# The conjectures of Artin–Tate and Birch–Swinnerton-Dyer

Stephen Lichtenbaum  Department of Mathematics, Brown University, Providence,
RI 02912<EMAIL_ADDRESS>,  Niranjan Ramachandran  Department
of Mathematics, University of Maryland, College Park, MD 20742 USA.
<EMAIL_ADDRESS>and  Takashi Suzuki  Department of Mathematics, Chuo
University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
<EMAIL_ADDRESS>
  *     * scAbstract. We provide two proofs that the conjecture of Artin–Tate for a fibered surface is equivalent to the conjecture of Birch–Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the orders of the Brauer group and the Tate–Shafarevich group.

scKeywords. Birch–Swinnerton-Dyer conjecture; finite fields; zeta functions;
Tate conjecture

sc2020 Mathematics Subject Classification. 11G40, 14G10, 19F27

  *
cDecember 27, 2021Received by the Editors on May 14, 2021\.
Accepted on January 29, 2022.

Department of Mathematics, Brown University, Providence, RI 02912

sce-mail<EMAIL_ADDRESS>
Department of Mathematics, University of Maryland, College Park, MD 20742 USA.

sce-mail<EMAIL_ADDRESS>
Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo
112-8551, Japan

sce-mail<EMAIL_ADDRESS>

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction and statement of results
  2. 2 Preparations
  3. 3 First proof of Theorem 1.1
  4. 4 Second proof of Theorem 1.1

## 1\. Introduction and statement of results

Let $k=\mathbb F_{q}$ be a finite field of characteristic $p$ and let $S$ be a
smooth projective (geometrically connected) curve over $T=\leavevmode\nobreak\
$Spec$\leavevmode\nobreak\ k$ and let $F=k(S)=\mathbb F_{q}(S)$ be the
function field of $S$. Let $X$ be a smooth proper surface over $T$ with a flat
proper morphism $\pi:X\to S$ with smooth geometrically connected generic fiber
$X_{0}$ over Spec $F$. The Jacobian $J$ of $X_{0}$ is an Abelian variety over
$F$.

Our first main result is a proof of the following statement conjectured by
Artin and Tate [Tat66, Conjecture (d)]:

######  Theorem 1.1.

The Artin–Tate conjecture for $X$ is equivalent to the Birch–Swinnerton-Dyer
conjecture for $J$.

Recall that these conjectures concern two (conjecturally finite) groups: the
Tate–Shafarevich group $\Sha(J/F)$ of $J$ and the Brauer group
$\textrm{Br}(X)$ of $X$. A result of Artin–Grothendieck [Gor79, Theorem 2.3]
[Gro68, §4] is that $\Sha(J/F)$ is finite if and only if $\textrm{Br}(X)$ is
finite.

Our second main result is a new proof of a beautiful result (2.18) of Geisser
[Gei20, Theorem 1.1] that relates the conjectural finite orders of $\Sha(J/F)$
and $\textrm{Br}(X)$; special cases of (2.18) are due to Milne–Gonzales-Aviles
[Mil81, GA03].

We actually provide two proofs of Theorem 1.1; while our first proof uses
Geisser’s result (2.18), the second (and very short) proof in §4, completely
due to the third-named author, does not.

### 1.1. History

Artin and Tate regarded Theorem 1.1 as easier to prove as opposed to the other
conjectures in [Tat66]. They proved Theorem 1.1 when $\pi$ is smooth and has a
section ([Tat66, p.427]) using the equality

(1.1) $[\Sha(J/F)]=[\textrm{Br}(X)]$

between the orders of the groups $\Sha(J/F)$ and $\textrm{Br}(X)$ which
follows from Artin’s theorem [Tat66, Theorem 3.1], [Gor79, Theorem 2.3]: if
$\pi$ is generically smooth with connected fibers and admits a section, then
$\Sha(J/F)\cong\textrm{Br}(X)$. Gordon [Gor79, Theorem 6.1] used (1.1) to
prove Theorem 1.1 when111There is another proof (up to $p$-torsion) in this
case due to Z. Yun [Yun15]. $\pi$ is cohomologically flat with a section (see
[Gor79, Theorem 2.3]). Building on Gordon [Gor79], Liu–Lorenzini–Raynaud
[LLR04] proved several new cases of Theorem 1.1 by eliminating the condition
of cohomological flatness of $\pi$; their proof [LLR04, Theorem 4.3] proceeds
by proving that Theorem 1.1 is equivalent to a precise relation generalizing
(1.1) between $[\textrm{Br}(X)]$ and $[\Sha(J/F)]$ which in their case had
been proved by Milne and Gonzales-Aviles [Mil81, GA03].

As Liu–Lorenzini–Raynaud (and Milne) point out [LLR05, Theorem 2], Theorem 1.1
follows by combining [Tat66, Gro68, Mil75, KT03]:

$AT(X)\xLeftrightarrow{\ \text{Artin--Tate--Milne}\
}\textrm{Br}(X)\leavevmode\nobreak\ \textrm{finite}\xLeftrightarrow{\
\text{Artin--Grothendieck}\ }\Sha(J/F)\leavevmode\nobreak\
\textrm{finite}\xLeftrightarrow{\ \text{Kato--Trihan}\ }BSD(J).$

In 2018, Geisser pointed out that a slight correction is necessary in the
relation [LLR04, Theorem 4.3] between $[\textrm{Br}(X)]$ and $[\Sha(J/F)]$;
Liu–Lorenzini–Raynaud [LLR18, Corrected Theorem 4.3] showed that Theorem 1.1
holds if and only if this slightly corrected version holds. This precise
relation (Theorem 2.11) was then proved by Geisser [Gei20, Theorem 1.1]
without using Theorem 1.1. Thus, combining [LLR18, Corrected Theorem 4.3] and
[Gei20, Theorem 1.1] gives the second known proof of Theorem 1.1. But this
proof relies heavily on the work of Gordon222Known to have several
inaccuracies; see [LLR18, §3.3]. [Gor79] as can be seen from [LLR18, §3,
(3.9)].

### 1.2. Our approach

Our first proof depends on [Gor79] only for the elementary result (2.9). As in
[Gor79, LLR04, LLR18], this proof also follows the strategy in [Tat66, §4]. We
use the localization sequence to record a short proof333This is similar to the
ideas of Hindry–Pacheco and Kahn in [Kah09, §§3.2-3.3]. of the Tate–Shioda
relation (Corollary 2.2). In turn, this gives a quick calculation (2.17) of
the height pairing ${}_{\mathrm{ar}}(\operatorname{NS}(X))$ on the
Néron–Severi group of $X$. The same calculation in [Gor79, LLR18] requires a
detailed analysis of various subgroups of $\operatorname{NS}(X)$. A beautiful
introduction to these results is [Ulm14]; see [Lic83, Lic05, GS20] for Weil-
étale analogues.

The second proof (§4) of Theorem 1.1 uses only (2.5) and the Weil-étale
formulations of the two conjectures. In this proof, we do not compare each
term of the two special value formulas and entirely work in derived
categories.

### Notations

Throughout, $k=\mathbb F_{q}$ is a finite field of characteristic $p$ and
$T=\mathrm{Spec}\leavevmode\nobreak\ k$; if $\bar{k}$ is an algebraic closure
of $k$, let $\bar{T}=\mathrm{Spec}\leavevmode\nobreak\ \bar{k}$. The function
field of $S$ is $F=k(S)$. Let $X$ be a smooth proper surface over $T$ with a
flat proper morphism $\pi:X\to S$ with smooth geometrically connected generic
fiber $X_{0}$ over Spec $F$. The Jacobian $J$ of $X_{0}$ is an Abelian variety
over $F$.

### 1.3. The Artin–Tate conjecture

Let $k=\mathbb F_{q}$ and $F=k(S)$. For any scheme $V$ of finite type over
$T$, the zeta function $\zeta(V,s)$ is defined as

$\zeta(V,s)=\prodop\displaylimits_{v\in V}\frac{1}{(1-q_{v}^{-s})};$

the product is over all closed points $v$ of $V$ and $q_{v}$ is the size of
the finite residue field $k(v)$ of $v$. If $V$ is smooth proper (geometrically
connected) of dimension $d$, then the zeta function $\zeta(V,s)$ factorizes as

$\zeta(V,s)=\frac{P_{1}(V,q^{-s})\cdots
P_{2d-1}(V,q^{-s})}{P_{0}(V,q^{-s})\cdots P_{2d}(V,q^{-s})},\quad
P_{0}=(1-q^{-s}),\quad P_{2d}=(1-q^{d-s}),$

where $P_{i}(V,t)\in\mathbb Z[t]$ is the characteristic polynomial of
Frobenius acting on the $\ell$-adic étale cohomology
$H^{i}(V\times_{T}\bar{T},\mathbb Q_{\ell})$ for any prime $\ell$ not dividing
$q$; by Grothendieck and Deligne, $P_{j}(V,t)$ is independent of $\ell$. One
has the factorization [Tat66, (4.1)] (the second equality uses Poincaré
duality)

(1.2) $\zeta(X,s)=\frac{P_{1}(X,q^{-s})\cdot P_{3}(X,q^{-s})}{(1-q^{-s})\cdot
P_{2}(X,q^{-s})\cdot(1-q^{2-s})}=\frac{P_{1}(X,q^{-s})\cdot
P_{1}(X,q^{1-s})}{(1-q^{-s})\cdot P_{2}(X,q^{-s})\cdot(1-q^{2-s})}.$

Let $\rho(X)$ be the rank of the finitely generated Néron–Severi group
$\operatorname{NS}(X)$. The intersection $D\cdot E$ of divisors $D$ and $E$
provides a symmetric non-degenerate bilinear pairing on
$\operatorname{NS}(X)$; the height pairing $\langle D,E\rangle_{\mathrm{ar}}$
[LLR18, Remark 3.11] on $\operatorname{NS}(X)$ is related to the intersection
pairing as follows:

$\operatorname{NS}(X)\times\operatorname{NS}(X)\to\mathbb Q(\log q),\qquad
D,E\mapsto\langle D,E\rangle_{\mathrm{ar}}=(D\cdot E)\log q.$

Let $A$ be the reduced identity component
$\operatorname{Pic}^{\mathrm{red},0}_{X/k}$ of the Picard scheme
$\operatorname{Pic}_{X/k}$ of $X$. Let

(1.3) $\alpha(X)=\chi(X,\mathcal{O}_{X})-1+\dim(A).$

We write $[G]$ for the order of a finite group $G$.

######  Conjecture 1.2 (Artin–Tate [Tat66, Conjecture (C)]).

The Brauer group $\operatorname{Br}(X)$ is finite,
$\operatorname{ord}_{s=1}P_{2}(X,q^{-s})=\rho(X)$, and the special value

$P^{*}_{2}(X,q^{-1}):=\lim_{s\to 1}\frac{P_{2}(X,q^{-s})}{(s-1)^{\rho(X)}}$

of $P_{2}(X,t)$ at $t=1/q$ $($this corresponds to $s=1)$ satisfies

(1.4)
$P^{*}_{2}(X,q^{-1})=[\operatorname{Br}(X)]\cdot{}_{\mathrm{ar}}(\operatorname{NS}(X))\cdot
q^{-\alpha(X)}.$

Here ${}_{\mathrm{ar}}(\operatorname{NS}(X))$ is the discriminant $($see §1.4
$)$ of the height pairing on $\operatorname{NS}(X)$.

######  Remark.

The discriminant ${}_{\mathrm{ar}}(\operatorname{NS}(X))$ of the height
pairing on $\operatorname{NS}(X)$ is related to the discriminant
$\Delta(\operatorname{NS}(X))$ of the intersection pairing as follows:
${}_{\mathrm{ar}}(\operatorname{NS}(X))=\Delta(\operatorname{NS}(X))\cdot(\log
q)^{\rho(X)}$.

### 1.4. Discriminants

For more details on the basic notions recalled next, see [Yun15, §2.8] and
[Blo87]. Let $N$ be a finitely generated Abelian group $N$ and let
$\psi:N\times N\to K$ be a symmetric bilinear form with values in any field
$K$ of characteristic zero. If $\psi:N/{\mathrm{tor}}\times
N/{\mathrm{tor}}\to K$ is non-degenerate, the discriminant $\Delta(N)$ is
defined as the determinant of the matrix $\psi(b_{i},b_{j})$ divided by
$(N:N^{\prime})^{2}$ where $N^{\prime}$ is the subgroup of finite index
generated by a maximal linearly independent subset $\\{b_{i}\\}$ of $N$. Note
that $\Delta(N)$ is independent of the choice of the subset $\\{b_{i}\\}$ and
the subgroup $N^{\prime}$ and incorporates the order of the torsion subgroup
of $N$. For us, $K=\mathbb Q$ or $\mathbb Q(\log q)$.

Given a short exact sequence $0\to N^{\prime}\to N\to N^{\prime\prime}\to 0$
which splits over $\mathbb Q$ as an orthogonal direct sum $N_{\mathbb Q}\cong
N^{\prime}_{\mathbb Q}\oplus N^{\prime\prime}_{\mathbb Q}$ with respect to a
definite pairing $\psi$ on $N$, one has the following standard relation

(1.5) $\Delta(N)=\Delta(N^{\prime})\cdot\Delta(N^{\prime\prime}).$

Given a map $f:C\to C^{\prime}$ of Abelian groups with finite kernel and
cokernel, the invariant $z(f)=\frac{[\textrm{Ker}(f)]}{[\textrm{Coker}(f)]}$
[Tat66] extends to the derived category $\mathcal{D}$ of complexes in Abelian
groups with bounded and finite homology: given any such complex $C_{\bullet}$,
the invariant

$z(C_{\bullet})=\prodop\displaylimits_{i}[H_{i}(C_{\bullet})]^{(-1)^{i}}$

is an Euler characteristic; for any triangle $K\to L\to M\to K[1]$ in
$\mathcal{D}$, the following relation holds

(1.6) $z(K)\cdot z(M)=z(L).$

One recovers $z(f)$ viewing $f:C\to C^{\prime}$ as a complex in degrees zero
and one. For any pairing $\psi:N\times N\to\mathbb Z$, the induced map $N\to
R\textrm{Hom}(N,\mathbb Z)$ recovers $\Delta(N)$ above:

$\Delta(N)=z(N\to R\textrm{Hom}(N,\mathbb Z))^{-1}.$

∎

### 1.5. The Birch–Swinnerton-Dyer conjecture

For more details on the basic notions recalled next, see [GS20]. Let $J$ be
the Jacobian of $X_{0}$. Recall that the complete L-function [Ser70, Mil72],
[GS20, §4] of $J$ is defined as a product of local factors

(1.7) $L(J,s)=\prodop\displaylimits_{v\in S}\frac{1}{L_{v}(J,q_{v}^{-s})}.$

For any closed point $v$ of $S$, the local factor $L_{v}(J,t)$ is the
characteristic polynomial of Frobenius on

(1.8) $H^{1}_{\mathrm{\acute{e}t}}(J\times F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell})^{I_{v}},$

where $F_{v}$ is the complete local field corresponding to $v$ and $I_{v}$ is
the inertia group at $v$. By [GS20, Proposition 4.1], $L_{v}(J,t)$ has
coefficients in $\mathbb Z$ and is independent of $\ell$, for any prime $\ell$
distinct from the characteristic of $k$. Let $\Sha(J/F)$ be the
Tate–Shafarevich group of $J$ over Spec $F$ and let $r$ be the rank of the
finitely generated group $J(F)$. Let ${}_{\mathrm{NT}}(J(F))$ be the
discriminant of the Néron–Tate pairing [Tat66, p. 419], [KT03, §1.5] on
$J(F)$:

(1.9) $J(F)\times J(F)\to\mathbb Q(\log\leavevmode\nobreak\
q),\quad(\gamma,\kappa)\mapsto\langle\gamma,\kappa\rangle_{\mathrm{NT}}.$

Let $\mathcal{J}\to S$ be the Néron model of $J$; for any closed point $v\in
S$, define $c_{v}=[{}_{v}(k_{v})]$ where v is the group of connected
components of $\mathcal{J}_{v}$ and put $c(J)=\prodop\displaylimits_{v\in
S}c_{v}$; this is a finite product as $c_{v}=1$ for all but finitely many $v$.
Let $\operatorname{Lie}\,\mathcal{J}$ be the locally free sheaf on $S$ defined
by the Lie algebra of $\mathcal{J}$. Recall the444By [GS20, Corollary 4.5],
this is equivalent to the formulation in [Tat66].

######  Conjecture 1.3 (Birch–Swinnerton-Dyer).

The group $\Sha(J/F)$ is finite, $\operatorname{ord}_{s=1}L(J,s)=r$, and the
special value

$L^{*}(J,1):=\lim_{s\to 1}\frac{L(J,s)}{(s-1)^{r}}$

satisfies

(1.10) $L^{*}(J,1)=[\Sha(J/F)]\cdot{{}_{\mathrm{NT}}(J(F))}\cdot c(J)\cdot
q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}.$

The proof of Theorem 1.1, _i.e._ the equivalence of Conjectures 1.2 and 1.3,
naturally divides into four parts:

  * •

$\operatorname{Br}(X)$ is finite if and only if $\Sha(J/F)$ is finite. This is
known [Gro68, (4.41), Corollaire (4.4)].

  * •

Comparison of $\chi(S,\operatorname{Lie}\,\mathcal{J})$ and $\alpha(X)$ given
in (2.5). This is known [LLR04, p. 483]. For the convenience of the reader, we
recall it in §2.2.

  * •

(Proposition 2.4) $\operatorname{ord}_{s=1}P_{2}(X,q^{-s})=\rho(X)$ if and
only if $\operatorname{ord}_{s=1}L(J,s)=r$.

  * •

(§3) $P_{2}^{*}(X,1)$ satisfies (1.4) if and only if $L^{*}(J,1)$ satisfies
(1.10).

The first two parts are not difficult and we provide elementary proofs of the
last two parts.

### Acknowledgements

This paper would not exist without the inspiration provided by [FS21, Gor79,
LLR18, Gei20, Yun15] in terms of both mathematical ideas and clear exposition.
We thank Professors Liu, Lorenzini and K. Sato for their valuable comments on
an earlier draft. We heartily thank the referee for a valuable and detailed
report.

## 2\. Preparations

### 2.1. Elementary identities and known results

The Néron–Severi group $\operatorname{NS}(X)$ is the group of $k$-points of
the group scheme $\operatorname{NS}_{X/k}=\pi_{0}(\operatorname{Pic}_{X/k})$
of connected components of the Picard scheme $\operatorname{Pic}_{X/k}$ of
$X$. Let $A=\operatorname{Pic}^{\mathrm{red},0}_{X/k}$. The Leray spectral
sequence for the morphism $X\to\mathrm{Spec}\leavevmode\nobreak\ k$ and the
étale sheaf $\mathbb G_{m}$ provides the first exact sequences [BLR90,
Proposition 4, p. 204] below:

$0\longrightarrow\operatorname{Pic}(k)\longrightarrow\operatorname{Pic}(X)\longrightarrow\operatorname{Pic}_{X/k}(k)\longrightarrow\operatorname{Br}(k)\quad\text{and}\quad
0\longrightarrow\operatorname{Pic}^{0}_{X/k}\longrightarrow\operatorname{Pic}_{X/k}\longrightarrow\pi_{0}(\operatorname{Pic}_{X/k})\longrightarrow
0.$

Since $\operatorname{Br}(k)=0$,
$H^{1}_{\mathrm{\acute{e}t}}(\mathrm{Spec}\leavevmode\nobreak\
k,\operatorname{Pic}^{0}_{X/k})=H^{1}_{\mathrm{\acute{e}t}}(\mathrm{Spec}\leavevmode\nobreak\
k,\operatorname{Pic}^{\mathrm{red},0}_{X/k})$ and
$H^{1}_{\mathrm{\acute{e}t}}(\mathrm{Spec}\leavevmode\nobreak\ k,A)=0$ (Lang’s
theorem [Tat66, p. 209]), this provides

(2.1)
$\operatorname{Pic}_{X/k}(k)=\operatorname{Pic}(X)\quad\text{and}\quad\operatorname{NS}(X)=\operatorname{NS}_{X/k}(k)=\frac{\operatorname{Pic}(X)}{A(k)}.$

Let $P$ be the identity component of the Picard scheme
$\operatorname{Pic}_{S/k}$ of $S$. Let $B$ be the cokernel of the natural
injective map $\pi^{*}:P\to A$. So one has short exact sequences (using Lang’s
theorem [Tat66, p. 209] for the last sequence)

(2.2) $A=\operatorname{Pic}^{\mathrm{red},0}_{X/k},\quad
P=\operatorname{Pic}^{0}_{S/k},\quad 0\longrightarrow P\longrightarrow
A\longrightarrow B\longrightarrow 0,\quad\text{and}\quad 0\longrightarrow
P(k)\longrightarrow A(k)\longrightarrow B(k)\longrightarrow 0.$

It is known that [Tat66, p. 428]

(2.3) $P_{1}(S,q^{-s})=P_{1}(P,q^{-s}),\quad
P_{1}(X,q^{-s})=P_{1}(A,q^{-s}),\quad\text{and}\quad
P_{1}(A,q^{-s})=P_{1}(P,q^{-s})\cdot P_{1}(B,q^{-s}).$

For any Abelian variety $G$ of dimension $d$ over $k=\mathbb F_{q}$, it is
well known that [Tat66, p. 429, top line] (or [Gor79, 6.1.3])

(2.4) $P_{1}(G,1)=[G(k)]\quad\text{and}\quad P_{1}(G,q^{-1})=[G(k)]q^{-d}.$

### 2.2. Comparison of $\chi(S,\operatorname{Lie}\,\mathcal{J})$ and
$\alpha(X)$

It is known [LLR04, p. 483] that

(2.5) $\chi(S,\operatorname{Lie}\,\mathcal{J})-\dim(B)=-\alpha(X).$

We include their proof here for the convenience of the reader. A special case
of this is due to Gordon [Gor79, Proposition 6.5]. The Leray spectral sequence
for $\pi$ and $\mathcal{O}_{X}$ provides $H^{0}(S,\mathcal{O}_{S})\cong
H^{0}(X,\mathcal{O}_{X})$,

$0\to H^{1}(S,\mathcal{O}_{S})\to H^{1}(X,\mathcal{O}_{X})\to
H^{0}(S,R^{1}\pi_{*}\mathcal{O}_{X})\to 0,\quad H^{2}(X,\mathcal{O}_{X})\cong
H^{1}(S,R^{1}\pi_{*}\mathcal{O}_{X}).$

This proves
$\chi(X,\mathcal{O}_{X})=\chi(S,\mathcal{O}_{S})-\chi(S,R^{1}\pi_{*}\mathcal{O}_{X})$.
Recall that $\mathcal{J}$ is the Néron model of the Jacobian $J$ of $X_{0}$.
As the kernel and cokernel of the natural map555The map $\phi$ is obtained by
the composition of the maps
$R^{1}\pi_{*}\mathcal{O}_{X}\to\operatorname{Lie}\,P$ [LLR04, Proposition 1.3
(b)] and $\operatorname{Lie}\,P\to\operatorname{Lie}\,Q$ [LLR04, Theorem 3.1]
with $Q\xrightarrow{\sim}\mathcal{J}$ [LLR04, Facts 3.7 (a)]; it uses the fact
that $X$ is regular, $\pi:X\to S$ is proper flat, and
$\pi_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$.
$\phi:R^{1}\pi_{*}\mathcal{O}_{X}\to\operatorname{Lie}\,\mathcal{J}$ are
torsion sheaves on $S$ of the same length [LLR04, Theorem 4.2], we have
[LLR04, p. 483]

(2.6)
$\chi(S,R^{1}\pi_{*}\mathcal{O}_{X})=\chi(S,\operatorname{Lie}\,\mathcal{J}).$

Thus,

$\displaystyle\alpha(X)$
$\displaystyle\overset{(\ref{alphax})}{=}\chi(X,\mathcal{O}_{X})-1+\dim(A)=\chi(S,\mathcal{O}_{S})-\chi(S,R^{1}\pi_{*}\mathcal{O}_{X})-1+\dim(A)$
$\displaystyle=1-\dim(P)-\chi(S,\operatorname{Lie}\,\mathcal{J})-1+\dim(A)=-\chi(S,\operatorname{Lie}\,\mathcal{J})+\dim(A)-\dim(P)$
$\displaystyle\overset{(\ref{eq5})}{=}-\chi(S,\operatorname{Lie}\,\mathcal{J})+\dim(B).$

### 2.3. The Tate–Shioda relation about the Néron–Severi group

The structure of $\operatorname{NS}(X)$ depends on the singular fibers of the
morphism $\pi:X\to S$.

#### 2.3.1. Singular fibers

Let $Z=\\{v\in S\leavevmode\nobreak\ |\leavevmode\nobreak\
\pi^{-1}(v)=X_{v}\leavevmode\nobreak\ \textrm{is\leavevmode\nobreak\
not\leavevmode\nobreak\ smooth}\\}$. For any $v\in S$, let $G_{v}$ be the set
of irreducible components i of $X_{v}$, let $m_{v}$ be the cardinality of
$G_{v}$, and $m:=\sumop\displaylimits_{v\in Z}(m_{v}-1)$; for any $i\in
G_{v}$, let $r_{i}$ be the number of irreducible components of
${}_{i}\times\overline{k(v)}$. Let $R_{v}$ be the quotient

(2.7) $R_{v}=\frac{\mathbb Z^{G_{v}}}{\mathbb Z}$

of the free Abelian group generated by the irreducible components of $X_{v}$
by the subgroup generated by the cycle associated with $X_{v}=\pi^{-1}(v)$. If
$v\notin Z$, then $R_{v}$ is trivial.

Let $U=S-Z$; the map $X_{U}=\pi^{-1}(U)\to U$ is smooth. For any finite
$Z^{\prime}\subset S$ with $Z\subset Z^{\prime}$, we consider
$U^{\prime}=S-Z^{\prime}$ and $X_{U^{\prime}}=X-\pi^{-1}(U^{\prime})$. The
following proposition provides a description of
$\operatorname{NS}(X)\overset{(\ref{eq7})}{\cong}\operatorname{Pic}(X)/{A(k)}$.

######  Proposition 2.1.

  1. (i)

The natural maps $\pi^{*}:\operatorname{Pic}(S)\to\operatorname{Pic}(X)$ and
$\pi^{*}:\operatorname{Pic}(U^{\prime})\to\operatorname{Pic}(X_{U^{\prime}})$
are injective.

  2. (ii)

There is an exact sequence

(2.8) $0\longrightarrow\underset{v\in Z}{\oplus}\leavevmode\nobreak\
R_{v}\longrightarrow\frac{\operatorname{Pic}(X)}{\pi^{*}\operatorname{Pic}(S)}\longrightarrow\operatorname{Pic}(X_{0})\longrightarrow
0.$

###### Proof.

(i) From the Leray spectral sequence for $\pi:X\to S$ and the étale sheaf
$\mathbb G_{m}$ on $X$, we get the exact sequence

$0\longrightarrow H^{1}_{et}(S,\pi_{*}\mathbb G_{m})\longrightarrow
H^{1}_{et}(X,\mathbb G_{m})\longrightarrow H^{0}(S,R^{1}\pi_{*}\mathbb
G_{m})\longrightarrow\operatorname{Br}(S).$

Now $X_{0}$ being geometrically connected and smooth over $F$ implies [Mil81,
Remark 1.7a] that $\pi_{*}\mathbb G_{m}$ is the sheaf $\mathbb G_{m}$ on $S$.
This provides the injectivity of the first map. The same argument with
$U^{\prime}$ in place of $S$ provides the injectivity of the second.

(ii) The class group $\textrm{Cl}(Y)$ and the Picard group
$\operatorname{Pic}(Y)$ are isomorphic for regular schemes $Y$ such as $S$ and
$X$. The localization sequences for $X_{U^{\prime}}\subset X$ and
$U^{\prime}\subset S$ can be combined as

${0}$${\Gamma({S},\mathbb G_{m})}$${\Gamma({U^{\prime}},\mathbb
G_{m})}$${\underset{v\in Z^{\prime}}{\oplus}\mathbb
Z}$${\operatorname{Pic}(S)}$${\operatorname{Pic}(U^{\prime})}$${0}$${0}$${\Gamma({X},\mathbb
G_{m})}$${\Gamma(X_{U^{\prime}},\mathbb G_{m})}$${\underset{v\in
Z^{\prime}}{\oplus}\mathbb
Z^{G_{v}}}$${\operatorname{Pic}(X)}$${\operatorname{Pic}(X_{U^{\prime}})}$${0.}$

$\scriptstyle\sim$

$\scriptstyle\sim$

Here $\Gamma({X},\mathbb G_{m})=H^{0}_{et}({X},\mathbb
G_{m})=H^{0}_{Zar}({X},\mathbb G_{m})$. The induced exact sequence on the
cokernels of the vertical maps is

$0\longrightarrow\underset{v\in
Z^{\prime}}{\oplus}R_{v}\longrightarrow\frac{\operatorname{Pic}(X)}{\pi^{*}\operatorname{Pic}(S)}\longrightarrow\frac{\operatorname{Pic}(X_{U^{\prime}})}{\pi^{*}\operatorname{Pic}(U^{\prime})}\longrightarrow
0.$

In particular, we get this sequence for $Z$ and $U$. By assumption, $X_{v}$ is
geometrically irreducible for any $v\notin Z$; so $R_{v}=0$ for any $v\notin
Z$. So this means that, for any $U^{\prime}=S-Z^{\prime}$ contained in $U$,
the induced maps

$\frac{\operatorname{Pic}(X_{U})}{\pi^{*}\operatorname{Pic}(U)}\longrightarrow\frac{\operatorname{Pic}(X_{U^{\prime}})}{\pi^{*}\operatorname{Pic}(U^{\prime})}$

are isomorphisms. Taking the limit over $Z^{\prime}$ gives us the exact
sequence in the proposition.∎

######  Corollary 2.2.

  1. (i)

The Tate–Shioda relation [Tat66, (4.5)] $\rho(X)=2+r+m$ holds.

  2. (ii)

One has an exact sequence

$0\longrightarrow
B(k)\longrightarrow\frac{\operatorname{Pic}(X)}{\pi^{*}\operatorname{Pic}(S)}\longrightarrow\frac{\operatorname{NS}(X)}{\pi^{*}\operatorname{NS}(S)}\longrightarrow
0.$

###### Proof.

(i) Since $r$ is the rank of $J(F)$, the rank of $\operatorname{Pic}(X_{0})$
is $r+1$. Since $\operatorname{Pic}(S)$ has rank one, $A(k)$ is finite and
$m=\sumop\displaylimits_{v\in Z}(m_{v}-1)$, this follows from (2.1) and (2.8).

(ii) This follows from the diagram

${0}$${P(k)}$${A(k)}$${B(k)}$${0}$${0}$${\operatorname{Pic}(S)}$${\operatorname{Pic}(X)}$${\frac{\operatorname{Pic}(X)}{\pi^{*}\operatorname{Pic}(S)}}$${0}$${0}$${\operatorname{NS}(S)}$${\operatorname{NS}(X)}$${\frac{\operatorname{NS}(X)}{\pi^{*}\operatorname{NS}(S)}}$${0.}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi^{*}}$

∎

### 2.4. Relating the order of vanishing at $s=1$ of $P_{2}(X,q^{-s})$ and
$L(J,s)$

By666This proposition, first stated on Page 176 of [Gor79], has a typo in the
formula for $P_{2}$ which is corrected in its restatement on Page 193. We only
need the part about $P_{2}$ (and this is elementary). [Gor79, Proposition
3.3], one has

(2.9) $\zeta(X_{v},s)=\frac{P_{1}(X_{v},q_{v}^{-s})}{(1-q_{v}^{-s})\cdot
P_{2}(X_{v},q_{v}^{-s})},\quad\text{and}\quad
P_{2}(X_{v},q_{v}^{-s})=\left\\{\begin{array}[]{lr}(1-q_{v}^{1-s}),&\text{for
}v\notin Z\\\ \prodop\displaylimits_{i\in
G_{v}}(1-(q_{v})^{r_{i}(1-s)}),&\text{for }v\in Z\end{array}\right\\},$

see §2.3.1 for notation. Using

$Q_{2}(s)=\prodop\displaylimits_{v\in
Z}\frac{P_{2}(X_{v},q^{-s})}{(1-q_{v}^{1-s})},\quad\zeta(S,s)=\frac{P_{1}(S,q^{-s})}{(1-q^{-s})\cdot(1-q^{1-s})},\quad\text{and}\quad
Q_{1}(s)=\prodop\displaylimits_{v\in S}P_{1}(X_{v},q_{v}^{-s}),$

we can rewrite

$\zeta(X,s)=\prodop\displaylimits_{v\in
S}\zeta(X_{v},s)=\frac{1}{Q_{2}(s)}\cdot\prodop\displaylimits_{v\in
S}\frac{P_{1}(X_{v},q_{v}^{-s})}{(1-q_{v}^{-s})\cdot(1-q_{v}^{1-s})}=\frac{\zeta(S,s)\cdot\zeta(S,s-1)\cdot
Q_{1}(s)}{Q_{2}(s)}.$

The precise relation between $P_{2}(X,q^{-s})$ and $L(J,s)$ is given by
(2.11).

######  Proposition 2.3.

One has $\operatorname{ord}_{s=1}Q_{2}(s)=m$ and

(2.10) $Q_{2}^{*}(1)=\lim_{s\to
1}\frac{Q_{2}(s)}{(s-1)^{m}}=\prodop\displaylimits_{v\in
Z}\leavevmode\nobreak\ {\left((\log
q_{v})^{(m_{v}-1)}\cdot\prodop\displaylimits_{i\in G_{v}}r_{i}\right)},$
(2.11) $\frac{P_{2}(X,q^{-s})}{(1-q^{1-s})^{2}}=P_{1}(B,q^{-s})\cdot
P_{1}(B,q^{1-s})\cdot L(J,s)\cdot Q_{2}(s).$

###### Proof.

Observe that (2.10) is elementary: for any positive integer $r$, one has

$\lim_{s\to 1}\frac{(1-q_{v}^{r(1-s)})}{(s-1)}=\lim_{s\to
1}\frac{(1-q_{v}^{r(1-s)})}{(1-q_{v}^{1-s})}\cdot\frac{(1-q_{v}^{1-s})}{(s-1)}=\lim_{s\to
1}(1+q_{v}^{1-s}+\cdots+q_{v}^{(r-1)(1-s)})\cdot\log q_{v}=r\cdot\log q_{v}.$

For each $v\in Z$, this shows that

$\lim_{s\to 1}\frac{P_{2}(X_{v},q^{-s})}{(s-1)^{m_{v}}}=(\log
q_{v})^{m_{v}}\cdot\prodop\displaylimits_{i\in G_{v}}r_{i}.$

Therefore, we obtain that

$\lim_{s\to 1}\frac{Q_{2}(s)}{(s-1)^{m}}=\prodop\displaylimits_{v\in
Z}\lim_{s\to
1}\frac{\frac{P_{2}(X_{v},q^{-s})}{(1-q_{v}^{1-s})}}{(s-1)^{m_{v}-1}}=\prodop\displaylimits_{v\in
Z}\lim_{s\to
1}\frac{\frac{P_{2}(X_{v},q^{-s})}{(s-1)^{m_{v}}}}{\frac{(1-q_{v}^{1-s})}{s-1}}=\prodop\displaylimits_{v\in
Z}{\left(\frac{(\log q_{v})^{m_{v}}\cdot\prodop\displaylimits_{i\in
G_{v}}r_{i}.}{\log q_{v}}\right)}.$

We now prove (2.11). Simplifying the identity

$\frac{P_{1}(X,q^{-s})\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
P_{1}(X,q^{1-s})}{(1-q^{-s})\cdot
P_{2}(X,q^{-s})\cdot(1-q^{2-s})}=\zeta(X,s)=\frac{P_{1}(S,q^{-s})}{(1-q^{-s})\cdot(1-q^{1-s})}\cdot\frac{P_{1}(S,q^{1-s})}{(1-q^{1-s})\cdot(1-q^{2-s})}\cdot\frac{Q_{1}(s)}{Q_{2}(s)}$

from (1.2) using (2.3), one obtains

$\frac{P_{1}(B,q^{-s})\cdot
P_{1}(B,q^{1-s})}{P_{2}(X,q^{-s})}=\frac{1}{(1-q^{1-s})}\cdot\frac{1}{(1-q^{1-s})}\cdot\frac{Q_{1}(s)}{Q_{2}(s)}.$

On reordering, this becomes

$\frac{P_{2}(X,q^{-s})}{(1-q^{1-s})^{2}}=\frac{P_{1}(B,q^{-s})\cdot
P_{1}(B,q^{1-s})\cdot Q_{2}(s)}{Q_{1}(s)}.$

Let $T_{\ell}J$ be the $\ell$-adic Tate module of the Jacobian $J$ of $X$. For
any $v\in S$, the Kummer sequence on $X$ and $J$ provides a
$\textrm{Gal}(F_{v}^{\mathrm{sep}}/{F_{v}})$-equivariant isomorphism

$H^{1}_{\mathrm{\acute{e}t}}(X\times_{S}F_{v}^{\mathrm{sep}},\mathbb
Z_{\ell}(1))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{\ell}J\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}H^{1}_{\mathrm{\acute{e}t}}(J\times_{F}F_{v}^{\mathrm{sep}},\mathbb
Z_{\ell}(1)),$

as $J$ is a self-dual Abelian variety: this provides the isomorphisms

$H^{1}_{\mathrm{\acute{e}t}}(J\times_{F}F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell})\cong
H^{1}_{\mathrm{\acute{e}t}}(X\times_{S}F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell}),\quad
H^{1}_{\mathrm{\acute{e}t}}(J\times_{F}F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell})^{I_{v}}\cong
H^{1}_{\mathrm{\acute{e}t}}(X\times_{S}F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell})^{I_{v}}.$

From [Del80, Théorème 3.6.1, pp.213–214] (the arithmetic case is in [Blo87,
Lemma 1.2]), we obtain an isomorphism

$H^{1}_{\mathrm{\acute{e}t}}(X_{v}\times_{k(v)}\overline{k(v)},\mathbb
Q_{\ell})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\mathrm{\acute{e}t}}(X\times_{S}F_{v}^{\mathrm{sep}},\mathbb
Q_{\ell})^{I_{v}}.$

The definition of $L_{v}(J,t)$ in (1.8) now implies that
$P_{1}(X_{v},q_{v}^{-s})=L_{v}(J,q_{v}^{-s})$ and hence $Q_{1}(s)\cdot
L(J,s)=1$.∎

######  Proposition 2.4.

  1. (i)

$\operatorname{ord}_{s=1}P_{2}(X,q^{-s})=\rho(X)$ if and only if
$\operatorname{ord}_{s=1}L(J,s)=r$.

  2. (ii)

One has

(2.12) $P_{2}^{*}(X,\frac{1}{q})=P_{1}(B,q^{-1})\cdot P_{1}(B,1)\cdot
L^{*}(J,1)\cdot Q_{2}^{*}(1)\cdot(\log\leavevmode\nobreak\
q)^{2}\overset{(\ref{fq-points})}{=}\frac{[B(k)]^{2}}{q^{\dim(B)}}\cdot
L^{*}(J,1)\cdot Q_{2}^{*}(1)\cdot(\log\leavevmode\nobreak\ q)^{2}.$

###### Proof.

As $P_{1}(B,q^{-s})\cdot P_{1}(B,q^{1-s})$ does not vanish at $s=1$ by (2.4),
it follows from (2.11) that

$\operatorname{ord}_{s=1}P_{2}(X,q^{-s})-2=\operatorname{ord}_{s=1}L(J,s)+\operatorname{ord}_{s=1}Q_{2}(s).$

Corollary 2.2 says $\rho(X)=r+m+2$; (i) follows as
$\operatorname{ord}_{s=1}Q_{2}(s)=m$.

For (ii), use (2.4) and (2.11). ∎

### 2.5. Pairings on $\operatorname{NS}(X)$

Our next task is to compute $\Delta(\operatorname{NS}(X))$.

######  Definition 2.5.

  1. (i)

Let $\operatorname{Pic}^{0}(X_{0})$ be the kernel of the degree map
$\textrm{deg}:\operatorname{Pic}(X_{0})\to\mathbb Z$; the order $\delta$ of
its cokernel is, by definition, the index of $X_{0}$ over $F$.

  2. (ii)

Let $\alpha$ be the order of the cokernel of the natural map
$\operatorname{Pic}^{0}(X_{0})\hookrightarrow J(F)$.

  3. (iii)

Let $H$ (horizontal divisor on $X$) be the Zariski closure in $X$ of a divisor
$d$ on $X_{0}$, rational over $F$, of degree $\delta$.

  4. (iv)

The (vertical) divisor $V$ on $X$ is $\pi^{-1}(s)$ for a divisor $s$ of degree
one on $S$. Such a divisor $s$ exists as $k$ is a finite field and so the
index of the curve $S$ over $k$ is one. Writing $s=\sumop\displaylimits
a_{i}v_{i}$ as a sum of closed points $v_{i}$ on $S$ gives
$V=\sumop\displaylimits a_{i}\pi^{-1}(v_{i})$. Note that $V$ generates
$\pi^{*}\operatorname{NS}(S)\subset\operatorname{NS}(X)$.

######  Remark.

The definitions show that the intersections of the divisor classes $H$ and $V$
in $\operatorname{NS}(X)$ are given by

(2.13) $H\cdot V=\delta=V\cdot H\quad\text{and}\quad V\cdot V=0.$

Also, since $\pi:X\to S$ is a flat map between smooth schemes, the map
$\pi^{*}:CH(S)\to CH(X)$ on Chow groups is compatible with intersection of
cycles. Since $V=\pi^{*}(s)$ and the intersection $s\cdot s=0$ in $CH(S)$, one
has $V\cdot V=0$.

Let $\operatorname{NS}(X)_{0}=(\pi^{*}\operatorname{NS}(S))^{\perp}$; as $V$
generates $\pi^{*}\operatorname{NS}(S)$, we see that
$\operatorname{NS}(X)_{0}$ is the subgroup of divisor classes $Y$ such that
$Y\cdot X_{v}=0$ for any fiber $\pi^{-1}(v)=X_{v}$ of $\pi$; let
$\operatorname{Pic}(X)_{0}$ be the inverse image of $\operatorname{NS}(X)_{0}$
under the projection
$\operatorname{Pic}(X)\to\operatorname{NS}(X)\cong\frac{\operatorname{Pic}(X)}{A(k)}$.

######  Lemma 2.6.

$\operatorname{NS}(X)_{0}$ is the subgroup of $\operatorname{NS}(X)$ generated
by divisor classes whose restriction to $X_{0}$ is trivial.

###### Proof.

We need to show that $\operatorname{NS}(X)_{0}$ is equal to
$K:=\textrm{Ker}(\operatorname{NS}(X)\to\operatorname{NS}(X_{0}))$. If $D$ is
a vertical divisor ($\pi(D)\subset S$ is finite), then $D$ is clearly in $K$;
by [Liu02, §9.1, Proposition 1.21], $D$ is in $\operatorname{NS}(X)_{0}$.

If $D$ has no vertical components, then $D\cdot V=\textrm{deg}(D_{0})$. To see
this, clearly we may assume $D$ is reduced and irreducible (integral) and so
flat over $S$. So $\mathcal{O}_{D}$ is locally free over $\mathcal{O}_{S}$ of
constant degree $n$ since $S$ is connected. But then $\textrm{deg}(D_{0})$ is
equal to $n$ as is the integer $D\cdot V$.∎

######  Lemma 2.7.

Let us denote

$R=\underset{v\in Z}{\oplus}R_{v}\quad\text{and}\quad E=B(k)\cap
R\subset\frac{\operatorname{Pic}(X)_{0}}{\pi^{*}\operatorname{Pic}(S)}.$

One has the exact sequences

$\displaystyle 0\longrightarrow
R\longrightarrow\frac{\operatorname{Pic}(X)_{0}}{\pi^{*}\operatorname{Pic}(S)}\longrightarrow\operatorname{Pic}^{0}(X_{0})\longrightarrow
0,\quad\text{and}$ (2.14) $\displaystyle
0\longrightarrow\frac{R}{E}\longrightarrow\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\longrightarrow\frac{\operatorname{Pic}^{0}(X_{0})}{B(k)/E}\longrightarrow
0.$

###### Proof.

Lemma 2.6 shows that
$R\subset\frac{\operatorname{Pic}(X)_{0}}{\pi^{*}\operatorname{Pic}(S)}$. As
$A(k)$ is the kernel of the map
$\operatorname{Pic}(X)\to\operatorname{NS}(X)$, it follows that
$A(k)\subset\operatorname{Pic}(X)_{0}$. Thus, $B(k)$ is a subgroup of
$\frac{\operatorname{Pic}(X)_{0}}{\pi^{*}\operatorname{Pic}(S)}$.

The first exact sequence follows from Lemma 2.6; the second one follows from
Corollary 2.2 (ii). ∎

######  Lemma 2.8.

One has the equality

${}_{\mathrm{ar}}\left(\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\right)=[B(k)]^{2}\cdot\alpha^{2}\cdot{}_{\mathrm{NT}}(J(F))\cdot\prodop\displaylimits_{v\in
Z}\leavevmode\nobreak\ {}_{\mathrm{ar}}(R_{v}).$

###### Proof.

The exact sequence (2.7) splits orthogonally over $\mathbb Q$: for any divisor
$\gamma$ representing an element of $\operatorname{Pic}(X_{0})$, consider its
Zariski closure $\bar{\gamma}$ in $X$. Since the intersection pairing on
$R_{v}$ is negative-definite [Liu02, §9.1, Theorem 1.23], the linear map
$R_{v}\to\mathbb Z$ defined by $\beta\mapsto\beta\cdot\bar{\gamma}$ is
represented by a unique element

$\psi_{v}(\gamma)\in R_{v}\otimes\mathbb
Q\subset\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\otimes\mathbb
Q.$

Thus, the element

$\tilde{\gamma}:=\bar{\gamma}-\sumop\displaylimits_{v\in Z}\psi_{v}(\gamma)$

is good in the sense of [Gor79, §5, p. 185]: by construction, the divisor
$\tilde{\gamma}$ on $X$ intersects every irreducible component of every fiber
of $\pi$ with multiplicity zero. Fix
$\gamma,\kappa\in\operatorname{Pic}^{0}(X_{0})$: viewing them as elements of
$J(F)$, one computes their Neron–Tate pairing (1.9); also, one can compute the
height pairing of $\tilde{\gamma}$ and $\tilde{\kappa}$ in
$\operatorname{NS}(X)$. These two are related by the identity [Tat66, p. 429]
[LLR18, Remark 3.11]

$\langle\gamma,\kappa\rangle_{\mathrm{NT}}=-\langle\tilde{\gamma},\tilde{\kappa}\rangle_{\mathrm{ar}}=-(\tilde{\gamma}\cdot\tilde{\kappa})\cdot\log
q.$

This says that

(2.15)
${}_{\mathrm{ar}}\left(\operatorname{Pic}^{0}(X_{0})\right)={}_{\mathrm{NT}}\left(\operatorname{Pic}^{0}(X_{0})\right).$

The map

$\operatorname{Pic}^{0}(X_{0})\otimes\mathbb
Q\to\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\otimes\mathbb
Q,\qquad\gamma\mapsto\tilde{\gamma}$

provides an orthogonal splitting of (2.7) (over $\mathbb Q$). So

$\displaystyle{}_{\mathrm{ar}}\left(\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\right)$
$\displaystyle\overset{(\ref{deltann})}{=}{}_{\mathrm{ar}}\left(\frac{\operatorname{Pic}^{0}(X_{0})}{B(k)/E}\right)\cdot{}_{\mathrm{ar}}\left(\frac{R}{E}\right)=\frac{[B(k)]^{2}}{e^{2}}\cdot{}_{\mathrm{ar}}\left({\operatorname{Pic}^{0}(X_{0})}\right)\cdot
e^{2}{}_{\mathrm{ar}}(R)$
$\displaystyle\overset{(\ref{equality})}{=}[B(k)]^{2}\cdot{}_{\mathrm{NT}}\left({\operatorname{Pic}^{0}(X_{0})}\right)\cdot{}_{\mathrm{ar}}(R)$

where $e=[E]$ as the size of $E$. As

(2.16)
${}_{\mathrm{NT}}(\operatorname{Pic}^{0}(X_{0}))=\alpha^{2}\cdot{}_{\mathrm{NT}}(J(F))\quad\text{and}{}_{\mathrm{ar}}(R)=\prodop\displaylimits_{v\in
Z}{}_{\mathrm{ar}}(R_{v}),$

this proves the lemma. ∎

With Lemma 2.8 at hand we are almost ready to compute
${}_{\mathrm{ar}}(\operatorname{NS}(X))$. As the intersection pairing on
$\operatorname{NS}(X)$ is not definite (Hodge index theorem), we cannot apply
(1.5). Instead, we use a variant of a lemma of Z. Yun [Yun15].

#### 2.5.1. A lemma of Yun

Given a non-degenerate symmetric bilinear pairing
$\Lambda\times\Lambda\to\mathbb Z$ on a finitely generated Abelian group , an
isotropic subgroup , a subgroup ′ containing and with finite index in ⟂, let
${}_{0}=\frac{{}^{\prime}}{\Gamma}$. We recall from §1.4 that
$\Delta(\Lambda)=z(D)^{-1}$ where $D:=\Lambda\to R\textrm{Hom}(\Lambda,\mathbb
Z)$ and $\Delta({}_{0})={z(D_{0})}^{-1}$ where $D_{0}:={}_{0}\to
R\textrm{Hom}({}_{0},\mathbb Z)$. Let be the discriminant of the induced non-
degenerate pairing $\Gamma\times\frac{\Lambda}{{}^{\prime}}\to\mathbb Z$:

$\Delta=\frac{1}{z(C)}=\frac{1}{z(C^{\prime})},\quad C:=\Gamma\to
R\textrm{Hom}\left(\frac{\Lambda}{{}^{\prime}},\mathbb
Z\right),\quad\text{and}\quad C^{\prime}:=\frac{\Lambda}{{}^{\prime}}\to
R\textrm{Hom}(\Gamma,\mathbb Z).$

######  Lemma 2.9 (_cf._ [Yun15, Lemma 2.12]).

One has $\Delta(\Lambda)=\Delta({}_{0})\cdot{}^{2}$.

###### Proof.

Applying (1.6) to the maps of triangles

${\frac{\Lambda}{\Gamma}}$${\Gamma[1]}$${R\textrm{Hom}\left(\frac{\Lambda}{{}^{\prime}},\mathbb
Z\right)}$${R\textrm{Hom}(\Lambda,\mathbb
Z)}$${R\textrm{Hom}({}^{\prime},\mathbb
Z)}$${R\textrm{Hom}\left(\frac{\Lambda}{{}^{\prime}},\mathbb Z\right)[1]}$

and

${\frac{{}^{\prime}}{\Gamma}}$${\frac{\Lambda}{\Gamma}}$${\frac{\Lambda}{{}^{\prime}}}$${\frac{{}^{\prime}}{\Gamma}[1]}$${R\textrm{Hom}\left(\frac{{}^{\prime}}{\Gamma},\mathbb
Z\right)}$${R\textrm{Hom}({}^{\prime},\mathbb
Z)}$${R\textrm{Hom}(\Gamma,\mathbb
Z)}$${R\textrm{Hom}\left(\frac{{}^{\prime}}{\Gamma},\mathbb Z\right)[1]}$

shows that $z(D)\cdot z(C)^{-1}=z(D_{0})\cdot z(C^{\prime})$. ∎

We can finally compute ${}_{\mathrm{ar}}(\operatorname{NS}(X))$.

######  Proposition 2.10.

The following relations hold

${}_{\mathrm{ar}}(\operatorname{NS}(X))=\delta^{2}\cdot{}_{\mathrm{ar}}\left(\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\right)\cdot(\log
q)^{2}\quad\text{and}\quad\Delta(\operatorname{NS}(X))=\delta^{2}\cdot\Delta\left(\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}\right).$

###### Proof.

Let $\mathbb
Z\cong\Gamma=\pi^{*}\operatorname{NS}(S)\subset\operatorname{NS}(X)=\Lambda$
with ${}^{\prime}=\operatorname{NS}(X)_{0}$ and
${}_{0}=\frac{\operatorname{NS}(X)_{0}}{\pi^{*}\operatorname{NS}(S)}$. Lemma
2.6 implies that

$\frac{\Lambda}{{}^{\prime}}=\frac{\operatorname{NS}(X)}{\operatorname{NS}(X)_{0}}\cong\mathbb
Z\quad\text{and}\quad
C=\Gamma\to\textrm{Hom}\left(\frac{\operatorname{NS}(X)}{\operatorname{NS}(X)_{0}},\mathbb
Z\right),$

with $C$ as in Lemma 2.9. Now (2.13) shows that $\pi^{*}\operatorname{NS}(S)$
is isotropic and $\Delta=\delta$. The result follows from Lemma 2.9. ∎

Combining the previous proposition with Lemma 2.8 provides the identity

(2.17)
${}_{\mathrm{ar}}(\operatorname{NS}(X))=\delta^{2}\cdot[B(k)]^{2}\cdot\alpha^{2}\cdot{}_{\mathrm{NT}}(J(F))\cdot\prodop\displaylimits_{v\in
Z}\leavevmode\nobreak\ {}_{\mathrm{ar}}(R_{v})\cdot(\log q)^{2}.$

For $v\in S$, we put $\delta_{v}$ and $\delta^{\prime}_{v}$ for the (local)
index and period of $X\times F_{v}$ over the local field $F_{v}$.

######  Theorem 2.11.

[Gei20, Theorem 1.1] Assume that $\operatorname{Br}(X)$ is finite. The
following equality holds:

(2.18)
$[\operatorname{Br}(X)]\alpha^{2}\delta^{2}=[\Sha(J/F)]\prodop\displaylimits_{v\in
S}\delta^{\prime}_{v}\delta_{v}.$

######  Remark 2.12.

Note that for $v\in U$, one has $\delta_{v}=1=\delta^{\prime}_{v}$ [LLR18, p.
603], [FS21, (74)] (for $\delta_{v}=1$), [Gro68, Proposition (4.1) (a)]
($\delta^{\prime}_{v}$ divides $\delta_{v}$); the basic reason is that if
$v\in U$, then $X_{v}$ has a rational divisor of degree one as $k(v)$ is
finite; this divisor lifts to a rational divisor of degree one on $X\times
F_{v}$ by smoothness of $X_{v}$. Also, $c_{v}=1$ [BLR90, Theorem 1, §9.5 p.
264]. So $c(J):=\prodop\displaylimits_{v\in S}c_{v}$ satisfies

(2.19) $c(J)=\prodop\displaylimits_{v\in Z}c_{v}.$

######  Lemma 2.13.

One has

(2.20) $c(J)\cdot Q_{2}^{*}(1)=\prodop\displaylimits_{v\in
Z}\delta_{v}\cdot\delta^{\prime}_{v}\cdot{}_{\mathrm{ar}}(R_{v}).$

###### Proof.

By a result of Flach and Siebel [FS21, Lemma 17] (using Raynaud’s theorem
[Gor79, Theorem 5.2] in [BL99]), one has

${}_{\mathrm{ar}}(R_{v})=\frac{c_{v}}{\delta_{v}\cdot\delta^{\prime}_{v}}\cdot(\log
q_{v})^{m_{v}-1}\cdot\prodop\displaylimits_{i\in G_{v}}r_{i}.$

So we find that

$\displaystyle\prodop\displaylimits_{v\in
Z}\delta_{v}\cdot\delta^{\prime}_{v}\cdot{}_{\mathrm{ar}}(R_{v})$
$\displaystyle=\prodop\displaylimits_{v\in Z}\left({c_{v}}\cdot(\log
q_{v})^{m_{v}-1}\cdot\prodop\displaylimits_{i\in
G_{v}}r_{i}\right)=\prodop\displaylimits_{v\in
Z}{c_{v}}\cdot\prodop\displaylimits_{v\in Z}\left((\log
q_{v})^{m_{v}-1}\cdot\prodop\displaylimits_{i\in G_{v}}r_{i}\right)$
$\displaystyle\overset{(\ref{rm1})}{=}c(J)\cdot\prodop\displaylimits_{v\in
Z}\leavevmode\nobreak\ \left((\log
q_{v})^{m_{v}-1}\cdot\prodop\displaylimits_{i\in
G_{v}}r_{i}\right)\overset{(\ref{rm2})}{=}c(J)\cdot Q_{2}^{*}(1).$

∎

## 3\. First proof of Theorem 1.1

###### Proof of Theorem 1.1.

By (2.17) and (2.20), we have

${}_{\mathrm{ar}}(\operatorname{NS}(X))=\frac{\alpha^{2}\,\delta^{2}}{\prodop\displaylimits_{v\in
Z}\delta_{v}\cdot\delta^{\prime}_{v}}\cdot{}_{\mathrm{NT}}(J(F))\cdot
c(J)\cdot[B(k)]^{2}\cdot Q_{2}^{*}(1)\cdot(\log q)^{2}.$

From Theorem 2.11, we have

$[\operatorname{Br}(X)]\cdot{}_{\mathrm{ar}}(\operatorname{NS}(X))=[\Sha(J/F)]\cdot{}_{\mathrm{NT}}(J(F))\cdot
c(J)\cdot[B(k)]^{2}\cdot Q_{2}^{*}(1)\cdot(\log q)^{2}.$

Further with (2.5), we obtain

$[\operatorname{Br}(X)]\cdot{}_{\mathrm{ar}}(\operatorname{NS}(X))\cdot
q^{-\alpha(X)}=[\Sha(J/F)]\cdot{}_{\mathrm{NT}}(J(F))\cdot c(J)\cdot
q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}\leavevmode\nobreak\
.\leavevmode\nobreak\ [B(k)]^{2}\cdot Q_{2}^{*}(1)\cdot q^{-\dim(B)}\cdot(\log
q)^{2}.$

On the other hand, recall (2.12)

$P_{2}^{*}(X,\frac{1}{q})=L^{*}(J,1)\cdot[B(k)]^{2}\cdot Q_{2}^{*}(1)\cdot
q^{-\dim(B)}\cdot(\log q)^{2}.$

The ratio of the previous two equalities gives

$\frac{P_{2}^{*}(X,\frac{1}{q})}{[\operatorname{Br}(X)]\cdot{}_{\mathrm{ar}}(\operatorname{NS}(X))\cdot
q^{-\alpha(X)}}=\frac{L^{*}(J,1)}{[\Sha(J/F)]\cdot{}_{\mathrm{NT}}(J(F))\cdot
c(J)\cdot q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}}.$

This equality implies Theorem 1.1. ∎

## 4\. Second proof of Theorem 1.1

We will give another more direct proof of Theorem 1.1 using Weil-étale
cohomology. We refer the reader to [Lic05, Gei04, GS20] for basics about Weil-
étale cohomology over finite fields. Throughout this section, we assume that
$\operatorname{Br}(X)$ (and hence $\Sha(J/F)$) is finite.

### 4.1. Setup

Let $C\in D^{b}(T_{\mathrm{\acute{e}t}})$ be an object of the bounded derived
category of sheaves of Abelian groups on the small étale site
$T_{\mathrm{\acute{e}t}}$. Let $D\in D^{b}(\mathrm{FDVect}_{k})$ be an object
of the bounded derived category of finite-dimensional vector spaces over $k$.
Assume that the Weil-étale cohomology $H^{\ast}_{W}(T,C)$ is finitely
generated and the cohomology sheaf
$H^{\ast}(C\otimes^{L}\mathbb{Z}/l\mathbb{Z})$ is finite in all degrees for
all prime numbers $l\nmid q$. Let $e\colon H^{i}_{W}(T,C)\to H^{i+1}_{W}(T,C)$
be the map defined by cup product with the arithmetic Frobenius $\in
H^{1}_{W}(T,\mathbb{Z})$. It defines a complex

$\cdots\stackrel{{\scriptstyle
e}}{{\longrightarrow}}H^{i}_{W}(T,C)\stackrel{{\scriptstyle
e}}{{\longrightarrow}}H^{i+1}_{W}(T,C)\stackrel{{\scriptstyle
e}}{{\longrightarrow}}\cdots$

with finite cohomology. Set
$C_{\mathbb{Q}_{l}}=R\varprojlim_{n}(C\otimes^{L}\mathbb{Z}/l^{n}\mathbb{Z})\otimes_{\mathbb{Z}_{l}}\mathbb{Q}_{l}$,
whose cohomologies are finite-dimensional vector spaces over $\mathbb{Q}_{l}$
(by the finiteness of $H^{\ast}(C\otimes^{L}\mathbb{Z}/l\mathbb{Z})$) equipped
with an action of the geometric Frobenius $\varphi$ of $k$. Define

$\displaystyle Z(C,t)$ $\displaystyle=\prodop\displaylimits_{i}\det(1-\varphi
t\,|\,H^{i}(C_{\mathbb{Q}_{l}}))^{(-1)^{i+1}},$ $\displaystyle\rho(C)$
$\displaystyle=\sumop\displaylimits_{j}(-1)^{j+1}\cdot
j\cdot\operatorname{rank}H^{j}_{W}(T,C),$ $\displaystyle\chi_{W}(C)$
$\displaystyle=\chi(H^{\ast}_{W}(T,C),e),\quad\text{and}$
$\displaystyle\chi(D)$ $\displaystyle=\sumop\displaylimits_{j}(-1)^{j}\dim
H^{j}(D).$

Assume that $Z(C,t)\in\mathbb{Q}(t)$ and is independent of $l$. Define
$Q(C,D)\in\mathbb{Q}_{>0}^{\times}\times(1-t)^{\mathbb{Z}}$ to be the leading
term of the $(1-t)$-adic expansion of the function

$\pm\frac{Z(C,t)(1-t)^{\rho(C)}}{\chi_{W}(C)q^{\chi(D)}}$

(the sign is the one that makes the coefficient positive). It is the defect of
a zeta value formula of the form

$\lim_{t\to 1}Z(C,t)(1-t)^{\rho(C)}=\pm\chi_{W}(C)q^{\chi(D)}.$

We mention $Q(C,D)$ only when $H^{\ast}_{W}(T,C)$ is finitely generated,
$H^{\ast}(C\otimes^{L}\mathbb{Z}/l\mathbb{Z})$ is finite and
$Z(C,t)\in\mathbb{Q}(t)$ is independent of $l$. These conditions are satisfied
for the cases of interest below. We have

$Q(C[1],D[1])=Q(C,D)^{-1}.$

If $(C,D)$, $(C^{\prime},D^{\prime})$ and
$(C^{\prime\prime},D^{\prime\prime})$ are pairs as above, and $C\to
C^{\prime}\to C^{\prime\prime}\to C[1]$ and $D\to D^{\prime}\to
D^{\prime\prime}\to D[1]$ are distinguished triangles, then
$Q(C^{\prime},D^{\prime})=Q(C,D)Q(C^{\prime\prime},D^{\prime\prime})$.

### 4.2. Special cases

We give two special cases of the above constructions. First, let
$\pi_{X}\colon X_{\mathrm{\acute{e}t}}\to T_{\mathrm{\acute{e}t}}$ be the
structure morphism. Let $P_{2}^{\diamond}(X,1)(1-t)^{\rho(X)^{\prime}}$ be the
leading term of the $(1-t)$-adic expansion of $P_{2}(X,t/q)$.

######  Proposition 4.1.

Let $(C,D)=(R\pi_{X,\ast}\mathbb{G}_{m}[-1],R\Gamma(X,\mathcal{O}_{X}))$. Then
$H^{\ast}(C\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z})$ is finite,
$H^{\ast}_{W}(T,C)$ is finitely generated, $Z(C,q^{-s})=\zeta(X,s+1)$ and

$Q(C,D)^{-1}=\frac{P_{2}^{\diamond}(X,1)\cdot(1-t)^{\rho(X)^{\prime}-\rho(X)}}{[\operatorname{Br}(X)]\cdot\Delta(\operatorname{NS}(X))\cdot
q^{-\alpha(X)}}.$

In particular, the statement $Q(C,D)=1$ is equivalent to Conjecture 1.2.

###### Proof.

We have $H^{\ast}_{W}(T,C)\cong H^{\ast}_{W}(X,\mathbb{G}_{m}[-1])\cong
H^{\ast}_{W}(X,\mathbb{Z}(1))$. The finiteness assumption on
$\operatorname{Br}(X)$ implies the Tate conjecture for divisors on $X$ and
hence the finite generation of $H^{\ast}_{W}(X,\mathbb{Z}(1))$ by [Gei04,
Theorems 8.4 and 9.3]. The object
$C\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z}\cong
R\pi_{X,\ast}\mathbb{Z}/l\mathbb{Z}(1)\in D^{b}(T_{\mathrm{\acute{e}t}})$ is
constructible and hence its cohomologies are finite. We have
$H^{i}(C_{\mathbb{Q}_{l}})\cong R^{i}\pi_{X,\ast}\mathbb{Q}_{l}(1)$, which is
the vector space
$H_{\mathrm{\acute{e}t}}^{i}(X\times_{k}\bar{k},\mathbb{Q}_{l}(1))$ equipped
with the natural Frobenius action. It follows that $Z(C,q^{-s})=\zeta(X,s+1)$.

We calculate $Q(C,D)^{-1}$. By (1.2), (2.3) and (2.4), the leading term of the
$(1-t)$-adic expansion of $Z(C,t)$ is

(4.1) $-\frac{[A(k)]^{2}}{P_{2}^{\diamond}(X,1)\cdot(q-1)^{2}\cdot q^{\dim
A-1}\cdot(1-t)^{\rho(X)^{\prime}}}.$

By [Gei04, Theorems 7.5 and 9.1], we have

$\chi_{W}(C)=\prodop\displaylimits_{i}[H_{W}^{i}(X,\mathbb{Z}(1))_{\mathrm{tor}}]^{(-1)^{i}}\cdot
R^{-1},$

where $R$ is the determinant of the pairing

$H_{W}^{2}(X,\mathbb{Z}(1))\times
H_{W}^{2}(X,\mathbb{Z}(1))\stackrel{{\scriptstyle\cup}}{{\longrightarrow}}H_{W}^{4}(X,\mathbb{Z}(2))\longrightarrow
H_{\mathrm{\acute{e}t}}^{4}(X\times_{k}\bar{k},\mathbb{Z}(2))\cong\mathrm{CH}^{2}(X\times_{k}\bar{k})\stackrel{{\scriptstyle\deg}}{{\longrightarrow}}\mathbb{Z}.$

We have $H_{W}^{n}(X,\mathbb{Z}(1))=0$ for $n>5$ by [Gei04, Theorem 7.3] and
for $n<1$ obviously. Also

$H_{W}^{1}(X,\mathbb{Z}(1))\cong k^{\times},\quad
H_{W}^{2}(X,\mathbb{Z}(1))\cong\operatorname{Pic}(X),\quad\text{and}\quad
H_{W}^{3}(X,\mathbb{Z}(1))_{\mathrm{tor}}\cong\operatorname{Br}(X)$

by [Gei04, Proposition 7.4 (c) and (d)]. By [Gei18, Remark 3.3], the group
$H_{W}^{i}(X,\mathbb{Z}(1))_{\mathrm{tor}}$ is Pontryagin dual to
$H_{W}^{6-i}(X,\mathbb{Z}(1))_{\mathrm{tor}}$ for any $i$. The above pairing
defining $R$ can be identified with the intersection pairing
$\operatorname{Pic}(X)\times\operatorname{Pic}(X)\to\mathbb{Z}$. Thus, with
(2.1), we have

(4.2)
$\chi_{W}(C)=\frac{[A(k)]^{2}}{[\operatorname{Br}(X)]\cdot\Delta(\operatorname{NS}(X))\cdot(q-1)^{2}}.$

Since the rank of $H_{W}^{i}(X,\mathbb{Z}(1))$ is $\rho(X)$ for $i=2,3$ and
zero otherwise by [Gei04, Proposition 7.4 (c) and (d)], we have

(4.3) $\rho(C)=\rho(X).$

Combining (1.3), (4.1), (4.2) and (4.3), we get the desired formula for
$Q(C,D)^{-1}$. ∎

Next, let $\pi_{S}\colon S_{\mathrm{\acute{e}t}}\to T_{\mathrm{\acute{e}t}}$
be the structure morphism. Let $L^{\diamond}(J,1)(1-q^{-s})^{r^{\prime}}$ be
the leading term of the $(1-q^{-s})$-adic expansion of $L(J,s+1)$. Let
$\Delta(J(F))$ be the discriminant of the pairing
$(\gamma,\kappa)\mapsto\langle\gamma,\kappa\rangle_{\mathrm{NT}}/\log q$ on
$J(F)$.

######  Proposition 4.2.

Let
$(C,D)=(R\pi_{S,\ast}\mathcal{J}[-1],R\Gamma(S,\operatorname{Lie}\,\mathcal{J}))$.
Then $H^{\ast}(C\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z})$ is finite,
$H^{\ast}_{W}(T,C)$ is finitely generated, $Z(C,q^{-s})=L(J,s+1)$ and

$Q(C,D)=\frac{L^{\diamond}(J,1)\cdot(1-t)^{r^{\prime}-r}}{[\Sha(J/F)]\cdot\Delta(J(F))\cdot
c(J)\cdot q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}}.$

In particular, the statement $Q(C,D)=1$ is equivalent to Conjecture 1.3.

###### Proof.

We have $H^{\ast}_{W}(T,C)\cong H_{W}^{\ast-1}(S,\mathcal{J})$. The finiteness
assumption of $\Sha(J/F)$ implies the finite generation of
$H_{W}^{\ast}(S,\mathcal{J})$ by [GS20, Proposition 6.4]. We have
$C\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z}\cong
R\pi_{S,\ast}(\mathcal{J}\otimes^{L}\mathbb{Z}/l\mathbb{Z})[-1]$. By the
paragraph before the proof of [GS20, Proposition 9.2] and the first displayed
equation in the proof of [GS20, Proposition 9.2], we know that
$\mathcal{J}\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z}\in
D^{b}(S_{\mathrm{\acute{e}t}})$ is constructible. Hence
$H^{\ast}(C\otimes^{\mathrm{L}}\mathbb{Z}/l\mathbb{Z})$ is finite. We also
have $H^{i}(C_{\mathbb{Q}_{l}})\cong R^{i}\pi_{S,\ast}V_{l}(\mathcal{J})$
(where $V_{l}$ denotes the $l$-adic Tate modules tensored with
$\mathbb{Q}_{l}$), which is the vector space
$H_{\mathrm{\acute{e}t}}^{i}(S\times_{k}\bar{k},V_{l}(\mathcal{J}))$ equipped
with the natural Frobenius action. Hence we have $Z(C,q^{-s})=L(J,s+1)$ by
[Sch82, Satz 1]. We have

$\chi_{W}(C)=[\Sha(J/F)]\cdot\Delta(J(F))\cdot c(J)$

by [GS20, Proposition 8.3]. By [GS20, Proposition 7.1], the rank of
$H_{W}^{i}(S,\mathcal{J})$ is $r$ for $i=0,1$ and zero otherwise. Hence
$\rho(C)=-r$. The formula for $Q(C,D)$ follows. ∎

### 4.3. Comparison

Now Theorem 1.1 follows from the following

######  Proposition 4.3.

One has

$Q(R\pi_{X,\ast}\mathbb{G}_{m}[-1],R\Gamma(X,\mathcal{O}_{X}))^{-1}=Q(R\pi_{S,\ast}\mathcal{J}[-1],R\Gamma(S,\operatorname{Lie}\,\mathcal{J})).$

###### Proof.

We have $R^{i}\pi_{\ast}\mathbb{G}_{m}=0$ over $S_{\mathrm{\acute{e}t}}$ for
all $i\geq 2$ by [Gro68, Corollaire (3.2)]. Hence we have a distinguished
triangle

$R\pi_{S,\ast}\mathbb{G}_{m}\longrightarrow
R\pi_{X,\ast}\mathbb{G}_{m}\longrightarrow
R\pi_{S,\ast}\operatorname{Pic}_{X/S}[-1]\longrightarrow
R\pi_{S,\ast}\mathbb{G}_{m}[1]$

in $D(T_{\mathrm{\acute{e}t}})$.777 Here
$\operatorname{Pic}_{X/S}=R^{1}\pi_{\ast}\mathbb{G}_{m}$ is only an étale
sheaf. The fppf sheaf denoted by the same symbol is not an algebraic space in
general.  Similarly, we have a distinguished triangle

$R\Gamma(S,\mathcal{O}_{S})\longrightarrow
R\Gamma(X,\mathcal{O}_{X})\longrightarrow
R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X})[-1]\longrightarrow
R\Gamma(S,\mathcal{O}_{S})[1].$

We have $Q(R\pi_{S,\ast}\mathbb{G}_{m}[-1],R\Gamma(S,\mathcal{O}_{S}))=1$ by
the class number formula ([Gei04, Theorems 9.1 and 9.3], or [Lic05, Theorems
5.4 and 7.4] and the functional equation). Therefore

(4.4)
$Q(R\pi_{X,\ast}\mathbb{G}_{m}[-1],R\Gamma(X,\mathcal{O}_{X}))^{-1}=Q(R\pi_{S,\ast}\operatorname{Pic}_{X/S}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X})).$

For a closed point $v\in S$, let
$\iota_{v}\colon\operatorname{Spec}k(v)\hookrightarrow S$ be the inclusion.
For any $i\in G_{v}$, let $k(v)_{i}$ be the algebraic closure of $k(v)$ in the
function field of i. Let $\iota_{v,i}\colon\operatorname{Spec}k(v)_{i}\to S$
be the natural morphism. Set

$E=\bigoplusop\displaylimits_{v\in Z}\frac{\bigoplusop\displaylimits_{i\in
G_{v}}\iota_{v,i,\ast}\mathbb{Z}}{\iota_{v,\ast}\mathbb{Z}}.$

Let $j\colon\operatorname{Spec}F\hookrightarrow S$ be the inclusion. Then we
have a natural exact sequence

$0\longrightarrow E\longrightarrow\operatorname{Pic}_{X/S}\longrightarrow
j_{\ast}\operatorname{Pic}_{X_{0}/F}\longrightarrow 0$

over $S_{\mathrm{\acute{e}t}}$ by [Gro68, Equations (4.10 bis) and (4.21)]
(where the assumption [Gro68, Equation (4.13)] is satisfied since $k(v)$ is
finite and hence perfect for all closed $v\in S$). Therefore we have a
distinguished triangle

$R\pi_{S,\ast}E\longrightarrow
R\pi_{S,\ast}\operatorname{Pic}_{X/S}\longrightarrow
R\pi_{S,\ast}j_{\ast}\operatorname{Pic}_{X_{0}/F}\longrightarrow
R\pi_{S,\ast}E[1].$

Since $E$ is skyscraper, we have $Q(R\pi_{S,\ast}E,0)=1$ by [GS21, Theorem
3.1] (Step 3 of the proof is sufficient). Therefore

(4.5)
$Q(R\pi_{S,\ast}\operatorname{Pic}_{X/S}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X}))=Q(R\pi_{S,\ast}j_{\ast}\operatorname{Pic}_{X_{0}/F}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X})).$

Applying $j_{\ast}$ to the exact sequence

$0\longrightarrow
J\longrightarrow\operatorname{Pic}_{X_{0}/F}\longrightarrow\mathbb{Z}\longrightarrow
0$

over $\operatorname{Spec}F_{\mathrm{\acute{e}t}}$, we obtain an exact sequence

$0\longrightarrow\mathcal{J}\longrightarrow
j_{\ast}\operatorname{Pic}_{X_{0}/F}\longrightarrow\mathbb{Z}$

over $S_{\mathrm{\acute{e}t}}$. Let $I$ be the image of the last morphism, so
that we have an exact sequence

$0\longrightarrow\mathcal{J}\longrightarrow
j_{\ast}\operatorname{Pic}_{X_{0}/F}\longrightarrow I\longrightarrow 0.$

Then we have distinguished triangles

$\displaystyle R\pi_{S,\ast}\mathcal{J}\longrightarrow
R\pi_{S,\ast}j_{\ast}\operatorname{Pic}_{X_{0}/F}\longrightarrow
R\pi_{S,\ast}I\longrightarrow R\pi_{S,\ast}\mathcal{J}[1],\quad\text{and}$
$\displaystyle R\pi_{S,\ast}I\longrightarrow
R\pi_{S,\ast}\mathbb{Z}\longrightarrow
R\pi_{S,\ast}(\mathbb{Z}/I)\longrightarrow R\pi_{S,\ast}I[1].$

We have $Q(R\pi_{S,\ast}\mathbb{Z},0)=1$ again by the class number formula
([Gei04, Theorems 9.1 and 9.2] or [Lic05, Theorem 7.4]). Since $\mathbb{Z}/I$
is skyscraper with finite stalks, we have $Q(R\pi_{S,\ast}(\mathbb{Z}/I),0)=1$
by [GS21, Theorem 3.1] (Step 2 of the proof is sufficient). Therefore

(4.6)
$Q(R\pi_{S,\ast}j_{\ast}\operatorname{Pic}_{X_{0}/F}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X}))=Q(R\pi_{S,\ast}\mathcal{J}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X})).$

The complexes $R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X})$ and
$R\Gamma(S,\operatorname{Lie}\,\mathcal{J})$ have the same Euler
characteristic by (2.15). Hence

(4.7)
$Q(R\pi_{S,\ast}\mathcal{J}[-1],R\Gamma(S,R^{1}\pi_{\ast}\mathcal{O}_{X}))=Q(R\pi_{S,\ast}\mathcal{J}[-1],R\Gamma(S,\operatorname{Lie}\,\mathcal{J})).$

Combining (4.4)—(4.7), we get the desired equality. ∎

### 4.4. A new proof of Geisser’s formula

The above proposition, combined with the results of the previous sections,
also gives a new proof of Theorem 2.11 as follows.

###### Proof of Theorem 2.11.

By Proposition 4.3, we have

$\frac{P_{2}^{\diamond}(X,1)}{[\operatorname{Br}(X)]\cdot\Delta(\operatorname{NS}(X))\cdot
q^{-\alpha(X)}}=\frac{L^{\diamond}(J,1)}{[\Sha(J/F)]\cdot\Delta(J(F))\cdot
c(J)\cdot q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}}.$

By (2.12), we have

$P_{2}^{\diamond}(X,1)=L^{\diamond}(J,1)\cdot q^{-\dim B}\cdot[B(k)]^{2}\cdot
Q_{2}^{\diamond}(1),$

where $Q_{2}^{\diamond}(1)$ is the leading coefficient of the
$(1-q^{-s})$-adic expansion of $Q_{2}(s+1)$. By (2.17) and (2.20), we have

$\Delta(\operatorname{NS}(X))=\frac{\alpha^{2}\delta^{2}}{\prodop\displaylimits_{v\in
Z}\delta_{v}^{\prime}\delta_{v}}\cdot\Delta(J(F))\cdot
c(J)\cdot[B(k)]^{2}\cdot Q_{2}^{\diamond}(1).$

By (2.5), we have

$q^{-\alpha(X)}=q^{\chi(S,\operatorname{Lie}\,\mathcal{J})}\cdot q^{-\dim B}.$

Taking a suitable alternating product of these four equalities, we obtain
(2.18). ∎

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