By Lemma 2.14 we get the following short exact sequence of $\mathbb{R}$-vector
spaces equipped with compatible Haar measures

$0\to\hat{\mathbf{T}}_{\mathbb{R}}^{*}\to\hat{T}^{{}^{\prime}*}_{F_{v},\mathbb{R}}\times\prod_{v\in\Omega_{F}^{\infty}}\mathbb{R}[A_{v}/\Gamma_{v}]\to\mathbb{R}[E/\Gamma_{v}]\to
0$ (16)

The logarithmic norm maps the exact sequence (15) to (16) and these maps are
surjective by Lemma 3.27. Taking kernels we thus get an exact sequence

$1\to\mathbf{T}(\mathbb{A}_{F})^{1}\to
T^{\prime}(\mathbb{A}_{F})^{1}\times\prod_{v\in\Omega_{F}^{\infty}}\mathbb{G}_{m}^{A_{v}}(\mathcal{O}_{v})\to\mathbb{G}_{m}^{E}(\mathbb{A}_{v})^{1}\to\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T})\to\operatorname{H}^{1}(\mathbb{A}_{F},T^{\prime})\to
0$

The relevant Tamagawa measures are again compatible.

Qoutienting this by the exact sequence (12) we get an exact sequence of
compact groups with compatible measures. This implies the equality

$\begin{split}&\tau_{\mathbf{T}^{1}}\left(\mathbf{T}(\mathbb{A}_{F})^{1}/T(F)\right)\tau_{\mathbb{G}_{m}^{A,1}}\left(\mathbb{G}_{m}^{E}(\mathbb{A}_{F})^{1}/\mathbb{G}_{m}^{E}(F)\right)\frac{\tau(T^{\prime})}{\\#{}^{1}(F,X^{*}(T^{\prime}))}=\\\
&\tau_{T^{\prime
1}}\left(T^{\prime}(\mathbb{A}_{F})^{1}/T^{\prime}(F)\right)\prod_{v\in\Omega_{F}^{\infty}}\tau_{v}(\mathbb{G}_{m}^{A_{v}}(\mathcal{O}_{v}))\frac{\tau(\mathbf{T})}{\\#{}^{1}(F,X^{*}(\mathbf{T}))}.\end{split}$

Now
$\tau_{\mathbb{G}_{m}^{E,1}}\left(\mathbb{G}_{m}^{E}(\mathbb{A}_{F})^{1}/\mathbb{G}_{m}^{E}(F)\right)=1$
by [Ono61, Thm. 3.5.1] and $\tau_{T^{\prime
1}}\left(T^{\prime}(\mathbb{A}_{F})^{1}/T^{\prime}(F)\right)=\tau(T^{\prime})$
by [Ono63, Main Thm.] and Corollary 3.38.

Since $\mathbb{G}_{m}^{A_{v}}(\mathcal{O}_{v})$ is a product of complex unit
circles $S^{1}$ and the group $\\{1,-1\\}$, respectively equipped with the
Lebesgue measure and the counting measure it only remains to show that
$\\#{}^{1}(F,X^{*}(T^{\prime}))=\\#{}^{1}(F,X^{*}(\mathbf{T}))$.

To do this we use that
$\operatorname{H}^{1}(F,\mathbb{Z}[E])=\operatorname{H}^{1}(F_{v},\mathbb{Z}[E])=\operatorname{H}^{1}(F_{v},\mathbb{Z}[A_{v}])=0$
for all $v$ to deduce the exactness of the following sequences

$0\to\operatorname{H}^{1}(F,X^{*}(T^{\prime}))\to\operatorname{H}^{1}(F,X^{*}(T_{F}))\to\operatorname{H}^{2}(F,\mathbb{Z}[E])$
$0\to\operatorname{H}^{1}(F_{v},X^{*}(T^{\prime}))\to\operatorname{H}^{1}(F_{v},X^{*}(T_{v}))\to\operatorname{H}^{2}(F_{v},\mathbb{Z}[E])$

Homological algebra shows that the sequence
$0\to{}^{1}(F,X^{*}(T^{\prime}))\to{}^{1}(F,X^{*}(\mathbf{T}))\to{}^{2}(F,\mathbb{Z}[E])$
is exact, but ${}^{2}(F,\mathbb{Z}[E])$ is dual to
${}^{1}(F,\mathbb{G}_{m}^{E})\subset\operatorname{H}^{1}(F,\mathbb{G}_{m}^{E})=0$
by [NSW08, Thm. 8.6.7]. The proposition follows. ∎

##  5\. Universal torsors and the Brauer-Manin obstruction

###  5.1. The Brauer-Manin pairing

We refer to [CTS21, §12] for the usual Brauer-Manin pairing for varieties. We
will generalize this to adelic varieties. For this section fix a number field
$F$ and a geometrically integral adelic variety $\mathbf{X}$ over $F$. Recall
that for each place $v\in\Omega_{F}$ there exists an injection
$\text{inv}_{v}:\operatorname{Br}F_{v}\to\mathbb{Q}/\mathbb{Z}$ which is
surjective if $v$ is finite

######  Definition 5.1.

The _Brauer-Manin pairing_ is defined as the pairing

$\langle\cdot,\cdot\rangle_{\text{BM}}:\mathbf{X}(\mathbb{A}_{F})\times\operatorname{Br}\mathbf{X}\to\mathbb{Q}/\mathbb{Z}:((x_{v})_{v\in\Omega_{F}},(A_{F},(A_{v})_{v\in\Omega_{F}})\to\sum_{v\in\Omega_{F}}\text{inv}_{v}(A_{v}(x_{v})).$

It is linear in the second variable. It is well-defined and continuous for the
usual reason [CTS21, Prop. 12.3.1].

The constant Brauer elements $\operatorname{Br}_{0}\mathbf{X}$ are contained
in the right kernel of this pairing due to [CTS21, Thm. 12.1.8], we thus get
an induced pairing
$\mathbf{X}(\mathbb{A}_{F})\times\operatorname{Br}\mathbf{X}/\operatorname{Br}_{0}\mathbf{X}\to\mathbb{Q}/\mathbb{Z}$,
which we will also call the Brauer-Manin pairing.

The _Brauer-Manin set_ $\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}}$ is the
left kernel of this pairing. By the usual argument
$X_{F}(F)\subset\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}}$. More
generally, for any subset $B\subset\operatorname{Br}\mathbf{X}$ or
$B\subset\operatorname{Br}\mathbf{X}/\operatorname{Br}_{0}\mathbf{X}$ we
denote by $\mathbf{X}(\mathbb{A}_{F})^{B}$ the elements of
$\mathbf{X}(\mathbb{A}_{F})$ orthogonal to $B$ with respect to the Brauer-
Manin pairing. If $B=\operatorname{Br}_{1}\mathbf{X}$ then we write
$\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}_{1}}:=\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}_{1}\mathbf{X}}$
and call it the algebraic Brauer-Manin set.

####  5.1.1. Strong approximation

Let $X$ be a geometrically integral smooth proper variety over $F$, $D\subset
X$ a geometrically simple normal crossing divisor. It is known that integral
points on $U:=X\setminus D$ tend to concentrate near maximal faces of the
Clemens complex. The naive form of strong approximation, that $U(F)$ is dense
in $U(\mathbb{A}_{F})$, will thus often fail, the simplest example is is
$U=\mathbb{A}^{1}$. We propose the following alternative definition

######  Definition 5.2.

Let $\mathbf{A}\in\mathcal{C}^{\operatorname{an}}_{\Omega_{F}^{\infty}}(D)$ be
a face of the Clemens complex. We say that $(X;\mathbf{A})$ satisfies _strong
approximation_ if the closure of $U(F)$ in $(X;\mathbf{A})(\mathbb{A}_{F})$
contains $U(\mathbb{A}_{F})_{\mathbf{A}}$.

The Brauer-Manin set $(X;\mathbf{A})(\mathbb{A}_{F})^{\operatorname{Br}}$ is
closed since the Brauer-Manin pairing is continuous and
$U(F)\subset(X;\mathbf{A})(\mathbb{A}_{F})^{\operatorname{Br}}$. So if
$U(\mathbb{A}_{F})_{\mathbf{A}}^{\operatorname{Br}}:=U(\mathbb{A}_{F})_{\mathbf{A}}\cap(X;\mathbf{A})(\mathbb{A}_{F})^{\operatorname{Br}}\neq
U(\mathbb{A}_{F})_{\mathbf{A}}$ then $(X;\mathbf{A})$ cannot satisfy strong
approximation, in this case we will that that there is a _Brauer-Manin
obstruction to strong approximation_. If the closure of $U(F)$ in
$(X;\mathbf{A})(\mathbb{A}_{F})$ contains
$U(\mathbb{A}_{F})_{\mathbf{A}}^{\operatorname{Br}}$ then we say that the
Brauer-Manin obstruction is the _only obstruction to strong approximation_. If
the closure of $U(F)$ in $(X;\mathbf{A})(\mathbb{A}_{F})$ contains
$U(\mathbb{A}_{F})_{\mathbf{A}}^{\operatorname{Br}_{1}}$ then we say that the
_algebraic_ Brauer-Manin obstruction is the only obstruction to strong
approximation.

###  5.2. Universal torsors

Let us recall the definition of type for any field $F$.

######  Definition 5.3.

Let $T$ be a group of multiplicative type and $\pi:Y\to X$ a $T$-torsor. Its
_type_ is the $\Gamma_{F}$-equivariant morphism
$\texttt{type}(\pi):X^{*}(T)\to\operatorname{Pic}X_{F^{\operatorname{sep}}}$
which sends a character $\chi:T_{F^{\operatorname{sep}}}\to\mathbb{G}_{m}$ to
the image of the class $[Y]\in H^{1}(X_{F^{\operatorname{sep}}},T)$ under the
map $\chi_{*}:H^{1}(X_{F^{\operatorname{sep}}},T)\to
H^{1}(X_{F^{\operatorname{sep}}},\mathbb{G}_{m})=\operatorname{Pic}X_{F^{\operatorname{sep}}}$.

There are other equivalent definitions of the type, see [Sko01, Lemma 2.3.1].

We can extend this definition to adelic scheme. Let $F$ be a number field,
$\mathbf{X}$ an adelic scheme over $F$ and $\mathbf{T}$ an adelic group of
multiplicative type over $F$. The following is well-defined since the type is
functorial and invariant under base change of field.

######  Definition 5.4.

Let $\bm{\pi}:\mathbf{Y}\to\mathbf{X}$ be a $\mathbf{T}$-torsor. Its _type_ is
the morphism of adelic modules
$\texttt{type}(\bm{\pi}):X^{*}(\mathbf{T})\to\bm{\operatorname{Pic}}\mathbf{X}$
given by the tuple
$(\texttt{type}(\pi_{F}),(\texttt{type}(\pi_{v}))_{v\in\Omega_{F}})$.

If a variety has a rational point then torsors of every possible type exist.
This also holds for adelic varieties.

######  Lemma 5.5.

Let $\mathbf{X}$ be an adelic scheme, $\mathbf{L}$ a finitely generated adelic
module $\mathbf{T}:=D(\mathbf{L})$ the Cartier dual and
$\bm{\lambda}:\mathbf{L}\to\bm{\operatorname{Pic}}\mathbf{X}$ a morphism of
adelic modules.

If $X_{F}(F)\neq\emptyset$ then there exists a torsor of type $\lambda$.

###### Proof.

Let $x\in X_{F}(F)$. By [CTS87, p. 426] there exist a $T_{F}$ torsor
$\pi_{F}:Y_{F}\to X_{F}$ of type $\lambda_{F}$ such that $\pi_{F}^{-1}(x)\cong
T_{F}$ which is unique up to (non-unique) isomorphism.

Similarly, for every place $v$ there exists a $T_{v}$-torsor $\pi_{v}:Y_{v}\to
X_{v}$ of type $\lambda_{v}$ such that $\pi_{v}^{-1}(\xi_{v}(x))\cong T_{v}$
which is unique up to isomorphism.

It follows from the uniqueness up to isomorphism that
$(Y_{F})_{F_{v}}\times^{T}T_{v}\cong Y_{v}\times_{X_{v}}(X_{F})_{F_{v}}$ for
all $v$. We can thus form an adelic variety $\mathbf{Y}:=(Y_{F},(Y_{v})_{v})$
and $\bm{\pi}:=(\pi_{F},(\pi_{v})_{v})$ is a $\mathbf{T}$-torsor of type
$\bm{\lambda}$ ∎

We can now define the notion of _universal torsors_ , which will play a
crucial role in our argument.

######  Definition 5.6.

Assume that the adelic module $\bm{\operatorname{Pic}}\mathbf{X}$ is finitely
generated as an abelian group444One could remove this assumption by replacing
the geometric Picard group $\operatorname{Pic}X_{F^{\operatorname{sep}}}$ by
the geometric Néron–Severi group.. The _Néron–Severi torus_ 555We warn that
this is not necessarily a torus. of $X$ is the adelic group of multiplicative
type $\mathbf{T}_{\operatorname{NS}}:=D(\bm{\operatorname{Pic}}\mathbf{X})$.

A _universal torsor_ of $X$ is a $\mathbf{T}_{\operatorname{NS}}$-torsor
$\bm{\pi}:\mathbf{Y}\to\mathbf{X}$ whose type $\texttt{type}(\bm{\pi})$ is the
identity map
$\bm{\operatorname{Pic}}\mathbf{X}\to\bm{\operatorname{Pic}}\mathbf{X}$. In
other words, $Y_{F}\to X_{F}$ and $Y_{v}\to X_{v}$ are universal torsors for
all places $v$.

######  Remark 5.7.

Universal torsors are not unique, but they are unique up to twists, see
[Sko01, (2.22)].

One can recover the universal torsor on an open subvariety from the universal
torsors of the original variety. The following lemma and construction are
probably known, but we were unable to find a reference.

######  Construction 5.8.

Let $X$ be a smooth variety over a field $K$ such that
$\operatorname{Pic}X_{K^{\operatorname{sep}}}$ is finitely generated and
torsion-free and
$K^{\operatorname{sep}}[X]^{\times}=K^{\operatorname{sep},\times}$. Let
$T_{\operatorname{NS}X}$ be the Néron-Severi torus of $X$ and $\pi:Y\to X$ a
universal torsor. Let $D\subset X$ be a divisor.

Let $A$ be the $\Gamma_{F}$-set of irreducible components of the divisor
$D_{K^{\operatorname{sep}}}\subset X_{K^{\operatorname{sep}}}$, for $a\in A$
let $D_{a}$ be the corresponding irreducible component. The pullback of
$D_{a}$ along $Y_{K^{\operatorname{sep}}}$ is a principal divisor by [CTS87,
Prop. 2.1.1] so defines a map
$Y_{K^{\operatorname{sep}}}\to\mathbb{A}_{K^{\operatorname{sep}}}^{1}$ such
that the fiber at $0$ is $\pi^{-1}(D_{a})$. This map is unique up to
multiplication by a constant since by loc. cit.
$K^{\operatorname{sep}}[Y]^{\times}=K^{\operatorname{sep}}$.

Multiplying these maps for all $a\in A$ defines a map
$Y_{F^{\operatorname{sep}}}\to\mathbb{A}_{K^{\operatorname{sep}}}^{A}$, unique
up to multiplication by an element of
$\mathbb{G}_{m}^{A}(K^{\operatorname{sep}})$. The obstruction to this map
being $\Gamma_{K}$-invariant is thus given by a cocycle
$\Gamma_{K}\to\mathbb{G}_{m}^{A}(K^{\operatorname{sep}})$. But
$\operatorname{H}^{1}(K,\mathbb{G}_{m}^{A}(K^{\operatorname{sep}}))=0$ by
Lemma 2.7 so after multiplication by an element of
$\mathbb{G}_{m}^{A}(K^{\operatorname{sep}})$ the map becomes
$\Gamma_{K}$-invariant and thus descends to a map
$f_{D}:Y\to\mathbb{A}_{m}^{A}$, unique up to multiplication by
$\mathbb{G}_{m}^{A}(K)$.

Let $U:=X\setminus D$. Assume that
$K^{\operatorname{sep}}[U]^{\times}=K^{\operatorname{sep},\times}$ and let
$T_{\operatorname{NS}U}$ be the Néron-Severi torus of $U$. This implies that
the map of $\Gamma_{K}$-modules
$\mathbb{Z}[A]\to\operatorname{Pic}X_{K^{\operatorname{sep}}}:a\to[D_{a}]$ is
injective. Its cokernel is $\operatorname{Pic}U_{K^{\operatorname{sep}}}$.
Dually we get an exact sequence of groups of multiplicative type $1\to
T_{\operatorname{NS}U}\to T_{\operatorname{NS}X}\to\mathbb{G}_{m}^{A}\to 1$.

The map $f:Y\to\mathbb{A}_{m}^{A}$ is $T_{\operatorname{NS}X}$-equivariant,
where $T_{\operatorname{NS}X}$ acts on $\mathbb{G}_{m}^{A}$ through the
inverse of the map $T_{\operatorname{NS}X}\to\mathbb{G}_{m}^{A}$. Indeed this
can be checked after base change to $K^{\operatorname{sep}}$ in which case it
can be reduced to the case when $D$ is irreducible. The statement is an easy
computation.

It follows from the construction that $\pi^{-1}(U)\cong
f^{-1}(\mathbb{G}_{m}^{A})$. Each fiber of the map
$f:\pi^{-1}(U)\to\mathbb{G}_{m}^{A}$ has a $T_{\operatorname{NS}U}$-action.
Note also that if $t\in\mathbb{G}_{m}(A)(F)$ and
$g\in\text{im}(T_{\operatorname{NS}X}(K)\to\mathbb{G}_{m}^{A}(K))$ then $g$
there is an isomorphism between $f^{-1}(t)$ and $f^{-1}(g\cdot t)$.

######  Lemma 5.9.

Let $t\in\mathbb{G}_{m}^{A}(K)$. Then $f^{-1}(t)\to U$ is a universal torsor.

Moreover, if we fix for each
$a\in\operatorname{H}^{1}(K,T_{\operatorname{NS}})$ a map
$f_{a}:Y_{a}\to\mathbb{G}_{m}^{A}$ then every universal torsor of $U$ is of
the form $f^{-1}(t)$ for some $t\in\mathbb{G}_{m}^{A}(K)$. Moreover, the $t$
is unique up to multiplication by an element of
$\emph{im}(T_{\operatorname{NS}X}(K)\to\mathbb{G}_{m}^{A}(K))$.

###### Proof.

The quotient $\pi^{-1}(U)/T_{\operatorname{NS}U}\to U$ is a
$\mathbb{G}_{m}^{A}$-torsor because $\pi^{-1}(U)\to U$ is a
$T_{\operatorname{NS}}$-torsor. The map $\pi^{-1}(U)\to\mathbb{G}_{m}^{A}$
factors trough $\pi^{-1}(U)/T_{\operatorname{NS}U}\to\mathbb{G}_{m}^{A}$ and
is $\mathbb{G}_{m}^{A}$-equivariant. This implies that the torsor
$\pi^{-1}(U)/T_{\operatorname{NS}U}\to U$ is trivial so that the product map
$\pi^{-1}(U)/T_{\operatorname{NS}U}\to U\times\mathbb{G}_{m}^{A}$ is an
isomorphism. Taking fibers of $t$ we deduce that
$f^{-1}(t)/T_{\operatorname{NS}U}\to U$ is an isomorphism, i.e. $f^{-1}(t)\to
U$ is a $T_{\operatorname{NS}U}$-torsor.

Note that the above implies that $(f^{-1}(t)\times
T_{\operatorname{NS}X})/T_{\operatorname{NS}U}\cong\pi^{-1}(U)$. We deduce
that $\pi^{U}\to U$ is the image of $f^{-1}(t)\to U$ along the map
$\operatorname{H}^{1}(U,T_{\operatorname{NS}U})\to\operatorname{H}^{1}(U,T_{\operatorname{NS}X})$.

The $T_{\operatorname{NS}X}$-torsor $\pi^{-1}(U)\to U$ has type
$\operatorname{Pic}X_{K^{\operatorname{sep}}}\to\operatorname{Pic}U_{K^{\operatorname{sep}}}$
by functoriality of the type. We deduce from the above paragraph and the
definition of the type that $f^{-1}(t)\to U$ is a universal torsor.

To show the last statement it suffices to show that the twists of $f^{-1}(1)$
are of always of the form $f_{a}^{-1}(t)$. Note that diagram $X\leftarrow
Y\to\mathbb{A}_{m}^{A}$ where $Y\to X$ is a $T_{\operatorname{NS}X}$ and
$Y\to\mathbb{A}^{A}$ is $T_{\operatorname{NS}X}$-equivariant has automorphism
group
$\ker(T_{\operatorname{NS}X}\to\mathbb{G}_{m}^{A})=T_{\operatorname{NS}U}$.

If $b\in\operatorname{H}^{1}(K,T_{\operatorname{NS}U})$ then this implies that
the twist of $f^{-1}(1)\to U$ by $b$ is the fiber of the rightmost map of the
twist by $b$ of the entire diagram $X\leftarrow Y\to\mathbb{A}_{m}^{A}$. The
twist of underlying variety $Y$ is $Y_{a}$ where $a$ is the image of $b$ along
$\operatorname{H}^{1}(K,T_{\operatorname{NS}U}U)\to\operatorname{H}^{1}(K,T_{\operatorname{NS}X})$.
The group $T_{\operatorname{NS}U}$ acts trivially on $\mathbb{A}^{A}$ so the
twist of this diagram is $X\leftarrow Y_{a}\xrightarrow{f_{b}}\mathbb{A}^{A}$.

The map $f_{b}$ is not necessarily equal to $f_{a}$, the two maps can differ
by an element of $\mathbb{G}_{m}^{A}(K)$. The two diagrams $X\leftarrow
Y_{a}\xrightarrow{f_{b}}\mathbb{A}_{m}^{A}$ and $X\leftarrow
Y_{a}\xrightarrow{f_{b}}\mathbb{A}_{m}^{A}$ are isomorphic if and only they
differ by an element of $T_{\operatorname{NS}X}(K)$.

We deduce that the twists of $X\leftarrow Y\to\mathbb{G}_{m}^{A}$ are all of
the form $X\leftarrow Y_{a}\xrightarrow{t\cdot f_{a}}\mathbb{A}_{m}^{A}$ with
$t\in\mathbb{G}_{m}^{A}(K)$ unique up to multiplication by
$\text{im}(T_{\operatorname{NS}X}(K)\to\mathbb{G}_{m}^{A}(K))$. The
corresponding twist of $f^{-1}(1)$ is given by $(t\cdot
f_{a})^{-1}(1)=f_{a}^{-1}(t^{-1})$. ∎

Universal torsors can be used to study the Brauer-Manin obstruction, as was
first observed by Colliot-Thélène and Sansuc [CTS87] for proper varieties $X$.
This relation extends to adelic varieties

######  Theorem 5.10.

Let $\mathbf{X}$ be an adelic variety over a number field $F$ such that
$\overline{F}_{v}[X_{v}]^{\times}=\overline{F}_{v}$ and
$\operatorname{Pic}(X_{v})_{\overline{F}_{v}}$ is finitely generated for all
places $v\in\Omega_{F}$. Let $\bm{\pi}:\mathbf{Y}\to\mathbf{X}$ be a universal
torsor. We then have

$\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}_{1}\mathbf{X}}=\bigcup_{a\in
H^{1}(F,T_{\operatorname{NS},F})}\bm{\pi}_{a}(\mathbf{Y}_{a}(\mathbb{A}_{F})).$

We will prove this completely analogously to the case of varieties [Sko01,
§6.1].

For $x_{v}\in X(F_{v})$ let $Y_{v}(x_{v}):=\pi_{v}^{-1}(x_{v})$. This is a
$T_{v}$-torsor over $F_{v}$, namely the unique $T_{v}$-torsor such that
$x_{v}\in\pi_{Y_{v}(x_{v})}(F_{v})$, see [Sko01, p. 22]. If $T$ has good
reduction at $v$ and $\mathcal{Y}\to\mathcal{X}$ is an integral model of the
$T_{v}$ and $x_{v}\in\mathcal{X}(\mathcal{O}_{v})$ then
$Y_{v}(x_{v})\in\operatorname{H}^{1}(\mathcal{O}_{v},T_{v})$ so taking
restricted products of the $Y_{v}$ induces a continuous map

$\mathbf{Y}(\cdot):\mathbf{X}(\mathbb{A}_{F})\to\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}).$
(17)

Theorem 5.10 will follow from the following lemma.

######  Lemma 5.11.

For $x\in\mathbf{X}(\mathbb{A}_{F})$ and $A\in\operatorname{Br}_{1}\mathbf{X}$
we have $\langle
x,A\rangle_{\emph{BM}}=\langle\mathbf{Y}(x),r(A)\rangle_{\emph{PT}}$.

###### Proof.

The bilinear pairings
$T_{F}\times\operatorname{Pic}(X_{F})_{\overline{F}}\to\mathbb{G}_{m}$ and
$T_{v}\times\operatorname{Pic}(X_{v})_{\overline{F}_{v}}\to\mathbb{G}_{m}$ for
$v\in\Omega_{F}$ induces compatible cup product pairings

$\displaystyle\cdot\cup\;\cdot:\operatorname{H}^{1}(F,\operatorname{Pic}(X_{F})_{\overline{F}})\times\operatorname{H}^{1}(X,T_{F})\to\operatorname{H}^{2}(X_{F},\mathbb{G}_{m})=\operatorname{Br}(X_{F})$
$\displaystyle\cdot\cup\;\cdot:\operatorname{H}^{1}(F,\operatorname{Pic}(X_{v})_{\overline{F}_{v}})\times\operatorname{H}^{1}(X,T_{v})\to\operatorname{H}^{2}(X_{v},\mathbb{G}_{m})=\operatorname{Br}(X_{v}).$

These pairings factor through $\operatorname{Br}_{1}X_{F}$ and
$\operatorname{Br}_{1}X_{v}$ respectively since every cocycle in
$\operatorname{H}^{1}(F,\operatorname{Pic}(X_{F})_{\overline{F}})$ and
$\operatorname{H}^{1}(F,\operatorname{Pic}(X_{v})_{\overline{F}_{v}})$ becomes
trivial after base change to the algebraic closure.

Consider the product pairing

$\cdot\cup\;\cdot:\operatorname{H}^{1}(\mathbb{A}_{F},\bm{\operatorname{Pic}}\mathbf{X})\times\operatorname{H}^{1}(\mathbf{X},\mathbf{T})\to\operatorname{Br}_{1}(\mathbf{X}).$

Note that $r(r(A)\cup[\mathbf{Y}])=r(A)$. Indeed, this means that
$r([Y_{F}]\cup r(A_{F}))=r(A_{F})$ and $r(r(A_{v})\cup[Y_{v}])=r(A_{v})$ and
these follow from [Sko01, Thm. 4.1.1].

By Lemma 3.25 we deduce that $A=r(A)\cup[\mathbf{Y}]$ in
$\operatorname{Br}_{1}\mathbf{X}/\operatorname{Br}F$. We thus have

$\langle x,A\rangle_{\text{BM}}=\langle
x,r(A)\cup[\mathbf{Y}]\rangle_{\text{BM}}=\sum_{v\in\Omega_{F}}\text{inv}_{v}(r(A)\cup
Y_{v}(x_{v}))=\langle r(A),\mathbf{Y}(x)\rangle_{\text{PT}}.$

∎

###### Proof of Theorem 5.10.

Let $x\in\mathbf{X}(\mathbb{A}_{F})$. By Lemma 5.11 we have
$x\in\mathbf{X}(\mathbb{A}_{F})^{\operatorname{Br}_{1}\mathbf{X}}$ if and only
if $\langle\mathbf{Y}(x),r(A)\rangle_{\text{PT}}=0$ for all
$A\in\operatorname{Br}_{1}\mathbf{X}$. But $r$ is surjective by Lemma 3.25 so
this is equivalent to the equality
$\langle\mathbf{Y}(x),b\rangle_{\text{PT}}=0$ for all
$b\in\operatorname{H}^{1}(F,\bm{\operatorname{Pic}}\mathbf{X})$. We deduce
from Proposition 3.29 that this is equivalent to the existence of an
$a\in\operatorname{H}^{1}(F,T_{\operatorname{NS},F})$ such that
$\mathbf{Y}(x)=a$, i.e. $x\in\pi_{a}(\mathbf{Y}_{a}(\mathbb{A}_{F}))$. ∎

###  5.3. Tamagawa measures on universal torsors

In this subsection we will construct Tamagawa measures on universal torsors
and show that they are compatible with Tamagawa measures on the base. We start
with the local case.

####  5.3.1. Local Tamagawa measures on universal torsors

Let $F$ be a local field, $X$ be a geometrically integral smooth variety over
$F$. Assume that $\operatorname{Pic}X_{F^{\operatorname{sep}}}$ is finitely
generated and
$F^{\operatorname{sep}}[X]^{\times}=F^{\operatorname{sep},\times}$. Let
$T_{\operatorname{NS}}$ be the Néron-Severi torus and $\pi:Y\to X$ be a
universal torsor.

Let $D\subset X$ be a geometrically simple normal crossing divisor and let
$D_{1},\cdots,D_{n}$ be the irreducible components of
$D_{F^{\operatorname{sep}}}$. Let $Z:=\cap_{i=1}^{n}D_{i}$. This is
$\Gamma_{F}$-invariant and thus defined over $F$.

Let us first recall the structure of $\omega_{Y}$.

######  Lemma 5.12.

There is a canonical isomorphism
$\pi^{*}(\omega_{X}(D))\otimes\operatorname{H}^{0}(T,\omega_{T})^{T}\cong\omega_{Y}(\pi^{-1}(D))$
which is locally on $X$ given by taking wedge products of differential forms.

We also have $\omega_{Y}(\pi^{-1}(D))\cong\mathcal{O}_{Y}$ and
$\operatorname{H}^{0}(Y,\omega_{Y}(\pi^{-1}(D))^{\neq 0})\cong F^{\times}$.

###### Proof.

The first statement follows from the fact that
$\omega_{Y/X}\cong\mathcal{O}_{Y}\otimes\operatorname{H}^{0}(T,\omega_{T})^{T}$
by taking determinants in [Sal98, Prop. 3.8, 3.9].

The second follows from the first and the fact that
$F^{\operatorname{sep}}[Y]^{\times}\cong F^{\operatorname{sep},\times}$ and
that the map $\operatorname{Pic}X\to\operatorname{Pic}Y$ is zero, see [CTS87,
Prop. 2.1.1]. ∎

######  Construction 5.13.

Let $||\cdot||$ be an adelic metric on $\omega_{X}$.

By repeated applications of the adjunction formula there is a canonical
isomorphism $\omega_{\pi^{-1}(Z)}\cong\omega_{Y}(D)$, cf. §4.3.1.

Let $\omega\in\omega_{X}(D)^{\neq 0}(F)$. Then $\pi^{*}\omega\wedge
d\log_{T}\in\omega_{Y}(\pi^{-1}(D))^{\neq 0}$ by Lemma 5.12. We can then use
the isomorphism $\omega_{\pi^{-1}(Z)}\cong\omega_{Y}(D)$ to define a local
Tamagawa measure $\tau_{Y,||\cdot||}:=|\pi^{*}\omega\wedge d\log_{T}|/$ on
$\pi^{-1}(Z)(F)$. This is independent of $\omega$, cf. §4.3.1.

We will denote the pushforward of this measure along the closed inclusion
$\pi^{-1}(Z)(F)\subset Y(F)$ also by $\tau_{Y,\omega}$.

Since we can do the above construction for any universal torsor the above
construction can be applied to define a measure
$\cup_{a\in\operatorname{H}^{1}(F,T_{\operatorname{NS}})}Y_{a}(F)$ which we
will denote by $\tau_{\cup_{a}Y_{a},\omega}$.

This measure is related to measures on the base.

######  Lemma 5.14.

For every open $V\subset Z(F)$ for which $\pi^{-1}(V)\cong V\times
T_{\operatorname{NS}}(F)$ one has the equality of measures

$\tau_{Y,||\cdot||}(x)=(\tau_{X,||\cdot||}\times\tau_{T_{\operatorname{NS}}})(x).$

###### Proof.

This follows immediately after unfolding the definitions. ∎

####  5.3.2. Global Tamagawa measures on universal torsors

Let now $X$ be a geometrically integral smooth proper variety over a number
field $F$. Let $D\subset X$ be a divisor with geometrically strict normal
crossing and
$\mathbf{A}\in\mathcal{C}^{\operatorname{an}}_{\Omega_{F}^{\infty}}$ a face of
the analytic Clemens complex. Let $U:=X\setminus D$ and assume that
$\overline{F}[U]^{\times}=\overline{F}^{\times}$

Assume that
$\operatorname{H}^{1}(X,\mathcal{O}_{X})=\operatorname{H}^{2}(X,\mathcal{O}_{X})=0$.
In particular $\operatorname{Pic}X$ is torsion-free and finitely generated.
Let $\mathbf{T}_{\operatorname{NS}}$ be the Néron-Severi torus.

Let $\bm{\pi}:\mathbf{Y}\to(X,\mathbf{A})$ be a universal torsor and let
$(||\cdot||_{v})_{v\in\Omega_{F}}$ be an adelic metric on this torsor with
corresponding height $\mathbf{H}$. The line bundle
$\omega_{(X;\mathbf{A})}\in\operatorname{Pic}(X;\mathbf{A})$ defines a
character $\mathbf{T}_{\operatorname{NS}}\to\mathbb{G}_{m}$. It follows from
the definition of type that there exists a
$\mathbf{T}_{\operatorname{NS}}$-equivariant map
$\mathbf{Y}\to\omega_{(X;\mathbf{A})}^{\neq 0}$, unique up to multiplication
by $F^{\times}$.

The adelic metric $(||\cdot||_{v})_{v\in\Omega_{F}}$ thus defines an adelic
metric on $\omega_{(X;\mathbf{A})}^{\neq 0}$, which by abuse of notation we
will also denote by $(||\cdot||_{v})_{v\in\Omega_{F}}$. It is only well-
defined up to multiplication by $F^{\times}$, but this does not influence the
heights or the adelic Tamagawa measures.

We have the following bound on the local integrals.

######  Lemma 5.15.

Let $v$ be a non-archimedean place such that $T_{\operatorname{NS}}$ has good
reduction and the $T_{\operatorname{NS}}$-torsor $Y\to U$ has an
$\mathcal{O}_{v}$-integral model $\mathcal{Y}\to\mathcal{U}$ such that
$\mathcal{U}$ is smooth and the metric $||\cdot||_{v}$ is
$\mathcal{U}$-integral. Let
$a_{v}\in\operatorname{H}^{1}(\mathcal{O}_{v},T_{\operatorname{NS}})$. Let
$\pi_{v}:\mathcal{Y}_{a}\to\mathcal{U}$ be a twist of
$\mathcal{Y}\to\mathcal{U}$. We have uniformly in $v$ that

$\tau_{Y_{v,a_{v}},||\cdot||_{v}}(\mathcal{Y}_{a_{v}}(\mathcal{O}_{v}))=1+O(q_{v}^{-\frac{3}{2}}).$

###### Proof.

This follows from [CLT10a, Thm 2.4.7, Rem. 2.4.12c)]. ∎

######  Construction 5.16.

Given
$\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})$
we define a measure on $\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})$ as the
product

$\tau_{\mathbf{Y}_{\mathbf{a}},\mathbf{H}}=\prod_{v}\tau_{Y_{v,a_{v}},||\cdot||_{v}}.$

This product converges by the previous lemma.

We also define a measure on
$\coprod_{\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}(\\#\pi_{0}((T_{\operatorname{NS},v}))^{-1}\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})$
as the product measure

$\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}=\prod_{v}(\\#\pi_{0}(T)(\mathcal{O}_{v}))^{-1}\tau_{\coprod_{a_{v}}Y_{v,a_{v}},||\cdot||_{v}}.$

This measure converges by the previous Lemma and Lemma 3.31.

######  Remark 5.17.

We again use $\mathbf{H}$ in the notation since these measures only depend on
the height function $\mathbf{H}$ by the same reasoning as in Lemma 4.23.

Actually the same argument as in Lemma 4.23 shows in this case that
$\mathbf{H}^{-K_{(X;\mathbf{A})}}\tau_{\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$
and
$\mathbf{H}^{-K_{(X;\mathbf{A})}}\tau_{\mathbf{Y}_{\mathbf{a},\mathbf{\mathbf{missing}}{H}}}\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$
are independent of $\mathbf{H}$.

We have the following relation between these measures.

######  Proposition 5.18.

Let
$f:\coprod_{\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})\to\mathbb{C}$
be a continuous $L^{1}$ function with respect to the measure
$\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$. Then

$\int_{\coprod_{\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})}f(y)d\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}(y)=\int_{\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\int_{\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})}f(y)d\tau_{\mathbf{Y}_{\mathbf{a}},\mathbf{H}}(y)d\mu_{\mathbf{T}_{\operatorname{NS}}}(\mathbf{a})$

###### Proof.

Both sides are infinite product measures over the same factors, because the
convergence factor $\\#\pi_{0}((T_{\operatorname{NS},v})$ appears both in
$\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$ and
$\mu_{\mathbf{T}_{\operatorname{NS}}}(\mathbf{a})$. Both sides are thus
integrals over the same measure and thus equal. ∎

We can also relate the measure
$\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$ to the
Tamagawa measure on the base $(X;\mathbf{A})$.

######  Lemma 5.19.

Let $V\subset(X;\mathbf{A})(\mathbb{A}_{F})$ be an open such that
$\bm{\pi}^{-1}(V)\cong\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})\times V$.
Over $V$ we have the equality of measures

$c_{\mathbf{A}}\cdot\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}=(\tau_{(X;\mathbf{A}),\mathbf{H}}\times\tau_{\mathbf{N}_{\operatorname{NS}}}).$

###### Proof.

This follows from Lemma 5.14 and noticing that the convergence factors in
$\tau_{(X;\mathbf{A}),\mathbf{H}}$ and $\tau_{\mathbf{N}_{\operatorname{NS}}}$
cancel with each other, except for the factor $c_{\mathbf{A}}$. ∎

An important consequence of this lemma is the following proposition.

######  Proposition 5.20.

Let $f:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{C}$ be an $L^{1}$ continuous
function with respect to the measure $\tau_{(X;\mathbf{A}),\mathbf{H}}$ and
$g:\operatorname{Pic}(X;\mathbf{A})\to\mathbb{C}$ a continuous $L^{1}$
function with respect to the measure
$d\mathbf{z}_{\operatorname{Pic}(X;\mathbf{A})}$. Then
$f(\bm{\pi}(y))g(\log\mathbf{H}(y))$ is $L^{1}$ for
$y\in\coprod_{\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)$
with respect to the measure
$\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}$ and

$\begin{split}&\int_{\coprod_{\mathbf{a}\in\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\mathbf{Y}_{\mathbf{a}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}f(\bm{\pi}(y))g(\log\mathbf{H}(y))d\tau_{\coprod_{\mathbf{a}}\mathbf{Y}_{\mathbf{a}},\mathbf{H}}(y)=\\\
&\tau(\mathbf{T}_{\operatorname{NS}})\int_{(X;\mathbf{A})(\mathbb{A}_{F})}f(x)d\tau_{(X;\mathbf{A}),\mathbf{H}}(x)\int_{\operatorname{Pic}(X;\mathbf{A})}g(z)d\mathbf{z}_{\operatorname{Pic}(X;\mathbf{A})}.\end{split}$

###### Proof.

By a partition of unity argument we may reduce to the case that $f$ is
supported on a $V\subset(X;\mathbf{A})(\mathbb{A}_{F})$ such that
$\bm{\pi}^{-1}(V)\cong\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})\times V$.

By the previous lemma and Fubini we may then first integrate over the fibers
of $\bm{\pi}$. Because $\tau_{\mathbf{T}_{\operatorname{NS}}}$ is a Haar
measure the integral over the fiber is always equal to
$\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}h(|y|_{\mathbf{T}_{\operatorname{NS}}})d\tau_{\mathbf{T}_{\operatorname{NS}}}$.

The integrand is $\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}$
invariant so by Proposition 4.30 we have

$\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}g(|y|_{\mathbf{T}_{\operatorname{NS}}})d\tau_{\mathbf{T}_{\operatorname{NS}}}=c_{\mathbf{A}}\tau(\mathbf{T}_{\operatorname{NS}})\int_{\operatorname{Pic}(X;\mathbf{A})}g(z)d\mathbf{z}_{\operatorname{Pic}(X;\mathbf{A})}.$

The remaining integral over $V$ is then exactly
$\int_{(X;\mathbf{A})(\mathbb{A}_{F})}f(x)d\tau_{(X;\mathbf{A}),\mathbf{H}}(x)$
as expected. ∎

####  5.3.3. Gauge forms

Peyre showed that in the case of proper varieties the Tamagawa measure on the
universal torsor can be interpreted in terms of Gauge forms of the universal
torsor. In this part we will show how to extend this relation to the non-
proper case. We keep the notation of the previous part.

##  6\. Integral Manin conjecture

Let $F$ be a number field, $X$ a geometrically integral smooth proper variety
over $F$, $D\subset X$ a geometrically strict normal crossing divisor and
$U:=X\setminus D$. Let $H:U(F)\to\mathbb{R}$ be a height function.

Let
$\mathcal{C}^{\operatorname{an}}_{\Omega_{F}^{\infty}}(D)=\prod_{v\in\Omega_{F}^{\infty}}\mathcal{C}^{\operatorname{an}}_{v}(D)$
be the $\Omega_{F}^{\infty}$-analytic Clemens complex of the pair $(X,D)$.
Note that for every place $v$ we have an open cover

$X(F_{v})=\bigcup_{(A_{v},Z_{v})\in\mathcal{C}^{\operatorname{an}}_{v}}(X;A_{v},Z_{v})(F_{v})=\bigcup_{(A_{v},Z_{v})\in\mathcal{C}^{\operatorname{an},\max}_{v}}(X;A_{v},Z_{v})(F_{v}).$

Choose a compactly supported partition of unity with respect to this cover,
i.e. choose for each
$(A_{v},Z_{v})\in\mathcal{C}^{\operatorname{an},\max}_{v}(D)$ a compactly
supported function $f_{(A_{v},Z_{v})}:(X;A_{v},Z_{v})(F_{v})\to[0,1]$ such
that
$\sum_{(A_{v},Z_{v})\in\mathcal{C}^{\operatorname{an},\max}_{v}}f_{(A_{v},Z_{v})}(x_{v})=1$
for all $x_{v}\in X(F_{v})$.

Let $\mathcal{U}$ be an integral model of $U$ and for each non-archimedean
place $v$ let $f_{v}:U(F_{v})\to\mathbb{R}$ be the indicator function of the
set $\mathcal{U}(\mathcal{O}_{v})$.

For a maximal face
$\mathbf{A}=((A_{v},Z_{v})_{v\in\Omega_{F}}^{\infty})\in\mathcal{C}^{\operatorname{an}}_{\Omega_{F}}$
we define the product

$f_{\mathbf{A}}:=\prod_{v\in\Omega_{F}^{\infty}}f_{(A_{v},Z_{v})}:(X;\mathbf{A})(\mathbb{A}_{F})\to[0,1].$

This is a compactly supported continuous function.

By construction we have the following equality for all $T>0$

$\\#\\{P\in\mathcal{U}(\mathcal{O}_{F}):H(P)\leq
T\\}=\sum_{\mathbf{A}\in\mathcal{C}^{\operatorname{an},\max}_{\Omega_{F}}}\sum_{P\in
U(F),H(P)\leq T}f_{\mathbf{A}}(x).$ (18)

To study integral points of bounded height it will thus suffice to study the
sums $\sum_{P\in U(F),H(P)\leq T}f_{\mathbf{A}}(x).$ We will make a conjecture
on the asymptotic behavior of such sums.

If there is an analytic obstruction near $\mathbf{A}$ then the integral points
counted by weighting by $f_{\mathbf{A}}$ are not Zariski dense by Theorem
3.13. We will thus ignore such faces.

###  6.1. The conjecture

Let $X,D,U$ be as above and let
$\mathbf{A}\in\mathcal{C}^{\operatorname{an}}_{\Omega_{F}^{\infty}}(D)$ be a
face which has no analytic obstruction, i.e.
$\operatorname{H}^{0}((X;\mathbf{A}),\mathcal{O}_{(X;\mathbf{A})})=F$.

Assume that $\operatorname{Pic}X$ is finitely generated. Then
$\operatorname{Pic}(X;\mathbf{A})$ is finitely generated by Lemma 3.10. The
pseudo-effective cone
$\operatorname{\overline{Eff}}_{(X;\mathbf{A})}\subset(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{R}}$
satisfies
$\operatorname{\overline{Eff}}_{(X;\mathbf{A})}\cap-\operatorname{\overline{Eff}}_{(X;\mathbf{A})}=0$
by Proposition 3.16.

Let $K_{(X;\mathbf{A})}\in\operatorname{Pic}(X,\mathbf{A})$ be the image of
the log canonical divisor $K_{(X,D)}:=K_{X}+D$ along the map
$\operatorname{Pic}X\to\operatorname{Pic}(X,\mathbf{A})$. Assume that
$-K_{(X;\mathbf{A})}$ is _big_ , i.e.
$-K_{(X;\mathbf{A})}\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{\circ}$.
This is true for example if $-K_{(X,D)}\in\operatorname{Pic}X$ is big. Let
$L\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{\circ}$ be a big line
bundle. Define the _Fujita invariant_ as

$a:=a((X;\mathbf{A}),L):=\inf\\{t\in\mathbb{R}:K_{(X;\mathbf{A})}+tL\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}\\}.$

Assume moreover that the pseudo-effective cone
$\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ is a polyhedral cone. This is
true if $\operatorname{\overline{Eff}}_{X}$ is polyhedral, which holds if
$(X,D)$ is log Fano by [BCHM10, Cor. 1.3.5]. Then define
$b:=b((X;\mathbf{A}),L)$ as the codimension of the minimal face of the cone
$\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ which contains the _adjoint
divisor_ $K_{(X;\mathbf{A})}+a((X;\mathbf{A}),L)L$.

######  Conjecture 6.1.

Assume that $(X,D)$ is log Fano and
$\overline{F}[U]^{\times}=\overline{F}^{\times}$. Let
$L\in\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}^{\circ}\cap\operatorname{Pic}(X;\mathbf{A})$
be a big and nef line bundle, $(||\cdot||_{v})_{v\in\Omega_{F}}$ an adelic
metric on the $\mathbb{G}_{m}$-torsor corresponding to $L$ and
$H:U(F)\to\mathbb{R}_{>0}^{\times}$ the induced height function.

There exists a thin subset $Z\subset U(F)$ such that the following holds: if
$f:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{\geq 0}$ is a compactly
supported continuous function which is non-zero when restricted to the closure
$\overline{U(F)\setminus Z}\cap
U(\mathbb{A}_{F})_{\mathbf{A}}\subset(X;\mathbf{A})(\mathbb{A}_{F})$ then
there exists a positive constant $c((X;\mathbf{A}),f,H)$ such that as
$T\to\infty$ we have

$\sum_{\begin{subarray}{c}P\in U(F)\setminus Z\\\ H(P)\leq
T\end{subarray}}f(P)\sim c((X;\mathbf{A}),f,H)T^{a((X;\mathbf{A}),L)}(\log
T)^{b((X;\mathbf{A}),L)-1}.$ (19)

######  Remark 6.2.

  1. (1)

The assumption that the pair $(X,D)$ is log Fano can be weakened, this is
already true for the usual Manin conjecture. Since it is unclear what the
precise geometric condition one needs for $X$ in the usual Manin conjecture we
will just make the log Fano assumption.

  2. (2)

There are examples of varieties $U$ for which the integral Manin conjecture
holds and whose Picard group has torsion, e.g. Example 6.3.3. In this case $U$
has a finite étale cover which implies that for every compact set
$C\subset(X;\mathbf{A})(\mathbb{A}_{F})$ the intersection $U(F)\cap C$ is
thin. To get a non-trivial conjecture we thus have to assume that the thin set
$Z$ is independent of $f$.

  3. (3)

The assumption that the restriction of $f$ to the closure
$\overline{U(F)\setminus Z}$ is non-zero is necessary since the left-hand side
of (19) is $0$ otherwise.

  4. (4)

If $L=-a((X;\mathbf{A}),L)K_{(X;\mathbf{A})}$ then
$b((X;\mathbf{A}),L)=\text{rk}\operatorname{Pic}(X;\mathbf{A})=\text{rk}\operatorname{Pic}U+\dim\mathbf{A}+1$
by Corollary 3.12.

  5. (5)

We expect that there should also be a conjecture without assuming that
$\overline{F}[U]^{\times}=\overline{F}^{\times}$. But in this case one should
give a different definition of $\operatorname{Pic}(X;\mathbf{A})$, see [Wil22,
Def. 2.2.1] for a definition in the split case.

###  6.2. Leading constant

In this section we will give a conjectural interpretation of leading constant
in (19) assuming that the algebraic Brauer-Manin obstruction is the only
obstruction. We keep the notations of Conjecture 6.1.

Peyre’s constant is easiest to define in the case that $L$ is _adjoint rigid_.
In the case that $L$ is not adjoint rigid one has to take into account the
_Iitaka fibration_ 666The terminology Iitaka fibration is now standard in
Manin’s conjecture literature, but Batyrev and Tschinkel instead used the word
$\mathcal{L}$-primitive fibration. as in [BT98b]. It is unclear to us what the
analogue of the Iitaka fibration is for integral points so we will assume that
$L$ satisfies the analogue of adjoint rigidity.

######  Definition 6.3.

By Lemma 3.10 we can write $K_{(X;\mathbf{A})}+a((X;\mathbf{A}),L)L$ as a sum

$\sum_{i=1}^{n}c_{i}[E_{i}]+\sum_{v\in\Omega_{F}^{\infty}}\sum_{j_{v}=1}^{m}d_{j_{v}}[D_{j_{v},v}].$

Where $E_{i}\subset X$ are irreducible divisors not contained in $D$,
$D_{j_{v},v}\subset D_{F_{v}}$ are irreducible divisors containing $Z_{v}$ and
$c_{i},d_{j_{v}}\in\mathbb{Q}_{>0}$. We say that $L$ is _adjoint rigid_ if
this decomposition is unique.

Let $U_{L}:=U\setminus(\bigcup_{i=1}^{n}E_{i})$ and
$U_{A_{v},L}:=(X;A_{v},Z_{v})\setminus\Big{(}\bigcup_{i=1}^{n}E_{i}\cup(\bigcup_{j_{v}=1}^{m_{v}}D_{j_{v}})\Big{)}$
for all places $v$. Consider the adelic variety
$(X;\mathbf{A})_{L}:=(U_{L},(U_{A_{v},L})_{v})$. We will need the following
properties.

######  Lemma 6.4.

If $L$ is adjoint rigid then we have

  1. (1)

$\overline{F}[U_{L}]^{\times}=\overline{F}^{\times}$.

  2. (2)

The map
$\operatorname{Pic}U_{\overline{F}}\to\operatorname{Pic}U_{L,\overline{F}}$ is
an isomorphism on torsion subgroups.

  3. (3)

$\operatorname{H}^{0}((X;\mathbf{A})_{L},\mathcal{O}_{(X;\mathbf{A})_{L}})=F$.

###### Proof.

Let $\mathcal{E}$ denote the $\Gamma_{F}$-set of irreducible components of
$(\bigcup_{i=1}^{n}E_{i})_{\overline{F}}$. We have an exact sequence of
$\Gamma_{F}$-modules

$\overline{F}[U_{L}]^{\times}/\overline{F}[U]^{\times}\to\mathbb{Z}[\mathcal{E}]\to\operatorname{Pic}U_{\overline{F}}\to\operatorname{Pic}U_{L,\overline{F}}\to
0.$ (20)

The first two statements will thus follow as long as
$\mathbb{Q}[\mathcal{E}]\to(\operatorname{Pic}U_{\overline{F}})_{\mathbb{Q}}$
is injective. If the map is not injective then there exists two non-trivial
elements $e_{1},e_{2}\in\mathbb{Q}_{\geq 0}[\mathcal{E}]$ which becoming equal
in $\operatorname{Pic}U_{\overline{F}}$. Taking norms we get a non-trivial
relation in $\operatorname{Pic}U$ between the $[E_{i}]$ which contradicts
adjoint rigidity.

For the last statement let
$\mathbf{g}\in\operatorname{H}^{0}((X;\mathbf{A})_{L},\mathcal{O}_{(X;\mathbf{A})_{L}})$
be non-constant. Since
$\operatorname{H}^{0}((X;\mathbf{A}),\mathcal{O}_{(X;\mathbf{A})})=F$ the
function $\mathbf{g}$ must have a pole at one of the $E_{i}$ or $D_{j_{v},v}$.
But this implies that a non-trivial non-negative linear combination of the
$[E_{i}]$ and $[D_{j_{v},v}]$ is equal to $\mathbf{g}^{-1}(0)$. This
contradicts adjoint rigidity. ∎

Assume from now on that $L$ is adjoint rigid.

Let $\mathbf{T}_{\operatorname{NS}L}$ be the Néron-Severi torus of
$(X;\mathbf{A})_{L}$. We have a map
$\mathbf{T}_{\operatorname{NS}L}\to\mathbf{T}_{\operatorname{NS}}$ coming from
the inclusion $(X;\mathbf{A})_{L}\subset(X;\mathbf{A})$.

######  Lemma 6.5.

The induced map
$\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}L})\to\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})$
of topological groups is proper, i.e. inverse images of compact subsets are
compact.

###### Proof.

For the places $v$ of good reduction we will show that
$\operatorname{H}^{1}(\mathcal{O}_{v},T_{\operatorname{NS}L})\to\operatorname{H}^{1}(\mathcal{O}_{v},T_{\operatorname{NS}})$
is an isomorphism. This suffices for the lemma by the definition of the
topology on
$\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}L})$ and
$\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})$.

Both groups have the same size by Lemma 3.31 and Lemma 6.4:(2) The map is also
surjective, as one sees by combining the exactness of the sequence $1\to
T_{\operatorname{NS}L}\to
T_{\operatorname{NS}}\to\mathbb{G}_{m}^{\mathcal{E}}\to 1$ dual to (20) and
Lemma 2.7. ∎

Let $\bm{\pi}:\mathbf{Y}_{L}\to(X;\mathbf{A})_{L}$ be a universal torsor and

$\mathbf{H}:\coprod_{\mathbf{a}\in
H^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}L})}\mathbf{Y}_{L,\mathbf{a}}(\mathbb{A}_{F})\to\mathbf{T}_{\operatorname{NS}L}(\mathbb{A}_{F})/\mathbf{T}_{\operatorname{NS}L}(\mathbb{A}_{F})^{1}\cong(\operatorname{Pic}(X;\mathbf{A})_{L})_{\mathbb{R}}^{*}$

a height function extending $H$, it exists by Lemma 4.12.

Note that $aL=-K_{(X,\mathbf{A})}=-K_{(X,\mathbf{A})_{L}}$ in
$\operatorname{Pic}(X,\mathbf{A})_{L}$ by construction. So the Tamagawa
measure $\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}$ is well-defined.

######  Conjecture 6.6.

Assume that $L$ is adjoint rigid and that the only obstruction to strong
approximation for $(X;\mathbf{A})_{L}$ is the algebraic Brauer-Manin
obstruction.

The leading constant $c((X;\mathbf{A}),f,H)$ in Conjecture 6.1 is then equal
to

$\frac{1}{a(b-1)!}\sum_{\alpha\in\operatorname{H}^{1}(F,T_{\operatorname{NS}L})}\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))H^{-1}(y)f(\bm{\pi}_{\alpha}(y))d\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y).$

######  Remark 6.7.

  1. (1)

This sum is finite by Lemma 6.5.

  2. (2)

The equality between this constant and Peyre’s constant for rational points
and if $L=-K_{X}$ is implicit in Salberger’s work [Sal98, Assertion 5.25] and
follows from work of Peyre [Pey98, Thm. 5.3.1]. Note that the measure in
Peyre’s work is different to Salberger’s measure, we use the latter. See
[Pey98, Rmk. 4.4.1] for a comparison.

Let us note the following compatibility when changing height functions.

######  Lemma 6.8.

Let
$g:(X;\mathbf{A})(\mathbb{A}_{F})\to\operatorname{Hom}(\operatorname{Pic}(X;\mathbf{A})_{L},\mathbb{R}^{\times})$
be a continuous function with $\mathbf{H}:=g\cdot\mathbf{H}^{\prime}$ and
$H^{\prime}:=\mathbf{H}^{\prime L}$. Then

$\begin{split}&\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))H^{-1}(y)f(\bm{\pi}_{\alpha}(y))d\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y)=\\\
&\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}^{\prime}(y))H^{\prime-1}(y)f(\bm{\pi}_{\alpha}(y))g(\bm{\pi}_{\alpha}(y))^{-aL}d\tau_{\mathbf{Y}_{L,\alpha},H^{\prime-a}}(y).\end{split}$

###### Proof.

Now note that $\tau_{\mathbf{Y}_{L,\mathfrak{b}},H^{\prime-a}}$ is
$T_{\operatorname{NS}L}(\mathbb{A}_{F})$-invariant by Lemma 5.14 so we can
make the change of variables $y^{\prime}=g(\bm{\pi}_{\alpha}(y))y$ to see that

$\begin{split}&\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y^{\prime}))H^{-1}(y^{\prime})f(\bm{\pi}_{\alpha}(y^{\prime}))d\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y^{\prime})=\\\
&\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(yg(\bm{\pi}_{\alpha}(y))))H^{-1}(yg(\bm{\pi}_{\alpha}(y)))f(\bm{\pi}_{\alpha}(y))d\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y).\end{split}$

The lemma then follows because
$\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y)=g(\bm{\pi}_{\alpha}(y))^{-aL}\tau_{\mathbf{Y}_{L,\alpha},H^{\prime-a}}(y)$
by Lemma 4.23 and
$\mathbf{H}(yg(\bm{\pi}_{\alpha}(y))=g(\bm{\pi}_{\alpha}(y))^{-1}\mathbf{H}(y)=\mathbf{H}^{\prime}(y)$.
∎

######  Remark 6.9.

  1. (1)

The extension $\mathbf{H}$ only matters due to the presence of
$\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))$.
It follows from the previous lemma that the constant is independent of this
extension.

  2. (2)

If we replace $H$ by $CH$ for $C\in\mathbb{R}_{>0}$ then this corresponds to
replacing $\mathbf{H}$ by $g\mathbf{H}$ with $g$ constant and $g(L)=C$. It
follows that $c((X;\mathbf{A}),f,CH)=C^{-a}c((X;\mathbf{A}),f,H)$, so
Conjecture 6.6 is compatible with multiplication of heights by constants.

####  6.2.1. Brauer-Manin set

We can relate this leading constant to Peyre’s usual constant defined in terms
of the Tamagawa measure of the Brauer-Manin set in good cases. Let use recall
the definition of the effective cone constant

######  Definition 6.10.

Let $L\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{\circ}$ be a big and
adjoint rigid divisor. We define the _effective cone constant_ as
$\theta((X;\mathbf{A}),L):=\mathcal{X}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}}(L)$.

######  Theorem 6.11.

If the sum

$\tau((X;\mathbf{A}),f,H):=\sum_{\beta\in\operatorname{Br}_{1}(X;\mathbf{A})_{L}/\operatorname{Br}F}\int_{(X;\mathbf{A})_{L}(\mathbb{A}_{F})}e^{2\pi
i\langle x,\beta\rangle_{\emph{BM}}}f(x)d\tau_{(X;\mathbf{A})_{L},H^{-a}}(x).$

converges absolutely then
$c((X;\mathbf{A}),f,H)=\frac{\theta((X;\mathbf{A}),L)}{a(b-1)!}\tau((X;\mathbf{A}),f,H)$.

######  Remark 6.12.

In the case that the sum $\tau((X;\mathbf{A}),f,H)$ converges absolutely we
can rewrite it as the following limit over finite subgroups
$\operatorname{Br}_{1}(X;\mathbf{A})_{L}/\operatorname{Br}F$

$\lim_{B\subset\operatorname{Br}_{1}(X;\mathbf{A})_{L}/\operatorname{Br}F}\\#B\int_{(X;\mathbf{A})_{L}(\mathbb{A}_{F})^{B}}f(x)d\tau_{(X;\mathbf{A})_{L},H^{-a}}(x).$

This is true since $\sum_{\beta\in B}e^{2\pi i\langle
x,\beta\rangle_{\emph{BM}}}$ is equal to $\\#B$ if
$x\in(X;\mathbf{A})_{L}(\mathbb{A}_{F})^{B}$ and $0$ otherwise by
orthogonality of characters.

If $\operatorname{Br}_{1}(X;\mathbf{A})_{L}/\operatorname{Br}F$ is finite then
the sum is finite and equal to

$\int_{(X;\mathbf{A})_{L}(\mathbb{A}_{F})^{\operatorname{Br}_{1}(X;\mathbf{A})_{L}}}f(x)d\tau_{(X;\mathbf{A})_{L},H^{-a}}(x)$

which agrees with the prediction of Batyrev and Tschinkel when $U=X$ [BT98b].

###### Proof.

Using Lemma 3.25 and Lemma 5.11 we have

$\tau((X;\mathbf{A}),f,H):=\sum_{\beta\in\operatorname{H}^{1}(\operatorname{Pic}((X;\mathbf{A}),L))}\int_{(X;\mathbf{A})_{L}(\mathbb{A}_{F})}e^{2\pi
i\langle\mathbf{Y}_{L}(x),\beta\rangle_{\emph{BM}}}f(x)d\tau_{(X;\mathbf{A})_{L},H^{-a}}(x).$

By Proposition 5.20, Proposition 5.18 and the definition of
$\theta((X;\mathbf{A}),L)$ we can rewrite
$\theta((X;\mathbf{A}),L)\tau((X;\mathbf{A}),f,H)$ as

$\begin{split}\frac{1}{\tau(\mathbf{T}_{\operatorname{NS}})}&\sum_{\beta\in\operatorname{H}^{1}(\operatorname{Pic}((X;\mathbf{A}),L))}\int_{\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}L})}e^{2\pi
i\langle\mathbf{Y}_{L}(x),\beta\rangle_{\emph{BM}}}\\\
&\int_{\mathbf{Y}_{L,\mathbf{a}}(\mathbb{A}_{F})}f(\pi_{\mathbf{a}}(y))\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))\mathbf{H}(y)^{-L}d\tau_{\mathbf{Y}_{L,\mathbf{a}},H^{-a}}(y)d\mu_{\mathbf{T}_{\operatorname{NS}L}}(\mathbf{a}).\end{split}$

The inner integral is by definition the Fourier transform of the function

$\operatorname{H}^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}L})\to\mathbb{C}:\mathbf{a}\to\int_{\mathbf{Y}_{L,\mathbf{a}}(\mathbb{A}_{F})}f(\pi_{\mathbf{a}}(y))\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))\mathbf{H}(y)^{-L}d\tau_{\mathbf{Y}_{L,\mathbf{a}},H^{-a}}(y).$

This function is compactly supported by Lemma 6.5 so the Poisson summation
formula holds. By Corollary 3.37 we find that
$\theta((X;\mathbf{A}),L)\tau((X;\mathbf{A}),f,H)$ is equal to

$\sum_{\alpha\in\operatorname{H}^{1}(F,\mathbf{T}_{\operatorname{NS}L})}\int_{\mathbf{Y}_{L,\alpha}(\mathbb{A}_{F})}f(\pi_{a}(y))\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})_{L}}^{*}}(\log\mathbf{H}(y))\mathbf{H}(y)^{-L}d\tau_{\mathbf{Y}_{L,\alpha},H^{-a}}(y).$

This is exactly what we had to prove. ∎

###  6.3. Examples

If
$\operatorname{Br}_{1}(X;\mathbf{A})/\operatorname{Br}F\subset\operatorname{Br}_{1}U/\operatorname{Br}F$
is trivial then Theorem 6.11 shows that Conjecture 6.1 agrees with the
prediction in [Wil22, §2.5]. As explained in loc. cit. this shows that our
prediction agrees with the previously known results in [CLT12, TBT13, Cho19,
Wil22, DW22, Wil23].

####  6.3.1. Wilsch’s counterexample

Let us explain what happens in Wilsch’s counterexample [Wil22] to Chambert-
Loir and Tschinkel’s work [CLT10b]. We will use the notation of Wilsch, so we
have a toric variety $X$ and a boundary divisor $D\subset X$ with three
irreducible components $M,E_{1},E_{2}$. The Clemens complex has $4$ non-
trivial elements $M,E_{1},E_{2}$ and $A:=E_{1}\cap E_{2}$.

Wilsch shows in [Wil22, §3.3] that the pair $(X,D)$ has an analytic
obstruction near the face $A$ and thus also near the faces $E_{1}$ and
$E_{2}$. The face $M$ is thus the only non-trivial face with no analytic
obstruction. We have $\operatorname{Pic}(X;M)\cong\mathbb{Z}^{3}$ by [Wil22,
Prop. 3.3.1] so the expected order of growth is $B(\log B)^{2}$ which agrees
with [Wil22, Thm. 1.0.1].

We have that $(X;M)$ is $(\mathbb{P}^{1})^{3}$ minus the two lines
$\\{a_{1}=b_{1}=0\\}$ and $\\{a_{1}=c_{1}=0\\}$ which intersect in a point.
Wilsch only counts integral points for which one always has $a_{1}/a_{0}\geq
1$ so is implicitly weighting by a compactly supported function
$(X;M)(\mathbb{R})\to\mathbb{R}$. He shows in Lemma 3.3.4 that the leading
constant can be interpreted as a Tamagawa measure. By Theorem 6.11 this agrees
with Conjecture 6.6. Actually, the leading constant can be shown to agree with
Conjecture 6.6 directly from [Wil22, Lemma 3.2.1] without computing Tamagawa
measures on $U$. We leave this to the interested reader.

####  6.3.2. Counterexample when there are global invertible sections

Let $X=\mathbb{P}^{2}$ and $D$ a union of two lines. The pair $(X,D)$ is log
Fano but $U\cong\mathbb{G}_{m}\times\mathbb{A}^{1}$. If $k=\mathbb{Q}$ then
the integral points
$\mathbb{G}_{m}\times\mathbb{A}^{1}(\mathbb{Z})=\\{-1,1\\}\times\mathbb{Z}$
are not even Zariski-dense. One checks that this is not due to an analytic
obstruction.

The integral points will be Zariski-dense for number fields $F$ with at least
two archimedean places. In this case one can count integral points of bounded
anticanonical height on $U$. But to do this one has to count integral points
on the torus $\mathbb{G}_{m}$. The number of integral points on
$\mathbb{G}_{m}$ of bounded height has order of magnitude a power of $\log T$
by Dirichlet’s unit theorem. Such counting problems are not very similar to
those appearing in Manin’s conjecture, this is why we make the assumption that
$\overline{F}^{\times}[U]=\overline{F}^{\times}$ in Conjecture 6.1.

####  6.3.3. Infinite Brauer group

Let
$U:=\operatorname{Spec}\mathbb{Q}[x,y,z]/(xy=z^{2})\setminus\\{x=y=z=0\\}$.
This is a smooth toric variety with no invertible global sections,
$\operatorname{Pic}U\cong\mathbb{Z}/2\mathbb{Z}$ and
$\operatorname{Br}U\cong\operatorname{H}^{1}(\mathbb{Q},\mathbb{Z}/2\mathbb{Z})=\mathbb{Q}^{\times}/\mathbb{Q}^{\times
2}$ is infinite. One can count integral points on this variety. Indeed, if
$\mathcal{U}:=\operatorname{Spec}\mathbb{Z}[x,y,z]/(xy=z^{2})\setminus\\{x=y=z=0\\}$
is the obvious integral model then integral points on $\mathcal{U}$ correspond
to triples $(x,y,z)\in\mathbb{Z}^{3}$ with $\gcd(x,y,z)=1$. This implies that
integral points are of the form $(\pm u^{2},\pm v^{2},uv)$ for coprime
$u,v\in\mathbb{Z}$777The map $\mathbb{A}^{2}_{u,v}\to
U:(u,v)\to(u^{2},uv,v^{2})$ is a universal torsor of $U$., well-defined up to
multiplication by $-1$. The log-anticanonical height is $\max(u^{2},v^{2})$.

We see that both $x$ and $y$ have to be $\pm$ a square in $\mathbb{Z}_{p}$ for
all primes $p$. This gives an infinite number of local conditions so
$\overline{\mathcal{U}(\mathbb{Z})}\subset\prod_{v}\mathcal{U}(\mathbb{Z}_{p})$
will not be open. If $\tau$ is the usual Tamagawa measure on
$\prod_{v}\mathcal{U}(\mathbb{Z}_{p})$ then
$\tau(\overline{\mathcal{U}(\mathbb{Z})})=0$.

####  6.3.4. Effect of different faces

In the previous examples there was a only a single relevant face. In the
following example one can see the effect of counting near different faces.

Let $X$ be the projective plane $\mathbb{P}^{2}_{x,y,z}$ blown up at the point
$\\{x=z=0\\}$ and the point $\\{y=z=0\\}$. Let $E_{x},E_{y}\subset X$ be the
respective exceptional divisors and $D$ the proper transform of $\\{z=0\\}$.
Consider the pair $(X,E_{x}\cup E_{y}\cup D)$, it is not log Fano but the
anticanonical divisor is big and an analogue of Manin’s conjecture still
holds. In this case $U=\mathbb{A}^{2}_{x,y}$ so for the standard integral
model $\mathcal{U}=\mathbb{A}^{2}_{\mathbb{Z}}$ integral points correspond to
pairs $x,y\in\mathbb{Z}$.

The log anticanonical height is given by $\max(|xy|,1)$. The Clemens complex
has five non-trivial faces, let us discuss the integral points of bounded
height close to each of them.

  1. (1)

The maximal face $Z=E_{x}\cup D$ has dimension $1$. In this case
$U_{Z}=X\setminus E_{y}$. Weighting by a compactly supported function
$f:U_{Z}\to\mathbb{R}_{\geq 0}$ thus causes one to only count those $x,y$ for
which $x=O(y)$. The order of magnitude in this case is $T\log T$. The other
maximal face $E_{y}\cup D$ is analogous.

  2. (2)

Another face is $D$ which has dimension $0$. In this case
$U_{D}=X\setminus(E_{x}\cup
E_{y})=\mathbb{P}^{2}\setminus(\\{x=z=0\\}\cup\\{y=z=0\\})$. Weighting by a
compactly supported function $f:U_{Z}\to\mathbb{R}_{\geq 0}$ thus cause one to
only count pairs $x,y$ such that $x=O(y)$ and $y=O(x)$. The order of magnitude
is $T$.

  3. (3)

For the face $E_{x}$ we have $U_{E_{x}}=X\setminus(D\cup
E_{y})\cong\mathbb{P}^{1}\times\mathbb{A}^{1}$. Weighting by a compactly
supported function $f:U_{Z}\to\mathbb{R}_{\geq 0}$ corresponds to only
counting those $(x,y)$ with $y=O(1)$. The collection of such points is not
Zariski dense. This is due to an analytic obstruction.

Note that counting all integral points by only counting those close to faces
corresponds to separating the count into two cases, in the first one $x=O(y)$
and in the second one $y=O(x)$. This is known as the Dirichlet hyperbola
method and it is the standard method to count pairs $x,y\in\mathbb{Z}$ such
that $|xy|\leq T$.

####  6.3.5. The circle method

The circle method is a very powerful tool to count integral and rational
points on varieties of large relative dimension to the degree, as worked out
in Birch’s seminal work [Bir62]. It is well-known [Pey95, Prop. 4.2.1] that
the circle method is compatible with Manin’s conjecture for rational points,
including Peyre’s prediction for the leading constant. Similar arguments show
that the circle method is also compatible with Conjectures 6.1 and 6.6. Let us
work this out in detail in the work of Skinner [Ski97] on integral points on
affine complete intersections.

Let $g_{1},\cdots,g_{m}\in\mathbb{Z}[X_{1},\cdots,X_{s}]$ be a collection of
polynomials of degree $d$ with $s>m+3$. Let
$G_{i}\in\mathbb{Z}[X_{0},\cdots,X_{s}]$ the homogenization of $g_{i}$ and let
$h_{i}\in\mathbb{Z}[X_{1},\cdots,X_{s}]$ be the degree $d$ part of $g_{i}$. We
want to understand integral points of bounded height on
$U:=\\{g_{1}=\cdots=g_{l}=0\\}\subset\mathbb{A}_{\mathbb{Q}}^{s}$. We will
consider the compactification $U\subset
X:=\\{G_{1}=\cdots=G_{m}=0\\}\subset\mathbb{P}_{\mathbb{Q}}^{s}$. Let
$D:=X\setminus
U=\\{h_{1}=\cdots=h_{m}=0\\}\subset\\{X_{0}=0\\}=\mathbb{P}_{\mathbb{Q}}^{s-1}$.
We will assume that $X$ and $D$ are smooth complete intersections, $D$ is in
particular geometrically irreducible.

The analytic Clemens complex has a single non-trivial face $D$. Let
$\mathcal{U}$ be the obvious integral model of $U$. Let $f$ to be the
indicator function of $\prod_{p}\mathcal{U}(\mathbb{Z}_{p})\times
X(\mathbb{R})$.

One has $\operatorname{Pic}X=\mathbb{Z}$ generated by $[D]$ by the Lefschetz
hyperplane theorem so $\operatorname{Pic}(X;D)=\mathbb{Z}$. This implies that
$\operatorname{Br}_{1}(X;D)=\operatorname{Br}_{1}U=\operatorname{Br}\mathbb{Q}$.
The canonical divisor is $K_{X}=(md-s-1)[D]$ and the log anti-canonical
divisor is $-K_{(X,D)}=(s-md)[D]$, which is ample. The adelic universal torsor
$\mathbf{Y}$ of $(X;D)$ is given by $U$ at the primes and by
$\\{g_{1}=\cdots=g_{n}=0\\}\subset\mathbb{A}^{s+1}_{\mathbb{Q}}\setminus\\{0\\}$
at $\infty$. Let $\mathcal{B}\subset\mathbb{R}^{s}$ be a compact neighbourhoud
of the origin and define an adelic metric on the universal torsor by setting
$||\cdot||_{p}=1$ for all primes and
$||(x_{0},\cdots,x_{s})||_{\infty}=\inf(t\in\mathbb{R}:(x_{1},\cdots,x_{s})\in
t\mathcal{B})^{-1}$ for $(x_{0},\cdots,x_{s})\in
Y_{\infty}(\mathbb{R})\subset\mathbb{R}^{s+1}$. Let
$H=||(x_{0},\cdots,x_{s})||_{\infty}^{-1}$ be the corresponding height.

Skinner [Ski97, Main Thm.] shows888Skinner actually only works with
$\mathcal{B}$ which are box shaped, the case of arbitrary $\mathcal{B}$
follows from [Lou15, Lemma 4.9]. via the circle method that if $s\geq
m(m+1)(d-1)2^{d-1}$ then there exists a $c>0$ such that

$\sum_{\begin{subarray}{c}P\in U(\mathbb{Q})\\\ H(P)\leq
T\end{subarray}}f(P)=\\#\\{(x_{1},\cdots,x_{s})\in\mathcal{U}(\mathbb{Z})\cap
T\mathcal{B}\\}\sim cT^{s-md}.$ (21)

This agrees with Conjecture 6.1.

The adelic universal torsor of $(X;D)$ is given by $U$ at the non-archimedean
places and by
$\\{G_{1}=\cdots=G_{n}=0\\}\subset\mathbb{A}^{s+1}_{\mathbb{Q}}\setminus\\{0\\}$
at the archimedean places. Let $\mathcal{B}\subset\mathbb{R}^{s}$ be a compact
s Let $du\in\omega_{U}$ be the differential form on $U$ which is the image of
the differential form $g_{1}^{-1}\cdots g_{m}^{-1}dX_{1}\cdots dX_{s}$ on
$\mathbb{A}^{s}_{\mathbb{Q}}$ under the adjunction formula for the immersion
$U\subset\mathbb{A}_{\mathbb{Q}}^{s}$. Similarly let $dy$ be the differential
form on $Y\cap\\{X_{0}=0\\}$ which is the image of $G_{1}^{-1}\cdots
G_{m}^{-1}X_{0}^{-1}dX_{0}dX_{1}\cdots dX_{s}$ under the adjunction formula
for the immersion $Y\cap\\{X_{0}=0\\}\subset\mathbb{A}^{s+1}$. These
differential forms are clearly compatible.

The equality of $\int_{\mathcal{U}(\mathbb{Z}_{p})}|du|_{p}=\sigma_{p}$ where
$\sigma_{p}:=\lim_{k\to\infty}p^{-k\dim
U}\\#\\{\mathbf{x}\in(\mathbb{Z}/p^{k}\mathbb{Z})^{s}:g_{i}(\mathbf{x})=0\\}$
is the local density in the singular series is [BR95, Lemma 1.8.1]. The
singular integral [Ski97, p. 23] is more interesting. The complete
intersection is non-singular so this integral is by [Bir62, Lemma 6.3] equal
to

$\sigma_{\infty}:=\int_{\mathbb{R}^{m}}\int_{\mathcal{B}}e^{2i\pi\sum_{i=1}^{m}t_{i}f_{i}(\mathbf{x})}d\mathbf{x}dt_{1}\cdots
dt_{m}=\int_{Y(\mathbb{R})_{D}=(Y\cap\\{X_{0}=0\\})(\mathbb{R})}\mathbf{1}_{\mathcal{B}}(y)|dy|_{\infty}.$

We see that the leading constant predicted in Conjecture 6.6 is
$\frac{1}{s-md}\prod_{p}\sigma_{p}$ times
$\int_{Y(\mathbb{R})_{D}}\mathbf{1}_{\\{H(y)>1\\}}(y)H(y)^{-s+md-1}|dy|_{\infty}$.
To compare this with the singular integral we note that for
$t\in\mathbb{R}_{>0}$ we have $|dty|_{\infty}=t^{s-md}|dy|_{\infty}$ so
$\text{Vol}(Y(\mathbb{R})_{D}\cap
t\mathcal{B}):=\int_{Y(\mathbb{R})_{D}}\mathbf{1}_{t\mathcal{B}}(y)|dy|_{\infty}=t^{s-md}\sigma_{\infty}$.
We then have

$\displaystyle\int_{Y(\mathbb{R})_{D}}\mathbf{1}_{\\{H(y)>1\\}}(y)H(y)^{-s+md-1}|dy|_{\infty}=\int_{1}^{\infty}t^{-s+md-1}\frac{d}{dt}\text{Vol}(Y(\mathbb{R})_{D}\cap
t\mathcal{B})dt=$
$\displaystyle(s-md)\sigma_{\infty}\int_{1}^{\infty}t^{-2}dt=(s-md)\sigma_{\infty}.$

We conclude that the leading constant $c:=c_{\infty}\prod_{p}c_{p}$ in (21)
agrees with Conjecture 6.6.

######  Remark 6.13.

Skinner in [Ski97] proves a counting result which is valid over all number
fields and not only over $\mathbb{Q}$. But the height he uses is not a height
in our sense so we were unfortunately unable to interpret this result in the
framework of this paper.

###  6.4. Reductions

It is often convenient to first prove Manin’s conjecture for simpler $f$ and
particular heights $H$ and then deduce the general conjecture. Since these
reductions are rather formal we collect them here.

######  Proposition 6.14.

Assume that Conjecture 6.1 holds with the constant conjectured in Conjecture
6.6 for one particular height $H$ and for all $f=\prod_{v}f_{v}$ where for
each $v\in\Omega_{F}$ the function
$f_{v}:(X;A_{v},Z_{v})(F_{v})\to\mathbb{R}_{\geq 0}$ is compactly supported
and is locally constant, resp. smooth, if $v$ is non-archimedean, resp.
archimedean.

Conjecture 6.1 with the constant conjectured in Conjecture 6.6 then holds for
all possible heights on $L$ and all compactly supported functions $f$.

###### Proof.

If Conjecture 6.6 holds for $f$ of the form $\prod_{v}f_{v}$ then it also
holds for linear combinations of such functions. The vector space of such
linear combinations is dense in the vector space of compactly supported
continuous functions.

Let $f:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{\geq 0}$ be any compactly
supported continuous function. We may then uniformly approximate $f$ both from
below and from above by functions for which Conjecture 6.6 holds. Taking
limits of both these approximations gives the desired statement.

That Conjecture 6.6 then holds for all possible heights is essentially due to
Peyre [Pey95, Prop. 3.3(b)]. Let us repeat the argument in our context for
completeness. Any other height $H^{\prime}$ is equal to $gH$ for
$g:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{>0}$ a continuous function.
For every $\varepsilon>0$ we can write $f$ as a sum of continuous compactly
supported functions $f_{i}:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{\geq
0}$ such that there exists a $c_{i}>0$ and such that
$c_{i}-\varepsilon<g(x)<c_{i}+\varepsilon$ for all $x$ contained in the
support of $f_{i}$ by a partition of unity argument.

The following inequality follows from Conjecture 6.1 as $T\to\infty$

$\sum_{\begin{subarray}{c}P\in U(F)\setminus Z\\\ (c_{i}-\varepsilon)T\leq
H(P)\leq(c_{i}+\varepsilon)T\end{subarray}}f_{i}(P)\ll\varepsilon(c_{i}T)^{a-1}(\log
c_{i}T)^{b}.$

Letting $\varepsilon\to 0$ we are reduced to the case when $g$ is constant.
This case follows from Lemma 6.8, cf. Remark 6.9. ∎

##  7\. Counting

###  7.1. Set-up

Let $F$ be a number field, $X$ a smooth proper toric variety over $F$ with
corresponding torus $T$ and $\Gamma_{F}$-invariant fan $\Sigma$ on the lattice
$N:=\operatorname{Hom}(\mathbb{G}_{m},T)$. Let $Z\subset X$ be a toric divisor
corresponding to a set of rays $Z_{\min}\subset\Sigma_{\min}$. Let
$U:=X\setminus Z$. We will assume that
$\overline{F}[U]^{\times}=\overline{F}^{\times}$. The complement
$D:=\Sigma_{\min}\setminus Z_{\min}$ is the $\Gamma_{F}$-set of irreducible
toric divisors of $U$.

Let
$\mathbf{A}=(A_{v},Z_{v})_{v\in\Omega_{F}^{\infty}}\in\mathcal{C}^{\text{an}}_{\Omega^{\infty}}(D)$
be a face of the analytic Clemens complex. Note that $A_{v}\subset D_{\min}$
is the set of divisors containing $Z_{v}$ for all $v\in\Omega_{F}^{\infty}$.
We will write $U_{A_{v}}:=(X;A_{v},Z_{v})$. We assume that
$\operatorname{H}^{0}((X;\mathbf{A}),\mathcal{O}_{(X;\mathbf{A})})$. This
implies by Proposition 3.16 that the pseudo-effective cone
$\operatorname{\overline{Eff}}_{(X;\mathbf{A})}\subset(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{R}}$
does not contain a line.

Assume furthermore that
$-K_{(X;\mathbf{A})}\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$. Note
that by (3) we have

$-K_{(X;\mathbf{A})}=\sum_{D_{\rho}\in D}[D_{\rho}].$ (22)

####  7.1.1. Universal torsor

Let
$\mathbf{T}_{\operatorname{NS}}=(T_{\operatorname{NS}},(T_{\operatorname{NS}A_{v}})_{v})$
be the Néron-Severi torus of $(X;\mathbf{A})$.

Let $\pi:Y\to U$, resp. $\pi_{v}:Y_{A_{v}}\to U_{A_{v}}$ for each place $v$,
be the universal constructed in Construction 2.9.

There is a natural toric inclusions $Y_{F_{v}}\subset Y_{A_{v}}$ so
$\mathbf{Y}:=(Y,(Y_{A_{v}})_{v})$ is an adelic variety and
$\bm{\pi}:\mathbf{Y}\to(X,\mathbf{A})$ is a universal torsor.

Let $\Theta=\mathbb{G}_{m}^{D}$ and $\Theta_{A_{v}}=\mathbb{G}_{m}^{D\cup
A_{v}}$. Let $\Theta_{\mathbf{A}}:=(\Theta,(\Theta_{A_{v}})_{v})$ be an adelic
torus. It is by construction equal to $\bm{\pi}^{-1}(T)\subset\mathbf{Y}$.

Fix a smooth adelic metric $(||\cdot||_{v})_{v\in\Omega_{F}}$ on the torsor
$\bm{\pi}$. By enlarging $S$ we may assume that
$(||\cdot||_{v})_{v\in\Omega_{F}}$ has good reduction at all places $v\not\in
S$. Note that $\mathbf{T}_{\operatorname{NS}}$ acts via the inverse of the
inclusion $\mathbf{T}_{\operatorname{NS}}\to\Theta_{\mathbf{A}}$. So if
$t\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$ and
$x\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})$ then $||t\cdot
x||=t^{-1}||x||\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{\mathcal{O}})$.

Note that
$\hat{\Theta}_{\mathbf{A}}=\mathbb{Z}[D/\Gamma_{F}]\times\prod_{v\in\Omega_{F}^{\infty}}\mathbb{Z}[A_{v}/\Gamma_{v}]$.
Given $c,d\in\mathbb{C}$ we denote by
$(\mathbf{c},\mathbf{d})\in\hat{\Theta}_{\mathbf{A},\mathbb{R}}$ the element
$\sum_{\rho\in
D/\Gamma_{F}}c\cdot[\rho]+\sum_{v\in\Omega_{F}^{\infty},\rho_{v}\in
A_{v}/\Gamma_{v}}d\cdot[\rho_{v}]$ and $\mathbf{c}=(\mathbf{c},\mathbf{c})$.
We will use the same notation for the images of these elements in
$\hat{\mathbf{T}}_{\operatorname{NS}}$.

We will also use the analogous notation for elements
$\hat{\Theta}_{A_{v}}=\mathbb{Z}[(D\cup A_{v})/\Gamma_{v}]$ for all places
$v$.

Let $\mathbf{H}:\coprod_{\mathbf{a}\in
H^{1}(\mathbb{A}_{F},\mathbf{T}_{\operatorname{NS}})}\mathbf{Y}(\mathbb{A}_{F})\to\operatorname{Hom}(\operatorname{Pic}(X,\mathbf{A}),\mathbb{R}_{>0}^{\times})$
be the corresponding height function. If
$t\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$ and
$x\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})$ then $\mathbf{H}(t\cdot
x)=|t|_{\mathbf{T}_{\operatorname{NS}}}\mathbf{H}(x)$.

The toric divisors generate the effective cone so by Corollary 4.20 we may
assume without loss of generality that that
$\log\mathbf{H}(x)\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{*}$ for
all $x\in T(F)$ by multiplying the metric and thus the height by a constant.

####  7.1.2. The function

Let $S$ be a finite set of places containing all the archimedean places, all
ramified places of $K/\mathbb{Q}$ and all the places of bad reduction of
$T,\Theta$ and $T_{\operatorname{NS}}$. Let $\mathcal{Y}\to\mathcal{U}$ an
$\mathcal{O}_{K,S}$-integral model of the $T_{\operatorname{NS}}$ torsor $Y\to
U$ such that $\mathcal{U}$, resp. $\mathcal{Y}$, is equipped with an integral
model of the action of $T$, resp. $\Theta$. This exists by spreading out,
after potentially enlarging $S$.

For each place $v$ of $F$ let $f_{v}:U_{A_{v}}(F_{v})\to\mathbb{R}_{\geq 0}$
be a compactly supported function. Assume that $f_{v}$ is smooth if $v$ is
archimedean and locally constant if $v$ is non-archimedean. We will assume
that if $v\not\in S$ then $f_{v}$ is the indicator function of
$\mathcal{U}(\mathcal{O}_{v})$. The product
$f:=\prod_{v}f_{v}:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{\geq 0}$ is
then a well-defined compactly supported function.

####  7.1.3. Smoothing the height

To ensure that certain integrals converge absolutely it will be convenient to
smooth the height function.

######  Construction 7.1.

For $v\in\Omega_{F}^{\text{fin}}$ let
$\varphi_{v}:T_{\operatorname{NS}}(F_{v})\to\mathbb{R}$ be the indicator
function of $T_{\operatorname{NS}}(\mathcal{O}_{v})$ and for each place
$v\in\Omega_{F}^{\infty}$ fix a non-zero smooth compactly supported function
$\varphi_{v}:T_{\operatorname{NS}A_{v}}(F_{v})\to\mathbb{R}$. The product
function
$\varphi:=\prod_{v}\varphi_{v}:\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})\to\mathbb{R}$
is then compactly supported.

By shifting the $\varphi_{v}$ we may assume that if
$y\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$ is such that
$\varphi(y)\neq 0$ then
$\log|y|_{\mathbf{T}_{\operatorname{NS}}}\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{*}$.

We will consider the following smoothed form of the height function
$\mathbf{H}$. For $\bm{s}\in(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{C}}$
let
$\mathcal{H}_{\varphi}(\cdot;\bm{s}):\Theta_{\mathbf{A}}(\mathbb{A}_{F})\to\mathbb{C}$
be defined by the integral

$\mathcal{H}_{\varphi}(x;\bm{s}):=\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})\varphi(||x||y)|y|_{\mathbf{T}_{\operatorname{NS}}}^{-\bm{s}}d\tau_{\operatorname{NS}}(y).$
(23)

This integral converges absolutely because $\varphi$ is compactly supported.
Note also that the integral is independent of the chosen representative of
$||x||\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$ because
$\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{-\bm{s}}$
is invariant under multiplication by $\mathbf{T}(\mathbb{A}_{F})^{1}$.

Let
$(\mathcal{M}\varphi)(\bm{s}):=\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})}\varphi(z)|z|_{\mathbf{T}_{\operatorname{NS}}}^{\bm{s}}d\tau_{\operatorname{NS}}(z)$
be the Mellin transform of $\varphi$ where
$\bm{s}\in(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{C}}$.

######  Remark 7.2.

For any fixed choice of
$\bm{s}\in(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{C}}$ we can always
choose a $\varphi$ such that $(\mathcal{M}\varphi)(\bm{s})\neq 0$. Since
everything that follows will follow for any choice of $\varphi$ satisfying the
above assumptions we may always assume that $(\mathcal{M}\varphi)(\bm{s})\neq
0$.

######  Lemma 7.3.

  1. (1)

The function $\mathcal{H}_{\varphi}(x;\bm{s})$ is continuous in the variable
$x$ and holomorphic in $\bm{s}$.

  2. (2)

If $\log\mathbf{H}(x)\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ then
$\mathcal{H}_{\varphi}(x;\bm{s})=(\mathcal{M}\varphi)(-\bm{s})\mathbf{H}(x)^{-\bm{s}}$.

  3. (3)

There exists some $\ell\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ such
that $\mathcal{H}_{\varphi}(x;\bm{s})=0$ if
$\mathbf{H}(x)-\ell\not\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$.

###### Proof.

For the first part note that the integrand of (23) is continuous in $x$ and
holomorphic in $s$ and that the integral converges absolutely.

For the second, note that if $\varphi(||x||y)\neq 0$ then by taking log norms
we have that
$\log|y|_{\mathbf{T}_{\operatorname{NS}}}-\log\mathbf{H}(x)\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{*}$.
This implies that
$\log|y|_{\mathbf{T}_{\operatorname{NS}}}\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{*}$.
Making the change of variables $z=||x||y$ we get

$\mathcal{H}_{\varphi}(x;\bm{s})=\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})}\varphi(||x||y)|y|_{\mathbf{T}_{\operatorname{NS}}}^{-\bm{s}}d\tau_{\operatorname{NS}}(y)=\mathbf{H}(x)^{-\bm{s}}\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})}\varphi(z)|z|_{\mathbf{T}_{\operatorname{NS}}}^{-\bm{s}}d\tau_{\operatorname{NS}}(z).$

The last part follows from the compact support of $\varphi$. ∎

####  7.1.4. Height zeta function

The height zeta function is defined as

$Z(\bm{s},f):=\sum_{P\in T(F)}f(P)\mathbf{H}(P)^{-\bm{s}}$

for
$\bm{s}\in(\operatorname{Pic}(X;\mathbf{A}))_{\mathbb{C}}:=\operatorname{Pic}(X;\mathbf{A})\otimes\mathbb{C}$
for which this sum convergences absolutely. All of the following equalities
will follow if at least one of the sides converges absolutely. We will show at
the end that this holds if
$\mathrm{Re}(\bm{s})-\mathbf{2}\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$.

Twisting the torsor $\pi:Y\to U$ by
$\mathfrak{a}\in\operatorname{H}^{1}(a,T_{\operatorname{NS}})$ results in a
$T_{\operatorname{NS}}$ torsor $\pi_{\mathfrak{a}}:Y\to X$ by Lemma 2.11. It
follows from 2.2 that the Zeta function has the following decomposition

$Z(\bm{s},f)=\sum_{a\in
H^{1}(F,\mathbf{T}_{\operatorname{NS}})}Z_{\mathfrak{a}}(\bm{s},f)$ (24)

where

$Z_{\mathfrak{a}}(\bm{s},f):=\sum_{P\in\Theta(F)/T_{\operatorname{NS}}(F)}f(\pi_{\mathfrak{a}}(P))\mathbf{H}(P)^{-\bm{s}}.$

We note that 2.11 applied to the $\mathcal{O}_{v}$-integral model
$\mathcal{Y}_{\mathcal{O}_{v}}\to\mathcal{U}_{\mathcal{O}_{v}}$ implies that
$Z_{\mathfrak{a}}(\bm{s},f)=0$ for $\mathfrak{a}\not\in\prod_{v\not\in
S}\operatorname{H}^{1}(\mathcal{O}_{v},T_{\operatorname{NS}})$. There are thus
only finitely many $\mathbf{a}$ such that $Z_{\mathfrak{a}}(\bm{s},f)\neq 0$
so the sum (24) is finite.

By Lemma 7.3 and the product formula we get an equality

$(\mathcal{M}\varphi)(-\bm{s})Z_{\mathfrak{a}}(\bm{s},f):=\sum_{P\in\Theta(F)/T_{\operatorname{NS}}(F)}\mathcal{H}_{\varphi}(P;\bm{s})f(\pi_{\mathfrak{a}}(P))|P|^{\mathbf{2}}_{\Theta_{\mathbf{A}}}.$

This is thus a sum over the adelic function

$\mathcal{G}(\cdot;\bm{s}):\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)\to\mathbb{C}:x\to\mathcal{H}_{\varphi}(x;\bm{s})f(\pi_{\mathfrak{a}}(x))|x|^{\mathbf{2}}_{\Theta_{\mathbf{A}}}.$

Note that this is well-defined because $\mathcal{H}(\cdot;{\bm{s}})$ is
invariant under $\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}$.

We will evaluate this sum by applying Poisson summation (4) to the locally
compact Hausdorff group
$\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)$. Let us first
check that the condition in Theorem 2.17 holds.

######  Lemma 7.4.

There exists a pair $\psi$ and $\psi^{\prime}$ of functions as in Construction
4.25 with the following property.

For each place $v\in S$ there exists a compactly supported continuous function
$g_{v}:U_{A_{v}}(F_{v})\to\mathbb{R}_{\geq 0}$ which is locally constant if
$v$ is non-archimedean and which is smooth if $v$ is archimedean and a
neighbourhood $V\subset\Theta_{\mathbf{A}}(\mathbb{A}_{F})$ of $1$ such that
if $g_{v}:=f_{v}$ for $v\not\in S$ and
$g:=\prod_{v}g_{v}:(X;\mathbf{A})(\mathbb{A}_{F})\to\mathbb{R}_{\geq 0}$ then

$|\mathcal{H}_{\varphi}(x;{\bm{s}})f(\pi_{\mathfrak{a}}(x))|x|^{\mathbf{2}}_{\Theta_{\mathbf{A}}}|\to\mathcal{H}_{\varphi^{\prime}}(yx;{\mathrm{Re}(\bm{s})})g(\pi_{\mathfrak{a}}(yx))|yx|^{\mathbf{2}}_{\Theta_{\mathbf{A}}}.$

for all $x\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})$, $y\in V$ and
$\mathrm{Re}(\bm{s})\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$.

###### Proof.

Let $\psi$ and $\psi^{\prime}$ such that $\psi(x)\leq\psi^{\prime}(wx)$ for
all $x\in\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$ and $w\in W$ where
$W$ is an open neighbourhood of the
$\prod_{v}T_{\operatorname{NS}A_{v}}(\mathcal{O}_{v})$. The existence of such
functions reduces to the archimedean case where it is clear.

For each $v\in S$ choose an open neighbourhood of the identity
$V_{v}\subset\Theta_{A_{v}}(F_{v})$ with compact closure. Moreover, take
$V_{v}\subset\Theta(\mathcal{O}_{v})$ for $v$ a non-archimedean place. By
compactness we can choose $g_{v}:U_{A_{v}}(F_{v})\to\mathbb{R}_{\geq 0}$
satisfying the assumptions such that for all $y_{v}\in V_{v}$ and $x\in
U_{A_{v}}(F_{v})$ we have $|f_{v}(x_{v})|\leq g_{v}(x_{v}\pi_{a_{v}}(y_{v}))$.

For $v\not\in S$ we let $V_{v}=\Theta(\mathcal{O}_{v})$. Then
$f_{v}\circ\pi_{a}$ is $V_{v}$ invariant by assumption so
$|f_{v}(\pi_{a}(x_{v}))|\leq g_{v}(x_{v}y_{v})$ also for these places $v$.

The product
$V^{\prime}:=\prod_{v}V_{v}\subset\Theta_{\mathbf{A}}(\mathbb{A}_{F})$ has
compact closure. It follows that there exist a $L>0$ such that
$L^{-1}\leq|y|^{\mathbf{2}}_{\Theta_{\mathbf{A}}}$ for all $y\in V$.

Consider the map $\Delta:\mathbf{Y}(\mathbb{A}_{F})\times
V\to\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/\prod_{v}T_{\operatorname{NS}A_{v}}(\mathcal{O}_{v}):(x,v)\to\Delta(x,y)=||x||_{v}||yx||_{v}^{-1}$.
The map $\Delta$ factors through $(X;\mathbf{A})(\mathbb{A}_{F})\times
V^{\prime}$ since it is invariant under
$T_{\operatorname{NS}}(\mathbb{A}_{F})$.

The map $\Delta$ is continuous and sends $(x,1)$ to $1$ so by compact support
of $g$ there exists an neighbourhood of the identity $V\subset V^{\prime}$
such that $\Delta(x,y)\in W$ for all $y\in V$ for which
$g(\pi_{\mathfrak{a}}(x))\neq 0$. By construction we have
$\mathcal{H}_{\varphi}(x;{\bm{s}})\leq
L\cdot\mathcal{H}_{\varphi^{\prime}}(yx;{\mathrm{Re}(\bm{s})})$.

The desired bound in the lemma then follows by taking by replacing $g_{v}$ by
$Lg_{v}$ for one $v$. ∎

###  7.2. The Fourier transform of $\mathcal{G}$

In this subsection we will study the Fourier transform of the function
$\mathcal{G}(\cdot;\bm{s})$. Let
$\chi:\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(\mathbb{A}_{F})^{1}\to\mathbb{C}^{\times}$
be a quasi-character and consider the Fourier transform

$\hat{\mathcal{G}}(\chi;\bm{s}):=\int_{\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}\chi\cdot\mathcal{G}(\cdot;\bm{s})d\tau_{\Theta_{\mathbf{A}}}.$

As a first step we substitute the definition of
$\mathcal{H}_{\varphi}(x;\bm{s})$ to get that this is equal

$\begin{split}&\int_{\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}\chi(x)f(\pi_{a}(x))|x|_{\Theta_{\mathbf{A}}}^{\mathbf{2}}\\\
&\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})\varphi(||x||y)|y|_{\mathbf{T}_{\operatorname{NS}}}^{-\bm{s}}d\tau_{\operatorname{NS}}(y)d\tau_{\Theta_{\mathbf{A}}}(x).\end{split}$

This integral will only converge absolutely for $\bm{s}$ sufficiently large.
In this section we will always assume that $\bm{s}$ is sufficiently large so
that at least one of the sides of each equality converges absolutely. At the
end we will show that such an $\bm{s}$ does exist.

Make the change of variables $z=y^{-1}x$. Using that
$\tau_{\Theta_{\mathbf{A}}}$ is a Haar measure, Fubini, that $||z||=y||x||$
and the fact that $f$ is
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})$-invariant shows that
$\hat{\mathcal{G}}(\chi;\bm{s})$ is equal to

$\begin{split}&\int_{\Theta_{\mathbf{A}}(\mathbb{A}_{F})}\chi(z)f(\pi_{a}(z))\varphi(||z||)|z|_{\Theta_{\mathbf{A}}}^{\mathbf{2}}d\tau_{\Theta_{\mathbf{A}}}(z)\\\
&\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}\chi(y)\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{\mathbf{2}-\bm{s}}d\tau_{\operatorname{NS}}(y).\end{split}$
(25)

We will evaluate these two integrals separately. The inner integral can be
evaluated explicitly

######  Lemma 7.5.

The inner integral of (25) converges absolutely if
$\mathrm{Re}(\bm{s})-\bm{2}-\mathrm{Re}(\chi)\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{\circ}$.

If the restriction of $\chi$ to
$\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F)$ is non-
trivial then

$\int_{{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}}\mathbf{\chi}(y)\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{\mathbf{2}-\bm{s}}d\tau_{\operatorname{NS}}(y)=0.$

Otherwise $\chi=|\cdot|_{\mathbf{T}_{\operatorname{NS}}}^{\bm{z}}$ for some
$\bm{z}\in\operatorname{Pic}(X,\mathbf{A})_{\mathbb{R}}$ and

$\begin{split}\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}\mathbf{\chi}(y)\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{\mathbf{2}-\bm{s}}d\tau_{\operatorname{NS}}(y)=\\\
\tau_{\mathbf{T}_{\operatorname{NS}}^{1}}(\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F))\mathcal{X}_{\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}}(\bm{s}-\bm{2}-\bm{z}).\end{split}$

###### Proof.

For absolute convergence we may replace $\chi$ by $|\chi|$. This quasi-
character is trivial when restricted to
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F)$
since this group is compact by Lemma 3.28. We can thus reduce to the second
case.

We rewrite the relevant integral as

$\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}}{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{\mathbf{2}-\bm{s}}\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F)}\chi(yx)d\tau_{\mathbf{T}_{\operatorname{NS}}^{1}}(x)d\tau_{\mathbf{T}_{\operatorname{NS}}/\mathbf{T}^{1}_{\operatorname{NS}}}(y).$

The inner integral is $0$ if $\chi$ is non-trivial when restricted to the
compact group
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F)$
by orthogonality of characters. Otherwise it is equal to
$d\tau_{\mathbf{T}_{\operatorname{NS}}^{1}}(\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}/T_{\operatorname{NS}}(F))$.

The log norm $\log|\cdot|_{\mathbf{T}_{\operatorname{NS}}}$ induces an
isomorphism
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)^{1}\cong\operatorname{Pic}(\mathbf{X},\mathbf{A})^{*}_{\mathbb{R}}$
by Lemma 3.27, and this isomorphism sends the Haar measure
$d\tau_{\mathbf{T}_{\operatorname{NS}}/\mathbf{T}^{1}_{\operatorname{NS}}}$ to
the measure $du_{\operatorname{Pic}(X;\mathbf{A})}$ by construction.

This is a real vector space so every quasi-character
$\operatorname{Pic}(\mathbf{X},\mathbf{A})^{*}_{\mathbb{R}}\to\mathbb{C}^{\times}$
is given by $e^{\langle\bm{z},\cdot\rangle}$ for some
$\bm{z}\in(\operatorname{Pic}(\mathbf{X},\mathbf{A}))_{\mathbb{C}}$. Every
quasi-character of
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)^{1}$
is thus equal to
$e^{\langle\bm{z},\log|\cdot|_{\mathbf{T}_{\operatorname{NS}}}\rangle}=|\cdot|_{\mathbf{T}_{\operatorname{NS}}}^{\bm{z}}$.

We then compute that

$\begin{split}&\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}}{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{\mathbf{2}-\bm{s}+\bm{z}}d\tau_{\mathbf{T}_{\operatorname{NS}}/\mathbf{T}^{1}_{\operatorname{NS}}}=\\\
&\int_{\operatorname{Pic}(\mathbf{X},\mathbf{A})^{*}_{\mathbb{R}}}{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(u)e^{-\langle\bm{s}-\mathbf{2}-\mathbf{z},u\rangle}du_{\operatorname{Pic}(X;\mathbf{A})}.\end{split}$

This is exactly the definition of
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}}(\bm{s}-\bm{2}-\bm{z})$.
∎

The integrand of the other integral is a product of the following local
factors. For each quasi-characters
$\chi_{v}:\Theta_{A_{v}}(F_{v})\to\mathbb{C}^{\times}$ define the local
integral

$\hat{G}_{v}(\chi_{v}):=\int_{\Theta_{A_{v}}(F_{v})}\chi_{v}(x_{v})f_{v}(\pi_{a_{v}}(x_{v}))\varphi_{v}(||x_{v}||_{v})|x_{v}|^{\mathbf{2}}_{\Theta_{A_{v}}}d\tau_{\Theta_{A_{v}}}(x_{v}).$

Every quasi-character $\chi$ can be written as a product
$\chi=\prod_{v}\chi_{v}$. By the definition of $\tau_{\Theta_{\mathbf{A}}}$ we
then have

$\int_{\Theta_{\mathbf{A}}(\mathbb{A}_{F})}\chi(z)f(\pi_{a}(z))\varphi(||z||)|z|_{\Theta_{\mathbf{A}}}^{\mathbf{2}}d\tau_{\Theta_{\mathbf{A}}}(z)=\prod_{\rho\in
D/\Gamma_{F}}\zeta^{*}_{F_{\rho}}(1)^{-1}\prod_{v\in\Omega_{F}}\hat{G}_{v}(\chi_{v}).$
(26)

###  7.3. Local Fourier transforms

Let us first note the following fact

######  Lemma 7.6.

The product function $Y_{A_{v}}(F_{v})\to\mathbb{R}:x_{v}\to
f_{v}(\pi_{a}(x_{v}))\varphi(||x_{v}||_{v})$ is compactly supported for all
places $v\in\Omega_{F}$ and is smooth for $v\in\Omega_{F}^{\infty}$.

###### Proof.

The function
$(\pi_{a_{v}},||\cdot||):Y_{A_{v}}(F_{v})/T_{\operatorname{NS},v}(\mathcal{O}_{v})\to
X(F_{v})\times
T_{\operatorname{NS},v}(F_{v})/T_{\operatorname{NS},v}(\mathcal{O}_{v})$ is an
open and closed immersion by Lemma 4.5.

The function $X(F_{v})\times
T_{\operatorname{NS},v}(F_{v})/T_{\operatorname{NS},v}(\mathcal{O}_{v})\to\mathbb{R}:(x_{v},t_{v})\to
f_{v}(x_{v})\varphi_{v}(x)$ is a product of compactly supported functions
which are smooth if $v\in\Omega_{F}^{\infty}$ and thus has those same
properties. The lemma follows. ∎

Let $v\in\Omega_{F}$ be a place. The torus $\Theta_{A_{v}}$ is by definition
the Weil restriction of the étale algebra corresponding to the
$\Gamma_{v}$-set $D\cup A_{v}$. For $\rho_{v}\in(D\cup A_{v})/\Gamma_{v}$ let
$F_{\rho_{v}}$ be the field extension of $F_{v}$ corresponding to the
$\Gamma_{v}$-orbit $\rho_{v}$. We then have

$\Theta_{A_{v}}(F_{v})\cong\prod_{\rho_{v}\in(R\cup
A_{v})/\Gamma_{v}}F_{\rho_{v}}^{\times}.$

Under this identification the we have an equality of measures

$\tau_{\Theta_{A_{v}}}(x)=\prod_{\rho_{v}\in(D\cup
A_{v})/\Gamma_{v}}\zeta_{F_{\rho_{v}}}(1)|x_{\rho_{v}}|^{-1}dx_{\rho_{v}}.$
(27)

We can also make the following identification

$\operatorname{Hom}(\Theta_{A_{v}}(F_{v}),\mathbb{C}^{\times})\cong\prod_{\rho_{v}\in(D\cup
A_{v})/\Gamma_{v}}\operatorname{Hom}(F^{\times}_{\rho_{v}},\mathbb{C}^{\times}).$

This space naturally has the structure of a complex manifold999With a
countable number of components..

Every quasi-character $\chi_{v}:\Theta_{A_{v}}(F_{v})\to\mathbb{C}^{\times}$
can thus be uniquely written as a product $\chi_{v}=\prod_{\rho\in(D\cup
A_{v})/\Gamma_{v}}\chi_{\rho_{v}}$ where
$\chi_{\rho_{v}}:F_{\rho_{v}}^{\times}\to\mathbb{C}^{\times}$ is a quasi-
character.

We can also conclude that every quasi-character
$\chi_{v}:\Theta_{A_{v}}(F_{v})\to\mathbb{C}^{\times}$ can be written as a
product $\chi_{v}=\tilde{\chi}_{v}|\cdot|_{v}^{\mathbf{z}}$ for some
$\mathbf{z}\in\mathbb{C}[(D\cup A_{v})/\Gamma_{v}]$ and character
$\tilde{\chi}_{v}:\Theta_{A_{v}}(F_{v})\to S^{1}$. Note that
$\mathrm{Re}(\mathbf{z})\in\mathbb{R}[(D\cup A_{v})/\Gamma_{v}]$ is uniquely
determined by the quasi-character $\chi$. We can choose $\tilde{\chi}=1$ if
and only if $\chi$ is trivial when restricted to
$\Theta_{A_{v}}(\mathcal{O}_{v})$, in this case we say that $\chi$ is
_unramified_.

For $\mathbf{x}\in\mathbb{R}[(D\cup A_{v})/\Gamma_{v}]$ we define the _tube
domain_

$\mathbf{T}_{>\mathbf{x}}:=\\{\chi\in\operatorname{Hom}(\Theta_{A_{v}}(F_{v}),\mathbb{C}^{\times}):\mathrm{Re}(\chi)>\mathbf{x}\\}.$

The partial order on $\mathbb{R}[(D\cup A_{v})/\Gamma_{v}]$ is the components-
wise one.

####  7.3.1. Finite places

For non-archimedean places we will use the following lemma

######  Lemma 7.7.

Let $v\in\Omega_{F}^{\emph{fin}}$. The Fourier transform
$\hat{G}_{v}(\chi_{v})$ converges absolutely and is holomorphic for
$\chi_{v}\in\mathbf{T}_{>-\mathbf{2}}$. There exists a compact open subgroup
$\mathbf{K}_{v}\subset\Theta(F_{v})$ such that $\hat{G}_{v}(\chi_{v})=0$ if
$\chi_{v}$ is non-trivial when restricted to $\mathbf{K}_{v}$.

###### Proof.

Let $\mathbf{K}_{v}\subset\Theta(\mathcal{O}_{v})$ be the maximal subgroup
such that $f_{v}$ and $||\cdot||_{v}$ are invariant under $\mathbf{K}_{v}$.
This is an open subgroup because $f_{v}$ and $||\cdot||_{v}$ are locally
constant.

By the usual argument via orthogonality of characters we conclude that
$\hat{G}_{v}(\chi_{v})=0$ if $\chi_{v}$ is trivial when restricted to
$\mathbf{K}_{v}$.

Now write the quasi-character $\chi_{v}=\tilde{\chi}_{v}|\cdot|^{\mathbf{z}}$
for some $\mathbf{z}\in\mathbb{R}[E/\Gamma_{v}]$. The holomorphicity of
$\hat{G}_{v}(\chi_{v})$ will follow from absolute convergence so we may assume
that $\tilde{\chi}_{v}=1$. By (27) we have

$\int_{\Theta_{A_{v}}(F_{v})}f_{v}(\pi_{a_{v}}(x_{v}))\varphi_{v}(||x_{v}||_{v})|x_{v}|^{\mathbf{1}+\mathbf{z}}_{\Theta_{A_{v}}}\prod_{\rho_{v}\in
D/\Gamma_{v}}\zeta_{F_{\rho_{v}}}(1)dx_{\rho_{v}}.$

This integral converges absolutely as long as $\mathbf{z}+\mathbf{2}>0$ by
[CLT10a, Prop. 3.4] and Lemma 7.6. ∎

We can do much better than the above lemma at the places of good reduction. We
note the similarity of the following lemma to [CLT10b, Lemma 3.2.2].

######  Lemma 7.8.

Let $v\not\in S$. If $\chi_{v}$ is ramified then $\hat{G}_{v}(\chi_{v})=0$, if
$\chi_{v}$ is unramified and contained in the tube domain
$\mathbf{T}_{>\mathbf{2}}$ then

$\hat{G}_{v}(\chi_{v})=\prod_{\rho_{v}\in\Gamma_{v}\setminus
E}L_{F_{\rho_{v}}}(2;\chi_{\rho_{v}})\left(1+O\left(q_{v}^{-4-2\min_{\rho_{v}\in
D/\Gamma_{v}}\mathrm{Re}(\chi_{\rho_{v}})}\right)\right).$

###### Proof.

The assumption $v\not\in S$ implies that $x_{v}\to
f_{v}(\pi_{a}(x_{v}))\varphi_{v}(x_{v})$ is the indicator function of
$\mathcal{Y}(\mathcal{O}_{v})$, it is in particular invariant under the action
of $\Theta_{A_{v}}(\mathcal{O}_{v})$ so $\hat{G}_{v}(\chi_{v})=0$ if
$\chi_{v}$ if $\chi_{v}$ is ramified by orthogonality of characters.

If $\chi_{v}$ is unramified then it is equal to
$|\cdot|_{\Theta_{v}}^{\mathbf{z}}$ for some
$\mathbf{z}\in\mathbb{C}[\Gamma_{v}\setminus E]$. By (27) we have to compute

$\hat{G}_{v}(\chi_{v})=\prod_{\rho_{v}\in
D/\Gamma_{v}}\zeta_{F_{\rho_{v}}}(1)\int_{\Theta(F_{v})\cap\mathcal{Y}(\mathcal{O}_{v})}|x|^{\mathbf{1}+\mathbf{z}}\prod_{\rho_{v}\in
D/\Gamma_{v}}dx_{\rho_{v}}.$

For each $\Gamma_{v}$-invariant subset $B\subset D$ let
$\mathcal{Z}_{B}:=\cap_{\beta\in B}\mathcal{Z}_{\beta}$ be the intersection of
the corresponding varieties, it is defined over $F$. The closed subscheme
$\mathcal{Z}_{B}$ is an intersection of toric divisors so the open subscheme
$\mathcal{Z}_{B}^{\circ}:=\mathcal{Z}_{B}\setminus\cup_{\varepsilon\in
D\setminus B}\mathcal{Z}_{\varepsilon}$ is the torus
$\mathbb{G}_{m}^{D\setminus B}$ if $B$ is the set of rays of a cone in
$\Sigma$ and empty otherwise.

For $\rho_{v}\in D/\Gamma_{v}$ let $f_{\rho_{v}}$ be the degree of the
(necessarily unramified) extension $F_{\rho_{v}}/F$.

The étale extension $F_{v}[D\setminus B]/F_{v}$ is unramified so
$\\#\mathbb{G}_{m}^{D\setminus
B}(\mathbb{F}_{v})=\prod_{\rho_{v}\in(D\setminus
B)/\Gamma_{v}}\mathbb{F}_{\rho_{v}}^{\times}=\prod_{\rho_{v}\in
D/\Gamma_{v}}(q^{f_{\rho_{v}}}-1)=\prod_{\rho_{v}\in
D/\Gamma_{v}}q^{f_{\rho_{v}}}\zeta_{F_{\rho_{v}}}(1)$.

Substituting this into Denef’s formula [CLT10b, Prop. 4.5] and simplifying we
find the equality

$\hat{G}_{v}(\chi_{v})=\sum_{\sigma\in\Sigma^{\Gamma_{v}}}\prod_{\rho_{v}\in\sigma/\Gamma_{v}}(q_{v}^{f_{\rho_{v}}(2+z_{\rho_{v}})}-1)^{-1}.$

The contribution of the trivial cone is $1$ and the contribution of a ray
$\rho_{v}\in D^{\Gamma_{v}}$ is
$q^{-2-z_{\rho_{v}}}+O(q^{-4-2\mathrm{Re}(z_{\rho_{v}})})$. The contribution
of higher dimensional cones $\sigma\in\Sigma^{\Gamma_{v}}$ is
$O(q^{-4-2\min_{\rho_{v}\in\sigma/\Gamma_{v}}(\mathrm{Re}(z_{\rho}))})$. The
lemma follows since
$L_{\rho_{v}}(2,\chi_{v})=(1-q^{f_{\rho_{v}}(-2-z_{\rho_{v}})})^{-1}$. ∎

####  7.3.2. Infinite places

When $v$ is an archimedean then every quasi-character $\chi_{v}$ of
$F_{\rho_{v}}$ can be written as a product $y\to y^{n}|y|^{z}$ for unique
$z\in\mathbb{C}$ and $n\in\\{0,1\\}$ or $n\in\mathbb{Z}$ for
$F_{\rho_{v}}\cong\mathbb{R}$ or $F_{\rho_{v}}\cong\mathbb{C}$ respectively.
We define $||\chi_{v}||:=\max(|z|,n)$.

######  Lemma 7.9.

Let $v\in\Omega_{F}^{\infty}$. The Fourier transform $\hat{G}_{v}(\chi_{v})$
converges absolutely for $\mathbf{T}_{>\mathbf{-2}}^{D\cup A_{v}}$ and there
exists a holomorphic function
$\varpi_{A_{v}}:\mathbf{T}_{>\mathbf{-\frac{5}{2}}}^{D\cup
A_{v}}\to\mathbb{C}$ such that

$\hat{G_{v}}(\chi_{v})=\varpi_{A_{v}}(\chi_{v})\prod_{\begin{subarray}{c}\rho_{v}\in
D/\Gamma_{v}\cup A_{v}\\\
\chi_{\rho_{v}}=|\cdot|^{z_{\rho_{v}}}\end{subarray}}\frac{1}{2+z_{\rho_{v}}}.$

The function $\varpi_{A_{v}}$ is rapidly decreasing in vertical strips in the
sense that for every compact $C\subset\mathbb{R}_{>-\frac{5}{2}}[(D\cup
A_{v})/\Gamma_{v}]$ and for all $\kappa>0$ we have for $\chi_{v}$ with
$\mathrm{Re}(\chi_{v})\in C$ that

$\varpi_{A_{v}}(\chi_{v})\ll_{\kappa,C}(1+||\chi_{v}||)^{-\kappa}.$

###### Proof.

Write the quasi-character as $\chi_{v}=\tilde{\chi}_{v}|\cdot|^{\mathbf{z}}$
for $\mathbf{z}\in\mathbb{C}[D/\Gamma_{v}\cup A_{v}]$ and $\tilde{\chi}_{v}$ a
character. We may moreover assume that if $\chi_{\rho_{v}}$ is unramified then
$\tilde{\chi}_{\rho_{v}}=1$. We then have by (27) that

$\hat{G}_{v}(\chi_{v})=\int_{\Theta_{A_{v}}(F_{v})}f_{v}(\pi_{a}(x_{v}))\varphi_{v}(||x_{v}||_{v})\tilde{\chi}_{v}(x_{v})|x_{v}|^{\mathbf{1}+\mathbf{z}}_{\Theta_{A_{v}}}\prod_{\rho_{v}\in(D\cup
A_{v})/\Gamma_{v}}dx_{\rho_{v}}$

This is the Mellin transform of a smooth compactly supported function by Lemma
7.6. The lemma thus follows from properties of Mellin transforms, c.f.
[CLT10a, Lemma 3.4] and [CLT10b, Lemma 3.4.1]. ∎

####  7.3.3. The global Fourier transform

For each place $v\in\Omega_{F}^{\text{fin}}$ let
$\mathbf{K}_{v}\subset\Theta(F_{v})$ be a compact open group as in Lemma 7.7.
By Lemma 7.8 we may assume that $\mathbf{K}_{v}=\Theta(\mathcal{O}_{v})$ for
$v\not\in S$.

For $v\in\Omega_{F}^{\infty}$ put $\mathbf{K}_{v}=1$. Define
$\mathbf{K}:=\prod_{v\in\Omega_{F}}\mathbf{K}_{v}\subset\Theta_{\mathbf{A}}(\mathbb{A}_{F})$.

Any quasi-character
$\chi:\Theta_{\mathbf{A}}(\mathbb{A}_{F})\to\mathbb{C}^{\times}$ is equal to a
product $\prod_{v}\chi_{v}$ of quasi-characters
$\Theta_{A_{v}}(F_{v})\to\mathbb{C}^{\times}$, all but finitely many of which
are trivial when restricted to $\Theta_{A_{v}}(\mathcal{O}_{v})$. We define
$||\chi||=\max_{v\in\Omega_{F}^{\infty}}||\chi_{v}||$.

Also, every quasi-character
$\chi:\Theta_{\mathbf{A}}(\mathbb{A}_{F})\to\mathbb{C}^{\times}$ can be
written as a product $\prod_{\rho\in
D/\Gamma}\chi_{\rho}\prod_{v\in\Omega_{F}^{\infty}}\prod_{\rho_{v}\in
A_{v}/\Gamma_{v}}\chi_{\rho_{v}}$ where
$\chi_{\rho}:\mathbb{G}_{m}(\mathbb{A}_{F_{\rho}})\to\mathbb{C}^{\times}$ and
$\chi_{\rho_{v}}:\mathbb{G}_{m}(F_{\rho_{v}})\to\mathbb{C}^{\times}$ by the
definition of $\Theta_{\mathbf{A}}$.

We can combine the above lemmas to immediately deduce the following.

######  Lemma 7.10.

Let $\chi$ be a quasi-character
$\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(\mathbb{A}_{F})^{1}\to\mathbb{C}^{\times}$.
If
$\mathrm{Re}(\chi)\in\mathbf{T}_{>(\mathbf{-\frac{3}{2}},\mathbf{-\frac{5}{2}})}$
then the integral $\hat{\mathcal{G}}(\chi;\mathbf{s})$ converges absolutely.

If the restriction of $\chi$ to $\mathbf{K}$ or
$\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})^{1}$ is non-trivial then
$\hat{\mathcal{G}}(\chi;\mathbf{s})=0$.

In general there exist a holomorphic function $\eta_{\mathbf{A}}$ on the tube
domain $\mathbf{T}_{>-\frac{3}{2},>-\frac{5}{2}}$ such that

$\hat{\mathcal{G}}(\chi;\mathbf{s})=\prod_{\rho\in
D/\Gamma}L_{F_{\rho}}(2;\chi_{\rho})\eta_{\mathbf{A}}(\chi)\prod_{\begin{subarray}{c}v\in\Omega^{\infty}\\\
\rho_{v}\in A_{v}/\Gamma_{v}\\\
\chi_{\rho_{v}}=|\cdot|^{z_{\rho_{v}}}\end{subarray}}\frac{1}{2+z_{\rho_{v}}}\mathcal{X}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\mathbf{s}-\mathbf{2}-\bm{z}).$

The function $\eta_{\mathbf{A}}$ is rapidly decreasing in vertical strips in
the sense that for any compact
$C\subset\mathbb{R}_{>-\frac{3}{2}}[D/\Gamma]\times\prod_{v\in\Omega_{F}^{\infty}}\mathbb{R}_{>-\frac{5}{2}}[A_{v}/\Gamma_{v}]$
and for all $\kappa>0$ we have for $\chi$ with $\mathrm{Re}(\chi)\in C$ that

$\eta_{\mathbf{A}}(\chi)\ll_{\kappa,C}(1+||\chi||)^{-\kappa}.$ (28)

The function $\hat{\mathcal{G}}(\chi;\mathbf{s})$ thus has a meromorphic
continuation to the tube domain
$\mathbf{T}_{>-(\mathbf{\frac{3}{2}},\mathbf{\frac{5}{2}})}$.

###  7.4. Poisson summation

We will now apply the Poisson summation formula. Let us first study the
Pontryagin dual $(\Theta_{\mathbf{A}}(\mathbb{A}_{F})/\Theta(F))^{\vee}$.

By Lemma 3.27 we have an exact sequence

$1\to\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta(F)\to\Theta_{\mathbf{A}}(\mathbb{A}_{F})/\Theta(F)\xrightarrow{|\cdot|_{\Theta_{\mathbf{A}}}}=\mathbb{R}_{>0}^{\times}[D/\Gamma_{F}]\prod_{v\in\Omega_{F}^{\infty}}\mathbb{R}_{>0}^{\times}[A_{v}/\Gamma_{v}]\to
1$ (29)

The group $\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta(F)$ is compact so
its Pontryagin dual is discrete. The measure dual to
$\tau_{\Theta_{\mathbf{A}}^{1}}$ is
$\tau_{\Theta_{\mathbf{A}}^{1}}(\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta(F))^{-1}$
times the counting measure.

The volume of this group has been computed in Proposition 4.30, in the
notation of the proposition we have $T^{\prime}=\Theta$, $E=\emptyset$ and
$A_{v}=A_{v}$. We have $\tau(\Theta_{\mathbf{A}})=1$ by Lemma 2.7 and the
definition of $\tau(\Theta_{\mathbf{A}})$ so
$\tau_{\Theta_{\mathbf{A}}^{1}}(\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta(F))=c_{\mathbf{A}}$.

The group
$\mathbb{R}_{>0}^{\times}[D/\Gamma_{F}]\times\prod_{v\in\Omega_{F}^{\infty}}\mathbb{R}_{>0}^{\times}[A_{v}/\Gamma_{v}]$
is a power of $\mathbb{R}^{\times}_{>0}$. The Pontryagin dual of
$\mathbb{R}^{\times}_{>0}$ is $\mathbb{R}$, where $x\in\mathbb{R}$ corresponds
to the character $|\cdot|^{ix}$. The corresponding dual Haar measure on
$\mathbb{R}$ is $\frac{1}{2\pi}$ times the Lebesgue measure.

A formal application of the Poisson summation (4) formula thus leads to the
equality

$(\mathcal{M}\varphi)(-\bm{s})Z_{\mathfrak{a}}(\bm{s};f,\varphi)=c_{\mathbf{A}}^{-1}\sum_{\chi\in(\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta_{\mathbf{A}}(F))^{\vee}}\mathfrak{Z}(\chi;\bm{s}).$
(30)

where

$\mathfrak{Z}(\chi;\bm{s}):=\mathop{\int\cdots\int}_{\begin{subarray}{c}\rho\in
D/\Gamma_{F}\\\
\mathrm{Re}(z_{\rho})=0\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in A_{v}/\Gamma_{v}\\\
\mathrm{Re}(w_{\rho_{v}})=0\end{subarray}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{B}}}^{(\bm{z},\bm{w})};\bm{s})d\frac{\bm{z}}{2\pi
i}d\frac{\bm{w}}{2\pi i}$ (31)

and

$\displaystyle d\frac{\bm{z}}{2\pi i}:=\prod_{\rho\in
D/\Gamma_{F}}d\frac{z_{\rho}}{2\pi i}$ $\displaystyle d\frac{\bm{w}}{2\pi
i}:=\prod_{v\in\Omega_{F}^{\infty}}\prod_{\rho_{v}\in
A_{v}/\Gamma_{v}}d\frac{w_{\rho_{v}}}{2\pi i}.$

The subgroup of
$(\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta_{\mathbf{A}}(F))^{\vee}$
consisting of characters which are trivial when restricted to $\mathbf{K}$ is
a finitely generated group.

The integrand decays sufficiently fast in vertical strips to ensure absolute
convergence by (28) and since L-functions grow at worst polynomially in
vertical strips, see e.g. [Lou18, Lemma 3.1]. The sum over $\chi$ also
converges absolutely for the same reason, noting that the conductors of the
$\chi_{\rho}$ are bounded since $\chi$ is trivial when restricted to
$\mathbf{K}$. Theorem 2.17 then holds by noting that this absolute convergence
also holds if one replaces $\phi$ and $f$ by the $\phi^{\prime}$ and $g$ in
Lemma 7.4.

####  7.4.1. Moving contours

Let $L\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}^{\circ}$ be a big line
bundle, $a:=a((X,\mathbf{A});L)$ and
$C\subset\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ the minimal face
containing the adjoint divisor $K_{(X;\mathbf{A})}+aL$. Let
$b:=b((X,\mathbf{A});L)$ be the codimension of $C$.

Let $D_{\operatorname{adj}}\subset E$ be the $\Gamma_{F}$-set of toric
divisors of $U$ whose divisor class lies in the face $C$. Similarly, for each
place $v$ let $A_{v,\operatorname{adj}}\subset A_{v}$ be the
$\Gamma_{F_{v}}$-set of toric divisors contained in the face $C$. The divisors
$D_{\operatorname{adj}}$ and $A_{v,\operatorname{adj}}$ generate $C$ since the
toric divisors generate the effective cone.

Let $E:=D\setminus D_{\operatorname{adj}}$ and $B_{v}:=A_{v}\setminus
A_{v,\operatorname{adj}}$ for all $v$.

The following lemma shows that $\mathfrak{Z}(\chi;\bm{s})$ can be analytically
continued.

######  Lemma 7.11.

There exists an $\varepsilon>0$ independent of $\chi$ such that the function
$s\to(\mathcal{M}\varphi)(-sL)Z_{\mathfrak{a}}(sL,f)$ has a meromorphic
continuation to $\mathrm{Re}(s)>a-\varepsilon$. It is holomorphic in this
region except for possibly a single pole of order at most $b$ at $s=a$.

Let $\xi(f,s,\bm{z},\bm{w})$ be the following sum of integrals for
$\bm{z}\in\mathbb{C}[E/\Gamma_{F}]$,
$\bm{w}\in\prod_{v\in\Omega_{F}^{\infty}}\mathbb{C}[B_{v}/\Gamma_{v}]$

$c_{\mathbf{A}}^{-1}\sum_{\begin{subarray}{c}\chi\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta_{\mathbf{A}}(F))^{\vee}\\\
\chi_{\rho}=1:\rho\in E/\Gamma_{F}\\\ \chi_{\rho_{v}}=1:\rho_{v}\in
B/\Gamma_{v}\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}\rho\in
D_{\operatorname{adj}}/\Gamma_{F}\\\
\mathrm{Re}(z_{\rho})=0\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in A_{v,\operatorname{adj}}/\Gamma_{v}\\\
\mathrm{Re}(w_{\rho_{v}})=0\end{subarray}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})+(\bm{z}_{\operatorname{adj}},\bm{w}_{\operatorname{adj}})};sL)d\frac{\bm{z}_{\operatorname{adj}}}{2\pi
i}d\frac{\bm{w}_{\operatorname{adj}}}{2\pi i}.$ (32)

Where $\bm{z}_{\operatorname{adj}}=(z_{\rho})_{\rho\in
D_{\operatorname{adj}}/\Gamma_{F}}\in\mathbb{C}[D_{\operatorname{adj}}/\Gamma_{F}]$
and $\bm{w}_{\operatorname{adj}}=(w_{\rho_{v}})_{\rho_{v}\in
A_{v,\operatorname{adj}}}\in\prod_{v\in\Omega_{F}^{\infty}}\mathbb{C}[A_{v,\operatorname{adj}}/\Gamma_{v}]$.

It converges absolutely for $\mathrm{Re}(s)$ sufficiently large and
$\mathrm{Re}(z_{\rho})>-1,\mathrm{Re}(w_{\rho_{v}})>-2$ for $\rho\in
E/\Gamma_{F}$, $\rho_{v}\in B_{v}$ respectively. The function

$(s-a)^{b}\prod_{\rho\in
E/\Gamma_{F}}(z_{\rho}+1)\prod_{\begin{subarray}{c}v\in\Omega_{F}\\\
\rho_{v}\in
B_{v}/\Gamma_{v}\end{subarray}}(w_{\rho_{v}}+2)\xi(f,s,\bm{z},\bm{w})$

has a holomorphic continuation to the region
$\mathrm{Re}(s)>a-\varepsilon,\mathrm{Re}(z_{\rho})>-1-\varepsilon,\mathrm{Re}(w_{\rho_{v}})>-2-\varepsilon$
and its value at $s=a,z_{\rho}=-1,w_{\rho_{v}}=-2$ is equal to $\lim_{s\to
a}(s-a)^{b}(\mathcal{M}\varphi)(-sL)Z_{\mathfrak{a}}(sL,f)$.

###### Proof.

Choose $0<\delta<\frac{1}{4}$ such that

$K_{(X;\mathbf{A})}+aL-\sum_{\rho\in
D_{\operatorname{adj}}}2\delta[Z_{\rho}]-\sum_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in
A_{v,\operatorname{adj}}\end{subarray}}2\delta[Z_{\rho_{v}}]\in\operatorname{\overline{Eff}}_{(X;\mathbf{A})}.$
(33)

Such a $\delta$ exists by the definitions of $D_{\operatorname{adj}}$ and
$A_{v,\operatorname{adj}}$.

For each
$\chi\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta_{\mathbf{A}}(F))^{\vee}$
move in the definition of $\mathfrak{Z}(\chi;sL)$ the contours coming from
$\rho\in E/\Gamma_{F}$ and $\rho_{v}\in B_{v}/\Gamma_{v}$ to the lines
$\mathrm{Re}(z_{\rho})=-\frac{5}{4}$ and
$\mathrm{Re}(w_{\rho_{v}})=-\frac{9}{4}$ respectively. Moreover, also move the
contours corresponding to $\rho\in D_{\operatorname{adj}}/\Gamma_{F}$ and
$\rho_{v}\in A_{v,\operatorname{adj}}/\Gamma_{v}$ to the lines
$\mathrm{Re}(z_{\rho})=-1+\delta$ and $\mathrm{Re}(w_{\rho_{v}})=-2+\delta$
respectively.

By Lemma 7.10 each contour moved this way only meets at most a single pole at
$z_{\rho}=1$ and $w_{\rho_{v}}=0$ respectively. And it only meets such a pole
if $\chi_{\rho_{v}}=1$ and $\chi_{\rho_{v}}=1$ respectively.

By the residue theorem $\mathfrak{Z}(\chi;sL)$ is thus equal to the sum over
all possible subsets $I\subset E/\Gamma_{F}$ and $J_{v}\subset
B_{v}/\Gamma_{v}$ for all $v\in\Omega_{F}^{\infty}$ such that $\chi_{\rho}=1$
for all $\rho\in I$ and $\chi_{\rho_{v}}=1$ for all $\rho_{v}\in J_{v}$ of the
following terms

$\begin{split}&\prod_{\rho\in
I}\zeta^{*}_{{}_{F_{\rho}}}(1)\mathop{\int\cdots\int}_{\begin{subarray}{c}\mathrm{Re}(z_{\rho})=-\frac{5}{4}:\rho\in
I^{c}\\\ \mathrm{Re}(z_{\rho})=-1+\delta:\rho\in
D_{\operatorname{adj}}/\Gamma_{F}\end{subarray}}\prod_{\rho\in
I^{c}\cup(D_{\operatorname{adj}}/\Gamma_{F})}L(2+z_{\rho};\chi_{\rho})\mathop{\int\cdots\int}_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\mathrm{Re}(w_{\rho_{v}})=-\frac{9}{4}:\rho_{v}\in J_{v}^{c}\\\
\mathrm{Re}(w_{\rho_{v}})=-2+\delta:\rho\in
A_{v,\operatorname{adj}}/\Gamma_{v}\end{subarray}}\\\
&\prod_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\ \rho_{v}\in
J_{v}\cup(B_{v}/\Gamma_{v})\\\
\chi_{\rho_{v}}=1\end{subarray}}\frac{1}{2+w_{\rho_{v}}}\mathcal{X}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(sL-\mathbf{2}-(\bm{z},\bm{w}))\eta_{\mathbf{A}}(|\cdot|^{(\bm{z},\bm{w})}\cdot\chi)d\frac{\bm{z}}{2\pi
i}d\frac{\bm{w}}{2\pi i}.\end{split}$ (34)

Here $z_{\rho}=1$ and $w_{\rho_{v}}=1$ for $\rho\in I$ and $\rho_{v}\in J_{v}$
respectively.

The integrand decays sufficiently fast in vertical strips for the same reason
as for $\mathfrak{Z}(\chi;\bm{s})$. Similarly, the sum over characters $\chi$
of (34) converges absolutely.

The integrand is a meromorphic function in $s$ and the only part of the
integrand which depends on $s$ is the factor
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(sL-\mathbf{2}-(\bm{z},\bm{w}))$
so let us analyse this.

By the identity (22) $K_{(X;\mathbf{A})}=(-\mathbf{1},\mathbf{0})$ we have

$\begin{split}sL-\mathbf{2}-(\bm{z},\bm{w})=&(s-a)L+K_{(X;\mathbf{A})}+aL-\sum_{\rho\in
D_{\operatorname{adj}}}2\delta[Z_{\rho}]-\sum_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in A_{v,\operatorname{adj}}\end{subarray}}2\delta[Z_{\rho_{v}}]\;+\\\
&\sum_{\rho\in I^{c}}(-z_{\rho}-1)[Z_{\rho}]+\sum_{\rho\in
D_{\operatorname{adj}}/\Gamma_{F}}(-z_{\rho}-1+2\delta)[Z_{\rho}]\;+\\\
&\sum_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\ \rho_{v}\in
J_{v}^{c}\end{subarray}}(-w_{\rho_{v}}-2)[Z_{\rho_{v}}]+\sum_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in
A_{v,\operatorname{adj}}/\Gamma_{v}\end{subarray}}(-w_{\rho_{v}}-2+2\delta)[Z_{\rho_{v}}].\end{split}$

Let $C_{I,(J_{v})_{v}}\subset\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$
be the face generated by all $[Z_{\rho}]$ for $\rho\not\in I$ and
$[Z_{\rho_{v}}]$ for $\rho_{v}\not\in J_{v}$. We clearly have $C\subset
C_{I,(J_{v})_{v}}$. Moreover we have $C=C_{I,(J_{v})_{v}}$ if and only if
$I=\emptyset,J_{v}=\emptyset$ since $[Z_{\rho}],[Z_{\rho_{v}}]\not\in C$ if
$\rho\in E/\Gamma_{F},\rho_{v}\in B_{v}/\Gamma_{v}$ by the definition of $E$
and $B_{v}$ respectively.

Let $b_{I,(J_{v})_{v}}$ be the codimension of $C_{I,(J_{v})_{v}}$. By the
above we have $b\geq b_{I,(J_{v})_{v}}$ with equality if and only
$I=\emptyset,J_{v}=\emptyset$

Note that in the integral (34) we have
$\mathrm{Re}(-z_{\rho}-1),\mathrm{Re}(z_{\rho}-1+2\delta),\mathrm{Re}(-w_{\rho_{v}}-2),\mathrm{Re}(-w_{\rho_{v}}-2+2\delta)>0$
for $\rho\in I^{c},\rho\in E/\Gamma_{\operatorname{adj}},\rho_{v}\in
J_{v}^{c},\rho_{v}\in A_{v,\operatorname{adj}}/\Gamma_{v}$ respectively.

In the notation of Lemma 2.16 the linear function
$sL-\mathbf{2}-(\bm{z},\bm{w})$ is thus equal to a sum of $(s-a)L$ and an
element in $C_{I,(J_{v})_{v},\mathbb{C}}^{\circ}$. The lemma then implies that
there exists a $\varepsilon>0$ such
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(sL-\mathbf{2}-(\bm{z},\bm{w}))$
has a meromorphic continuation to the region $\mathrm{Re}(s)>a-\varepsilon$
which is holomorphic except for a single pole at $s=a$ of order at most
$b_{I,(J_{v})_{v}}$. It follows that the same is true for the integral (34).

The integral $\xi(f,s,\bm{z},\bm{w})$ converges and has a meromorphic
continuation for the same reason.

By moving the contours corresponding to $z_{\rho}$ for $\rho\in
D_{\operatorname{adj}}/\Gamma_{F}$ and $w_{\rho_{v}}$ for $\rho_{v}\in
A_{v,\operatorname{adj}}/\Gamma_{v}$ back we see that the contribution of (34)
for the case $I=J_{v}=\emptyset$ is equal to the value at $z_{\rho}=-1$ for
$\rho\in E/\Gamma_{v}$ and $w_{\rho_{v}}=-2$ for $\rho_{v}\in
B_{v}/\Gamma_{v}$ of the meromorphic continuation of

$\prod_{\rho\in
E/\Gamma_{F}}(z_{\rho}+1)\prod_{\begin{subarray}{c}v\in\Omega_{F}\\\
\rho_{v}\in
B_{v}/\Gamma_{v}\end{subarray}}(w_{\rho_{v}}+2)\mathop{\int\cdots\int}_{\begin{subarray}{c}\rho\in
D_{\operatorname{adj}}/\Gamma_{F}\\\
\mathrm{Re}(z_{\rho})=0\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in A_{v,\operatorname{adj}}/\Gamma_{v}\\\
\mathrm{Re}(w_{\rho_{v}})=0\end{subarray}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})+(\bm{z}_{\operatorname{adj}},\bm{w}_{\operatorname{adj}})};sL)d\frac{\bm{z}_{\operatorname{adj}}}{2\pi
i}d\frac{\bm{w}_{\operatorname{adj}}}{2\pi i}.$

We can switch the limit with the sum over characters by absolute convergence.
∎

######  Corollary 7.12.

The sum $Z_{\mathfrak{a}}(sL,f)$ is absolutely convergent for $s>a$ and has a
meromorphic continuation to the half place $\mathrm{Re}(s)>a-\varepsilon$ with
at most a single pole at $s=a$ of order $b$.

###### Proof.

By Remark 7.2 we may assume that $(\mathcal{M}\varphi)(-sL)\neq 0$ for any
particular $s$ by changing $\varphi$. This then follows from Lemma 7.11. ∎

##  8\. The leading constant

In this section we will finish our analysis by determining the value of
$\lim_{s\to a}(s-a)^{b}Z_{\mathfrak{a}}(sL,f)$. This is closely related to the
leading constant. We keep the notation of the last section.

###  8.1. Geometry

Consider the adelic tori
$\Theta_{\mathbf{A}_{\operatorname{adj}}}:=(\mathbb{G}_{m}^{D_{\operatorname{adj}}},(\mathbb{G}_{m}^{D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}})_{v})$ and
$\Theta_{\mathbf{B}}:=(\mathbb{G}_{m}^{E}),(\mathbb{G}_{m}^{E\cup
B_{v}})_{v})$. There are clearly split exact sequence of locally compact
groups

$\displaystyle
1\to\Theta_{\mathbf{B}}(\mathbb{A}_{F})\to\Theta_{\mathbf{A}}(\mathbb{A}_{F})\to\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})\to
1$ (35) $\displaystyle
1\to\mathbb{G}_{m}^{E}(F)\to\Theta(F)\to\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F)\to
1$ (36)

Let
$\mathbf{T}_{\operatorname{NS}L}:=\ker(\mathbf{T}_{\operatorname{NS}}\to\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F}))=\mathbf{T}_{\operatorname{NS}}\cap\Theta_{\mathbf{B}}$.

Let $N_{L}\subset N$, resp. $N_{B_{v}}$ be the full sublattice of $N$
generated by the rays $\rho$, resp. the rays $\rho_{v}$, for $\rho\in E$,
resp. $\rho_{v}\in E\cup B_{v}$. Let
$M_{L}:=\operatorname{Hom}(N_{L},\mathbb{Z})$, resp.
$M_{B_{v}}:=\operatorname{Hom}(N_{A_{v}})$, be the dual lattice. Let
$M_{\operatorname{adj}}:=\ker(M\to M_{L})$, resp.
$M_{v,\operatorname{adj}}:=\ker(M\to M_{B_{v}})$.

The following lemma can be compared with the diagram [BT96, p. 3225]

######  Lemma 8.1.

Let
$\operatorname{Pic}U_{L,B_{v}}:=\operatorname{coker}(\mathbb{Z}[D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}]\to\operatorname{Pic}U_{A_{v},\overline{F_{v}}})$.
The following commutative diagram is well-defined. It has exact rows, exact
columns, and the top vertical maps are injective.

${0}$${0}$${0}$${0}$${{M_{v,\operatorname{adj}}}}$${{\mathbb{Z}[D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}]}}$${{\emph{Im}(\mathbb{Z}[D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}]\to\operatorname{Pic}U_{A_{v},\overline{F_{v}}})}}$${0}$${0}$${M}$${{\mathbb{Z}[D\cup
A_{v}]}}$${{\operatorname{Pic}U_{A_{v},\overline{F_{v}}}}}$${0}$${0}$${{M_{B_{v}}}}$${{\mathbb{Z}[E\cup
B_{v}]}}$${{\operatorname{Pic}U_{L,B_{v},\overline{F_{v}}}}}$${0}$${0}$${0}$${0}$

Moreover, $M_{L}=M_{B_{v}},M_{\operatorname{adj}}=M_{v,\operatorname{adj}}$
and $N_{L}=N_{B_{v}}$ for all places $v$.

###### Proof.

Exactness of the columns is by definition.

Exactness of the second row is a special case of (2) since
$\overline{F}[U_{A_{v}}]^{\times}\subset\overline{F}[U]^{\times}=\overline{F}^{\times}$.

Note that $M_{B_{v}}$ is the image of $M$ along the projection
$\mathbb{Z}[D\cup A-v]\to\mathbb{Z}[E\cup B_{v}]$ by definition. It follows
that $M_{v,\operatorname{adj}}=\mathbb{Z}[D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}]\cap M$. This implies that the first row is exact.
The third is then exact by the snake lemma.

If $M_{\operatorname{adj}}\neq M_{v,\operatorname{adj}}$ then by the above
there exists an element $m\in M$ whose image in $\mathbb{Z}[D\cup A_{v}]$ lies
in $\mathbb{Z}[D_{\operatorname{adj}}\cup A_{v}]$ but not in
$\mathbb{Z}[D_{\operatorname{adj}}\cup A_{v,\operatorname{adj}}]$. This
implies that the rational function defined by $m$ has a pole or zero at a
divisor of $B_{v}$ and all other poles and divisors on $U_{A_{v}}$ are
contained in $D_{\operatorname{adj}}\cup A_{v,\operatorname{adj}}$.

Taking Galois norms and divisor classes we find that the image in
$\operatorname{Pic}(X,\mathbf{A})$ of a non-trivial linear combination of
elements in $B_{v}/\Gamma_{v}$ is contained in $C_{\mathbb{R}}$. Since $C$ is
a face of $\operatorname{\overline{Eff}}_{(X;\mathbf{A})}$ the pullback along
$\mathbb{Z}[A_{v}/\Gamma_{v}]\to\operatorname{Pic}(X;\mathbf{A})$ is a face of
the standard cone $\mathbb{R}_{\geq 0}[A_{v}/\Gamma_{v}]$. But this implies
that the divisor class of an element of $B_{v}$ lies in $C$, which contradicts
the definition of $C$.

We conclude that $M_{\operatorname{adj}}=M_{v,\operatorname{adj}}$. The rest
of the lemma follows. ∎

Let $T_{L}\subset T$ be the torus dual to $M_{L}=M_{B_{v}}$. Let $\Sigma_{L}$,
resp. $\Sigma_{B_{v}}$ be the subfan of $\Sigma$ of the cones all of whose
rays are elements of $E$, resp. $E\cup B_{v}$. This defines a regular fan on
$N_{L}$ and thus a smooth toric subvariety $T_{L}\subset U_{L}\subset U$,
resp. $U_{L}\subset U_{L,B_{v}}\subset U_{A_{v}}$ since Galois descent is
effective for closed immersions.

Note that the $\operatorname{Pic}U_{L,B_{v},\overline{F_{v}}}$ appearing in
Lemma 8.1 is the geometric Picard group of $U_{L,B_{v}}$ by (2).

Let $(U_{L};\mathbf{B}):=(U_{L},(U_{L,B_{v}})_{v})$. We have the following
properties.

######  Lemma 8.2.

  1. (1)

$\overline{F}[U_{L}]^{\times}=\overline{F}^{\times}$.

  2. (2)

$\operatorname{H}^{0}((U_{L};\mathbf{B}),\mathcal{O}_{(U_{L};\mathbf{B})})=F$.

  3. (3)

$\mathbf{T}_{\operatorname{NS}L}$ is the Néron-Severi torus of
$(U_{L};\mathbf{B})$.

  4. (4)

The image of $aL$ in $(\operatorname{Pic}(U_{L};\mathbf{B}))_{\mathbb{R}}$ is
equal to $-K_{(U_{L};\mathbf{B})}$.

###### Proof.

For the first statement use that $E$ generates $N_{L}$ by definition.

For the second statement it suffices to show that $N_{L}$ is generated by non-
negative linear combinations of elements of $E$ and $B_{v}$ for all places
$v$. If this was not true there would exist a non-zero $m\in M_{L}$ which maps
to a non-zero element $\mathbb{Z}_{\geq
0}[E\cup\bigcup_{v}B_{v}]\subset\mathbb{Z}_{\geq 0}[D\cup\bigcup_{v}A_{v}]$.
Taking Galois norms we get a non-zero element $f\in\mathbb{Z}_{\geq
0}[D/\Gamma_{v}\cup\bigcup_{v}A_{v}/\Gamma_{v}]$.

The element $f$ maps to $0$ in
$\operatorname{Pic}U_{L,B_{v},\overline{F_{v}}}$ for all $v$, so the image in
$\operatorname{Pic}(X;\mathbf{A})$ must be contained in $C_{\mathbb{R}}$ by
Lemma 8.1. Since $f$ is non-negative we must have $f\in C$. But
$f\in\mathbb{Z}_{\geq 0}[E/\Gamma_{v}\cup\bigcup_{v}B_{v}/\Gamma_{v}]$ so this
contradicts the definition of $E$ and $A_{v}$. We conclude by contradiction.

Because
$\operatorname{Pic}U_{L,B_{v},\overline{F_{v}}}=\operatorname{coker}(\mathbb{Z}[D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}]\to\operatorname{Pic}U_{A_{v},\overline{F_{v}}})$ we
have that its dual group of multiplicative type is
$\ker(T_{\operatorname{NS}A_{v}}\to\mathbb{G}_{m}^{D_{\operatorname{adj}}\cup
A_{v,\operatorname{adj}}})$. This agrees with the definition of
$\mathbf{T}_{\operatorname{NS}L}$.

The last statement follows from the definitions of
$a,D_{\operatorname{adj}},A_{v,\operatorname{adj}}$ and Lemma 8.1. ∎

Let $\mathbf{Y}_{L}:=\mathbf{Y}\cap(\mathbb{A}^{E},(\mathbb{A})^{E\cup
B_{v}})$. It follows directly from the definitions that the restriction of
$\pi$ to $\mathbf{Y}_{L}$ factors through $(U_{L},\mathbf{B})$ and that
$\mathbf{T}_{\operatorname{NS}L}$ acts on $\mathbf{Y}_{L}$. It is clear from
Proposition 2.10 that $\pi:\mathbf{Y}_{L}\to(U_{L},\mathbf{B})$ is a universal
torsor.

Let
$\mathcal{R}:=\operatorname{coker}(T_{\operatorname{NS}}(F)\to\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F))$.
For each $\mathfrak{a}\in\operatorname{H}^{1}(F,T_{\operatorname{NS}})$ and
$\alpha\in\mathcal{R}$ the map
$\pi_{\mathfrak{a},\alpha}:\mathbf{Y}_{L}\to(U_{L},\mathbf{B})\cdot\pi_{\mathfrak{a}}(\alpha):x\to\pi_{\mathfrak{a}}(x\alpha)$
is a universal torsor since it is just a shift of $\pi$.

###  8.2. Integral over universal torsor

We can rephrase the conditions $\chi_{\rho}=1$ for $\rho\in E/\Gamma_{F}$ and
$\chi_{b_{v}}=1$ for all $b_{v}\in B_{v}/\Gamma_{v}$ and
$v\in\Omega_{F}^{\infty}$ in the definition (32) of $\xi(s,f,\varphi)$ as the
statement that $\chi$ is trivial when restricted to
$\Theta_{\mathbf{B}}(\mathbb{A}_{F})$. This is equivalent to
$\chi\in\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})^{\vee}$ by
(35).

Completely analogously to the analysis in §7.4 one works out that

$\begin{split}&\int_{(\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})/\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F))^{\vee}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})};sL)d\tau_{\Theta_{\mathbf{A}_{\operatorname{adj}}}^{\vee}}(\chi)=\\\
&c_{\mathbf{A}_{\operatorname{adj}}}^{-1}\sum_{\begin{subarray}{c}\chi\in\Theta_{\mathbf{A}}(\mathbb{A}_{F})^{1}/\Theta_{\mathbf{A}}(F))^{\vee}\\\
\chi_{\rho}=1:\rho\in D/\Gamma_{F}\\\ \chi_{\rho_{v}}=1:\rho_{v}\in
B/\Gamma_{v}\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}\rho\in
E_{\operatorname{adj}}/\Gamma_{F}\\\
\mathrm{Re}(z_{\rho})=0\end{subarray}}\mathop{\int\cdots\int}_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
\rho_{v}\in A_{v,\operatorname{adj}}/\Gamma_{v}\\\
\mathrm{Re}(w_{\rho_{v}})=0\end{subarray}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})+(\bm{z}_{\operatorname{adj},\bm{w}_{\operatorname{adj}}})};sL)d\frac{\bm{z}_{\operatorname{adj}}}{2\pi
i}d\frac{\bm{w}_{\operatorname{adj}}}{2\pi i}.\end{split}$

Where
$c_{\mathbf{A}_{\operatorname{adj}}}:=\prod_{v\in\Omega_{F}^{\infty}}2^{\\#\\{a_{v}\in
A_{v,\operatorname{adj}}/\Gamma_{v}:F_{a_{v}}=\mathbb{R}\\}}(2\pi)^{\\#\\{a_{v}\in
A_{v,\operatorname{adj}}/\Gamma_{v}:F_{a_{v}}=\mathbb{C}\\}}$.

We can substitute this into (32) to deduce that

$\begin{split}\xi(s,f,\bm{z},\bm{w})=c_{\mathbf{B}}^{-1}\int_{(\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})/\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F))^{\vee}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})};sL)d\tau_{\Theta_{\mathbf{A}_{\operatorname{adj}}}^{\vee}}(\chi).\end{split}$
(37)

Where $c_{\mathbf{B}}:=\prod_{v\in\Omega_{F}^{\infty}}2^{\\#\\{b_{v}\in
B_{v}/\Gamma_{v}:F_{b_{v}}=\mathbb{R}\\}}(2\pi)^{\\#\\{b_{v}\in
B_{v}/\Gamma_{v}:F_{v_{v}}=\mathbb{C}\\}}$.

It follows from the exact sequences (35) and (36) that the following sequence
is exact

$1\to\Theta_{\mathbf{B}}(\mathbb{A}_{F})\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F)/T_{\operatorname{NS}}(F)\to\Theta_{\mathbf{A}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)\to\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})/\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F)\to
1$

The integral in (37) converges absolutely by Lemma 7.11, it also still
converges after replacing $\varphi$ and $f$ by the $\varphi^{\prime}$ and $g$
in Lemma 7.4. By applying the Poisson summation formula, i.e. Theorem 2.17 to
the function
$\Theta_{\mathbf{B}}(\mathbb{A}_{F})\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F)/T_{\operatorname{NS}}(F)\to\mathbb{C}:x\to\mathcal{G}(x;sL)|x|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})}$
we deduce that $\xi(s,f,\bm{z},\bm{w})$ is equal to

$\begin{split}&c_{\mathbf{B}}^{-1}\int_{(\Theta_{\mathbf{A}_{\operatorname{adj}}}(\mathbb{A}_{F})/\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F))^{\vee}}\hat{\mathcal{G}}(\chi\cdot|\cdot|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})};sL)d\tau_{\Theta_{\mathbf{A}_{\operatorname{adj}}}^{\vee}}(\chi)=\\\
&c_{\mathbf{B}}^{-1}\int_{\Theta_{\mathbf{B}}(\mathbb{A}_{F})\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F)/T_{\operatorname{NS}}(F)}\mathcal{G}(x;sL)|x|_{\Theta_{\mathbf{A}}}^{(\bm{z},\bm{w})}d\tau_{\Theta_{\mathbf{B}}}(x)\\\
&\sum_{\alpha\in\mathcal{R}}c_{\mathbf{B}}^{-1}\int_{\Theta_{\mathbf{B}}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}\mathcal{G}(\alpha
x;sL)|x|_{\Theta_{\mathbf{B}}}^{(\bm{z},\bm{w})}d\tau_{\Theta_{\mathbf{B}}}(x).\end{split}$
(38)

Where
$\mathcal{R}:=\operatorname{coker}(T_{\operatorname{NS}L}(F)\to\mathbb{G}_{m}^{D_{\operatorname{adj}}}(F))$
and the last equality follows since
$|x\alpha|_{\Theta_{\mathbf{A}}}=|x|_{\Theta_{\mathbf{B}}}|\alpha|_{\Theta_{\mathbf{A}_{\operatorname{adj}}}}=|x|_{\Theta_{\mathbf{B}}}$
by the product formula. This sum is thus absolutely convergent for
$\mathrm{Re}(s)$ sufficiently large and $\mathrm{Re}(z_{\rho})>-1$,
$\mathrm{Re}(w_{\rho_{v}})>-2$.

We will relate these integrals to geometric integrals, the following lemma is
inspired by [CLT10a, Prop. 4.3.4]

######  Lemma 8.3.

Let $g:\mathbf{Y}_{L}(\mathbb{A}_{F})\to\mathbb{C}$ be a compactly supported
continuous function. We then have

$\begin{split}&c_{\mathbf{B}}\int_{\mathbf{Y}_{L}(\mathbb{A}_{F})}g(x)dx=\\\
&\prod_{\rho\in
E/\Gamma_{F}}\lim_{z_{\rho}\to-1}(z_{\rho}+1)\prod_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
w_{\rho_{v}}\in
B_{v}/\Gamma_{v}\end{subarray}}\lim_{w_{\rho_{v}}\to-2}(w_{\rho_{v}}+2)\int_{\Theta_{\mathbf{B}}(\mathbb{A}_{F})/T_{\operatorname{NS}L}(F)}g(x)|x|_{\Theta_{\mathbf{B}}}^{\mathbf{2}+(\bm{z},\bm{w})}d\tau_{\Theta_{\mathbf{B}}}(x).\end{split}$

###### Proof.

By a standard density argument we may assume that $g=\prod_{v}g_{v}$ where
$g_{v}:Y_{L,B_{v}}(F_{v})\to\mathbb{R}_{\geq 0}$ is a compactly supported
function which is smooth, resp. locally constant, if $v$ is archimedean, resp.
non-archimedean.

For $v\in\Omega_{F}^{\infty}$ the local contribution from the right-hand side
is

$\prod_{\begin{subarray}{c}v\in\Omega_{F}\\\ \rho_{v}\in
A_{v}/\Gamma_{v}\end{subarray}}\lim_{w_{\rho_{v}}\to-2}(w_{\rho_{v}}+1)\int_{\mathbb{G}_{m}^{E\cup
B_{v}}(F_{v})}g_{v}(x_{v})|x_{v}|_{\mathbb{G}_{m}^{E\cup
B_{v}}(F_{v})}^{\mathbf{1}+(\bm{z},\bm{w})}\prod_{\rho_{v}\in(E\cup
B_{v})/\Gamma_{v}}dx_{\rho_{v}}.$

This is equal to
$c_{B_{v}}\int_{Y_{L,v}(F_{v})_{B_{v}}}g(x_{v})|x_{v}|_{\mathbb{G}_{m}^{E\cup
B_{v}}}^{\mathbf{1}+\bm{z}}\prod_{\rho_{v}\in E/\Gamma_{v}}dx_{\rho_{v}}$ by
[CLT10a, Prop. 3.4], where

$c_{B_{v}}=2^{\\#\\{b_{v}\in
B_{v}/\Gamma_{v}:F_{b_{v}}=\mathbb{R}\\}}(2\pi)^{\\#\\{b_{v}\in
B_{v}/\Gamma_{v}:F_{v_{v}}=\mathbb{C}\\}}.$

For $v\in\Omega_{F}^{\text{fin}}$ the local contribution in the right-hand
side is

$\int_{\mathbb{G}_{m}^{E}(F_{v})}g_{v}(x_{v})|x_{v}|_{\mathbb{G}_{m}^{E}}^{\mathbf{1}+\bm{z}}\prod_{\rho_{v}\in
E/\Gamma_{v}}\zeta_{F_{\rho_{v}}}(1)dx_{\rho_{v}}.$

Taking the product over all places and taking into account the factor
$\prod_{\rho\in E/\Gamma}\zeta^{*}_{F_{\rho}}(1)$ shows that the right-hand
side is the limit

$\begin{split}c_{\mathbf{B}}\prod_{\rho\in
E/\Gamma_{F}}\lim_{z_{\rho}\to-1}&\prod_{v\in\Omega_{F}^{\infty}}\int_{Y_{L,v}(F_{v})_{B_{v}}}g(x_{v})|x_{v}|_{\mathbb{G}_{m}^{E\cup
B_{v}}}^{\mathbf{1}+\bm{z}}\prod_{\rho_{v}\in E/\Gamma_{v}}dx_{\rho_{v}}\\\
&\prod_{v\in\Omega_{F}^{\text{fin}}}\int_{Y_{L}(F_{v})}g_{v}(x_{v})|x_{v}|_{\mathbb{G}_{m}^{E}}^{\mathbf{1}+\bm{z}}\prod_{\rho_{v}\in
E/\Gamma_{v}}\frac{\zeta_{F_{\rho_{v}}}(1)}{\zeta_{F_{\rho_{v}}}(2+z_{\rho_{v}})}dx_{\rho_{v}}.\end{split}$

This limit converges, which proves exactly what we had to prove. ∎

This allows us to compute the residue of $\xi(f,s,\bm{z},\bm{w})$.

######  Lemma 8.4.

We have the equality

$\begin{split}\prod_{\rho\in
E/\Gamma_{F}}\lim_{z_{\rho}\to-1}(z_{\rho}+1)\prod_{\begin{subarray}{c}v\in\Omega_{F}^{\infty}\\\
w_{\rho_{v}}\in
B_{v}/\Gamma_{v}\end{subarray}}\lim_{w_{\rho_{v}}\to-2}(w_{\rho_{v}}+2)\xi(f,s,\bm{z},\bm{w})=\\\
\sum_{\alpha\in\mathcal{R}}\int_{\mathbf{Y}_{L}(\mathbb{A}_{F})}\mathcal{H}_{\varphi}(\alpha
x;sL)f(\pi_{\mathfrak{a}}(\alpha x))dx.\end{split}$

###### Proof.

Apply Lemma 8.3 to each term of (38). This application is valid since after
expanding the definition of $\mathcal{H}_{\varphi}(x;sL)$ the remaining
integral over $\Theta_{\mathbf{B}}$ has as integrand
$f(\bm{\pi}_{\mathfrak{a}}(\alpha x))\varphi(y||\alpha
x||)|x|_{\Theta_{\mathbf{B}}}^{\mathbf{2}+(\bm{z},\bm{w})}$ which is compactly
supported on $\mathbf{Y}(\mathbb{A}_{F})$ and thus also on the closed subspace
$\mathbf{Y}_{L}(\mathbb{A}_{F})$. ∎

###  8.3. Anticanonical height

######  Lemma 8.5.

Assume that $K_{(X;\mathbf{A})}=-aL$. In this case $\lim_{s\to
a}(s-a)^{b}Z_{\mathfrak{a}}(sL,f)$ is equal to

$\int_{\mathbf{Y}(\mathbb{A}_{F})}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log\mathbf{H}(z))f(\pi_{\mathfrak{a}}(z))\mathbf{H}(z)^{-L}d\tau_{\mathbf{Y},\mathbf{H}^{-aL}}(z).$

###### Proof.

We have $D=E$ and $B_{v}=A_{v}$ for all $v$ since $K_{(X;\mathbf{A})}=-aL$ so
$\mathbf{Y}_{L}=\mathbf{Y}$ and $\mathcal{R}=1$. We will study the integral in
the right-hand side of Lemma 8.4.

Expanding out the definitions and making a change of variables as in §7.2 we
see that this is equal to

$\int_{\mathbf{T}_{\operatorname{NS}}(\mathbb{A}_{F})/T_{\operatorname{NS}}(F)}\mathbf{1}_{\operatorname{\overline{Eff}}_{(X;\mathbf{A})}}(\log|y|_{\mathbf{T}_{\operatorname{NS}}})|y|_{\mathbf{T}_{\operatorname{NS}}}^{-sL-
K_{(X;\mathbf{A})}}d\tau_{\mathbf{T}_{\operatorname{NS}}}(y)\int_{\mathbf{Y}(\mathbb{A}_{F})_{\mathbf{A}}}f(\pi_{\mathfrak{a}}(x))\varphi(||x||)dx.$

The first integral is the only part which depends on $s$. This integral was
computed in Lemma 7.5, it is a constant times
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}}(sL-
K_{(X;\mathbf{A})})$. The residue of
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}}(sL-
K_{(X;\mathbf{A})})$ at $a$ is
$\mathcal{X}_{\operatorname{\overline{Eff}}_{(\mathbf{X};\mathbf{A})}}(L)$ by
Lemma 2.16 since $aL=K_{(X;\mathbf{A})}$. We deduce that $\lim_{s\to