i,v},\beta_{i,v}$ such that

$g_{v}(\mathbf{x})\leq\alpha_{i,v}\max\bigg{(}-\log\bigg{(}\frac{|P(\mathbf{x})|^{\deg_{\mathbf{a}}(Q)/\deg_{\mathbf{a}}(P)}_{v}}{|Q(\mathbf{x})|_{v}}\bigg{)}^{\beta_{i,v}},1\bigg{)}$

for every $\mathbf{x}\in D(PQ_{i})(F_{v})$. Fix an index $i$. For
$\mathbf{x}\in\mathcal{D}(Q_{i}P)(F)$, using the fact that $f_{v}=f_{v}^{\\#}$
for $v\not\in S$ we deduce that

$\displaystyle H(\mathbf{x})$ $\displaystyle=\prod_{v\in
M_{F}}f_{v}(\widetilde{\mathbf{x}})$ $\displaystyle=\prod_{v\in
M_{F}}f_{v}^{\\#}(\widetilde{\mathbf{x}})\prod_{v\in
S}g_{v}(\widetilde{\mathbf{x}})^{-1}$ $\displaystyle\geq
H^{\\#}(\mathbf{x})\prod_{v\in
S}\alpha_{v,i}^{-1}\max\bigg{(}-\log\bigg{(}\frac{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}}{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}\bigg{)},1\bigg{)}^{-\beta_{v,i}}$
$\displaystyle\geq H^{\\#}(\mathbf{x})\prod_{v\in
S}\alpha_{v,i}^{-1}\max\bigg{(}\log\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}}\bigg{)},1\bigg{)}^{-\beta_{v,i}}$
$\displaystyle=H^{\\#}(\mathbf{x})\prod_{v\in
S}\alpha_{v,i}^{-1}\log\bigg{(}\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}}\bigg{)},e\bigg{)}^{-\beta_{v,i}}$
$\displaystyle\geq\alpha_{i}H^{\\#}(\mathbf{x})\prod_{v\in
S}\log\bigg{(}\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}}\bigg{)},e\bigg{)}^{-\beta_{v,i}},$

where $\alpha=\prod_{v\in S}\alpha_{v,i}^{-1}$. Lemma 8 gives that for every
$v\in S$, there exists $\gamma_{v}>1$ such that for every
$\mathbf{x}\in\mathcal{D}(Q_{i}P)(F)$ one has that :

$\displaystyle\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}},e\bigg{)}$
$\displaystyle\leq
e\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})/\deg_{\mathbf{a}}(P)}},1\bigg{)}$
$\displaystyle\leq
e\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|^{\deg_{\mathbf{a}}(P)}_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})}},1\bigg{)}^{1/\deg_{\mathbf{a}}(P)}$
$\displaystyle\leq e\prod_{v\in
M_{F}}\max\bigg{(}\frac{|Q_{i}(\widetilde{\mathbf{x}})|^{\deg_{\mathbf{a}}(P)}_{v}}{|P(\widetilde{\mathbf{x}})|_{v}^{\deg_{\mathbf{a}}(Q_{i})}},1\bigg{)}^{1/\deg_{\mathbf{a}}(P)}$
$\displaystyle=eH_{\mathbb{P}^{1}}(\overline{J}(Q^{\deg_{\mathbf{a}}(P)}_{i},P^{\deg_{\mathbf{a}}(Q_{i})})(\mathbf{x}))^{1/\deg_{\mathbf{a}}(P)}$
$\displaystyle\leq\gamma_{v}H^{\\#}(\mathbf{x})^{\deg_{\mathbf{a}}(Q_{i})}.$

For $v\in S$ let us set
$\delta_{v}=\max(\log(\gamma_{v}),{\deg_{\mathbf{a}}(Q_{i})})$. We have for
every $\mathbf{x}\in\mathcal{D}(Q_{i}P)(F)$ that

$\displaystyle H(\mathbf{x})$
$\displaystyle\geq\alpha_{i}H^{\\#}(\mathbf{x})\prod_{v\in
S}\log(\gamma_{v}H^{\\#}(\mathbf{x})^{\deg_{\mathbf{a}}(Q_{i})})^{-\beta_{v,i}}$
$\displaystyle=\alpha_{i}H^{\\#}(\mathbf{x})\prod_{v\in
S}\big{(}\log(\gamma_{v})+{\deg_{\mathbf{a}}(Q_{i})}\log(H^{\\#}(\mathbf{x}))\big{)}^{-\beta_{v,i}}$
$\displaystyle\geq\alpha_{i}H^{\\#}(\mathbf{x})\prod_{v\in
S}\delta_{v}^{-\beta_{v,i}}\big{(}1+\log(H^{\\#}(\mathbf{x}))\big{)}^{-\beta_{v,i}}$
$\displaystyle=C_{i}H^{\\#}(\mathbf{x})\log(1+H^{\\#}(\mathbf{x}))^{-\beta_{i}},$

where we have set $C_{i}=\alpha_{i}\prod_{v\in S}\delta_{v}^{-\beta_{v,i}}$
and $\beta_{i}=\sum_{v\in S}\beta_{v,i}$. We have that
$\bigcup_{i}D(Q_{i}P)=D(P).$ Thus for $C=\min_{i}C_{i}$ and
$\beta=\max\beta_{i}$ we get

$H(\mathbf{x})\geq CH^{\\#}\log(1+H^{\\#}(\mathbf{x}))^{-\beta},$

for every $\mathbf{x}\in D(P)(F)$. ∎

######  Corollary 3.

Let $(f_{v})_{v}$ be as above and let $H$ be the corresponding height. The
height $H$ is a Northcott height. Moreover, for every $\epsilon>0$, there
exists $C=C(\epsilon)>0$ such that for every $B>1$ one has that

$|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]|H(\mathbf{x})\leq B\\}|\leq
CB^{1+\epsilon}$

and that

$|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]\cap([\mathscr{P}(\mathbf{a})(F)]-[\mathscr{T}({\mathbf{a}})(F)])|H(\mathbf{x})\leq
B\\}|\\\ \leq
CB^{\frac{(1+\epsilon)(|\mathbf{a}|-\min_{j}a_{j})}{|\mathbf{a}|}}.$

###### Proof.

By 8, there exist $C^{\prime},N>0$, such that for every
$\mathbf{x}\in[\mathcal{D}(P)(F)]$ one has that

$H(\mathbf{x})\geq
C^{\prime}H^{\\#}(\mathbf{x})\log(1+H^{\\#}(\mathbf{x}))^{-N}.$

and thus there exists $A>0$ and $\frac{\epsilon}{1+\epsilon}>\delta>0$ such
that

$H(\mathbf{x})\geq AH^{\\#}(\mathbf{x})^{1-\delta}\hskip
28.45274pt\forall\mathbf{x}\in[\mathcal{D}(P)(F)].$

Using 2, one has hence that there exists $C>0$ such that

$\displaystyle|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]|H(\mathbf{x})\leq
B\\}|\hskip-85.35826pt$
$\displaystyle\leq|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]|AH^{\\#}(\mathbf{x})^{1-\delta}\leq
B\\}|$
$\displaystyle=|\\{\mathbf{x}\in[\mathcal{D}((P)(F))]|H^{\\#}(\mathbf{x})\leq
A^{-1/(1-\delta)}B^{1/(1-\delta)}\\}|$ $\displaystyle\leq
A^{-1/(1-\delta)}B^{1/(1-\delta)}\log(2+A^{-1/(1-\delta)}B^{1/(1-\delta)})^{n^{2}(r_{1}+r_{2})+n-1}$
$\displaystyle\leq CB^{1+\epsilon}$

and such that

$\displaystyle|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]\cap([\mathscr{P}(\mathbf{a})(F)]-[\mathscr{T}({\mathbf{a}})(F)])|H(\mathbf{x})\leq
B\\}|\hskip-284.52756pt$
$\displaystyle\leq|\\{\mathbf{x}\in[\mathcal{D}(P)(F)]|AH^{\\#}(\mathbf{x})^{1-\delta}\leq
B\\}|$
$\displaystyle=|\\{\mathbf{x}\in[\mathcal{D}((P)(F))]\cap([\mathscr{P}(\mathbf{a})(F)]-[\mathscr{T}({\mathbf{a}})(F)])|H^{\\#}(\mathbf{x})\leq
A^{-\frac{1}{1-\delta}}B^{\frac{1}{1-\delta}}\\}|$
$\displaystyle\leq(A^{-1}B)^{\frac{(1+\epsilon)(|\mathbf{a}|-\min_{j}a_{j})}{(1-\delta)|\mathbf{a}|}}\log(2+(A^{-1}B)^{\frac{|\mathbf{a}|-\min_{j}a_{j}}{(1-\delta)|\mathbf{a}|}})^{n^{2}(r_{1}+r_{2})+n-1}$
$\displaystyle\leq
CB^{\frac{(1+\epsilon)(|\mathbf{a}|-\min_{j}a_{j})}{|\mathbf{a}|}}.$

The statement is proven.∎

##  Chapter 3 Measures

Let $n\geq 1$ be an integer. Let $\mathbf{a}\in\mathbb{Z}^{n}_{>0}$. Recall
that $\mathscr{P}(\mathbf{a})$ is the quotient stack for the action

$\mathbb{G}_{m}\times(\mathbb{A}^{n}-\\{0\\})\to\mathbb{A}^{n}-\\{0\\}\hskip
28.45274ptt\cdot\mathbf{x}=(t^{a_{j}}x_{j})_{j}.$

The open subscheme $\mathbb{G}_{m}^{n}\subset\mathbb{A}^{n}-\\{0\\}$ is
$\mathbb{G}_{m}$-invariant for this action and $\mathscr{T}({\mathbf{a}})$ is
defined to be the quotient $\mathbb{G}_{m}^{n}/\mathbb{G}_{m}$. In this
chapter we define measures on $[\mathscr{P}(\mathbf{a})(F_{v})]$ and
$[\mathscr{T}({\mathbf{a}})(F_{v})]$ for $v\in M_{F}$. We use these measures
to define Peyre’s constant as in [47]. Later in the section we define a Haar
measure on the adelic space $[\mathscr{T}({\mathbf{a}})(\mathbb{A}_{F})]$ and
we define and calculate the corresponding Tamagawa number.

###  1 Quotient measures

#### 1

We make several conventions on measures that will be used throughout the
chapter. Let $X$ be a locally compact topological space. Let
$\mathscr{C}^{0}_{c}(X,\mathbb{C})$ be the set of continuous compactly
supported functions on $X$. We endow $\mathscr{C}^{0}_{c}(X,\mathbb{C})$ by
the uniform convergence topology. By a measure on $X$ [9, Definition 2, n0 3,
§1, Chapter III], we mean a continuous linear functional
$\mu\mathrel{\mathop{\ordinarycolon}}\mathscr{C}^{0}_{c}(X,\mathbb{C})\to\mathbb{C}$
.

Let $\mu$ be a measure on $X$. Let $L^{1}(X,\mu)$ be the Banach space of
absolutely $\mu$-integrable complex valued functions modulo negligible
functions [9, Definition 2, n0 4, §3, Chapter IV]. By the abuse of the
terminology, we may call an element $f\in L^{1}(X,\mu)$ a function and for a
function $g\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{C}$ which is
$\mu$-absolutely integrable we may write $g\in L^{1}(X,\mu)$. For $f\in
L^{1}(X,\mu)$, we denote by $\int_{X}f\mu$ the integral of $f$ against $\mu$
[9, Definition 1, n0 1, §4, Chapter IV]. If $U\subset X$ is a subset, such
that $\mathbf{1}_{U,X}\in L^{1}(X,\mu)$, we write $\mu(U)$ for
$\mu(\mathbf{1}_{U,X})$ (where $\mathbf{1}_{U,X}$ stands for the
characteristic function of $U$ in $X$, written sometimes as $\mathbf{1}_{U}$).
Such $U$ will be said to be $\mu$-measurable.

#### 2

We recall some facts about quotient measures from §2, Chapter 7 of [10]. Let
$X$ be a locally compact Hausdorff topological space. Let $G$ be a locally
compact topological group acting on the right on $X$ continuously and properly
(that is the action $X\times G\to X$ is continuous and proper). The quotient
topological space $X/G$ is separated [7, Proposition 3, n0 2, §2, Chapter III]
and locally compact [7, Proposition 10, n0 4, §10, Chapter I].

Let $dg$ be a left Haar measure on $G$. Let
$\Delta_{G}\mathrel{\mathop{\ordinarycolon}}G\to\mathbb{R}_{>0}$ be the
modular function of $G$ (we recall the definition of the modular function:
according to [10, Formula (11), n0 1, §1, Chapter VII], for every $h\in G$,
the association $A\mapsto dg(Ah)$, for $dg$-measurable subset $A$ of $G$, is a
left Haar measure on $G$, hence, by the unicity of the Haar measure there
exists a unique positive real number $\Delta_{G}(h)$ such that for every
$dg$-measurable subset $A$ of $G$, one has that $dg(Ag)=\Delta_{G}(g)dg(A)$;
further, it does not depend on the choice of the Haar measure $dg$).

######  Proposition 1 ([10, Proposition 1, n0 2, §2, Chapter VII]).

For $x\in X/G$, let $\widetilde{x}\in X$ be a lift of $x$. Let
$\phi\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{C}$ be a compactly supported
continuous function. For every $y\in X$, one has that $g\mapsto\phi(yg)\in
L^{1}(G,\mathbb{C}).$ Moreover, for $x\in X/G$, the value of
$\int_{G}\phi(\widetilde{x}g)dg$ does not depend on the choice of
$\widetilde{x}$. The function
$\phi^{*}\mathrel{\mathop{\ordinarycolon}}x\mapsto\int_{G}\phi(\widetilde{x}g)dg$
is continuous and compactly supported.

######  Proposition 2 ([10, Proposition 4, n0 2, §2, Chapter VII]).

  1. 1.

Let $\mu$ be a measure on $X$ such that for every $\mu$-measurable $U$ one has
that $\mu(Ug)=\Delta_{G}(g)\mu(U)$ (such measures will be called $G$-invariant
measures). There exists a unique measure $\mu/dg$ on $X/G$ such that for every
compactly supported function
$\phi\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{C}$ one has that

$\int_{X}\phi\mu=\int_{X/G}(\phi^{*})(\mu/dg).$

  2. 2.

Let $\mu^{\prime}$ be a measure on $X/G$. There exists a unique $G$-invariant
measure $\mu$ on $X$ such that $\mu/dg=\mu^{\prime}$.

Let $\pi\mathrel{\mathop{\ordinarycolon}}X\to X/G$ be the quotient map. We
quote following two propositions from [10].

######  Proposition 3 ([10, Proposition 8, n0 4, §2, Chapter VII]).

Suppose that $X/G$ is paracompact. Let $\lambda>0$. There exists a continuous
function $k\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{R}_{\geq 0}$, the
support of which has a compact intersection with the preimage under $\pi$ of
any compact of $X/G$ and such that for any $x\in X$ one has that

$\int_{G}k(xg)dg=\lambda.$

Of course, when $X/G$ is compact, the condition on the support of $k$ becomes
that it is compact.

######  Lemma 4.

Let $f\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{C}$ be a continuous
$G$-invariant function. Let
$\overline{f}\mathrel{\mathop{\ordinarycolon}}X/G\to\mathbb{C}$ be the
function induced from $f$. One has that $(f\mu)/dg=(\overline{f})(\mu/dg).$

###### Proof.

Let $\phi\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{C}$ be a compactly
supported continuous function. The function $\phi f$ is compactly supported.
Let $x\in X/G$ and let $\widetilde{x}\in X$ be its lift. One has that

$(\phi\cdot f)^{*}(x)=\int_{G}\phi
f(\widetilde{x}g)dg=\int_{G}\phi(\widetilde{x}g)f(\widetilde{x}g)dg=\overline{f}(x)\int_{G}\phi(\widetilde{x}g)dg\\\
=(\phi^{*})\cdot\overline{f})(x).$

It follows that: $(\phi\cdot f)^{*}=\phi^{*}\cdot\overline{f}$. We deduce
that:

$\displaystyle\int_{X/G}(\phi^{*})(f\mu/dg)$
$\displaystyle=\int_{X}(\phi)(f\mu)$ $\displaystyle=\int_{X}(\phi f)\mu$
$\displaystyle=\int_{X/G}((\phi\cdot f)^{*})(\mu/dg)$
$\displaystyle=\int_{X/G}(\phi^{*}\overline{f})(\mu/dg)$
$\displaystyle=\int_{X/G}(\phi^{*})((\overline{f})(\mu/dg)).$

It follows that $(f\mu/dg)=(\overline{f})(\mu/dg)$. ∎

######  Proposition 5 ([10, Proposition 9, n0 4, §2, Chapter VII]).

Let us suppose that $X/G$ is paracompact. Let
$k\mathrel{\mathop{\ordinarycolon}}X\to\mathbb{R}_{\geq 0}$ be a continuous
function, the support of which has compact intersection with the preimage
under $\pi$ of any compact of $X/G$ and such that for every $x\in X$ one has

$\int_{G}k(xg)dg=1.$

Then for any function $h\mathrel{\mathop{\ordinarycolon}}X/G\to\mathbb{C}$ one
has that $h\in L^{1}(X/G,\mu/dg)$ if and only if $k\cdot(h\circ\pi)\in
L^{1}(X,\mu)$ and if $h\in L^{1}(X/G,\mu/dg)$ then

$\int_{X/G}(h)(\mu/dg)=\int_{X}k\cdot(h\circ\pi)\mu.$

#### 3

We recall the notion of the quotient measure when we have inclusion of locally
compact abelian groups.

Let $G$ is an abelian locally compact Hausdorff topological group. The action
of $G\times G\to G$ of $G$ on $G$ given by the multiplication is proper
(because the induced map $G\times G\xrightarrow{(m_{G},p_{2})}G\times G$ is a
homeomorphism, where $m_{G}$ is the multiplication map and $p_{2}$ the
projection to the second coordinate). Thus if $A$ a closed subgroup, the
action of $A$ on $G,$ given by the restriction of the action of $G$, is proper
by [7, Example 1, n0 1, §4, Chapter III]. Let $dg$ be a Haar measure on $G$
and let $da$ be a Haar measure on $A$. By [10, Proposition 10, n0 7, §2,
Chapter VII], the quotient measure $dg/da$ is a Haar measure on $G/A$.

#### 4

In this paragraph we recall the theory of [43] on the Euler characteristics of
complexes of locally compact abelian groups endowed with Haar measures.

A homomorphism of topological groups is said to be strict (see [7, Definition
1, n0 8, §2, Chapter III]), if the induced homomorphism to its image is open.
By [8, Corollary of Proposition 6, n0 3, §5, Chapter IX], any morphism of
locally compact abelian groups which are countable at infinity is strict if
and only if its image is closed. Consider a complex $C_{\bullet}$ of locally
compact abelian groups which are countable at infinity

$\cdots\to C_{n+1}\xrightarrow{d_{n+1}}C_{n}\xrightarrow{d_{n}}C_{n-1}\cdots$

such that the following conditions are satisfied:

  * •

the homomorphisms $d_{n}$ are continuous for every $n\in\mathbb{Z}$,

  * •

the complex is bounded and of finite homology (i.e.
$\ker(d_{n})/\operatorname{Im}(d_{n+1})$ is a finite group for every
$n\in\mathbb{Z}$.)

Suppose that for every $n\in\mathbb{Z}$, we are given a Haar measure
$\lambda_{n}$ on $C_{n}$ and that for almost all $n$ the measure $\lambda_{n}$
normalized by $\lambda_{n}(C_{n})=1$. For every $n\in\mathbb{Z}$, let us
choose Haar measures $\nu_{n}$ and $\theta_{n}$ on $\ker(d_{n})$ and
$\operatorname{Im}(d_{n}),$ respectively. Let $\alpha_{n}$ be the volume of
the finite set $\ker(d_{n})/\operatorname{Im}(d_{n+1})$ for the quotient
measure $\nu_{n}/\theta_{n+1}$. Let $\beta_{n}>0$ be the unique real number
such that $\lambda_{n}/\nu_{n}=\beta_{n}\theta_{n}$ (after identifying
$C_{n}/\ker(d_{n})$ and $\operatorname{Im}(d_{n})$ via the isomorphism induced
from $d_{n}$). Oesterlé defines the number

$\chi(C_{\bullet})=\chi(C_{\bullet},(\lambda_{n})_{n})\mathrel{\mathop{\ordinarycolon}}=\prod_{n\in\mathbb{Z}}(\alpha_{n}\beta_{n})^{(-1)^{n}},$

which does not depend on the choice of $\nu_{n}$ and $\theta_{n}$. We may say
that $C_{\bullet}$ is of the trivial measure Euler-Poincaré characteristic if
$\chi(C_{\bullet})=1$.

######  Lemma 1 (Oesterlé, [43, Examples 2 and 3]).

The following claims are valid:

  1. 1.

Suppose that every $C_{n}$ is compact. Then

$\chi(C_{\bullet})=\prod_{n\in\mathbb{Z}}\lambda_{n}(C_{n})^{(-1)^{n}}.$

  2. 2.

Suppose that every $C_{n}$ is discrete and that $\lambda_{n}$ are counting
measures. One has that

$\chi(C_{\bullet})=\prod_{n\in\mathbb{Z}}[\ker(d_{n})\mathrel{\mathop{\ordinarycolon}}\operatorname{Im}(d_{n+1})]^{(-1)^{n}}.$

######  Proposition 2 (Oesterlé, [43, Proposition 1, n0 4]).

Consider a bicomplex

${\vdots}$${\vdots}$${\cdots}$${{C_{n,m-1}}}$${{C_{n-1,m-1}}}$${\cdots}$${\cdots}$${{C_{n,m}}}$${{C_{n-1,m}}}$${\cdots}$${\vdots}$${\vdots}$$\scriptstyle{d^{\prime}_{n,m-1}}$$\scriptstyle{d^{\prime}_{n,m}}$$\scriptstyle{d^{\prime\prime}_{n,m}}$$\scriptstyle{d^{\prime\prime}_{n-1,m}}$

of locally compact abelian groups endowed with Haar measures, such that
$C_{n,m}$ is the trivial group for $|n|+|m|$ big enough and such that for
every $n\in\mathbb{Z}$ and every $m\in\mathbb{Z}$, the complexes
$C_{n,\bullet}$ and $C_{\bullet,m}$ satisfy the above conditions. One has that

$\prod_{n\in\mathbb{Z}}\chi(C_{n,\bullet})^{(-1)^{n}}=\prod_{m\in\mathbb{Z}}\chi(C_{\bullet,m})^{(-1)^{m}}.$

We end the paragraph by a lemma that will be used on several occasions.

######  Lemma 3.

Let $\epsilon\mathrel{\mathop{\ordinarycolon}}H\to G$ be a proper continuous
homomorphism of locally compact Hausdorff abelian topological groups. Let $dg$
and $dh$ be Haar measures on $G$ and $H$. Consider the left action of $H$ on
$G$

$H\times G\to G\hskip 28.45274pth\cdot g=\epsilon(h)g.$

  1. 1.

The action is continuous and proper. The quotient $G/H$ identifies with the
quotient group $G/\epsilon(H)$.

  2. 2.

The group $K\mathrel{\mathop{\ordinarycolon}}=\ker(\epsilon)$ is compact and
let $dk$ be the probability Haar measure on $K$. We have an equality of
measures on $\epsilon(H)$:

$\epsilon_{*}(dh)=dh/dk.$

  3. 3.

The subgroup $\epsilon(H)$ is closed in $G$. The measure $dg/dh$ is the
quotient Haar measure $dg/\epsilon_{*}(dh)$.

  4. 4.

Suppose that $H$ and $G$ are countable at infinity. The exact sequence $1\to
K\to H\to G\to G/H\to 1$ is of the trivial measure Euler-Poincaré
characteristics, when the measures on $K,$ $H$, $G$ and $G/H$ are $dk$, $dh$,
$dg$ and $dg/dh$.

###### Proof.

  1. 1.

The continuity of the group action follows from the continuity of the
multiplication by an element in $G$. The action is proper, as one has a
Cartesian diagram

${G\times H}$${G\times
G}$${H}$${G,}$$p_{2}$$(g,h)\mapsto(g,g\epsilon(h))$$\epsilon$

where the right vertical homomorphism is given by $(g_{1},g_{2})\mapsto
g_{1}^{-1}g_{2}$. The canonical homomorphism $G\to G/\epsilon(H)$ is
continuous and open and induces a continuous and open map $G/H\rightarrow
G/\epsilon(H).$ The map is bijective because one has an equality of sets
$G/H=G/\epsilon(H)$. We deduce a topological identification
$G/H=G/\epsilon(H)$. The claim follows.

  2. 2.

The map $\epsilon$ is proper and it follows that $K$ is a compact subgroup of
$H$. The quotient space $\epsilon(H)=H/\ker(\epsilon)$ is paracompact [9,
Proposition 13, n0 6, §4, Chapter III]. Note that the constant function
$H\to\mathbb{C}$ given by $h\mapsto 1,$ satisfies the condition that its
support (i.e. the whole of $H$) has compact intersection with the preimage
$\epsilon^{-1}(A)$ for every compact $A\subset\epsilon(H),$ because $\epsilon$
is proper. Now, by 5, for every
$r\mathrel{\mathop{\ordinarycolon}}\epsilon(H)\to\mathbb{C}$ with $r\in
L^{1}(\epsilon(H),dh/dk)$ we have

$\int_{\epsilon(H)}(r)(dh/dk)=\int_{H}(r\circ\epsilon)\cdot
1dh=\int_{H}(r\circ\epsilon)dh,$

which means precisely that $\epsilon_{*}(dh)=dh/dk$.

  3. 3.

The subgroup $\epsilon(H)$ is closed as $\epsilon$ is a proper map. Let
$\phi\mathrel{\mathop{\ordinarycolon}}G/H\to\mathbb{C}$ be a compactly
supported function. By [10, Proposition 2, n0 1, §2, Chapter VII], there
exists a compactly supported continuous function
$\Phi\mathrel{\mathop{\ordinarycolon}}G\to\mathbb{C}$, such that for any $x\in
G/H$ and any lift $\widetilde{x}\in G$ of $x$, we have

$\phi(x)=\int_{H}\Phi(h\widetilde{x})dh.$

Note that

$\phi(x)=\int_{\epsilon(H)}\Phi(h\widetilde{x})\epsilon_{*}(dh).$

We have that

$\int_{G/H}(\phi)(dg/dh)=\int_{G}\Phi\hskip
2.84544ptdg=\int_{G/\epsilon(H)}(\phi)(dg/\epsilon_{*}(dh)).$

It follows that $dg/dh=dg/\epsilon_{*}(dh)$ as claimed.

  4. 4.

The short exact sequences

${1}$${K}$${H}$${{\epsilon(H)}}$${1}$${1}$${{\epsilon(H)}}$${G}$${{G/H}}$${1}$

are of the trivial measure Euler-Poincaré characteristics by (2) and (3). Thus
$1\to K\to H\to G\to G/H\to 1$ is of the trivial measure Euler-Poincaré
characteristics.

∎

###  2 Measures on $[\mathscr{P}(\mathbf{a})(F_{v})]$

For $v\in M_{F}$, the abelian locally compact group $F_{v}^{\times}$ acts on
$F^{n}_{v}-\\{0\\}$ by $t\cdot\mathbf{x}=(t^{a_{j}}x_{j})_{j}$. This action is
proper by 1. By 2, the quotient $(F^{n}_{v}-\\{0\\})/F_{v}^{\times}$
identifies with $[\mathscr{P}(\mathbf{a})(F_{v})].$ The goal of this section
is to define measures on
$[\mathscr{P}(\mathbf{a})_{F_{v}}(F_{v})]=[\mathscr{P}(\mathbf{a})(F_{v})]$
for $v\in M_{F}$. The topological spaces $[\mathscr{P}(\mathbf{a})(F_{v})]$
are Hausdorff and compact by 5. Let us denote by $q^{\mathbf{a}}_{v}$ the
quotient maps $F^{n}_{v}-\\{0\\}\to[\mathscr{P}(\mathbf{a})(F_{v})]$ for $v\in
M_{F}$.

#### 1

In this paragraph, we define measure on $F_{v}$.

For $v\in M_{F}$, let $dx_{v}$ be the Haar measure on $F_{v}$ normalized by

  * •

$dx_{v}(\mathcal{O}_{v})=1$ if $v$ is finite,

  * •

$dx_{v}$ is the Lebesgue measure on $F_{v}\cong\mathbb{R}$ if $v$ is real,

  * •

$dx_{v}=2dxdy$ on $F_{v}\cong\mathbb{C}\cong\mathbb{R}^{2}$ if $v$ is complex.

For $v\in M_{F}$, let $d^{*}x_{v}$ be the Haar measure
$\frac{dx_{v}}{|x_{v}|_{v}}$ on $F_{v}^{\times}.$ For $v\in M_{F}^{0}$, it
satisfies that

$d^{*}x_{v}(\mathcal{O}_{v}^{\times})=\int_{\mathcal{O}_{v}^{\times}}1d^{*}x_{v}=\int_{\mathcal{O}_{v}^{\times}}1dx_{v}=1-|\pi_{v}|_{v}.$

When $v$ is real, the measure $d^{*}x_{v}$ identifies with the measure

$d^{*}x\mathrel{\mathop{\ordinarycolon}}=\frac{dx}{|x|}$

on $\mathbb{R}^{\times}$ and when $v$ is complex, the measure $d^{*}x_{v}$
identifies with the measure

$\frac{2dxdy}{x^{2}+y^{2}}$

on $\mathbb{C}^{2}-\\{0\\}\cong\mathbb{R}^{2}-\\{0\\}$. Let us set

$F_{v,1}\mathrel{\mathop{\ordinarycolon}}=\\{x\in F_{v}|\hskip
2.84544pt|x|_{v}=1\\},$

so that $F_{v,1}=\\{\pm 1\\}$ when $v$ is real and

$F_{v,1}=S^{1}=\\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}=1\\}$

when $v$ is complex. Recall that we have set $n_{v}=1$ if $v$ is a real place
and let $n_{v}=2$ if $v$ is complex. The exact sequence

$\\{1\\}\to
F_{v,1}\xrightarrow{i_{F_{v,1}}}F_{v}^{\times}\xrightarrow{x\mapsto|x|_{v}}\mathbb{R}_{>0}\to\\{1\\},$

where $i_{F_{v,1}}$ is the inclusion map, admits a section

$\rho_{v}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}_{>0}\to
F_{v}^{\times}\hskip 28.45274ptr\mapsto r^{1/n_{v}}.$

The section induces a continuous isomorphism

$\widetilde{\rho}_{v}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}_{>0}\times
F_{v,1}\to F_{v}^{\times}\hskip 28.45274pt(r,z)\mapsto\rho_{v}(r)z.$ (1)

The inverse of this homomorphism is given by

$F_{v}^{\times}\to\mathbb{R}_{>0}\times F_{v,1}\hskip
28.45274ptx\mapsto(|x|_{v},x\rho_{v}(|x|_{v}))$ (2)

and is also continuous. We deduce that $\widetilde{\rho}_{v}$ is an
isomorphism of topological groups. Let us set $\lambda_{v,1}$ to be the
counting measure on $\\{\pm 1\\}$ when $v$ is real and let us set
$\lambda_{v,1}$ to be the Haar measure on $F_{v,1}$ normalized by
$\lambda_{v,1}(F_{v,1})=2\pi$ when $v$ is complex.

######  Lemma 3.

Let $v\in M_{F}^{\infty}$. One has that and that
$(\widetilde{\rho}_{v})_{*}\big{(}dr\times\lambda_{v,1}\big{)}=dx_{v}|_{F_{v}^{\times}}$
and that
$(\widetilde{\rho}_{v})_{*}\big{(}d^{*}r\times\lambda_{v,1}\big{)}=d^{*}x_{v}$.

###### Proof.

We will firstly verify that
$(\widetilde{\rho}_{v})_{*}\big{(}d^{*}r\times\lambda_{v,1}\big{)}=d^{*}x_{v}$.
Suppose $v$ is real. Both measures are Haar measures on $F_{v}^{\times}$, so
it suffices to check their equality on a single Borel subset of
$F_{v}^{\times}\cong\mathbb{R}^{\times},$ and we will verify it on $[1,2]$. We
have that $[1,2]=\widetilde{\rho}_{v}(\\{1\\}\times[1,2])$ and

$\displaystyle{d^{*}x_{v}}([1,2])={d^{*}x}([1,2])$
$\displaystyle=1\cdot{d^{*}x}([1,2])$
$\displaystyle=\lambda_{v,1}(\\{1\\})\cdot d^{*}r([1,2])$
$\displaystyle=\big{(}\lambda_{v,1}\times d^{*}r\big{)}(\\{1\\}\times[1,2])$
$\displaystyle=\big{(}(\widetilde{\rho}_{v})_{*}\big{(}d^{*}r\times\lambda_{v,1}\big{)}\big{)}(\widetilde{\rho}_{v}(\\{1\\}\times[1,2])).$

Suppose $v$ is complex. Note that $\lambda_{v,1}$ is the pushforward measure
for the map $[0,2\pi[\to S^{1}$ given by

$\theta\mapsto e^{i\theta}=(\cos(\theta),\sin(\theta)).$

Hence, the map

$S^{1}\times\mathbb{R}_{>0}\to\mathbb{R}^{2}-\\{0\\}\hskip
28.45274pt(z,r)\mapsto zr^{1/2}$

is measure preserving if and only if the map

$[0,2\pi[\times\mathbb{R}_{>0}\to\mathbb{R}^{2}-\\{0\\}\hskip
28.45274pt(\theta,r)\mapsto(r^{1/2}\cos(\theta),r^{1/2}\sin(\theta))$

is measure preserving. The corresponding Jacobian matrix is

$\begin{pmatrix}-r^{1/2}\sin(\theta)&\frac{1}{2}r^{-1/2}\cos(\theta)\\\
r^{1/2}\cos(\theta)&\frac{1}{2}r^{-1/2}\sin(\theta)\end{pmatrix}$

and its determinant equals $-\frac{1}{2}$. It follows that

$dxdy=(\widetilde{\rho}_{v})_{*}(\lvert-1/2\big{\rvert}drd\theta)=(\widetilde{\rho}_{v})_{*}((1/2)drd\theta),$

and hence that

$d^{*}x_{v}=\frac{2dxdy}{x^{2}+y^{2}}=(\widetilde{\rho}_{v})_{*}\bigg{(}\frac{2\cdot(1/2)drd\theta}{r}\bigg{)}=(\widetilde{\rho}_{v})_{*}(d^{*}rd\theta).$

We have proven that
$(\widetilde{\rho}_{v})_{*}\big{(}d^{*}r\times\lambda_{v,1}\big{)}=d^{*}x_{v}$.

Let us now verify that
$(\widetilde{\rho}_{v})_{*}\big{(}dr\times\lambda_{v,1}\big{)}=dx_{v}|_{F_{v}^{\times}}$.
One has that $|\widetilde{\rho}_{v}(r,z)|_{v}=r$ for every
$(r,z)\in\mathbb{R}_{>0}\times F_{v,1}$. For a Borel subset $U\subset
F_{v}^{\times}$, one has that

$\displaystyle dx_{v}|_{F_{v}^{\times}}(U)=\int_{U}|x_{v}|_{v}d^{*}x_{v}$
$\displaystyle=\int_{U}|\widetilde{\rho}_{v}(r,z)|_{v}(\rho_{v})_{*}(d^{*}r\times\lambda_{v,1})$
$\displaystyle=\int_{\widetilde{\rho}_{v}^{-1}(U)}r\hskip
2.84544pt(d^{*}r\times\lambda_{v,1})$
$\displaystyle=\int_{\widetilde{\rho}_{v}^{-1}(U)}dr\times\lambda_{v,1}$
$\displaystyle=(dr\times\lambda_{v,1})(U).$

It follows that
$dx_{v}|_{F_{v}^{\times}}=(\widetilde{\rho}_{v})_{*}(dr\times\lambda_{v,1})$.
The statement is proven. ∎

We will often write $dx$ and $d^{*}x$ for $dx_{v}$ and $d^{*}x_{v}$,
respectively.

#### 2

In this paragraph we will define compactly supported continuous functions
$k^{\mathbf{a}}_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{R}_{\geq
0}$ which satisfy that their integrals in every orbit for the weighted action
of $\mathbb{G}_{m}(F_{v})$ on $F^{n}_{v}-\\{0\\}$ are equal to $1$. These
functions will enable us to use 5.

For every $v\in M_{F}^{0}$, let

$\mathcal{D}^{\mathbf{a}}_{v}\mathrel{\mathop{\ordinarycolon}}=\\{\mathbf{y}\in(\mathcal{O}_{v})^{n}|\exists
j\mathrel{\mathop{\ordinarycolon}}v(y_{j})<a_{j}\\}=(\mathcal{O}_{v})^{n}-\pi_{v}^{a_{1}}\mathcal{O}_{v}\times\cdots\times\pi_{v}^{a_{n}}\mathcal{O}_{v}$

and set

$k^{\mathbf{a}}_{v}\mathrel{\mathop{\ordinarycolon}}=\frac{\mathbf{1}_{\mathcal{D}^{\mathbf{a}}_{v},F^{n}_{v}-\\{0\\}}}{1-|\pi_{v}|_{v}}.$

######  Lemma 1.

Let $v\in M_{F}^{0}$ and let $\mathbf{x}\in F^{n}_{v}-\\{0\\}$. The function
$k^{\mathbf{a}}_{v}$ is compactly supported, locally constant and one has that

$\int_{F_{v}^{\times}}k^{\mathbf{a}}_{v}(t\cdot\mathbf{x})d^{*}t=1.$

###### Proof.

We have seen in 4 that the subset $\mathcal{D}^{\mathbf{a}}_{v}$ is open,
closed and compact subset of $F^{n}_{v}-\\{0\\}$. We conclude that
$\mathbf{1}_{\mathcal{D}^{\mathbf{a}}_{v}}$ is locally constant and compactly
supported, hence is such $k^{\mathbf{a}}_{v}$. In 1 we have defined
$r_{v}\mathrel{\mathop{\ordinarycolon}}F^{n}_{v}-\\{0\\}\to\mathbb{Z}$ by

$r_{v}(\mathbf{y})=\sup_{\begin{subarray}{c}j=1,\dots,n\\\ y_{j}\neq
0\end{subarray}}\bigg{\lceil}-\frac{v(y_{j})}{a_{j}}\bigg{\rceil}.$

One has that

$t\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}\iff\mathbf{x}\in
t^{-1}\cdot\mathcal{D}^{\mathbf{a}}_{v}\iff\mathbf{x}\in
t^{-1}\pi_{v}^{r_{v}(\mathbf{x})}\cdot(\pi_{v}^{-r_{v}(\mathbf{x})}\cdot\mathcal{D}^{\mathbf{a}}_{v}).$

Lemma 1 gives that
$\mathbf{x}\in\pi_{v}^{-r_{v}(\mathbf{x})}\cdot\mathcal{D}^{\mathbf{a}}_{v}$
and that if
$(u\cdot(\pi_{v}^{-r_{v}(\mathbf{x})}\cdot\mathcal{D}^{\mathbf{a}}_{v}))\cap(\pi_{v}^{-r_{v}(\mathbf{x})}\cdot\mathcal{D}^{\mathbf{a}}_{v})\neq\emptyset$
then $v(u)=0.$ We conclude $v(t^{-1}\pi_{v}^{r_{v}(\mathbf{x})})=0$ and hence

$\\{t|t\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}\\}=\\{t|v(t)=r_{v}(\mathbf{x})\\}.$

We deduce

$d^{*}t\\{t\in
F_{v}^{\times}|t\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}\\}=d^{*}t(\pi_{v}^{r_{v}(\mathbf{x})}\mathcal{O}_{v}^{\times})=1-|\pi_{v}|_{v}.$

One has further that

$\int_{F_{v}^{\times}}k^{\mathbf{a}}_{v}(t\cdot\mathbf{x})d^{*}t=\int_{F_{v}^{\times}}\frac{\mathbf{1}_{\mathcal{D}^{\mathbf{a}}_{v}}(t\cdot\mathbf{x})}{1-|\pi_{v}|_{v}}d^{*}t=\frac{d^{*}t(\\{t\in
F_{v}^{\times}|t\cdot\mathbf{x}\in\mathcal{D}^{\mathbf{a}}_{v}\\})}{1-|\pi_{v}|_{v}}=1.$

The claim is proven. ∎

The following auxiliary lemma will be used in the definition of
$k^{\mathbf{a}}_{v}$ for $v$ infinite.

######  Lemma 2.

Let $a_{G_{1}}$ and $a_{G_{2}}$ be continuous actions of topological groups
$G_{1}$ and $G_{2}$ on a topological space $X.$ Suppose the actions are
permutable, that is for every $g_{1}\in G_{1},$ every $g_{2}\in G_{2}$ and
every $x\in X$ one has $g_{1}g_{2}x=g_{2}g_{1}x$.

  1. 1.

The map

$\displaystyle a_{G_{1}\times
G_{2}}\mathrel{\mathop{\ordinarycolon}}G_{1}\times G_{2}\times X$
$\displaystyle\to X$ $\displaystyle(g_{1},g_{2},x)$ $\displaystyle\mapsto
g_{1}(g_{2}x)$

defines a continuous action of $G_{1}\times G_{2}$ on $X$.

  2. 2.

There exists a continuous action of $G_{2}$ on $X/G_{1}$ such that

$g_{2}\cdot[x]_{G_{1}}=[g_{2}x]_{G_{1}}$

for every $x\in X$, where $[x]_{G_{1}}$ is the image of $x$ in $X/G_{1}$ for
the quotient map.

  3. 3.

The canonical map $X/G_{1}\to X/(G_{1}\times G_{2})$ is open, continuous,
surjective and $G_{2}$-invariant. The induced map $(X/G_{1})/G_{2}\to
X/(G_{1}\times G_{2})$ is a homeomorphism.

###### Proof.

  1. 1.

The map $a_{G_{1}\times G_{2}}$ factorizes as
$a_{G_{2}}\circ(\operatorname{Id}_{G_{1}}\times a_{G_{1}})$ thus is
continuous. If $e_{G_{1}},e_{G_{2}}$ are neutral elements of $G_{1}$ and
$G_{2}$, respectively, by definition one has $(e_{G_{1}},e_{G_{2}})\cdot
x=e_{G_{1}}e_{G_{2}}x=x.$ Moreover, if $(g_{1},g_{2})$ and
$(g_{1}^{\prime},g_{2}^{\prime})$ are elements of $G_{1}\times G_{2}$, then
for $x\in X$ one has

$(g_{1}g_{1}^{\prime},g_{2}g_{2}^{\prime})x=(g_{1}g_{1}^{\prime})(g_{2}g_{2}^{\prime}x)=g_{1}((g_{1}^{\prime}(g_{2}g_{2}^{\prime}x)))=g_{1}(g_{2}(g_{1}^{\prime}g_{2}^{\prime}x))\\\
=(g_{1},g_{2})(g_{1}^{\prime},g_{2}^{\prime})x.$

We deduce that $a_{G_{1}\times G_{2}}$ is a continuous action of $G_{1}\times
G_{2}$ on $X$.

  2. 2.

This is proven in [7, Remark to Proposition 11, n0 4, §4, Chapter III].

  3. 3.

The map $X/G_{1}\to X/(G_{1}\times G_{2})$ is the induced map from
$G_{1}$-invariant continuous, open and surjective map $X\to X/(G_{1}\times
G_{2})$, thus itself is continuous, open and surjective. If $g_{2}\in G_{2}$
and $x\in X$, one has that

$[g_{2}\cdot[x]_{G_{1}}]_{G_{1}\times G_{2}}=[[g_{2}x]_{G_{1}}]_{G_{1}\times
G_{2}}=[g_{2}x]_{G_{1}\times G_{2}}=[x]_{G_{1}\times G_{2}},$

where $[\cdot]_{G_{1}\times G_{2}}$ is the image in $X/(G_{1}\times G_{2})$.
Hence, $X/G_{1}\to X/(G_{1}\times G_{2})$ is continuous. Suppose now elements
$[x]_{G_{1}}$ and $[y]_{G_{1}}$ have the same image in $X/(G_{1}\times
G_{2})$. This means precisely that there exists $(g_{1},g_{2})\in G_{1}\times
G_{2}$ such that $(g_{1},g_{2})\cdot x=g_{1}g_{2}x=y$. We deduce that

$[y]_{G_{1}}=[g_{2}g_{1}x]_{G_{1}}=[g_{2}x]_{G_{1}}=g_{2}[x]_{G_{1}},$

i.e. $[x]_{G_{1}}$ and $[y]_{G_{2}}$ are in the same orbit of $G_{2}$. We
deduce that the induced map $((X/G_{1})/G_{2})\to(X/(G_{1}\times G_{2}))$ is
bijective. It follows that it is a homeomorphism.

∎

######  Lemma 3.