#  $2$-descent for Bloch–Kato Selmer groups and rational points on
hyperelliptic curves II

Netan Dogra

###### Abstract.

We give refined methods for proving finiteness of the Chabauty–Coleman–Kim set
$X(\mathbb{Q}_{2})_{2}$, when $X$ is a hyperelliptic curve with a rational
Weierstrass point. The main developments are methods for computing Selmer
conditions at $2$ and $\infty$ for the mod $2$ Bloch–Kato Selmer group
associated to the higher Chow group $\mathrm{CH}^{2}(\mathrm{Jac}(X),1)$. As a
result we show that most genus 2 curves in the LMFDB of Mordell–Weil rank 2
with exactly one rational Weierstrass point satsify
$\\#X(\mathbb{Q}_{2})_{2}<\infty$. We also obtain a field-theoretic
description of second descent on the Jacobian of a hyperelliptic curve (under
some conditions).

###### Contents

  1. 1 Introduction
  2. 2 A review of $2$-descent for Bloch–Kato Selmer groups
  3. 3 Boundary maps for $J[4]$
  4. 4 The boundary map for $\wedge^{2}J[4]$
  5. 5 Lifting obstructions at $\infty$
  6. 6 Finiteness criteria
  7. 7 Examples

##  1\. Introduction

This paper is concerned with the question of extending the applicability of
the Chabauty–Coleman–Kim method for studying rational points on higher genus
curves $X/\mathbb{Q}$. This method produces sets $X(\mathbb{Q}_{p})_{n}\subset
X(\mathbb{Q}_{p})$ for each $n>0$, which should provide successive
approximations to $X(\mathbb{Q})$:

$X(\mathbb{Q}_{p})\supset X(\mathbb{Q}_{p})_{1}\supset
X(\mathbb{Q}_{p})_{2}\supset\ldots\supset X(\mathbb{Q}).$

Kim has conjectured [BDCKW18] that for $n\gg 0$,
$X(\mathbb{Q}_{p})_{n}=X(\mathbb{Q})$. In the context we work in (that of
genus two curves) it is often enough to produce a finite set, since then the
Mordell–Weil sieve can rule out ‘fake rational points’.

When $n=1$, the problem of finiteness of $X(\mathbb{Q}_{p})_{1}$ is well
understood, as this is exactly the set produced by the Chabauty–Coleman
method: a sufficient condition is that $r<g$, where $r$ is the Mordell–Weil
rank of the Jacobian of $X$ and $g$ is the genus of $X$. When $n=2$, it is
known that $X(\mathbb{Q}_{p})_{2}$ is finite when $r<g+\rho(J)-1$, where
$\rho(J):=\operatorname{rk}\mathrm{NS}(J)$ is the rank of the Néron–Severi
group of $J$ (over $\mathbb{Q}$).

The obstruction to extending this result is the rank of the mysterious
$\mathbb{Z}_{p}$-module
$H^{1}_{f}(\operatorname{Gal}(\overline{\mathbb{Q}}|\mathbb{Q}),\wedge^{2}T_{p}J)$,
which is the Bloch–Kato Selmer group of the higher Chow group
$\mathrm{CH}^{2}(\mathrm{Jac}(X),1)$, or of the Galois representation
$H^{2}_{\mathrm{\acute{e}t}}(\mathrm{Jac}(X)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p}(2))\simeq\wedge^{2}T_{p}J$.
The Bloch–Kato conjectures [BK90] give a precise formula for this number,
which implies that $X(\mathbb{Q}_{p})_{2}$ should be finite whenever
$r<g^{2}+\rho(J)-1$. In this series of papers we introduce $2$-descent methods
for proving bounds on
$H^{1}_{f}(\operatorname{Gal}(\overline{\mathbb{Q}}|\mathbb{Q}),\wedge^{2}T_{2}J)$.
The first paper in this series [Dog23] applied these to provably determine the
set of rational points on the curve $y^{2}-y=x^{5}-x$, answering a question of
Bugeaud, Mignotte, Siksek, Stoll and Tengely [BMS+08]. In this paper we extend
these methods to provide fairly robust criteria for finiteness of
$X(\mathbb{Q}_{2})_{2}$ for a genus $2$ curve $X$ with a rational Weierstrass
point. As an application we prove the following.

######  Theorem 1.

Of the 6,603 curves in the LMFDB with Mordell–Weil rank 2 and exactly one
rational Weierstrass point, at least $3,323$ satisfy
$\\#X(\mathbb{Q}_{2})_{2}<\infty$. Moreover, for each such $X$,
$X^{\prime}(\mathbb{Q}_{2})_{2}$ is finite whenever $X^{\prime}$ is a
quadratic twist of $X$ with Jacobian of Mordell–Weil rank $\leq 2$.

By contrast, only 8 of the the 6,603 curves in question satisfy the condition
$\rho(J)>1$ usually associated with quadratic Chabauty.

###  1.1. Relation to elliptic curve Chabauty

A number of authors including Bruin [Bru03], Flynn and Wetherell [FW99] and
recently Hast [Has22] have studied rational points on genus two curves using a
method called elliptic curve Chabauty. Specifically, given a genus 2 curve $X$
over $\mathbb{Q}$ with a rational Weierstrass point one can find a finite set
of covers $f_{\alpha}:Y_{\alpha}\to X$ (all isomorphic over
$\overline{\mathbb{Q}}$) such that $X(\mathbb{Q})=\cup f_{\alpha}(Y_{\alpha})$
and such that $\mathrm{Jac}(Y_{\alpha})$ contains an isogeny factor isomorphic
to the Weil restriction of an elliptic curve. This elliptic curve will be the
Jacobian of the genus one curve defined by the quartic obtained by removing a
root of a degree 5 polynomial defining $X$.

This method may also be thought of as a nonabelian version of the
Chabauty–Coleman method, in the sense that the Tate module of the Jacobian of
$Y_{\alpha}$ is a subquotient of the étale fundamental group of
$X_{\overline{\mathbb{Q}}}$. In fact the analogy is stronger: as we explain in
section 6.1, the Galois cohomology of $\wedge^{2}J[2]$ and
$\mathrm{Jac}(Y_{\alpha})[2]$ are closed related. Intriguingly, however, the
Selmer conditions can be different, meaning that one can instances where the
$2$-descent methods described in this paper can prove finiteness of
$X(\mathbb{Q}_{2})_{2}$, but $2$-descent methods for elliptic curve Chabauty
are insufficient, and vice versa.

###  1.2. Arithmetic statistics of Bloch–Kato Selmer groups

Although this paper only considers the Bloch–Kato Selmer groups
$H^{1}_{f}(\mathbb{Q},\wedge^{2}T_{2}J)$ for a hyperelliptic Jacobian $J$, it
seems reasonable to believe that the general idea (bounding ranks of
Bloch–Kato Selmer groups by ‘elementary’ descent methods) has a larger realm
of applicability. In considering the merits of such an approach, one may
compare with the case of Selmer groups of elliptic curves. Here two notable
applications of elementary $2$-descent are explicit computations on a given
elliptic curve, and proving results in arithmetic statistics. This paper, and
the previous paper in this series, demonstrate the analogue of the first
application for the Selmer group of a higher Chow group. It would be
interesting to further explore the analogue of the second application, i.e. to
explore the extent to which it possible to prove results about Bloch–Kato
Selmer groups ‘on average’, or for a positive proportion of motives in a
family.

The simplest (new) case would seem to be the Galois representation
$T_{2}J(n)$, where $J$ is a $g$-dimensional hyperelliptic Jacobian and $n$ is
a (nonzero) integer. Then the Bloch–Kato conjectures imply that the rank of
$H^{1}_{f}(\mathbb{Q},T_{2}J(n))$ should be $g$ when $n>0$ and $0$ when $n<0$.
In proposition 8 we explain a relation between this conjecture and certain
results of Ho, Shankar and Varma [HSV18].

###  1.3. Notation

We will sometimes denote the Galois cohomology of a field $K$ with values in a
$\operatorname{Gal}(K)$-module $M$ by $H^{i}(K,M)$. We will sometimes also use
the notation $H^{i}(K,M)$ when $K=\prod K_{i}$ is merely an étale algebra with
field factors $K_{i}$, and $M=(M_{i})$ is a tuple of
$\operatorname{Gal}(K)$-module $M_{i}$. In this case one can take $H^{i}(K,M)$
to denote the étale cohomology of $\mathrm{Spec}(K)$ with values in the
corresponding sheaf on $\mathrm{Spec}(K)_{\mathrm{\acute{e}t}}$.

Given a finite extension $L|K$ of étale algebras, corresponding to a product
$L_{ij}|K_{i}$ of field extensions, and a tuple $(M_{ij})$ of
$\operatorname{Gal}(L_{ij})$-modules, we write $\mathrm{Ind}^{L}_{K}M$ to mean
the tuple
$(\mathrm{Ind}^{\operatorname{Gal}(K_{i})}_{\operatorname{Gal}(L_{ij})}M_{ij})_{i}$
of $\operatorname{Gal}(K_{i})$-modules.

### Acknowledgements

I am grateful to Lee Berry for many helpful discussions about boundary maps.
This research was supported by a Royal Society University Research Fellowship.

##  2\. A review of $2$-descent for Bloch–Kato Selmer groups

We first (very briefly) recall the context of this paper. More details can be
found in [Dog23] (although there are usually better references for what we
describe below, this reference will usually prove convenient because this
paper is a continuation of it). Let $X/\mathbb{Q}$ be a smooth projective
geometrically irreducible curve of genus $g$ with Jacobian $J$. We have the
following sufficient condition for finiteness of $X(\mathbb{Q}_{p})_{2}$.

######  Lemma 1 ([Dog23], Lemma 25).

The set $X(\mathbb{Q}_{p})_{2}$ is finite whenever

$\dim
H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}V_{p}J)<\frac{(3g-2)(g+1)}{2}-\operatorname{rk}J(\mathbb{Q}).$

The goal of this paper is to find ways for bounding the dimension of
$H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}V_{2}J)$. The dimension of
$H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}V_{2}J)$ is predicted by the following
special case of the Bloch–Kato conjectures (see [Dog23, §6] for an explanation
of the deduction).

######  Conjecture 1.

[[BK90]] Let $Z/\mathbb{Q}$ be a smooth projective variety with good reduction
outside $S$, suppose $2j\geq i$. Then

$\dim
H^{1}_{f}(G_{\mathbb{Q}},H^{j}(Z_{\overline{\mathbb{Q}}},\mathbb{Q}_{p}(i)))=0.$

The dimension of $H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}V_{2}J)$ is
(unconditionally) equal, via the short exact sequence

$0\to\wedge^{2}T_{2}J\to\wedge^{2}V_{2}J\to\wedge^{2}V_{2}J/\wedge^{2}T_{2}J\to
0,$

to the rank of $H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$.

In this paper, unlike [Dog23], we do not attempt to use the crystalline
condition. Instead, we will try to estimate the rank of the larger space
$H^{1}_{f,\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$. By definition this is
simply the subspace of cohomology classes which are unramified outside a
finite set of primes and whose restriction to each prime away from $2$ is
torsion. We will sometimes denote this subspace simply by
$H^{1}_{\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$.

We first explain how much information working with
$H^{1}_{f,\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$. throws away. We have

$\dim H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)\leq\dim
H^{1}_{f,\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)+1.$

In general, we would expect this inequality to be strict, as explained by the
following lemma, which follows straightforwardly from standard facts about
Galois cohomology and $p$-adic Hodge theory (see e.g. [Ber13]).

######  Lemma 2.

We have

$\dim H^{1}(G_{\mathbb{Q}_{p}},\wedge^{2}V_{p}J)-\dim
H^{1}_{f}(G_{\mathbb{Q}_{p}},\wedge^{2}V_{p}J)=\binom{g}{2}+\dim_{\mathbb{Q}_{p}}\mathrm{Hom}(\mathbb{Q}_{p}(1),\wedge^{2}V_{p}J).$

Hence Conjecture 1 predicts that, replacing $H^{1}_{f}$ with
$H^{1}_{f,\\{2\\}}$, we should be able to prove finiteness of
$X(\mathbb{Q}_{2})_{2}$ whenever

$\operatorname{rk}J(\mathbb{Q})<\binom{g+1}{2}.$

For example, when $g=2$, we expect $X(\mathbb{Q}_{p})_{2}$ to be finite
whenever the Mordell–Weil rank of the Jacobian is less than $4$, but we expect
to be able to prove finiteness without using the crystalline condition
whenever the rank is less than $3$.

The basic strategy for bounding the rank of
$H^{1}_{f}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$, or
$H^{1}_{f,\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$, is to try to describe
their image in $H^{1}(G_{\mathbb{Q}},\wedge^{2}J[2])$. Recall that by the
short exact sequence

$0\to\wedge^{2}T_{2}J\stackrel{{\scriptstyle\cdot
2}}{{\longrightarrow}}\wedge^{2}T_{2}J\to\wedge^{2}J[2]\to 0$

we have a short exact sequence

$0\to H^{1}(K,\wedge^{2}T_{2}J)\otimes\mathbb{F}_{2}\to
H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}T_{2}J)[2]\to 0.$

Hence our goal is to find a subvector space of
$H^{1}(\mathbb{Q},\wedge^{2}T_{2}J)\otimes\mathbb{F}_{2}$ of dimension at
least that of $H^{1}_{f}(\mathbb{Q},\wedge^{2}V_{2}J)$, and explicitly bound
the dimension of its image in $H^{1}(\mathbb{Q},\wedge^{2}J[2])$.

We now recall some constructions from [Dog23] which sometimes achieve this.
The starting point is an explicit field-theoretic description of
$H^{1}(\mathbb{Q},\wedge^{2}J[2])$. Let $K$ be a field of characteristic
different from $2$, and let $X$ be a hyperelliptic curve defined by a
separable polynomial $f\in K[x]$ of odd degree. Let $K_{f}$ denote the étale
algebra $K[x]/(f)$. Let $K_{f,2}$ denote the étale algebra
$K[x,y,\frac{1}{x-y}]/(f(x),f(y))$. Let $K_{f}^{(2)}\subset K_{f,2}$ denote
the subalgebra fixed by the involution swapping $\overline{x}$ and
$\overline{y}$. Let

$\mathrm{Nm}:K_{f}^{(2)}\to K_{f}$

be the composite of the inclusion $K_{f}^{(2)}\to K_{f,2}$ with the norm map
$K_{f,2}\to K_{f}$.

######  Proposition 1 ([Dog23], Proposition 1).

We have an isomorphism

$H^{1}(K,\wedge^{2}J[2])\simeq\mathrm{Ker}(K_{f}^{(2),\times}\otimes\mathbb{F}_{2}\stackrel{{\scriptstyle\mathrm{Nm}}}{{\longrightarrow}}K_{f}^{\times}\otimes\mathbb{F}_{2}).$

######  Proposition 2 ([Dog23]).

Suppose that $X$ has semistable reduction outside $S\cup\\{2\\}$, where $S$ is
a finite set of odd primes.

  1. (1)

Under the isomorphism from Lemma 2, the image of
$H^{1}_{\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}T_{2}J)$ in
$\mathrm{Ker}(\mathbb{Q}_{f}^{(2),\times}\otimes\mathbb{F}_{2}\to\mathbb{Q}_{f}^{\times}\otimes\mathbb{F}_{2})$
is contained in

$M:=\mathrm{Ker}(\mathbb{Q}_{f}^{(2),\times}\otimes\mathbb{F}_{2}\stackrel{{\scriptstyle\mathrm{val}}}{{\longrightarrow}}\oplus_{v\notin
S}\mathbb{F}_{2}^{\\#\mathrm{Spec}(\mathbb{Q}_{v,f}^{(2)})}).$

Here $\mathrm{val}$ is the valuation map sending an element of
$\mathbb{Q}_{f}^{(2),\times}\otimes\mathbb{F}_{2}$ to its valuation mod $2$
under all embeddings from a field factor of $\mathbb{Q}_{f}$ to a finite
extension of $\mathbb{Q}_{v}$ for a prime $v$ not in $S\cup\\{2\\}$.

  2. (2)

If

$\dim_{\mathbb{F}_{2}}\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}})[2]=\dim_{\mathbb{F}_{2}}\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}})[2]$

then the image of $H^{1}_{\\{2\\}}(\mathbb{Q},\wedge^{2}T_{2}J)$ in $M$ is
contained in the subspace of

$\mathrm{Ker}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}}\left[\frac{1}{2\prod_{v\in
S}v}\right]^{\times}\otimes\mathbb{F}_{2}\to\mathbb{Q}_{f}^{\times}\otimes\mathbb{F}_{2}$

whose image in
$\mathbb{R}_{f}^{(2),\times}\otimes\mathbb{F}_{2}\oplus\mathbb{Q}_{2,f}^{(2),\times}\otimes\mathbb{F}_{2}$
is in the image of $H^{1}(\mathbb{Q}_{2},\wedge^{2}T_{2}J)\oplus
H^{1}(\mathbb{R},\wedge^{2}T_{2}J)$.

###### Proof.

We briefly explain how to deduce this statement from results in [Dog23]. As
explained in [Dog23, Lemma 19], the condition that $X$ has semistable
reduction at $v\neq 2$ allows us to ensure
$H^{1}_{f,\\{2\\}}(\mathbb{Q},\wedge^{2}T_{2}J)\otimes\mathbb{F}_{2}$ maps
into

$H^{1}_{\mathrm{ur}}(\mathbb{Q}_{v},\wedge^{2}J[2])=\mathrm{Ker}(\mathbb{Q}_{f}^{(2),\times}\otimes\mathbb{F}_{2}\to\mathbb{Q}_{f}^{\times}\otimes\mathbb{F}_{2}\oplus\oplus_{w\in\mathrm{Spec}(\mathbb{Q}_{f}^{(2)})}\mathbb{F}_{2}),$

where the second map is given by valuation. The condition on the class group
allows us to ensure that $M$ can be identified with a subspace of
$\mathrm{Ker}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}}\left[\frac{1}{2\prod_{v\in
S}v}\right]^{\times}\otimes\mathbb{F}_{2}$ by a standard Galois cohomological
argument [PS97, Proposition 12.6] (see [Dog23, Lemma 26] for details of the
deduction). ∎

##  3\. Boundary maps for $J[4]$

The main result in this paper is a method for computing appropriate ‘Selmer
conditions’ on $H^{1}(\mathbb{Q},\wedge^{2}J[2])$ at the prime 2. We have two
approaches to describing the image of
$H^{1}_{f}(G_{\mathbb{Q}_{2}},\wedge^{2}T_{2}J)$ in
$H^{1}(G_{\mathbb{Q}_{2}},\wedge^{2}J[2])$. The first, which is needed in
[Dog23], is to construct classes using the nonabelian $(x-T)$ map. Since it is
easy to compute the rank of $H^{1}_{f}(G_{\mathbb{Q}_{2}},\wedge^{2}T_{2}J)$,
if one can find enough crystalline classes one can verify that one has found a
basis for the image of $H^{1}_{f}(G_{\mathbb{Q}_{2}},\wedge^{2}T_{2}J)$ in
$H^{1}(G_{\mathbb{Q}_{2}},\wedge^{2}J[2])$ by a dimension count.

The second approach, taken in this paper, is to construct liftability
obstructions. Specifically, in the next section we try to compute the boundary
map

$H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$

associated to the short exact sequence

$0\to\wedge^{2}J[2]\to\wedge^{2}J[4]\to\wedge^{2}J[2]\to 0.$

In this section, as a step towards this we describe the boundary map

$\delta:H^{1}(K,J[2])\to H^{2}(K,J[2])$

associated to the short exact sequence

$0\to J[2]\to J[4]\to J[2]\to 0.$

The split short exact sequence

$0\to J[2]\to\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}\to\mathbb{F}_{2}\to 0$

gives isomorphisms

$H^{1}(K,J[2])\simeq\mathrm{Ker}(K_{f}^{\times}\otimes\mathbb{F}_{2}\stackrel{{\scriptstyle\mathrm{Nm}}}{{\longrightarrow}}K^{\times}\otimes\mathbb{F}_{2})$

and

$H^{2}(K,J[2])\simeq\mathrm{Ker}(H^{2}(K_{f},\mathbb{F}_{2})\stackrel{{\scriptstyle\mathrm{Nm}}}{{\longrightarrow}}H^{2}(K,\mathbb{F}_{2})).$

Our characteristation of $\delta$ is in terms of these isomorphisms. We let
$i_{1}$ and $i_{2}$ denote the two inclusions of $K_{f}$ into $K_{f,2}$,
letting $\alpha$ and $\beta$ denote the images of $x$ under the corresponding
maps

$K[x]\to K_{f,2}.$

Let $\mathrm{cores}^{(1)}$ and $\mathrm{cores}^{(2)}$ denote the corestriction
maps corresponding to the inclusions of $K_{f}$ into $K_{f,2}$ via $i_{1}$ and
$i_{2}$ respectively.

######  Proposition 3.

Suppose that $X$ descends to a subfield $K_{0}$ for which the Galois group of
$K_{0}$ acts $2$-transitively on the set of roots of $f$ and for which $X$ has
a $K_{0}$-rational non-Weierstrass point. Then the boundary map

$H^{1}(K,J[2])\to H^{2}(K,J[2])$

associated to $J[4]$ is given on $z\in H^{1}(K,J[2])\subset
K_{f}^{\times}\otimes\mathbb{F}_{2}$ by

$z\mapsto\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(z)\cup((\beta-\alpha)f^{\prime}(\beta)))$

######  Remark 1.

We expect that the formula for the boundary map should still hold without the
condition on the existence of a rational non-Weierstrass point or the
condition on the $2$-transitivity of the Galois group of $f$. The condition of
a rational non-Weierstrass point is innocuous for our applications, and
enables us to quote certain explicit models for abelian covers of
$X_{\overline{K}}$ from [Dog23]. In loc. cit. it was important that these were
pointed $K$-covers lying over $(X,b)$ where $b$ was a rational non-Weierstrass
point. The condition on $2$-transitivity of the Galois group is more
restrictive, and allows us to apply Shapiro’s lemma in a very direct way when
describing boundary maps.

In general, the computation of the boundary map involves working over a large
étale algebra $K_{f}\otimes K_{f}$. However, for local fields these
computations can actually be reduced to $K_{f}$, by local class field theory.

######  Lemma 3.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$. Let $L|K$ be a finite
extension. Then the norm map

$\mathrm{cores}:H^{2}(K,\mathrm{Ind}^{L}_{K}\mu_{2})\to H^{2}(K,\mu_{2})$

is an isomorphism of one dimensional $\mathbb{F}_{2}$-vector spaces.

Let $f=c\cdot\prod_{i=1}^{m}f_{i}$ be a factorisation of $f$ into monic
irreducibles, and $c\in K$. Let $K_{i}:=K[x]/(f_{i})$, and let $\alpha_{i}$ be
the image of $x$ in $K_{i}$. Finally, let

$\langle,\rangle_{K_{i}}:K_{i}^{\times}\otimes\mathbb{F}_{2}\times
K_{i}\times\otimes\mathbb{F}_{2}\to\mathbb{F}_{2}\simeq
H^{2}(K_{i},\mathbb{F}_{2})$

be the Hilbert symbol map. Note that we have an isomorphism

$H^{2}(K,J[2])\simeq\mathrm{Ker}(\mathbb{F}_{2}^{m}\to\mathbb{F}_{2})$

by Tate duality. Hence the boundary map may be viewed as a map

$\mathrm{Ker}(\oplus_{i=1}^{m}K_{i}^{\times}\otimes\mathbb{F}_{2}\stackrel{{\scriptstyle\mathrm{Nm}}}{{\longrightarrow}}K^{\times}\otimes\mathbb{F}_{2})\to\mathbb{F}_{2}^{m}.$

######  Lemma 4.

Suppose $X$ is defined over a field $K_{0}$, and that the Galois group of
$K_{0}$ acts $2$-transitively on the roots of $f$ and $X$ has a
$K_{0}$-rational non-Weierstrass point. If $K$ is a finite extension of
$\mathbb{Q}_{p}$ and an extension of $K_{0}$, then the boundary map

$H^{1}(K,J[2])\to
H^{2}(K,J[2])\simeq(\mathrm{Ker}(\oplus_{i}\mathbb{F}_{2}\to\mathbb{F}_{2}))^{*}$

is given by sending a tuple
$(z_{i})\in\prod_{i}K_{i}^{\times}\otimes\mathbb{F}_{2}$ to the class of the
tuple $(w_{i})\in\mathbb{F}_{2}^{m}$, where

$w_{i}:=\langle-\prod_{j\neq
i}f_{j}(\alpha_{i}),z_{i}\rangle_{K_{i}}+\sum_{j\neq i}\langle
cf_{i}(\alpha_{j}),z_{j}\rangle_{K_{j}}$

###### Proof.

By Proposition 3, it is enough to show that $w_{i}$ is equal to the
$H^{2}(K_{i},\mu_{2})$ component of

$\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}((z_{j}),(\beta-\alpha)f^{\prime}(\beta)).$

We have

$\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}((z_{j}),-f^{\prime}(\beta))=(\mathrm{Nm}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(z_{j}))\cup-f^{\prime}(\beta))$

by the compatibility of cup products with corestrictions. For any $(z)\in
H^{1}(K_{f},\mu_{2})$, we have

$i(\mathrm{Nm}_{K_{f}|K}(z))=\mathrm{Nm}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(z))+i(z)$

using the decomposition $K_{f}\otimes K_{f}\simeq K_{f,2}\times K_{f}$. In
particular, if $z$ is in $\mathrm{Ker}(\mathrm{Nm})$, we obtain

$\mathrm{Nm}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(z))=z,$

hence for $(z_{j})$ as above

$\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}((z_{j}),-f^{\prime}(\beta))=(\langle
z_{i},-f^{\prime}(\alpha_{i})\rangle_{K_{i}})_{i}$

To compute $\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}((z_{j})\cup(\alpha-\beta))$,
we decompose $K_{f,2}$ into fields $K_{ijk}$ lying above $K_{i}$ and $K_{j}$,
then we have

$\sum_{j,k}\mathrm{cores}_{K_{jik}|K_{i}}(z_{j}\cup(\alpha_{j}-\beta_{i})),$

which by Lemma 3 is equal to

$\sum_{j,k}\mathrm{cores}_{K_{jik}|K_{i}}(z_{j}\cup(\alpha_{j}-\beta_{i})).$

Applying Lemma 3 again, we see that this is equal to

$\sum_{j,k}\mathrm{cores}_{K_{jik}|K}(z_{j}\cup(\alpha_{j}-\beta_{i})).$

We have

$\sum_{k}\mathrm{cores}_{K_{jik}|K_{j}}(z_{j}\cup(\alpha_{j}-\beta_{i}))=\begin{cases}\langle
z_{j},f_{i}(\alpha_{j})\rangle_{K_{j}},&i\neq j\\\ \langle
z_{j},f_{i}^{\prime}(\alpha_{i})\rangle_{K_{i}},&i=j.\end{cases}.$

Hence the lemma follows from noting that
$f^{\prime}(\alpha_{i})=cf_{i}^{\prime}(\alpha_{i})\prod_{j\neq
i}f_{j}(\alpha_{i})$. ∎

###  3.1. Strategy of proof of Proposition 3

We will reduce the proof of Proposition 3 to certain explicit calculations
with $H^{1}(K,\mathrm{End}(J[2]))$, via the following well known
characterisation of boundary maps.

######  Lemma 5.

Let $G$ be a group, $M$ and $N$ finite $G$-modules, and let $E$ be an
extension of $M$ by $N$. Let $E^{c}$ be a twist of this extension by $c\in
H^{1}(G,\mathrm{Hom}(M,N))$. Let $\delta$ and $\delta_{c}$ be the boundary
maps $H^{1}(G,M)\to H^{2}(G,N)$ associated to the extensions $E$ and $E^{c}$
respectively. Then, for all $\alpha\in H^{1}(G,M)$,

$\delta_{c}(\alpha)=\delta(\alpha)+c\cup\alpha.$

###### Proof.

Let $\rho:G\to\mathrm{Aut}(E)$ and $\rho_{c}:G\to\mathrm{Aut}(E^{c})$ denote
the $G$-actions on $E$ and $E^{c}$ respectively. Let $\alpha\in Z^{1}(G,M)$.
Choose a section $s$ of $E\to M$, which we may also think of as a section of
$E^{c}\to M$. Then

$\delta(\alpha)(g_{1},g_{2})=\widetilde{\alpha}(g_{1}g_{2})-\rho(g_{1})\cdot\widetilde{\alpha}(g_{2})-\widetilde{\alpha}(g_{1}),$

and similarly

$\delta_{c}(\alpha)(g_{1},g_{2})=\widetilde{\alpha}(g_{1}g_{2})-\rho_{c}(g_{1})\cdot\widetilde{\alpha}(g_{2})-\widetilde{\alpha}(g_{1}).$

Hence

$\delta_{c}(\alpha)(g_{1},g_{2})-\delta(g_{1},g_{2})=(\rho(g_{1})-\rho_{c}(g_{1}))\cdot\widetilde{\alpha}(g_{2}).$

On the other hand, for any $e\in E$ and $g\in G$,
$\rho_{c}(g)(e)=\rho(g)(e)+c(g)(g\cdot(e))$, from which Lemma 5 follows. ∎

In particular, if $\delta(\alpha)=0$, we can describe the $\delta_{c}$ purely
in terms of the cup product. If $E_{2}$ is the twist of $E_{1}$ by $c$, we
shall sometimes denote the class of $c$ in $H^{1}(K,\mathrm{Hom}(M,N))$ by
$[E_{2}]-[E_{1}]$. We note that $c$ admits the following description.

######  Lemma 6.

If $E_{1}$ and $E_{2}$ are extensions of $M$ by $N$ which are isomorphic as
abelian groups, then $E_{2}$ is the twist of $E_{1}$ by the $G$-equivariant
$\mathrm{Hom}(M,N)$-torsor of isomorphisms $E_{1}\simeq E_{2}$ of extensions
of $M$ by $N$.

It remains to a give an explicit description of the cup product map. To apply
Lemma 5, we need to identify a self-extension of $J[2]$ for which the boundary
map is trivial. The norm map

$N:\mathrm{Ind}^{K_{f}}_{K}\mu_{4}\to\mu_{4}$

is split by plus or minus the inclusion
$\mu_{4}\to\mathrm{Ind}^{K_{f}}_{K}\mu_{4}$, depending on the parity of $g$.
It follows that $\mathrm{Ker}(N)$ is an extension of $J[2]$ by $J[2]$ for
which the boundary map $\delta_{\mathrm{Ker}(N)}$ vanishes.

######  Proposition 4.

Suppose $f$ is defined over a subfield $K_{0}$ of $K$ such that
$\operatorname{Gal}(K_{0})$ acts $2$-transitively on the roots of $f$.

  1. (1)

We have an isomorphism

$H^{1}(K,\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))\simeq
K_{f,2}^{\times}\otimes\mathbb{F}_{2}.$

With respect to this isomorphism, the cup product map

$H^{1}(K,J[2])\times
H^{1}(K,\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\to
H^{2}(K,\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$

is given by

$(g(\alpha),h(\alpha,\beta))\mapsto\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(g(\alpha))\cup
h(\alpha,\beta)).$

  2. (2)

Suppose that $X$ has a rational non-Weierstrass point over $K_{0}$. Then with
respect to this isomorphism, the class of the torsor of isomorphisms between
$J[4]$ and $\mathrm{Ker}(N)$ is equal to $(\alpha-\beta)f^{\prime}(\beta).$

###  3.2. Field-theoretic description of $H^{1}(K,\mathrm{End}(J[2]))$ and the
cup product map

We will prove Proposition 4 in several stages. In this subsection we discuss
the proof of part (1). We first recall the explicit form of Shapiro’s lemma,
which we will make repeated use of. Let $H<G$ be a finite index subgroup, and
let $M$ be a finite $H$-module. Let
$\pi:\mathrm{Res}^{G}_{H}\mathrm{Ind}^{G}_{H}M\to M$ be the map induced by
adjunction.

######  Lemma 7.

The composite map

$H^{1}(G,\mathrm{Ind}^{G}_{H}M)\stackrel{{\scriptstyle\mathrm{Res}}}{{\longrightarrow}}H^{1}(H,\mathrm{Res}^{G}_{H}\mathrm{Ind}^{G}_{H}M)\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}H^{1}(H,M)$

is equal to the isomorphism from Shapiro’s lemma.

###### Proof.

See [Sti10], [Sti13], or [NSW08] for more general statements (for nonabelian
cohomology and cohomology in arbitrary degree respectively). ∎

For a root $\gamma$ of $f$, let $D_{\gamma}\subset J[2]$ denote the kernel of
the norm map

$\mathbb{F}_{2}[\mathrm{Roots}(f)-\\{\gamma\\}]\to\mathbb{F}_{2}.$

######  Lemma 8.

Suppose $\operatorname{Gal}(K^{\mathrm{sep}}|K)$ acts transitively on the
roots of $f$. Then we have an isomorphism

(1)
$H^{1}(K,\mathrm{End}(J[2]))\simeq\mathrm{Ker}(H^{1}(K_{f},\mathrm{Hom}(J[2],\mu_{2}))\to
H^{1}(K,\mathrm{Hom}(J[2],\mu_{2})))$

from Shapiro’s lemma, which is given by the composite map

$H^{1}(K,\mathrm{End}(J[2]))\to H^{1}(K_{f},\mathrm{End}(J[2]))\to
H^{1}(K_{f},\mathrm{Hom}(J[2],\mu_{2}))$

where the first map is restriction and the second map is the projection
induced by quotienting $J[2]$ by $D_{\alpha}$.

###### Proof.

Since the exact sequence

(2) $0\to J[2]\to\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}\to\mathbb{F}_{2}\to 0$

splits, we have an isomorphism

$H^{1}(K,\mathrm{End}(J[2]))\simeq\mathrm{Ker}(H^{1}(\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))\to
H^{1}(K,\mathrm{Hom}(J[2],\mathbb{F}_{2}))).$

Via the $\operatorname{Gal}(K^{\mathrm{sep}}|K)$-equivariant isomorphism

$\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\simeq\mathrm{Ind}^{K_{f}}_{K}\mathrm{Hom}(J[2],\mathbb{F}_{2}),$

together with Shapiro’s lemma, we obtain (1). ∎

If we restrict to $K_{f}$, we obtain a
$\mathrm{Hom}(J[2],\mathbb{F}_{2})$-torsor of unipotent isomorphisms

$J[4]/2D_{\alpha}\simeq\mathrm{Ker}(\mathbb{Z}/4\mathbb{Z}[\mathrm{Roots}(f)]\to\mathbb{Z}/4\mathbb{Z})/2D_{\alpha}$

Suppose $\operatorname{Gal}(f)$ acts $2$-transitively on the roots of $f$.
Then we have an isomorphism

$\mathrm{Ind}^{H}_{H^{(2)}}\mathbb{F}_{2}\simeq J[2]$

induced by the isomorphism

$\mathbb{F}_{2}[\mathrm{Roots}(f(x)/(x-\alpha))]\simeq\mathrm{Ker}(\mathbb{F}_{2}[\mathrm{Roots}(f)]\to\mathbb{F}_{2})$

given by sending $[\gamma]$ to $[\gamma]-[\alpha]$.

######  Lemma 9.

We have a commutative diagram

(3)
${H^{1}(K_{f},\mathrm{Hom}(J[2],\mu_{2}))}$${H^{1}(K,\mathrm{Hom}(J[2],\mu_{2}))}$${H^{1}(K_{f,2},\mathbb{F}_{2})}$${H^{1}(K_{f},\mathbb{F}_{2})/H^{1}(K,\mathbb{F}_{2})}$

whose vertical maps are isomorphisms, where the top horizontal map is
corestriction, and the bottom horizontal map is the composite of

$\mathrm{Nm}^{(1)}_{K_{f,2}|K_{f}}:K_{f,2}^{\times}\otimes\mathbb{F}_{2}\to
K_{f}^{\times}\otimes\mathbb{F}_{2}$

with the projection $K_{f}^{\times}\otimes\mathbb{F}_{2}\to
H^{1}(K_{f},\mathbb{F}_{2})/H^{1}(K,\mathbb{F}_{2})$.

We deduce the following.

######  Lemma 10.

Suppose $\operatorname{Gal}(K)$ acts $2$-transitively on the roots of $f$.
Then we have an isomorphism

$H^{1}(K,\mathrm{End}(J[2]))\simeq\mathrm{Ker}(H^{1}(K_{f,2},\mathbb{F}_{2})\stackrel{{\scriptstyle\mathrm{Nm}^{(1)}}}{{\longrightarrow}}H^{1}(K_{f},\mathbb{F}_{2})/H^{1}(K,\mathbb{F}_{2}))$

given by the composite of the isomorphism from Lemma 8 with the map

$\displaystyle\mathrm{Ker}(H^{1}(K_{f},\mathrm{Hom}(J[2],\mu_{2}))\to$
$\displaystyle H^{1}(K,\mathrm{Hom}(J[2],\mu_{2}))$
$\displaystyle\mathrm{Ker}(H^{1}(K_{f,2},\mathbb{F}_{2})\to$ $\displaystyle
H^{1}(K_{f},\mathbb{F}_{2})/H^{1}(K,\mathbb{F}_{2}))$

from Lemma 9.

###### Proof.

This again comes from Shapiro’s lemma and the direct sum decomposition
$\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}\simeq J[2]\oplus\mathbb{F}_{2}$. ∎

Since we are adopting the convention of describing the boundary map for $J[4]$
as a map from norm one elements of $K(\alpha)$ to $2$-torsion in the Brauer
group of $K(\beta)$, it will be convenient to swap the roles of $\alpha$ and
$\beta$ in the above discussion (i.e. to use the involution generating
$\mathrm{Aut}(K_{f,2}|K_{f}^{(2)})$) giving an isomorphism

(4)
$H^{1}(K,\mathrm{End}(J[2]))\simeq\mathrm{Ker}(H^{1}(K_{f,2},\mathbb{F}_{2})\stackrel{{\scriptstyle\mathrm{Nm}^{(2)}}}{{\longrightarrow}}H^{1}(K_{f},\mathbb{F}_{2})/H^{1}(K,\mathbb{F}_{2})).$

It remains to check the second claim in part (1) of Proposition 4, which
specifies how the cup product map

$H^{1}(K,J[2])\times H^{1}(K,\mathrm{End}(J[2]))\to H^{2}(K,J[2])$

relates to the field theoretic description of the cohomology groups involved.
This amounts to describing how Shapiro’s lemma behaves with respect to cup
products.

######  Lemma 11.

Let $H,G,M$ be as in Lemma 7, and $N$ a $G$-module. Then the diagram

${H^{1}(G,\mathrm{Ind}^{G}_{H}M)\otimes
H^{1}(G,N)}$${H^{2}(G,\mathrm{Ind}^{G}_{H}(M)\otimes N)}$${H^{1}(H,M)\otimes
H^{1}(H,N)}$${H^{2}(H,M\otimes N)}$

commutes, where the right vertical map is the composite of Shapiro’s
isomorphism with the isomorphism $\mathrm{Ind}^{G}_{H}(M)\otimes
N\simeq\mathrm{Ind}^{G}_{H}(M\otimes N)$.

###### Proof.

By Lemma 7, this reduces to commutativity of

${H^{1}(G,\mathrm{Ind}^{G}_{H}M)\otimes
H^{1}(G,N)}$${H^{2}(G,\mathrm{Ind}^{G}_{H}(M)\otimes
N)}$${H^{1}(H,\mathrm{Res}^{G}_{H}\mathrm{Ind}^{G}_{H}M)\otimes
H^{1}(H,N)}$${H^{2}(H,\mathrm{Res}^{G}_{H}\mathrm{Ind}^{G}_{H}(M)\otimes
N)}$${H^{1}(H,M)\otimes H^{1}(H,N)}$${H^{2}(H,M\otimes N).}$

commutativity of the top part is just compatibility of cup products with
restriction maps. Commutativity of the second part amounts to the fact that
the projection map

$\mathrm{Res}^{G}_{H}\mathrm{Ind}^{G}_{H}(M\otimes N)\to M\otimes N$

is equal to the tensor product of the projection map on $M$ with the identity
on $N$. ∎

######  Proposition 5.

Let $H<G$ be a finite index subgroup and let $R$ be a ring with trivial
$G$-action (and discrete topology). Let $M$ be a discrete $R[G]$-module. Let
$\theta\in H^{1}(H,M)$ and $\psi\in H^{1}(H,R)$ map to $\widetilde{\theta}$
and $\widetilde{\psi}$ in $H^{1}(G,\mathrm{Ind}^{G}_{H}M)$ and
$H^{1}(G,\mathrm{Ind}^{G}_{H}R)$ under the isomorphism of Shapiro’s lemma.
Then we have an equality

$\widetilde{\theta}\cup\widetilde{\psi}=\mathrm{cores}(\theta\cup\psi)$

of classes in $H^{2}(G,M)$.

###### Proof.

In the absence of a conceptual proof, we write out cochains. Let
$g_{1},\ldots,g_{m}$ be a set of coset representatives for $G$, and define
$\gamma:G\to H$ and $n:G\to\\{1,\ldots,m\\}$ by the property that
$g=\gamma(g)\cdot g_{n(g)}$.

Choose coset representatives for $\psi$ and $\theta$, which we will also
denote by $\psi$ and $\theta$. Since $G$ and $H$ act trivially on $R$, the cup
product of $\theta$ and $\psi$ is given on cocycles by

$(\theta\cup\psi)(x_{1},x_{2})=\theta(x_{1})\cdot\psi(x_{2}).$

Hence the corestriction of $\theta\cup\psi$ is given by

$(x_{1},x_{2})\mapsto\sum_{i}g_{i}^{-1}\theta(\gamma(g_{i})^{-1}\gamma(g_{i}x_{1}))\otimes\psi(\gamma(g_{i}x)^{-1}\gamma(g_{i}x_{1}x_{2})).$

The cup product of $\widetilde{\theta}$ and $\psi$ is the $2$-cocycle

$G\times G\to\mathrm{Ind}^{G}_{H}M\otimes\mathrm{Ind}^{G}_{H}R$

given by

$(x_{1},x_{2})\mapsto\widetilde{\theta}(x_{1})\otimes
x_{1}\widetilde{\psi}(x_{2}).$

We have

$\widetilde{\theta}(x_{1})=\sum_{i}\theta(u(g_{i}x_{1}))\cdot[g_{i}]$

and

$\widetilde{\psi}(x_{2})=\sum_{i}\psi(u(g_{i}x_{2}))\cdot[g_{i}].$

When we project $\widetilde{\theta}\cup\widetilde{\psi}$ to $H^{2}(G,M)$, we
see that it is given by

$(x_{1},x_{2})\mapsto\sum_{i}\theta(\gamma(g_{i}x_{1}))\cdot\psi(\gamma(g_{m_{i}}x_{2}))$

where $m_{i}$ has the property that $Hg_{i}x_{1}=Hg_{m_{i}}$. The proposition
now follows from the identity

$\gamma(g_{i}x_{1}x_{2})=\gamma(g_{i}x)\gamma(g_{m_{i}}y).$

∎

To complete the proof of part (1) of Proposition 4, it is enough to show that
the diagram

${H^{1}(K,J[2])\times
H^{1}(K,\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))}$${H^{2}(K,\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$${\mathrm{Ker}(K_{f}^{\times}\otimes\mathbb{F}_{2}\stackrel{{\scriptstyle\mathrm{Nm}}}{{\longrightarrow}}K^{\times}\otimes\mathbb{F}_{2})\times
K_{f,2}^{\times}\otimes\mathbb{F}_{2}}$${H^{2}(K_{f},\mathbb{F}_{2})}$

commutes, where the vertical maps are the isomorphisms above, the top
horizontal map is the cup product and the bottom horizontal map is

$(x,y)\mapsto\mathrm{cores}^{(2)}_{K_{f,2}|K_{f}}(i_{1}(x)\cup y).$

Via the split exact sequence (2), it is enough to prove that the diagram

${H^{1}(K,\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\times
H^{1}(K,\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))}$${H^{2}(K,\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$${K_{f}^{\times}\otimes\mathbb{F}_{2}\times\left(K_{f,2}^{\times}\otimes\mathbb{F}_{2}\times
K_{f}^{\times}\otimes\mathbb{F}_{2}\right)}$${H^{2}(K_{f},\mathbb{F}_{2})}$

commutes. From Proposition 5, we deduce that, for $c_{1}\in
H^{1}(K,\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$ and $c_{2}\in
H^{1}(K,\mathrm{Ind}^{K_{f}}_{K}(\mathrm{Res}^{K_{f}}_{K}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))$
corresponding to $\overline{c}_{1}$ and $\overline{c}_{2}$ in
$H^{1}(K_{f},\mathbb{F}_{2})$ and
$H^{1}(K_{f},\mathrm{Res}^{K_{f}}_{K}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$
respectively, we have

$c_{1}\cup
c_{2}=\mathrm{cores}_{K_{f}|K}(\overline{c}_{1}\cup\overline{c}_{2}).$

Now suppose that $\overline{c}_{2}$ corresponds to $(c_{3},c_{4})\in
H^{1}(K_{f,2},\mathbb{F}_{2})\times H^{1}(K_{f},\mathbb{F}_{2})$ under the
isomorphism

$H^{1}(K,\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))\simeq
K_{f,2}^{\times}\otimes\mathbb{F}_{2}\times
K_{f}^{\times}\otimes\mathbb{F}_{2}.$

Then, by Lemma 11, under the isomorphism
$H^{2}(K_{f},\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})\simeq
H^{2}(K_{f},\mathrm{Ind}^{K_{f,2}}_{K_{f}}\mathbb{F}_{2})\oplus
H^{2}(K_{f},\mathbb{F}_{2})$ we have

$\overline{c}_{1}\cup\overline{c}_{2}=(c_{3}\cup i_{1}(c_{2}),c_{4}\cup
c_{2}).$

Applying Proposition 5, we deduce part (1) of Proposition 4.

###  3.3. Describing the class of $J[4]$

We now prove part (2) of Proposition 4, namely that the class of
$[J[4]]-[(\mathrm{Ind}^{K_{f}}_{K}\mu_{4})/\mu_{4}]$ in
$H^{1}(K,\mathrm{End}(J[2]))$ is equal to $(\alpha-\beta)f^{\prime}(\beta)$
with respect to the isomorphism from part (1) of Proposition 4. First, note
that the class of

$[(\mathrm{Ind}^{K_{f}}_{K}\mu_{4})/\mu_{4}]-[(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z})/\mathbb{Z}/4\mathbb{Z}]$

is equal to $-1$ in $K_{f,2}$. Hence it will be enough to show that the class
of
$[J[4]]-[(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z})/\mathbb{Z}/4\mathbb{Z}]$
is $(\beta-\alpha)f^{\prime}(\beta)$.

To do this, we recall some results from [Dog23, §3]. The
$\mathbb{Z}/2\mathbb{Z}$ cover corresponding to a root $\alpha$ of $f$ is
given on function fields by $\overline{K}(X)(u_{\alpha})$, where
$c_{\alpha}u_{\alpha}^{2}=x-\alpha$, and $c_{\alpha}=x(b)-\alpha$, where $b\in
X(K)$ is a non-Weierstrass point.

Let $z_{\beta}$ be a square root of

$\prod_{\gamma\in\mathrm{Roots}(f)-\\{\beta\\}}\left(1+\frac{c_{\beta}u_{\beta}-c_{\gamma}u_{\gamma}}{\gamma-\beta}\right).$

By [Dog23, Lemma 11] and [Dog23, Lemma 12], the extension of
$\overline{\mathbb{Q}}(X)$ corresponding to $J[4]$ is equal to
$L_{0}=\overline{\mathbb{Q}}(X)(u_{\alpha},z_{\alpha}:\alpha\in\mathrm{Roots}(f))$.
In particular we have an isomorphism of Galois modules

$J[4]\simeq\operatorname{Gal}(L_{0}|\overline{\mathbb{Q}}(X))$

where the action of $\operatorname{Gal}(K)$ on the latter is via the outer
action of $\operatorname{Gal}(K(X))$ via conjugation. We recall [Dog23, Lemma
10] that the étale algebra
$\bigotimes_{\overline{\mathbb{Q}}(X),\alpha\in\mathrm{Roots}(f)}\overline{\mathbb{Q}}(X)[u_{\alpha}]$
is not a field, and we are identifying $L_{0}$ with the field in which

(5) $\prod_{\alpha\in\mathrm{Roots}(f)}u_{\alpha}=y/y(b).$

Let $L_{1}=K(X)(u_{\gamma}:\gamma\neq\alpha,\beta)$, and
$L_{2}=L_{1}(u_{\alpha},u_{\beta},z_{\beta})$. Then we have an outer action of
$\operatorname{Gal}(K_{f,2}(X))$ on $\operatorname{Gal}(L_{2}|L_{1})$.

######  Lemma 12.

Suppose the Galois group of $K$ acts $2$-transitively on the roots of $f$ and
$X$ has a $K$-rational non-Weierstrass point. Then under the isomorphism

$H^{1}(K,\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))\simeq
H^{1}(K_{f,2},\mathbb{F}_{2}),$

the class of the $\mathrm{End}(J[2])$-torsor of isomorphisms

$J[4]\simeq(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z})/\mathbb{Z}/4\mathbb{Z}$

is sent to the $\mathbb{Z}/2\mathbb{Z}$-torsor of isomorphisms

$\operatorname{Gal}(L_{2}|L_{1})\simeq\mathbb{Z}/4\mathbb{Z}.$

###### Proof.

This follows from twice applying the explicit description of Shapiro’s lemma
from Lemma 7 to the isomorphism (4) obtained from Lemma 10. ∎

The main calculation in the proof of part (2) of Proposition 4 is the
following.

######  Lemma 13.

There is a $\operatorname{Gal}(K_{f,2})$-equivariant bijection

$\mathrm{Iso}(\operatorname{Gal}(L_{2}|L_{1}),\mathbb{Z}/4\mathbb{Z})\simeq\mathrm{Roots}(x^{2}-f^{\prime}(\beta)(\beta-\alpha))$

###### Proof.

A generator of $\operatorname{Gal}(L_{2}|L_{1})$ is given by a lift of the
generator of $\operatorname{Gal}(L_{1}(u_{\alpha})|L_{1})$ to
$\operatorname{Gal}(L_{2}|L_{1})$. The generator of
$\operatorname{Gal}(L_{1}(u_{\alpha})|L_{1})$ is the automorphism
$u_{\alpha}\mapsto-u_{\alpha}$, and a lift $\sigma$ to
$\operatorname{Gal}(L_{2}|L_{1})$ is uniquely determined by where it sends
$z_{\beta}$. Note that by (5), $\sigma$ sends $u_{\beta}$ to $-u_{\beta}$.
Hence we see that

$\displaystyle z_{\beta}^{2}\sigma(z_{\beta})^{2}$ $\displaystyle=$
$\displaystyle(1+\frac{c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha}}{\alpha-\beta})(1-\frac{c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha}}{\alpha-\beta})\cdot\prod_{\gamma\neq\alpha,\beta}(1+\frac{c_{\beta}u_{\beta}-c_{\gamma}u_{\gamma}}{\gamma-\beta})(1+\frac{-c_{\beta}u_{\beta}-c_{\gamma}u_{\gamma}}{\gamma-\beta})$

We have

$(1+\frac{c_{\beta}u_{\beta}-c_{\gamma}u_{\gamma}}{\gamma-\beta})(1+\frac{-c_{\beta}u_{\beta}-c_{\gamma}u_{\gamma}}{\gamma-\beta})=\frac{c_{\gamma}}{\beta-\gamma}(u_{\gamma}+1)^{2},$

and

$(1+\frac{c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha}}{\alpha-\beta})(1-\frac{c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha}}{\alpha-\beta})=-c_{\beta}c_{\alpha}\left(\frac{(u_{\beta}-1)(u_{\alpha}+1)}{(\alpha-\beta-
c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha})}\right)^{2}.$

We deduce that

$\sigma(z_{\beta})=\sqrt{\frac{f^{\prime}(\beta)}{\beta-\alpha}}\frac{\left(\frac{(u_{\beta}-1)(u_{\alpha}+1)}{(\alpha-\beta-
c_{\beta}u_{\beta}-c_{\alpha}u_{\alpha})}\right)y(b)\prod_{\gamma\neq\alpha,\beta}(u_{\gamma}+1)}{z_{\beta}}$

for some choice of square root, giving the desired Galois-equivariant
bijection. ∎

##  4\. The boundary map for $\wedge^{2}J[4]$

We now apply the above calculations to describe the boundary map

$H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$

associated to the short exact sequence

$0\to\wedge^{2}J[2]\to\wedge^{2}J[4]\to\wedge^{2}J[2]\to 0.$

and

$0\to\wedge^{2}J[2]\to\mathrm{Ker}(\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mu_{4}\to\mathrm{Ind}^{K_{f}}_{K}\mu_{4})\to\wedge^{2}J[4]\to
0$

###  4.1. A field-theoretic description of
$H^{1}(K,\mathrm{End}(\wedge^{2}J[2]))$

We have an isomorphism

$H^{1}(K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\simeq(K_{f}^{(2)}\otimes
K_{f}^{(2)})^{\times}\otimes\mathbb{F}_{2}.$

Using the direct sum decomposition of
$\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}$, we can obtain a field-theoretic
description of $H^{1}(K,\mathrm{End}(\wedge^{2}J[2]))$. To do this, we need to
introduce some notation for various étale algebras obtained from tensor powers
of $K_{f}$. For any $n<\deg(f)$, we define $K_{f,n}$ to be the étale algebra
obtained by adjoining $n$ distinct roots of $f$, i.e.

$K_{f,n}:=K[t_{1},\ldots,t_{n}][\frac{1}{t_{i}-t_{j}}:i\neq
j]/(f(t_{1}),\dots,f(t_{n})).$

We define $K_{f}^{(2,1,1)}$ to be the subfield of $K_{f,4}$ fixed by the
involution swapping $\overline{t}_{1}$ and $\overline{t}_{2}$ and define
$K_{f,2}^{(2)}$ to be the subfield of $K_{f}^{(2,1,1)}$ fixed by the
involution swapping $\overline{t}_{3}$ and $\overline{t}_{4}$. We have
isomorphisms

$\displaystyle K_{f}^{(2)}\otimes K_{f}^{(2)}\simeq K_{f,2}^{(2)}\times
K_{f,3}\times K_{f}^{(2)}$ $\displaystyle K_{f}^{(2)}\otimes K_{f,2}\simeq
K_{f}^{(2,1,1)}\times K_{f,3}\times K_{f,3}\times K_{f,2}$ $\displaystyle
K_{f}^{(2)}\otimes K_{f}\simeq K_{f}^{(2,1)}\times K_{f,2}$ $\displaystyle
K_{f,2}\otimes K_{f}\simeq K_{f,3}\times K_{f,2}\times K_{f,2}$

Informally, thinking of $K_{f,n}$ as the algebra obtained by adjoining
$n$-distinct roots of $f$ to $K$, these isomorphisms can be obtained from
breaking $K_{f,n}\otimes K_{f,m}$ into a product of algebras of the form
$K_{f,e}$, by considering the possible roots of $f$ that the two algebras have
in common. On the level of Galois cohomology, we obtain the following.

######  Lemma 14.

We have an isomorphism

$H^{1}(K,\mathrm{End}(\wedge^{2}J[2]))\simeq\mathrm{Ker}\left((K_{f}^{(2)}\otimes
K_{f}^{(2)})^{\times}\otimes\mathbb{F}_{2}\to(K_{f}^{(2)}\otimes
K_{f})^{\times}\otimes\mathbb{F}_{2}/(K_{f}\otimes
K_{f})^{\times}\otimes\mathbb{F}_{2}\right).$

###### Proof.

This follows from an iterated application of Shapiro’s lemma:

$\displaystyle
H^{(}K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))$
$\displaystyle\simeq
H^{1}(K,\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2}\otimes\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2})$
$\displaystyle\simeq
H^{1}(K,\mathrm{Ind}^{K_{f}^{(2)}}_{K}(\mathrm{Res}^{K_{f}^{(2)}}_{K}\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2})$
$\displaystyle\simeq
H^{1}(K_{f}^{(2)},\mathrm{Res}^{K_{f}^{(2)}}_{K}\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2})$
$\displaystyle\simeq
H^{1}(K_{f}^{(2)},\mathrm{Ind}^{K_{f,2}^{(2)}}_{K_{f}^{(2)}}\mathbb{F}_{2}\oplus\mathrm{Ind}^{K_{f,3}}_{K_{f}^{(2)}}\mathbb{F}_{2}\oplus\mathbb{F}_{2})$
$\displaystyle\simeq K_{f,2}^{(2),\times}\otimes\mathbb{F}_{2}\oplus
K_{f,3}^{\times}\otimes\mathbb{F}_{2}\oplus
K_{f}^{(2),\times}\otimes\mathbb{F}_{2}$

∎

We define maps $\pi$ and $\iota$ by the following commutative diagrams

${K_{f}^{(2)}\otimes K_{f}^{(2)}}$${K_{f}^{(2)}\otimes
K_{f,2}}$${K_{f,2}^{(2)}\times K_{f,3}\times
K_{f}^{(2)}}$${K_{f}^{(2,1,1)}\times K_{f,3}\times K_{f,3}\times
K_{f,2}}$$\scriptstyle{1\otimes i}$$\scriptstyle{\iota}$

and

${K_{f}^{(2)}\otimes K_{f,2}}$${K_{f}^{(2)}\otimes
K_{f}}$${K_{f}^{(2,1,1)}\times K_{f,3}\times K_{f,3}\times
K_{f,2}}$${K_{f}^{(2,1)}\times
K_{f,2},}$$\scriptstyle{1\otimes\mathrm{Nm}}$$\scriptstyle{\pi}$

where the vertical maps are the isomorphisms above. Then $\iota$ is the map

$(f_{1},f_{2},f_{3})\mapsto(f_{1},f_{2},f_{2},f_{3})$

and $\pi$ is the map

$(\mathrm{Nm}(g_{1})\mathrm{Nm}(g_{2}),g_{4}\mathrm{Nm}(g_{3})).$

###  4.2. Field-theoretic description of the map
$H^{1}(K,\mathrm{End}(J[2]))\to H^{1}(K,\mathrm{End}(\wedge^{2}J[2]))$

The direct sum decompositions $\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}\simeq
J[2]\oplus\mathbb{F}_{2}$ and
$\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}\simeq\wedge^{2}J[2]\oplus
J[2]$ induce maps

$\displaystyle\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\to\mathrm{End}(J[2])\to\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$
$\displaystyle\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\to\mathrm{End}(\wedge^{2}J[2])\to\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$

######  Lemma 15.

The map

$H^{1}(K,\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))\to
H^{1}(K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2}))$

sends $(g(\alpha,\beta),h(\alpha))$ to
$(1,g(\alpha,\gamma),h(\alpha)h(\beta))$.

###### Proof.

The map

(6)
$\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\to\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})$

sends $[\alpha_{i}]^{*}\otimes[\alpha_{j}]$ to $\sum_{k\neq
i,j}[\\{\alpha_{i},\alpha_{k}\\}]^{*}\otimes[\\{\alpha_{k},\alpha_{j}\\}]$.
Under the identifications

$\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\simeq\mathrm{Ind}^{K_{f,2}}_{K}\mathbb{F}_{2}\oplus\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}$

and

$\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\simeq\mathrm{Ind}^{K_{f,2}^{(2)}}_{K}\mathbb{F}_{2}\oplus\mathrm{Ind}^{K_{f,3}}_{K}\mathbb{F}_{2}\oplus\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2},$

(6) corresponds to the direct sum $f_{1}\oplus f_{2}$, where $f_{1}$ is the
map induced by the inclusion of $K_{f,2}$ into $K_{f,3}$ via sending
$\overline{s}$ and $\overline{t}$ to the first and third parameters in
$K_{f,3}$, and $f_{2}$ is the composite of the inclusion of $K_{f}$ into
$K_{f,2}$ with the norm from $K_{f,2}$ to $K_{f}^{(2)}$. ∎

Given a vector space $V$, we have a map

$\Theta_{V}:\mathrm{End}(V)\to\mathrm{End}(\wedge^{2}V)$

given by sending $\theta\in\mathrm{End}(V)$ to

$v\wedge w\mapsto\theta(v)\wedge w-v\wedge\theta(w).$

From the following lemma we deduce that the extension class of
$\wedge^{2}J[4]$ is essentially given by $\Theta_{J[2]}$ of the class of
$J[4]$.

######  Lemma 16.

Let $M$ be a Galois module, and suppose that $E_{1}$ and $E_{2}$ are self-
extensions of $M$, such that $E_{1}$ is isomorphic to the twist of $E_{2}$ by
$c\in Z^{1}(K,\mathrm{End}(M))$. Then $\wedge^{2}E_{1}$ is isomorphic to the
twist of $\wedge^{2}E_{2}$ by $\Theta(c)\in
Z^{1}(K,\mathrm{End}(\wedge^{2}M))$. In particular, if $\delta_{i}$ denotes
the boundary map for $\wedge^{2}(E_{i})$, then

$\delta_{1}=\Theta(c)\cup(.)+\delta_{2}.$

###### Proof.

The first claim can be seen on the level of cocycles. The second claim is a
consequence of Lemma 5. ∎

To give an explicit formula for the cup product, it is enough to give an
explicit formula for the class in
$H^{1}(K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}))$.
There is a minor subtlety here, since the diagram

${\mathrm{End}(J[2])}$${\mathrm{End}(\wedge^{2}J[2])}$${\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$${\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})}$$\scriptstyle{\Theta_{J[2]}}$$\scriptstyle{\Theta_{\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}}}$

does not commute. However this issue disappears if we replace
$\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})$ with
$\mathrm{Hom}(\wedge^{2}J[2],\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})$
via projection.

######  Lemma 17.

The diagram

${H^{1}(K,\mathrm{End}(J[2])\times
H^{1}(K,\wedge^{2}J[2])}$${H^{1}(K,\mathrm{End}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\times
H^{1}(K,\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}])}$${H^{1}(K,\mathrm{End}(\wedge^{2}J[2])\times
H^{1}(K,\wedge^{2}J[2])}$${H^{1}(K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\times
H^{1}(K,\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$${H^{2}(K,\wedge^{2}J[2])}$${H^{2}(K,\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$$\scriptstyle{\Theta_{J[2]}\times
1}$$\scriptstyle{\Theta_{\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}}\times 1}$

commutes.

###### Proof.

The map

$\wedge^{2}J[2]\otimes\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})\to\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2}$

factors through
$\wedge^{2}J[2]\otimes\mathrm{Hom}(\wedge^{2}J[2],\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})$,
so it is enough to check commutativity of

${\mathrm{End}(J[2])}$${\mathrm{End}(\wedge^{2}J[2])}$${\mathrm{Hom}(J[2],\mathrm{Ind}^{K_{f}}_{K}\mathbb{F}_{2})}$${\mathrm{Hom}(\wedge^{2}J[2],\wedge^{2}\mathrm{Ind}^{K_{f}}\mathbb{F}_{2})}$

which follows from the definition of $\Theta$. ∎

###  4.3. The boundary map for $\wedge^{2}J[4]$ in terms of cup products

We now wish to describe the boundary map for $\wedge^{2}J[4]$ in terms of the
cup product with $\Theta$ applied to the class of $J[4]$. Let
$G<\mathrm{Sym}(\mathrm{Roots}(f))$ denote the subgroup of the symmetric group
on the set of roots of $f$ stabilising the set $\\{\alpha,\beta\\}$, and let
$H<G$ denote the stabiliser of the ordered pair $(\alpha,\beta)$. Let $\chi$
denote the $G$-module which is a free rank one $\mathbb{Z}/4\mathbb{Z}$-module
with the unique nontrivial action of $G/H$. Abusing notation, we shall denote
by $\chi$ the associated sheaf on
$\mathrm{Spec}(K_{f}^{(2)})_{\mathrm{\acute{e}t}}$.

######  Lemma 18.

We have isomorphism of Galois modules

$\displaystyle\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}\simeq(\mathrm{Ind}^{K_{f,2}}_{K}\mathbb{Z}/4\mathbb{Z})/\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{Z}/4\mathbb{Z}$
$\displaystyle\simeq\mathrm{Ind}^{K_{f}^{(2)}}_{K}\chi.$

###### Proof.

We prove each of the isomorphisms in turn. For the first isomorphism, we take
the map

$\mathrm{Ind}^{K_{f,2}}_{K}\mathbb{Z}/4\mathbb{Z}\to\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}$

sending $[\\{\epsilon_{1},\epsilon_{2}\\}]$ to
$[\\{\epsilon_{1}\\}]\wedge[\\{\epsilon_{2}\\}]$. The second isomorphism is
the induction from $K_{f}^{(2)}$ to $K$ of the isomorphism of
$\operatorname{Gal}(K_{f}^{(2)})$-modules

$(\mathrm{Ind}^{K_{f,2}}_{K_{f}^{(2)}}\mathbb{Z}/4\mathbb{Z})/\mathbb{Z}/4\mathbb{Z}\simeq\chi$

which is purely a statement about the group algebra of
$\mathbb{Z}/4\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]$. ∎

We deduce the following lemma.

######  Lemma 19.

The boundary map $H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$
associated to
$\wedge^{2}\mathrm{Ker}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z})$
is given by

$z\mapsto\langle-(\alpha^{2}+\beta^{2}-2\alpha\beta),z\rangle_{K_{f}^{(2)}}.$

###### Proof.

By Lemma 18, the class of
$[\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}]-[\mathrm{Ind}^{K_{f}^{(2)}}\mathbb{Z}/4\mathbb{Z}]$
in
$H^{1}(K,\mathrm{End}(\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/2\mathbb{Z}))$
is the image of the class of $(\alpha-\beta)^{2}$ in
$K_{f}^{(2),\times}\otimes\mathbb{F}_{2}$ under the diagonal map

$H^{1}(K_{f}^{(2)},\mathbb{Z}/2\mathbb{Z})\to
H^{1}(K,\mathrm{End}(\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{F}_{2})).$

Via the field-theoretic descriptions of the source and target this corresponds
to the map

$K_{f}^{(2),\times}\otimes\mathbb{Z}/2\mathbb{Z}\to
K_{f,2}^{(2),\times}\otimes\mathbb{F}_{2}\oplus
K_{f,3}^{\times}\otimes\mathbb{F}_{2}\oplus
K_{f}^{(2),\times}\otimes\mathbb{F}_{2}$

given by $w\mapsto(1,1,w)$. Similarly, the class of
$[\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{Z}/4\mathbb{Z}]-[\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mu_{4}]$
corresponds to $(1,1,-1)$. Since the boundary map for
$\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mu_{4}$ is zero, we deduce that the boundary
map for $\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}$ is given as
above. We have a direct sum decomposition

$\wedge^{2}\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}\simeq
I\oplus\wedge^{2}I,$

where
$I:=\mathrm{Ker}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z})$.
Hence the boundary map for $\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}$ is
also the direct sum of the boundary maps for $\wedge^{2}I$ and $I$. ∎

We have a surjection

$\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mathbb{Z}/4\mathbb{Z}\to\mathrm{Ker}(\mathrm{Ind}^{K_{f}}_{K}\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z})$

given by sending $\\{\epsilon_{1},\epsilon_{2}\\}$ to
$[\epsilon_{1}]+[\epsilon_{2}]+2\sum_{\epsilon_{3}\in\mathrm{Roots}(f)}[\epsilon_{3}]$.
This has a section given by

$[\epsilon_{1}]-[\epsilon_{2}]\mapsto(-1)^{g+1}\sum_{\epsilon_{3}\neq\epsilon_{1},\epsilon_{2}}\\{\epsilon_{1},\epsilon_{3}\\}-\\{\epsilon_{2},\epsilon_{3}\\}$

We deduce that

$A:=\mathrm{Ker}(\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mu_{4}\to\mathrm{Ker}(\mathrm{Ind}^{K_{f}}_{K}\mu_{4}\to\mu_{4}))$

is a direct summand of $\mathrm{Ind}^{K_{f}^{(2)}}_{K}\mu_{4}$ and hence that
$A$ is a self-extension of $\wedge^{2}J[2]$ for which the boundary map
$H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$ is zero. Let
$\alpha,\beta,\gamma$ denote the images of $t_{1},t_{2},t_{3}$ in $K_{f,3}$.
Let $i_{1}$ and $i_{2}$ denote the inclusions $K_{f}^{(2)}\to K_{f,3}$ sending
$(t_{1},t_{2})$ to $(t_{1},t_{2})$ and $(t_{2},t_{3})$ respectively. We will
also denote by $i_{1}$ and $i_{2}$ the composite with $K_{f,3}\to
K_{f}^{(2)}\otimes K_{f}^{(2)}$. We let $i_{3}$ denote the inclusion
$K_{f}^{(2)}\to K_{f}^{(2)}\otimes K_{f}^{(2)}$ defined above.

######  Proposition 6.

The boundary map

$H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$

associated to $\wedge^{2}J[4]$ is given by

$z\mapsto\mathrm{cores}_{K_{f,3}|K_{f}^{(2)}}^{(2)}\langle(\gamma-\alpha)f^{\prime}(\gamma),i_{1}(z)\rangle_{K_{f,3}}+\langle-(\alpha^{2}+\beta^{2}-2\alpha\beta),z\rangle_{K_{f}^{(2)}}.$

###### Proof.

By the direct sum decomposition above, the boundary map sends $z$ to

$\mathrm{cores}^{(2)}_{K_{f}^{(2)}\otimes
K_{f}^{(2)}|K_{f}^{(2)}}\langle\kappa,i_{2}(z)\rangle_{K_{f}^{(2)}\otimes
K_{f}^{(2)}},$

where $\kappa\in K_{f}^{(2),\times}\otimes K_{f}^{(2),\times}$ is the image of
$[A]-[\wedge^{2}J[4]]$ under the map

$H^{1}(K,\mathrm{End}(\wedge^{2}J[2]))\to(K_{f}^{(2),\times}\otimes
K_{f}^{(2),\times})^{\times}\otimes\mathbb{F}_{2}.$

Hence it is enough to compute $\kappa$. By Proposition 3, together with Lemmas
15, 17,16, 18 and 19, we have

$\kappa=i_{3}(-(\alpha^{2}+\beta^{2}-2\alpha\beta))\cdot\Theta((\beta-\alpha)f^{\prime}(\beta)).$

The proposition follows. ∎

###  4.4. The boundary map for local fields

As for $J[4]$, the boundary map for $\wedge^{2}J[4]$ admits a simpler
description when the field $K$ is $p$-adic. Given a decomposition of $K_{f,3}$
into fields $\prod L_{i}$, we let $\alpha_{i},\beta_{i}$ and $\gamma_{i}$
denote the images of $t_{1},t_{2}$ and $t_{3}$ in $L_{i}$.

######  Proposition 7.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$. Let
$K_{f}^{(2)}=\prod_{i=1}^{m}K_{i}$, and let $K_{f,3}=\prod_{i=1}^{n}L_{i}$ be
decompositions into products of fields. Define
$\pi_{1},\pi_{2}:\\{1,\ldots,n\\}\to\\{1,\ldots,m\\}$ by the property that the
two maps $K_{f}^{(2)}\to K_{f,3}$ send $K_{\pi_{1}(i)}$ and $K_{\pi_{2}(i)}$
respectively to $L_{i}$. Then the boundary map

$H^{1}(K,\wedge^{2}J[2])\to H^{2}(K,\wedge^{2}J[2])$

sends $(z_{i})\in
H^{1}(K,\wedge^{2}J[2])\subset\prod_{i}K_{i}^{\times}\otimes\mathbb{F}_{2}$ to
$(w_{i})\in H^{2}(K,\wedge^{2}J[2])\subset\prod_{i}\mathrm{Br}(K_{i})[2]$,
where

$w_{i}=\langle-(\alpha_{i}^{2}+\beta_{i}^{2}-2\alpha_{i}\beta_{i}),z_{j}\rangle_{K_{j}}+\sum_{\pi_{2}(k)=i}\langle\mathrm{Nm}_{L_{k}|K_{\pi_{2}(k)}}(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),z_{\pi_{1}(k)}\rangle_{K_{\pi_{1}(k)}}.$

###### Proof.

By Proposition 6, the boundary map sends $(z_{i})$ to
$\mathrm{cores}_{K_{f,3}|K_{f}^{(2)}}^{(2)}\langle(\gamma-\alpha)f^{\prime}(\gamma),i_{1}(z)\rangle_{K_{f,3}}+\langle-(\alpha^{2}+\beta^{2}-2\alpha\beta),z\rangle_{K_{f}^{(2)}}$.
By definition, this is equal to $(w_{i})$, where
$w_{i}\in\mathrm{Br}(K_{i})[2]$ is given by

$\langle-(\alpha_{i}^{2}+\beta_{i}^{2}-2\alpha_{i}\beta_{i}),z_{i}\rangle_{K_{i}}+\sum_{\pi_{2}(k)=i}\mathrm{cores}^{(2)}_{L_{k}|K_{i}}\langle(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),\iota_{k}(z_{\pi_{1}(k)})\rangle_{L_{k}}.$

By Lemma 4, we have

$\displaystyle\mathrm{cores}^{(2)}_{L_{k}|K_{i}}\langle(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),\iota_{k}(z_{\pi_{1}(k)})\rangle_{L_{k}}$
$\displaystyle=\mathrm{cores}^{(2)}_{L_{k}|K}\langle(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),\iota_{k}(z_{\pi_{1}(k)})\rangle_{L_{k}}$
$\displaystyle=\mathrm{cores}^{(2)}_{L_{k}|K_{\pi_{1}(k)}}\langle(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),\iota_{k}(z_{\pi_{1}(k)})\rangle_{L_{k}},$

and the last expression is equal to

$\langle\mathrm{Nm}_{L_{k}|K_{\pi_{2}(k)}}(\gamma_{k}-\alpha_{k})f^{\prime}(\gamma_{k}),z_{\pi_{1}(k)}\rangle_{K_{\pi_{1}(k)}}.$

∎

For a prime $p$ we denote by $\theta_{p}$ the map

$\mathrm{Ker}(\mathbb{Q}_{p,f}^{(2),\times}\otimes\to\mathbb{Q}_{p,f}^{\times})\to\mathrm{Ker}(\mathrm{Br}(\mathbb{Q}_{p,f}^{(2)})[2]\to\mathrm{Br}(\mathbb{Q}_{p,f})[2])$

from Proposition 7.

##  5\. Lifting obstructions at $\infty$

We now apply the description of these boundary maps to obtain lifting
obsructions at $\infty$, i.e. to compute the boundary map

$H^{1}(\mathbb{R},\wedge^{2}J[2])\to H^{2}(\mathbb{R},\wedge^{2}T_{2}J).$

###  5.1. Local aspects at infinity: dimensions

Suppose $M$ is a torsion-free abelian group with an indecomposable action of
$\mathbb{Z}/2\mathbb{Z}=\langle c\rangle$. Then there are exactly three
options for $M$. Either $M=\mathbb{Z}$ with the trivial action,
$M=\mathbb{Z}(1)$ (i.e. rank 1, and $c$ acts as $-1$) or
$M=M_{c}:=\mathbb{Z}[\mathbb{Z}/2\mathbb{Z}]$. It is straightforward to verify
the following computations.

######  Lemma 20.

  1. (1)

$M_{c}\otimes\mathbb{Z}(1)\simeq M_{c}$,

  2. (2)

$M_{c}\otimes M_{c}\simeq M_{c}^{\oplus 2}$,

  3. (3)

$\wedge^{2}M_{c}\simeq\mathbb{Z}(1)$,

  4. (4)

$H^{1}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}_{2})=0.$

  5. (5)

$H^{1}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}_{2}(1))=\mathbb{Z}/2\mathbb{Z}.$

  6. (6)

$H^{1}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}_{2}\otimes M_{c})=0.$

  7. (7)

$H^{1}(\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2}\otimes M_{c})=0$.

######  Lemma 21.

Let $f$ be an odd degree separable polynomial in $\mathbb{R}[x]$ with exactly
$r_{1}(f)$ real roots. Let $J$ denote the Jacobian of

$T_{2}(\mathrm{Jac}(X))\simeq(\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}(1))^{(r_{1}(f)-1)/2}\oplus
M_{c}\otimes\mathbb{Z}_{2}^{g-(r_{1}(f)-1)/2}.$

###### Proof.

It is enough to prove the corresponding claim for
$H_{1}(X(\mathbb{C}),\mathbb{Z})$. By the classification of indecomposable
$\mathbb{Z}/2\mathbb{Z}$-modules, $H_{1}(\mathbb{X}(\mathbb{C}),\mathbb{Z})$
is isomorphic to $\mathbb{Z}^{a}\oplus\mathbb{Z}(1)^{b}\oplus M_{c}^{d}$,where
$a+b+2c=2g$. Furthermore we must have $a+d=g$, and by the Lemma we have

$a+b+d=\dim H^{0}(\mathbb{R},J[2]).$

On ther other hand we also have

$\dim H^{0}(\mathbb{R},J[2])=r_{1}(f)+r_{2}(f)-1.$

The lemma follows from rearranging these formulas. ∎

We arrive at the following Lemma, which gives a liftability obstruction when
$f$ has more than one real root.

######  Lemma 22.

For $f$ and $J$ as above,

$\dim_{\mathbb{F}_{2}}H^{1}(\operatorname{Gal}(\mathbb{C}|\mathbb{R}),\wedge^{2}T_{2}J)=\binom{(r_{1}(f)-1)/2}{2}+g,$

and

$\dim_{\mathbb{F}_{2}}H^{1}(\operatorname{Gal}(\mathbb{C}|\mathbb{R}),\wedge^{2}J[2])=\binom{2(r_{1}(f)-1)/2}{2}+g.$

###### Proof.

By Lemma 20, we deduce

$\wedge^{2}T_{2}J\simeq\mathbb{Z}(1)^{\binom{(r_{1}(f)-1)/2}{2}+g}\oplus\mathbb{Z}^{\binom{(r_{1}(f)-1)/2}{2}}\oplus
M_{c}^{(r_{1}(f)-1+g)(5r_{1}(f)+g-1)/2},$

and conclude from Lemma 20. ∎

###  5.2. Local aspects at infinity: boundary maps

In this subsection we use the description of boundary maps for
$\wedge^{2}J[4]$ given in section 4 when $K=\mathbb{R}$.

We note that if $X$ is a genus $g$ hyperelliptic curve with a rational
Weierstrass point over $\mathbb{R}$, defined by an polynomial
$f(x)\in\mathbb{R}[x]$ of degree $2g+1$, we can describe
$H^{1}(\mathbb{R},J[2])$ and the image of $H^{1}(\mathbb{R},T_{2}J)$ in
$H^{1}(\mathbb{R},J[2])$ explicitly in terms of the polynomial $f$. Namely,
suppose $f$ has $2d+1$ real roots. Write

$f=c\cdot\prod_{i=1}^{2d+1}(x-\alpha_{i})\cdot\prod_{i=1}^{g-d}q_{i}(x)$

where $c\in\mathbb{R}^{\times}$, $q_{i}$ are monic irreducible quadratics
polynomials, and $\alpha_{1}<\alpha_{2}<\ldots<\alpha_{n}$. Then
$H^{1}(\mathbb{R},J[2])\simeq\mathrm{Ker}(\mathrm{Nm}:\\{\pm
1\\}^{2d+1}\to\\{\pm 1\\})$, where the norm map is just multiplication. Given
non-Weierstrass points $b,z$, the class of $z-b$ in $H^{1}(\mathbb{R},J[2])$
is given by the tuple
$(\mathrm{sgn}\left(\frac{x(z)-\alpha_{i}}{x(b)-\alpha_{i}}\right))_{i=1}^{2d+1}$.
If $c>0$, then $x=\lambda$ defines a real point of $X$ if and only if
$\lambda>\alpha_{i}$ for an even number of $\alpha_{i}$. It follows that the
subgroup of $H^{1}(\mathbb{R},J[2])$ generated by
$X(\mathbb{R})-X(\mathbb{R})$ has dimension $d$ (and hence equals the image of
$H^{1}(\mathbb{R},T_{2}J)$) and is equal to the subgroup of
$\mathbb{F}_{2}^{2d+1}$ generated by

$e_{2}+e_{3},e_{4}+e_{5},\ldots,e_{2d}+e_{2d+1}.$

Similarly if $c<0$ one deduces that the subgroup generated by
$X(\mathbb{R})-X(\mathbb{R})$ is equal to the image of
$H^{1}(\mathbb{R},T_{2}J)$, and is equal to the subgroup of
$\mathbb{F}_{2}^{2d+1}$ generated by

$e_{1}+e_{2},e_{3}+e_{4},\ldots,e_{2d-1}+e_{2d}.$

Let’s say $c>0$. Fix $\alpha$. Then $-f^{\prime}(\alpha)<0$ iff the number of
roots $>\alpha$ is even. So the condition on a tuple $(\epsilon_{i})$ is that,
for $i$ odd,

$\epsilon_{i}\cdot\prod_{j<i}\epsilon_{j}=1,$

and for $i$ even

$\prod_{j<i}\epsilon_{j}=1.$

So the condition is for all $i$,

(7) $\epsilon_{2i}\epsilon_{2i+1}=1.$

Similarly when $c<0$ the condition is

(8) $\epsilon_{2i-1}\epsilon_{2i}=1.$

This gives another way of seeing that the boundary obstruction exactly
recovers the image of $H^{1}(\mathbb{R},T_{2}J)$.

If we now look at the wedge square, we see that a basis of the
$\mathbb{Z}_{2}$-summand is given by
$(e_{2i-1}+e_{2i})\wedge(e_{2j-1}+e_{2j})$ and
$(e_{2i}+e_{2i+1})\wedge(e_{2j}+e_{2j+1})$ for $i<j$.

Write a factorisation of $f(x)$ over $\mathbb{R}[x]$ into irreducibles as

$f(x)=c\cdot\prod_{i=1}^{2d+1}(x-\alpha_{i})\cdot\prod_{j=1}^{g-d}(x^{2}+a_{i}x+b_{i})$

Then, as explained above, we have an embedding

$\iota_{ij}:\mathbb{Q}_{f}^{(2)}\hookrightarrow\mathbb{R}$

for each pair of real roots $\alpha_{i},\alpha_{j}$, given by sending
$g(\alpha,\beta)$ to $g(\alpha_{i},\beta_{i})$ for each symmetric polynomial
in $\alpha$ and $\beta$. For each $1\leq i\leq g-d$, we also have an embedding
$\rho_{i}$ sending $\alpha\beta$ to $b_{i}$ and $\alpha+\beta$ to $-a_{i}$.
For notational convenience for all $1\leq i\leq 2d+1$ we will also define
$\iota_{ii}$ to be the map

$\mathbb{Q}_{f}^{(2)}\to\\{1\\}.$

######  Lemma 23.

Let $z$ be an element of
$\mathrm{Ker}(\mathbb{Q}_{f}^{(2),\times}\otimes\mathbb{F}_{2}\to\mathbb{Q}_{f}^{\times}\otimes\mathbb{F}_{2})$
$\simeq H^{1}(\mathbb{Q},\wedge^{2}J[2])$ whose image in
$H^{1}(\mathbb{R},\wedge^{2}J[2])$ lifts to
$H^{1}(\mathbb{R},\wedge^{2}T_{2}J)$. Then for all $i<j\leq d$,

$\iota_{2i-1,2j-1}(z)\iota_{2i-1,2j}(z)\iota_{2i,2j-1}(z)\iota_{2i,2j}(z)>0.$

and

$\iota_{2i,2j}(z)\iota_{2i+1,2j}(z)\iota_{2i,2j+1}(z)\iota_{2i+1,2j+1}(z)>0.$

###### Proof.

This follows from (7) and (8). ∎

We denote the map

(9)
$((\rho_{i})_{i},\iota_{2i,2j-1}\cdot\iota_{2i+1,2j-1}\cdot\iota_{2i,2j}\cdot\iota_{2i+1,2j}):K_{f}^{(2),\times}\otimes\mathbb{F}_{2}\to(\mathbb{R}^{\times}/\mathbb{R}_{>0})^{\frac{d(d-1)}{2}}$

by $\theta_{\mathbb{R}}$.

##  6\. Finiteness criteria

The results recalled in section 2 and proved in sections 4 and 5 combine to
give the following criterion for finiteness of $X(\mathbb{Q}_{2})_{2}$.

######  Lemma 24.

Let $X$ be a genus $g$ hyperelliptic curve with a rational Weierstrass point,
given by a polynomial $f(x)\in\mathbb{Q}[x]$ whose Galois group is a
$2$-transitive subgroup of $S_{2g+1}$. Let $\theta_{\mathbb{R}}$ be the map
from (9). Let $\theta_{2}$ be the map from Proposition 7 (with $p=2$). Suppose

  1. (1)

$\\#\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}})[2]=\\#\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}})[2].$

  2. (2)

$X$ has semistable reduction at all primes away from $2$.

  3. (3)

The rank of $\mathrm{Ker}(\theta_{\mathbb{R}})\cap\mathrm{Ker}(\theta_{2})$ is
less than $\frac{3g^{2}+g}{2}-\operatorname{rk}J(\mathbb{Q})$.

Then $X(\mathbb{Q}_{2})_{2}$ is finite.

###### Proof.

Let

$\pi:H^{1}(\mathbb{R},\wedge^{2}T_{2}J)\oplus
H^{1}(\mathbb{Q}_{2},\wedge^{2}T_{2}J)\to
H^{1}(\mathbb{R},\wedge^{2}J[2])\oplus H^{1}(\mathbb{Q}_{2},\wedge^{2}J[2])$

and

$\mathrm{loc}:H^{1}_{f,\\{2\\}}(\mathbb{Q},\wedge^{2}J[2])\to
H^{1}(\mathbb{R},\wedge^{2}J[2])\oplus H^{1}(\mathbb{Q}_{2},\wedge^{2}J[2])$

be the obvious maps. By Proposition 2, conditions (1) and (2) imply that the
dimension of $H^{1}_{\\{}2\\}(G_{\mathbb{Q}},\wedge^{2}V_{2}J)$ is bounded by
the dimension of $\mathrm{loc}^{-1}(\mathrm{Image}(\pi))$. By Proposition 7
and Lemma 23, we have the containment

$\mathrm{loc}^{-1}(\mathrm{Image}(\pi))\subset\mathrm{Ker}(\theta_{\mathbb{R}})\cap\mathrm{Ker}(\theta)_{2}.$

Note that we have not assumed that $X$ has a rational non-Weierstrass point
over $\mathbb{Q}$, but if $\operatorname{Gal}(\mathbb{Q})$ acts
$2$-transitively on $f$ then we can always find a quadratic extension
$K_{0}|\mathbb{Q}$ which splits completely at $2$ and for which $X(K_{0})$
contains a non-Weierstrass point and $\operatorname{Gal}(K_{0})$ acts
$2$-transitively on the roots of $f$.

Since $H^{1}_{f,\\{2\\}}(\mathbb{Q},\wedge^{2}V_{2}J)$ contains the one-
dimensional image of $H^{1}_{\\{2\\}}(\mathbb{Q},\mathbb{Q}_{2}(1))$ of
dimension one, the dimension of the complement
$H^{1}_{f,\\{2\\}}(G_{\mathbb{Q}},\wedge^{2}V_{2}J/\mathbb{Q}_{2}(1))$ is less
than $\frac{(3g-2)(g+1)}{2}-\operatorname{rk}J(\mathbb{Q})$. Hence the lemma
follows from Lemma 1. ∎

Note that in the case when the number of primes above $2$ in
$\mathbb{Q}_{f}^{(2)}$ is equal to the number of primes above $2$ in
$\mathbb{Q}_{f}$, condition $3$ is empty.

Although this is independent of the Diophantine results we consider in this
paper, we note that $2$-descent methods are particularly effective at
verifying the dimension conjectures of Bloch and Kato when one considers
suitably large Tate twists of a Galois representations. More precisely, if $W$
is a crystalline representation of negative weight with
$F^{0}D_{\mathrm{dR}}(W)=0$, then the crystalline condition on Galois
cohomology is empty, i.e. $H^{1}(\mathbb{Q}_{2},W)=H^{1}_{f}(\mathbb{Q},W)$.
This means that we can hope to have a field-theoretic

######  Proposition 8.

Suppose that $f$ is degree $2g+1$ polynomial over $\mathbb{Q}$ which is
irreducible over $\mathbb{Q}_{2}$, that $K_{f}$ has odd class number, and the
discriminant of $f$ is squarefree up to a power of $2$. Then for all $n>0$,

$\dim H^{1}_{f}(G_{\mathbb{Q}},V_{2}J(n))\geq g.$

###### Proof.

The assumption on $n$ implies that
$H^{1}_{f}(G_{\mathbb{Q}_{2}},V_{2}J(n))=H^{1}(G_{\mathbb{Q}_{2}},V_{2}J(n))$.
The assumptions on the discriminant imply that inertia acts unipotently on
$V_{2}J(n)$ for all primes away from $2$. Putting these things together, we
deduce that the submodule of classes in $H^{1}(G_{\mathbb{Q},2},J[2])$ which
lift to $H^{1}(\mathbb{R},T_{2}J(n))$ has dimension $g$, from which the
proposition follows. ∎

The statistics for number fields $K_{f}$ of the form above were recently
considered by Ho, Shankar and Varma [HSV18]. In particular they show that,
subject to certain ‘tail estimates’ a positive proportion (in an appropriate
sense) of $f$ should satisfy these conditions. This suggests the possibility
that one may apply the methods of arithmetic statistics to prove (parts of)
the Bloch–Kato conjectures for a positive proportion of hyperelliptic curves.

###  6.1. The relation with elliptic curve Chabauty

We now explain the relation with elliptic curve Chabauty, as discussed in the
introduction. Let $X$ be a genus $2$ curve over a field $K$ of characteristic
different from $2$ with a $K$-rational Weierstrass point and let $f(x)\in
K[x]$ be a degree $5$ polynomial giving a model of $X$. If $Y\to X$ is an
étale $\mathbb{Z}/2\mathbb{Z}$-cover of $X$, then the Prym variety of $Y\to X$
is an elliptic curve. This elliptic curve may be realised as the Jacobian of a
genus one quotient of $Y$. To explain this, we may work more generally /
universally over $K_{f}$, and consider the curve

$Y_{\alpha}:v_{\alpha}^{2}=cc_{\alpha}\prod_{\beta\neq\alpha}(u_{\alpha}^{2}-(\beta-\alpha)/c_{\alpha})$

over $K_{f}$. Then

$v_{\alpha}^{2}=cc_{\alpha}\prod_{\beta\neq\alpha}(t_{\alpha}-(\beta-\alpha)/c_{\alpha})$

defines a genus one curve which is a quotient of $Y_{\alpha}$ given by
quotienting out by the involution $u_{\alpha}\mapsto-u_{\alpha}$.

Note that the $2$-Selmer group of $\mathrm{Jac}(Y_{\alpha})$ is a subspace of

$H^{1}(K_{f},\mathrm{Jac}(Y_{\alpha})[2])\simeq
H^{1}(K,\mathrm{Ind}^{K_{f}}_{K}\mathrm{Jac}(Y_{\alpha})[2]).$

The $\operatorname{Gal}(K^{\mathrm{sep}}|K)$-module
$\mathrm{Ind}^{K_{f}}_{K}\mathrm{Jac}(Y_{\alpha})[2]$ is isomorphic to
$\mathrm{Ind}^{K_{f,2}}_{K}\mathbb{F}_{2}$, hence calculations in elliptic
curve Chabauty are essentially happening in
$K_{f,2}^{\times}\otimes\mathbb{F}_{2}$ rather than
$K_{f}^{(2),\times}\otimes\mathbb{F}_{2}$. A more significant difference is
the Selmer conditions: the conditions of locally lifting to points on twists
of $\mathrm{Jac}(Y_{\alpha})$ are not the same as the condition of lifting to
$H^{1}_{f}(G_{K_{v}},\wedge^{2}T_{2}J)$. In particular, it can happen that the
$2$-descent methods from this paper prove finiteness of
$X(\mathbb{Q}_{2})_{2}$, but that there is a twist of the cover $Y_{2}\to X$
(coming from a rational point of $X$) for which the corresponding $2$-Selmer
group has large rank.

On the other hand, a common obstruction to both methods is $2$-torsion in the
class group of $K_{f,2}$. If one were to compare four-descents for
$\wedge^{2}T_{2}J$ and the elliptic curves above presumably in some cases here
one would again see a difference between the Tate–Shafarevich classes which
survive.

##  7\. Examples

###  7.1. Statistics of rank two curves satisfying the conditions of Lemma 24

We tested the applicability of these algorithms on a list of genus two curves
recently produced by the LMFDB [BSS+16]. Using some of the formulas described
in this paper, we found that at least 3,323 of the 7,224 genus 2 curves in the
LMFDB of rank 2 with at least one rational Weierstrass point satisfied the
condition $X(\mathbb{Q}_{2})_{2}$ is finite. In fact we did not use all the
conditions. We first restricted to curves whose Jacobians had no rational
$2$-torsion (equivalently those admitting an odd-degree model with an
irreducible Weierstrass polynomial). This reduces to 6,342 curves. Of those,
we searched for curves satisfying the following conditions (note that some of
these conditions depend on the choice of $f$ defining $X$, which was chosen
based on the model of the curve provided by the LMFDB).

  1. (1)

$f$ is not irreducible over $\mathbb{Q}$.

  2. (2)

$f$ is in $\mathbb{Z}[\frac{1}{2}][x]$.

  3. (3)

Let $g\in\mathbb{Q}_{f}[x]$ be an irreducible factor of $f$ not equal to
$x-\alpha$. Let $\mathbb{Q}_{f}(\beta)$ be the field obtained by adjoining a
root of $g$ to $\mathbb{Q}_{f}$. Let $H$ be an irreducible factor of
$\mathrm{Nm}_{\mathbb{Q}_{f}[x]|\mathbb{Q}[x]}(\alpha+\beta)$. Then
$\deg(H)=10$.

  4. (4)

For all primes $p>2$ dividing the discriminant of $f$, $p$ does not divide the
$x^{5}$ coordinate of $f$, and $f$ is separable modulo $p$.

  5. (5)

$\mathrm{val}_{2}(\mathrm{Cl}(\mathbb{Q}_{f}^{(2)}))=\mathrm{val}_{2}(\mathrm{Cl}(\mathbb{Q}_{f})).$

  6. (6)

$\operatorname{rk}(\theta_{2}\oplus\theta_{\mathbb{R}})<\\#\mathrm{Spec}(\mathbb{Q}_{2,f}^{(2)})-\mathrm{Spec}(\mathbb{Q}_{2,f})+d(d-1).$

Note that if $f$ satisfies all these conditions, then it satisfies the
conditions of Lemma 24, and hence $X(\mathbb{Q}_{2})_{2}$ is finite.

Next we explain how these conditions were computed. All computations were
carried out using magma [BCP97]. The first five conditions can easily be
checked using functions in the magma library. For condition 6, we use
Proposition 7 to reduce the computation of $\theta_{2}$ to the problem of
calculating Hilbert symbols over finite extensions of $\mathbb{Q}_{2}$, which
can be done using magma’s Hilbert Symbol function.

In the table below, we record the implementation of this algorithm on the
6,603 hyperelliptic curves in the LMFDB with rank 2 and exactly one rational
Weierstrass point. The numbers indicate the number of curves where the
algorithm failed at this step (but passed all previous steps).

$\\#\mathcal{S}$ | 6603
---|---
$f$ not irreducible | 259
Coefficient issues | 22
$\mathbb{Q}_{f}^{(2)}\neq\mathbb{Q}(\alpha+\beta)$ | $14$
Bad primes | $843$
$(\mathrm{Cl}(\mathbb{Q}_{f}^{(2)})/\mathrm{\mathrm{Cl}}(\mathbb{Q}_{f}))[2]\neq 0$ | $762$
$\operatorname{rk}(\theta_{2}\oplus\theta_{\mathbb{R}})<\\#\mathrm{Spec}(\mathbb{Q}_{2,f}^{(2)})-\mathrm{Spec}(\mathbb{Q}_{2,f})+d(d-1)$ | $1380$
Verified $\\#X(\mathbb{Q}_{2})_{2}<\infty$ | 3323

The most significant obstacle is the last one. In fact one can further break
this down: for most of the curves is question $d\leq 1$, so
$\theta_{\mathbb{R}}=0$ and the issue is the rank of $\theta_{2}$.

###  7.2. Future developments

There are a number of possible ways to extend these results to prove
finiteness of $X(\mathbb{Q}_{2})_{2}$ for more curves. Conditions (2) and (4)
are sufficient but not necessary to guarantee semistable reduction away from
$2$. Condition (3) is sufficient but not necessary for $2$-transitivity of the
action of $\operatorname{Gal}(\mathbb{Q})$ on the roots of $f$. In fact
conditions (1) and (3) could be removed completely if the condition of
$2$-transitivity of the action of $\operatorname{Gal}(\mathbb{Q})$ on the
roots of $f$ were removed from Proposition 7.

The following more serious obstacles remain.

  1. (1)

Local obstructions away from 2: the discriminant of $f(X)$ is divisible by
$p^{2}$ for some $p\neq 2$.

  2. (2)

Local obstructions at 2: the number of primes above $2$ in
$\mathbb{Q}_{f}^{(2)}$ is greater than the number of primes above $2$ in
$\mathbb{Q}_{f}$, and this obstruction is not dealt with by
$\mathrm{Ker}(\theta_{2})$.

  3. (3)

Class group obstructions: the order of
$\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}^{(2)}})[2]$ is greater than the order
of $\mathrm{Cl}(\mathcal{O}_{\mathbb{Q}_{f}})[2]$.

For the class group obstructions, one approach might be the following. Section
3 gives a description of the boundary map

$H^{1}(G_{K,S},\wedge^{2}J[2])\to H^{2}(G_{K,S},\wedge^{2}J[2])$

in terms of cup products of Galois cohomology with values in $\mu_{2}$. This
gives an obstruction to elements in $H^{1}(G_{K,S},\wedge^{2}J[2])$ coming
from the class group lifting to $H^{1}(G_{K,S},\wedge^{2}J[4])$, given in
terms of cup products

$H^{1}(G_{L,S},\mu_{2})\otimes H^{1}(G_{L,S},\mu_{2})\to
H^{2}(G_{L,S},\mu_{2})$

for various number fields $L$. Since the classes come from the class group,
these cup products will give classes in $H^{2}(G_{L,S},\mu_{2})$ which locally
vanish, so the computational methods above do not apply. Is it possible to use
the explicit formula for such cup products in [MS03] to compute these
obstructions in practice?

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