# Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two
Torsion Points

Jiali Yan

###### Abstract

In this paper, we give an explicit formula as well as a practical algorithm
for computing the Cassels-Tate pairing on
$\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$ where $J$ is the Jacobian variety
of a genus two curve under the assumption that all points in $J[2]$ are
$K$-rational. We also give an explicit formula for the Obstruction map
$\text{Ob}:H^{1}(G_{K},J[2])\rightarrow\text{Br}(K)$ under the same
assumption. Finally, we include a worked example demonstrating we can indeed
improve the rank bound given by a 2-descent via computing the Cassels-Tate
pairing.

##  1 Introduction

For any principally polarized abelian variety $A$ defined over a number field
$K$, Cassels and Tate [cassels1] [cassels2] and [tate] constructed a pairing

$\Sh(A)\times\Sh(A)\rightarrow\mathbb{Q}/\mathbb{Z},$

that is nondegenerate after quotienting out the maximal divisible subgroup of
$\Sh(A)$. This pairing is called the Cassels-Tate pairing and it naturally
lifts to a pairing on Selmer groups. One application of this pairing is in
improving the bound on the Mordell-Weil rank $r(A)$ obtained by performing a
standard descent calculation. More specifically, if $\Sh(A)$ is finite or if
all the $n$-torsion points of $A$ are defined over $K$, the kernel of the
Cassels-Tate pairing on $\text{Sel}^{n}(A)\times\text{Sel}^{n}(A)$ is equal to
the image of the natural map
$\text{Sel}^{n^{2}}(A)\rightarrow\text{Sel}^{n}(A)$ induced from the map
$A[n^{2}]\xrightarrow{n}A[n]$, see [thesis, Proposition 1.9.3] for details.
This shows that carrying out an $n$-descent and computing the Cassels-Tate
pairing on $\text{Sel}^{n}(A)\times\text{Sel}^{n}(A)$ gives the same rank
bound as obtained from $n^{2}$-descent where $\text{Sel}^{n^{2}}(A)$ needs to
be computed.

There have been many results on computing the Cassels-Tate pairing in the case
of elliptic curves, such as [cassels98] [steve] [binaryquartic] [monique]
[3isogeny] [platonic] [3selmer]. We are interested in the natural problem of
generalizing the different algorithms for computing the Cassels-Tate pairing
for elliptic curves to compute the pairing for abelian varieties of higher
dimension.

In Section 2, we give the preliminary results needed for the later sections,
including the homogeneous space definition of the Cassels-Tate pairing. In
Section 3, we state and prove an explicit formula for the pairing
$\langle\;,\;\rangle_{CT}$ on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$ where
$J$ is the Jacobian variety of a genus two curve under the assumption that all
points in $J[2]$ are $K$-rational. This formula is analogous to that in the
elliptic curve case in [cassels98]. In Section 4, we describe a practical
algorithm for computing the pairing $\langle\;,\;\rangle_{CT}$ using the
formula in Section 3. In section 5, we also give an explicit formula for the
Obstruction map $\text{Ob}:H^{1}(G_{K},J[2])\rightarrow\text{Br}(K)$ under the
assumption that all points in $J[2]$ are defined over $K$ generalizing the
result in the elliptic curve case [oneil, Proposition 3.4], [clark, Theorem
6]. Finally, in Section 7, we include a worked example demonstrating that
computing the Cassels-Tate pairing can indeed turn a 2-descent to a 4-descent
and improve the rank bound given by a 2-descent. The content of this paper is
based on Chapter 4 of the thesis of the author [thesis].

### Acknowledgements

I express my sincere and deepest gratitute to my PhD supervisor, Dr. Tom
Fisher, for his patient guidance and insightful comments at every stage during
my research.

##  2 Preliminary Results

###  2.1 The set-up

In this paper, we are working over a number field $K$. For any field $k$, we
let $\bar{k}$ denote its algebraic closure and let $\mu_{n}\subset\bar{k}$
denote the $n^{th}$ roots of unity in $\bar{k}$. We let $G_{k}$ denote the
absolute Galois group $\text{Gal}(\bar{k}/k)$.

Let $\mathcal{C}$ be a general _genus two curve_ defined over $K$, which is a
smooth projective curve. It can be given in the following hyperelliptic form:

$\mathcal{C}:y^{2}=f(x)=f_{6}x^{6}+f_{5}x^{5}+f_{4}x^{4}+f_{3}x^{3}+f_{2}x^{2}+f_{1}x+f_{0},$

where $f_{i}\in K$, $f_{6}\neq 0$ and the discriminant $\triangle(f)\neq 0$,
which implies that $f$ has distinct roots in $\bar{K}$.

We let $J$ denote the _Jacobian variety_ of $\mathcal{C}$, which is an abelian
variety of dimension two defined over $K$ that can be identified with
$\text{Pic}^{0}(\mathcal{C})$. We denote the identity element of $J$ by
$\mathcal{O}_{J}$. Via the natural isomorphism
$\text{Pic}^{2}(\mathcal{C})\rightarrow\text{Pic}^{0}(\mathcal{C})$ sending
$[P_{1}+P_{2}]\mapsto[P_{1}+P_{2}-\infty^{+}-\infty^{-}]$, a point $P\in J$
can be identified with an unordered pair of points of $\mathcal{C}$,
$\\{P_{1},P_{2}\\}$. This identification is unique unless $P=\mathcal{O}_{J}$,
in which case it can be represented by any pair of points on $\mathcal{C}$ in
the form $\\{(x,y),(x,-y)\\}$ or $\\{\infty^{+},\infty^{-}\\}$. Suppose the
roots of $f$ are denoted by $\omega_{1},...,\omega_{6}$. Then
$J[2]=\\{\mathcal{O}_{J},\\{(\omega_{i},0),(\omega_{j},0)\\}\text{ for }i\neq
j\\}$. Also, for a point $P\in J$, we let $\tau_{P}:J\rightarrow J$ denote the
translation by $P$ on $J$.

As described in [thebook, Chapter 3, Section 3], suppose
$\\{P_{1},P_{2}\\}$and $\\{Q_{1},Q_{2}\\}$ represent $P,Q\in J[2]$ where
$P_{1},P_{2},Q_{1},Q_{2}$ are Weierstrass points, then

$e_{2}(P,Q)=(-1)^{|\\{P_{1},P_{2}\\}\cap\\{Q_{1},Q_{2}\\}|}.$

###  2.2 Theta divisor and Kummer surface

The _theta divisor_ , denoted by $\Theta$, is defined to be the divisor on $J$
that corresponds to the divisor
$\\{P\\}\times\mathcal{C}+\mathcal{C}\times\\{P\\}$ on
$\mathcal{C}\times\mathcal{C}$ under the birational morphism
$\text{Sym}^{2}\mathcal{C}\rightarrow J$, for some Weierstrass point
$P\in\mathcal{C}$. The Jacobian variety $J$ is principally polarized abelian
variety via $\lambda:J\rightarrow J^{\vee}$ sending $P$ to
$[\tau_{P}^{*}\Theta-\Theta].$

The _Kummer surface_ , denoted by $\mathcal{K}$, is the quotient of $J$ via
the involution $[-1]:P\mapsto-P$. The fixed points under the involution are
the 16 points of order 2 on $J$ and these map to the 16 nodal singular points
of $\mathcal{K}$ (the nodes). General theory, as in [abelianvarieties, Theorem
11.1], [theta, page 150], shows that the linear system of $n\Theta$ of $J$ has
dimension $n^{2}$. Moreover, $|2\Theta|$ is base point free and $|4\Theta|$ is
very ample.

###  2.3 Explicit embeddings of $J$ and $\mathcal{K}$

Denote a generic point on the Jacobian $J$ of $\mathcal{C}$ by
$\\{(x,y),(u,v)\\}$. Then, following [thebook, Chapter 3, Section 1], the
morphism from $J$ to $\mathbb{P}^{3}$ is given by

$k_{1}=1,k_{2}=(x+u),k_{3}=xu,k_{4}=\beta_{0},$

where

$\beta_{0}=\frac{F_{0}(x,u)-2yv}{(x-u)^{2}}$

with
$F_{0}(x,u)=2f_{0}+f_{1}(x+u)+2f_{2}(xu)+f_{3}(x+u)(xu)+2f_{4}(xu)^{2}+f_{5}(x+u)(xu)^{2}+2f_{6}(xu)^{3}.$

We denote the above morphism by
$J\xrightarrow{|2\Theta|}\mathcal{K}\subset\mathbb{P}^{3}$ and it maps
$\mathcal{O}_{J}$ to $(0:0:0:1)$. It is known that its image in
$\mathbb{P}^{3}$ is precisely the Kummer surface $\mathcal{K}$ and is given by
the vanishing of the quartic $G(k_{1},k_{2},k_{3},k_{4})$ with explicit
formula given in [thebook, Chapter 3, Section 1]. Therefore, the Kummer
surface $\mathcal{K}\subset\mathbb{P}^{3}_{k_{i}}$ is defined by
$G(k_{1},k_{2},k_{3},k_{4})=0.$

######  Remark 2.1.

Suppose $P\in J[2]$. We know $\tau_{P}^{*}(2\Theta)\sim 2\Theta$ via the
polarization. This implies that translation by $P$ on $J$ induces a linear
isomorphism on $\mathcal{K}\subset\mathbb{P}^{3}$.

We now look at the embedding of $J$ in $\mathbb{P}^{15}$ induced by
$|4\Theta|$. Let $k_{ij}=k_{i}k_{j}$, for $1\leq i\leq j\leq 4$. Since
$\mathcal{K}$ is irreducible and defined by a polynomial of degree 4,
$k_{11},k_{12},...,k_{44}$ are 10 linearly independent even elements in
$\mathcal{L}(2\Theta^{+}+2\Theta^{-})$. The six odd basis elements in
$\mathcal{L}(2\Theta^{+}+2\Theta^{-})$ are given explicitly in [explicittwist,
Section 3]. A function $g$ on $J$ is _even_ when it is invariant under the
involution $[-1]:P\mapsto-P$ and is _odd_ when $g\circ[-1]=-g$.

Unless stated otherwise, we will use the basis
$k_{11},k_{12},...,k_{44},b_{1},...,b_{6}$ for
$\mathcal{L}(2\Theta^{+}+2\Theta^{-})$, to embed $J$ in $\mathbb{P}^{15}$. The
following theorem gives the defining equations of $J$.

######  Theorem 2.2.

([72theorem, Therorem 1.2], [thegplawpaper, Therorem 1.2]) Let $J$ be the
Jacobian variety of the genus two curve $\mathcal{C}$ defined by
$y^{2}=f_{6}x^{6}+...+f_{1}x+f_{0}$. The $72$ quadratic forms over
$\mathbb{Z}[f_{0},...,f_{6}]$ given in [72theorem, Appendix A] are a set of
defining equations for the projective variety given by the embedding of $J$ in
$\mathbb{P}^{15}$ induced by the basis of
$\mathcal{L}(2\Theta^{+}+2\Theta^{-})$ with explicit formulae given in
[72theorem, Definition 1.1] or [thegplawpaper, Definition 1.1]. The change of
basis between this basis of $\mathcal{L}(2\Theta^{+}+2\Theta^{-})$ and
$k_{11},k_{12},...,k_{44},b_{1},...,b_{6}$ is given in [explicittwist, Section
3].

###  2.4 Principal homogeneous space and 2-coverings

A _principal homogeneous space_ or _torsor_ for $J$ defined over a field $K$
is a variety $V$ together with a morphism $\mu:J\times V\rightarrow V$, both
defined over $K$, that induces a simply transitive action on the
$\bar{K}$-points.

We say $(V_{1},\mu_{1})$ and $(V_{2},\mu_{2})$ are isomorphic over a field
extension $K_{1}$ of $K$ if there is an isomorphism $\phi:V_{1}\rightarrow
V_{2}$ defined over $K_{1}$ that respects the action of $J$.

A _2-covering_ of $J$ is a variety $X$ defined over $K$ together with a
morphism $\pi:X\rightarrow J$ defined over $K$, such that there exists an
isomorphism $\phi:X\rightarrow J$ defined over $\bar{K}$ with
$\pi=[2]\circ\phi$. An isomorphism $(X_{1},\pi_{1})\rightarrow(X_{2},\pi_{2})$
between two 2-coverings is an isomorphism $h:X_{1}\rightarrow X_{2}$ defined
over $K$ with $\pi_{1}=\pi_{2}\circ h$. We sometimes denote $(X,\pi)$ by $X$
when the context is clear.

It can be checked that a 2-covering is a principal homogeneous space. The
short exact sequence $0\rightarrow J[2]\rightarrow
J\xrightarrow{2}J\rightarrow 0$ induces the connecting map in the long exact
sequence

$\delta:J(K)\rightarrow H^{1}(G_{K},J[2]).$ (2.1)

The following two propositions are proved in [explicittwist].

######  Proposition 2.3.

[explicittwist, Lemma 2.14] Let $(X,\pi)$ be a $2$-covering of an abelian
variety $J$ defined over $K$ and choose an isomorphism $\phi:X\rightarrow J$
such that $\pi=[2]\circ\phi$. Then for each $\sigma\in G_{K}$, there is a
unique point $P\in J[2](\bar{K})$ satisfying
$\phi\circ\sigma(\phi^{-1})=\tau_{P}$. The map $\sigma\mapsto P$ is a cocycle
whose class in $H^{1}(G_{K},J[2])$ does not depend on the choice of $\phi$.
This yields a bijection between the set of isomorphism classes of
$2$-coverings of $J$ and the set $H^{1}(G_{K},J[2])$.

######  Proposition 2.4.

[explicittwist, Proposition 2.15] Let $X$ be a $2$-covering of $J$
corresponding to the cocycle class $\epsilon\in H^{1}(G_{K},J[2])$. Then $X$
contains a $K$-rational point (equivalently $X$ is a trivial principle
homogeneous space) if and only if $\epsilon$ is in the image of the connecting
map $\delta$ in (2.1).

We also state and prove the following proposition which is useful for the
computation of the Cassels-Tate pairing later in Sections 4.2 and 4.3. A
_Brauer-Severi_ variety is a variety that is isomorphic to a projective space
over $\bar{K}$.

######  Proposition 2.5.

Let $(X,\pi)$ be a $2$-covering of J, with $\phi\circ[2]=\pi$. Then the linear
system $|\phi^{*}(2\Theta)|$ determines a map $X\rightarrow S$ defined over
$K$, where $S$ is a Brauer-Severi variety. Also, there exists an isomorphism
$\psi$ defined over $\bar{K}$ making the following diagram commute:

${X}$${S}$${J}$${\mathbb{P}^{3}.}$$\scriptstyle{|\phi^{*}(2\Theta)|}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\scriptstyle{|2\Theta|}$
(2.2)

In particular, if $(X,\pi)$ corresponds to a Selmer element via the
correspondence in Proposition 2.3, then the Brauer-Severi variety $S$ is
isomorphic to $\mathbb{P}^{3}$.

###### Proof.

Since $(X,\pi)$ is a $2$-covering of $J$, by Proposition 2.3, we have that for
each $\sigma\in G_{K}$, $\phi\circ(\phi^{-1})^{\sigma}=\tau_{P}$ for some
$P\in J[2]$. The principal polarization gives $\tau_{P}^{*}(2\Theta)\sim
2\Theta$ which implies that
$\phi^{*}(2\Theta)\sim(\phi^{\sigma})^{*}(2\Theta)$, hence the morphism
induced by $|\phi^{*}(2\Theta)|$ is defined over $K$.

Now if $(X,\pi)$ corresponds to a Selmer element, then $X$ everywhere locally
has a point by Proposition 2.4, and hence $S$ everywhere locally has a point.
Since the Hasse principle holds for Brauer-Severi varieties by [bshasse,
Corollary 2.6], we know that $S$ has a point over $K$ and hence it is
isomorphic to $\mathbb{P}^{3}$ by [QA, Therem 5.1.3].

∎

We now make some observations and give some notation.

######  Remark 2.6.

Let $\epsilon\in\text{Sel}^{2}(J)$, and let $(J_{\epsilon},\pi_{\epsilon})$
denote the 2-covering corresponding to $\epsilon$. There exists an isomorphism
$\phi_{\epsilon}$ defined over $\bar{K}$ such that
$[2]\circ\phi_{\epsilon}=\pi_{\epsilon}$. Then, by Proposition 2.5, we have
the following commutative diagram:

${J_{\epsilon}}$${\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}}$${J}$${\mathcal{K}\subset\mathbb{P}^{3}.}$$\scriptstyle{|\phi_{\epsilon}^{*}(2\Theta)|}$$\scriptstyle{\phi_{\epsilon}}$$\scriptstyle{\psi_{\epsilon}}$$\scriptstyle{|2\Theta|}$
(2.3)

The image of $J_{\epsilon}$ under the morphism induced by
$|\phi_{\epsilon}^{*}(2\Theta)|$ is a surface, denoted by
$\mathcal{K}_{\epsilon}$, which we call the _twisted Kummer surface_
corresponding to $\epsilon$. Also $\psi_{\epsilon}$ is a linear isomorphism
$\mathbb{P}^{3}\rightarrow\mathbb{P}^{3}$ defined over $\bar{K}$.

######  Notation 2.7.

Suppose $(J_{\epsilon},\pi_{\epsilon})$ is the 2-covering of $J$ corresponding
to $\epsilon\in H^{1}(G_{K},J[2])$. The involution $[-1]:P\mapsto-P$ on $J$
induces an involution $\iota_{\epsilon}$ on $J_{\epsilon}$ such that
$\phi_{\epsilon}\circ\iota_{\epsilon}=[-1]\circ\phi_{\epsilon}$, where
$[2]\circ\phi_{\epsilon}=\pi_{\epsilon}$. Moreover, the degree 2 morphism
$J_{\epsilon}\xrightarrow{|\phi_{\epsilon}^{*}(2\Theta)|}\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}$
in (2.3) is precisely the quotient by $\iota_{\epsilon}$ and so an alternative
definition of $\mathcal{K}_{\epsilon}$ is as the quotient of $J_{\epsilon}$ by
$\iota_{\epsilon}$. We call a function $g$ on $J_{\epsilon}$ even if it is
invariant under $\iota_{\epsilon}$ and odd if $g\circ\iota_{\epsilon}=-g$.

###  2.5 Definition of the Cassels-Tate Pairing

There are four equivalent definitions of the Cassels-Tate pairing stated and
proved in [poonenstoll]. In this paper we will only be using the homogeneous
space definition of the Cassels-Tate pairing. Suppose $a,a^{\prime}\in\Sh(J)$.
Via the polarization $\lambda$, we get $a^{\prime}\mapsto b$ where
$b\in\Sh(J^{\vee}).$ Let $X$ be the (locally trivial) principal homogeneous
space defined over $K$ representing $a$. Then $\text{Pic}^{0}(X_{\bar{K}})$ is
canonically isomorphic as a $G_{K}$-module to
$\text{Pic}^{0}(J_{\bar{K}})=J^{\vee}(\bar{K}).$ Therefore,
$b\in\Sh(J^{\vee})\subset H^{1}(G_{K},J^{\vee})$ represents an element in
$H^{1}(G_{K},\text{Pic}^{0}(X_{\bar{K}}))$.

Now consider the exact sequence:

$0\rightarrow\bar{K}(X)^{*}/\bar{K}^{*}\rightarrow\text{Div}^{0}(X_{\bar{K}})\rightarrow\text{Pic}^{0}(X_{\bar{K}})\rightarrow
0.$

We can then map $b$ to an element $b^{\prime}\in
H^{2}(G_{K},\bar{K}(X)^{*}/\bar{K}^{*})$ using the long exact sequence
associated to the short exact sequence above. Since
$H^{3}(G_{K},\bar{K}^{*})=0$, $b^{\prime}$ has a lift $f^{\prime}\in
H^{2}(G_{K},\bar{K}(X)^{*})$ via the long exact sequence induced by the short
exact sequence
$0\rightarrow\bar{K}^{*}\rightarrow\bar{K}(X)^{*}\rightarrow\bar{K}(X)^{*}/\bar{K}^{*}\rightarrow
0:$

$H^{2}(G_{K},\bar{K}^{*})\rightarrow H^{2}(G_{K},\bar{K}(X)^{*})\rightarrow
H^{2}(G_{K},\bar{K}(X)^{*}/\bar{K}^{*})\rightarrow
H^{3}(G_{K},\bar{K}^{*})=0.$ (2.4)

Next we show that $f^{\prime}_{v}\in H^{2}(G_{K_{v}},\bar{K_{v}}(X)^{*})$ is
the image of an element $c_{v}\in H^{2}(G_{K_{v}},\bar{K_{v}}^{*}).$ This is
because $b\in\Sh(J^{\vee})$ is locally trivial which implies its image
$b^{\prime}$ is locally trivial. Then the statement is true by the exactness
of local version of sequence (2.4).

We then can define

$\langle a,b\rangle=\sum_{v}\text{inv}_{v}(c_{v})\in\mathbb{Q}/\mathbb{Z}.$

The Cassels-Tate pairing $\Sh(J)\times\Sh(J)\rightarrow\mathbb{Q}/\mathbb{Z}$
is defined by

$\langle a,a^{\prime}\rangle_{CT}:=\langle a,\lambda(a^{\prime})\rangle.$

We sometimes refer to $\text{inv}_{v}(c_{v})$ above as the local Cassels-Tate
pairing between $a,a^{\prime}\in\Sh(J)$ for a place $v$ of $K$. Note that the
local Cassels-Tate pairing depends on the choice of $f^{\prime}\in
H^{2}(G_{K},\bar{K}(X)^{*})$. We make the following remarks that are useful
for the computation for the Cassels-Tate pairing.

######  Remark 2.8.

  1. (i)

By [poonenstoll], we know the homogeneous space definition of the Cassels-Tate
pairing is independent of all the choices we make.

  2. (ii)

Via the map $\text{Sel}^{2}(J)\rightarrow\Sh(J)[2]$, the definition of the
Cassels-Tate pairing on $\Sh(J)[2]\times\Sh(J)[2]$ naturally lifts to a
pairing on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$. In fact, from now on,
we will only be considering $\langle\epsilon,\eta\rangle_{CT}$ for
$\epsilon,\eta\in\text{Sel}^{2}(J)$. The principal homogeneous space $X$ in
the definition is always taken to be the $2$-covering of $J$ corresponding to
$\epsilon$. One can compute $c_{v}$ by evaluating $f_{v}^{\prime}$ at a point
in $X(K_{v})$ provided that one avoids the zeros and poles of
$f_{v}^{\prime}$. Note that $X(K_{v})\neq\emptyset$ by Proposition 2.4.

###  2.6 Explicit 2-coverings of $J$

Let $\Omega$ represent the set of 6 roots of $f$, denoted by
$\omega_{1},...,\omega_{6}$. Recall, as in Proposition 2.3, the isomorphism
classes of 2-coverings of $J$ are parameterized by $H^{1}(G_{K},J[2])$. For
the explicit computation of the Cassels-Tate pairing, we need the following
result on the explicit 2-coverings of $J$ corresponding to elements in
$\text{Sel}^{2}(J)$. We note that this theorem in fact works over any field of
characteristic different from 2.

######  Theorem 2.9.

[explicittwist, Proposition 7.2, Theorem 7.4, Appendix B] Let $J$ be the
Jacobian variety of a genus two curve defined by $y^{2}=f(x)$ where $f$ is a
degree 6 polynomial and $\epsilon\in\mathrm{Sel}^{2}(J)$. Embed $J$ in
$\mathbb{P}^{15}$ via the coordinates
$k_{11},k_{12},...,k_{44},b_{1},...,b_{6}$. There exists
$J_{\epsilon}\subset\mathbb{P}^{15}$ defined over $K$ with Galois invariant
coordinates $u_{0},...,u_{9},v_{1},...,v_{6}$ and a linear isomorphism
$\phi_{\epsilon}:J_{\epsilon}\rightarrow J$ such that
$(J_{\epsilon},[2]\circ\phi_{\epsilon})$ is a $2$-covering of $J$ whose
isomorphism class corresponds to the cocycle class $\epsilon$. Moreover,
$\phi_{\epsilon}$ can be explicitly represented by the $16\times 16$ matrix
$R=\begin{bmatrix}R_{1}&0\\\ 0&R_{2}\\\ \end{bmatrix}$ for some $10\times 10$
matrix $R_{1}$ and some $6\times 6$ matrix $R_{2}$.

######  Remark 2.10.

The explicit formula for $\phi_{\epsilon}$ is given in the beginning of
[explicittwist, Section 7] and depends only on $\epsilon$ and the underlying
genus two curve. Note that the coordinates $u_{0},...,u_{9},v_{1},...,v_{6}$
are derived from another set of coordinates $c_{0},...,c_{9},d_{1},...,d_{6}$
defined in [explicittwist, Definitions 6.9, 6.11] where $c_{0},...,c_{9}$ are
even and $d_{1},...,d_{6}$ are odd. This set of coordinates are in general not
Galois invariant, however, they are in the case where all points of $J[2]$ are
defined over the base field.

##  3 Formula for the Cassels-Tate Pairing

From now on, we always assume that the genus two curve $\mathcal{C}$ is
defined by $y^{2}=f(x)$ such that all roots of $f$ are defined over $K$. Note
that this implies that all points in $J[2]$ are defined over $K$ which is
equivalent to all the Weierstrass points defined over $K$. In this section,
under the above assumption, we state and prove an explicit formula for the
Cassels-Tate pairing on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$.

Let the genus two curve $\mathcal{C}$ be of the form

$\mathcal{C}:y^{2}=\lambda(x-\omega_{1})(x-\omega_{2})(x-\omega_{3})(x-\omega_{4})(x-\omega_{5})(x-\omega_{6}),$

where $\lambda,\omega_{i}\in K$ and $\lambda\neq 0$. Its Jacobian variety is
denoted by $J$.

The 2-torsion subgroup $J[2]$ has basis

$\displaystyle P=\\{(\omega_{1},0),(\omega_{2},0)\\},$
$\displaystyle\;\;\;Q=\\{(\omega_{1},0),(\omega_{3},0)\\},$ $\displaystyle
R=\\{(\omega_{4},0),(\omega_{5},0)\\},$
$\displaystyle\;\;\;S=\\{(\omega_{4},0),(\omega_{6},0)\\}.$

By the discussion at the end of Section 2.1, the Weil pairing is given
relative to this basis by:

$W=\begin{bmatrix}1&-1&1&1\\\ -1&1&1&1\\\ 1&1&1&-1\\\ 1&1&-1&1\\\
\end{bmatrix}.$ (3.1)

More explicitly, $W_{ij}$ denotes the Weil pairing between the $i^{th}$ and
$j^{th}$ generators.

We now show that this choice of basis determines an isomorphism
$H^{1}(G_{K},J[2])\cong(K^{*}/(K^{*})^{2})^{4}$. Consider the map
$J[2]\xrightarrow{w_{2}}(\mu_{2}(\bar{K}))^{4},$ where $w_{2}$ denotes taking
the Weil pairing with $P,Q,R,S$. Since $P,Q,R,S$ form a basis for $J[2]$ and
the Weil pairing is a nondegerate bilinear pairing, we get that $w_{2}$ is
injective. This implies that $w_{2}$ is an isomorphism as
$|J[2]|=|(\mu_{2}(\bar{K}))^{4}|=16$. We then get

$H^{1}(G_{K},J[2])\xrightarrow{w_{2,*}}H^{1}(G_{K},(\mu_{2}(\bar{K}))^{4})\cong(K^{*}/(K^{*})^{2})^{4},$
(3.2)

where $w_{2,*}$ is induced by $w_{2}$ and $\cong$ is the Kummer isomorphism
derived from Hilbert’s Theorem 90. Since the map (3.2) is an isomorphism, we
can represent elements in $H^{1}(G_{K},J[2])$ by elements in
$(K^{*}/(K^{*})^{2})^{4}$.

Before stating and proving the formula for the Cassels-Tate pairing, we first
state and prove the following lemma.

######  Lemma 3.1.

For $\epsilon\in\mathrm{Sel}^{2}(J)$, let $(J_{\epsilon},\pi_{\epsilon})$
denote the corresponding $2$-covering of $J$. Hence, there exists an
isomorphism $\phi_{\epsilon}:J_{\epsilon}\rightarrow J$ defined over $\bar{K}$
such that $[2]\circ\phi_{\epsilon}=\pi_{\epsilon}$. Suppose $T\in J(K)$ and
$T_{1}\in J(\bar{K})$ satisfy $2T_{1}=T$. Then

  1. (i)

There exists a $K$-rational divisor $D_{T}$ on $J_{\epsilon}$ which represents
the divisor class of $\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$.

  2. (ii)

Let $D$ and $D_{T}$ be $K$-rational divisors on $J_{\epsilon}$ representing
the divisor class of $\phi_{\epsilon}^{*}(2\Theta)$ and
$\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$ respectively. Then
$D_{T}-D\sim\phi_{\epsilon}^{*}(\tau_{T}^{*}\Theta-\Theta)$. Suppose $T$ is a
two torsion point. Then $2D_{T}-2D$ is a $K$-rational principal divisor.
Hence, there exists a $K$-rational function $f_{T}$ on $J_{\epsilon}$ such
that $\text{div}(f_{T})=2D_{T}-2D$.

###### Proof.

By definition of a 2-covering, $[2]\circ\phi_{\epsilon}=\pi_{\epsilon}$ is a
morphism defined over $K$. Also, by Proposition 2.3,
$\phi_{\epsilon}\circ(\phi_{\epsilon}^{-1})^{\sigma}=\tau_{\epsilon_{\sigma}}$
for all $\sigma\in G_{K}$, where $(\sigma\mapsto\epsilon_{\sigma})$ is a
cocycle representing $\epsilon$. Since
$[2]\circ\tau_{T_{1}}\circ\phi_{\epsilon}=\tau_{T}\circ[2]\circ\phi_{\epsilon}=\tau_{T}\circ\pi_{\epsilon}$
and $\tau_{T}$ is defined over $K$,
$(J_{\epsilon},\tau_{T}\circ\pi_{\epsilon})$ is also a 2-covering of $J$. We
compute
$\tau_{T_{1}}\circ\phi_{\epsilon}\circ((\tau_{T_{1}}\circ\phi_{\epsilon})^{-1})^{\sigma}=\tau_{T_{1}}\circ\phi_{\epsilon}\circ(\phi_{\epsilon}^{-1})^{\sigma}\circ\tau_{-\sigma(T_{1})}=\tau_{\epsilon_{\sigma}}\circ\tau_{T_{1}}\circ\tau_{-\sigma(T_{1})},$
for all $\sigma\in G_{K}$. This implies the 2-covering
$(J_{\epsilon},\tau_{T}\circ\pi_{\epsilon})$ corresponds to the element in
$H^{1}(G_{K},J[2])$ that is represented by the cocycle
$(\sigma\mapsto\epsilon_{\sigma}+T_{1}-\sigma(T_{1}))$. Hence,
$(J_{\epsilon},\tau_{T}\circ\pi_{\epsilon})$ is the 2-covering of $J$
corresponding to $\epsilon+\delta(T)$, where $\delta$ is the connecting map as
in (2.1). By Proposition 2.5, there exists a commutative diagram:

${J_{\epsilon}}$${\mathbb{P}^{3}}$${J}$${\mathbb{P}^{3},}$$\scriptstyle{|\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))|}$$\scriptstyle{\tau_{T_{1}}\circ\phi_{\epsilon}}$$\scriptstyle{\psi_{\epsilon}}$$\scriptstyle{|2\Theta|}$

where the morphism
$J_{\epsilon}\xrightarrow{|\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))|}\mathbb{P}^{3}$
is defined over $K$. So the pull back of a hyperplane section via this
morphism gives us a rational divisor $D_{T}$ representing the divisor class of
$\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$ as required by (i).

Since the polarization $\lambda:J\rightarrow J^{\vee}$ is an isomorphism and
$2T_{1}=T$, we have
$\phi_{\epsilon}^{*}(\lambda(T))=[\phi_{\epsilon}^{*}(\tau_{T}^{*}\Theta-\Theta)]=[\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))]-[\phi_{\epsilon}^{*}(2\Theta)]=[D_{T}]-[D]$.
The fact that $T$ is a two torsion point implies that
$2\phi_{\epsilon}^{*}(\lambda(P))=0$. Hence, $2D_{T}-2D$ is a $K$-rational
principal divisor which gives (ii).

∎

The following remark explains how we will use Lemma 3.1 in the formula for the
Cassels-Tate pairing on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$.

######  Remark 3.2.

Applying Lemma 3.1(i) with $T=\mathcal{O}_{J},P,Q,R,S\in J[2]$ gives divisors
$D=D_{\mathcal{O}_{J}}$ and $D_{P},D_{Q},D_{R}D_{S}$. Then by Lemma 3.1(ii),
there exist $K$-rational functions $f_{P},f_{Q},f_{R},f_{S}$ on $J_{\epsilon}$
such that $\text{div}(f_{T})=2D_{T}-2D$ for $T=P,Q,R,S$.

######  Theorem 3.3.

Let $J$ be the Jacobian variety of a genus two curve $\mathcal{C}$ defined
over a number field $K$ where all points in $J[2]$ are defined over $K$. For
any $\epsilon,\eta\in\mathrm{Sel}^{2}(J)$, let
$(J_{\epsilon},[2]\circ\phi_{\epsilon})$ be the $2$-covering of $J$
corresponding to $\epsilon$ where $\phi_{\epsilon}:J_{\epsilon}\rightarrow J$
is an isomorphism defined over $\bar{K}$. Fix a choice of basis $P,Q,R,S$ for
$J[2]$, with the Weil pairing given by matrix (3.1). Let $(a,b,c,d)$ denote
the image of $\eta$ via $H^{1}(G_{K},J[2])\cong(K^{*}/(K^{*})^{2})^{4}$, where
this is the isomorphism induced by taking the Weil pairing with $P,Q,R,S$. Let
$f_{P},f_{Q},f_{R},f_{S}$ be the $K$-rational functions on $J_{\epsilon}$
defined in Remark 3.2. Then the Cassels-Tate pairing
$\langle\;,\;\rangle_{CT}:\mathrm{Sel}^{2}(J)\times\mathrm{Sel}^{2}(J)\rightarrow\\{\pm
1\\}$ is given by

$\langle\epsilon,\eta\rangle_{CT}=\prod_{\text{place
}v}(f_{P}(P_{v}),b)_{v}(f_{Q}(P_{v}),a)_{v}(f_{R}(P_{v}),d)_{v}(f_{S}(P_{v}),c)_{v},$

where $(\;,\;)_{v}$ denotes the Hilbert symbol for a given place $v$ of $K$
and $P_{v}$ is an arbitrary choice of a local point on $J_{\epsilon}$ avoiding
the zeros and poles of $f_{P},f_{Q},f_{R},f_{S}$.

###### Proof.

We know $\eta\in H^{1}(G_{K},J[2])$ corresponds to
$(a,b,c,d)\in(K^{*}/(K^{*})^{2})^{4}$ via taking the Weil pairing with
$P,Q,R,S$. Hence, $\eta$ is represented by the cocycle

$\sigma\mapsto\tilde{b}_{\sigma}P+\tilde{a}_{\sigma}Q+\tilde{d}_{\sigma}R+\tilde{c}_{\sigma}S,$

where $\sigma\in G_{K}$ and for each element $x\in K^{*}/(K^{*})^{2}$, we
define $\tilde{x}_{\sigma}\in\\{0,1\\}$ such that
$(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$.

Then the image of $\eta$ in $H^{1}(G_{K},\text{Pic}^{0}(J_{\epsilon}))$ is
represented by the cocycle that sends $\sigma\in G_{K}$ to

$\tilde{b}_{\sigma}\phi_{\epsilon}^{*}[\tau_{P}^{*}\Theta-\Theta]+\tilde{a}_{\sigma}\phi_{\epsilon}^{*}[\tau_{Q}^{*}\Theta-\Theta]+\tilde{d}_{\sigma}\phi_{\epsilon}^{*}[\tau_{R}^{*}\Theta-\Theta]+\tilde{c}_{\sigma}\phi_{\epsilon}^{*}[\tau_{S}^{*}\Theta-\Theta].$

By Remark 3.2, there exist $K$-rational divisors $D_{P},D_{Q},D_{R},D_{S}$ on
$J_{\epsilon}$ such that the above cocycle sends $\sigma\in G_{K}$ to

$\tilde{b}_{\sigma}[D_{P}-D]+\tilde{a}_{\sigma}[D_{Q}-D]+\tilde{d}_{\sigma}[D_{R}-D]+\tilde{c}_{\sigma}[D_{S}-D].$

We need to map this element in $H^{1}(G_{K},\text{Pic}^{0}(J_{\epsilon}))$ to
an element in $H^{2}(G_{K},\bar{K}(J_{\epsilon})^{*}/\bar{K}^{*})$ via the
connecting map induced by the short exact sequence

$0\rightarrow\bar{K}(J_{\epsilon})^{*}/\bar{K}^{*}\rightarrow\text{Div}^{0}(J_{\epsilon})\rightarrow\text{Pic}^{0}(J_{\epsilon})\rightarrow
0.$

Hence, by the formula for the connecting map and the fact that the divisors
$D,D_{P},D_{Q},D_{R},$ $D_{S}$ are all $K$-rational, we get that the
corresponding element in $H^{2}(G_{K},\bar{K}(J_{\epsilon})^{*}/\bar{K}^{*})$
has image in $H^{2}(G_{K},\text{Div}^{0}(J_{\epsilon}))$ represented by the
following cocycle:

$\displaystyle(\sigma,\tau)\mapsto$
$\displaystyle(\tilde{b}_{\tau}-\tilde{b}_{\sigma\tau}+\tilde{b}_{\sigma})(D_{P}-D)+(\tilde{a}_{\tau}-\tilde{a}_{\sigma\tau}+\tilde{a}_{\sigma})(D_{Q}-D)$
$\displaystyle+(\tilde{d}_{\tau}-\tilde{d}_{\sigma\tau}+\tilde{d}_{\sigma})(D_{R}-D)+(\tilde{c}_{\tau}-\tilde{c}_{\sigma\tau}+\tilde{c}_{\sigma})(D_{S}-D),$

for $\sigma,\tau\in G_{K}$.

It can be checked that, for $x\in K^{*}/(K^{*})^{2}$ and $\sigma,\tau\in
G_{K}$, we get $\tilde{x}_{\tau}-\tilde{x}_{\sigma\tau}+\tilde{x}_{\sigma}=2$
if both $\sigma$ and $\tau$ flip $\sqrt{x}$ and otherwise it is equal to zero.
Define $\iota_{\sigma,\tau,x}=1$ if both $\sigma$ and $\tau$ flip $\sqrt{x}$
and otherwise $\iota_{\sigma,\tau,x}=0$. Note that the map that sends $x\in
K^{*}/(K^{*})^{2}$ to the class of $(\sigma,\tau)\mapsto\iota_{\sigma,\tau,x}$
explicitly realizes the map $K^{*}/(K^{*})^{2}\cong
H^{1}(G_{K},\frac{1}{2}\mathbb{Z}/\mathbb{Z})\subset
H^{1}(G_{K},\mathbb{Q}/\mathbb{Z})\rightarrow H^{2}(G_{K},\mathbb{Z})$. Then,
for $\sigma,\tau\in G_{K}$, the cocycle in the last paragraph sends
$(\sigma,\tau)$ to

$\iota_{\sigma,\tau,b}\cdot 2(D_{P}-D)+\iota_{\sigma,\tau,a}\cdot
2(D_{Q}-D)+\iota_{\sigma,\tau,d}\cdot 2(D_{R}-D)+\iota_{\sigma,\tau,c}\cdot
2(D_{S}-D).$

Hence, by Remark 3.2, the corresponding element in
$H^{2}(G_{K},\bar{K}(J_{\epsilon})^{*}/\bar{K}^{*})$ is represented by

$(\sigma,\tau)\mapsto[f_{P}^{\iota_{\sigma,\tau,b}}\cdot
f_{Q}^{\iota_{\sigma,\tau,a}}\cdot f_{R}^{\iota_{\sigma,\tau,d}}\cdot
f_{S}^{\iota_{\sigma,\tau,c}}],$

for all $\sigma,\tau\in G_{K}$.

For each place $v$ of $K$, following the homogeneous space definition of
$\langle\epsilon,\eta\rangle_{CT}$ as given in Section 2.5, we obtain an
element in $H^{2}(G_{K_{v}},\bar{K_{v}^{*}})$ from the long exact sequence
induced by the short exact sequence
$0\rightarrow\bar{K_{v}^{*}}\rightarrow\bar{K_{v}}(J_{\epsilon})^{*}\rightarrow\bar{K_{v}}(J_{\epsilon})^{*}/\bar{K_{v}}^{*}\rightarrow
0$. The long exact sequence is the local version of (2.4) with $X$ replaced by
$J_{\epsilon}$. By Remark 2.8(ii), this element in
$H^{2}(G_{K_{v}},\bar{K_{v}}^{*})$ can be represented by

$(\sigma,\tau)\mapsto f_{P}(P_{v})^{\iota_{\sigma,\tau,b}}\cdot
f_{Q}(P_{v})^{\iota_{\sigma,\tau,a}}\cdot
f_{R}(P_{v})^{\iota_{\sigma,\tau,d}}\cdot
f_{S}(P_{v})^{\iota_{\sigma,\tau,c}},$

for all $\sigma,\tau\in G_{K}$ and some local point $P_{v}\in
J_{\epsilon}(K_{v})$ avoiding the zeros and poles of
$f_{P},f_{Q},f_{R},f_{S}$.

Hence, the above element in $\text{Br}(K_{v})\cong
H^{2}(G_{K_{v}},\bar{K_{v}}^{*})$ is the class of the tensor product of
quaternion algebras

$(f_{P}(P_{v}),b)+(f_{Q}(P_{v}),a)+(f_{R}(P_{v}),d)+(f_{S}(P_{v}),c).$

Then, we have that

inv
$\displaystyle\big{(}(f_{P}(P_{v}),b)+(f_{Q}(P_{v}),a)+(f_{R}(P_{v}),d)+(f_{S}(P_{v}),c)\big{)}$
$\displaystyle=(f_{P}(P_{v},b)_{v}(f_{Q}(P_{v}),a)_{v}(f_{R}(P_{v}),d)_{v}(f_{S}(P_{v}),c)_{v},$

where $(\;,\;)_{v}$ denotes the Hilbert symbol: $K_{v}^{*}\times
K_{v}^{*}\rightarrow\\{1,-1\\}$, as required.
∎

######  Remark 3.4.

In Section 6, we will directly show that the formula for the Cassels-Tate
pairing on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$ given in Theorem 3.3 is
a finite product.

##  4 Explicit Computation

In this section, we explain how we explicitly compute the Cassels-Tate pairing
on $\text{Sel}^{2}(J)\times\text{Sel}^{2}(J)$ using the formula given in
Theorem 3.3, under the assumption that all points in $J[2]$ are defined over
$K$. We fix $\epsilon\in\text{Sel}^{2}(J)$ and
$(J_{\epsilon},[2]\circ\phi_{\epsilon})$, the 2-covering of $J$ corresponding
to $\epsilon$ with
$\phi_{\epsilon}:J_{\epsilon}\subset\mathbb{P}^{15}\rightarrow
J\subset\mathbb{P}^{15}$ given in Theorem 2.9. The statement of Theorem 3.3
suggests that we need to compute the $K$-rational divisors
$D,D_{P},D_{Q},D_{R},D_{S}$ on $J_{\epsilon}$ and the $K$-rational function
$f_{P},f_{Q},f_{R},f_{S}$ on $J_{\epsilon}$, as in Remark 3.2.

###  4.1 Computing the twist of the Kummer surface

We describe a practical method for computing a linear isomorphism
$\psi_{\epsilon}:\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}\rightarrow\mathcal{K}\subset\mathbb{P}^{3}$
corresponding to $\epsilon$. More explicitly, we need to compute
$\psi_{\epsilon}$ such that $\psi_{\epsilon}(\psi_{\epsilon}^{-1})^{\sigma}$
is the action of translation by $\epsilon_{\sigma}\in J[2]$ on $\mathcal{K}$
and $(\sigma\mapsto\epsilon_{\sigma})$ is a cocycle representing $\epsilon$.
Since all points in $J[2]$ are defined over $K$, the coboundaries in
$B^{1}(G_{K},J[2])$ are trivial. Therefore, these conditions determine
$\psi_{\epsilon}$ uniquely up to a change of linear automorphism of
$\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}$ over $K$.

For each $T\in J[2]$, we have an explicit formula for
$M_{T}\in\text{GL}_{4}(K)$, given in [thebook, Chapter 3 Section 2],
representing the action of translation by $T\in J[2]$ on the Kummer surface
$\mathcal{K}\subset\mathbb{P}^{3}$. It can be checked that they form a basis
of $\text{Mat}_{4}(K)$ and we suppose
$M_{P}^{2}=c_{P}I,M_{Q}^{2}=c_{Q}I,M_{R}^{2}=c_{R}I,\;\text{and}\;M_{S}^{2}=c_{S}I.$
The explicit formulae for $c_{P},c_{Q},c_{R},c_{S}$ can also be found in
[thebook, Chapter 3 Section 2]. Moreover, by [thebook, Chapter 3 Section 3]
and the Weil pairing relationship among the generators $P,Q,R,S$ of $J[2]$
specified by (3.1), we know that $[M_{P},M_{Q}]=[M_{R},M_{S}]=-I$ and the
commutators of the other pairs are trivial.

Suppose $(a,b,c,d)\in(K^{*}/(K^{*})^{2})^{4}$ represents $\epsilon$. Let
$A\in\text{GL}_{4}(\bar{K})$ represent the linear isomorphism
$\psi_{\epsilon}$ and let
$M_{T}^{\prime}=A^{-1}M_{T}A\in\text{GL}_{4}(\bar{K})$ represent the action of
$T$ on the twisted Kummer $\mathcal{K}_{\epsilon}$. It can be checked, see
[thesis, Lemma 3.2.1] for details, that the set of matrices in
$\text{PGL}_{4}(\bar{K})$ that commute with $M_{T}$ in
$\text{PGL}_{4}(\bar{K})$ for all $T\in J[2]$ is $\\{[M_{T}],T\in J[2]\\}$.
This implies that any $B\in\text{GL}_{4}(\bar{K})$ such that
$[M_{T}]^{\prime}=[B^{-1}M_{T}B]\in\text{PGL}_{4}(\bar{K})$ for any $T\in
J[2]$ is equal to a multiple of $M_{T}$ composed with $A$ and so can also
represents $\psi_{\epsilon}$. Hence, it will suffice to compute the matrices
$M_{T}^{\prime}$.

Consider $[M_{T}^{\prime}]\in\text{PGL}_{4}(\bar{K})$ and $\sigma\in G_{K}$.
We have

$[M_{T}^{\prime}]([M_{T}^{\prime}]^{-1})^{\sigma}=[A^{-1}M_{T}A(A^{-1})^{\sigma}M_{T}^{-1}A^{\sigma}]\in\text{PGL}_{4}(\bar{K}).$

Recall that for each element $x\in K^{*}/(K^{*})^{2}$, we define
$\tilde{x}_{\sigma}\in\\{0,1\\}$ such that
$(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$. Since
$[A(A^{-1})^{\sigma}]=[M_{P}^{\tilde{b}_{\sigma}}M_{Q}^{\tilde{a}_{\sigma}}M_{R}^{\tilde{d}_{\sigma}}M_{S}^{\tilde{c}_{\sigma}}]$,
we have $[M_{T}^{\prime}]$ is in $\text{PGL}_{4}(K)$. This means that we can
redefine $M_{T}^{\prime}=\lambda_{T}A^{-1}M_{T}A$ for some
$\lambda_{T}\in\bar{K}$ such that $M_{T}^{\prime}\in\text{GL}_{4}(K)$ by
Hilbert Theorem 90.

Let
$N_{P}=1/\sqrt{c_{P}}M_{P},N_{Q}=1/\sqrt{c_{Q}}M_{Q},N_{R}=1/\sqrt{c_{R}}M_{R},N_{S}=1/\sqrt{c_{S}}M_{S}$.
Then $N_{T}^{2}=I$ for $T=P,Q,R,S$. Define
$N_{T}^{\prime}=A^{-1}N_{T}A\in\text{GL}_{4}(\bar{K})$ for $T=P,Q,R,S$. We
note that $N_{T}^{\prime},M_{T}^{\prime}$ represent the same element in
$\text{PGL}_{4}(K)$ and $N_{T}^{\prime 2}=I$ for each $T=P,Q,R,S$. Suppose
$M_{P}^{\prime 2}=\alpha_{P}I,M_{Q}^{\prime 2}=\alpha_{Q}I,M_{R}^{\prime
2}=\alpha_{R}I,M_{S}^{\prime 2}=\alpha_{S}I.$ Then
$N_{T}^{\prime}=1/\sqrt{\alpha_{T}}M_{T}^{\prime}$ for each $T=P,Q,R,S$. Note
that there might be some sign issues here but they will not affect the later
computation. Since

$N_{P}^{\prime}(N_{P}^{\prime-1})^{\sigma}=A^{-1}N_{P}A(A^{-1})^{\sigma}(N_{P}^{-1})^{\sigma}A^{\sigma},$

using $N_{P}=1/\sqrt{c_{P}}M_{P}$ and
$N_{P}^{\prime}=1/\sqrt{\alpha_{P}}M_{P}^{\prime}$ with
$M_{P},M_{P}^{\prime}\in\text{GL}_{4}(K)$, we compute that

$\frac{\sigma(\sqrt{\alpha_{P}})}{\sqrt{\alpha_{P}}}=\frac{\sigma(\sqrt{a})}{\sqrt{a}}\frac{\sigma(\sqrt{c_{P}})}{\sqrt{c_{P}}},$

and similar equations for $Q,R,S.$

This implies that $\alpha_{P}=c_{P}a$ up to squares in $K$ and so via
rescaling $M_{P}^{\prime}$ by elements in $K$, we have $M_{P}^{\prime
2}=c_{P}aI$. Similarly, $M_{Q}^{\prime 2}=c_{Q}bI,M_{R}^{\prime
2}=c_{R}cI,M_{S}^{\prime 2}=c_{S}dI$. We note that we also have
$[M_{P}^{\prime},M_{Q}^{\prime}]=[M_{R}^{\prime},M_{S}^{\prime}]=-I$ and the
commutators of the other pairs are trivial. This implies that

$\text{Mat}_{4}(K)\cong(c_{P}a,c_{Q}b)\otimes(c_{R}c,c_{S}d)$
$M_{P}^{\prime}\mapsto i_{1}\otimes 1,M_{Q}^{\prime}\mapsto j_{1}\otimes
1,M_{R}^{\prime}\mapsto 1\otimes i_{2},M_{S}^{\prime}\mapsto 1\otimes j_{2},$

where $(c_{P}a,c_{Q}b)$ and $(c_{R}c,c_{S}d)$ are quaternion algebras with
generators $i_{1},j_{1}$ and $i_{2},j_{2}$ respectively. In Section5, we will
interpret this isomorphism as saying that the image of $\epsilon$ via the
obstruction map is trivial.

Let $A=(c_{P}a,c_{Q}b),B=(c_{R}c,c_{S}d)$. By the argument above, we know
$A\otimes B$ represents the trivial element in $\text{Br}(K)$ and an explicit
isomorphism $A\otimes B\cong\text{Mat}_{4}(K)$ will give us the explicit
matrices $M_{P}^{\prime},M_{Q}^{\prime},M_{R}^{\prime},M_{S}^{\prime}$ we
seek. Since the classes of $A,B$ are in $\text{Br}[2]$, we have $A,B$
representing the same element in $\text{Br}(K)$. This implies that $A\cong B$
over $K$, by Wedderburn’s Theorem. We have the following lemma.

######  Lemma 4.1.

Consider a tensor product of two quaternion algebras $A\otimes B$, where
$A=(\alpha,\beta),\;B=(\gamma,\delta),$ with generators $i_{1},j_{1}$ and
$i_{2},j_{2}$ respectively. Suppose there is an isomorphism
$\psi:B\xrightarrow{\sim}A$ given by

$\displaystyle i_{2}\mapsto a_{1}\cdot 1+b_{1}\cdot i_{1}+c_{1}\cdot
j_{1}+d_{1}\cdot i_{1}j_{1},$ $\displaystyle j_{2}\mapsto a_{2}\cdot
1+b_{2}\cdot i_{1}+c_{2}\cdot j_{1}+d_{2}\cdot i_{1}j_{1}.$

Then there is an explicit isomorphism

$A\otimes B\cong\text{Mat}_{4}(K)$

given by

$i_{1}\otimes 1\mapsto M_{i_{1}}:=\begin{bmatrix}0&\alpha&0&0\\\ 1&0&0&0\\\
0&0&0&\alpha\\\ 0&0&1&0\\\ \end{bmatrix}$

$j_{1}\otimes 1\mapsto M_{j_{1}}:=\begin{bmatrix}0&0&\beta&0\\\
0&0&0&-\beta\\\ 1&0&0&0\\\ 0&-1&0&0\\\ \end{bmatrix}$

$1\otimes i_{2}\mapsto
M_{i_{2}}:=\begin{bmatrix}a_{1}&b_{1}\cdot\alpha&c_{1}\cdot\beta&-d_{1}\cdot\alpha\beta\\\
b_{1}&a_{1}&-d_{1}\cdot\beta&c_{1}\cdot\beta\\\
c_{1}&d_{1}\cdot\alpha&a_{1}&-b_{1}\cdot\alpha\\\ d_{1}&c_{1}&-b_{1}&a_{1}\\\
\end{bmatrix}$

$1\otimes j_{2}\mapsto
M_{j_{2}}:=\begin{bmatrix}a_{2}&b_{2}\cdot\alpha&c_{2}\cdot\beta&-d_{2}\cdot\alpha\beta\\\
b_{2}&a_{2}&-d_{2}\cdot\beta&c_{2}\cdot\beta\\\
c_{2}&d_{2}\cdot\alpha&a_{2}&-b_{2}\cdot\alpha\\\ d_{2}&c_{2}&-b_{2}&a_{2}\\\
\end{bmatrix}$

###### Proof.

We have that $A\otimes A^{op}$ is isomorphic to a matrix algebra. More
specifically, $A\otimes A^{op}\cong\text{End}_{K}(A)$ via $u\otimes
v\mapsto(x\mapsto uxv)$, which makes $A\otimes A^{op}\cong\text{Mat}_{4}(K)$
after picking a basis for $A$. Hence,

$\begin{array}[]{ccc}A\otimes B^{op}&\cong&\text{Mat}_{4}(K)\\\ u\otimes
v&\mapsto&(x\mapsto ux\psi(v)).\\\ \end{array}\;$

More explicitly, fixing the basis of $A$ to be
$\\{1,i_{1},j_{1},i_{1}j_{1}\\}$, the isomorphism is as given in the statement
of the lemma.

∎

By taking $A=(c_{P}a,c_{Q}b),B=(c_{R}c,c_{S}d)$ in Lemma 4.1, we know that the
matrices $M_{P}^{\prime},M_{Q}^{\prime},M_{S}^{\prime},M_{R}^{\prime}$ and
$M_{i_{1}},M_{j_{1}},M_{i_{2}},M_{j_{2}}$ are equal up to conjugation by a
matrix $C\in\text{GL}_{4}(K)$ via the Noether Skolem Theorem. After a change
of coordinates for $\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}$ according to
$C$, we have that
$M_{P}^{\prime},M_{Q}^{\prime},M_{S}^{\prime},M_{R}^{\prime}$ are equal to
$M_{i_{1}},M_{j_{1}},M_{i_{2}},M_{j_{2}}$. Lemma 4.1 therefore reduces the
problem of computing the $M_{T}^{\prime}$ to that of computing an isomorphism
between two quaternion algebras. See [thesis, Corollary 4.2.3 ] for the
description of an explicit algorithm. Finally we solve for a matrix $A$ such
that $M_{T}^{\prime}=\lambda_{T}A^{-1}M_{T}A$ for some $\lambda_{T}\in\bar{K}$
and $T=P,Q,R,S$ by linear algebra.

###  4.2 Explicit computation of $D$

In this section, we explain a method for computing the $K$-rational divisor
$D$ on $J_{\epsilon}$ representing the divisor class
$\phi_{\epsilon}^{*}(2\Theta)$. The idea is to compute it via the commutative
diagram (2.3) in Remark 2.6.

By Theorem 2.9, there is an explicit isomorphism
$J_{\epsilon}\subset\mathbb{P}^{15}\xrightarrow{\phi_{\epsilon}}J\subset\mathbb{P}^{15}$.
We write $u_{0},...,u_{9},v_{1},...,v_{6}$ for the coordinates on the ambient
space of $J_{\epsilon}\subset\mathbb{P}^{15}$ and write
$k_{11},k_{12},...,k_{44},b_{1},...,b_{6}$ for the coordinates on the ambient
space of $J\subset\mathbb{P}^{15}$. By the same theorem, $\phi_{\epsilon}$ is
represented by a block diagonal matrix consisting of a block of size 10
corresponding to the even basis elements and a block of size 6 corresponding
to the odd basis elements. Following Section 4.1, we can compute an explicit
isomorphism
$\psi_{\epsilon}:\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}\rightarrow\mathcal{K}\subset\mathbb{P}^{3}$
corresponding to $\epsilon$. We write $k_{1}^{\prime},...,k_{4}^{\prime}$ for
the coordinates on the ambient space of
$\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}$. Recall that since all points in
$J[2]$ are defined over $K$, all coboundaries in $B^{1}(G_{K},J[2])$ are
trivial. So, we have that $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and
$\psi_{\epsilon}(\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of
translation by some $\epsilon_{\sigma}\in J[2]$ such that
$(\sigma\mapsto\epsilon_{\sigma})$ represents $\epsilon\in\text{Sel}^{2}(J)$.

Define $k_{ij}^{\prime}=k_{i}^{\prime}k_{j}^{\prime}$. The isomorphism
$\psi_{\epsilon}:\mathcal{K}_{\epsilon}\subset\mathbb{P}_{k_{i}^{\prime}}^{3}\rightarrow\mathcal{K}\subset\mathbb{P}_{k_{i}}^{3}$,
induces a natural isomorphism
$\tilde{\psi_{\epsilon}}:\mathbb{P}^{9}_{k_{ij}^{\prime}}\rightarrow\mathbb{P}^{9}_{k_{ij}}$.
More explicitly, suppose $\psi_{\epsilon}$ is represented by the $4\times 4$
matrix $A$ where
$(k_{1}^{\prime}:...,k_{4}^{\prime})\mapsto(\sum_{i=1}^{4}A_{1i}k_{i}^{\prime}:...:\sum_{i=1}^{4}A_{4i}k_{i}^{\prime})$.
Then
$\tilde{\psi_{\epsilon}}:\mathbb{P}^{9}_{k_{ij}^{\prime}}\rightarrow\mathbb{P}^{9}_{k_{ij}}$
is given by
$(k_{11}^{\prime}:k_{12}^{\prime}:...:k_{44}^{\prime})\mapsto(\sum_{i,j=1}^{4}A_{1i}A_{1j}k_{ij}^{\prime}:\sum_{i,j=1}^{4}A_{1i}A_{2j}k_{ij}^{\prime}:...:\sum_{i,j=1}^{4}A_{4i}A_{4j}k_{ij}^{\prime})$.

On the other hand, the isomorphism
$\phi_{\epsilon}:J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}\rightarrow
J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}$ induces a natural isomorphism
$\tilde{\phi_{\epsilon}}:\mathbb{P}^{9}_{u_{i}}\rightarrow\mathbb{P}^{9}_{k_{ij}}$
represented by the $10\times 10$ block of the matrix representing
$\phi_{\epsilon}$. Since $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and
$\psi_{\epsilon}(\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of
translation by some $\epsilon_{\sigma}\in J[2]$, we get
$\tilde{\phi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}=\tilde{\psi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}$.
Therefore, $\tilde{\psi_{\epsilon}}^{-1}\tilde{\phi_{\epsilon}}$ is defined
over $K$ and we obtain the following commutative diagram that decomposes the
standard commutative diagram (2.3):

${J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}}$${\mathbb{P}^{9}_{u_{i}}}$${\mathbb{P}^{9}_{k_{ij}^{\prime}}}$${\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}_{k_{i}^{\prime}}}$${J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}}$${\mathbb{P}^{9}_{k_{ij}}}$${\mathcal{K}\subset\mathbb{P}^{3}_{k_{i}},}$$\scriptstyle{proj}$$\scriptstyle{\phi_{\epsilon}}$$\scriptstyle{(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}}$$\scriptstyle{\tilde{\phi_{\epsilon}}}$$\scriptstyle{\tilde{\psi_{\epsilon}}}$$\scriptstyle{g_{2}}$$\scriptstyle{\psi_{\epsilon}}$$\scriptstyle{proj}$$\scriptstyle{g_{1}}$
(4.1)

where $g_{1}:(k_{11}:...:k_{44})\rightarrow(k_{1}:...:k_{4})$ and
$g_{2}:(k^{\prime}_{11}:...:k^{\prime}_{44})\rightarrow(k^{\prime}_{1}:...:k^{\prime}_{4})$
are the projection maps. The composition of the morphisms on the bottom row
gives the standard morphism
$J\xrightarrow{|2\Theta|}\mathcal{K}\subset\mathbb{P}^{3}$ and the composition
of the morphisms on the top row gives
$J_{\epsilon}\xrightarrow{|\phi_{\epsilon}^{*}(2\Theta)|}\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}$.

Let $D$ be the pull back on $J_{\epsilon}$ via
$J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}\xrightarrow{proj}\mathbb{P}^{9}_{u_{i}}\xrightarrow{(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}}\mathbb{P}^{9}_{k_{ij}^{\prime}}\xrightarrow{proj}\mathbb{P}^{3}_{k_{i}^{\prime}}$
of the hyperplane section given by $k_{1}^{\prime}=0$. This implies that $D$
is a $K$-rational divisor on $J_{\epsilon}$ representing the class of
$\phi_{\epsilon}^{*}(2\Theta)$. Moreover, the pull back on $J_{\epsilon}$ via
$J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}\xrightarrow{proj}\mathbb{P}^{9}_{u_{i}}\xrightarrow{(\tilde{\psi_{\epsilon}})^{-1}\tilde{\phi_{\epsilon}}}\mathbb{P}^{9}_{k_{ij}^{\prime}}$
of the hyperplane section given by $k_{11}^{\prime}=0$ is $2D$.

###  4.3 Explicit computation of $D_{P},D_{Q},D_{R},D_{S}$

In this section, we explain how to compute the $K$-rational divisors
$D_{P},D_{Q},D_{R},D_{S}$ defined in Remark 3.2. More explicitly, for $T\in
J[2]$, we give a method for computing a $K$-rational divisor $D_{T}$ on
$J_{\epsilon}$ representing the divisor class of
$\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$ for some $T_{1}$ on $J$ such
that $2T_{1}=T$. Recall that we assume all points in $J[2]$ are defined over
$K$ and we have an explicit isomorphism
$\phi_{\epsilon}:J_{\epsilon}\rightarrow J$ such that
$(J_{\epsilon},[2]\circ\phi_{\epsilon})$ is the 2-covering of $J$
corresponding to $\epsilon\in\text{Sel}^{2}(J)$. Recall
$\delta:J(K)\rightarrow H^{1}(G_{K},J[2])$ in (2.1). We first prove the
following lemma.

######  Lemma 4.2.

Let $T\in J(K)$. Suppose
$\phi_{\epsilon+\delta(T)}:J_{\epsilon+\delta(T)}\rightarrow J$ is an
isomorphism and $(J_{\epsilon+\delta(T)},[2]\circ\phi_{\epsilon+\delta(T)})$
is the 2-covering of $J$ corresponding to $\epsilon+\delta(T)\in
H^{1}(G_{K},J[2])$. Let $T_{1}\in J$ such that $2T_{1}=T$. Then,
$\phi_{\epsilon+\delta(T)}^{-1}\circ\tau_{T_{1}}\circ\phi_{\epsilon}:J_{\epsilon}\rightarrow
J_{\epsilon+\delta(T)}$ is defined over $K$.

###### Proof.

Using the same argument as in the proof of Lemma 3.1(i), we know that
$(J_{\epsilon},[2]\circ\tau_{T_{1}}\circ\phi_{\epsilon})$ is the 2-covering of
$J$ corresponding to $\epsilon+\delta(T)\in H^{1}(G_{K},J[2])$. Since all
points in $J[2]$ are defined over $K$, we have
$\tau_{T_{1}}\circ\phi_{\epsilon}\circ((\tau_{T_{1}}\circ\phi_{\epsilon})^{-1})^{\sigma}=\phi_{\epsilon+\delta(T)}\circ(\phi_{\epsilon+\delta(T)}^{-1})^{\sigma}$,
as required.
∎

Let $T\in J(K)$ with $2T_{1}=T$. Consider the commutative diagram below which
is formed by two copies of the standard diagram (2.3). Note that all the
horizontal maps are defined over $K$ as is the composition of the vertical map
on the left by Lemma 4.2. Hence, the composition of the thick arrows is
defined over $K$. Then the pull back on $J_{\epsilon}$ via the thick arrows of
a hyperplane section on
$\mathcal{K}_{\epsilon+\delta(T)}\subset\mathbb{P}^{3}$ is a $K$-rational
divisor $D_{T}$ on $J_{\epsilon}$ representing the divisor class
$\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$. We note that in the case
where $T\in J[2]$, the composition of the vertical maps on the left hand side
of the the diagram below is in fact given by a $16\times 16$ matrix defined
over $K$ even though the individual maps are not defined over $K$.

${J_{\epsilon}\subset\mathbb{P}^{15}}$${\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}}$${J\subset\mathbb{P}^{15}}$${\mathcal{K}\subset\mathbb{P}^{3}}$${J\subset\mathbb{P}^{15}}$${\mathcal{K}\subset\mathbb{P}^{3}}$${J_{\epsilon+\delta(T)}\subset\mathbb{P}^{15}}$${\mathcal{K}_{\epsilon+\delta(T)}\subset\mathbb{P}^{3}.}$$\scriptstyle{|\phi_{\epsilon}^{*}(2\Theta)|}$$\scriptstyle{\phi_{\epsilon}}$$\scriptstyle{\psi_{\epsilon}}$$\scriptstyle{|2\Theta|}$$\scriptstyle{\tau_{T_{1}}}$$\scriptstyle{|2\Theta|}$$\scriptstyle{\phi_{\epsilon+\delta(T)}^{-1}}$$\scriptstyle{|\phi^{*}_{\epsilon+\delta(T)}(2\Theta)|}$$\scriptstyle{\psi_{\epsilon+\delta(T)}}$
(4.2)

The bottom horizontal morphism
$J_{\epsilon+\delta(T)}\xrightarrow{|\phi^{*}_{\epsilon+\delta(T)}(2\Theta)|}\mathcal{K}_{\epsilon+\delta(T)}\subset\mathbb{P}^{3}$
can be explicitly computed using the algorithm in Section 4.2 with the Selmer
element $\epsilon$ replaced by $\epsilon+\delta(T)$. Also, by Theorem 2.9, we
have explicit formulae for $\phi_{\epsilon}$ and $\phi_{\epsilon+\delta(T)}.$
Hence, to explicitly compute $D_{T}$, we need to find a way to deal with
$\tau_{T_{1}}$, for some $T_{1}$ such that $2T_{1}=T$.

Since we need to apply the above argument to $T=P,Q,R,S$, the basis for
$J[2]$, it would suffice to compute the translation maps
$\tau_{T_{1}}:J\subset\mathbb{P}^{15}\rightarrow J\subset\mathbb{P}^{15}$ when
$T_{1}\in J[4]$. This map is given by a $16\times 16$ matrix. We show how to
compute a $10\times 16$ matrix in the following proposition. We then explain
below why this is sufficient for our purposes.

######  Proposition 4.3.

Suppose $T_{1}\in J[4]$. Given the coordinates of $T_{1}\in
J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}$, we can compute the following
composition of morphisms:

$\Psi:J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}\xrightarrow{\tau_{T_{1}}}J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}\xrightarrow{proj}\mathbb{P}^{9}_{k_{ij}}.$

###### Proof.

Let $T=2T_{1}\in J[2]$. Recall that we let $M_{T}$ denote the action of
translation by $T$ on $\mathcal{K}\subset\mathbb{P}^{3}$. Then for any $P\in
J$, we have $k_{i}(P+T)=\sum_{j=1}^{4}(M_{T})_{ij}k_{j}(P)$ projectively as a
vector of length 4, and projectively as a vector of length 10,
$k_{ij}(P+T_{1})$ is equal to

$k_{i}(P+T_{1})k_{j}(P+T_{1})=k_{i}(P+T_{1})\sum_{l=1}^{4}(M_{T})_{jl}k_{l}(P-T_{1})=\sum_{l=1}^{4}(M_{T})_{jl}k_{l}(P-T_{1})k_{i}(P+T_{1}).$

By [thegplawpaper, Theorem 3.2], there exists a $4\times 4$ matrix of bilinear
forms $\phi_{ij}(P,T_{1})$, with explicit formula, that is projectively equal
to the matrix $k_{i}(P-T_{1})k_{j}(P+T_{1})$. Since we have an explicit
formula for $M_{T}$ in [thebook, Chapter 3, Section 2], we can partially
compute the linear isomorphism $\tau_{T_{1}}$:

$\Psi:J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}\xrightarrow{\tau_{T_{1}}}J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}\xrightarrow{proj}\mathbb{P}^{9}_{k_{ij}},$

as required.
∎

######  Remark 4.4.

Suppose $2T_{1}=T\in J[2]$. From the doubling formula on $\mathcal{K}$ as in
[thegplawpaper, Appendix C], we can compute the coordinates of the image of
$T_{1}$ on $\mathcal{K}\subset\mathbb{P}^{3}$ from the coordinates of the
image of $T$ on $\mathcal{K}\subset\mathbb{P}^{3}$. This gives the 10 even
coordinates, $k_{ij}(T_{1})$ and we can solve for the odd coordinates by the
72 defining equations of $J$ given in Theorem 2.2. Note that by Lemma 4.2, we
know the field of definition of $T_{1}$ is contained in the composition of the
field of definition of $\phi_{\epsilon}$ and $\phi_{\epsilon+\delta(T)}$.
Hence, we can compute this field explicitly which helps solving for this point
using MAGMA [magma].

Consider $T\in J[2]$ with $T_{1}\in J[4]$ such that $2T_{1}=T$. We follow the
discussion in Section 4.2 with $\epsilon$ replaced by $\epsilon+\delta(T)$.
This gives a diagram analogous to (4.1). Let
$k_{1,T}^{\prime},...,k_{4,T}^{\prime}$ be the coordinates on the ambient
space of $\mathcal{K}_{\epsilon+\delta(T)}\subset\mathbb{P}^{3}$ and let
$u_{0,T},...,u_{9,T},v_{1,T},...,v_{6,T}$ be the coordinates on the ambient
space of $J_{\epsilon+\delta(T)}\subset\mathbb{P}^{15}$. Let
$k_{ij,T}^{\prime}=k_{i,T}^{\prime}k_{j,T}^{\prime}$. Decomposing the lower
half of the diagram (4.2) gives the commutative diagram below:

${J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}}$${\mathcal{K}_{\epsilon}\subset\mathbb{P}^{3}_{k_{i}^{\prime}}}$${J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}}$${\mathcal{K}\subset\mathbb{P}_{k_{i}}^{3}}$${J\subset\mathbb{P}^{15}_{\\{k_{ij},b_{i}\\}}}$${\mathbb{P}_{k_{ij}}^{9}}$${\mathcal{K}\subset\mathbb{P}_{k_{i}}^{3}}$${J_{\epsilon+\delta(T)}\subset\mathbb{P}^{15}_{\\{u_{i,T},v_{i,T}\\}}}$${\mathbb{P}_{k_{ij,T}^{\prime}}^{9}}$${\mathcal{K}_{\epsilon+\delta(T)}\subset\mathbb{P}_{k_{i,T}^{\prime}}^{3}.}$$\scriptstyle{|\phi_{\epsilon}^{*}(2\Theta)|}$$\scriptstyle{\phi_{\epsilon}}$$\scriptstyle{\psi_{\epsilon}}$$\scriptstyle{|2\Theta|}$$\scriptstyle{\tau_{T_{1}}}$$\scriptstyle{\Psi}$$\scriptstyle{proj}$$\scriptstyle{g_{1}}$$\scriptstyle{(\tilde{\psi}_{\epsilon+\delta(T)})^{-1}}$$\scriptstyle{{\phi}_{\epsilon+\delta(T)}}$$\scriptstyle{g_{2}}$$\scriptstyle{\psi_{\epsilon+\delta(T)}}$
(4.3)

Recall Proposition 4.3 explains how $\Psi$ can be explicitly computed and the
composition of the thick arrows in (4.3) is defined over $K$ by Lemma 4.2. Let
$D_{T}$ be the pull back on $J_{\epsilon}$ via the thick arrows in (4.3) of
the hyperplane section given by $k_{1,T}^{\prime}=0$. This implies that
$D_{T}$ is a $K$-rational divisor on $J_{\epsilon}$ representing the class of
$\phi_{\epsilon}^{*}(\tau_{T_{1}}^{*}(2\Theta))$. Moreover, the pull back on
$J_{\epsilon}$ via

$J_{\epsilon}\subset\mathbb{P}^{15}_{\\{u_{i},v_{i}\\}}\xrightarrow{\phi_{\epsilon}}J\subset\mathbb{P}^{15}_{\\{k_{ij,b_{i}}\\}}\xrightarrow{\Psi}\mathbb{P}^{9}_{k{ij}}\xrightarrow{(\tilde{\psi}_{\epsilon+\delta(T)})^{-1}}\mathbb{P}^{9}_{k_{ij,T}^{\prime}}$

of the hyperplane section given by $k_{11,T}^{\prime}=0$ is $2D_{T}$.

We now apply the above discussion with $T=P,Q,R,S$ and get that the divisors
$D_{P},D_{Q},D_{R},D_{S}$ on $J_{\epsilon}$ described in Remark 3.2 as
required.

######  Remark 4.5.

From the above discussion and the discussion in Section 4.2, the $K$-rational
functions $f_{P},f_{Q},f_{R},f_{S}$ in the formula for the Cassels-Tate
pairing in Theorem 3.3 are quotients of linear forms in the coordinates of the
ambient space of $J_{\epsilon}\subset\mathbb{P}^{15}$. They all have the same
denominator, this being the linear form that cuts out the divisor $2D$.

##  5 The Obstruction Map

In this section, we will state and prove an explicit formula for the
obstruction map $\text{Ob}:H^{1}(G_{K},J[2])\rightarrow\text{Br}(K)$. See
below for the definition of this map. This generalizes a formula in the
elliptic curve case due to O’Neil [oneil, Proposition 3.4], and later refined
by Clark [clark, Theorem 6]. Although this is not needed for the computation
of the Cassels-Tate pairing, it explains why we needed to work with quaternion
algebras in Section 4.1.

######  Definition 5.1.

The obstruction map

$\text{Ob}:H^{1}(G_{K},J[2])\rightarrow
H^{2}(G_{K},\bar{K}^{*})\cong\text{Br}(K)$

is the composition of the map $H^{1}(G_{K},J[2])\rightarrow
H^{1}(G_{K},\text{PGL}_{4}(\bar{K}))$ induced by the action of translation of
$J[2]$ on $\mathcal{K}\subset\mathbb{P}^{3}$, and the injective map
$H^{1}(G_{K},\text{PGL}_{4}(\bar{K}))\rightarrow H^{2}(G_{K},\bar{K}^{*})$
induced from the short exact sequence
$0\rightarrow\bar{K}^{*}\rightarrow\text{GL}_{4}(\bar{K})\rightarrow\text{PGL}_{4}(\bar{K})\rightarrow
0$.

######  Theorem 5.2.

Let $J$ be the Jacobian variety of a genus two curve defined over a field $K$
with $char(K)\neq 2$. Suppose all points in $J[2]$ are defined over $K$. For
$\epsilon\in H^{1}(G_{K},J[2])$, represented by
$(a,b,c,d)\in(K^{*}/(K^{*})^{2})^{4}$ as in Section 3, the obstruction map
$\mathrm{Ob}:H^{1}(G_{K},J[2])\rightarrow\mathrm{Br}(K)$ sends $\epsilon$ to
the class of the tensor product of two quaternion algebras:

$\mathrm{Ob}(\epsilon)=(c_{P}a,c_{Q}b)+(c_{R}c,c_{S}d),$

where $c_{P},c_{Q},c_{R},c_{S}\in K$ are such that
$M_{P}^{2}=c_{P}I,M_{Q}^{2}=c_{Q}I,M_{R}^{2}=c_{R}I,\;\text{and}\;M_{S}^{2}=c_{S}I$
as defined in Section 4.1.

###### Proof.

Let
$N_{P}=1/\sqrt{c_{P}}M_{P},N_{Q}=1/\sqrt{c_{Q}}M_{Q},N_{R}=1/\sqrt{c_{R}}M_{R},N_{S}=1/\sqrt{c_{S}}M_{S}\in\text{GL}_{4}(\bar{K})$.
Then $N_{P}$ is a normalized representation in $\text{GL}_{4}(\bar{K})$ of
$[M_{P}]\in\text{PGL}_{4}(K)$. Similar statements are true for $Q,R,S$. Notice
that $N_{P}^{2}=N_{Q}^{2}=N_{R}^{2}=N_{S}^{2}=I$. So there is a uniform way of
picking a representation in $\text{GL}_{4}(\bar{K})$ for the translation
induced by $\alpha_{1}P+\alpha_{2}Q+\alpha_{3}R+\alpha_{4}S$ for
$\alpha_{i}\in\mathbb{Z}$, namely
$N_{P}^{\alpha_{1}}N_{Q}^{\alpha_{2}}N_{R}^{\alpha_{3}}N_{S}^{\alpha_{4}}.$

Since $\epsilon\in H^{1}(K,J[2])$ is represented by
$(a,b,c,d)\in(K^{*}/{K^{*}}^{2})^{4}$ and $P,Q,R,S$ satisfy the Weil pairing
matrix (3.1), a cocycle representation of $\epsilon$ is:

$\sigma\mapsto\tilde{b}_{\sigma}P+\tilde{a}_{\sigma}Q+\tilde{d}_{\sigma}R+\tilde{c}_{\sigma}S,$

where for each element $x\in K^{*}/(K^{*})^{2}$, we define
$\tilde{x}_{\sigma}\in\\{0,1\\}$ such that
$(-1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$.

Now consider the following commutative diagram of cochains:

${C^{1}(G_{K},\bar{K}^{*})}$${C^{1}(G_{K},\text{GL}_{4})}$${C^{1}(G_{K},\text{PGL}_{4})}$${C^{2}(G_{K},\bar{K}^{*})}$${C^{2}(G_{K},\text{GL}_{4})}$${C^{2}(G_{K},\text{PGL}_{4}).}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$

Defining
$N_{\sigma}=N_{P}^{\tilde{b}_{\sigma}}N_{Q}^{\tilde{a}_{\sigma}}N_{R}^{\tilde{d}_{\sigma}}N_{S}^{\tilde{c}_{\sigma}}$,
we have

$\begin{array}[]{ccc}H^{1}(K,J[2])&\rightarrow&H^{1}(G_{K},\text{PGL}_{4})\\\
(a,b,c,d)&\mapsto&(\sigma\mapsto[N_{\sigma}]).\end{array}$

Then $(\sigma\mapsto[N_{\sigma}])\in C^{1}(G_{K},\text{PGL}_{4})$ lifts to
$(\sigma\mapsto N_{\sigma})\in C^{1}(G_{K},\text{GL}_{4})$ which is then
mapped to

$((\sigma,\tau)\mapsto(N_{\tau})^{\sigma}N_{\sigma\tau}^{-1}N_{\sigma})\in
C^{2}(G_{K},\text{GL}_{4}).$

Note that

$N_{P}^{\sigma}=(\frac{1}{\sqrt{c_{P}}}M_{P})^{\sigma}=\frac{1}{\sigma(\sqrt{c_{P}})}M_{P}=\frac{\sqrt{c_{P}}}{\sigma(\sqrt{c_{P}})}N_{P}=(-1)^{(\widetilde{c_{P}})_{\sigma}}N_{P},$

treating $c_{P}$ in $K^{*}/(K^{*})^{2}$. Similar results also hold for
$Q,R,S$. Observe that for any $x\in K^{*}/(K^{*})^{2}$ and $\sigma,\tau\in
G_{K}$, we have $\tilde{x}_{\sigma}-\tilde{x}_{\sigma\tau}+\tilde{x}_{\sigma}$
is equal to 0 or 2. Since $N_{P}^{2}=N_{Q}^{2}=N_{R}^{2}=N_{S}^{2}=I$,
$[N_{P},N_{Q}]=[N_{R},N_{S}]=-I$ and the commutators of the other pairs are
trivial, we have

$\displaystyle(N_{\tau})^{\sigma}N_{\sigma\tau}^{-1}N_{\sigma}=$
$\displaystyle(N_{P}^{\tilde{b}_{\tau}}N_{Q}^{\tilde{a}_{\tau}}N_{R}^{\tilde{d}_{\tau}}N_{S}^{\tilde{c}_{\tau}})^{\sigma}\cdot
N_{S}^{-\tilde{c}_{\sigma\tau}}N_{R}^{-\tilde{d}_{\sigma\tau}}N_{Q}^{-\tilde{a}_{\sigma\tau}}N_{P}^{-\tilde{b}_{\sigma\tau}}\cdot
N_{P}^{\tilde{b}_{\sigma}}N_{Q}^{\tilde{a}_{\sigma}}N_{R}^{\tilde{d}_{\sigma}}N_{S}^{\tilde{c}_{\sigma}}$
$\displaystyle=$
$\displaystyle(-1)^{(\widetilde{c_{P}})_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{(\widetilde{c_{Q}})_{\sigma}\cdot\tilde{a}_{\tau}}\cdot(-1)^{(\widetilde{c_{R}})_{\sigma}\cdot\tilde{d}_{\tau}}\cdot(-1)^{(\widetilde{c_{S}})_{\sigma}\cdot\tilde{c}_{\tau}}$
$\displaystyle\cdot N_{P}^{\tilde{b}_{\tau}}N_{Q}^{\tilde{a}_{\tau}}\cdot
N_{R}^{\tilde{d}_{\tau}}N_{S}^{\tilde{c}_{\tau}}N_{S}^{-\tilde{c}_{\sigma\tau}}N_{R}^{-\tilde{d}_{\sigma\tau}}\cdot
N_{Q}^{-\tilde{a}_{\sigma\tau}}N_{P}^{-\tilde{b}_{\sigma\tau}}N_{P}^{\tilde{b}_{\sigma}}N_{Q}^{\tilde{a}_{\sigma}}\cdot
N_{R}^{\tilde{d}_{\sigma}}N_{S}^{\tilde{c}_{\sigma}}$ $\displaystyle=$
$\displaystyle(-1)^{(\widetilde{c_{P}})_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{(\widetilde{c_{Q}})_{\sigma}\cdot\tilde{a}_{\tau}}\cdot(-1)^{(\widetilde{c_{R}})_{\sigma}\cdot\tilde{d}_{\tau}}\cdot(-1)^{(\widetilde{c_{S}})_{\sigma}\cdot\tilde{c}_{\tau}}$
$\displaystyle\cdot
N_{P}^{\tilde{b}_{\tau}}N_{Q}^{\tilde{a}_{\tau}}N_{Q}^{-\tilde{a}_{\sigma\tau}}N_{P}^{-\tilde{b}_{\sigma\tau}}N_{P}^{\tilde{b}_{\sigma}}N_{Q}^{\tilde{a}_{\sigma}}\cdot
N_{R}^{\tilde{d}_{\tau}}N_{S}^{\tilde{c}_{\tau}}N_{S}^{-\tilde{c}_{\sigma\tau}}N_{R}^{-\tilde{d}_{\sigma\tau}}N_{R}^{\tilde{d}_{\sigma}}N_{S}^{\tilde{c}_{\sigma}}$
$\displaystyle=$
$\displaystyle(-1)^{(\widetilde{c_{P}})_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{(\widetilde{c_{Q}})_{\sigma}\cdot\tilde{a}_{\tau}}\cdot(-1)^{(\widetilde{c_{R}})_{\sigma}\cdot\tilde{d}_{\tau}}\cdot(-1)^{(\widetilde{c_{S}})_{\sigma}\cdot\tilde{c}_{\tau}}\cdot(-1)^{\tilde{a}_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{\tilde{c}_{\sigma}\cdot\tilde{d}_{\tau}}\cdot
I.$

On the other hand, $(c_{P},c_{Q})\otimes(c_{R},c_{S})$ is isomorphic to
$\langle M_{P},M_{Q},M_{R},M_{S}\rangle=\text{Mat}_{4}(K)$ which represents
the identity element in the Brauer group. Hence, we have

$(c_{P}a,c_{Q}b)+(c_{R}c,c_{S}d)=(a,b)+(c,d)+(c_{P},b)+(c_{Q},a)+(c_{R},d)+(c_{S},c),$

which is precisely represented by a cocycle that sends $(\sigma,\tau)$ to

$(-1)^{(\widetilde{c_{P}})_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{(\widetilde{c_{Q}})_{\sigma}\cdot\tilde{a}_{\tau}}\cdot(-1)^{(\widetilde{c_{R}})_{\sigma}\cdot\tilde{d}_{\tau}}\cdot(-1)^{(\widetilde{c_{S}})_{\sigma}\cdot\tilde{c}_{\tau}}\cdot(-1)^{\tilde{a}_{\sigma}\cdot\tilde{b}_{\tau}}\cdot(-1)^{\tilde{c}_{\sigma}\cdot\tilde{d}_{\tau}},$

for all $\sigma,\tau\in G_{K}$ as required.

∎

##  6 Bounding the Set of Primes

In this section, we directly show that the formula for
$\langle\epsilon,\eta\rangle_{CT}$ in Theorem 3.3 is actually always a finite
product, as mentioned in Remark 3.4. Since for a local field with odd residue
characteristic, the Hilbert symbol between $x$ and $y$ is trivial when the
valuations of $x$, $y$ are both 0, it suffices to find a finite set $S$ of
places of $K$, such that outside $S$ the first arguments of the Hilbert
symbols in the formula for $\langle\epsilon,\eta\rangle_{CT}$ have valuation 0
for some choice of the local point $P_{v}$.

Let $\mathcal{O}_{K}$ be the ring of integers for the number field $K$. By
rescaling the variables, we assume the genus two curve is defined by
$y^{2}=f(x)=f_{6}x^{6}+...+f_{0}$ where the $f_{i}$ are in $\mathcal{O}_{K}$.

The first arguments of the Hilbert symbols in the formula for
$\langle\epsilon,\eta\rangle_{CT}$ are $f_{P}(P_{v})$, $f_{Q}(P_{v})$,
$f_{R}(P_{v})$ or $f_{S}(P_{v}),$ where $f_{P},f_{Q},f_{R},f_{S}$ can be
computed as the quotients of two linear forms in $\mathbb{P}^{15}$ with the
denominators being the same, as explained in Remark 4.5. Since we know that
the Cassels-Tate pairing is independent of the choice of the local points
$P_{v}$ as long as these are chosen to avoid all the zeros and poles, it
suffices to make sure that there exists at least one local point $P_{v}$ on
$J_{\epsilon}$ for which the values of the quotients of the linear forms all
have valuation 0 for all $v$ outside $S$. The idea is to first reduce the
problem to the residue field.

By Theorem 2.9 and Remark 2.10, we have an explicit formula for the linear
isomorphism

$J_{\epsilon}\subset\mathbb{P}^{15}\xrightarrow{\phi_{\epsilon}}J\subset\mathbb{P}^{15},$

which is defined over $K^{\prime}=K(\sqrt{a},\sqrt{b},\sqrt{c},\sqrt{d})$
where $\epsilon=(a,b,c,d)\in(K^{*}/(K^{*})^{2})^{4}$. Suppose
$\phi_{\epsilon}$ is represented by
$M_{\epsilon}\in\text{GL}_{16}(K^{\prime})$. Note we can assume that all
entries of $M_{\epsilon}$ are in $\mathcal{O}_{K^{\prime}}$, the ring of
integers of $K^{\prime}$.

######  Notation 6.1.

Let $K$ be a local field with valuation ring $\mathcal{O}_{K}$, uniformizer
$\pi$ and residue field $k$. Let $X\subset\mathbb{P}^{N}$ be a variety defined
over $K$ and $I(X)\subset K[x_{0},...,x_{N}]$ be the ideal of $X$. Then the
reduction of $X$, denoted by $\bar{X}$, is the variety defined by the
polynomials $\\{\bar{f}:f\in I(X)\cap\mathcal{O}_{K}[x_{0},...,x_{N}]\\}$.
Here $\bar{f}$ is the polynomial obtained by reducing all the coefficients of
$f$ modulo $\pi$. Note that this definition of the reduction of a variety
$X\subset\mathbb{P}^{N}$ defined over a local field $K$ is equivalent to
taking the special fibre of the closure of $X$ in $\mathbb{P}^{N}_{S}$, where
$S=\operatorname{Spec}\mathcal{O}_{K}$.

Let $S_{0}=\\{\text{places of bad reduction for
}\mathcal{C}\\}\cup\\{\text{places dividing 2}\\}\cup\\{\text{infinite
places}\\}$. Fix a place $v\notin S_{0}$ and suppose it is above the prime
$p$. We now treat $J,J_{\epsilon}$ and $\mathcal{C}$ as varieties defined over
the local field $K_{v}$. Let $\mathcal{O}_{v}$ denote the valuation ring of
$K_{v}$ and $\mathbb{F}_{q}$ denote its residue field, where $q$ is some power
of $p$. It can be shown that $\bar{J}$ is also an abelian variety as the
defining equations of $J$ are defined over $\mathcal{O}_{v}$ and are derived
algebraically in terms of the coefficients of the defining equation of the
genus two curve $\mathcal{C}$ by Theorem 2.2. In fact, $\bar{J}$ is the
Jacobian variety of $\bar{\mathcal{C}}$, the reduction of $\mathcal{C}$.

Now fix a place $v^{\prime}$ of $K^{\prime}$ above the place $v$ of $K$. Let
$\mathcal{O}_{v^{\prime}}$ and $\mathbb{F}_{q^{r}}$ denote the valuation ring
and the residue field of $K^{\prime}_{v^{\prime}}$. It can be checked that as
long as $v^{\prime}$ does not divide $\det
M_{\epsilon}\in\mathcal{O}_{K^{\prime}}$, the following diagram commutes and
$\bar{M_{\epsilon}}$ is a well defined linear isomorphism defined over the
residue field $\mathbb{F}_{q^{r}}$ between two varieties defined over
$\mathbb{F}_{q}$:

${J_{\epsilon}\subset\mathbb{P}^{15}}$${J\subset\mathbb{P}^{15}}$${\bar{J_{\epsilon}}\subset\mathbb{P}^{15}}$${\bar{J}\subset\mathbb{P}^{15},}$$\scriptstyle{M_{\epsilon}}$reductionreduction$\scriptstyle{\bar{M_{\epsilon}}}$

where $\bar{M_{\epsilon}}$ denotes the reduction of the matrix $M_{\epsilon}$
over the residue field $\mathbb{F}_{q^{r}}$.

This linear isomorphism $\bar{M_{\epsilon}}$ implies that $\bar{J_{\epsilon}}$
is smooth whenever $\bar{J}$ is. In this case, $\bar{J_{\epsilon}}$ is a twist
of $\bar{J}$ and it in fact a 2-covering of $\bar{J}$. Indeed, the
surjectivity of the natural map
$\text{Gal}(K^{\prime}_{v^{\prime}}/K_{v})\rightarrow\text{Gal}(\mathbb{F}_{q^{r}}/\mathbb{F}_{q})$
shows that $M_{\epsilon}(M_{\epsilon}^{-1})^{\sigma}=\tau_{P_{\sigma}}$ for
all $\sigma\in\text{Gal}(K^{\prime}_{v^{\prime}}/K_{v})$ implies that
$\bar{M_{\epsilon}}(\bar{M_{\epsilon}}^{-1})^{\bar{\sigma}}=\tau_{\bar{P_{\bar{\sigma}}}}$
for all $\bar{\sigma}\in\text{Gal}(\mathbb{F}_{q^{r}}/\mathbb{F}_{q})$. We
know any principal homogeneous space of $\bar{J}$ over a finite field has a
point by [lang, Theorem 2] and so is trivial by Proposition 2.4. Therefore,
there exists an isomorphism $\bar{J_{\epsilon}}\xrightarrow{\psi}\bar{J}$
defined over $\mathbb{F}_{q}$. Hence, as long as $v\notin S_{0}$ and $v$ does
not divide $N_{K^{\prime}/K}(\det M_{\epsilon})$, $\bar{J_{\epsilon}}$ has the
same number of $\mathbb{F}_{q}$-points as $\bar{J}$. By the Hasse-Weil bound,
we know the number of $\mathbb{F}_{q}$-points on $\mathcal{C}$ is bounded
below by $q-1-4\sqrt{q}$. Since we can represent points on $\bar{J}$ by pairs
of points on $\bar{\mathcal{C}}$ and this representation is unique other than
the identity point on $\bar{J}$. The number of $\mathbb{F}_{q}$-points on
$\bar{J}$ is bounded below by $(q-1-4\sqrt{q})(q-3-4\sqrt{q})/2$.

On the other hand, let $l_{1},...,l_{5}$ be the 5 linear forms that appear as
numerator or denominator of $f_{P},f_{Q},f_{R},f_{S}$. We can assume that the
coefficients of $l_{i}$ are in $\mathcal{O}_{K}$ by scaling, for all
$i=1,...,5$. Fix a place $v$ of $K$ that does not divide all the coefficients
of $l_{i}$, for any $i=1,...,5$. Let $H_{i}$ be the hyperplane defined by the
linear form $l_{i}$ and $\bar{H_{i}}$ be its reduction, which is a hyperplane
defined over the residue field $\mathbb{F}_{q}$, We need to bound the number
of $\mathbb{F}_{q}$-points of $\bar{J_{\epsilon}}$ that lie on one of the
hyperplanes $\bar{H_{i}}$. Let $r_{i}$ be the number of irreducible components
of $\bar{J_{\epsilon}}\cap\bar{H_{i}}$. By [hartshorne, Chapter 1, Theorem 7.2
(Projective Dimension Theorem) and Theorem 7.7], we know that each irreducible
component $C^{i}_{j}$ of $\bar{J_{\epsilon}}\cap\bar{H_{i}}$, where
$j=1,...,r_{i}$, is a curve and the sum of degrees of all the irreducible
components counting intersection multiplicity is $\deg\bar{J_{\epsilon}}=32.$
Leting $d^{i}_{j}=\deg C^{i}_{j}$, we have $\sum_{j=1}^{r_{i}}d^{i}_{j}\leq
32$ for all $i$.

######  Lemma 6.2.

Let $C\subset\mathbb{P}^{N}$ be a curve of degree $d$. Then
$\\#C(\mathbb{F}_{q})\leq d(q+1)$.

###### Proof.

We may assume that $C$ is contained in no hyperplane. Then projection to the
first two coordinates gives a nonconstant morphism
$C\rightarrow\mathbb{P}^{1}$ of degree $\leq d$. Since
$\\#\mathbb{P}^{1}(\mathbb{F_{q}})=q+1$, this gives the required bound.
∎

By applying the above lemma to each $C_{j}^{i}$, we get the number of
$\mathbb{F}_{q}$-points of $\bar{J_{\epsilon}}$ that lie on one of the
hyperplanes $\bar{H_{i}},i=1,...,5,$ is no more than

$\sum_{i=1}^{5}\sum_{j=1}^{r_{i}}d^{i}_{j}\cdot(q+1)\leq 160(q+1).$

We compute that for any $x>500$, we have
$(x-1-4\sqrt{x})(x-3-4\sqrt{x})/2>160(x+1)$. Recall $q$ is a power of $p$.
Hence, if $v$ is a place of $K$ above the prime $p>500$ such that $v\notin
S_{0}$ and $v$ does not divide $N_{K^{\prime}/K}(\det M_{\epsilon})$ or all
the coefficients of $l_{i}$ for some $i$, we have a smooth
$\mathbb{F}_{q}$-point on $\bar{J_{\epsilon}}$ which by Hensel’s Lemma
[hensel, Exercise C.9(c)] lifts to the point $P_{v}$ as required. This implies
that the first arguments of the Hilbert symbols in the formula for the local
Cassels-Tate pairing of $\langle\epsilon,\eta\rangle_{CT}$ have valuation 0.
It can be checked that since $v\notin S_{0}$, the second arguments of these
Hilbert symbols also have valuation 0. Hence, the formula for the Cassels-Tate
is indeed always a finite product.

Note that in the case where $K=\mathbb{Q}$ or more generally if $K$ has class
number 1, we can always make the linear forms primitive by scaling. Therefore,
in this case, the subset $\\{$places dividing all the coefficients of the
denominator or the numerator of $f_{P},f_{Q},f_{R}\text{ or }f_{S}\\}$ is
empty.

##  7 Worked Example

Now we demonstrate the algorithm with a worked example computed using MAGMA
[magma]. In particular, we will see with this example, that computing the
Cassels-Tate pairing on $\text{Sel}^{2}(J)$ does improve the rank bound
obtained via a 2-descent. This genus two curve was kindly provided by my PhD
supervisor, Tom Fisher, along with a list of other genus two curves for me to
test the algorithm.

Consider the following genus two curve

$\mathcal{C}:y^{2}=-10x(x+10)(x+5)(x-10)(x-5)(x-1).$

Its Jacobian variety $J$ has all its two torsion points defined over
$\mathbb{Q}$. A set of generators of $J[2]$ compatible with the Weil pairing
matrix (3.1) are
$P=\\{(0,0),(-10,0)\\},Q=\\{(0,0),(-5,0)\\},R=\\{(10,0),(5,0)\\},S=\\{(10,0),(1,0)\\}.$
We identify $H^{1}(G_{K},J[2])=(\mathbb{Q}^{*}/(\mathbb{Q}^{*})^{2})^{4}$ as
in Section 3. Consider $\epsilon,\eta\in\text{Sel}^{2}(J)$ represented by
$(-33,1,-1,-11)$ and $(11,1,-1,-11)$ respectively. The images of
$[P],[Q],[R],[S]$ via $\delta:J(\mathbb{Q})/2J(\mathbb{Q})\rightarrow
H^{1}(G_{\mathbb{Q}},J[2])$, computed via the explicit formula as in [thebook,
Chapter 6, Section 1], are
$\delta([P])=(-66,1,6,22),\delta([Q])=(-1,1,3,1),\delta([R])=(6,3,1,3),\delta([S])=(22,1,-3,-11).$
Now following the discussions in Sections 4.2 and 4.3, we can compute, using
the coordinates $c_{0},...,c_{9},d_{1},...,d_{6}$ for
$J_{\epsilon}\in\mathbb{P}^{15}$ as described in Remark 2.10. we have

$\displaystyle k_{11}^{\prime}$
$\displaystyle=618874080c_{0}-496218440c_{1}-390547052c_{3}+205551080c_{4}$
$\displaystyle+384569291c_{6}+52868640c_{8};$ $\displaystyle
k_{11,P}^{\prime}$
$\displaystyle=-36051078800000c_{2}+8111492730000c_{3}+265237150000c_{7}$
$\displaystyle-196928587500c_{8}-6786529337500c_{9}+22531924250d_{2}$
$\displaystyle-126449158891d_{4}-117221870375d_{5}+937774963000d_{6};$
$\displaystyle k_{11,Q}^{\prime}$
$\displaystyle=134800c_{1}+235600c_{3}+62000c_{4}+52235c_{6}+60016d_{1}-5456d_{5};$
$\displaystyle k_{11,R}^{\prime}$
$\displaystyle=-30223125c_{6}+4050000c_{8}-49750d_{3}+709236d_{4}$
$\displaystyle k_{11,S}^{\prime}$
$\displaystyle=4724524800c_{1}+8557722360c_{3}+13102732800c_{4}+1258642935c_{6}$
$\displaystyle+7291944000c_{9}-2709362304d_{1}+97246845d_{2}+8475710d_{3}$
$\displaystyle+30788208d_{5}.$

Hence, we have explicit formulae for

$f_{P}=\frac{k_{11,P}^{\prime}}{k_{11}^{\prime}},f_{Q}=\frac{k_{11,Q}^{\prime}}{k_{11}^{\prime}},f_{R}=\frac{k_{11,R}^{\prime}}{k_{11}^{\prime}},f_{S}=\frac{k_{11,S}^{\prime}}{k_{11}^{\prime}}.$

In particular, they are defined over $\mathbb{Q}$ as claimed. From Section 6,
we compute that only primes below 500 can potentially contribute to
$\langle\epsilon,\eta\rangle_{CT}$. Then, it turns out that the only
nontrivial local Cassels-Tate pairings between $\epsilon$ and $\eta$ are at
places $11,19,\infty$ and $\langle\epsilon,\eta\rangle_{CT}=-1$.

Under the isomorphism
$H^{1}(G_{\mathbb{Q}},J[2])\rightarrow(\mathbb{Q}^{*}/(\mathbb{Q}^{*})^{2})^{4}$,
$\text{Sel}^{2}(J)$ has size $2^{6}$ and is generated by
$(-33,1,-1,-11),(11,1,-1,-11),(66,1,2,22),(11,1,2,22),(3,3,3,3),(3,1,3,1).$
Since $\mathcal{C}$ has rational points, the Cassels-Tate pairing can be shown
to be alternating using [poonenstoll, Corollary 7]. Since all the two torsion
points on $J$ are rational and $\langle\epsilon,\eta\rangle_{CT}=-1$, we get
$|\ker\langle\;,\;\rangle_{CT}|=2^{4}$.

Indeed, we verified that the Cassels-Tate pairing matrix, with the generators
of $\text{Sel}^{2}(J)$ listed above, is

$\begin{bmatrix}1&-1&1&1&-1&-1\\\ -1&1&1&-1&1&-1\\\ 1&1&1&1&1&1\\\
1&-1&1&1&-1&-1\\\ -1&1&1&-1&1&-1\\\ -1&-1&1&-1&-1&1\\\ \end{bmatrix},$

which is a rank 2 matrix.

As shown in [thesis, Remark 1.9.4(ii)], in the case where all points in $J[2]$
are defined over the base field, computing the Cassels-Tate pairing on
$\text{Sel}^{2}(J)$ gives the same rank bound as obtained from carrying out a
$4$-descent, i.e. computing $\text{Sel}^{4}(J)$, which can potentially give a
better rank bound than the one given by a 2-descent. In this example, the rank
bound coming from 2-decent was $\operatorname{rank}(J(\mathbb{Q}))\leq 2$. Our
calculations of the Cassels-Tate pairing on $\text{Sel}^{2}(J)$ improves this
bound and in fact shows that $\operatorname{rank}(J(\mathbb{Q}))=0$.

## References