# Hasse principle for intersections of two quadrics via Kummer surfaces

Adam Morgan and Alexei Skorobogatov

###### Abstract

We prove new cases of the Hasse principle for Kummer surfaces constructed from
2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant
Tate–Shafarevich groups. Under the same assumption, we deduce the Hasse
principle for quartic del Pezzo surfaces with trivial Brauer group and
irreducible or completely split characteristic polynomial, hence the Hasse
principle for smooth complete intersections of two quadrics in the projective
space of dimension at least 5.

###### Contents

  1. 1 Preliminaries
    1. 1.1 Algebraic notation
    2. 1.2 Transitive subgroups of ${\rm S}_{5}$
    3. 1.3 Quartic del Pezzo and Kummer surfaces
    4. 1.4 Severi–Brauer fibrations
    5. 1.5 Number fields
  2. 2 Projective geometry of pencils of quadrics in ${\mathbb{P}}^{4}_{k}$
    1. 2.1 Canonical equations of quartic del Pezzo surfaces
    2. 2.2 The associated Kummer fibration
    3. 2.3 The associated Severi–Brauer fibration
  3. 3 From Kummer to del Pezzo
  4. 4 Hasse principle for Kummer surfaces
    1. 4.1 $2$-Selmer group in quadratic twist families
    2. 4.2 The Galois group ${\rm S}_{5}$, ${\rm A}_{5}$, or ${\rm F}_{20}$
    3. 4.3 The Galois group ${\rm D}_{10}$

## Introduction

The study of the Hasse principle for geometrically rational varieties has a
long history. Fundamental contributions to this subject were made by Manin who
introduced the Brauer–Manin obstruction, and by Colliot-Thélène, Sansuc and
Swinnerton-Dyer who pioneered the use of descent and fibration methods.

By the Castelnuovo–Manin–Iskovskikh classification, smooth and projective
geometrically rational surfaces that are minimal in the sense of birational
geometry (every birational morphism onto a smooth surface must be an
isomorphism) fall into two families: conic bundles and del Pezzo surfaces
[Isk79, Theorem 1]. The existence of a conic fibration greatly simplifies the
study of rational points, essentially because universal torsors over conic
bundle surfaces have a simple description as certain complete intersections of
quadrics. As a result, when the number of singular geometric fibres of the
conic fibration is small, a combination of descent and fibration methods can
be used to prove that the Brauer–Manin obstruction is the only obstruction to
the Hasse principle and weak approximation (see the survey [CT92] as well as
[Sko01, §4.4 and Chapter 7]). By contrast, no unconditional results of this
kind are known for del Pezzo surfaces without a conic fibration, if we exclude
the well understood case of del Pezzo surfaces of degree larger than $4$. The
most interesting open cases are smooth cubic surfaces in
${\mathbb{P}}^{3}_{k}$ and quartic del Pezzo surfaces, which are smooth
intersections of two quadrics in ${\mathbb{P}}^{4}_{k}$. (Universal torsors
over arbitrary del Pezzo surfaces were described in [SS07], but this
description looks too difficult to work with directly.)

Attempts were made to prove the Hasse principle conditionally on the
conjecture of Tate and Shafarevich about finiteness of the Tate–Shafarevich
group of abelian varieties. This is known for many elliptic curves and for
some higher-dimensional abelian varieties. A conceptual reason comes from a
theorem of Manin which says that the Brauer–Manin obstruction is the only
obstruction to the Hasse principle on torsors for an abelian variety provided
its Tate–Shafarevich group is finite, see [Sko01, Theorem 6.2.3]. Thus it
seems reasonable to build on this conjecture when dealing with surfaces that
do not have a pencil of rational curves.

For diagonal cubic surfaces with coefficients satisfying certain divisibility
properties, Swinnerton-Dyer proved the Hasse principle conditionally on
finiteness of the Tate–Shafarevich group of elliptic curves [SD01]. He
crucially used the geometric fact that any diagonal cubic surface is
birational to a quotient of a torsor for an abelian surface, namely the
product of two diagonal cubic curves. Extending this approach, Swinnerton-
Dyer’s method was developed (and later much simplified) to prove the Hasse
principle for many Kummer surfaces constructed from 2-coverings of abelian
surfaces, assuming finiteness of the relevant Tate–Shafarevich groups, see
[SSD05, HS16, Har19, Mor23b].

In his thesis, Wittenberg developed an earlier version of this method
applicable to surfaces with a pencil of genus 1 curves [CSS98a] to prove the
Hasse principle for many classes of quartic del Pezzo surfaces, including the
most general case [Wit07, Théorème 3.2]. In addition to the Tate–Shafarevich
conjecture he had to assume Schinzel’s Hypothesis (H), a notoriously difficult
conjecture on prime values of polynomials, deemed to be inaccessible in the
current state of analytic number theory.

In this paper we return to the philosophy of [SD01]. We study the Hasse
principle on quartic del Pezzo surfaces using their representation as
quotients of Kummer surfaces which are themselves quotients of torsors for the
Jacobians of genus 2 curves.

For a quartic del Pezzo surface $X$ represented as the zero set of two
quadratic forms in 5 variables $\Phi_{1}=\Phi_{2}=0$, one calls
$\det(\Phi_{1}-t\Phi_{2})\in k[t]$ the characteristic polynomial of $X$. Our
main result is the following theorem.

Theorem A Let $k$ be a number field. Assume that for every genus $2$ curve
over every finite extension of $k$ the $2$-primary torsion subgroup of the
Tate–Shafarevich group of its Jacobian is finite. Then the quartic del Pezzo
surfaces $X$ over $k$ whose characteristic polynomial is irreducible, or whose
characteristic polynomial is completely split and the natural map
${\rm{Br}}(k)\to{\rm{Br}}(X)$ is surjective, satisfy the Hasse principle.

To prove Theorem A, we use a new geometric idea that allows us to reduce the
study of rational points on quartic del Pezzo surfaces to Kummer surfaces
attached to 2-coverings of Jacobians of genus 2 curves, by exploiting a
relation between these surfaces spelled out in [Sko10]. Namely, we show that
the projectivised tangent bundle of a quartic del Pezzo surface $X$ is
birationally equivalent to a family of Kummer surfaces over
${\mathbb{P}}^{1}_{k}$, each of which is a double cover of $X$. Although
classical in spirit, this construction appears to be new. Applying the
fibration theorem for zero-cycles of Harpaz and Wittenberg [HW16, Theorem 6.2]
to this family, we prove the main unconditional result of this paper, Theorem
3.4. It says that the Hasse principle with Brauer–Manin obstruction for $X$
over $k$ follows from the Hasse principle with algebraic Brauer–Manin
obstruction for Kummer surfaces defined over finite extensions of $k$ of odd
degree and satisfying certain local conditions. In the proof, one obtains a
closed point of odd degree on $X$, but since $X$ is an intersection of two
quadrics, a theorem of Amer–Brumer implies that $X$ has a $k$-point. The
required results on the existence of rational points on Kummer surfaces are
proved conditionally on finiteness of the Tate–Shafarevich groups of Jacobians
of genus 2 curves. In the completely split case we use that the algebraic
Brauer–Manin obstruction is the only obstruction to the Hasse principle on the
relevant Kummer surfaces, as proved by Harpaz in [Har19, Theorem 1.3]. In the
irreducible case, we establish the Hasse principle for the relevant Kummer
surfaces in Theorem 3.7, which we prove by extending the technique of [HS16,
Har19, Mor23b] using work of the first named author [Mor19, Mor23a].

Although in this paper we only deal with quartic del Pezzo surfaces with
trivial Brauer group, our method is applicable to arbitrary quartic del Pezzo
surfaces. Instances where the Brauer group is non-trivial will be the subject
of a forthcoming work of Julian Lyczak and the first named author.

The following statement is a consequence of Theorem A, see [Wit07, §3.5.2] for
this implication.

Theorem B Let $k$ be a number field. Assume that $\textcyr{Sh}({\rm
Jac}(C))[2^{\infty}]$ is finite for every genus $2$ curve $C$ over every
finite extension of $k$. Then the smooth complete intersections of two
quadrics in ${\mathbb{P}}^{n}_{k}$, $n\geq 5$, satisfy the Hasse principle.

Previously, this was known under the assumption of finiteness of the
Tate–Shafarevich groups of elliptic curves over number fields and Schinzel’s
Hypothesis (H), see [Wit07, Théorème 3.3]. The best unconditional result in
this direction is due to Heath-Brown [HB18] who proved that smooth complete
intersections of two quadrics in ${\mathbb{P}}^{n}_{k}$ satisfy the Hasse
principle for $n\geq 7$. See [CT24] for a different proof.

We briefly outline the structure of the paper. Section 1 contains various
preliminary results. In Section 2 we introduce the main geometric construction
of the paper, namely a stable birational equivalence between a quartic del
Pezzo surface $X$ and a threefold $Z$ fibred into Kummer surfaces over
${\mathbb{P}}^{1}_{k}$. Using a natural realisation of
$Z/{\mathbb{P}}^{1}_{k}$ inside a fibration into 3-dimensional Severi–Brauer
varieties, we show the triviality of the vertical Brauer group of
$Z/{\mathbb{P}}^{1}_{k}$. This result is not used in the proof of Theorem A,
but it is interesting in its own right. In Section 3 we deduce the Hasse
principle with Brauer–Manin obstruction for $X$ from the Hasse principle with
algebraic Brauer–Manin obstruction of a special type for certain closed fibres
of $Z/{\mathbb{P}}^{1}_{k}$, see Theorem 3.4 for a precise statement. This
reduces Theorem A (=Theorem 3.8) to the relevant cases of the Hasse principle
for Kummer surfaces, see Theorem 3.7. Section 4 contains the proof of Theorem
3.7.

The work on this paper started at the BIRS–CMI workshop “New directions in
rational points” at the Chennai Mathematical Institute. The authors are very
grateful to Julian Lyczak for helpful discussions and for suggesting to use
the relation between quartic del Pezzo and Kummer surfaces to study rational
points on quartic del Pezzo surfaces. We thank Olivier Wittenberg for
explaining to us the relevant results of [HW16].

Throughout this work, the first named author was supported by the Engineering
and Physical Sciences Research Council (EPSRC) grant EP/V006541/1 ‘Selmer
groups, Arithmetic Statistics and Parity Conjectures’.

##  1 Preliminaries

###  1.1 Algebraic notation

Let $k$ be a field of characteristic zero with algebraic closure $\bar{k}$ and
absolute Galois group $\Gamma={\rm{Gal}}(\bar{k}/k)$.

For a variety $X$ over $k$ we write ${\rm{Br}}(X)$ for the cohomological
Brauer group ${\rm H}^{2}_{\rm{\acute{e}t}}(X,{\mathbb{G}}_{m})$. We use the
standard notation ${\rm{Br}}_{0}(X)$ for the image of the natural map
${\rm{Br}}(k)\to{\rm{Br}}(X)$ and ${\rm{Br}}_{1}(X)$ for the kernel of the
natural map ${\rm{Br}}(X)\to{\rm{Br}}(X_{\bar{k}})$.

Let ${\rm S}_{5}$ be the group of permutations of $\\{1,2,3,4,5\\}$. Define
the ${\rm S}_{5}$-module $G$ as the zero-sum submodule of the permutation
module $\mathbb{F}_{2}[{\rm S}_{5}/{\rm S}_{4}]\simeq({\mathbb{F}}_{2})^{5}$.
Since $5$ is odd, we have a direct sum decomposition of ${\rm S}_{5}$-modules
$\mathbb{F}_{2}[{\rm S}_{5}/{\rm S}_{4}]\cong G\oplus{\mathbb{F}}_{2}$. It is
clear that $G$ is a faithful ${\rm S}_{5}$-module.

In all of this paper, $P(t)$ is a monic separable polynomial in $k[t]$ of
degree $5$. It gives rise to an étale $k$-algebra $A=k[t]/(P(t))$ and an
embedding of the $k$-scheme $\Sigma={\rm{Spec}}(A)$ into the affine line
${\mathbb{A}}^{1}_{k}$. We have $A=\oplus_{\sigma\in\Sigma}k_{\sigma}$, where
$k_{\sigma}$ is the residue field of a closed point $\sigma\in\Sigma$.

Let ${\rm{Gal}}(P)$ be the Galois group of $P(t)$, and let
$\varphi:\Gamma\to{\rm{Gal}}(P)$ be the natural surjection. The action of
${\rm{Gal}}(P)$ on the roots of $P(t)$ gives rise to an embedding of
${\rm{Gal}}(P)$ into ${\rm S}_{5}$ (defined up to conjugation). The Galois
group $\Gamma$ acts on $G$ via $\varphi$, making $G$ a finite commutative
group $k$-scheme. Namely, we have

$G\cong R_{A/k}^{1}(\mu_{2}):={\rm{Ker}}[R_{A/k}(\mu_{2})\to\mu_{2}],$

where $R_{A/k}$ is the Weil restriction of scalars and the map is induced by
the norm map $A\to k$. Since ${\rm{dim}}_{k}(A)=5$ is odd, the exact sequence

$1\to G\to R_{A/k}(\mu_{2})\to\mu_{2}\to 1$

is split, giving an isomorphism

${\rm H}^{1}(k,G)\cong(A^{\times}/A^{\times
2})_{1}=\big{(}\prod_{\sigma\in\Sigma}k_{\sigma}^{\times}/k_{\sigma}^{\times
2}\big{)}_{1},$

where the subscript 1 denotes the subgroup of elements of norm $1\in
k^{\times}/k^{\times 2}$. Note that there is also an isomorphism $G\cong
R_{A/k}(\mu_{2})/\mu_{2}$, which induces an isomorphism ${\rm H}^{1}(k,G)\cong
A^{\times}/(k^{\times}A^{\times 2})$.

The fundamental datum in this paper is a pair $(P(t),\delta)$, where
$\delta\in{\rm H}^{1}(k,G)$. We have $\delta=(\delta_{\sigma})$, where
$\delta_{\sigma}\in k_{\sigma}^{\times}/k_{\sigma}^{\times 2}$ are such that

$\prod_{\sigma\in\Sigma}{\rm N}_{k_{\sigma}/k}(\delta_{\sigma})=1\in
k^{\times}/k^{\times 2}.$

We choose a lifting $\delta^{\prime}\in A^{\times}$ of $\delta$ such that
${\rm N}_{A/k}(\delta^{\prime})\in k^{\times 2}$ and define
$\Delta:={\rm{Spec}}(A[x]/(x^{2}-\delta^{\prime})).$ It is clear that the
isomorphism class of $\Delta$ does not depend on the lifting
$\delta^{\prime}\in A^{\times}$ of $\delta$. The obvious morphism of
$k$-schemes $\Delta\to\Sigma$ gives rise to a $\Gamma$-equivariant 2-to-1 map
$\Delta(\bar{k})\to\Sigma(\bar{k})$. We choose a set-theoretic section
$s:\Sigma(\bar{k})\to\Delta(\bar{k})$ to this map. This identifies the group
of permutations of the 10-element set $\Delta(\bar{k})$ that respect the map
$\Delta(\bar{k})\to\Sigma(\bar{k})$ with the semidirect product
$({\mathbb{Z}}/2)^{5}\rtimes{\rm S}_{5}$. The group $k$-scheme
$R_{A/k}(\mu_{2})$ acts faithfully on $\Delta$ preserving the fibres of
$\Delta\to\Sigma$, thus $R_{A/k}(\mu_{2})(\bar{k})$ is naturally identified
with $({\mathbb{Z}}/2)^{5}$ in $({\mathbb{Z}}/2)^{5}\rtimes{\rm S}_{5}$. It
follows that $G(\bar{k})=R_{A/k}^{1}(\mu_{2})(\bar{k})$ is identified with the
zero sum subgroup $({\mathbb{Z}}/2)^{4}\subset({\mathbb{Z}}/2)^{5}$.

For each $r\in\Sigma(\bar{k})$, define $\tilde{\delta}_{r}:\Gamma\to\\{\pm
1\\}$ by setting $\tilde{\delta}_{r}(g)=1$ if $gs(r)=s(gr)$, and
$\tilde{\delta}_{r}(g)=-1$ if $gs(r)\neq s(gr)$.

###### Lemma 1.1

The function
$\tilde{\delta}=(\tilde{\delta}_{r}):\Gamma\to({\mathbb{Z}}/2)^{5}$ is a
$1$-cocycle $\Gamma\to G$ with class $\delta$.

Proof. We have canonical isomorphisms of $\Gamma$-sets
$\Sigma(\bar{k})={\rm{Hom}}_{k-{\rm alg}}(A,\bar{k})$ and
$\Delta(\bar{k})={\rm{Hom}}_{k-{\rm alg}}(A^{\prime},\bar{k})$, where
$A^{\prime}=A[x]/(x^{2}-\delta^{\prime})$. Let $r\in\Sigma(\bar{k})$. The
choice of a section $s:\Sigma(\bar{k})\to\Delta(\bar{k})$ is equivalent to the
choice of a lifting of each homomorphism of $k$-algebras $r\colon A\to\bar{k}$
to a homomorphism of $k$-algebras $\tilde{r}\colon A^{\prime}\to\bar{k}$, that
is, to the choice of a square root of $r(\delta^{\prime})$ in $\bar{k}$. By
the definition of $\tilde{\delta}_{r}$ we have

$\tilde{\delta}_{r}(g)=g(\sqrt{r(\delta^{\prime})})/\sqrt{gr(\delta^{\prime})}.$
(1)

Since ${\rm N}_{A/k}(\delta^{\prime})\in k^{\times 2}$, we have
$\prod_{r\in\Sigma(\bar{k})}\tilde{\delta}_{r}=1$. For each closed point
$\sigma\in\Sigma$, formula (1) with $r\in\sigma$ describes the character
${\rm{Gal}}(\bar{k}/k_{\sigma})\to\\{\pm 1\\}$ of the extension
$k_{\sigma}(\sqrt{\delta^{\prime}_{\sigma}})/k_{\sigma}$. Thus
$\tilde{\delta}=(\tilde{\delta}_{r})\colon\Gamma\to G(\bar{k})$ is a 1-cocycle
with class $\delta=(\delta_{\sigma})$. $\Box$

We fix the lifting $\delta^{\prime}$, the set-theoretic section $s$ and hence
the cocycle $\tilde{\delta}$, once and for all.

Let us recall a useful construction from [HS16, Section 3]. Suppose that we
are given a finite subset $T=\\{\alpha_{1},...,\alpha_{t}\\}\subset{\rm
H}^{1}(k,G)$. Let us represent each $\alpha_{i}$ by a $1$-cocycle
$\tilde{\alpha}_{i}:\Gamma\to G$. Then the map $\Gamma\to
G^{T}\rtimes\textup{Gal}(P)$ given by

$g\mapsto\big{(}\tilde{\alpha}_{1}(g),\ldots,\tilde{\alpha}_{t}(g),\varphi(g)\big{)}$

is a homomorphism. Denote by $k_{T}$ the fixed field of the kernel of this
homomorphism, so that we have an injection
$\varphi_{T}\colon\textup{Gal}(k_{T}/k)\hookrightarrow
G^{T}\rtimes{\rm{Gal}}(P)$. Choosing different cocycle representatives for
$\alpha_{1},...,\alpha_{t}$ conjugates $\varphi_{T}$ by an element of $G^{T}$.
The extension $k_{T}/k$ contains the splitting field of $P(t)$ and is the
smallest Galois extension of $k$ through which each of the cocycles
$\tilde{\alpha}_{1},\ldots,\tilde{\alpha}_{t}$ factors.

From Lemma 1.1 we see that that the natural action of $\Gamma$ on
$\Delta(\bar{k})$ factors through

$\varphi_{\delta}\colon{\rm{Gal}}(k_{\delta}/k)\hookrightarrow
G(\bar{k})\rtimes{\rm{Gal}}(P)\hookrightarrow({\mathbb{Z}}/2)^{5}\rtimes{\rm
S}_{5}.$

###### Lemma 1.2

The subgroup ${\rm{Gal}}(k_{\delta}/k)\subset({\mathbb{Z}}/2)^{4}\rtimes{\rm
S}_{5}$ is conjugate to a subgroup of ${\rm S}_{5}$ in
$({\mathbb{Z}}/2)^{5}\rtimes{\rm S}_{5}$ if and only if $\delta=0$.

Proof. It is clear that $\delta=0$ if and only if the morphism
$\Delta\to\Sigma$ has a section. The stabiliser of a subset of
$\Delta(\bar{k})$ that maps bijectively onto $\Sigma(\bar{k})$ is $h{\rm
S}_{5}h^{-1}$ for some $h\in({\mathbb{Z}}/2)^{5}$. $\Box$

Let us denote by $\Lambda_{\delta}$ the $k$-torsor for $G$ defined by the
cocycle $\tilde{\delta}$. The isomorphism class of $\Lambda_{\delta}$ depends
only on $\delta=[\Lambda_{\delta}]\in{\rm H}^{1}(k,G)$. As above, the actions
of $\Gamma$ on $\Lambda_{\delta}(\bar{k})$ and on $\Delta(\bar{k})$ determine
each other. In both cases, $\Gamma$ acts through the faithfully acting finite
group ${\rm{Gal}}(k_{\delta}/k)$. The relation between $\Lambda_{\delta}$ and
$\Delta$ can also be described as follows: the image of $\delta$ in ${\rm
H}^{1}(k,R_{A/k}(\mu_{2}))$ induced by $G\hookrightarrow R_{A/k}(\mu_{2})$
corresponds to $[\Delta]\in{\rm H}^{1}(A,\mu_{2})$ under the isomorphism ${\rm
H}^{1}(A,\mu_{2})\cong{\rm H}^{1}(k,R_{A/k}(\mu_{2}))$ given by the Weil
restriction.

Let $({\mathbb{F}}_{2})^{A,\delta}$ be the vector space over
${\mathbb{F}}_{2}$ with a basis whose vectors bijectively correspond to the
closed points $\sigma\in\Sigma$ such that $\delta_{\sigma}\neq 1\in
k_{\sigma}^{\times}/k_{\sigma}^{\times 2}$. Let
${\mathbb{F}}_{2}\subset({\mathbb{F}}_{2})^{A,\delta}$ be the subspace spanned
by the vector $(1,\ldots,1)$. Define ${\rm B}_{A,\delta}$ as the subgroup of
$({\mathbb{F}}_{2})^{A,\delta}/{\mathbb{F}}_{2}$ represented by the vectors
$\gamma=(\gamma_{\sigma})\in({\mathbb{F}}_{2})^{A,\delta}$ such that

$\prod_{\sigma\in\Sigma}{\rm
N}_{k_{\sigma}/k}(\delta_{\sigma})^{\gamma_{\sigma}}=1\in k^{\times}/k^{\times
2}.$

If $\delta=0$, we define ${\rm B}_{A,\delta}=0$.

###  1.2 Transitive subgroups of ${\rm S}_{5}$

Recall that a transitive subgroup of ${\rm S}_{5}$ is isomorphic to the cyclic
group $\mathbb{Z}/5$, the dihedral group ${\rm D}_{10}$, the Frobenius group
${\rm F}_{20}$, the alternating group ${\rm A}_{5}$, or the full symmetric
group ${\rm S}_{5}$. Thus it contains a $5$-cycle, and contains a double
transposition if it is not cyclic.

###### Lemma 1.3

Let ${\cal G}$ be a transitive subgroup of ${\rm S}_{5}$. Then $G$ is a simple
and faithful ${\cal G}$-module, and ${\rm H}^{1}({\cal G},G)=0$. Moreover,
$\textup{End}_{\cal G}(G)\cong\mathbb{F}_{2^{r}}$, where $r=4$ if ${\cal
G}\cong{\mathbb{Z}}/5$, $r=2$ if ${\cal G}\cong{\rm D}_{10}$, and $r=1$
otherwise.

Proof. Let ${\mathcal{H}}\subset{\cal G}$ denote the stabiliser of an element
of $\\{1,2,3,4,5\\}$. Then we have

${\rm H}^{1}({\cal G},G)\oplus\textup{Hom}({\cal G},\mathbb{F}_{2})\cong{\rm
H}^{1}({\cal G},\mathbb{F}_{2}[{\rm S}_{5}/{\rm
S}_{4}])\cong\textup{Hom}\big{(}{\mathcal{H}},\mathbb{F}_{2}\big{)},$

where the final equality follows from Shapiro’s lemma. Considering each
transitive subgroup of ${\rm S}_{5}$ in turn, we check from this that ${\rm
H}^{1}({\cal G},G)=0$.

Let $\tau\in{\rm S}_{5}$ be a $5$-cycle. The action of $\tau$ on $G$ has
minimal polynomial $m_{\tau}(t)=(t^{5}-1)/(t-1)$ which is irreducible of
degree $4$ over $\mathbb{F}_{2}$. Thus

${\mathbb{F}}_{2}[\tau]\cong{\mathbb{F}}_{2}[t]/(t^{4}+t^{3}+t^{2}+t+1)\cong{\mathbb{F}}_{16},$

so that $G$ is an ${\mathbb{F}}_{16}$-vector space of dimension 1, hence is
simple as a ${\mathbb{Z}}/5$-module where ${\mathbb{Z}}/5$ is the cyclic group
generated by $\tau$. Replacing ${\cal G}$ by a conjugate subgroup in ${\rm
S}_{5}$, we can assume that $\tau\in{\cal G}$. Thus $G$ is a simple ${\cal
G}$-module.

The ${\mathbb{F}}_{2}$-algebra $\textup{End}_{\cal G}(G)$ is the centraliser
of ${\cal G}$ in ${\rm{End}}_{{\mathbb{Z}}/5}(G)$, so it is the invariant
subfield ${\mathbb{F}}_{2}(\tau)^{\cal
G}\cong{\mathbb{F}}_{2^{r}}\subset{\mathbb{F}}_{16}$. Thus $r=4$ when ${\cal
G}\cong\mathbb{Z}/5$. In other cases, we have $\textup{End}_{\cal G}(G)$ is
$\mathbb{F}_{2}$ or $\mathbb{F}_{2}(\tau+\tau^{-1})$. The latter case occurs
precisely when ${\cal G}\cong{\rm D}_{10}$. $\Box$

###### Lemma 1.4

Let $T$ be a finite subset of ${\rm H}^{1}(k,G)$. If $P(t)$ is irreducible,
then $\varphi_{T}$ is an isomorphism if and only if the elements of $T$ are
$\mathbb{F}_{q}$-linearly independent, where
${\mathbb{F}}_{q}={\rm{End}}_{{\rm{Gal}}(P)}(G)$. When this is the case, every
abelian subextension of $k_{T}/k$ is contained in the splitting field of
$P(t)$.

Proof. By Lemma 1.3 we have ${\rm H}^{1}(\textup{Gal}(P),G)=0$. Given this,
the first part is [HS16, Proposition 3.6] and the second part is [HS16,
Corollary 3.9]. $\Box$

###  1.3 Quartic del Pezzo and Kummer surfaces

To a pair $(P(t),\delta)$ as above one associates a quartic del Pezzo surface,
as follows. Let $X_{0}$ be the blowing-up of ${\mathbb{P}}^{2}_{k}$ in the
image of $\Sigma\subset{\mathbb{A}}^{1}_{k}\subset{\mathbb{P}}^{1}_{k}$ in
${\mathbb{P}}^{2}_{k}$ under the Veronese embedding
${\mathbb{P}}^{1}_{k}\hookrightarrow{\mathbb{P}}^{2}_{k}$. Thus $X_{0}$ is a
del Pezzo surface of degree 4 with a $k$-line (the strict transform of the
image of ${\mathbb{P}}^{1}_{k}$). The group $k$-scheme $G$ acts on $X_{0}$ by
automorphisms. Let $X$ be the twisting of $X_{0}$ by the 1-cocycle
$\tilde{\delta}$. By [Sko10, Thm. 2.3] associating $X$ to $(P(t),\delta)$
gives a bijection between isomorphism classes of quartic del Pezzo surfaces
over $k$ and pairs $(\Sigma,\delta)$, where $\Sigma$ is a reduced
0-dimensional closed subscheme of ${\mathbb{P}}^{1}_{k}$ of degree 5
considered up to the action of ${\rm{PGL}}(2,k)$, and $\delta\in{\rm
H}^{1}(k,G)$, where $G=R^{1}_{A/k}(\mu_{2})$ with $A={\rm
H}^{0}(\Sigma,O_{\Sigma})$. Under this correspondence, the action of $G$ on
the $k$-scheme of 16 lines of $X_{\bar{k}}$ makes it a $k$-torsor for $G$
isomorphic to $\Lambda_{\delta}$. The action of $G$ on the $k$-scheme of 10
pencils of conics of $X_{\bar{k}}$ identifies it with the $G$-scheme $\Delta$.
(See the end of Section 2.2 for more details.)

Now let $b\in k$ be such that $P(b)\neq 0$. We denote by $C_{b}$ the smooth
projective hyperelliptic curve of genus 2 with affine equation
$y^{2}=(t-b)P(t)$. Let $J_{b}$ be the Jacobian of $C_{b}$. We denote by
$d_{b}=P(b)^{2}\,{\rm discr\,}(P(t))$ the discriminant of $(t-b)P(t)$.

A crucial property for us in this paper is that for every $b\in k$, $P(b)\neq
0$, the group $k$-scheme $G$ is canonically isomorphic to $J_{b}[2]$.
Moreover, the Weil pairing $G\times G\to\mu_{2}$ does not depend on $b$.

For an extension $F=k(\sqrt{a})/k$ (we allow $F=k$), let $J_{b}^{F}$ be the
quadratic twist of $J_{b}$ by $F/k$. It is the Jacobian of the hyperelliptic
curve $C_{b}^{F}$ with affine equation $y^{2}=a(t-b)P(t)$.

Let $J_{b,\delta}$ denote the $2$-covering of $J_{b}$ given by a cocycle with
class $\delta\in{\rm H}^{1}(k,G)$. The natural morphism $J_{b,\delta}\to
J_{b}$ is a torsor for $G$ whose specialisation at the origin of the group law
on $J_{b}$ is a $k$-torsor isomorphic to $\Lambda_{\delta}$. The antipodal
involution $[-1]\colon J_{b}\to J_{b}$ gives rise to an involution on
$J_{b,\delta}$ fixing $\Lambda_{\delta}$. We write ${\cal K}_{b,\delta}$ for
the Kummer surface ${\rm Kum}(J_{b,\delta})$ that can be defined as the
minimal desingularisation of the quotient of $J_{b,\delta}$ by this
involution. Thus the $k$-torsor of 16 lines of ${\cal K}_{b,\delta,\bar{k}}$
is canonically isomorphic to $\Lambda_{\delta}$. We have ${\cal
K}_{b,\delta}\cong{\rm Kum}(J_{b,\delta,}^{F})$ for any $F=k(\sqrt{a})/k$ with
$a\in k^{\times}$.

The quartic del Pezzo surfaces and Kummer surfaces associated to a pair
$(P(t),\delta)$ are related as follows: there is a $G$-invariant double
covering ${\cal K}_{b,\delta}\to X$ that sends the above 16 lines on ${\cal
K}_{b,\delta}$ isomorphically onto the 16 lines on $X$, see [Sko10,
Proposition 3.2]. The branch locus of this covering is a canonical curve of
genus 5.

###  1.4 Severi–Brauer fibrations

We shall need to consider the following situation. Let ${\mathcal{S}}$ be a
smooth, projective, geometrically integral variety over $k$ with a dominant
morphism $\pi\colon{\mathcal{S}}\to{\mathbb{P}}^{1}_{k}$ such that the generic
fibre ${\mathcal{S}}_{k(t)}$ is a Severi–Brauer variety. Assume, moreover,
that the restriction of $\pi$ to ${\mathbb{P}}^{1}_{k}\setminus\Sigma$ is
smooth, and that each closed fibre $\pi^{-1}(\sigma)$, $\sigma\in\Sigma$,
consists of two geometric components of multiplicity 1 individually defined
over $k_{\sigma}(\sqrt{\varepsilon_{\sigma}})$ for some
$\varepsilon_{\sigma}\in k_{\sigma}^{\times}$. Let $\varepsilon\in
A^{\times}/A^{\times
2}=\prod_{\sigma\in\Sigma}k_{\sigma}^{\times}/k_{\sigma}^{\times 2}$ be the
image of $(\varepsilon_{\sigma})$.

We write ${\rm{Br}}_{\rm vert}({\mathcal{S}})$ for the vertical Brauer group
associated to the morphism $\pi\colon{\mathcal{S}}\to{\mathbb{P}}^{1}_{k}$.
The following statement is well-known; we give the proofs for the convenience
of the reader.

###### Proposition 1.5

Let $\alpha\in{\rm{Br}}(k(t))$ be the class of the Severi–Brauer variety
${\mathcal{S}}_{k(t)}$. Then we have the following statements.

(i) The residue of $\alpha$ at $\sigma\in\Sigma$ is the class of
$\varepsilon_{\sigma}$ in $k_{\sigma}^{\times}/k_{\sigma}^{\times 2}$. The
residue of $\alpha$ at any closed point of
${\mathbb{P}}^{1}_{k}\setminus\Sigma$ is zero. We have
$\prod_{\sigma\in\Sigma}{\rm N}_{k_{\sigma}/k}(\varepsilon_{\sigma})\in
k^{\times 2}$.

(ii) The natural map ${\rm{Br}}(k)\to{\rm{Br}}({\mathcal{S}})$ is injective
unless $\varepsilon=1\in A^{\times}/A^{\times 2}$, in which case its kernel is
generated by $\alpha$.

(iii) The inclusion ${\rm{Br}}_{\rm
vert}({\mathcal{S}})\subset{\rm{Br}}({\mathcal{S}})$ is an equality.

(iv) There is a natural isomorphism
${\rm{Br}}({\mathcal{S}})/{\rm{Br}}_{0}({\mathcal{S}})\cong{\rm
B}_{A,\varepsilon}$.

(v) There is a natural isomorphism
${\rm{Br}}({\mathcal{S}})/{\rm{Br}}_{0}({\mathcal{S}})\cong{\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{S}}_{\bar{k}}))$.

Proof. (i) By Lichtenbaum’s theorem [GS17, Theorem 5.4.11], the natural map
${\rm{Br}}(k(t))\to{\rm{Br}}({\mathcal{S}}_{k(t)})$ is surjective with kernel
generated by $\alpha$. Thus $\alpha$ goes to zero under the restriction map to
${\rm{Br}}(k({\mathcal{S}}))$, hence the residue of $\alpha$ at the generic
point of every closed fibre ${\mathcal{S}}_{x}$ of $\pi$ is trivial. If
$x\notin\Sigma$, then ${\mathcal{S}}_{x}$ is geometrically integral, hence the
residue field $k(x)$ is separably closed in $k({\mathcal{S}}_{x})$. Using
[CTS21, Lemma 11.1.2] we conclude that the residue of $\alpha$ at $x$ is zero.
A similar argument gives that the residue of $\alpha$ at a closed point
$\sigma\in\Sigma$ belongs to the subgroup of ${\rm
H}^{1}(k_{\sigma},{\mathbb{Z}}/2)$ generated by the class of
$\varepsilon_{\sigma}$. The triviality of this residue in the case when
$\varepsilon_{\sigma}\notin k_{\sigma}^{\times 2}$ implies the existence of a
Severi–Brauer scheme over the local ring of $\sigma$ in ${\mathbb{P}}^{1}_{k}$
with generic fibre ${\mathcal{S}}_{k(t)}$, but this contradicts the birational
invariance of $\varepsilon_{\sigma}$, see [Sko15, Corollary 2.3]. The product
formula comes from the Faddeev reciprocity law [CTS21, Theorem 1.5.2].

(ii) We adapt the argument in the proof of [CTS21, Lemma 11.3.3]. Restricting
to the generic point gives an injective map
${\rm{Br}}({\mathcal{S}})\hookrightarrow{\rm{Br}}(k({\mathcal{S}}))$, since
$S$ is smooth over $k$. The kernel of
${\rm{Br}}(k)\to{\rm{Br}}({\mathcal{S}})$ is thus equal to the kernel of the
composition

${\rm{Br}}(k)\hookrightarrow{\rm{Br}}(k(t))\to{\rm{Br}}({\mathcal{S}}_{k(t)})\hookrightarrow{\rm{Br}}(k({\mathcal{S}})),$

where the last map is injective since ${\mathcal{S}}_{k(t)}$ is smooth over
$k(t)$. We know that the kernel of
${\rm{Br}}(k(t))\to{\rm{Br}}({\mathcal{S}}_{k(t)})$ is generated by $\alpha$.
By (i) and the Faddeev reciprocity law [CTS21, Theorem 1.5.2], $\alpha$
belongs to ${\rm{Br}}(k)$ if and only if $\varepsilon=1\in
A^{\times}/A^{\times 2}$.

(iii) This follows from the surjectivity of the natural map
${\rm{Br}}(k(t))\to{\rm{Br}}({\mathcal{S}}_{k(t)})$.

(iv) The notation ${\rm B}_{A,\varepsilon}$ makes sense by virtue of (i). In
view of Lichtenbaum’s theorem, (iii) follows from (ii) and the standard
calculation of the vertical Brauer group, see [CTS21, Proposition 11.1.8].

(v) Since ${\mathcal{S}}_{\bar{k}}$ is a rational variety, we have
${\rm{Br}}({\mathcal{S}}_{\bar{k}})=0$. The statement follows from [CTS21,
Proposition 5.4.2] and [CTS21, Remark 5.4.3 (2)]. $\Box$

The following well-known statement is given for the ease of reference.

###### Corollary 1.6

Let $X$ be a quartic del Pezzo surface defined by a pair $(P(t),\delta)$. Then
${\rm H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))\cong{\rm B}_{A,\delta}$.

Proof. Let $K=k(X)$. The generic point of $X$ gives rise to a $K$-point on
$X_{K}=X\times_{k}K$ not contained in any of the 16 lines. Projecting from
this point gives a birational equivalence of $X_{K}$ with a cubic surface $Y$
over $K$ containing a $K$-line $\ell\cong{\mathbb{P}}^{1}_{K}$. Residual
conics of the plane sections through $\ell$ make $Y$ a conic bundle over
${\mathbb{P}}^{1}_{K}$ satisfying the assumptions of Proposition 1.5 with
$\varepsilon=\delta$. Thus ${\rm
H}^{1}(K,{\rm{Pic}}(Y_{\overline{K}}))\cong{\rm B}_{A,\delta}$. The natural
isomorphism of ${\rm{Gal}}(\overline{K}/K)$-modules
${\rm{Pic}}(Y_{\overline{K}})\cong{\rm{Pic}}(X_{\overline{K}})\oplus{\mathbb{Z}}$
implies that ${\rm H}^{1}(K,{\rm{Pic}}(X_{\overline{K}}))\cong{\rm
B}_{A,\delta}$. Using that $k$ is separably closed in $K$, we deduce the
required isomorphism. $\Box$

###  1.5 Number fields

When $k$ is a number field, we denote by $O_{k}$ the ring of integers of $k$.
For a set of places $S$ of $k$, let $O_{k,S}$ be the subring of $k$ consisting
of the elements that are integral ouside of $S$. We denote by $k_{v}$ the
completion of $k$ at a place $v$. When $v$ is non-archimedean, we denote the
ring of integers of $k_{v}$ by $O_{v}$ and the residue field of $O_{v}$ by
${\mathbb{F}}_{v}$.

In all of this paper we write $S_{0}$ for a finite set of places of $k$ that
contains all the ‘bad’ places for the pair $(P(t),\delta)$: the archimedean
places, the places above the prime 2, the places $v$ where $P(t)\not\in
O_{v}[t]$, and the places dividing ${\rm discr\,}(P(t))$. We arrange that
$\delta$ is unramified away from $S_{0}$. Increasing $S_{0}$ by a fixed finite
set of primes, we ensure that any quartic del Pezzo surface over
${\mathbb{F}}_{v}$ has more ${\mathbb{F}}_{v}$-points than any genus 1 curve.

For a place $v\notin S_{0}$ we have a Frobenius element ${\rm
Frob}_{v}\in{\rm{Gal}}(k_{\delta}/k)$ defined up to conjugation.

###### Lemma 1.7

Assume that $P(t)$ is irreducible and
$\textup{End}_{\textup{Gal}(P)}(G)=\mathbb{F}_{q}$. Let $T$ be a finite,
$\mathbb{F}_{q}$-linearly independent subset of ${\rm H}^{1}(k,G)$. Let $S$ be
a finite set of places of $k$ containing $S_{0}$ and all primes ramified in
$k_{T}/k$. Let $\sigma\in\textup{Gal}(k_{T}/k)$ be such that every
homomorphism $\textup{Gal}(P)\to\mathbb{Z}/2$ vanishes on the image of
$\sigma$. Then there is a prime $\mathfrak{p}\notin S$ such that
$\textup{Frob}_{\mathfrak{p}}\in\textup{Gal}(k_{T}/k)$ lies in the conjugacy
class of $\sigma$, and a quadratic extension $F/k$ that is ramified at
$\mathfrak{p}$, is split at all $v\in S$ and is unramified at all other
primes.

Proof. Let $\mathfrak{m}$ be the formal product of $8$ and all places in $S$.
Let $k_{\mathfrak{m}}$ be the ray class field associated to the modulus
$\mathfrak{m}$, and let $k_{\mathfrak{m},2}/k$ denote its maximal
multiquadratic subextension. By Lemma 1.4, the intersection of
$k_{\mathfrak{m},2}/k$ with $k_{T}/k$ is contained in the splitting field of
$P(t)$. Let $L$ be the compositum of $k_{T}$ and $k_{\mathfrak{m},2}$. Since
every homomorphism $\textup{Gal}(P)\to\mathbb{Z}/2$ vanishes on the image of
$\sigma$, we can find $\sigma^{\prime}\in\textup{Gal}(L/k)$ with image
$\sigma$ in $\textup{Gal}(k_{T}/k)$ and trivial image in
$\textup{Gal}(k_{\mathfrak{m},2}/k)$.

Let $\mathfrak{p}$ be any prime outside $S$ such that
$\textup{Frob}_{\mathfrak{p}}\in\textup{Gal}(L/k)$ lies in the conjugacy class
of $\sigma^{\prime}$. By construction, the image of
$\textup{Frob}_{\mathfrak{p}}$ in $\textup{Gal}(k_{T}/k)$ lies in the
conjugacy class of $\sigma$. Since the image of $\textup{Frob}_{\mathfrak{p}}$
in $\textup{Gal}(k_{\mathfrak{m},2}/k)$ is trivial, there is a fractional
ideal $I$ supported on primes outside $S$ such that $\mathfrak{p}I^{2}=(a)$
for some totally positive element $a\in k^{\times}$ congruent to $1$ modulo
$8$ and modulo all odd primes in $S$. Then $F=k(\sqrt{a})$ satisfies the
requirements of the statement. $\Box$

##  2 Projective geometry of pencils of quadrics in ${\mathbb{P}}^{4}_{k}$

###  2.1 Canonical equations of quartic del Pezzo surfaces

A quartic del Pezzo surface $X$ over $k$ is a smooth complete intersection of
two quadrics in ${\mathbb{P}}^{4}_{k}$. We now explain why this definition is
equivalent to the definition in terms of $(P(t),\delta)$ given in Section 1.3.

We have $X=Q_{1}\cap Q_{2}$, where $Q_{1}$ and $Q_{2}$ are any two distinct
quadrics containing $X$. Recall that a pencil of quadrics in
${\mathbb{P}}^{4}_{k}$ is a projective line in the projective space of all
quadrics in ${\mathbb{P}}^{4}_{k}$. Let us call the pencil spanned by $Q_{1}$
and $Q_{2}$ by ${\mathcal{P}}$.

Let $\Phi_{i}$ be the quadratic form in 5 variables such that $Q_{i}$ is given
by $\Phi_{i}=0$, for $i=1,2$. Let $\Sigma\subset{\mathbb{P}}^{1}_{k}$ be the
closed $k$-subscheme of zeros of $\det(\mu\Phi_{1}-\nu\Phi_{2})\in
k[\mu,\nu]$. The smoothness of $X$ implies that $\Sigma={\rm{Spec}}(A)$, where
$A$ is an étale $k$-algebra of degree ${\rm{dim}}_{k}(A)=5$. Choose a
$k$-point in ${\mathbb{P}}^{1}_{k}\setminus\Sigma$ and let
${\mathbb{A}}^{1}_{k}={\rm{Spec}}(k[t])\subset{\mathbb{P}}^{1}_{k}$ be its
complement. There is a unique separable monic polynomial $P(t)$ such that
$A=k[t]/(P(t))$.

We have $A=\oplus_{\sigma\in\Sigma}k_{\sigma}$, where $k_{\sigma}$ is the
residue field of a closed point $\sigma\in\Sigma$. Let $\theta_{\sigma}$ be
the image of $t$ in $k_{\sigma}$, and let $\theta=(\theta_{\sigma})$ be the
image of $t$ in $A=\oplus_{\sigma\in\Sigma}k_{\sigma}$.

Let ${\mathcal{Q}}\subset{\mathbb{P}}^{4}_{k}\times_{k}{\mathbb{P}}^{1}_{k}$
be the closed subvariety given by $\mu\Phi_{1}-\nu\Phi_{2}=0$, where
$(\mu:\nu)$ are homogeneous coordinates in ${\mathbb{P}}^{1}_{k}$. The
projection $p\colon{\mathcal{Q}}\to{\mathbb{P}}^{1}_{k}$ is a quadric bundle
of relative dimension 3 which is smooth over
${\mathbb{P}}^{1}_{k}\setminus\Sigma$. The fibre ${\mathcal{Q}}_{m}$, where
$m=(\mu:\nu)\in{\mathbb{P}}^{1}_{k}(\bar{k})$, is the zero set of the
quadratic form $\mu\Phi_{1}-\nu\Phi_{2}=0$. The closed fibres
${\mathcal{Q}}_{\sigma}$, for $\sigma\in\Sigma$, are given by quadratic forms
of rank 4. The determinants of these quadratic forms are well-defined up to a
square, thus give rise to well-defined elements $\delta_{\sigma}\in
k_{\sigma}^{\times}/k_{\sigma}^{\times 2}$. Let $\delta=(\delta_{\sigma})\in
A^{\times}/A^{\times 2}$. We can lift $\delta$ to an element
$\delta^{\prime}\in A^{\times}=\prod_{\sigma\in\Sigma}k_{\sigma}^{\times}$.

It can be shown that $\delta\in(A^{\times}/A^{\times 2})_{1}$, so that
$N_{A/k}(\delta^{\prime})\in k^{\times 2}$ see Lemma 2.4 below. Thus we have
$\delta\in{\rm H}^{1}(k,G)$.

Recall that the del Pezzo surface $X$ can be given by certain canonical
equations (that only depend on the choice of the coordinate $t$), as follows.
The kernels of the five degenerate quadratic forms ${\mathcal{Q}}_{m}$, for
$m\in\Sigma(\bar{k})$, give rise to a Galois-equivariant decomposition of
${\rm H}^{0}(X_{\bar{k}},{\cal O}_{X}(1))$ into a direct sum of 1-dimensional
subspaces. Thus we have
${\mathbb{P}}^{4}_{k}={\mathbb{P}}(R_{A/k}({\mathbb{A}}^{1}_{A}))$. Choose a
coordinate $u$ on $A$; it is well-defined up to multiplication by an element
of $A^{\times}$. By [Sko10, §2] the del Pezzo surface $X$ is isomorphic to the
surface in ${\mathbb{P}}(R_{A/k}({\mathbb{A}}^{1}_{A}))$ with equations

${\rm
Tr}_{A/k}\left(\frac{1}{P^{\prime}(\theta)}\delta^{\prime}u^{2}\right)={\rm
Tr}_{A/k}\left(\frac{\theta}{P^{\prime}(\theta)}\delta^{\prime}u^{2}\right)=0.$
(2)

These equations show that the action of $R_{A/k}({\mathbb{G}}_{m})$ on
$R_{A/k}({\mathbb{A}}^{1}_{A})$ by multiplication on $u$ gives rise to an
action of $G\cong R_{A/k}(\mu_{2})/\mu_{2}$ on $X$ by automorphisms.

Let us call $X_{0}\subset{\mathbb{P}}^{4}_{k}$ the surface given by (2) with
$\delta^{\prime}=1$. It contains the projective $k$-line $\lambda_{0}$ with
homogeneous coordinates $(r:s)$ given by $u=r+s\theta$, cf. [Sko10, Remark, p.
76]. In fact, $X_{0}$ is isomorphic to the blowing-up of the Veronese image of
$\Sigma\subset{\mathbb{P}}^{1}_{k}$ in ${\mathbb{P}}^{2}_{k}$. The surface $X$
is then the twist of $X_{0}$ by $\tilde{\delta}$, so $X$ is indeed obtained
from the pair $(P(t),\delta)$ by the process explained in Section 1.3.

The following lemma will be used in the next section. We say that a hyperplane
$H\subset{\mathbb{P}}^{4}_{k}$ is tangent to a quadric
$Q\subset{\mathbb{P}}^{4}_{k}$ if $Q\cap H$ is singular. In particular, if $Q$
is singular, then any $H$ passing through a singular point of $Q$ is tangent
to $Q$.

###### Lemma 2.1

Let $x\in X(\bar{k})$ and let $Q\in{\mathcal{P}}(\bar{k})$. Assume that $x$ is
not the intersection point of two $\bar{k}$-lines in $X$, or that $Q$ is
smooth. Then $Q$ intersects the tangent plane to $X$ at $x$ in a double line
if and only if the tangent space at $x$ to a quadric in ${\mathcal{P}}$ other
than $Q$ is tangent to $Q$.

Proof. Let $R$ be any $\bar{k}$-quadric in ${\mathcal{P}}$ other than $Q$. We
denote by $T_{R,x}\subset{\mathbb{P}}^{4}_{\bar{k}}$ the tangent space to $R$
at $x$, and likewise for $Q$. Since $X$ is smooth, $T_{R,x}$ and $T_{Q,x}$ are
distinct hyperplanes in ${\mathbb{P}}^{4}_{\bar{k}}$. Under our assumptions,
$Q$ does not contain $T_{X,x}=T_{Q,x}\cap T_{R,x}$. Indeed, a smooth quadric
in ${\mathbb{P}}^{4}_{\bar{k}}$ does not contain planes. If $Q$ is singular,
then every plane contained in $Q$ meets $R$ in a smooth conic or a pair of
lines meeting at a point, so $x$ must be the intersection point of two
$\bar{k}$-lines in $X$.

Suppose that the intersection of $Q$ with $T_{X,x}$ is a line with
multiplicity 2. Then $Q\cap T_{R,x}$ is a quadric in
${\mathbb{P}}^{3}_{\bar{k}}$ that intersects the projective plane $T_{X,x}$ in
a line with multiplicity 2. This is not possible if $Q\cap T_{R,x}$ is a
smooth quadric in ${\mathbb{P}}^{3}_{\bar{k}}$ because in this case the plane
sections can be either smooth conics or pairs of lines intersecting
transversally.

To prove the converse, suppose that $Q\cap T_{R,x}$ is singular. Choose a
$\bar{k}$-point $y$ in the singular locus of $Q\cap T_{R,x}$. Note that $x$ is
a smooth point of $Q\cap T_{R,x}$, so $y\neq x$. The line $(xy)\subset
T_{R,x}$ passes through a singular point of $Q\cap T_{R,x}$ and another point
of $Q\cap T_{R,x}$, hence $(xy)\subset Q\cap T_{R,x}$ so that $(xy)\subset
Q\cap T_{X,x}$.

We claim that $Q$ intersects $T_{X,x}$ in the line $(xy)$ with multiplicity 2.
Otherwise $Q\cap T_{X,x}$ contains another line $L$. Let $z$ be any point of
$L$. The line $(yz)$ meets $Q\cap T_{R,x}$ in two points, one of them
singular, hence $L\subset Q$. We conclude that $Q$ contains the tangent plane
$T_{X,x}\cong{\mathbb{P}}^{2}_{\bar{k}}$, a contradiction. $\Box$

###  2.2 The associated Kummer fibration

Let ${\mathbb{P}}({\cal T}_{X})$ be the projectivisation of the tangent bundle
on $X$. Thus ${\mathbb{P}}({\cal T}_{X})$ has a smooth projective morphism to
$X$ all of whose fibres are projective lines. In particular,
${\mathbb{P}}({\cal T}_{X})$ is birationally equivalent to
$X\times_{k}{\mathbb{P}}^{1}_{k}$.

Let us write ${\rm Gr}({\mathbb{P}}^{1},{\mathbb{P}}^{4})$ for the
Grassmannian of projective lines in ${\mathbb{P}}^{4}_{k}$. Define
$Y\subset{\rm Gr}({\mathbb{P}}^{1},{\mathbb{P}}^{4})\times{\mathcal{P}}$ to be
the variety parameterising pairs $(\ell,m)$, where
$\ell\subset{\mathbb{P}}^{4}_{\bar{k}}$ is a tangent line to $X$ contained in
the quadric ${\mathcal{Q}}_{m}$, where $m\in{\mathcal{P}}(\bar{k})$.

Let $Z\subset{\mathbb{P}}({\cal T}_{X})\times{\mathcal{P}}$ be the variety of
triples $(x,\ell,m)$, where $x\in X(\bar{k})$,
$\ell\subset{\mathbb{P}}^{4}_{\bar{k}}$ is a line tangent to $X$ at $x$
contained in the quadric ${\mathcal{Q}}_{m}$, where
$m\in{\mathcal{P}}(\bar{k})$. Note that the restriction of ${\mathbb{P}}({\cal
T}_{X})\to X$ to a $\bar{k}$-line $\lambda\subset X$ has a canonical section
sending a point $x\in\lambda(\bar{k})$ to $(x,\lambda)$. This gives rise to 16
disjoint $\bar{k}$-lines $\tilde{\lambda}$ in ${\mathbb{P}}({\cal T}_{X})$
which project onto the 16 $\bar{k}$-lines $\lambda$ in $X$. The inverse image
of each $\tilde{\lambda}$ in $Z$ is $\tilde{\lambda}\times{\mathcal{P}}$,
because $\lambda$ is contained in every quadric of the pencil ${\mathcal{P}}$.

When $\ell$ is a tangent line to $X$ not contained in $X$, there is exactly
one quadric in ${\mathcal{P}}$ through $\ell$, thus the restriction of the
projection $Z\to{\mathbb{P}}({\cal T}_{X})$ to ${\mathbb{P}}({\cal
T}_{X})\setminus\coprod\tilde{\lambda}$ is an isomorphism. Associating to a
point $(x,\ell)$ of ${\mathbb{P}}({\cal
T}_{X})\setminus\coprod\tilde{\lambda}$ the unique quadric in ${\mathcal{P}}$
that contains $\ell$ defines a surjective morphism ${\mathbb{P}}({\cal
T}_{X})\setminus\coprod\tilde{\lambda}\to{\mathbb{P}}^{1}_{k}$ that extends to
a dominant rational map from ${\mathbb{P}}({\cal T}_{X})$ to
${\mathbb{P}}^{1}_{k}$. From the definition of $Z$ we see that $Z$ resolves
the indeterminacy of this rational map, so $Z$ is isomorphic to the blowing-up
of ${\mathbb{P}}({\cal T}_{X})$ in the closed subvariety
$\coprod\tilde{\lambda}$. Since ${\mathbb{P}}({\cal T}_{X})$ and
$\coprod\tilde{\lambda}$ are smooth, we conclude that $Z$ is smooth.

The projection $f\colon Z\to{\mathcal{P}}\cong{\mathbb{P}}^{1}_{k}$ sending
$(x,\ell,m)$ to $m$ factors through the projection
$Y\to{\mathcal{P}}\cong{\mathbb{P}}^{1}_{k}$. The morphism $f$ restricted to
$\tilde{\lambda}\times{\mathcal{P}}$ is the projection to
${\mathcal{P}}\cong{\mathbb{P}}^{1}_{k}$. We thus have the following
commutative diagram:

$\textstyle{(\coprod\tilde{\lambda})\times{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}\cong{\mathbb{P}}^{1}_{k}}$$\textstyle{\coprod\tilde{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}({\cal
T}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Gr}({\mathbb{P}}^{1},{\mathbb{P}}^{4})}$$\textstyle{\cup\lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$

Note that the action of $G$ preserves all varieties in this diagram and all
morphisms are $G$-equivariant.

We would like to understand the fibre of $f\colon Z\to{\mathbb{P}}^{1}_{k}$
corresponding to a smooth quadric in ${\mathcal{P}}$. Let
$R\subset{\mathbb{P}}^{4}_{k}$ be the quadric given by the first equation in
(2). Every $\bar{k}$-quadric in the pencil ${\mathcal{P}}$ other than $R$ is
the quadric ${\mathcal{Q}}_{b}\subset{\mathbb{P}}^{4}_{k}$ given by

${\rm
Tr}_{A/k}\left(\frac{b-\theta}{P^{\prime}(\theta)}\delta^{\prime}u^{2}\right)=0$

for some $b\in\bar{k}$; it is smooth if and only if $P(b)\neq 0$. The fibre
$Z_{b}=f^{-1}(b)$ is the set of pairs $(x,\ell)$, where $x\in X(\bar{k})$ and
$\ell$ is a line in ${\mathcal{Q}}_{b}$ tangent to $X$ at $x$. For any $x\in
X(\bar{k})$, the intersection of ${\mathcal{Q}}_{b}$ with
$T_{X,x}\cong{\mathbb{P}}^{2}_{\bar{k}}\subset{\mathbb{P}}^{4}_{\bar{k}}$ is a
conic with a singular point at $x$, that is, two lines meeting at $x$ or a
double line. Thus the projection $Z_{b}\to X$ is a double cover ramified at
the points $x\in X(\bar{k})$ such that ${\mathcal{Q}}_{b}$ intersects
$T_{X,x}$ in a double line. A geometric description of this subvariety of $X$
is given by Lemma 2.1. We now give an equation for it.

###### Lemma 2.2

Let $b\in\bar{k}$ be such that $P(b)\neq 0$. The branch locus of the double
cover $Z_{b}\to X$ is the closed subset of $X$ given by

${\rm
Tr}_{A/k}\left(\frac{1}{(b-\theta)P^{\prime}(\theta)}\delta^{\prime}u^{2}\right)=0.$

Proof. In dual coordinates, the tangent space $T_{R,x}$ is the vector
$(P\textquoteright(\theta)^{-1}\delta^{\prime}x)$. The dual quadric
${\mathcal{Q}}_{b}^{*}$ has equation ${\rm
Tr}_{A/k}((b-\theta)^{-1}P^{\prime}(\theta)\delta^{\prime-1}u^{2})=0$. By
Lemma 2.1, the quadric ${\mathcal{Q}}_{b}\subset{\mathbb{P}}^{4}_{k}$
intersects $T_{X,x}$ in a double line if and only if $T_{R,x}$ belongs to
${\mathcal{Q}}_{b}^{*}$. This gives the desired equation. $\Box$

Let $U={\mathbb{A}}^{1}_{k}\setminus\Sigma$. Let $C\to U$ be a smooth
projective morphism whose fibres are genus 2 curves with affine equation
$y^{2}=(x-t)P(x)$, where $t$ is the coordinate on ${\mathbb{A}}^{1}_{k}$ fixed
above. Let $J\to U$ be the relative Jacobian of $C\to U$. We note that the
2-torsion group subscheme $J[2]$ is constant: it is the pullback to $U$ of the
group $k$-scheme $G$ defined in Section 1.1.

A $k$-scheme on which $G$ acts by automorphisms can be twisted by the
1-cocycle $\tilde{\delta}\colon{\rm{Gal}}(\bar{k}/k)\to G(\bar{k})$. Thus we
obtain a smooth projective morphism $J_{\delta}\to U$ whose fibres are
2-coverings of the fibres of $J\to U$. Passing to the fibrewise quotient by
the antipodal involution we get a projective morphism
${\mathcal{K}}^{\prime}_{\delta}\to U$ whose fibres are singular Kummer
surfaces (with 16 singular points). Its fibrewise desingularisation is a
smooth projective morphism ${\cal K}_{\delta}\to U$ whose fibres are
desingularised Kummer surfaces (with 16 lines). When $\delta\in{\rm
H}^{1}(k,G)$ is zero, we denote ${\cal K}_{\delta}$ by ${\cal K}$.

Let us write $Z_{U}=Z\times_{{\mathbb{P}}^{1}_{k}}U$.

###### Theorem 2.3

There is an isomorphism of $U$-schemes $Z_{U}\cong{\cal K}_{\delta}$. In
particular, ${\cal K}_{\delta}$ is birationally equivalent to
$X\times_{k}{\mathbb{P}}^{1}_{k}$.

Proof. Recall that $t$ is a coordinate on $U\subset{\mathbb{A}}^{1}_{k}$. It
is easy to deduce from [Sko10, Theorem 3.1, Corollary 3.3] that ${\cal
K}_{\delta}$ is the closed subset of ${\mathbb{P}}({\mathbb{A}}^{1}_{k}\times
R_{A/k}({\mathbb{A}}^{1}_{A}))\times U$ given by equations $(\ref{dp4})$
together with the equation

$x^{2}={\rm
Tr}_{A/k}\left(\frac{P(t)}{(t-\theta)P^{\prime}(\theta)}\delta^{\prime}u^{2}\right),$
(3)

where $(x,u)$ is a coordinate in ${\mathbb{A}}^{1}_{k}\times
R_{A/k}({\mathbb{A}}^{1}_{A})$. Forgetting the coordinate $x$ gives rise to a
double cover ${\cal K}_{\delta}\to X\times U$, cf. [Sko10, Proposition 3.2].
This double cover is $G$-equivariant, and is the twist by $\tilde{\delta}$ of
the double cover ${\cal K}\to X_{0}\times U$.

By the discussion preceding Lemma 2.2, the natural projection $Z\to X$ makes
$Z_{U}$ a double cover of $X\times U$. Since $P(t)\neq 0$ on $U$, it follows
from Lemma 2.2 that the branch locus of $Z_{U}\to X\times U$ is given by the
right hand side of (3). Thus it is the same as the branch locus of ${\cal
K}_{\delta}\to X\times U$. This implies that the two double covers are
isomorphic after twisting by an $X\times U$-torsor for $\mu_{2}$. We claim
that they are in fact isomorphic. Since both are $G$-equivariant, they are
isomorphic if and only if their twists by $\tilde{\delta}$ are isomorphic, so
we can assume without loss of generality that $\delta=1$ and $X=X_{0}$. In
particular, we can assume that $X$ has a $k$-line $\lambda_{0}$.

Let $p\colon X\times U\to U$ be the natural projection. We have
$(R^{0}p_{*})(\mu_{2})\cong\mu_{2}$ and $(R^{1}p_{*})(\mu_{2})=0$, using
smooth and proper base change together with the fact that $X$ is geometrically
rational. Then the spectral sequence

${\rm H}^{p}(U,(R^{q}p_{*})(\mu_{2}))\Rightarrow{\rm H}^{p+q}(X\times
U,\mu_{2})$

gives an isomorphism ${\rm
H}^{1}(U,\mu_{2})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
H}^{1}(X\times U,\mu_{2})$. Let us denote by $\eta={\rm{Spec}}(k(t))$ the
generic point of $U$ and write $X_{\eta}=X\times\eta$ for the generic fibre of
$p$.

The natural map ${\rm H}^{1}(U,\mu_{2})\to{\rm H}^{1}(k(t),\mu_{2})$ is
injective. Let $K=k(t,t^{\prime})$. The restriction map ${\rm
H}^{1}(k(t),\mu_{2})\to{\rm H}^{1}(K,\mu_{2})$ is injective. Thus it is enough
to check that $Z_{\eta}$ and ${\cal K}_{\eta}$ restrict to trivial $K$-torsors
for $\mu_{2}$ at some $K$-point of $X_{\eta}$.

Let $K$ be the function field of $\lambda_{0}\cong{\mathbb{P}}^{1}_{k}$ and
let $\eta^{\prime}\in X(K)$ be the generic point of $\lambda_{0}$. We know
from [Sko10, §3] that the inverse image of $\lambda_{0}\times\eta$ in ${\cal
K}_{\eta}$ is the union of two copies of ${\mathbb{P}}^{1}_{k(t)}$, hence the
restriction of ${\cal K}_{\eta}\to X_{\eta}$ to $\eta^{\prime}$ has a section.
The same holds for the double cover $Z_{\eta}\to X_{\eta}$. Indeed,
$\lambda_{0}$ is not contained in the branch locus of $Z_{\eta}\to X_{\eta}$
(which is a smooth curve of genus 5, by [Sko10, Proposition 3.2]), thus
$T_{X,\eta^{\prime}}\cap{\mathcal{Q}}_{\eta}$ is the union of
$\lambda_{0}\times\eta$ and another $k(\eta)$-line. This finishes the proof.
$\Box$

We have chosen ${\mathbb{A}}^{1}_{k}$ in the beginning of Section 2.1 as the
complement to an arbitrary $k$-point in ${\mathbb{P}}^{1}_{k}\setminus\Sigma$.
Applying Theorem 2.3 with a different choice of ${\mathbb{A}}^{1}_{k}$ shows
that the fibre of $f\colon Z\to{\mathbb{P}}^{1}_{k}$ at infinity is a smooth
Kummer surface.

We now analyse the structure of the fibre $Z_{\sigma}$, where $\sigma$ is a
closed point of $\Sigma$. The quadric ${\mathcal{Q}}_{\sigma}$ is given by a
quadratic form of rank 4 of discriminant $\delta_{\sigma}\in
k_{k(\sigma)}^{\times}$ (well-defined up to multiplication by a square), so it
is a cone over a non-degenerate quadric
${\mathcal{Q}}_{\sigma}^{\sharp}\subset{\mathbb{P}}^{3}_{k(\sigma)}$. Let
$v_{\sigma}$ be the vertex of ${\mathcal{Q}}_{\sigma}$, and let $\pi$ be the
projection
${\mathcal{Q}}_{\sigma}\setminus\\{v_{\sigma}\\}\to{\mathcal{Q}}_{\sigma}^{\sharp}$
from $v_{\sigma}$. Recall that ${\mathcal{Q}}_{\sigma}^{\sharp}$ is isomorphic
to the Weil descent of the projective line over
$k_{\sigma}(\sqrt{\delta_{\sigma}})$. Let $\phi^{+}$ and $\phi^{-}$ be the two
(disjoint) families of $\bar{k}$-lines on ${\mathcal{Q}}_{\sigma}^{\sharp}$.
Each of $\phi^{+}$ and $\phi^{-}$ is defined over
$k_{\sigma}(\sqrt{\delta_{\sigma}})$. As we vary $\sigma$, these families give
rise to the ten families of conics on $X$ which are parameterised by the
$\bar{k}$-points of $\Delta$.

The quadric ${\mathcal{Q}}_{\sigma}$ has two families of lines, $F^{+}$ and
$F^{-}$, consisting of the lines whose image under $\pi$ belongs to a line in
$\phi^{+}$ or in $\phi^{-}$, respectively. Each of $F^{+}$ and $F^{-}$ is
defined over $k_{\sigma}(\sqrt{\delta_{\sigma}})$. The intersection $F^{+}\cap
F^{-}$ consists of the lines through $v_{\sigma}$.

The morphism $\pi\colon X\to{\mathcal{Q}}_{\sigma}^{\sharp}$ is a double
covering ramified in a smooth curve $E_{\sigma}$ of genus 1 (a hyperplane
section of $X$). If $x$ is a point of $X$ away from $E_{\sigma}$, then there
is exactly one line in $F^{+}$ which is tangent to $X$ at $x$ (respectively,
in $F^{-}$).

Over $k_{\sigma}(\sqrt{\delta_{\sigma}})$, we have two projections
${\mathcal{Q}}_{\sigma}^{\sharp}\to{\mathbb{P}}^{1}$. Each of them represents
$E_{\sigma}$ as a double cover of ${\mathbb{P}}^{1}$ ramified in four points.
Each of these four points is the intersection point of a pair of lines
contained in $X$. When $x$ is a point of $E_{\sigma}$ other than these eight
points (four for each projection), then the line $(v_{\sigma}x)$ is the only
line in ${\mathcal{Q}}_{\sigma}$ tangent to $X$ at $x$.

Finally, when $x$ is one of the four points, say defined by the family
$\phi^{+}$, the tangent plane $T_{X,x}$ passes through $v_{\sigma}$ and
projects onto a line of $\phi^{+}$. In this case, the family of lines in
$F^{+}$ tangent to $X$ at $x$ is parameterised by the projective line
${\mathbb{P}}(T_{X,x})$.

We conclude that the irreducible components of the fibres of $f\colon
Z_{\bar{k}}\to{\mathbb{P}}^{1}_{\bar{k}}$ over the points of $\Sigma(\bar{k})$
correspond bijectively and Galois-equivariantly to the points of
$\Delta(\bar{k})$. Each of these is isomorphic to $X_{\bar{k}}$ blown-up at
four points, namely the singular points of the four singular conics in the
family of conics given by the point of $\Delta(\bar{k})$. The two components
of the fibre above each point of $\Sigma(\bar{k})$ intersect in a smooth curve
of genus 1.

###  2.3 The associated Severi–Brauer fibration

Recall that for a non-degenerate quadratic form $\Phi$ in 5 variables, the
even Clifford algebra $C^{+}(\Phi)$ is a central simple algebra over $k$. The
class of this algebra in ${\rm{Br}}(k)$ is called the Clifford invariant of
$\Phi$ and is denoted by $c(\Phi)$. In fact, $c(\Phi)$ does not change under
multiplication of $\Phi$ by an element of $k^{\times}$, so $c(\Phi)$ depends
only on the projective quadric $Q\subset{\mathbb{P}}^{4}_{k}$ with equation
$\Phi=0$. Explicitly, when $\Phi$ is a diagonal quadratic form
$\sum_{i=1}^{5}a_{i}x_{i}^{2}$, where $a_{i}\in k^{\times}$, we have

$c(Q)=(-1,-1)\prod_{i<j}(a_{i},a_{j})\in{\rm{Br}}(k),$

see [Lam05, Chapter V, §3].

The variety parameterising projective lines contained in a smooth
3-dimensional quadric $Q\subset{\mathbb{P}}^{4}_{k}$ is a Severi–Brauer
variety ${\rm SB}(Q)$ of dimension 3, whose class $[{\rm
SB}(Q)]\in{\rm{Br}}(k)$ is annihilated by 2, see [Kol18, Lemma 61 (1), (2)].
In fact, we have $[{\rm SB}(Q)]=c(Q)$, as easily follows from [Kol18, Lemma 61
(6)].

Define ${\mathcal{S}}\subset{\rm
Gr}({\mathbb{P}}^{1},{\mathbb{P}}^{4})\times{{\mathcal{P}}}$ to be the closed
subscheme of pairs $(\ell,m)$ such that the projective line
$\ell\subset{\mathbb{P}}^{4}_{k}$ is contained in the quadric
${\mathcal{Q}}_{m}=p^{-1}(m)$. By [Reid72, Theorem 1.10], the variety
${\mathcal{S}}$ is smooth over $k$. The projection
$\pi\colon{\mathcal{S}}\to{\mathbb{P}}^{1}_{k}$ restricts to a smooth morphism
above ${\mathbb{P}}^{1}_{k}\setminus\Sigma$ such that the fibres are
Severi–Brauer varieties of dimension 3.

As promised, we can now prove that $\delta$, as defined in terms of the
singular quadrics in the pencil, lies in $(A^{\times}/A^{\times 2})_{1}$.

###### Lemma 2.4

We have

${\rm N}_{A/k}(\delta)=\prod_{\sigma\in\Sigma}{\rm
N}_{k_{\sigma}/k}(\delta_{\sigma})=1\in k^{\times}/k^{\times 2},$ (4)

where ${\rm N}_{A/k}\colon A^{\times}/A^{\times 2}\to k^{\times}/k^{\times 2}$
is induced by the norm map $A\to k$.

Proof. We note that the morphism
$\pi\colon{\mathcal{S}}\to{\mathbb{P}}^{1}_{k}$ satisfies the assumptions of
Proposition 1.5. One immediately checks that the fibre $\pi^{-1}(\sigma)$,
$\sigma\in\Sigma$, consists of two geometric components of multiplicity 1
individually defined over $k_{\sigma}(\sqrt{\delta_{\sigma}})$. The statement
thus follows from Proposition 1.5 (i). $\Box$

###### Remark 2.5

Let $X^{[2]}$ be the Hilbert scheme of $0$-dimensional subschemes of $X$ of
length 2. This is a smooth variety that can be constructed as the quotient of
the blow-up of $X\times X$ along the diagonal ${\rm diag}(X)$, by the unique
involution lifting the one on $X\times X$ swapping the factors. Define
$\tilde{\mathcal{S}}\subset X^{[2]}\times{\mathcal{S}}$ as the set of triples
$(z,\ell,m)$, where $z\in X^{[2]}$, $(\ell,m)\in{\mathcal{S}}$, and
$z\subset\ell$. The projection $\tilde{\mathcal{S}}\to{\mathcal{S}}$ is an
isomorphism outside of the 16 lines in ${\mathcal{S}}$ given by $(\lambda,m)$,
where $\lambda\cong{\mathbb{P}}^{1}_{\bar{k}}$ is a line contained in $X$ and
$m\in{\mathcal{P}}(\bar{k})$. Indeed, if a line $\ell$ in ${\mathbb{P}}^{4}$
is contained in a quadric of ${\mathcal{P}}$ but not in $X$, then intersecting
$\ell$ with any quadric of ${\mathcal{P}}$ other than ${\mathcal{Q}}_{m}$ we
get a closed subscheme $z$ of $X$ of degree 2. The projection
$\tilde{\mathcal{S}}\to X^{[2]}$ is an isomorphism outside of the 16 planes
$\lambda^{[2]}=\textup{Sym}^{2}\lambda\cong{\mathbb{P}}^{2}_{\bar{k}}$,
because then the line through $z$ is contained in a unique quadric of
${\mathcal{P}}$. Thus both $\tilde{\mathcal{S}}\to{\mathcal{S}}$ and
$\tilde{\mathcal{S}}\to X^{[2]}$ are birational morphisms.

We now study the relation between the Brauer groups of the Severi–Brauer
fibration ${\mathcal{S}}/{\mathbb{P}}^{1}_{k}$ and the Kummer fibration
$Z/{\mathbb{P}}^{1}_{k}$. Recall that $Z$ is a smooth subvariety of
${\mathbb{P}}({\cal T}_{X})\times{\mathcal{P}}$. We have a natural morphism
$Z\to{\mathbb{P}}({\cal T}_{X})\to X$ whose generic fibre is a projective
line. Thus $Z$ is birationally equivalent to $X\times{\mathbb{P}}^{1}_{k}$,
and the induced map ${\rm{Br}}(X)\to{\rm{Br}}(Z)$ is an isomorphism.

The natural map $Z\to Y\subset{\rm
Gr}({\mathbb{P}}^{1},{\mathbb{P}}^{4})\times{\mathcal{P}}$ gives rise to a
morphism $\varphi\colon Z\to{\mathcal{S}}$ compatible with projections to
${\mathcal{P}}\cong{\mathbb{P}}^{1}_{k}$, that is, $f=\pi\circ\varphi$. By
Theorem 2.3 the fibres of $f\colon Z\to{\mathbb{P}}^{1}_{k}$ above the points
of $U$ are smooth Kummer surfaces.

###### Proposition 2.6

The homomorphism

${\rm{Br}}({\mathcal{S}})/{\rm{Br}}(k)\to{\rm{Br}}(Z)/{\rm{Br}}_{0}(Z)$

induced by $\varphi$ is trivial.

Proof. By Proposition 1.5, we have
${\rm{Br}}({\mathcal{S}})/{\rm{Br}}(k)\cong{\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{S}}_{\bar{k}}))$ and this group is annihilated
by 2. We will show that the homomorphism

$\varphi^{*}:{\rm H}^{1}(k,{\rm{Pic}}({\mathcal{S}}_{\bar{k}}))[2]\to{\rm
H}^{1}(k,{\rm{Pic}}(Z_{\bar{k}}))$

induced by $\varphi$ is trivial, which implies the statement as a consequence
of [CTS21, Proposition 5.4.2].

Recall that if $V\to V^{\prime}$ is a birational morphism between smooth
projective varieties, then the induced map ${\rm
H}^{1}(k,{\rm{Pic}}(V^{\prime}_{\bar{k}}))\to{\rm
H}^{1}(k,{\rm{Pic}}(V_{\bar{k}}))$ is an isomorphism. Consider the following
commutative diagram, in which each variety is smooth and projective:

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathcal{S}}$$\textstyle{\tilde{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}({\cal
T}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Bl}_{{\rm
diag}(X)}(X\times
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\quad\
\
q}$$\scriptstyle{p}$$\textstyle{X^{[2]}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
diag}\ \ }$$\textstyle{X\times X}$

In the diagram, ${\rm diag}$ is the diagonal embedding, $\textup{Bl}_{{\rm
diag}(X)}(X\times X)$ is the blow-up of $X\times X$ along the diagonal, the
map ${\mathbb{P}}({\cal T}_{X})\to\textup{Bl}_{{\rm diag}(X)}(X\times X)$ is
the inclusion of the exceptional divisor of the blow-up, and the maps $p$ and
$q$ are the natural contraction and quotient maps respectively. Each vertical
arrow in the diagram is a birational morphism, with the exception of the
projection ${\mathbb{P}}({\cal T}_{X})\to X$ which nevertheless induces an
isomorphism between ${\rm H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))$ and ${\rm
H}^{1}(k,{\rm{Pic}}({\mathbb{P}}({\cal T}_{X})_{\bar{k}}))$.

From consideration of the diagram, it suffices to show that the map

${\rm diag}^{*}\circ(p^{*})^{-1}\circ q^{*}:{\rm
H}^{1}(k,{\rm{Pic}}(X^{[2]}_{\bar{k}}))[2]\to{\rm
H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))$

is trivial. Since $X_{\bar{k}}$ is rational, the two projections $X\times X\to
X$ induce an isomorphism ${\rm{Pic}}((X\times
X)_{\bar{k}})\cong{\rm{Pic}}(X_{\bar{k}})\oplus{\rm{Pic}}(X_{\bar{k}})$, after
which the map ${\rm diag}^{*}$ identifies with the sum map
${\rm{Pic}}(X_{\bar{k}})\oplus{\rm{Pic}}(X_{\bar{k}})\to{\rm{Pic}}(X_{\bar{k}})$.
Consequently, it suffices to show that the image of the map

$(p^{*})^{-1}\circ q^{*}:{\rm H}^{1}(k,{\rm{Pic}}(X^{[2]}_{\bar{k}}))\to{\rm
H}^{1}(k,{\rm{Pic}}((X\times X)_{\bar{k}}))$

is invariant under the action of the involution $\iota$ of $X\times X$
swapping the factors. This last statement is a formal consequence of the fact
that $X^{[2]}$ is the quotient of $\textup{Bl}_{{\rm diag}(X)}(X\times X)$ by
an involution lifting $\iota$. $\Box$

Let ${\rm{Br}}_{\rm vert}(Z)$ be the vertical Brauer group associated to the
morphism $f\colon Z\to{\mathbb{P}}^{1}_{k}$.

###### Corollary 2.7

The natural map ${\rm{Br}}(k)\to{\rm{Br}}_{\rm vert}(Z)$ is surjective.

Proof. The vertical Brauer group ${\rm{Br}}_{\rm vert}(Z)$ is the subgroup of
${\rm{Br}}(Z_{k(t)})$ defined as the intersection of ${\rm{Br}}(Z)$ with the
image of ${\rm{Br}}(k(t))\to{\rm{Br}}(Z_{k(t)})$. The morphism $\varphi\colon
Z\to{\mathcal{S}}$ is compatible with the projections $f$ and $\pi$ to
${\mathbb{P}}^{1}_{k}$, hence it induces a homomorphism
$\varphi^{*}\colon{\rm{Br}}_{\rm vert}({\mathcal{S}})\to{\rm{Br}}_{\rm
vert}(Z)$. Just like the fibres of
$\pi\colon{\mathcal{S}}\to{\mathbb{P}}^{1}_{k}$, the fibres of $f$ above
${\mathbb{P}}^{1}_{k}\setminus\Sigma$ are split and the fibre $Z_{\sigma}$,
$\sigma\in\Sigma$, decomposes as the union of two geometrically irreducible
components individually defined over $k_{\sigma}(\sqrt{\delta_{\sigma}})$.
This implies that the map $\varphi^{*}\colon{\rm{Br}}_{\rm
vert}({\mathcal{S}})\to{\rm{Br}}_{\rm vert}(Z)$ is surjective. Since the
composition

${\rm{Br}}_{\rm vert}({\mathcal{S}})/{\rm{Br}}(k)\to{\rm{Br}}_{\rm
vert}(Z)/{\rm{Br}}_{0}(Z)\hookrightarrow{\rm{Br}}(Z)/{\rm{Br}}_{0}(Z)$

is trivial by Proposition 2.6, the result follows. $\Box$

###### Remark 2.8

In spite of Proposition 2.6, there are canonical isomorphisms

${\rm H}^{1}(k,{\rm{Pic}}(Z_{\bar{k}}))\cong{\rm
H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))\cong{\rm B}_{A,\delta}\cong{\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{S}}_{\bar{k}}))\cong{\rm{Br}}({\mathcal{S}})/{\rm{Br}}(k).$

The first one is a consequence of stably birational equivalence of $X$ and
$Z$, and the rest follow from the combination of Proposition 1.5 and Corollary
1.6.

##  3 From Kummer to del Pezzo

The following proposition is a particular case of the general fibration
theorem for zero-cycles of Harpaz and Wittenberg [HW16, Theorem 6.2]. The
statement is adapted for our needs in this paper and does not aim at maximum
generality.

###### Proposition 3.1

Let $k$ be a number field. Let $Z$ be a smooth, projective, geometrically
integral variety. Let $f\colon Z\to{\mathbb{P}}^{1}_{k}$ be a dominant
morphism with geometrically integral generic fibre such that every geometric
fibre of $f$ contains an irreducible component of multiplicity $1$. Let
$U\subset{\mathbb{P}}^{1}_{k}$ be a non-empty open subset such that the
restriction of $f$ to $Z_{U}=f^{-1}(U)$ is a smooth morphism $Z_{U}\to U$. Let
$B\subset{\rm{Br}}(Z_{U})$ be a finite subgroup and let $m$ be any positive
integer. Assume that there is an adelic point $(x_{v})\in Z({\bf A}_{k})$
orthogonal to $(B+f^{*}{\rm{Br}}(k({\mathbb{P}}^{1}_{k})))\cap{\rm{Br}}(Z)$.
Then $U$ contains a closed point $R={\rm{Spec}}(K)$, of degree coprime to $m$,
such that $Z_{R}({\bf A}_{K})^{B}\neq\emptyset$. Moreover, for any non-
archimedean places $v_{1},...,v_{n}$ of $k$, we can choose $R$ such that, for
each $i=1,...,n$, there is a place $w_{i}$ of $K$ over $v_{i}$ with
$K_{w_{i}}\cong k_{v_{i}}$ and $R$ is arbitrarily close to $f(x_{v_{i}})$ in
${\mathbb{A}}^{1}_{k}(K_{w_{i}})$.

Proof. Note that $(B+f^{*}{\rm{Br}}(k({\mathbb{P}}^{1}_{k})))\cap{\rm{Br}}(Z)$
is finite modulo ${\rm{Br}}(k)$, by [HW16, Lemma 5.3], hence is generated by
finitely many elements $\alpha_{1},\ldots,\alpha_{r}$ modulo ${\rm{Br}}(k)$.
Each of $\alpha_{1},\ldots,\alpha_{r}$ has trivial evaluation map at
$Z(k_{v})$ for almost all places $v$.

Choose an arbitrary closed point $N$ in $Z_{U}$.

For each real place $v$, by a version of Bertini’s theorem we can choose a
smooth, projective, geometrically irreducible curve $C_{v}\subset Z_{k_{v}}$
passing through $x_{v}$ and $N$ such that $f(C_{v})={\mathbb{P}}^{1}_{k_{v}}$.
We choose a positive integer $r$ large enough so that if
$y\in{\rm{Pic}}({\mathbb{P}}^{1}_{k})$ is a divisor class of degree
$1+mr\,{\rm{deg}}_{k}(N)$, then the inequality in condition (ii) of [HW16,
Theorem 5.1] holds and the inequality in condition (iii) of [HW16, Theorem
5.1] holds for every real place $v$ of $k$.

For each place $v$, define $z_{v}$ to be an effective zero-cycle in
$Z_{k_{v}}$ that is a small deformation of $x_{v}+mrN$ such that $f(z_{v})$ is
reduced and is contained in $U_{k_{v}}$. For each real place $v$ we take
$z_{v}$ to lie on $C_{v}$. Since $\alpha_{1},\ldots,\alpha_{r}$ have trivial
evaluation maps at $Z(k_{v})$ for almost all places $v$, at these places we
can take $z_{v}$ to be an arbitrary effective zero-cycle of degree
$1+mr\,{\rm{deg}}_{k}(N)$ such that $f(z_{v})$ is reduced and contained in
$U_{k_{v}}$, and so arrange that $z_{v}$ is integral with respect to some
integral model of $Z_{U}$. (An integral model of $Z_{U}$ is a faithfully flat
$O_{k,S}$-scheme of finite type with generic fibre $Z_{U}$, where $S$ is a
finite set of primes of $k$. A zero-cycle $z_{v}$ is integral if there is an
integral model of $Z_{U}$ such that each closed point in the support of
$z_{v}$ extends to a finite $O_{v}$-subscheme of the model.)

Thus, in the notation of [HW16], we have $(z_{v})\in{\rm Z}_{0,{\bf A}}^{{\rm
eff,red,}y}(Z_{U})$, where $y\in{\rm{Pic}}({\mathbb{P}}^{1}_{k})$ has degree
$1+mr\,{\rm{deg}}_{k}(N)$. The adelic zero-cycle $(z_{v})$ satisfies
conditions (i), (ii), (iii) of [HW16, Theorem 5.1]. An application of [HW16,
Theorems 5.1, 6.2] gives a closed point $R$ as in the theorem. $\Box$

###### Definition 3.2

Let ${\cal G}$ be a subgroup of $({\mathbb{Z}}/2)^{4}\rtimes{\rm S}_{5}$ and
let $(C_{1},\ldots,C_{n})$ be conjugacy classes in ${\cal G}$. When ${\cal
G}={\rm{Gal}}(k_{\delta}/k)$, we call $(C_{1},\ldots,C_{n})$ local conditions
for $(P(t),\delta)$ over $k$. Local conditions $(C_{1},\ldots,C_{n})$ are
called admissible if any $g_{i}\in C_{i}$ has at least one fixed point in
$\Delta(\bar{k})$. We say that a pair $(b,T)$, where $b\in k$ and
$T=\\{v_{1},\ldots,v_{n}\\}$ is an ordered set of distinct non-archimedean
places of $k$, satisfies local conditions $(C_{1},\ldots,C_{n})$ for
$(P(t),\delta)$ over $k$ if $T\cap S_{0}=\emptyset$, $P(b)\neq 0$, and for
each $i=1,\ldots,n$ we have $b\in O_{v_{i}}$, ${\rm val}_{v_{i}}(P(b))=1$, and
${\rm Frob}_{v_{i}}\in C_{i}$.

###### Remark 3.3

For any admissible local conditions for $(P(t),\delta)$ there is a pair
$(b,T)$ as above that satisfies them over $k$. Indeed, by the Chebotarev
density theorem we can find $T=\\{v_{1},\ldots,v_{n}\\}$ such that $T\cap
S_{0}=\emptyset$ and ${\rm Frob}_{v_{i}}\in C_{i}$ for $i=1,\ldots,n$.
Admissibility implies that $P(t)$ has a simple root in $k_{v_{i}}$ for
$i=1,\ldots,n$, call it $\theta_{i}$. Since $P(t)\in O_{v_{i}}[t]$, we have
$\theta_{i}\in O_{v_{i}}$. Let $b_{i}\in O_{v_{i}}$ be such that ${\rm
val}_{v_{i}}(b_{i}-\theta_{i})=1$. Then $P(b_{i})\neq 0$ and ${\rm
val}_{v_{i}}(P(b_{i}))=1$. By weak approximation, we can find an element $b\in
k$ close enough to $b_{i}$ in $k_{v_{i}}$ for $i=1,\ldots,n$. The pair $(b,T)$
does the job.

Using notation of [SZ17], we denote by $\Pi$ the Kummer lattice, i.e. the
saturation, in the geometric Picard group of a Kummer surface, of the free
abelian group freely generated by the classes of the 16 lines. The group $G$
naturally acts on $\Pi$. Thus the Kummer lattice in
${\rm{Pic}}(Z_{b,\bar{k}})$ is a Galois module $\Pi_{\delta}$ obtained by
twisting the Galois module $\Pi$ by the cocycle $\tilde{\delta}\colon\Gamma\to
G(\bar{k})$ (in particular, its Galois module structure does not depend on
$b$).

###### Theorem 3.4

Let $X$ be a quartic del Pezzo surface over a number field $k$ given by a pair
$(P(t),\delta)$. Then $X({\bf A}_{k})^{\rm{Br}}\neq\emptyset$ implies
$X(k)\neq\emptyset$, provided the following holds:

There exist admissible local conditions for $(P(t),\delta)$ such that, for any
finite extension $K/k$ linearly disjoint from $k_{\delta}/k$ and any pair
$(b,T)$ satisfying these local conditions over $K$, the Kummer surfaces
${\mathcal{K}}_{b,\delta}={\rm Kum}(J_{b,\delta})$ have the property that the
algebraic Brauer–Manin obstruction attached to the Kummer lattice
$\Pi_{\delta}\subset{\rm{Pic}}({\mathcal{K}}_{b,\delta,\bar{k}})$ is the only
obstruction to the Hasse principle on ${\mathcal{K}}_{b,\delta}$.

Proof. Let $X$ be a quartic del Pezzo surface given by $(P(t),\delta)$. Then
$X$ is stably birational to the associated Kummer fibration $f\colon
Z\to{\mathbb{P}}^{1}_{k}$, see Section 2.2. By Theorem 2.3,
$Z_{U}=Z\times_{{\mathbb{P}}^{1}_{k}}U$ is a relative Kummer surface over
$U={\mathbb{A}}^{1}_{k}\setminus\Sigma$. In particular, the morphism $Z_{U}\to
U$ is smooth and projective with geometrically integral fibres. This implies
that the restriction to the generic fibre induces an isomorphism
${\rm{Pic}}(Z_{U_{\bar{k}}})\to{\rm{Pic}}(Z_{\bar{k}(t)})$. In particular,
${\rm{Pic}}(Z_{U_{\bar{k}}})$ is a finitely generated free abelian group,
hence ${\rm H}^{1}(k,{\rm{Pic}}(Z_{U_{\bar{k}}}))$ is finite. The proof of the
following lemma is well known and is given here for the convenience of the
reader.

###### Lemma 3.5

The natural map ${\rm{Br}}_{1}(Z_{U})\to{\rm
H}^{1}(k,{\rm{Pic}}(Z_{U_{\bar{k}}}))$ is surjective.

Proof. The Leray spectral sequence ${\rm H}^{p}(k,{\rm
H}^{q}(Z_{U_{\bar{k}}},{\mathbb{G}}_{m}))\Rightarrow{\rm
H}^{p+q}(Z_{U},{\mathbb{G}}_{m})$ gives an exact sequence

${\rm{Br}}_{1}(Z_{U})\to{\rm H}^{1}(k,{\rm{Pic}}(Z_{U_{\bar{k}}}))\to{\rm
H}^{3}(k,{\rm H}^{0}(Z_{U_{\bar{k}}},{\mathbb{G}}_{m})),$

so it suffices to show that ${\rm H}^{3}(k,{\rm
H}^{0}(Z_{U_{\bar{k}}},{\mathbb{G}}_{m}))=0$. Since the morphism
$Z_{U_{\bar{k}}}\to U_{\bar{k}}$ is proper with geometrically integral fibres,
we have ${\rm H}^{0}(Z_{U_{\bar{k}}},{\mathbb{G}}_{m})\cong{\rm
H}^{0}(U_{\bar{k}},{\mathbb{G}}_{m})\cong\bar{k}[U]^{\times}$. There is an
exact sequence of $\Gamma$-modules

$0\to\bar{k}^{\times}\to\bar{k}[U]^{\times}\to{\mathbb{Z}}[\Sigma]\to 0,$

where the third arrow sends an invertible regular function on
$U_{\bar{k}}={\mathbb{A}}^{1}_{\bar{k}}\setminus\Sigma_{\bar{k}}$ to its
divisor in ${\mathbb{A}}^{1}_{\bar{k}}$. We obtain an exact sequence of Galois
cohomology groups

${\rm H}^{3}(k,\bar{k}^{\times})\to{\rm H}^{3}(k,\bar{k}[U]^{\times})\to{\rm
H}^{3}(k,{\mathbb{Z}}[\Sigma]).$

Since $k$ is a number field, we have ${\rm H}^{3}(k,\bar{k}^{\times})=0$
[CF67, Ch. VII, §11.4]. We also have ${\rm H}^{3}(k,{\mathbb{Z}}[\Sigma])=0$,
which is a consequence of Shapiro’s lemma and the vanishing of ${\rm
H}^{3}(k^{\prime},{\mathbb{Z}})\cong{\rm
H}^{2}(k^{\prime},{\mathbb{Q}}/{\mathbb{Z}})$ for any number field
$k^{\prime}$ [Sch79, §4, Satz 1 (ii)]. $\Box$

We have natural embeddings of $\Gamma$-modules
$\Pi_{\delta}\to{\rm{Pic}}(Z_{U_{\bar{k}}})$ and
$\Pi_{\delta}\to{\rm{Pic}}(Z_{b,\bar{k}})$ compatible under the specialisation
map ${\rm{Pic}}(Z_{U_{\bar{k}}})\to{\rm{Pic}}(Z_{b,\bar{k}})$. By Lemma 3.5,
we can choose lifts to ${\rm{Br}}_{1}(Z_{U})$ of each element of the image of
the finite group ${\rm H}^{1}(k,\Pi_{\delta})$ in ${\rm
H}^{1}(k,{\rm{Pic}}(Z_{U_{\bar{k}}}))$. Define $B$ as the subgroup of
${\rm{Br}}_{1}(Z_{U})$ generated by these lifts. Thus $B$ is a finite subgroup
of ${\rm{Br}}_{1}(Z_{U})$ such that for any $b\in k$, $P(b)\neq 0$, the image
of the specialisation map

$B\hookrightarrow{\rm{Br}}_{1}(Z_{U})\to{\rm{Br}}_{1}(Z_{b})\to{\rm
H}^{1}(k,{\rm{Pic}}(Z_{b,\bar{k}}))$

contains the image of ${\rm H}^{1}(k,\Pi_{\delta})\to{\rm
H}^{1}(k,{\rm{Pic}}(Z_{b,\bar{k}}))$.

The variety $Z$ is stably birationally equivalent to $X$, hence if $X$ is
everywhere locally soluble, then so is $Z$. For the same reason, the morphism
$Z\to X$ induces an isomorphism
${\rm{Br}}(X)\tilde{\longrightarrow}{\rm{Br}}(Z)$, thus if $X({\bf
A}_{k})^{\rm{Br}}\neq\emptyset$, then $Z({\bf A}_{k})^{\rm{Br}}\neq\emptyset$.

For any sufficiently large finite set of places $S$, the projective
${\mathbb{P}}^{1}_{k}$-scheme $Z$ extends to a projective
${\mathbb{P}}^{1}_{O_{k,S}}$-scheme ${\mathcal{Z}}$ such that the morphism
${\mathcal{Z}}\to{\mathbb{P}}^{1}_{O_{k,S}}$ is smooth away from $P(t)=0$. We
choose $S$ such that $S_{0}\subset S$ and the Brauer classes
$\alpha_{1},\ldots,\alpha_{r}$ from the proof of Proposition 3.1 have zero
evaluation maps outside of $S$.

Using Remark 3.3 we find a pair $(b,T)$ satisfying the local conditions in the
statement of the theorem over $k$.

We claim that $Z_{b}(k_{v_{i}})\neq\emptyset$ for $i=1,\ldots,n$. Indeed,
$Z_{b}$ naturally extends to a projective scheme
${\mathcal{Z}}_{b}\to{\rm{Spec}}(O_{v_{i}})$ with special fibre isomorphic to
the reduction of the singular fibre $Z_{\theta_{i}}$, where $\theta_{i}$ is a
root of $P(t)$ in $k_{v_{i}}$. Moreover, since smoothness is stable under base
change, the morphism ${\mathcal{Z}}_{b}\to{\rm{Spec}}(O_{v_{i}})$ is smooth
away from the singular locus of the special fibre. Since ${\rm Frob}_{v_{i}}$
fixes a point of $\Delta(\bar{k})$ above the root $\theta_{i}$ of $P(t)$, the
reduction of $Z_{\theta_{i}}$ is the union of two copies of the reduction of
$X$ intersecting in a smooth curve of genus 5. By the definition of $S_{0}$,
there are ${\mathbb{F}}_{v_{i}}$-points in $X$ outside this curve. Then the
reduction of $Z_{\theta_{i}}$ has a smooth ${\mathbb{F}}_{v_{i}}$-point. By
Hensel’s lemma it can be lifted by a point $x_{v_{i}}\in Z_{b}(k_{v_{i}})$.

An application of Proposition 3.1 gives a closed point $R={\rm{Spec}}(K)$ in
$U$ such that the fibre $Z_{R}$ is a Kummer surface over $K$ with an adelic
point orthogonal to $B$. We can ensure that $[K:k]$ is coprime to $30$, hence
$K$ and $k_{\delta}$ are linearly disjoint over $k$. Moreover, we can ensure
that $K$ has a place $w_{i}$ over $v_{i}$ such that $K_{w_{i}}\cong k_{v_{i}}$
and $R$ is arbitrarily close to $b$ at $w_{i}$, for $i=1,\ldots,n$. It follows
that the pair consisting of the point $R\in{\mathbb{A}}^{1}_{k}(K)$ and the
set of places $\\{w_{1},\ldots,w_{n}\\}$ of $K$ satisfies the local conditions
in the statement of the theorem over $K$.

The condition in the theorem now implies that $Z_{R}$ has a $K$-point. It
follows that $X$ has a $K$-point. Since $[K:k]$ is odd, the Amer–Brumer
theorem [Amer76, Bru78] then implies that $X$ has a $k$-point. $\Box$

We shall use the following restatement of the main result of [Har19].

######  Proposition 3.6 (Harpaz)

Let $k$ be a number field. Assume that a pair $(P(t),\delta)$ is such that
$P(t)$ is completely split over $k$ and
$\delta=(\delta_{1},\ldots,\delta_{5})$, where $\prod_{i=1}^{5}\delta_{i}=1$
is the only relation satisfied by the images of $\delta_{i}$ in
$k^{\times}/k^{\times 2}$. Let $b\in k$ be such that $P(b)\neq 0$. Assume that
$\textcyr{Sh}(J^{F}_{b})[2^{\infty}]$ is finite for every quadratic extension
$F/k$. Then the Kummer surfaces ${\rm Kum}(J_{b,\delta})$ have the property
that the algebraic Brauer–Manin obstruction attached to the Kummer lattice
$\Pi_{\delta}$ is the only obstruction to the Hasse principle, provided there
exist primes $w_{1},\ldots,w_{5}$ of $k$ outside of $S_{0}$ such that, for
$i=1,\ldots,5$, we have $b\in O_{w_{i}}$, ${\rm val}_{w_{i}}(P(b))=1$, and
$\delta_{i}\in k_{w_{i}}^{\times 2}$ but ${\rm res}_{w_{i}}(\delta)\neq 0$.

Proof. From [Har19, Theorem 1.3, Remark 2.9] we know that under the
assumptions of the proposition the algebraic Brauer–Manin obstruction is the
only obstruction to the Hasse principle on ${\rm Kum}(J_{b,\delta})$. In fact,
Harpaz proves the stronger statement concerning the algebraic Brauer–Manin
obstruction attached to the Kummer lattice $\Pi_{\delta}$. Let us explain
this. Write ${\mathcal{K}}_{b,\delta}={\rm Kum}(J_{b,\delta})$ and let
$W={\mathcal{K}}_{b,\delta}\setminus\Lambda_{\delta}$ be the complement to the
16 lines. The diagram (7) from [SZ17] is a commutative diagram of
$\Gamma$-modules with exact rows and columns

$\begin{array}[]{ccccccccc}&&&&0&&0&&\\\ &&&&\uparrow&&\uparrow&&\\\ &&&&{\rm
NS\,}(J_{b,\delta,\bar{k}})&=&{\rm NS\,}(J_{b,\delta,\bar{k}})&&\\\
&&&&\uparrow&&\uparrow&&\\\
0&\longrightarrow&{\mathbb{Z}}[\Lambda_{\delta}]&\longrightarrow&{\rm{Pic}}({\mathcal{K}}_{b,\delta,\bar{k}})&\longrightarrow&{\rm{Pic}}(W_{\bar{k}})&\longrightarrow&0\\\
&&||&&\uparrow&&\uparrow&&\\\
0&\longrightarrow&{\mathbb{Z}}[\Lambda_{\delta}]&\longrightarrow&\Pi_{\delta}&\longrightarrow&{\rm{Pic}}(W_{\bar{k}})_{\rm
tors}&\longrightarrow&0\\\ &&&&\uparrow&&\uparrow&&\\\ &&&&0&&0&&\end{array}$
(5)

The $\Gamma$-module ${\mathbb{Z}}[\Lambda_{\delta}]$ is permutational, hence
${\rm H}^{1}(k,{\mathbb{Z}}[\Lambda_{\delta}])=0$. Thus (5) gives rise to a
commutative diagram with exact columns

$\begin{array}[]{ccccccccc}{\rm H}^{1}(k,{\rm
NS\,}(J_{b,\delta,\bar{k}}))&=&{\rm H}^{1}(k,{\rm
NS\,}(J_{b,\delta,\bar{k}}))\\\ \uparrow&&\uparrow\\\ {\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{K}}_{b,\delta,\bar{k}}))&\hookrightarrow&{\rm
H}^{1}(k,{\rm{Pic}}(W_{\bar{k}}))\\\ \uparrow&&\uparrow\\\ {\rm
H}^{1}(k,\Pi_{\delta})&\hookrightarrow&{\rm
H}^{1}(k,{\rm{Pic}}(W_{\bar{k}})_{\rm tors})&\\\ \end{array}$ (6)

From this diagram we conclude that the subgroup of ${\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{K}}_{b,\delta,\bar{k}}))$ consisting of the
elements whose images in ${\rm H}^{1}(k,{\rm{Pic}}(W_{\bar{k}}))$ come from
${\rm H}^{1}(k,{\rm{Pic}}(W_{\bar{k}})_{\rm tors})$ is just the image of ${\rm
H}^{1}(k,\Pi_{\delta})$. It follows that the group denoted by

${\mathcal{C}}({\mathcal{K}}_{b,\delta})\subset{\rm
H}^{1}(k,{\rm{Pic}}({\mathcal{K}}_{b,\delta,\bar{k}}))={\rm{Br}}_{1}({\mathcal{K}}_{b,\delta})/{\rm{Br}}_{0}({\mathcal{K}}_{b,\delta})$

in [Har19, Definition 4.3] is the image of ${\rm H}^{1}(k,\Pi_{\delta})$. By
[Har19, Remark 2.9] the obstruction given by
${\mathcal{C}}({\mathcal{K}}_{b,\delta})$ is the only obstruction to the Hasse
principle on ${\mathcal{K}}_{b,\delta}$. $\Box$

The following theorem gives the Hasse principle for Kummer surfaces with
irreducible polynomial $P(t)$, assuming that $b\in k$ satisfies certain local
conditions. The proof will be given in Sections 4.2 and 4.3.

###### Theorem 3.7

Let $k$ be a number field. Assume that a pair $(P(t),\delta)$ is such that
$P(t)$ is irreducible. Let $b\in k$ be such that $P(b)\neq 0$. Assume that
$\textcyr{Sh}(J^{F}_{b})[2^{\infty}]$ is finite for every quadratic extension
$F/k$.

(a) Suppose that the Galois group of $P(t)$ is ${\rm S}_{5}$, ${\rm A}_{5}$,
or ${\rm F}_{20}$. Then the Hasse principle holds for the Kummer surfaces
${\rm Kum}(J_{b,\delta})$ for which there is a prime $w$ outside of $S_{0}$
such that $b\in O_{w}$, ${\rm val}_{w}(P(b))=1$, and ${\rm
res}_{w}(\delta)=0$.

(b) Suppose that the Galois group of $P(t)$ is ${\rm D}_{10}$. Then the Hasse
principle holds for the Kummer surfaces ${\rm Kum}(J_{b,\delta})$ for which
there exist primes $w_{1}$ and $w_{2}$ outside of $S_{0}$ such that for
$i=1,2$ we have $b\in O_{w_{i}}$, ${\rm val}_{w_{i}}(P(b))=1$, the Frobenius
element at $w_{i}$ acts on the roots of $P(t)$ as a double transposition, and
$\textup{res}_{w_{1}}(\delta)\neq 0$, $\textup{res}_{w_{2}}(\delta)=0$.

Now we are ready to prove Theorem A of the Introduction.

###### Theorem 3.8

Let $k$ be a number field. Assume that $\textcyr{Sh}({\rm
Jac}(C))[2^{\infty}]$ is finite for every genus $2$ curve $C$ over every
finite extension of $k$. Then the quartic del Pezzo surfaces $X$ over $k$ such
that the polynomial $P(t)\in k[t]$ is completely split and
${\rm{Br}}(X)={\rm{Br}}_{0}(X)$, or $P(t)\in k[t]$ is irreducible, satisfy the
Hasse principle.

Proof. Without loss of generality we can assume that $\delta\neq 0$. Indeed,
in the opposite case $X$ contains a $k$-line, so there is nothing to prove.

Suppose that $P(t)\in k[t]$ is completely split and
${\rm{Br}}(X)={\rm{Br}}_{0}(X)$. Then ${\rm{Gal}}(k_{\delta}/k)$ is a subgroup
of $({\mathbb{Z}}/2)^{4}\rtimes{\rm S}_{5}$ contained in
$({\mathbb{Z}}/2)^{4}$. We can assume without loss of generality that
${\rm{Gal}}(k_{\delta}/k)$ does not fix a point of $\Delta(\bar{k})$ or,
equivalently, that no $\delta_{i}$ is a square in $k$. Indeed, otherwise $X$
has a conic bundle defined over $k$, so there is a morphism
$X\to{\mathbb{P}}^{1}_{k}$ making $X$ a conic bundle with 4 geometric
degenerate fibres. In this case $X$ satisfies the Hasse principle
unconditionally by a theorem of Salberger [Sal88, Corollary 7.8].

We claim that ${\rm{Gal}}(k_{\delta}/k)$ is the normal subgroup
$({\mathbb{Z}}/2)^{4}$ of $({\mathbb{Z}}/2)^{4}\rtimes{\rm S}_{5}$. Indeed, if
${\rm{Gal}}(k_{\delta}/k)$ is a proper subgroup of $({\mathbb{Z}}/2)^{4}$ and
no $\delta_{i}$ is a square in $k$, then ${\rm B}_{A,\delta}\neq 0$. By
Corollary 1.6 we have ${\rm H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))\cong{\rm
B}_{A,\delta}\neq 0$. Since $k$ is a number field, we have ${\rm
H}^{3}(k,\bar{k}^{\times})=0$. Then the exact sequence of low degree terms of
the spectral sequence ${\rm H}^{p}(k,{\rm
H}^{q}(X_{\bar{k}},{\mathbb{G}}_{m}))\Rightarrow{\rm
H}^{p+q}(X,{\mathbb{G}}_{m})$ gives an isomorphism
${\rm{Br}}(X)/{\rm{Br}}_{0}(X)\cong{\rm H}^{1}(k,{\rm{Pic}}(X_{\bar{k}}))$,
hence we get a contradiction.

For $i=1,\ldots,5$ take $g_{i}$ to be the product of all generators
$c_{j}\in({\mathbb{Z}}/2)^{5}$ except $c_{i}$. These local conditions are
admissible. Let $w_{1},\ldots,w_{5}$ be primes not in $S_{0}$ such that ${\rm
Frob}_{w_{i}}=g_{i}\in({\mathbb{Z}}/2)^{4}$. By Lemma 1.2 we have ${\rm
res}_{w_{i}}(\delta)\neq 0$. Then Proposition 3.6 implies that the assumption
of Theorem 3.4 is satisfied.

Assume now that $P(t)$ is irreducible over $k$.

Let ${\rm{Gal}}(P)$ be one of ${\rm S}_{5}$, ${\rm A}_{5}$, ${\rm F}_{20}$. By
Theorem 3.7 (a) the assumption of Theorem 3.4 is satisfied for the admissible
local condition at one prime $w\notin S_{0}$ where the Frobenius is the
identity in ${\rm{Gal}}(k_{\delta}/k)$. Lemma 1.2 implies that ${\rm
res}_{w}(\delta)=0$.

Let ${\rm{Gal}}(P)\cong{\rm D}_{10}$. By Theorem 3.7 (b) the assumption of
Theorem 3.4 is satisfied for the admissible local conditions at two primes
$w_{1},w_{2}\notin S_{0}$ whose Frobenius elements in
$({\mathbb{Z}}/2)^{4}\rtimes{\rm D}_{10}$ are $g_{1}=c_{1}c_{3}(12)(34)$ and
$g_{2}=(12)(34)$, respectively. Indeed, Lemma 1.2 implies that ${\rm
res}_{w_{1}}(\delta)\neq 0$ and ${\rm res}_{w_{2}}(\delta)=0$, as required.

Finally, let ${\rm{Gal}}(P)\cong{\mathbb{Z}}/5$. From Lemma 1.4 we see that
${\rm{Gal}}(k_{\delta}/k)$ coincides with
$({\mathbb{Z}}/2)^{4}\rtimes{\mathbb{Z}}/5$ in
$({\mathbb{Z}}/2)^{4}\rtimes{\rm S}_{5}$. Let $K$ be the splitting field of
$P(t)$. Then $X_{K}$ is a quartic del Pezzo surface such that $P(t)$ is
completely split over $K$ and ${\rm{Gal}}(K_{\delta}/K)$ is the subgroup
$({\mathbb{Z}}/2)^{4}$ of $({\mathbb{Z}}/2)^{4}\rtimes{\rm S}_{5}$. This
implies ${\rm{Br}}(X_{K})={\rm{Br}}_{0}(X_{K})$. Applying an already proved
case of the theorem, we obtain that $X$ has a $K$-point. Since $[K:k]=5$ is
odd, the Amer–Brumer theorem [Amer76, Bru78] implies that $X$ has a $k$-point.
$\Box$

###### Remark 3.9

The argument in the above proof works equally well for quartic del Pezzo
surfaces such that
${\rm{Gal}}(k_{\delta}/k)=({\mathbb{Z}}/2)^{4}\rtimes{\mathbb{Z}}/3$.

###### Remark 3.10

For quartic del Pezzo surfaces $X$ over $k={\mathbb{Q}}$ such that the
polynomial $P(t)\in k[t]$ is completely split and
${\rm{Br}}(X)={\rm{Br}}_{0}(X)$, one can prove Theorem 3.8 using the fibration
theorem of Harpaz–Wittenberg for rational points [HW16, Theorem 9.17] (see
also [CTS21, Theorem 14.2.21]). Indeed, since $P(t)$ completely splits, the
complement ${\mathbb{P}}^{1}_{k}\setminus U$ is a union of $k$-points, so that
Hypothesis (HW) holds for $k={\mathbb{Q}}$, see [HW16, Theorem 9.14] based on
work of L. Matthiesen in additive combinatorics [Mat18]. Thus [HW16, Theorem
9.17] implies the existence of a point $b\in k$, $P(b)\neq 0$, such that
$Z_{b}$ has an adelic point orthogonal to $B$. This proof avoids replacing $k$
by a finite extension. As a consequence, the assumption that
$\textcyr{Sh}(J^{F}_{b})[2^{\infty}]$ is finite for every $b\in k$, $P(b)\neq
0$, and every quadratic extension $F/k$, is enough to conclude that $X$
satisfies the Hasse principle. When ${\mathbb{P}}^{1}_{k}\setminus U$ is a
union of $k$-points, Hypothesis (HW) is expected to hold for an arbitrary
number field $k$, see the forthcoming paper of C. Frei and L. Matthiesen. Thus
the restriction $k={\mathbb{Q}}$ can be dropped.

###### Remark 3.11

When $P(t)$ is irreducible, to prove Theorem 3.8, one can use the older
fibration results [CSS98b, Theorems 3.1, 4.1] in place of Proposition 3.1.

##  4 Hasse principle for Kummer surfaces

###  4.1 $2$-Selmer group in quadratic twist families

Let $k$ be a number field. Let $b\in k$ be such that $P(b)\neq 0$. We choose a
finite set $S_{0,b}$ of places of $k$ containing $S_{0}$, all primes where $b$
is not integral and all primes dividing the discriminant $d_{b}=P(b)^{2}\,{\rm
discr\,}(P(t))$ of $(t-b)P(t)$.

Let $v$ be a place of $k$. Composition of cup-product and the local invariant
map induces a non-degenerate pairing

${\rm H}^{1}(k_{v},G)\times{\rm
H}^{1}(k_{v},G)\stackrel{{\scriptstyle\cup}}{{\longrightarrow}}\textup{Br}(k_{v})[2]\stackrel{{\scriptstyle\textup{inv}_{v}}}{{\longrightarrow}}\tfrac{1}{2}\mathbb{Z}/\mathbb{Z},$
(7)

which we refer to as the local Tate pairing (recall from Section 1.3 that $G$
is canonically isomorphic to $J_{b}[2]$ and that we equip $G$ with the Weil
pairing). Write ${\partial}:J_{b}(k_{v})\to{\rm H}^{1}(k_{v},G)$ for the
Kummer map. By local Tate duality, ${\partial}(J_{b}(k_{v}))$ is its own
orthogonal complement under the pairing (7), see [PR12, Proposition 4.11].

If $v$ is an odd place, then
${\rm{dim}}_{\mathbb{F}_{2}}{\partial}(J_{b}(k_{v}))={\rm{dim}}_{\mathbb{F}_{2}}J_{b}(k_{v})[2]$
by [Mil06, Lemma I.3.3]. Thus ${\rm{dim}}_{\mathbb{F}_{2}}{\rm
H}^{1}(k_{v},G)=2\,{\rm{dim}}_{{\mathbb{F}}_{2}}J_{b}(k_{v})[2].$ Let
$k_{v}^{\rm nr}$ be the maximal unramified extension of $k_{v}$ and let ${\rm
H}^{1}_{\rm nr}(k_{v},G)$ be the kernel of ${\rm H}^{1}(k_{v},G)\to{\rm
H}^{1}(k_{v}^{\rm nr},G)$. Provided $v$ is unramified in the splitting field
of $P(t)$, evaluation of cocycles at the Frobenius element
$\textup{Frob}_{v}\in\textup{Gal}(k_{v}^{\textup{nr}}/k_{v})$ induces an
isomorphism

${\rm H}^{1}_{\rm nr}(k_{v},G)\cong G/(\textup{Frob}_{v}-1).$ (8)

Let $F/k$ be a quadratic (or trivial) extension of $k$. For each place $v$ of
$k$, we view the Kummer map for $J_{b}^{F}$ as taking values in ${\rm
H}^{1}(k_{v},G)$ too, and denote by ${\partial}^{F}:J_{b}^{F}(k_{v})\to{\rm
H}^{1}(k_{v},G)$ the resulting map. The Weil pairing on $J_{b}[2]$ is
unchanged under quadratic twist, so ${\partial}^{F}(J_{b}^{F}(k_{v}))$ is its
own orthogonal complement with respect to the local Tate pairing also.

Suppose that $v\notin S_{0,b}$. If $F/k$ is unramified at $v$, then
$J_{b}^{F}$ has good reduction at $v$ and

${\partial}^{F}(J_{b}^{F}(k_{v}))={\rm H}^{1}_{\textup{nr}}(k_{v},G).$

On the other hand, if $v$ is ramified in $F/k$, then by [HS16, Lemma 4.3] we
have

${\partial}^{F}(J_{b}^{F}(k_{v}))\cap{\rm H}^{1}_{\textup{nr}}(k_{v},G)=0.$

We now turn to the global picture. The $2$-Selmer group ${\rm
Sel}_{2}(J_{b}^{F})$ is the subgroup of ${\rm H}^{1}(k,G)$ consisting of
elements $x$ such that
$\textup{res}_{v}(x)\in{\partial}^{F}(J_{b}^{F}(k_{v}))$ for all places $v$ of
$k$. It sits in a short exact sequence

$0\to J_{b}^{F}(k)/2\to{\rm
Sel}_{2}(J_{b}^{F})\to\textcyr{Sh}(J_{b}^{F})[2]\to 0,$ (9)

where $\textcyr{Sh}(J_{b}^{F})$ is the Shafarevich–Tate group. We denote by

$\left\langle~{},~{}\right\rangle_{F}:\textcyr{Sh}(J_{b}^{F})\times\textcyr{Sh}(J_{b}^{F})\to\mathbb{Q}/\mathbb{Z}$

the Cassels–Tate pairing. Since $C_{b}^{F}$ has a rational Weierstrass point,
$\left\langle~{},~{}\right\rangle_{F}$ is alternating by [PS99, Theorem 8 and
Corollary 12]. If the $2$-primary subgroup of $\textcyr{Sh}(J_{b}^{F})$ is
finite then the same result shows that
${\rm{dim}}_{\mathbb{F}_{2}}\textcyr{Sh}(J_{b}^{F})[2]$ is even.

For a place $v$ of $k$, we denote by ${\rm Sel}_{2}(J_{b}^{F})^{(v)}$ the
subgroup of ${\rm H}^{1}(k,G)$ consisting of elements $x$ such that
$\textup{res}_{w}(x)$ lies in ${\partial}^{F}(J_{b}^{F}(k_{w}))$ for all
places $w\neq v$. That is, we impose the usual Selmer conditions at all places
other than $v$ but impose no condition at $v$. The following result has its
origin in work of Mazur–Rubin [MR10].

###### Lemma 4.1

Let $F/k$ be a quadratic extension and $v$ an odd place of $k$.

  * (i)

The image of $\textup{res}_{v}\big{(}{\rm Sel}_{2}(J_{b}^{F})^{(v)}\big{)}$ in
${\rm H}^{1}(k_{v},G)/{\partial}^{F}(J_{b}^{F}(k_{v}))$ is the orthogonal
complement of $\textup{res}_{v}\big{(}{\rm Sel}_{2}(J_{b}^{F})\big{)}$ with
respect to the non-degenerate pairing

$\frac{{\rm
H}^{1}(k_{v},G)}{{\partial}^{F}(J_{b}^{F}(k_{v}))}\times{\partial}^{F}(J_{b}^{F}(k_{v}))\to\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$

induced by (7).

  * (ii)

Suppose that $v$ is unramified in $F/k$ and $J_{b}^{F}$ has good reduction at
$v$. Let $F^{\prime}/k$ be a quadratic extension ramified at $v$ such that
${\partial}^{F}(J_{b}^{F}(k_{w}))={\partial}^{F^{\prime}}(J_{b}^{F^{\prime}}(k_{w}))$
for each place $w\neq v$. Then

${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})-{\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})=n_{1}-n_{2},$

where $n_{1}={\rm{dim}}_{\mathbb{F}_{2}}\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\big{)}$ and
$n_{2}={\rm{dim}}_{\mathbb{F}_{2}}\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}$. Moreover, writing
$r={\rm{dim}}_{\mathbb{F}_{2}}\big{(}G/(\textup{Frob}_{v}-1)\big{)}$, we have

$n_{1}+n_{2}\leq r\quad\textup{ and }\quad n_{1}+n_{2}\equiv r\pmod{2}.$

Proof. (i) This is a consequence of Poitou–Tate duality; see [Rub00, Theorem
I.7.3].

(ii) This is a special case of [Har19, Lemma 3.27]. To deduce it from that
result, take $T=\\{v\\}$ in the statement, noting that by [Har19, Lemma 3.24]
the resulting spaces $W_{T}$ and $\overline{W}_{T}$ (defined at the start of
[Har19, Section 3.6]) coincide. $\Box$

The following lemma is a variant of [Mor19, Theorem 10.12] and will be used to
control the parity of the dimension of the $2$-Selmer group in the family of
quadratic twists of $J_{b}$.

###### Lemma 4.2

Let $F=k(\sqrt{a})/k$ be a quadratic extension. For $v\in S_{0,b}$, denote by
$F_{v}$ the completion of $F$ at any place extending $v$ (the choice of which
is not important). Then we have

$(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})}=(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b})}\cdot\prod_{v\in{S_{0,b}}}\left(\big{(}d_{b},a\big{)}_{k_{v}}\cdot(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}J_{b}(k_{v})/N(J_{b}(F_{v}))}\right),$
(10)

where $N\colon J_{b}(F_{v})\to J_{b}(k_{v})$ is the norm map and
$(~{},~{})_{k_{v}}$ is the Hilbert symbol.

Proof. If $v\notin S_{0,b}$ is a prime of $k$, then by [Mor23a, Theorem 1.8]
the quantity

$(d_{b},a)_{k_{v}}\cdot(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}J_{b}(k_{v})/N(J_{b}(F_{v}))}$

is equal to the local root number of $J_{b}$ over $F_{v}$, which is $1$ since
$J_{b}$ has good reduction at $v$ (since $C_{b}$ and $C^{F}_{b}$ have a
rational Weierstrass point, the terms $\epsilon(C_{b}/k_{v})$ and
$\epsilon(C^{F}_{b}/k_{v})$ appearing in [Mor23a, Conjecture 1.7] are equal to
$0$ for all places $v$). Consequently, we may replace the product over $v\in
S_{0,b}$ in (10) with the product over all places $v$ of $k$. The result now
follows from [Mor19, Theorem 10.12] and the product formula for the Hilbert
symbol. $\Box$

###  4.2 The case when $P(t)$ has Galois group ${\rm S}_{5}$, ${\rm A}_{5}$,
or ${\rm F}_{20}$

In this section we prove Theorem 3.7 (a). The proof starts with the following
key proposition.

###### Proposition 4.3

Under the assumptions of Theorem 3.7 (a), if the Kummer surface ${\rm
Kum}(J_{b,\delta})$ is everywhere locally soluble, then there is a quadratic
extension $F/k$ such that $J_{b,\delta}^{F}({\bf A}_{k})\neq\emptyset$ and
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})$ is odd.

Proof. Take $w$ as in Theorem 3.7 (a). We first show that if $F=k(\sqrt{a})$
is a quadratic extension in which $w$ is inert, then $w$ contributes non-
trivially to the right hand side of (10), that is,

$\big{(}d_{b},a\big{)}_{k_{w}}\cdot(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}J_{b}(k_{w})/N(J_{b}(F_{w}))}=-1.$
(11)

Indeed, our assumptions on $w$ entail that the equation $y^{2}=(x-b)P(x)$,
viewed over $O_{w}$ and considered along with the usual chart at infinity,
defines a semistable model of $C_{b}$. Since its special fibre has a single
node (at $x=b$), it follows that $J_{b}$ has semiabelian reduction at $w$ and
conductor exponent $1$. In particular, by [Čes18, Corollary 4.6] the local
root number of $J_{b}$ over $F_{w}$ is $-1$, so the claimed result follows
from [Mor23a, Theorem 1.8]. (One can also compute the left hand side of (11)
directly by verifying that the assumptions on $w$ imply that the group of
connected components of the special fibre of the Néron model of $J_{b}$ over
$O_{w}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then use [Maz72,
Proposition 4.3]).

We now adapt [HS16, Proposition 6.2] (which has its origin in [SSD05, Section
5]) to construct a desired quadratic extension $F/k$. For convenience, we
recall the setup of that result. Denote by $\tilde{J}_{b,\delta}$ the blow-up
of $J_{b,\delta}$ in the $16$ fixed points of its antipodal involution
$\iota$. Let $\mathcal{Y}$ denote the quotient of
$\tilde{J}_{b,\delta}\times\mathbb{G}_{m,k}$ by the diagonal action of
$\mu_{2}$ (acting by the extension of $\iota$ on the first factor, and by $-1$
on the second factor). The fibre of $\mathcal{Y}$ over $a\in\mathbb{G}_{m}(k)$
identifies with the quadratic twist of $\tilde{J}_{b,\delta}$ by
$k(\sqrt{a})$. Let $\mathcal{Y}\subseteq\mathcal{X}$ be a smooth
compactification fitting into a commutative diagram

$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{m,k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}_{k}}$

Since $\textup{Gal}(P)$ is ${\rm S}_{5}$, ${\rm A}_{5}$ or ${\rm F}_{20}$, it
follows from Lemma 1.3 that $J_{b}$ satisfies conditions (a) and (b) of [HS16,
Section 2]. Consequently, [HS16, Proposition 6.1] applies to show that the
vertical Brauer group of $\mathcal{X}$ over $\mathbb{P}^{1}_{k}$ is the image
of $\textup{Br}(k)$ in $\textup{Br}(\mathcal{X})$.

Local solubility of ${\rm Kum}(J_{b,\delta})$ at each place $v\in
S_{0,b}\setminus\\{w\\}$ ensures the existence of a local extension
$F_{v}=k_{v}(\sqrt{a_{v}})/k_{v}$ of degree at most $2$ such that
$\tilde{J}^{F_{v}}_{b,\delta}$ has a $k_{v}$-point, say $P_{v}$. Set

$\kappa=(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b})}\cdot\prod_{v\in
S_{0,b}\setminus\\{w\\}}\left(\big{(}d_{b},a_{v}\big{)}_{k_{v}}\cdot(-1)^{{\rm{dim}}_{\mathbb{F}_{2}}J_{b}(k_{v})/N(J_{b}(F_{v}))}\right).$

If $\kappa=-1$ then take $a_{w}=1$, while if $\kappa=1$ then take $a_{w}\in
k_{w}^{\times}$ to be any element generating the unique degree $2$ unramified
extension of $k_{w}$. Let $F_{w}=k_{w}(\sqrt{a_{w}})$. Since the restriction
of $\delta$ to ${\rm H}^{1}(k_{w},G)$ is zero, we have
$\tilde{J}^{F_{w}}_{b,\delta}(k_{w})\neq\emptyset$, irrespective of the choice
of $a_{w}$. Fix $P_{w}\in\tilde{J}^{F_{w}}_{b,\delta}(k_{w})$. Using local
solubility of ${\rm Kum}(J_{b,\delta})$ at primes outside $S_{0,b}$, we extend
the collection of local points $(P_{v},\sqrt{a_{v}})_{v\in S_{0,b}}$ to an
adelic point $(y_{v})\in\mathcal{Y}({\bf A}_{k})\subseteq\mathcal{X}({\bf
A}_{k})$.

Since the quotient of the vertical Brauer group of
$\mathcal{X}\to\mathbb{P}^{1}_{k}$ by the image of $\textup{Br}(k)$ is
trivial, we can use the fibration method to produce an adelic point
$(y^{\prime}_{v})\in\mathcal{X}({\bf A}_{k})$, arbitrarily close to $(y_{v})$,
and such that the image of $(y_{v}^{\prime})$ in $\mathbb{P}^{1}({\bf A}_{k})$
is a $k$-point. Specifically, we apply [HW16, Theorem 9.17] with $B=0$ and
$U=\mathbb{G}_{m,k}$, noting that the cited result applies unconditionally
thanks to [HW16, Theorem 9.11].

Denote by $a\in\mathbb{G}_{m}(k)$ the image of $(y_{v}^{\prime})$ and let
$F=k(\sqrt{a})$. Then $J_{b,\delta}^{F}$ is everywhere locally soluble.
Further, for each $v\in S_{0,b}$, we can ensure that $a$ and $a_{v}$ are close
enough that their square roots generate the same extension of $k_{v}$. Having
done this, if $\kappa=-1$ then $w$ splits in $F$ by construction and we see
from Lemma 4.2 that ${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})$ is
odd. On the other hand, if $\kappa=1$ then $w$ is inert in $F$ and combining
Lemma 4.2 with (11) gives the same conclusion. $\Box$

Proof of Theorem 3.7 (a). We can assume without loss of generality that
$\delta\neq 0$, since otherwise ${\rm Kum}(J_{b,\delta})$ contains a $k$-line.

Suppose that ${\rm Kum}(J_{b,\delta})$ is everywhere locally soluble and let
$F=k(\sqrt{a_{0}})$ be as in the statement of Proposition 4.3. Then
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})$ is odd and
$\delta\in{\rm Sel}_{2}(J_{b}^{F})$ since $J_{b,\delta}^{F}$ is everywhere
locally soluble.

If ${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})=1$ then we are done.
Indeed, as in Section 4.1, finiteness of the $2$-primary part of
$\textcyr{Sh}(J_{b}^{F})$ implies that the $\mathbb{F}_{2}$-dimension of
$\textcyr{Sh}(J_{b}^{F})[2]$ is even. From the exact sequence (9) we conclude
that $\delta$ maps to $0$ in $\textcyr{Sh}(J_{b}^{F})$, hence the $2$-covering
$J_{b,\delta}^{F}$ has a $k$-point. Thus so does
$\textup{Kum}(J^{F}_{b,\delta})\cong\textup{Kum}(J_{b,\delta})$.

Suppose ${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})\geq 3$. We will
construct another quadratic extension $F^{\prime}/k$ such that
$\delta\in\textup{Sel}(J_{b}^{F^{\prime}})$ and

${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})={\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})-2.$

Let $T=\\{\delta=\alpha_{0},\alpha_{1},...,\alpha_{t}\\}$ be an
$\mathbb{F}_{2}$-basis for ${\rm Sel}_{2}(J_{b}^{F})$. The resulting map
$\varphi_{T}:\textup{Gal}(k_{T}/k)\to G^{t+1}\rtimes\textup{Gal}(P)$ is an
isomorphism by Lemmas 1.3 and 1.4, where $q=2$. Fix a finite set $S$ of places
of $K$ containing $S_{0,b}$ and all primes ramified in $F/k$. Fix
$\sigma_{0}\in\textup{Gal}(P)$ acting on the roots of $P(t)$ as a double
transposition, noting that

${\rm{dim}}_{\mathbb{F}_{2}}G/(\sigma_{0}-1)=2.$

Further, fix elements $x_{1}$ and $x_{2}$ of $G$ whose images in
$G/(\sigma_{0}-1)$ are $\mathbb{F}_{2}$-linearly independent, and let
$\sigma\in\textup{Gal}(k_{T}/k)$ be the preimage of
$((0,0,...,x_{1},x_{2}),\sigma_{0})$ under $\varphi_{T}$. Our assumption that
$\textup{Gal}(P)$ is ${\rm S}_{5}$, ${\rm A}_{5}$, or ${\rm F}_{20}$ implies
that every homomorphism $\textup{Gal}(P)\to\mathbb{Z}/2$ vanishes on double
transpositions. Thus by Lemma 1.7 we can find an odd place $v\notin S$, whose
Frobenius element in $\textup{Gal}(k_{T}/k)$ lies in the conjugacy class of
$\sigma$, and $a\in k^{\times}$ such that $k(\sqrt{a})/k$ is ramified at $v$,
split at all places in $S$, and is unramified otherwise. Recall that
$F=k(\sqrt{a_{0}})$. Set $F^{\prime}=k(\sqrt{aa_{0}})$. By construction, we
have
${\partial}^{F}(J_{b}^{F}(k_{w}))={\partial}^{F^{\prime}}(J_{b}^{F^{\prime}}(k_{w}))$
for each place $w\neq v$. Our choice of $\sigma$ implies both that
$\textup{res}_{v}(\delta)=0$, hence $\delta\in{\rm
Sel}_{2}(J_{b}^{F^{\prime}})$, and that the image of ${\rm
Sel}_{2}(J_{b}^{F})$ in

${\rm H}^{1}_{\textup{nr}}(k_{v},G)\cong G/(\textup{Frob}_{v}-1)$

is $2$-dimensional. By Lemma 4.1 (ii) we have

${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})={\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})-2.$

We now relabel $F^{\prime}$ as $F$ and iterate the above argument, eventually
arriving at a quadratic extension $F^{\prime}/k$ such that ${\rm
Sel}_{2}(J_{b}^{F^{\prime}})$ is generated by $\delta$. We can then conclude
as before. $\Box$

###  4.3 The case when $P(t)$ has Galois group ${\rm D}_{10}$

In this section we prove Theorem 3.7 (b). The proof proceeds along roughly the
same lines as the proof of Theorem 3.7 (a). However, there are two
complications. Firstly, as the action of the Galois group $\Gamma$ on $G$
factors through ${\rm{Gal}}(P)$, the fact that
${\rm{End}}_{k}(G)=\textup{End}_{{\rm D}_{10}}(G)=\mathbb{F}_{4}$ implies
that, given $\mathbb{F}_{2}$-linearly independent elements
$\alpha_{0},...,\alpha_{t}$ of ${\rm H}^{1}(k,G)$, it need no longer be the
case that the smallest Galois extension through which all the $\alpha_{i}$
factor has Galois group $G^{t+1}\rtimes\textup{Gal}(P)$. By Lemma 1.4, one
needs $\alpha_{0},...,\alpha_{t}$ to be $\mathbb{F}_{4}$-linearly independent
to guarantee this. Secondly, it is no longer the case that every homomorphism
$\textup{Gal}(P)\to\mathbb{Z}/2\mathbb{Z}$ vanishes on double transpositions.
The result is that, upon twisting, we are only able to guarantee that the
$\mathbb{F}_{2}$-dimension of the Selmer group decreases by $4$ at each step.
We then need an additional argument to treat the case where the twisting
process terminates in a $3$-dimensional Selmer group. To handle this, we
introduce a second descent step.

We begin with two preparatory results which help handle the first difficulty.
From now on we take all the notation from the statement of Theorem 3.7 (b).

###### Lemma 4.4

Let $w\in\\{w_{1},w_{2}\\}$ and suppose that $F/k$ is a quadratic extension
unramified at $w$. If $\alpha\in{\rm Sel}_{2}(J_{b}^{F})$ has
$\textup{res}_{w}(\alpha)\neq 0$, then

$\mathbb{F}_{4}\alpha\cap{\rm Sel}_{2}(J_{b}^{F})=\\{0,\alpha\\}.$

Proof. By assumption, ${\rm{Frob}}_{w}$ acts on the roots of $P(t)$ as a
double transposition. In particular, we have

${\rm{dim}}_{\mathbb{F}_{2}}{\partial}^{F}(J_{b}^{F}(k_{w}))={\rm{dim}}_{\mathbb{F}_{2}}J_{b}^{F}(k_{w})[2]=2.$

The natural maps ${\rm H}^{1}(k,G)\to{\rm H}^{1}(k_{w},G)\to{\rm
H}^{1}(k_{w}^{\rm nr},G)$ are linear transformations of
${\mathbb{F}}_{4}$-vector spaces. Suppose for contradiction that
$\mathbb{F}_{4}\alpha\subset{\rm Sel}_{2}(J_{b}^{F})$. Since
$\textup{res}_{w}(\alpha)\neq 0$, we then have
${\partial}^{F}(J_{b}^{F}(k_{w}))=\mathbb{F}_{4}\textup{res}_{w}(\alpha)$. As
${\rm H}^{1}_{\textup{nr}}(k_{w},G)$ is an $\mathbb{F}_{4}$-vector space,
${\partial}^{F}(J_{b}^{F}(k_{w}))\cap{\rm H}^{1}_{\textup{nr}}(k_{w},G)$ is an
${\mathbb{F}}_{4}$-vector space too. We will show however that this
intersection is isomorphic to ${\mathbb{Z}}/2$, giving the sought
contradiction.

As before, the conditions on $w$ imply that the equation $y^{2}=(x-b)P(x),$
viewed over $O_{w}$ and considered along with the usual chart at infinity,
defines a semistable model of $C_{b}$ whose special fibre has a single node at
$x=b$. The minimal proper regular model of $C_{b}$ over $O_{w}$ is obtained by
blowing up once at the node. Let $\mathcal{J}_{b}\to\textup{Spec}(O_{w})$ be
the Néron model of $J_{b}$ and let $\pi_{0}$ be the group
${\mathbb{F}}_{w}$-scheme of connected components of
$\mathcal{J}_{b}\times_{O_{w}}{\mathbb{F}}_{w}$. Since the special fibre of
the minimal regular model of $C_{b}$ consists of two components intersecting
in two ordinary double points, [BLR90, Proposition 9.6.10] shows that
$\pi_{0}$ has order $2$, hence $\pi_{0}\cong{\mathbb{Z}}/2$. Néron models
commute with unramified base change, so the same is true for $J_{b}^{F}$.

Consider the commutative diagram with exact rows and columns

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}_{\rm
nr}(k_{w},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}_{\rm
nr}(k_{w},J^{F}_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\partial}^{F}(J_{b}^{F}(k_{w}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}(k_{w},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}(k_{w},J_{b}^{F})}$

By [Mil06, Proposition I.3.8], we have ${\rm H}^{1}_{\rm
nr}(k_{w},J^{F}_{b})\cong{\rm H}^{1}_{\rm nr}(k_{w},\pi_{0})\cong\pi_{0}$,
hence $\delta^{F}(J_{b}^{F}(k_{w}))\cap{\rm H}^{1}_{\textup{nr}}(k_{w},G)$ is
identified with the kernel of the natural map

$J_{b}(k_{w}^{\rm nr})[2]/(\textup{Frob}_{w}-1)\to\pi_{0}\cong{\mathbb{Z}}/2.$
(12)

The natural reduction map $J_{b}(k_{w}^{\textup{nr}})\to\pi_{0}$ is surjective
by Hensel’s lemma. The kernel of this map is ${\mathcal{J}}_{b}^{0}(O_{w}^{\rm
nr})$, where ${\mathcal{J}}_{b}^{0}$ is the identity component of
${\mathcal{J}}_{b}$. The morphism
$[2]\colon{\mathcal{J}}_{b}^{0}\to{\mathcal{J}}_{b}^{0}$ is étale and
surjective since $w$ is an odd prime, thus ${\mathcal{J}}_{b}^{0}(O_{w}^{\rm
nr})$ is a $2$-divisible abelian group. Applying the snake lemma to the exact
sequence

$0\to{\mathcal{J}}_{b}^{0}(O_{w}^{\rm nr})\to
J_{b}(k_{w}^{\textup{nr}})\to\pi_{0}\to 0$

we obtain the surjectivity of the reduction map $J_{b}(k_{w}^{\rm
nr})[2]\to\pi_{0}$. Thus the kernel of the map in (12) is ${\mathbb{Z}}/2$,
hence $\delta^{F}(J_{b}^{F}(k_{v}))\cap{\rm
H}^{1}_{\textup{nr}}(k_{v},G)\cong{\mathbb{Z}}/2$. $\Box$

###### Lemma 4.5

Let $F/k$ be a quadratic extension unramified at $w_{1}$ and $w_{2}$. Suppose
that $\delta\in{\rm Sel}_{2}(J_{b}^{F})$ and ${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})=3$. Then there is a quadratic extension $F^{\prime}/k$
unramified at $w_{1}$ and $w_{2}$, such that $\delta\in{\rm
Sel}_{2}(J_{b}^{F^{\prime}})$ and such that either
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F^{\prime}})=1$, or
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F^{\prime}})=3$ and the
vectors of every $\mathbb{F}_{2}$-basis of ${\rm Sel}_{2}(J_{b}^{F^{\prime}})$
are $\mathbb{F}_{4}$-linearly independent in ${\rm H}^{1}(k,G)$.

Proof. Suppose that ${\rm Sel}_{2}(J_{b}^{F})$ has an $\mathbb{F}_{2}$-basis
which is not $\mathbb{F}_{4}$-linearly independent in ${\rm H}^{1}(k,G)$. Let
$V={\rm Sel}_{2}(J_{b}^{F})$ and let $W$ be the ${\mathbb{F}}_{4}$-linear span
of $V$ in ${\rm H}^{1}(k,G)$. We have
${\rm{dim}}_{{\mathbb{F}}_{2}}(W)=2{\rm{dim}}_{{\mathbb{F}}_{4}}(W)\leq 4$.
Let $x\in\mathbb{F}_{4}\setminus\mathbb{F}_{2}$. As
${\rm{dim}}_{{\mathbb{F}}_{2}}(V)=3$ we see that $V\cap xV$ is non-zero and
invariant under multiplication by $x$, so it is a non-zero
${\mathbb{F}}_{4}$-vector space. Thus ${\rm Sel}_{2}(J_{b}^{F})$ contains
${\mathbb{F}}_{4}\alpha_{1}$ for some $\alpha_{1}\neq 0$. Lemma 4.4 implies
that ${\rm res}_{w_{1}}(\alpha_{1})=0$ and
$\textup{res}_{w_{2}}(\alpha_{1})=0$. By assumption
$\textup{res}_{w_{1}}(\delta)\neq 0$ and $\textup{res}_{w_{2}}(\delta)=0$,
hence ${\rm
Sel}_{2}(J_{b}^{F})=\mathbb{F}_{4}\alpha_{1}\oplus\mathbb{F}_{2}\delta$ and so
$\textup{res}_{w_{2}}\big{(}{\rm Sel}_{2}(J_{b}^{F})\big{)}=0$.

By part (i) of Proposition 4.1, a consequence of the vanishing of
$\textup{res}_{w_{2}}\big{(}{\rm Sel}_{2}(J_{b}^{F})\big{)}$ is the existence
of an element $\alpha_{2}\in{\rm Sel}_{2}(J_{b}^{F})^{(w_{2})}$ with ${\rm
res}_{w_{2}}(\alpha_{2})\neq 0$. The set
$T=\\{\delta,\alpha_{1},\alpha_{2}\\}$ is $\mathbb{F}_{4}$-linearly
independent, thus by Lemma 1.4 the associated map
$\varphi_{T}:\textup{Gal}(k_{T}/k)\to G^{3}\rtimes\textup{Gal}(P)$ is an
isomorphism.

Choose an $\mathbb{F}_{4}$-basis $\\{x_{1},x_{2}\\}$ of $G$. Let
$\sigma\in\textup{Gal}(k_{T}/k)$ be the preimage of $((0,x_{1},x_{2}),1)$
under $\varphi_{T}$. Let $S$ be a finite set of places of $k$ containing
$S_{0,b}$ and all primes ramified in $F/k$. By Lemma 1.7 we can find a place
$v\notin S$ whose Frobenius element in $\textup{Gal}(k_{T}/k)$ lies in the
conjugacy class of $\sigma$, and $a\in k^{\times}$ such that $k(\sqrt{a})/k$
is ramified at $v$, split at all places in $S$ and is unramified otherwise.
Write $F=k(\sqrt{a_{0}})$ and set $F^{\prime}=k(\sqrt{aa_{0}})$, noting that
$F^{\prime}/k$ is unramified at $w_{1}$ and $w_{2}$. Since $\textup{Frob}_{v}$
has trivial image in $\textup{Gal}(P)$ by construction, evaluation of cocycles
at $\textup{Frob}_{v}$ gives an isomorphism ${\rm
H}^{1}_{\textup{nr}}(k_{v},G)\cong G.$ Our choice of $\sigma$ ensures that
$\textup{res}_{v}(\delta)=0$, hence $\delta\in{\rm
Sel}_{2}(J_{b}^{F^{\prime}})$. It also ensures that ${\rm
H}^{1}_{\textup{nr}}(k_{v},G)={\mathbb{F}}_{4}{\rm
res}_{v}(\alpha_{1})\oplus{\mathbb{F}}_{4}{\rm res}_{v}(\alpha_{2})$. In
particular, we have ${\rm{dim}}_{\mathbb{F}_{2}}\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}=2$.

By part (ii) of Proposition 4.1 we see that $\textup{dim}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})=1+n_{1},$ where
$n_{1}={\rm{dim}}_{\mathbb{F}_{2}}\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\big{)}$ is either $0$ or $2$. If $n_{1}=0$ then
we are done, so suppose that $n_{1}=2$. By construction, ${\rm
Sel}_{2}(J_{b}^{F^{\prime}})$ is contained in ${\rm
Sel}_{2}(J_{b}^{F})^{(v)}$, hence $\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\big{)}$ and $\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}$ are orthogonal with respect to the local Tate
pairing at $v$. Since ${\rm H}^{1}_{\textup{nr}}(k_{v},G)$ is its own
orthogonal complement with respect to this pairing, and since
$\delta^{F^{\prime}}(J_{b}^{F^{\prime}}(k_{v}))$ and ${\rm
H}^{1}_{\textup{nr}}(k_{v},G)$ have trivial intersection, the restriction of
(7) to a pairing

${\rm
H}^{1}_{\textup{nr}}(k_{v},G)\times\delta^{F^{\prime}}(J_{b}^{F^{\prime}}(k_{v}))\to\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$
(13)

is non-degenerate. From this we conclude that the $2$-dimensional
$\mathbb{F}_{2}$-vector spaces $\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}$ and $\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\big{)}$ are orthogonal complements with respect
to the pairing (13). Since $\textup{res}_{v}(\alpha_{2})\in{\rm
H}^{1}_{\textup{nr}}(k_{v},G)$ does not lie in $\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}$, we deduce the existence of an element
$\eta\in{\rm Sel}_{2}(J_{b}^{F^{\prime}})$ such that $\textup{res}_{v}(\eta)$
pairs non-trivially with $\textup{res}_{v}(\alpha_{2})$. Since both $\eta$ and
$\alpha_{2}$ satisfy the Selmer conditions for $J_{b}^{F}$ away from $w_{2}$
and $v$, we conclude from reciprocity for the Brauer group of $k$ that
$\textup{res}_{w_{2}}(\eta)$ pairs non-trivially with
$\textup{res}_{w_{2}}(\alpha_{2})$ under the local Tate pairing at $w_{2}$. In
particular, $\textup{res}_{w_{2}}(\eta)\neq 0$. Thus ${\rm
Sel}_{2}(J_{b}^{F^{\prime}})\cong({\mathbb{F}}_{2})^{3}$ contains
${\mathbb{F}}_{2}\delta\oplus{\mathbb{F}}_{2}\eta$ and we have
$\textup{res}_{w_{2}}(\eta)\neq 0$. By the argument in the beginning of the
proof, if ${\rm Sel}_{2}(J_{b}^{F^{\prime}})$ has an $\mathbb{F}_{2}$-basis
which is not $\mathbb{F}_{4}$-linearly independent in ${\rm H}^{1}(k,G)$, then
we have $\textup{res}_{w_{2}}\big{(}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\big{)}=0$, a contradiction. $\Box$

We now give the analogue of Proposition 4.3.

###### Proposition 4.6

Under the assumptions of Theorem 3.7 (b), if the Kummer surface ${\rm
Kum}(J_{b,\delta})$ is everywhere locally soluble, then there is a quadratic
extension $F/k$, unramified at $w_{1}$ and $w_{2}$, such that
$J_{b,\delta}^{F}({\bf A}_{k})\neq\emptyset$ and
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})$ is odd.

Proof. We follow the proof of Proposition 4.3 verbatim, with $w_{2}$ in place
of $w$, until it comes to showing that the vertical Brauer group of
$\mathcal{X}$ over $\mathbb{P}^{1}_{k}$ is the image of $\textup{Br}(k)$ in
$\textup{Br}(\mathcal{X})$. Since $\textup{Gal}(P)$ is now ${\rm D}_{10}$,
condition (b) of [HS16, Proposition 6.1] is satisfied, but condition (a) is
not. However, writing $T=\\{\delta\\}$, by Lemma 1.4 it is nevertheless the
case that the resulting map $\varphi_{T}:\textup{Gal}(k_{T}/k)\to
G\rtimes\textup{Gal}(P)$ is an isomorphism, which is enough for the proof of
[HS16, Proposition 6.1] to carry over unchanged to show that the vertical
Brauer group of $\mathcal{X}\to\mathbb{P}^{1}_{k}$ is the image of
$\textup{Br}(k)$. We then continue to follow the proof of Proposition 4.3 to
construct the sought extension $F/k$, with the only additional input needed
being the existence of an unramified local extension $F_{w_{1}}/k_{w_{1}}$
such that $\tilde{J}^{F_{w_{1}}}_{b,\delta}$ has a $k_{w_{1}}$-point. Such an
extension is afforded by [Har19, Lemma 3.16]. $\Box$

Proof of Theorem 3.7 (b). Suppose that ${\rm Kum}(J_{b,\delta})$ is everywhere
locally soluble and let $F/k$ be as in the statement of Proposition 4.6,
noting that $\delta\in{\rm Sel}_{2}(J_{b}^{F})$. Suppose that
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})\geq 5$. We claim that we
can find a further quadratic extension $F^{\prime}/k$, unramified at $w_{1}$
and $w_{2}$, such that $\delta\in{\rm Sel}_{2}(J_{b}^{F^{\prime}})$ and

${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})={\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})-4.$

To begin, denote by $V$ the subgroup of ${\rm Sel}_{2}(J_{b}^{F})$ consisting
of elements $\alpha$ for which $\mathbb{F}_{4}\alpha\subseteq{\rm
Sel}_{2}(J_{b}^{F})$. This is an $\mathbb{F}_{4}$-vector space. Let
$\alpha_{1},...,\alpha_{n}$ be an $\mathbb{F}_{4}$-basis for $V$. Lemma 4.4
and the assumption that $\textup{res}_{w_{1}}(\delta)\neq 0$ means that
$\delta\notin V$. Fix $x\in\mathbb{F}_{4}\setminus\mathbb{F}_{2}$, write
$\alpha_{0}=\delta$, and extend
$\alpha_{0},\alpha_{1},x\alpha_{1},...,\alpha_{n},x\alpha_{n}$ to an
$\mathbb{F}_{2}$-basis for ${\rm Sel}_{2}(J_{b}^{F})$, adding in elements
$\alpha_{n+1},...,\alpha_{t}$. The set
$T=\\{\alpha_{0},\alpha_{1},...,\alpha_{t}\\}$ is then
$\mathbb{F}_{4}$-linearly independent. By Lemma 1.4, the resulting map
$\varphi_{T}:\textup{Gal}(k_{T}/k)\to G^{t+1}\rtimes\textup{Gal}(P)$ is an
isomorphism. Since ${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F})\geq
5$, we can thus find $\sigma\in\textup{Gal}(k_{T}/k)$ with trivial image in
$\textup{Gal}(P)$, such that $\alpha_{0}(\sigma)=0$ and the elements

$\alpha_{1}(\sigma),x\alpha_{1}(\sigma),...,\alpha_{n}(\sigma),x\alpha_{n}(\sigma),\alpha_{n+1}(\sigma),...,\alpha_{t}(\sigma)$

span $G$ (note that since $\sigma$ has trivial image in $\textup{Gal}(P)$ we
can unambiguously evaluate cocycle classes at $\sigma$). For example, if
$n\geq 2$ then we can take $x_{1},x_{2}$ to be an $\mathbb{F}_{4}$-basis for
$G$ and take $\sigma$ to be the preimage of $((0,x_{1},x_{2},0,...,0),1)$
under $\varphi_{T}$.

As usual, we now fix a finite set $S$ of places of $k$ containing $S_{0,b}$
and all primes ramified in $F/k$. By Lemma 1.7 there is a place $v\notin S$
whose Frobenius element in $\textup{Gal}(k_{T}/k)$ lies in the conjugacy class
of $\sigma$, and $a\in k^{\times}$ such that $k(\sqrt{a})/k$ is ramified at
$v$, split at all places in $S$, and is unramified otherwise. Write
$F=k(\sqrt{a_{0}})$ and set $F^{\prime}=k(\sqrt{aa_{0}})$, noting that
$F^{\prime}/k$ is unramified at $w_{1}$ and $w_{2}$. Then
$\textup{res}_{v}(\delta)=0$ and
${\rm{dim}}_{\mathbb{F}_{2}}\textup{res}_{v}\big{(}{\rm
Sel}_{2}(J_{b}^{F})\big{)}=4.$ We conclude that $\delta\in{\rm
Sel}_{2}(J_{b}^{F^{\prime}})$ and that, by Proposition 4.1 (ii), we have

${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})={\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F})-4.$

We now relabel $F^{\prime}$ as $F$ and iterate the above argument, eventually
arriving at a quadratic extension $F^{\prime}/k$, unramified at $w_{1}$ and
$w_{2}$, such that $\delta\in{\rm Sel}_{2}(J_{b}^{F^{\prime}})$ and
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F^{\prime}})\in\\{1,3\\}$. If
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F^{\prime}})=1$ then we are
done, so suppose that ${\rm{dim}}_{\mathbb{F}_{2}}{\rm
Sel}_{2}(J_{b}^{F^{\prime}})=3$. By Lemma 4.5 we can further assume that every
$\mathbb{F}_{2}$-basis for ${\rm Sel}_{2}(J_{b}^{F^{\prime}})$ is
$\mathbb{F}_{4}$-linearly independent.

To ease notation, let us once more relabel $F^{\prime}$ as $F$. For any
further quadratic extension $F^{\prime}/k$, pulling back the Cassels–Tate
pairing along ${\rm
Sel}_{2}(J_{b}^{F^{\prime}})\to\textcyr{Sh}(J_{b}^{F^{\prime}})[2]$ gives an
alternating pairing

$\left\langle~{},~{}\right\rangle_{F^{\prime},2}:{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\times{\rm
Sel}_{2}(J_{b}^{F^{\prime}})\to\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}.$

Having arranged that the elements of $\textup{Sel}_{2}(J_{b}^{F})$ are
$\mathbb{F}_{4}$-linearly independent, we can follow [Mor23b, Sections 3-5] to
find a final quadratic extension $F^{\prime}/k$ such that ${\rm
Sel}_{2}(J_{b}^{F^{\prime}})={\rm Sel}_{2}(J_{b}^{F})$ as subgroups of ${\rm
H}^{1}(k,G)$, and such that $\delta$ is the unique non-trivial element of
${\rm Sel}_{2}(J_{b}^{F^{\prime}})$ in the kernel of
$\left\langle~{},~{}\right\rangle_{F^{\prime},2}$. Assuming finiteness of
$\textcyr{Sh}(J_{b}^{F^{\prime}})[2^{\infty}]$, this is sufficient to conclude
that $\textup{Kum}(J_{b,\delta})$ has a $k$-point. Indeed, then
${\rm{dim}}_{\mathbb{F}_{2}}{\rm Sel}_{2}(J_{b}^{F^{\prime}})=3$ while
${\rm{dim}}_{\mathbb{F}_{2}}\textcyr{Sh}(J_{b}^{F^{\prime}})[2]$ is even. From
the exact sequence (9), some element of $J_{b}^{F^{\prime}}(k)$ must have non-
trivial image $\alpha$ in ${\rm Sel}_{2}(J_{b}^{F^{\prime}})$. Then $\alpha$
is in the kernel of $\left\langle~{},~{}\right\rangle_{F^{\prime},2}$, hence
$\alpha=\delta$ and $\delta$ maps to $0$ in
$\textcyr{Sh}(J_{b}^{F^{\prime}})$.

To construct the sought extension $F^{\prime}/k$, fix an
$\mathbb{F}_{2}$-basis $T=\\{\alpha_{0}=\delta,\alpha_{1},\alpha_{2}\\}$ for
$\textup{Sel}_{2}(J_{b}^{F})$. Pick cocycles
$\tilde{\alpha}_{0},\tilde{\alpha}_{1}$ and $\tilde{\alpha}_{2}$ representing
these classes. By Lemma 1.4, having arranged that $T$ is
$\mathbb{F}_{4}$-linearly independent, the resulting map
$\varphi_{T}:\textup{Gal}(k_{T}/k)\to G^{3}\rtimes\textup{Gal}(P)$ is an
isomorphism. Since $\alpha_{0},\alpha_{1}$ and $\alpha_{2}$ lie in ${\rm
Sel}_{2}(J_{b}^{F})$, it follows from local Tate duality and reciprocity for
${\rm{Br}}(k)$ that the cup products $\alpha_{i}\cup\alpha_{j}$ are trivial in
${\rm H}^{2}(k,\mu_{2})$ for each $i,j$. For each $i<j$, choose $1$-cochains
$\gamma_{ij}:\Gamma\to\mu_{2}$ such that
$d\gamma_{ij}=\tilde{\alpha}_{i}\cup\tilde{\alpha}_{j}$. The elements
$\chi_{ij}=\textup{res}_{k_{T}}(\gamma_{ij})$ are quadratic characters
$G_{k_{T}}\to\mu_{2}$. As in [Mor23b, Proposition 4.5], a formal consequence
of the fact that $\varphi_{T}$ is an isomorphism is that $\chi_{01}$,
$\chi_{02}$ and $\chi_{12}$ are $\mathbb{F}_{2}$-linearly independent and that
the smallest extension $L/k_{T}$ through which they all factor is Galois over
$k$ with Galois group a central extension of $\textup{Gal}(k_{T}/k)$ by
$(\mu_{2})^{3}$.

Denote by $\mathcal{C}\subseteq\textup{Gal}(L/k)$ the set of elements acting
on the roots of $P(t)$ as a $5$-cycle. For each $\sigma\in\mathcal{C}$ we have
$G/(\sigma-1)=0$, hence for $i=0,1,2$ there is a unique element
$x_{i,\sigma}\in G$ such that
$\tilde{\alpha}_{i}(\sigma)=(\sigma-1)x_{i,\sigma}$. Writing $\mu_{2}$
additively and identifying it with $\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$ in the
unique way, we follow [Mor23b, Section 3.1] and define, for each $i<j$, a
class function $\psi_{ij}:\mathcal{C}\to\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$ by
setting

$\psi_{ij}(\sigma)=(x_{i,\sigma}\cup\tilde{\alpha}_{j}+\gamma_{ij})(\sigma).$

Let $B$ be the unique $\tfrac{1}{2}\mathbb{Z}/\mathbb{Z}$-valued alternating
pairing on ${\rm Sel}_{2}(J_{b}^{F})$ such that

$B(\alpha_{0},\alpha_{1})=0=B(\alpha_{0},\alpha_{2})\quad\textup{ and }\quad
B(\alpha_{1},\alpha_{2})=\tfrac{1}{2},$

noting the $\delta$ is the unique non-trivial element in the kernel of $B$. As
in [Mor23b, Corollary 4.17], linear independence of the characters
$\chi_{01}$, $\chi_{02}$ and $\chi_{12}$ means that we can find
$\sigma\in\mathcal{C}$ such that, for each $i<j$, we have

$\psi_{ij}(\sigma)=\left\langle\alpha_{i},\alpha_{j}\right\rangle_{F,2}+B(\alpha_{i},\alpha_{j}).$

Fix a finite set $S$ of places of $k$ containing $S_{0,b}$ and all primes
ramified in $LF/k$. As in [Mor23b, Proposition 4.13], any homomorphism
$\textup{Gal}(L/k)\to\mathbb{Z}/2$ factors through $\textup{Gal}(P)$, hence
vanishes on $\sigma$. Thus, arguing as in Lemma 1.7, we can find a place
$v\notin S$ whose Frobenius element in $\textup{Gal}(L/k)$ lies in the
conjugacy class of $\sigma$, and $a\in k^{\times}$ such that $k(\sqrt{a})/k$
is ramified at $v$, split at all places in $S$, and is unramified otherwise.
Write $F=k(\sqrt{a_{0}})$ and set $F^{\prime}=k(\sqrt{aa_{0}})$. By [Mor23b,
Lemma 2.8] we have ${\rm Sel}_{2}(J_{b}^{F^{\prime}})={\rm
Sel}_{2}(J_{b}^{F})$ as subgroups of ${\rm H}^{1}(k,G)$ and by [Mor23b,
Proposition 3.3] we have
$\left\langle\alpha_{i},\alpha_{j}\right\rangle_{F^{\prime},2}=B(\alpha_{i},\alpha_{j})$
for each $i<j$. Since both pairings $B$ and
$\left\langle~{},~{}\right\rangle_{F^{\prime},2}$ are alternating, they must
be equal. Thus $\delta$ is the unique non-trivial element in the kernel of
$\left\langle~{},~{}\right\rangle_{F^{\prime},2}$. $\Box$

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