# Computing the Cassels-Tate pairing
on the $2$-Selmer group of a genus 2 Jacobian

Tom Fisher  University of Cambridge, DPMMS, Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WB, UK<EMAIL_ADDRESS>and  Jiali
Yan<EMAIL_ADDRESS>
(Date: 9th June 2023)

###### Abstract.

We describe a method for computing the Cassels-Tate pairing on the $2$-Selmer
group of the Jacobian of a genus $2$ curve. This can be used to improve the
upper bound coming from $2$-descent for the rank of the group of rational
points on the Jacobian. Our method remains practical regardless of the Galois
action on the Weierstrass points of the genus $2$ curve. It does however
depend on being able to find a rational point on a certain twisted Kummer
surface. The latter does not appear to be a severe restriction in practice. In
particular, we have used our method to unconditionally determine the ranks of
all genus $2$ Jacobians in the $L$-functions and modular forms database
(LMFDB).

###### Contents

  1. 1 Introduction
  2. 2 Explicit 2-descent on genus 2 Jacobians
    1. 2.1 The Kummer surface and its dual
    2. 2.2 The desingularised Kummer
    3. 2.3 The Selmer and fake Selmer groups
    4. 2.4 Models for 2-coverings
    5. 2.5 Recovering the Selmer group element
    6. 2.6 Twisted Kummer surfaces
  3. 3 The Cassels-Tate pairing on the $2$-Selmer group
    1. 3.1 Definition of the Cassels-Tate pairing
    2. 3.2 A formula for the Cassels-Tate pairing
    3. 3.3 Remnants of the group law
    4. 3.4 The Heisenberg group
    5. 3.5 A formula for the untwisted $(2,2,2)$-form
    6. 3.6 A formula for the twisted $(2,2,2)$-form
  4. 4 Implementation and Examples
    1. 4.1 Controlling the bad primes
    2. 4.2 First examples over ${\mathbb{Q}}$
    3. 4.3 Examples from an experiment of Bruin and Stoll
    4. 4.4 Examples from the LMFDB

##  1\. Introduction

Let ${\mathcal{C}}$ be a smooth projective curve defined over a number field
$k$. Its Jacobian ${\mathcal{J}}=\operatorname{Jac}({\mathcal{C}})$ is a
principally polarised abelian variety, also defined over $k$. We are
interested in the group ${\mathcal{J}}(k)$ of $k$-rational points on
${\mathcal{J}}$. By the Mordell-Weil theorem this is a finitely generated
abelian group. We may therefore write

${\mathcal{J}}(k)\cong\Delta\times{\mathbb{Z}}^{r}$

where $\Delta$ is a finite abelian group, called the torsion subgroup, and $r$
is a non-negative integer, called the rank.

We may compute an upper bound for the rank by carrying out a $2$-descent.
Specifically, there is a short exact sequence of ${\mathbb{F}}_{2}$-vector
spaces

(1) $0\to{\mathcal{J}}(k)/2{\mathcal{J}}(k)\to
S^{(2)}({\mathcal{J}}/k)\to\mbox{\wncyr Sh}({\mathcal{J}}/k)[2]\to 0$

where the $2$-Selmer group $S^{(2)}({\mathcal{J}}/k)$ is finite and
effectively computable. Computing $S^{(2)}({\mathcal{J}}/k)$ gives an upper
bound for the rank, but this will only be sharp if the Tate-Shafarevich group
$\mbox{\wncyr Sh}({\mathcal{J}}/k)$ contains no elements of order $2$.

In principle, the upper bound for the rank can be improved by carrying out a
$4$-descent, that is, by computing the $4$-Selmer group
$S^{(4)}({\mathcal{J}}/k)$. This sits in a commutative diagram with exact rows

(2)
$\textstyle{{\mathcal{J}}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times
4}$$\scriptstyle{\times
2}$$\textstyle{{\mathcal{J}}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{(4)}({\mathcal{J}}/k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mbox{\wncyr
Sh}({\mathcal{J}}/k)[4]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times
2}$$\textstyle{0}$$\textstyle{{\mathcal{J}}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times
2}$$\textstyle{{\mathcal{J}}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{(2)}({\mathcal{J}}/k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mbox{\wncyr
Sh}({\mathcal{J}}/k)[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

from which we read off the inclusions
${\mathcal{J}}(k)/2{\mathcal{J}}(k)\subset\operatorname{Im}(\alpha)\subset
S^{(2)}({\mathcal{J}}/k)$. However, by results of Cassels [C2] in the case of
elliptic curves, generalised to abelian varieties by Tate [Ta], [Mi], there is
a symmetric pairing of ${\mathbb{F}}_{2}$-vector spaces

$\langle~{},~{}\rangle_{{\operatorname{CT}}}:S^{(2)}({\mathcal{J}}/k)\times
S^{(2)}({\mathcal{J}}/k)\to{\mathbb{F}}_{2}$

whose kernel is $\operatorname{Im}(\alpha)$. Thus a $2$-descent followed by
computing the Cassels-Tate pairing
$\langle~{},~{}\rangle_{{\operatorname{CT}}}$ gives the same information (as
regards bounding the rank) as a $4$-descent, but with potentially very much
less effort.

Cassels [C5] gave a practical method for computing the pairing in the case of
elliptic curves (that is, when ${\mathcal{C}}$ has genus $1$). His method
involves solving a conic over the field of definition of each $2$-torsion
point of the elliptic curve. This was subsequently improved by Donnelly [Do],
who found a method that only requires that we solve conics over the ground
field.

The second author [Y2] generalised the method of Cassels to curves of genus 2.
However the analogue of solving a conic is now the problem of finding an
explicit isomorphism between an algebra given by structure constants and the
algebra of $4\times 4$ matrices. So even taking $k={\mathbb{Q}}$ the method is
only practical if the genus $2$ curve has all its Weierstrass points defined
over ${\mathbb{Q}}$, or at least over a very small number field. In this paper
we develop a new method that avoids these restrictions. To do this we
generalise the method of Donnelly [Do], or more specifically the variant in
[Fi2], to curves of genus $2$.

Although we are able to compute the pairing for curves of genus $2$ regardless
of the Galois action on the Weierstrass points, our method depends on being
able to find rational points on certain twisted Kummer surfaces. In general
such points need not exist, as shown by Logan and van Luijk [LvL]. However
this does not appear to be a severe restriction in practice. Indeed we have
implemented our method in Magma [BCP], and in running our program on thousands
of random examples, and on every genus 2 Jacobian in the L-functions and
modular forms database [LMFDB] with non-trivial analytic order of Sh, we are
yet to find an example where a lack of rational points on the twisted Kummer
surfaces prevents us from computing the pairing.

There are also good theoretical reasons why the twisted Kummer surfaces should
typically have rational points. The surfaces of interest are always everywhere
locally soluble, and so have rational points if they satisfy the Hasse
principle. However Kummer surfaces are examples of $K3$ surfaces, and for
these it is conjectured that all counterexamples to the Hasse principle are
explained by the Brauer-Manin obstruction. Moreover there is work of Harpaz
and Skorobogatov [HS], and more recently Morgan [Mo], showing that if we
assume the finiteness of all relevant Tate-Shafarevich groups, then twisted
Kummer surfaces, satisfying certain additional mild assumptions, do indeed
satisfy the Hasse principle.

Another approach to computing the Cassels-Tate pairing on the $2$-Selmer group
of a genus $2$ Jacobian (in fact, more generally, of a hyperelliptic Jacobian)
is currently being developed by Shukla and Stoll. This is based on the
Albanese-Albanese definition of the Cassels-Tate pairing, due to Poonen and
Stoll [PSt]. The practicality of their method is not yet clear, but unlike our
method, it only applies to curves with a rational Weierstrass point.

In the case of elliptic curves, algorithms for $4$-descent have been
developed, but computing the Cassels-Tate pairing on the $2$-Selmer group
requires significantly less effort. For genus $2$ Jacobians the contrast is
even more stark as there is, to our knowledge, no practical way of carrying
out a $4$-descent. Computing the Cassels-Tate pairing on the $2$-Selmer group
therefore gives us information that could not otherwise be obtained. In
particular we have used our methods to compute the ranks of some genus $2$
Jacobians, taken from the LMFDB and from an experiment of Bruin and Stoll
[BS], in cases where the rank had only previously been computed conditional on
the Birch Swinnerton-Dyer conjecture. In fact prior to our work there were
$69$ genus $2$ curves in the LMFDB that were unresolved in this sense, and we
have resolved all of them.

This paper has its origins in Chapters 5 and 6 of the second author’s PhD
thesis [Y1]. However in the intervening two years we have succeeded in making
the method significantly more practical. Some of the main improvements concern
explicit $2$-descent on genus $2$ Jacobians, by which we mean the problem of
converting elements of the $2$-Selmer group, represented algebraically (as
units modulo squares in suitable étale algebras), to explicit equations for
the corresponding $2$-coverings. This problem was previously addressed by
Flynn, Testa and van Luijk [FTvL], extending earlier work of Gordon and Grant
[GG]. However we find it convenient to take a different approach, based on the
observation that, at least over the complex numbers, the Jacobian of a genus 2
curve may be realised as the variety of lines on the intersection of two
quadrics in ${\mathbb{P}}^{5}$. This “neoclassical approach” is discussed in
the book of Cassels and Flynn [CF, Chapter 17], and has its origins in the
geometric literature in works of Newstead [N], Narasimhan and Ramanan [NR],
Reid [R] and Donagi [D]. More recently it has found important applications in
arithmetic statistics; see for example [BG], [SW].

The upshot for us is that by representing the $2$-Selmer group elements as
pairs of quadratic forms in $6$ variables, we obtain simple elegant formulae
for the $2$-coverings in ${\mathbb{P}}^{15}$, the twisted Kummer surfaces in
${\mathbb{P}}^{3}$, the twisted desingularised Kummer surfaces in
${\mathbb{P}}^{5}$, the maps between these, and (various maps induced by) the
covering map to the Jacobian. We have also found invariant-theoretic formulae,
analogous to those in [Fi2], that allow us to directly compute the
$(2,2,2)$-form that appears in our formula for the Cassels-Tate pairing.

We concentrate in this paper in giving practical methods for curves of genus
$2$. The question as to how much of our work generalises to hyperelliptic
curves of higher genus is left to future work. One reason why such a
generalisation might not be automatic is that we make implicit use of the
exceptional isomorphism of Lie algebras
$\mathfrak{so}_{6}\cong\mathfrak{sl}_{4}$. The practicality of searching for
rational points on higher dimensional Kummer varieties could also be a
problem.

Throughout this paper ${\mathcal{C}}$ will be a genus $2$ curve, defined over
a field $k$ of characteristic not $2$, with equation

(3) $y^{2}=f(x)=f_{6}x^{6}+f_{5}x^{5}+\ldots+f_{1}x+f_{0}.$

We assume that $f_{6}\not=0$, noting that if $\deg f=5$ and $|k|>5$ then we
can reduce to this case by applying a suitable Möbius map.

We have divided the paper into three main sections, each with its own
introduction. Section 2 gives the background on $2$-descent, including the
refinements mentioned above. In Section 3 we develop our method for computing
the Cassels-Tate pairing, and in Section 4 we give examples and applications.

We plan to make some Magma code accompanying this article (checking some of
the formulae and examples) available from the first author’s website.

Acknowledgements. We thank Victor Flynn, Jef Laga, August Liu, Adam Morgan,
Ross Paterson, Himanshu Shukla, Michael Stoll, Drew Sutherland and Jack Thorne
for useful conversations.

##  2\. Explicit 2-descent on genus 2 Jacobians

We give an account of $2$-descent on the Jacobian of a genus $2$ curve. The
emphasis is on explicit formulae, some of which are taken from the existing
literature and some of which are new.

In Sections 2.1 and 2.2 we give formulae relating to the Kummer surface, the
dual Kummer surface, the desingularised Kummer, the embedding of the Jacobian
in ${\mathbb{P}}^{15}$, and the action of the $2$-torsion points. We are
particularly interested in constructions that carry over to the twisted Kummer
surfaces considered in Section 2.6. In Section 2.3 we briefly recall the
standard algebraic description of the $2$-Selmer group, and define its
canonical element. Section 2.4 introduces the “neoclassical approach” where we
represent Selmer group elements as pairs of quadratics forms. We call these
pairs of quadratic forms models. In Section 2.5 we describe some methods for
recovering a Selmer group element from a model. In Section 2.6 we explain how
a model determines an equation for the corresponding twisted Kummer surface,
and a wealth of other information.

###  2.1. The Kummer surface and its dual

Let ${\mathcal{C}}$ be a genus $2$ curve with equation $y^{2}=f(x)$ where $f$
is a polynomial of degree $6$, with coefficients labelled as in (3). Let
${\mathcal{J}}$ be the Jacobian of ${\mathcal{C}}$, and let
${\mathcal{K}}={\mathcal{J}}/[-1]$ be its Kummer surface. We may represent
points on ${\mathcal{J}}$ as

$[(x,y)+(x^{\prime},y^{\prime})-\kappa]$

where $(x,y),(x^{\prime},y^{\prime})\in{\mathcal{C}}$ and $\kappa$ is the
canonical divisor on ${\mathcal{C}}$. A point on ${\mathcal{K}}$ may therefore
by represented by $x,x^{\prime}$ and a choice of square root of
$f(x)f(x^{\prime})$. It follows that ${\mathcal{K}}$ is the double cover of
the projective plane ${\mathbb{P}}^{2}_{a,b,c}$ with equation

(4) $z^{2}=\operatorname{Res}(ax^{2}-bx+c,f(x)).$

If we specialise $a,b,c$ to $1,2t,t^{2}$ then this resultant equals
$f(t)^{2}$. This suggests putting

$\displaystyle P_{2}(a,b,c)$ $\displaystyle=b^{2}-4ac,$ $\displaystyle
P_{3}(a,b,c)$
$\displaystyle=2(2f_{0}a^{3}+f_{1}a^{2}b+2f_{2}a^{2}c+f_{3}abc+2f_{4}ac^{2}+f_{5}bc^{2}+2f_{6}c^{3}),$

so that $P_{2}(1,2t,t^{2})=0$ and $P_{3}(1,2t,t^{2})=4f(t)$. We can then solve
for the homogeneous polynomial $P_{4}(a,b,c)$ of degree $4$ satisfying

$P_{3}^{2}-4P_{2}P_{4}=16\operatorname{Res}(ax^{2}-bx+c,f(x)).$

The equation (4) for the Kummer surface becomes

${\mathcal{K}}=\\{P_{2}d^{2}-P_{3}d+P_{4}=0\\}\subset{\mathbb{P}}^{3}_{a,b,c,d}.$

Renaming the coordinates $a,b,c,d$ as $x_{1},\ldots,x_{4}$ we have the
following equation for ${\mathcal{K}}\subset{\mathbb{P}}^{3}$, in agreement
with [CF, page 19].

(5) $\displaystyle F$
$\displaystyle=(x_{2}^{2}-4x_{1}x_{3})x_{4}^{2}-2(2f_{0}x_{1}^{3}+f_{1}x_{1}^{2}x_{2}+2f_{2}x_{1}^{2}x_{3}+f_{3}x_{1}x_{2}x_{3}+2f_{4}x_{1}x_{3}^{2}$
$\displaystyle+f_{5}x_{2}x_{3}^{2}+2f_{6}x_{3}^{3})x_{4}+(f_{1}^{2}-4f_{0}f_{2})x_{1}^{4}-4f_{0}f_{3}x_{1}^{3}x_{2}-2f_{1}f_{3}x_{1}^{3}x_{3}$
$\displaystyle-4f_{0}f_{4}x_{1}^{2}x_{2}^{2}+4(f_{0}f_{5}-f_{1}f_{4})x_{1}^{2}x_{2}x_{3}+(f_{3}^{2}-4f_{0}f_{6}+2f_{1}f_{5}-4f_{2}f_{4})x_{1}^{2}x_{3}^{2}$
$\displaystyle-4f_{0}f_{5}x_{1}x_{2}^{3}+4(2f_{0}f_{6}-f_{1}f_{5})x_{1}x_{2}^{2}x_{3}+4(f_{1}f_{6}-f_{2}f_{5})x_{1}x_{2}x_{3}^{2}-2f_{3}f_{5}x_{1}x_{3}^{3}$
$\displaystyle-4f_{0}f_{6}x_{2}^{4}-4f_{1}f_{6}x_{2}^{3}x_{3}-4f_{2}f_{6}x_{2}^{2}x_{3}^{2}-4f_{3}f_{6}x_{2}x_{3}^{3}+(f_{5}^{2}-4f_{4}f_{6})x_{3}^{4}.$

This quartic surface is also the image of ${\mathcal{J}}$ by the linear system
$|2\Theta|$, where $\Theta$ is the theta divisor.111We abuse notation
(suppressing the choice of Weierstrass point) as in [FTvL, page 397]. It has
exactly 16 nodes, these being the images of the $2$-torsion points on
${\mathcal{J}}$.

The dual Kummer ${\mathcal{K}}^{\vee}\subset({\mathbb{P}}^{3})^{\vee}$ is
obtained by mapping each smooth point on
${\mathcal{K}}\subset{\mathbb{P}}^{3}$ to its tangent plane. We write
$x^{*}_{1},\ldots,x^{*}_{4}$ for the coordinates on
$({\mathbb{P}}^{3})^{\vee}$ dual to the coordinates $x_{1},\ldots,x_{4}$ on
${\mathbb{P}}^{3}$. According to [CF, page 33], the dual Kummer has equation

$\left|\begin{array}[]{cccc}2f_{0}x^{*}_{4}&f_{1}x^{*}_{4}&x^{*}_{1}&x^{*}_{2}\\\
f_{1}x^{*}_{4}&2f_{2}x^{*}_{4}-2x^{*}_{1}&f_{3}x^{*}_{4}-x^{*}_{2}&x^{*}_{3}\\\
x^{*}_{1}&f_{3}x^{*}_{4}-x^{*}_{2}&2f_{4}x^{*}_{4}-2x^{*}_{3}&f_{5}x^{*}_{4}\\\
x^{*}_{2}&x^{*}_{3}&f_{5}x^{*}_{4}&2f_{6}x^{*}_{4}\end{array}\right|=0.$

We identify $\operatorname{Pic}^{d}{\mathcal{C}}$ and
$\operatorname{Pic}^{d+2}{\mathcal{C}}$ via addition of the canonical divisor,
so that $\operatorname{Pic}^{d}{\mathcal{C}}$ only depends on the parity of
$d$. It is a double cover of ${\mathcal{K}}$ if $d$ is even and a double cover
of ${\mathcal{K}}^{\vee}$ if $d$ is odd. Adding a point $P\in{\mathcal{C}}$
defines a map
$\operatorname{Pic}^{d}{\mathcal{C}}\to\operatorname{Pic}^{d+1}{\mathcal{C}}$.
If $P=(\theta,0)$ is a Weierstrass point then these maps descend to a pair of
inverse maps ${\mathcal{K}}\to{\mathcal{K}}^{\vee}$ and
${\mathcal{K}}^{\vee}\to{\mathcal{K}}$. According to [CF, pages 38 and 184],
the map ${\mathcal{K}}\to{\mathcal{K}}^{\vee}$ is given by

(6) $\begin{pmatrix}x^{*}_{1}\\\ x^{*}_{2}\\\ x^{*}_{3}\\\
x^{*}_{4}\end{pmatrix}=\begin{pmatrix}0&h_{5}(\theta)&h_{4}(\theta)&\theta^{2}\\\
-h_{5}(\theta)&0&h_{3}(\theta)&-\theta\\\ -h_{4}(\theta)&-h_{3}(\theta)&0&1\\\
-\theta^{2}&\theta&-1&0\end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\ x_{3}\\\
x_{4}\end{pmatrix}$

where

(7) $\displaystyle h_{3}(\theta)$
$\displaystyle=2f_{6}\theta^{3}+f_{5}\theta^{2},$ $\displaystyle
h_{4}(\theta)$
$\displaystyle=2f_{6}\theta^{4}+2f_{5}\theta^{3}+2f_{4}\theta^{2}+f_{3}\theta,$
$\displaystyle h_{5}(\theta)$
$\displaystyle=2f_{6}\theta^{5}+2f_{5}\theta^{4}+2f_{4}\theta^{3}+2f_{3}\theta^{2}+2f_{2}\theta+f_{1}.$

For $A=(A_{ij})$ a $4\times 4$ skew symmetric matrix we put222In basis free
language, we are identifying the alternating square of a $4$-dimensional
vector space with its dual.

(8) $A^{*}=\begin{pmatrix}0&A_{43}&A_{24}&A_{32}\\\ A_{34}&0&A_{41}&A_{13}\\\
A_{42}&A_{14}&0&A_{21}\\\ A_{23}&A_{31}&A_{12}&0\\\ \end{pmatrix}.$

This is again a skew symmetric matrix. The Pfaffian $\operatorname{Pf}(A)$
then satisfies

(9) $AA^{*}=A^{*}A=\operatorname{Pf}(A)I_{4}.$

We note that $\operatorname{Pf}(A)^{2}=\det(A)$,
$\operatorname{Pf}(A^{*})=\operatorname{Pf}(A)$ and $A^{**}=A$. Moreover for
any $4\times 4$ matrix $P$ we have

(10) $\operatorname{Pf}(P^{T}AP)=\det(P)\operatorname{Pf}(A).$

Writing $A$ for the matrix in (6), we have

(11) $\operatorname{Pf}(A)=h_{5}(\theta)+\theta
h_{4}(\theta)+\theta^{2}h_{3}(\theta)=f^{\prime}(\theta).$

Let $f$ have roots $\theta_{1},\ldots,\theta_{6}$, and write $A_{i}$ for the
matrix obtained by taking $\theta=\theta_{i}$ in (6). As explained in [CF,
page 38], for $i\not=j$ the action of
$T_{ij}=[(\theta_{i},0)+(\theta_{j},0)-\kappa]\in{\mathcal{J}}[2]$ by
translation on ${\mathcal{K}}\subset{\mathbb{P}}^{3}$ is given by

(12) $A_{i}^{*}A_{j}=-A_{j}^{*}A_{i}.$

Since $T_{12}+T_{34}+T_{56}=0$ it follows that
$A_{1}^{*}A_{2}A_{3}^{*}A_{4}A_{5}^{*}A_{6}=\lambda I_{4}$ for some scalar
$\lambda$. Squaring both sides and using (9), (11) and (12) shows that
$\lambda^{2}=f_{6}^{6}\prod_{i<j}(\theta_{i}-\theta_{j})^{2}$. Making the
correct choice of square root it turns out that

(13)
$A_{1}^{*}A_{2}A_{3}^{*}A_{4}A_{5}^{*}A_{6}=f_{6}^{3}\prod_{i<j}(\theta_{i}-\theta_{j})I_{4}.$

An alternative, less symmetric, way to write this identity is as

(14)
$\frac{A_{1}A_{2}^{*}A_{3}}{(\theta_{1}-\theta_{2})(\theta_{2}-\theta_{3})(\theta_{3}-\theta_{1})}=\frac{A_{4}A_{5}^{*}A_{6}}{(\theta_{4}-\theta_{5})(\theta_{5}-\theta_{6})(\theta_{6}-\theta_{4})}.$

######  Remark 2.1.

The skew symmetric matrices $A_{i}$ define $6$ linear isomorphisms
${\mathcal{K}}\to{\mathcal{K}}^{\vee}$. A further $10$ such isomorphisms are
defined by the symmetric matrices of the form (14). These matrices are indexed
by the $(1,5)$-partitions and $(3,3)$-partitions of the Weierstrass points.

###  2.2. The desingularised Kummer

Blowing up the $16$ nodes of ${\mathcal{K}}$ (or equally of
${\mathcal{K}}^{\vee}$) gives a surface $\widetilde{{\mathcal{K}}}$ called the
desingularised Kummer. Equations for $\widetilde{{\mathcal{K}}}$ may be
obtained as follows; see for example [FTvL, Section 4]. Let
$L=k[x]/(f)=k[\theta]$ and define quadratic forms $Q_{0},Q_{1},\ldots,Q_{5}\in
k[u_{0},u_{1},\ldots,u_{5}]$ by

(15)
$(u_{0}+u_{1}\theta+\ldots+u_{5}\theta^{5})^{2}=Q_{0}+Q_{1}\theta+\ldots+Q_{5}\theta^{5}.$

Then
$\widetilde{{\mathcal{K}}}=\\{Q_{3}=Q_{4}=Q_{5}=0\\}\subset{\mathbb{P}}^{5}$.
The quadratic form $Q_{5}$ only involves the monomials $u_{i}u_{j}$ with
$i+j\geqslant 5$, and so has a 3-dimensional isotropic subspace given by
$u_{3}=u_{4}=u_{5}=0$. This suggest making a change of coordinate on
${\mathbb{P}}^{5}$ transforming $Q_{5}$ to a scalar multiple of
$u_{0}u_{5}+u_{1}u_{4}+u_{2}u_{3}$. Such a change of coordinates is given by
replacing the basis $1,\theta,\ldots,\theta^{5}$ for $k[\theta]$ on the left
hand side of (15) by the basis

$1,\theta,\theta^{2},h_{3}(\theta),h_{4}(\theta),h_{5}(\theta).$

where the $h_{j}(\theta)$ were defined in (7). Making the same change of basis
on the right hand side of (15), we find that $Q_{5}=2G$, $Q_{4}=2H$ and
$Q_{3}=2S$ where

$\displaystyle G$ $\displaystyle=u_{0}u_{5}+u_{1}u_{4}+u_{2}u_{3},$
$\displaystyle H$
$\displaystyle=u_{0}u_{4}+u_{1}u_{3}-f_{0}u_{5}^{2}-f_{1}u_{4}u_{5}-f_{2}u_{4}^{2}-f_{3}u_{3}u_{4}-f_{4}u_{3}^{2}+\tfrac{1}{4}f_{6}^{-1}(u_{2}-f_{5}u_{3})^{2},$
$\displaystyle S$
$\displaystyle=u_{0}u_{3}-2f_{0}u_{4}u_{5}-f_{1}(u_{3}u_{5}+u_{4}^{2})-2f_{2}u_{3}u_{4}-f_{3}u_{3}^{2}$
$\displaystyle\qquad\qquad+\tfrac{1}{2}f_{6}^{-1}(u_{1}-f_{3}u_{4}-2f_{4}u_{3})(u_{2}-f_{5}u_{3})-\tfrac{1}{4}f_{6}^{-2}f_{5}(u_{2}-f_{5}u_{3})^{2}.$

The desingularised Kummer is now given by
$\widetilde{{\mathcal{K}}}=\\{G=H=S=0\\}\subset{\mathbb{P}}^{5}$.

Writing ${\mathbf{G}}$, ${\mathbf{H}}$, ${\mathbf{S}}$ for the symmetric
matrices with $G(u)=\frac{1}{2}{\mathbf{u}}^{T}{\mathbf{G}}{\mathbf{u}}$ etc.,
where ${\mathbf{u}}=(u_{0},\ldots,u_{5})^{T}$, we find that
${\mathbf{S}}={\mathbf{H}}{\mathbf{G}}^{-1}{\mathbf{H}}$ and

$\det(x{\mathbf{G}}-{\mathbf{H}})=-f_{6}^{-1}f(x).$

In particular the $6\times 6$ matrix $\theta{\mathbf{G}}-{\mathbf{H}}$ is
singular. In fact this matrix has rank 5 and its kernel is spanned by the
vector with entries

(16) $h_{5}(\theta),h_{4}(\theta),h_{3}(\theta),\theta^{2},\theta,1.$

We may associate to each point $(x_{1}:x_{2}:x_{3}:x_{4})\in{\mathbb{P}}^{3}$
a $3$-dimensional isotropic subspace for $G$ spanned by the rows of the matrix

(17) $\Lambda=\begin{pmatrix}0&0&x_{4}&0&x_{3}&x_{2}\\\
0&-x_{4}&0&x_{3}&0&-x_{1}\\\ x_{4}&0&0&-x_{2}&-x_{1}&0\\\
-x_{3}&x_{2}&-x_{1}&0&0&0\end{pmatrix}.$

The “neoclassical approach” (see the introduction for references) suggests we
look for lines contained in $\\{G=H=0\\}\subset{\mathbb{P}}^{5}$. We obtain
two such lines if the restriction of $H$ to a $3$-dimensional isotropic
subspace for $G$ has rank $2$. This suggests that
${\mathcal{K}}\subset{\mathbb{P}}^{3}$ should be defined by the $3\times 3$
minors of $B=\Lambda{\mathbf{H}}\Lambda^{T}$. This is confirmed by a direct
calculation, which more precisely shows that the equation $F$ for
${\mathcal{K}}$ in (5) satisfies

(18)
$-2f_{6}\operatorname{Adj}B=F(x_{1},\ldots,x_{4}){\mathbf{x}}{\mathbf{x}}^{T}$

where ${\mathbf{x}}=(x_{1},x_{2},x_{3},x_{4})^{T}$.

The equations for $\widetilde{{\mathcal{K}}}$ may be written in terms of
Pfaffians. To do this we define skew symmetric matrices of linear forms in
$u_{0},\ldots,u_{5}$ by

(19) $Z=\begin{pmatrix}0&u_{0}&u_{1}&u_{3}\\\ &0&u_{2}&-u_{4}\\\ &&0&u_{5}\\\
&&&0\end{pmatrix}\quad\text{ and }\quad
W=\begin{pmatrix}0&H_{0}&H_{1}&H_{3}\\\ &0&H_{2}&-H_{4}\\\ &&0&H_{5}\\\
&&&0\end{pmatrix}^{*}$

where $H_{i}=\partial H/\partial u_{i}$, and the superscript $*$ has the
meaning explained in (8). Then for indeterminates $\lambda$ and $\mu$ we have

$\operatorname{Pf}(\lambda Z+\mu W)=\lambda^{2}G-2\lambda\mu H+\mu^{2}S.$

######  Remark 2.2.

The birational map between ${\mathcal{K}}\subset{\mathbb{P}}^{3}$ and
$\widetilde{{\mathcal{K}}}\subset{\mathbb{P}}^{5}$ may be described in the
following ways.

  1. (i)

The graph of this map in ${\mathbb{P}}^{3}\times{\mathbb{P}}^{5}$ is defined
by the 8 bilinear forms

$(x_{1}\,\,x_{2}\,\,x_{3}\,\,x_{4})(Z|W)=0.$

  2. (ii)

The $4\times 4$ matrix of quadratic forms $ZW^{*}$ has rank at most $1$ on
$\widetilde{{\mathcal{K}}}$. Any row defines the map
$\widetilde{{\mathcal{K}}}\to{\mathcal{K}}$ and any column defines the map
$\widetilde{{\mathcal{K}}}\to{\mathcal{K}}^{\vee}$.

  3. (iii)

The $2\times 2$ minors of $B$ may be arranged in a $6\times 6$ symmetric
matrix of quartic forms that has rank at most $1$ on ${\mathcal{K}}$. Any row
or column defines the map ${\mathcal{K}}\to\widetilde{{\mathcal{K}}}$. Setting
any diagonal entry of the $6\times 6$ matrix equal to $-f_{6}$ times a square
defines the double cover ${\mathcal{J}}\to{\mathcal{K}}$.

######  Remark 2.3.

The previous remark is closely related to the 72 quadratic equations defining
${\mathcal{J}}\subset{\mathbb{P}}^{15}$, as originally computed by Flynn [F1],
and revisited in [FTvL]. We write our coordinates on ${\mathbb{P}}^{15}$ as
$x_{ij}$ and $z_{ij}$ where these are the entries of a generic $4\times 4$
symmetric matrix $X$, and a generic $4\times 4$ skew symmetric matrix $Z$. We
identify the $z_{ij}$ with the $u_{i}$ via (19). Our first $21$ quadratic
equations are the $2\times 2$ minors of $X$. Next the entries of $XZ$ and $XW$
give $32$ quadratic equations, in fact only spanning a space of dimension
$30$, since these matrices have trace zero. Finally, generalising the last
part of Remark 2.2(iii), there are $21$ quadratic equations
$-f_{6}z_{ij}z_{kl}=B_{ik}B_{jl}-B_{il}B_{jk}$ where each quartic on the right
is rewritten as a quadratic using $x_{rs}=x_{r}x_{s}$. In view of the first
$21$ equations, the choices here do not matter. In total this gives
$21+30+21=72$ equations.

###  2.3. The Selmer and fake Selmer groups

We continue to take ${\mathcal{C}}$ a genus $2$ curve with equation
$y^{2}=f(x)$ where $f\in k[x]$ is a polynomial of degree $6$. We now suppose
that $k$ is a number field. Let ${\mathcal{J}}$ be the Jacobian of
${\mathcal{C}}$. The $2$-Selmer group is

(20)
$S^{(2)}({\mathcal{J}}/k)=\ker\left(H^{1}(k,{\mathcal{J}}[2])\to\prod_{v}H^{1}(k_{v},{\mathcal{J}})\right).$

Let $W$ be the set of Weierstrass points on ${\mathcal{C}}$, that is, the
points $(\theta,0)$ where $\theta$ is a root of $f$. Let $\Phi$ be the set of
all ways of partitioning $W$ into two subsets. There is a natural Galois
action on $\Phi$ induced by the Galois action on $W$, and a group law given by
symmetric difference, equivalently by pointwise addition of indicator
functions in $\operatorname{Map}(W,{\mathbb{F}}_{2})/{\mathbb{F}}_{2}$. We
write $\Phi=\Phi_{0}\sqcup\Phi_{1}$ where $\Phi_{0}$ (resp. $\Phi_{1}$) is the
set of decompositions into subsets of even (resp. odd) size. It is well known
that $\Phi_{0}$ is isomorphic to ${\mathcal{J}}[2]$ as a Galois module. It
follows that $\Phi_{1}$ is a torsor under ${\mathcal{J}}[2]$. We write $c\in
H^{1}(k,{\mathcal{J}}[2])$ for the class of $\Phi_{1}$.

######  Lemma 2.4.

The canonical element $c$ belongs to $S^{(2)}({\mathcal{J}}/k)$.

###### Proof.

The canonical element $c$ is represented by the cocycle $\sigma\mapsto[\sigma
P-P]$ where $P\in{\mathcal{C}}$ is a Weierstrass point. It follows that the
natural map $H^{1}(k,{\mathcal{J}}[2])\to H^{1}(k,{\mathcal{J}})$ sends $c$ to
the class of $\operatorname{Pic}^{1}{\mathcal{C}}$. The lemma is therefore
equivalent to showing that ${\mathcal{C}}$ admits a $k_{v}$-rational divisor
class of degree $1$ for each place $v$ of $k$. As noted by Poonen and Stoll
[PSt, Lemma 1] this is a consequence of Tate local duality, and is originally
due to Lichtenbaum [L, Theorem 7]. ∎

######  Definition 2.5.

The fake Selmer group is $S_{\rm
fake}^{(2)}({\mathcal{J}}/k)=S^{(2)}({\mathcal{J}}/k)/\langle c\rangle$.

The analogue of (20) is

$S_{\rm
fake}^{(2)}({\mathcal{J}}/k)=\ker\left(\frac{H^{1}(k,{\mathcal{J}}[2])}{\langle
c\rangle}\to\prod_{v}H^{1}(k_{v},{\mathcal{J}})\right).$

Let $L=k[x]/(f)=k[\theta]$ be the étale algebra of $W$. As shown by Poonen and
Schaefer [PSc] there is an isomorphism

(21) $\frac{H^{1}(k,{\mathcal{J}}[2])^{\cup c=0}}{\langle
c\rangle}\cong\ker\left(L^{\times}/(k^{\times}(L^{\times})^{2})\stackrel{{\scriptstyle
N_{L/k}}}{{\longrightarrow}}k^{\times}/(k^{\times})^{2}\right)$

where the superscript ${\cup c=0}$ indicates the subgroup of elements that
pair trivially with $c$ under the pairing

$H^{1}(k,{\mathcal{J}}[2])\times
H^{1}(k,{\mathcal{J}}[2])\to\operatorname{Br}(k)$

given by cup product and the Weil pairing. The composite of the connecting map
${\mathcal{J}}(k)/2{\mathcal{J}}(k)\to H^{1}(k,{\mathcal{J}}[2])$ and (21) is
the Cassels map (see e.g. [C4], [FPS]), given for $yy^{\prime}\not=0$ by

(22)
$[(x,y)+(x^{\prime},y^{\prime})-\kappa]\mapsto(x-\theta)(x^{\prime}-\theta).$

The fake Selmer group $S_{\rm fake}^{(2)}({\mathcal{J}}/k)$ naturally arises
as the subgroup of the right hand side of (21) consisting of elements that are
everywhere locally in the image of the Cassels map. This group can be computed
in Magma [BCP], using programs originally due to Stoll [St].

It is easy to compute the dimension of $S^{(2)}({\mathcal{J}}/k)$ as an
${\mathbb{F}}_{2}$-vector space from that of $S_{\rm
fake}^{(2)}({\mathcal{J}}/k)$, simply by adding $0$ or $1$ according as the
canonical element $c$ is trivial or non-trivial. However we shall need a more
explicit way of representing elements of $S^{(2)}({\mathcal{J}}/k)$. As
explained by Stoll and van Luijk [SvL], this is given by augmenting elements
of the right hand side of (21) with a choice of square root of the norm. Thus
$S^{(2)}({\mathcal{J}}/k)$ is a subgroup of

(23) $H^{1}(k,{\mathcal{J}}[2])^{\cup c=0}\cong\frac{\\{(\xi,m)\in
L^{\times}\times
k^{\times}|N_{L/k}(\xi)=m^{2}\\}}{\\{(r\nu^{2},r^{3}N_{L/k}(\nu))|r\in
k^{\times},\nu\in L^{\times}\\}}.$

###  2.4. Models for 2-coverings

Following the “neoclassical approach” (see the introduction for references) we
explain how elements of the $2$-Selmer group $S^{(2)}({\mathcal{J}}/k)$ may be
represented by certain pairs of quadratic forms, which we call models.

We only need the following proposition in the case $n=6$, but the extra
generality costs us nothing.

######  Proposition 2.6.

Let $f(x)=f_{n}x^{n}+\ldots+f_{1}x+f_{0}\in k[x]$ be a square-free polynomial
of degree $n$. Let $L=k[\theta]=k[x]/(f)$ and fix $\xi\in L^{\times}$.
Consider the quadratic forms $Q_{j}\in k[u_{0},\ldots,u_{n-1}]$ defined by

(24)
$\xi(u_{0}+u_{1}\theta+\ldots+u_{n-1}\theta^{n-1})^{2}=\sum_{j=0}^{n-1}Q_{j}(u_{0},\ldots,u_{n-1})\theta^{j}.$

  1. (i)

There are $n\times n$ symmetric matrices ${\mathbf{G}}$ and ${\mathbf{H}}$,
with ${\mathbf{G}}$ invertible, such that

$\qquad
Q_{j}(u_{0},\ldots,u_{n-1})=f_{n}^{-1}\sum_{i=j+1}^{n}f_{i}{\mathbf{u}}^{T}{\mathbf{G}}({\mathbf{G}}^{-1}{\mathbf{H}})^{i-j-1}{\mathbf{u}}$

for all $0\leqslant j\leqslant n-1$, where
${\mathbf{u}}=(u_{0},\ldots,u_{n-1})^{T}$.

  2. (ii)

We have

$\det(x{\mathbf{G}}-{\mathbf{H}})=(-1)^{\binom{n}{2}}f_{n}^{-1}N_{L/k}(\xi)f(x).$

  3. (iii)

If ${\mathbf{v}}={\mathbf{G}}^{-1}(1,\theta,\ldots,\theta^{n-1})^{T}$ then
$(\theta{\mathbf{G}}-{\mathbf{H}}){\mathbf{v}}=0$ and
${\mathbf{v}}^{T}{\mathbf{G}}{\mathbf{v}}=f_{n}^{-1}f^{\prime}(\theta)/\xi$.

  4. (iv)

We have

$\qquad\left(\frac{1}{2^{n}}\,\frac{\partial(Q_{0},\ldots,Q_{n-1})}{\partial(u_{0},\ldots,u_{n-1})}\right)^{2}=(-1)^{\binom{n}{2}}(\det{\mathbf{G}})N_{L/k}\left(\sum_{j=0}^{n-1}Q_{j}(u_{0},\ldots,u_{n-1})\theta^{j}\right).$

###### Proof.

Let $\theta_{1},\ldots,\theta_{n}$ be the roots of $f$. A well known identity
that is used in the theory of the different (see for example [Sw, page 45])
states that

$\sum_{r=1}^{n}\frac{f(x)}{x-\theta_{r}}\,\frac{\theta_{r}^{i}}{f^{\prime}(\theta_{r})}=x^{i}.$

for all $0\leqslant i\leqslant n-1$. The proof is simply that each side is a
polynomial of degree less than $n$ taking the same values at
$\theta_{1},\ldots,\theta_{n}$. Writing

(25) $f(x)=(x-\theta)(\beta_{n-1}x^{n-1}+\ldots+\beta_{1}x+\beta_{0})$

it follows that the $\theta^{i}$ and $\beta_{i}/f^{\prime}(\theta)$ are dual
bases for $L$ with respect to the trace pairing
$(x,y)\mapsto\operatorname{Tr}_{L/k}(xy)$. In particular for any $a\in L$ we
have

(26)
$N_{L/k}(a)=\det\left(\operatorname{Tr}_{L/k}\left(\frac{a\theta^{i}\beta_{j}}{f^{\prime}(\theta)}\right)_{i,j=0,\ldots,n-1}\right)$

and (24) is satisfied with

(27)
$Q_{j}(u_{0},\ldots,u_{n-1})=\operatorname{Tr}_{L/k}\left(\frac{\xi\beta_{j}(u_{0}+\ldots+u_{n-1}\theta^{n-1})^{2}}{f^{\prime}(\theta)}\right).$

(i) Let ${\mathbf{G}}$ represent the pairing
$(x,y)\mapsto\operatorname{Tr}_{L/k}(f_{n}\xi xy/f^{\prime}(\theta))$, and
${\mathbf{\Theta}}$ represent the multiplication-by-$\theta$ map, both with
respect to the basis $1,\theta,\ldots,\theta^{n-1}$. Comparing coefficients in
(25) shows that $\beta_{j}=\sum_{i=j+1}^{n}f_{i}\theta^{i-j-1}$. Rewriting
(27) using this notation and putting
${\mathbf{H}}={\mathbf{G}}{\mathbf{\Theta}}$ gives the stated formula.

(ii) We have
$\det(x{\mathbf{G}}-{\mathbf{H}})=\det{\mathbf{G}}\det(xI_{n}-{\mathbf{\Theta}})$
where

(28)
$\det{\mathbf{G}}=N_{L/k}(\xi)\Delta(1,\theta,\ldots,\theta^{n-1})/N_{L/k}(f_{n}^{-1}f^{\prime}(\theta))=(-1)^{\binom{n}{2}}N_{L/k}(\xi)$

and $\det(xI_{n}-{\mathbf{\Theta}})=f_{n}^{-1}f(x)$.

(iii) We have
${\mathbf{v}}^{T}(\theta{\mathbf{G}}-{\mathbf{H}})=(1,\theta,\ldots,\theta^{n-1})(\theta
I_{n}-{\mathbf{\Theta}})=0$. On the other hand
${\mathbf{G}}{\mathbf{v}}=(1,\theta,\ldots,\theta^{n-1})^{T}$ and
${\mathbf{v}}=(f_{n}\xi)^{-1}(\beta_{0},\beta_{1},\ldots,\beta_{n-1})^{T}$.
Therefore

${\mathbf{v}}^{T}{\mathbf{G}}{\mathbf{v}}=(\beta_{0}+\beta_{1}\theta+\ldots+\beta_{n-1}\theta^{n-1})/(f_{n}\xi)=f^{\prime}(\theta)/(f_{n}\xi).$

(iv) We differentiate (27) to get

$\frac{\partial Q_{j}}{\partial
u_{i}}(u_{0},\ldots,u_{n-1})=\operatorname{Tr}_{L/k}\left(\frac{2\xi\theta^{i}\beta_{j}(u_{0}+\ldots+u_{n-1}\theta^{n-1})}{f^{\prime}(\theta)}\right).$

It follows by (26) that

$\frac{\partial(Q_{0},\ldots,Q_{n-1})}{\partial(u_{0},\ldots,u_{n-1})}=N_{L/k}\left(2\xi(u_{0}+u_{1}\theta+\ldots+u_{n-1}\theta^{n-1})\right).$

Squaring both sides and then using (24) and (28) gives the stated formula. ∎

We now return to considering a genus $2$ curve ${\mathcal{C}}$ with equation
$y^{2}=f(x)$ where $f\in k[x]$ is a polynomial of degree $6$. Let
${\mathcal{J}}$ be the Jacobian of ${\mathcal{C}}$.

######  Definition 2.7.

(i) A model (for a $2$-covering of ${\mathcal{J}}$) is a pair of quadratic
forms $(G,H)$ in variables $u_{0},\ldots,u_{5}$, where
$G=u_{0}u_{5}+u_{1}u_{4}+u_{2}u_{3}$ and the corresponding symmetric matrices
(given by
$G({\mathbf{u}})=\frac{1}{2}{\mathbf{u}}^{T}{\mathbf{G}}{\mathbf{u}}$ and
$H({\mathbf{u}})=\frac{1}{2}{\mathbf{u}}^{T}{\mathbf{H}}{\mathbf{u}}$) satisfy

(29) $\det(x{\mathbf{G}}-{\mathbf{H}})=-f_{6}^{-1}f(x).$

Since $G$ is fixed, we sometimes refer to the model $(G,H)$ simply as $H$.
(ii) Two models are $k$-equivalent if they are in the same orbit for the
action of $\operatorname{PGL}_{4}(k)$ defined as follows. We identify the
$4\times 4$ skew symmetric matrices $Z$ with the column vectors
${\mathbf{u}}=(u_{0},\ldots,u_{5})^{T}$ via (19), so that
$G=\operatorname{Pf}(Z)$. Then the action of $P\in\operatorname{GL}_{4}$ on
the space of such matrices via $Z\mapsto PZP^{T}$ is described by a matrix
$\wedge^{2}P\in\operatorname{GL}_{6}$. We let $P\in\operatorname{GL}_{4}$ act
on a model $(G,H)$ first by changing coordinates using $\wedge^{2}P$ and then
dividing both $G$ and $H$ by $\det P$. By (10) this preserves $G$. Since the
scalar matrices act trivially, this is an action of $\operatorname{PGL}_{4}$.

######  Remark 2.8.

(i) If we change our equation $y^{2}=f(x)$ for ${\mathcal{C}}$ by applying a
Möbius map to the $x$-coordinate then, provided $f$ still has degree $6$, we
may update our models by applying the inverse Möbius map to the transformation
${\mathbf{\Theta}}={\mathbf{G}}^{-1}{\mathbf{H}}$ in the proof of Proposition
2.6.
(ii) Any model for a $2$-covering of
${\mathcal{J}}=\operatorname{Jac}({\mathcal{C}})$ is also a model for a
$2$-covering of ${\mathcal{J}}_{d}=\operatorname{Jac}({\mathcal{C}}_{d})$
where ${\mathcal{C}}_{d}:dy^{2}=f(x)$ is any quadratic twist of
${\mathcal{C}}$.

Starting from $(\xi,m)\in S^{(2)}({\mathcal{J}}/k)$ we compute a model as
follows. First, analogous to (15), we define quadratic forms
$Q_{0},Q_{1},\ldots,Q_{5}\in k[u_{0},u_{1},\ldots,u_{5}]$ by

(30)
$\xi(u_{0}+u_{1}\theta+\ldots+u_{5}\theta^{5})^{2}=Q_{0}+Q_{1}\theta+\ldots+Q_{5}\theta^{5}.$

If $\xi$ is in the image of the Cassels map (22) then, after multiplying by a
square, it is quadratic in $\theta$. This motivates our interest in rational
points on the twisted desingularised Kummer surface
$\\{Q_{3}=Q_{4}=Q_{5}=0\\}\subset{\mathbb{P}}^{5}$. As suggested by
Proposition 2.6(i) with $n=6$ (see also [CF, page 167] or [FTvL, page 403]),
we rewrite our equations for this surface as

$\displaystyle G$ $\displaystyle=Q_{5}/2,$ $\displaystyle H$
$\displaystyle=(f_{6}Q_{4}-f_{5}Q_{5})/(2f_{6}),$ $\displaystyle S$
$\displaystyle=(f_{6}^{2}Q_{3}-f_{5}f_{6}Q_{4}+(f_{5}^{2}-f_{4}f_{6})Q_{5})/(2f_{6}^{2}),$

The corresponding symmetric matrices (with
$G({\mathbf{u}})=\frac{1}{2}{\mathbf{u}}^{T}{\mathbf{G}}{\mathbf{u}}$ etc.)
are then related by ${\mathbf{S}}={\mathbf{H}}{\mathbf{G}}^{-1}{\mathbf{H}}$.
By Proposition 2.6(ii) we have

(31) $\det(x{\mathbf{G}}-{\mathbf{H}})=-f_{6}^{-1}N_{L/k}(\xi)f(x).$

If $\xi$ is in the image of the Cassels map (22) then, after multiplying by a
square, it is quadratic in $\theta$. If we further take
$u_{2}=u_{3}=u_{4}=u_{5}=0$ then the left hand side of (30) has no term
$\theta^{5}$. It follows that the quadratic form $G$ has a $2$-dimensional
isotropic subspace, and is therefore the orthogonal direct sum of two
hyperbolic planes and a binary quadratic form. As a special case of (31) we
have $\det{\mathbf{G}}=-N_{L/k}(\xi)$. Since $N_{L/k}(\xi)=m^{2}$ the binary
quadratic form has discriminant a square. Therefore $G$ has a $3$-dimensional
isotropic subspace.333Our argument is specific to curves of genus $2$. However
corresponding results are known for hyperelliptic curves in general; see [T1],
[Wa].

We are interested in $(\xi,m)$ representing an element of the Selmer group.
This means that $\xi$ is everywhere locally in the image of the Cassels map.
The previous paragraph shows that $G$ has a $3$-dimensional isotropic subspace
everywhere locally. By a suitable form on the the Hasse principle (for
example, the weak Hasse principle in [C3, Chapter 6]) it follows that $G$ has
a $3$-dimensional isotropic subspace globally. In the case $k={\mathbb{Q}}$ we
use the existing function IsotropicSubspace in Magma [BCP] to find such a
subspace. It is then easy to find a change of coordinates putting $G$ in the
standard form specified in Definition 2.7(i). We can and do choose this change
of coordinates so that it has determinant $m$ (rather than $-m$). This gives
the desired model $(G,H)$.

It is shown in [CF, Lemma 17.1.1] that the $3$-dimensional isotropic subspaces
of $G$ fall into two algebraic families. This may conveniently be seen as
follows. We write $G=\operatorname{Pf}(Z)$ as in Definition 2.7(ii). Setting
all entries in the $j$th row and column to zero makes $G$ vanish, as does
setting all entries outside the $j$th row and column to zero. Repeating these
observations for linear combinations of the rows and columns we see that the
$3$-dimensional isotropic subspaces of $G$ are parametrised by
${\mathbb{P}}^{3}\sqcup({\mathbb{P}}^{3})^{\vee}$.

Following [FH, Section 19.1], the subgroup
$\operatorname{PO}(G)\subset\operatorname{PGL}_{6}$ preserving $G$ up to
scalars sits in an exact sequence

$0\to\operatorname{PSO}(G)\to\operatorname{PO}(G)\to\\{\pm 1\\}\to 0$

where $\operatorname{PGL}_{4}\cong\operatorname{PSO}(G)$ via
$P\mapsto\wedge^{2}P$, and $\\{\pm 1\\}$ swaps over the two families of
isotropic subspaces. It follows that our procedure (noting in particular the
condition on the determinant of the change of coordinates) associates to each
Selmer group element $(\xi,m)$ a unique equivalence class of models.

######  Remark 2.9.

If we started with an arbitrary pair $(\xi,m)$ in (23), not necessarily a
Selmer group element, then the $3$-dimensional isotropic subspaces of $G$
would be parametrised by $S\sqcup S^{\vee}$, where $S$ is a $3$-dimensional
Brauer-Severi variety, and $S^{\vee}$ is its dual. The obstruction map
$H^{1}(k,{\mathcal{J}}[2])\to\operatorname{Br}(k)$, as defined in [Cl, Section
5], is then realised by $(\xi,m)\mapsto[S]$.

###  2.5. Recovering the Selmer group element

We explain how to recover a Selmer group element $(\xi,m)$ from a model
$(G,H)$. This not only helps us check that the models we computed in the last
section are correct, but is also useful for testing equivalence of models, and
for realising the group law.

As in Section 2.2 the matrix $\theta{\mathbf{G}}-{\mathbf{H}}$ has rank $5$.
We pick a vector spanning its kernel, and rewrite it using (19) as a $4\times
4$ skew symmetric matrix $A$ with entries in $L=k[x]/(f)=k[\theta]$. We claim
that

(32)
$\xi\equiv\operatorname{Pf}(A)/f^{\prime}(\theta)\mod{k^{\times}(L^{\times})^{2}}.$

In the untwisted case (i.e., when $\xi=1$) this claim follows by (6), (11) and
(16). It is also easy to see that the class (32) only depends on the
equivalence class of $(G,H)$. Indeed picking a different basis for the kernel
of $\theta{\mathbf{G}}-{\mathbf{H}}$ has the effect of multiplying $A$ by an
element of $L^{\times}$ and hence $\operatorname{Pf}(A)$ by an element of
$(L^{\times})^{2}$. By (10) the only further effect of replacing $(G,H)$ by an
equivalent model is to multiply $\operatorname{Pf}(A)$ by an element of
$k^{\times}$.

If we take $\xi=\operatorname{Pf}(A)/f^{\prime}(\theta)$ then the matrix $A$
also determines a square root $m$ of $N_{L/k}(\xi)$ via the formula

(33)
$A_{1}^{*}A_{2}A^{*}_{3}A_{4}A_{5}^{*}A_{6}=mf_{6}^{3}\prod_{i<j}(\theta_{i}-\theta_{j})I_{4},$

which generalises (13).

Proposition 2.6(iii) shows that the class of $\xi$ in
$L^{\times}/k^{\times}(L^{\times})^{2}$ is given by evaluating the quadratic
form $G$ at a vector in the kernel of $\theta{\mathbf{G}}-{\mathbf{H}}$, and
then dividing by $f^{\prime}(\theta)$. Since in computing a model we change
coordinates so that $G$ becomes the Pfaffian, this proves (32). Following this
proof gives another formula for $m$, which improves on (33) in that it avoids
the need for field extensions. Indeed we may take $m=(\det N)/(2f_{6})^{3}$
where $N$ is the $6\times 6$ matrix that writes the entries of $A$ above the
diagonal in terms of the basis $1,\theta,\ldots,\theta^{5}$ for $L$.

######  Remark 2.10.

If we change $(G,H)$ by reversing the coordinates $u_{0},\ldots,u_{5}$ then
$A$ is replaced by $A^{*}$. Since
$\operatorname{Pf}(A)=\operatorname{Pf}(A^{*})$ this does not change $\xi$,
but using (12) and (33), or the formula in the last paragraph, it does change
the sign of $m$. This in turn corresponds to adding the canonical element $c$
(as defined in Section 2.3).

The following alternative way to recover a Selmer group element $(\xi,m)$ from
a model $(G,H)$ will be useful in Sections 2.6 and 3.6.

######  Lemma 2.11.

Let $(G,H)$ be a model and let $Q_{0},\ldots,Q_{5}$ be the quadratic forms
defined by the formula in Proposition 2.6(i). Then the forms

$\Xi(u_{0},\ldots,u_{5})=\sum_{j=0}^{5}Q_{j}(u_{0},\ldots,u_{5})\theta^{j}\,\,\,\,\text{
and
}\,\,\,\,M(u_{0},\ldots,u_{5})=\frac{1}{2^{6}}\,\frac{\partial(Q_{0},\ldots,Q_{5})}{\partial(u_{0},\ldots,u_{5})}$

satisfy $N_{L/k}(\Xi)=M^{2}$. Moreover specialising these forms at any $a\in
k^{6}$ with $M(a)\not=0$ gives a pair $(\xi,m)$ representing the equivalence
class of the model.

###### Proof.

The first statement is proved by checking that Proposition 2.6(iv) is
invariant under all changes of coordinates, and then noting that for $G$ in
the standard form (as specified in Definition 2.7(i)) we have
$\det{\mathbf{G}}=-1$. The second statement follows by (24). ∎

###  2.6. Twisted Kummer surfaces

We may also represent elements of the $2$-Selmer group as $2$-coverings. In
fact any ${\varepsilon}\in H^{1}(k,{\mathcal{J}}[2])$ may be represented by a
$2$-covering $({\mathcal{J}}_{\varepsilon},\pi_{\varepsilon})$. By definition
this fits in a commutative diagram

(34)
$\textstyle{{\mathcal{J}}_{\varepsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\varepsilon}}$$\scriptstyle{\pi_{\varepsilon}}$$\textstyle{{\mathcal{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times
2}$$\textstyle{\mathcal{J}}$

where $\phi_{\varepsilon}$ is an isomorphism defined over $\overline{k}$. We
emphasise that $\phi_{\varepsilon}$ is not part of the data defining the
$2$-covering. Indeed we are free to change it by composing with translation by
$T$ for any $T\in{\mathcal{J}}[2]$.

We give ${\mathcal{J}}_{\varepsilon}$ the structure of principal homogeneous
space under ${\mathcal{J}}$ via

(35) $\displaystyle{\mathcal{J}}\times{\mathcal{J}}_{\varepsilon}$
$\displaystyle\to{\mathcal{J}}_{\varepsilon}$ $\displaystyle(P,Q)$
$\displaystyle\mapsto\phi_{\varepsilon}^{-1}(P+\phi_{\varepsilon}(Q)).$

This action is independent of the choice of $\phi_{\varepsilon}$ and hence
defined over $k$. The natural map $H^{1}(k,{\mathcal{J}}[2])\to
H^{1}(k,{\mathcal{J}})$ is then realised by
$({\mathcal{J}}_{\varepsilon},\pi_{\varepsilon})\mapsto{\mathcal{J}}_{\varepsilon}$.

The isomorphism $\phi_{\varepsilon}$ identifies the map $[-1]$ on
${\mathcal{J}}$ with an involution $\iota_{\varepsilon}$ on
${\mathcal{J}}_{\varepsilon}$. Again it may be checked that
$\iota_{\varepsilon}=\phi_{\varepsilon}^{-1}\circ[-1]\circ\phi_{\varepsilon}$
is independent of the choice of $\phi_{\varepsilon}$ and hence defined over
$k$.

We have ${\mathcal{J}}/[-1]\cong{\mathcal{K}}\subset{\mathbb{P}}^{3}$, where
${\mathcal{K}}$ is the Kummer surface as described in Section 2.1. Twisting by
${\varepsilon}\in S^{(2)}({\mathcal{J}}/k)$ gives
${\mathcal{J}}_{\varepsilon}/\iota_{\varepsilon}\cong{\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$
where ${\mathcal{K}}_{\varepsilon}$ is a twisted Kummer surface. (To recap
Remark 2.9: For general ${\varepsilon}\in H^{1}(k,{\mathcal{J}}[2])$ we might
need to replace ${\mathbb{P}}^{3}$ by a Brauer-Severi variety $S$, but for
Selmer group elements the local-global principle for the Brauer group shows
that $S\cong{\mathbb{P}}^{3}$.)

We record this set-up in a commutative diagram

(36)
$\textstyle{{\mathcal{J}}_{\varepsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\varepsilon}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathcal{K}}_{\varepsilon}}$$\scriptstyle{\psi_{\varepsilon}}$$\textstyle{{\mathcal{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{|2\Theta|}$$\textstyle{\mathcal{K}}$

where $\phi_{\varepsilon}$ is as in (34) and
$\psi_{\varepsilon}\in\operatorname{GL}_{4}(\overline{k})$. The divisor
$H_{\varepsilon}$ on ${\mathcal{J}}_{\varepsilon}$ obtained by pulling back a
hyperplane section on ${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$
therefore satisfies

(37) $H_{\varepsilon}\sim\phi_{\varepsilon}^{*}(2\Theta)$

where $\Theta$ is the theta divisor on ${\mathcal{J}}$.

In [Y1, Chapter 3] the second author gave two different methods for computing
the twisted Kummer surface ${\mathcal{K}}_{\varepsilon}$, both of which worked
by first computing the change of coordinates $\psi_{\varepsilon}$. This is
typically defined over a degree $16$ number field. Using the models introduced
in Section 2.4 we are now able to compute an equation for
${\mathcal{K}}_{\varepsilon}$ without making any field extensions. We start by
finding a model $(G,H)$ that represents ${\varepsilon}$. We then put
$B=\Lambda{\mathbf{H}}\Lambda^{T}$ where $\Lambda$ is defined by (17). Then,
exactly as in Section 2.2, a quartic form defining
${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$ is obtained as the GCD of
the $3\times 3$ minors of $B$. Moreover all the formulae we gave in that
section for ${\mathcal{K}}$, ${\mathcal{K}}^{\vee}$,
$\widetilde{{\mathcal{K}}}$, ${\mathcal{J}}$ and the maps between them carry
over immediately to their twisted counterparts ${\mathcal{K}}_{\varepsilon}$,
$({\mathcal{K}}_{\varepsilon})^{\vee}$,
$\widetilde{{\mathcal{K}}}_{\varepsilon}$ and ${\mathcal{J}}_{\varepsilon}$.
In particular the double cover
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}$ is defined by
setting any leading $2\times 2$ minor of $B$ equal to $-f_{6}$ times a square.
As mentioned in the introduction, we find these formulae more convenient to
use than those in [FTvL].

######  Remark 2.12.

The model $(G,H)$ that represents ${\varepsilon}$ also determines a $4\times
4$ skew symmetric matrix $A=A_{\varepsilon}$ as in Section 2.5. Just as in the
untwisted case (see Section 2.1) this matrix defines an isomorphism
${\mathcal{K}}_{\varepsilon}\to({\mathcal{K}}_{\varepsilon})^{\vee}$. In the
context of testing equivalence of models, these matrices help us solve for the
corresponding transformation when the models are equivalent. This is analogous
to the treatment of $3$-coverings of elliptic curves in [Fi1]. The matrices
$A_{\varepsilon}$ also give, by means of the formula (12), the matrices
$M_{T}$ describing the action of $T\in{\mathcal{J}}[2]$ on
${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$. As explained in Remark
4.3 below, this is useful for reduction.

######  Remark 2.13.

Let $c\in S^{(2)}({\mathcal{J}}/k)$ be the canonical element defined in
Section 2.3. Then continuing with Remark 2.10 we see that
$({\mathcal{K}}_{\varepsilon})^{\vee}\cong{\mathcal{K}}_{{\varepsilon}+c}$.

The covering map
$\pi_{\varepsilon}:{\mathcal{J}}_{\varepsilon}\to{\mathcal{J}}$ in (34)
induces a map ${\mathcal{K}}_{\varepsilon}\to{\mathcal{K}}$, which we also
call the covering map. We now explain how to compute this map explicitly.
Starting with a model $(G,H)$ we use the formula in Proposition 2.6(i) to
define quadratic forms $Q_{0},\ldots,Q_{5}$. Then Lemma 2.11 gives a formula
for the covering map from the twisted desingularised Kummer
$\widetilde{{\mathcal{K}}}_{\varepsilon}=\\{Q_{3}=Q_{4}=Q_{5}=0\\}\subset{\mathbb{P}}^{5}$
to the Kummer surface ${\mathcal{K}}$ written as a double cover of
${\mathbb{P}}^{2}$ as in (4).

We modify this to give the covering map
${\mathcal{K}}_{\varepsilon}\to{\mathcal{K}}$ where both surfaces are quartic
surfaces in ${\mathbb{P}}^{3}$. As suggested by Remark 2.2(iii) our
construction involves the $2\times 2$ minors of
$B=\Lambda{\mathbf{H}}\Lambda^{T}$. For $Q=\sum c_{ijkl}z_{ij}z_{kl}$ a
quadratic form we put

$Q\star H=\sum c_{ijkl}(B_{ik}B_{jl}-B_{il}B_{jk}).$

Then we define quartic forms
$E_{r}=\operatorname{Adj}(\Lambda{\mathbf{G}}({\mathbf{G}}^{-1}{\mathbf{H}})^{r}\Lambda^{T})_{44}/x_{4}^{2}$
and

$\displaystyle F_{0}$ $\displaystyle=(2f_{6})^{-1}E_{1},$ $\displaystyle
F_{1}$ $\displaystyle=Q_{2}\star H-2f_{4}F_{0},$ $\displaystyle F_{2}$
$\displaystyle=-Q_{1}\star H+f_{3}F_{0}=-E_{2}-f_{3}F_{0},$ $\displaystyle
F_{3}$ $\displaystyle=Q_{0}\star H,$ $\displaystyle F_{4}$
$\displaystyle=f_{6}E_{3}-(f_{2}Q_{2}-f_{3}Q_{1}+f_{4}Q_{0})\star
H-(f_{1}f_{5}-4f_{2}f_{4}+2f_{3}^{2})F_{0}.$

The quartic forms $F_{i}=F_{i}(H)$ are covariants of the model $H$. By this we
mean that for some integers $d_{0},\ldots,d_{4}$ we have

$F_{i}(\lambda H\circ\wedge^{2}P)=(\lambda\det P)^{d_{i}}F_{i}(H)\circ P$

for all $\lambda\in k^{\times}$ and $P\in\operatorname{GL}_{4}(k)$. In fact
$(d_{0},\ldots,d_{4})=(3,5,6,7,9)$. The next lemma shows that these covariants
define the covering map. This generalises an observation of Weil [We] in the
case of $2$-coverings of elliptic curves.

######  Lemma 2.14.

Let ${\varepsilon}\in S^{(2)}({\mathcal{J}}/k)$ be represented by a model
$(G,H)$ and let the $F_{i}=F_{i}(H)$ be the quartic polynomials defined above.

  1. (i)

${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$ has equation $F_{0}=0$.

  2. (ii)

Let the matrices $M_{T}$ be as defined in Remark 2.12. Then
$F_{0},\ldots,F_{4}$ are a basis for the space of Heisenberg invariant
quartics

$\qquad\\{F\in k[x_{1},\ldots,x_{4}]_{4}:F\circ M_{T}=\det(M_{T})F\text{ for
all }T\in{\mathcal{J}}[2]\\}.$

  3. (iii)

The covering map ${\mathcal{K}}_{\varepsilon}\to{\mathcal{K}}$ is given by

$(x_{1}:\ldots:x_{4})\mapsto(F_{1}:\ldots:F_{4}).$

###### Proof.

Since we are free to extend our field, and the $F_{i}$ are covariants, it
suffices to prove this for $H$ as given in Section 2.2. We already checked (i)
in Section 2.2. For (ii) we checked by computer algebra that
$F_{0},\ldots,F_{4}$ are linearly independent and then used Lemmas 3.11 and
3.13 below. See Proposition 3.19 below for further details of the proof of
(iii). ∎

######  Remark 2.15.

For the purposes of Lemma 2.14 it would not make any difference if we changed
$F_{1},\ldots,F_{4}$ by adding multiples of $F_{0}$. The reason for making the
choices we did is to simplify the statement of Proposition 3.19 below.

##  3\. The Cassels-Tate pairing on the $2$-Selmer group

Let ${\mathcal{A}}$ be an abelian variety defined over a number field $k$. The
Cassels-Tate pairing is a bilinear map

(38) $\mbox{\wncyr Sh}({\mathcal{A}}/k)\times\mbox{\wncyr
Sh}({\mathcal{A}}^{\vee}/k)\to{\mathbb{Q}}/{\mathbb{Z}}.$

It has the property that for any integer $n\geqslant 2$ the image of
multiplication-by-$n$ on $\mbox{\wncyr Sh}({\mathcal{A}}/k)$ and kernel of
multiplication-by-$n$ on $\mbox{\wncyr Sh}({\mathcal{A}}^{\vee}/k)$ are exact
annihilators. This follows from [Mi, Chapter I, Theorem 6.13] or more directly
from the injectivity of the middle vertical map in the diagram on page 88 of
[Mi]. Flach [Fl] showed that if we identify
${\mathcal{A}}={\mathcal{A}}^{\vee}$ using a principal polarisation, then the
pairing is skew-symmetric.

We take ${\mathcal{A}}={\mathcal{J}}$ a genus $2$ Jacobian, and identify
${\mathcal{J}}={\mathcal{J}}^{\vee}$ using the principal polarisation

(39) $X\mapsto[\tau_{X}^{*}\Theta-\Theta].$

By composing with the map $S^{(2)}({\mathcal{J}}/k)\to\mbox{\wncyr
Sh}({\mathcal{J}}/k)[2]$ in (1), the pairing (38) lifts to a pairing of
${\mathbb{F}}_{2}$-vector spaces

(40) $\langle~{},~{}\rangle_{\operatorname{CT}}:S^{(2)}({\mathcal{J}}/k)\times
S^{(2)}({\mathcal{J}}/k)\to{\mathbb{F}}_{2}.$

Specialising the results cited in the last paragraph, we see that this pairing
is symmetric and bilinear and, by chasing around the diagram (2), has kernel
the image of the natural map $S^{(4)}({\mathcal{J}}/k)\to
S^{(2)}({\mathcal{J}}/k)$. The pairing is not in general alternating, but
rather has the following properties.

######  Lemma 3.1 (Poonen and Stoll).

Let $c\in S^{(2)}({\mathcal{J}}/k)$ be the canonical element as defined in
Section 2.3. Then

  1. (i)

$\langle x,x+c\rangle_{\operatorname{CT}}=0$ for all $x\in
S^{(2)}({\mathcal{J}}/k)$.

  2. (ii)

$\langle c,c\rangle_{\operatorname{CT}}=0$ if and only if the number of
deficient places is even, where a place $v$ is deficient if ${\mathcal{C}}$
has no $k_{v}$-rational divisor of degree $1$.

  3. (iii)

The rank of the pairing $\langle~{},~{}\rangle_{\operatorname{CT}}$ is even if
and only if $\langle c,c\rangle_{\operatorname{CT}}=0$.

###### Proof.

(i) and (ii). See [PSt, Theorem 5 and Corollary 12]. Note that Poonen and
Stoll write $c$ for what in our notation is the image of $c$ in $\mbox{\wncyr
Sh}({\mathcal{J}}/k)$.
(iii) This follows by (i) and linear algebra. ∎

One consequence of Lemma 3.1 is that if the number of deficient places is odd
then the upper bound for ${\operatorname{rank}}\,{\mathcal{J}}(k)$ coming from
$2$-descent improves by $1$, without even having to compute the pairing. Our
focus in Section 4 is therefore in giving examples where the rank bound
improves by $2$ or more.

######  Remark 3.2.

Further partial information about the pairing is given by work of Creutz [Cr]
who (assuming there are no deficient places) gave a criterion for whether the
canonical element $c$ is in the kernel of the pairing, equivalently, for
whether the pairing on $S^{(2)}({\mathcal{J}}/k)$ is alternating.

######  Remark 3.3.

More generally, fixing a prime $p$, we may write $S_{m}$ for the image of the
$p^{m}$-Selmer group in the $p$-Selmer group. Then, exactly following Cassels’
treatment [C1] in the case of elliptic curves, we have inclusions of
${\mathbb{F}}_{p}$-vector spaces

${\mathcal{J}}(k)/2{\mathcal{J}}(k)\subset\ldots\subset S_{3}\subset
S_{2}\subset S_{1}=S^{(p)}({\mathcal{J}}/k)$

and a skew-symmetric pairing on each $S_{m}$ with kernel $S_{m+1}$. It is
interesting to note (paraphrasing the work of Poonen and Stoll) that all of
these pairings are alternating, except for the one we compute in this paper,
that is, the pairing on $S_{1}$ when $p=2$. In particular, assuming finiteness
of $\mbox{\wncyr Sh}({\mathcal{J}}/k)[2^{\infty}]$, the rank bounds we compute
(by a combination of $2$-descent and computing the pairing (40)) always have
the same parity as the true rank.

In Section 3.1 we briefly review the definition of the Cassels-Tate pairing
(38) that is relevant to our work. Then in Section 3.2 we give our new formula
for the pairing (40). In outline it works as follows. Given
${\varepsilon},\eta\in S^{(2)}({\mathcal{J}}/k)$ we compute equations for the
twisted Kummer surfaces ${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$
and ${\mathcal{K}}_{\eta}\subset{\mathbb{P}}^{3}$. If we can find a $k$-point
on ${\mathcal{K}}_{\eta}$ then the image of $\eta$ in $\mbox{\wncyr
Sh}({\mathcal{J}}/k)$ splits over a quadratic extension. We can then represent
$\eta$ as a Brauer class on ${\mathcal{K}}_{\varepsilon}$ split by the same
quadratic extension. We evaluate this Brauer class at suitable local points on
${\mathcal{K}}_{\varepsilon}$, and then compute
$\langle{\varepsilon},\eta\rangle_{{\operatorname{CT}}}$ as the sum of the
local invariants.

To turn this formula into a practical method for computing the pairing, we
must compute a rational function on ${\mathcal{K}}_{\varepsilon}$ that
represents the required Brauer class. In the case of elliptic curves, the
first author’s solution to this problem [Fi2], was based on the well known
formulae relating the $x$-coordinates of the points $P,Q,P+Q,P-Q$ on an
elliptic curve. The analogue of these formulae in genus $2$ are the
biquadratic forms computed by Flynn [F2]. These describe remnants of the group
law on ${\mathcal{J}}$ on the Kummer surface
${\mathcal{K}}={\mathcal{J}}/[-1]$. This leads in Section 3.3 to the
definition of a $(2,2,2)$-form on
${\mathbb{P}}^{3}\times{\mathbb{P}}^{3}\times({\mathbb{P}}^{3})^{\vee}$, a
twisted version of which specialises to the rational function we need. In
Sections 3.4–3.6 we give equations for the twisted $(2,2,2)$-form in terms of
our invariant-theoretic formulae for the $2$-covering map. Interestingly, we
discover that the coefficients of the original $(2,2,2)$-form may be viewed as
the structure constants for an étale algebra.

###  3.1. Definition of the Cassels-Tate pairing

We briefly review the homogeneous space definition of the Cassels-Tate pairing
(38), following Poonen and Stoll [PSt] and Milne [Mi, Chapter 1, Remark 6.11].

We start with $a\in\mbox{\wncyr Sh}({\mathcal{A}}/k)$ and $b\in\mbox{\wncyr
Sh}({\mathcal{A}}^{\vee}/k)$. Let $X$ be a (locally trivial) principal
homogeneous space over $k$ representing $a$. When we write
$\operatorname{Div}^{0}(X)$ and $\operatorname{Pic}^{0}(X)$ we shall mean
these groups for $X$ over $\overline{k}$. Since $\operatorname{Pic}^{0}(X)$ is
canonically isomorphic as a $\operatorname{Gal}(\overline{k}/k)$-module to
$\operatorname{Pic}^{0}({\mathcal{A}})={\mathcal{A}}^{\vee}(\overline{k})$,
the element $b\in\mbox{\wncyr Sh}({\mathcal{A}}^{\vee}/k)\subset
H^{1}(k,{\mathcal{A}}^{\vee})$ corresponds to an element of
$H^{1}(k,\operatorname{Pic}^{0}(X))$.

The short exact sequence of Galois modules

$0\to\overline{k}(X)^{\times}/\overline{k}^{\times}\to\operatorname{Div}^{0}(X)\to\operatorname{Pic}^{0}(X)\to
0,$

induces a long exact sequence of Galois cohomology

(41) $\ldots\to H^{1}(k,\operatorname{Pic}^{0}(X))\to
H^{2}(k,\overline{k}(X)^{\times}/\overline{k}^{\times})\to\ldots$

Let $b^{\prime}\in H^{2}(k,\overline{k}(X)^{\times}/\overline{k}^{\times})$ be
the image of $b$. Next we consider the short exact sequence

$0\to\overline{k}^{\times}\to\overline{k}(X)^{\times}\to\overline{k}(X)/\overline{k}^{\times}\to
0,$

and its associated long exact sequence

(42) $\ldots\to H^{2}(k,\overline{k}^{\times})\to
H^{2}(k,\overline{k}(X)^{\times})\to
H^{2}(k,\overline{k}(X)^{\times}/\overline{k}^{\times})\to
H^{3}(k,\overline{k}^{\times})\to\ldots$

Since $k$ is a number field we have $H^{3}(k,\overline{k}^{\times})=0$.
Therefore $b^{\prime}$ lifts to some $f^{\prime}\in
H^{2}(k,\overline{k}(X)^{\times})$. In the local analogue of (42), each
$f^{\prime}_{v}$ maps to zero (since $b$ is locally trivial) and so is the
image of some $c_{v}\in
H^{2}(k_{v},\overline{k}_{v}^{\times})=\operatorname{Br}(k_{v})$. The Cassels-
Tate pairing is defined by

$(a,b)\mapsto\sum_{v}\operatorname{inv}_{v}(c_{v})\in{\mathbb{Q}}/{\mathbb{Z}}.$

######  Remark 3.4.

(i) Although the sum is over all places $v$ of $k$, it may be shown that only
finitely many places contribute and so the sum is in fact finite.
(ii) We can compute $c_{v}$ by evaluating $f^{\prime}_{v}$ at any local point
$P_{v}\in X(k_{v})$. Since $b$ is locally trivial, the choice of $P_{v}$ does
not matter.
(iii) The pairing is independent of the choice of $f^{\prime}$ since by class
field theory the sum of the local invariants of an element in
$\operatorname{Br}(k)$ is always zero.

###  3.2. A formula for the Cassels-Tate pairing

In this section we give our new formula for the pairing (40). Accordingly, we
start with ${\varepsilon},\eta\in S^{(2)}({\mathcal{J}}/k)$ and aim to compute
$\langle{\varepsilon},\eta\rangle_{\operatorname{CT}}$. We write
${\mathcal{K}}_{\varepsilon}$, ${\mathcal{K}}_{\eta}$ and
${\mathcal{K}}_{{\varepsilon}+\eta}$ for the corresponding twisted Kummer
surfaces. Our method depends on finding a rational point
$P\in{\mathcal{K}}_{\eta}(k)$. In general there is no guarantee that such a
point exists, but as discussed in the introduction this does not appear to be
a severe restriction in practice. Since
${\mathcal{J}}_{\eta}\to{\mathcal{K}}_{\eta}$ is a double cover, the point $P$
lifts to a point defined over $k(\sqrt{a})$ for some $a\in k$. We may suppose
that $P$ is not a node, and that $a$ is not a square, otherwise
${\mathcal{J}}_{\eta}$ is trivial as a homogeneous space, in which case $\eta$
has trivial image in $\mbox{\wncyr Sh}({\mathcal{J}}/k)$ and
$\langle{\varepsilon},\eta\rangle_{\operatorname{CT}}=0$. As in Section 2.6,
we have
${\mathcal{K}}_{\varepsilon}={\mathcal{J}}_{\varepsilon}/\iota_{\varepsilon}$.

######  Theorem 3.5.

Suppose that $P\in{\mathcal{K}}_{\eta}(k)$ lifts to a point on
${\mathcal{J}}_{\eta}$ defined over $k(\sqrt{a})$. Let $H_{\varepsilon}$ and
$H_{{\varepsilon}+\eta}$ be divisors on ${\mathcal{J}}_{\varepsilon}$ and
${\mathcal{J}}_{{\varepsilon}+\eta}$ obtained by pulling back hyperplane
sections on
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$
and
${\mathcal{J}}_{{\varepsilon}+\eta}\to{\mathcal{K}}_{{\varepsilon}+\eta}\subset{\mathbb{P}}^{3}$.
Then

  1. (i)

The point $P$ determines an isomorphism
$\phi:{\mathcal{J}}_{\varepsilon}\to{\mathcal{J}}_{{\varepsilon}+\eta}$
defined over $k(\sqrt{a})$.

  2. (ii)

There is a $k$-rational function $g$ on ${\mathcal{J}}_{\varepsilon}$ with
divisor

$\operatorname{div}(g)=\phi^{*}H_{{\varepsilon}+\eta}+\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})-2H_{\varepsilon}.$

  3. (iii)

The Cassels-Tate pairing (40) is given by

$\langle{\varepsilon},\eta\rangle_{\operatorname{CT}}=\sum_{v}(a,g(P_{v}))_{v}$

where for each place $v$ of $k$ we pick a local point
$P_{v}\in{\mathcal{J}}_{\varepsilon}(k_{v})$, avoiding the zeros and poles of
$g$, and $(~{},~{})_{v}:k_{v}^{\times}/(k_{v}^{\times})^{2}\times
k_{v}^{\times}/(k_{v}^{\times})^{2}\to{\mathbb{F}}_{2}$ is the Hilbert norm
residue symbol.

######  Remark 3.6.

(i) The theorem is only useful if we have a practical method for computing the
rational function $g$. This is the subject of Sections 3.3–3.6.
(ii) The scaling of $g$ is not important for the same reason as in Remark
3.4(iii).
(iii) Since $g$ factors via ${\mathcal{K}}_{\varepsilon}$ we may evaluate it
at local points on ${\mathcal{K}}_{\varepsilon}$, provided that we are careful
to choose local points that can be lifted to local points on
${\mathcal{J}}_{\varepsilon}$.

We start the proof of Theorem 3.5 by describing a twisted form of the group
law on ${\mathcal{J}}$.

######  Lemma 3.7.

Let ${\varepsilon},\eta\in H^{1}(k,{\mathcal{J}}[2])$. Let
$\phi_{\varepsilon}:{\mathcal{J}}_{\varepsilon}\to{\mathcal{J}}$,
$\phi_{\eta}:{\mathcal{J}}_{\eta}\to{\mathcal{J}}$ and
$\phi_{{\varepsilon}+\eta}:{\mathcal{J}}_{{\varepsilon}+\eta}\to{\mathcal{J}}$
be isomorphisms defined over $\overline{k}$ making the diagram (34) and its
analogues for $\eta$ and ${\varepsilon}+\eta$ commute. Then, after possibly
adjusting our choice of $\phi_{{\varepsilon}+\eta}$, there is a morphism $\mu$
defined over $k$ making the following diagram commute

$\textstyle{{\mathcal{J}}_{\varepsilon}\,\,\,\times\,\,\,{\mathcal{J}}_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\phi_{\varepsilon}}$$\scriptstyle{\phi_{\eta}}$$\textstyle{{\mathcal{J}}_{{\varepsilon}+\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{{\varepsilon}+\eta}}$$\textstyle{{\mathcal{J}}\,\,\,\times\,\,\,{\mathcal{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+}$$\textstyle{{\mathcal{J}}\hbox
to0.0pt{.\hss}}$

###### Proof.

We know that ${\varepsilon}$ is represented by a cocycle
$(\sigma\mapsto{\varepsilon}_{\sigma})$ where
$\sigma(\phi_{\varepsilon})\phi_{\varepsilon}^{-1}$ is translation by
${\varepsilon}_{\sigma}\in{\mathcal{J}}[2]$. The corresponding statements hold
for $\eta$ and ${\varepsilon}+\eta$. Since the cocycles
$(\sigma\mapsto({\varepsilon}+\eta)_{\sigma})$ and
$(\sigma\mapsto{\varepsilon}_{\sigma}+\eta_{\sigma})$ differ by a coboundary,
we may adjust our choice of $\phi_{{\varepsilon}+\eta}$ so that
$({\varepsilon}+\eta)_{\sigma}={\varepsilon}_{\sigma}+\eta_{\sigma}$ for all
$\sigma\in\operatorname{Gal}(\overline{k}/k)$.

Let
$\mu:{\mathcal{J}}_{\varepsilon}\times{\mathcal{J}}_{\eta}\to{\mathcal{J}}_{{\varepsilon}+\eta}$
be the morphism that makes the diagram in the statement of the lemma commute.
For any $P\in{\mathcal{J}}_{\varepsilon}$, $Q\in{\mathcal{J}}_{\eta}$ and
$\sigma\in\operatorname{Gal}(\overline{k}/k)$ we have

$\displaystyle\sigma(\mu(P,Q))$
$\displaystyle=\sigma(\phi_{{\varepsilon}+\eta}^{-1}(\phi_{\varepsilon}(P)+\phi_{\eta}(Q)))$
$\displaystyle=\phi_{{\varepsilon}+\eta}^{-1}(\sigma(\phi_{\varepsilon}(P))+\sigma(\phi_{\eta}(Q))-{\varepsilon}_{\sigma}-\eta_{\sigma})$
$\displaystyle=\phi_{{\varepsilon}+\eta}^{-1}(\phi_{\varepsilon}(\sigma
P)+\phi_{\eta}(\sigma Q))$ $\displaystyle=\mu(\sigma P,\sigma Q).$

This proves that $\mu$ is defined over $k$. ∎

######  Remark 3.8.

We are still free to replace $\phi_{{\varepsilon}+\eta}$ by
$P\mapsto\phi_{{\varepsilon}+\eta}(P)+T$ for $T\in{\mathcal{J}}(k)[2]$, and
for this reason there are $\\#{\mathcal{J}}(k)[2]$ choices for the map $\mu$.

Proof of Theorem 3.5. Let $\\{Q,Q^{\prime}\\}\subset{\mathcal{J}}_{\eta}$ be
the inverse image of $P\in{\mathcal{K}}_{\eta}$, and let $\mu$ be as defined
in Lemma 3.7. Then $\phi=\mu(-,Q)$ is an isomorphism
${\mathcal{J}}_{\varepsilon}\to{\mathcal{J}}_{{\varepsilon}+\eta}$ defined
over $k(Q)=k(\sqrt{a})$. This proves (i). From Lemma 3.7 we also get a
commutative diagram

(43)
$\textstyle{{\mathcal{J}}_{\varepsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\phi_{\varepsilon}}$$\textstyle{{\mathcal{J}}_{{\varepsilon}+\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{{\varepsilon}+\eta}}$$\textstyle{{\mathcal{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{R}}$$\textstyle{{\mathcal{J}}\hbox
to0.0pt{,\hss}}$

where $\tau_{R}$ is translation by $R=\phi_{\eta}(Q)$.

Let $\psi:{\mathcal{J}}_{\eta}\to{\mathcal{J}}$ be the isomorphism given by
$X\mapsto X-Q$. Then $\sigma(\psi)\psi^{-1}$ is translation by $Q-\sigma(Q)$.
By (35) we have $Q-\sigma(Q)=\phi_{\eta}(Q)-\phi_{\eta}(\sigma(Q))$. Since
$\phi_{\eta}(Q)=R$ and $\iota_{\eta}$ swaps over $Q$ and $Q^{\prime}$ we have
$\phi_{\eta}(Q^{\prime})=-R$. It follows that the image of $\eta$ in
$H^{1}(k,{\mathcal{J}})$ is represented by the cocycle

$\sigma\mapsto\left\\{\begin{array}[]{cl}0_{\mathcal{J}}&\text{ if
}\sigma(\sqrt{a})=\sqrt{a},\\\ 2R&\text{ if
}\sigma(\sqrt{a})=-\sqrt{a}.\end{array}\right.$

We follow the definition of the Cassels-Tate pairing in Section 3.1, with $a$
and $b$ the images of ${\varepsilon}$ and $\eta$ in $\mbox{\wncyr
Sh}({\mathcal{J}}/k)$ and $\mbox{\wncyr Sh}({\mathcal{J}}^{\vee}/k)$, in the
latter case using the principal polarisation (39). Now $b$ corresponds to the
element in $H^{1}(k,\operatorname{Pic}^{0}{\mathcal{J}}_{\varepsilon})$
represented by the cocycle

$\sigma\mapsto\left\\{\begin{array}[]{ll}{\rm id}&\text{ if
}\sigma(\sqrt{a})=\sqrt{a},\\\
\\![\phi_{\varepsilon}^{*}(\tau_{2R}^{*}\Theta-\Theta)]&\text{ if
}\sigma(\sqrt{a})=-\sqrt{a}.\end{array}\right.$

Next we claim that

$\phi_{\varepsilon}^{*}(\tau_{2R}^{*}\Theta-\Theta)\sim\phi_{\varepsilon}^{*}(\tau_{R}^{*}(2\Theta)-2\Theta)\sim\phi^{*}H_{{\varepsilon}+\eta}-H_{\varepsilon}.$

Indeed the first linear equivalence follows from the fact the polarisation
(39) is a group homomorphism, whereas the second follows from (37) and (43).

To apply the connecting homomorphism in (41) we lift to a cochain

$\sigma\mapsto\left\\{\begin{array}[]{ll}{\rm id}&\text{ if
}\sigma(\sqrt{a})=\sqrt{a},\\\
\phi^{*}H_{{\varepsilon}+\eta}-H_{\varepsilon}&\text{ if
}\sigma(\sqrt{a})=-\sqrt{a},\end{array}\right.$

and then take its differential to get

$(\sigma_{1},\sigma_{2})\mapsto\left\\{\begin{array}[]{ll}\phi^{*}H_{{\varepsilon}+\eta}+\sigma_{1}(\phi^{*}H_{{\varepsilon}+\eta})-2H_{\varepsilon}&\text{
if }\sigma_{1}(\sqrt{a})=\sigma_{2}(\sqrt{a})=-\sqrt{a},\\\ {\rm id}&\text{
otherwise. }\end{array}\right.$

For any $\sigma_{1}\in\operatorname{Gal}(\overline{k}/k)$ with
$\sigma_{1}(\sqrt{a})=-\sqrt{a}$ we have
$\sigma_{1}(\phi)=\iota_{{\varepsilon}+\eta}\circ\phi\circ\iota_{\varepsilon}$
and hence
$\sigma_{1}(\phi^{*}H_{{\varepsilon}+\eta})=\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})$.
Thus there exists a $k$-rational function $g$ on ${\mathcal{J}}_{\varepsilon}$
as specified in (ii), and we can take $c_{v}\in
H^{2}(k_{v},\overline{k}_{v}^{\times})=\operatorname{Br}(k_{v})$ to be the
class of

$(\sigma_{1},\sigma_{2})\mapsto\left\\{\begin{array}[]{cl}g(P_{v})&\text{ if
}\sigma_{1}(\sqrt{a})=\sigma_{2}(\sqrt{a})=-\sqrt{a},\\\ 1&\text{ otherwise,
}\end{array}\right.$

where $P_{v}\in{\mathcal{J}}_{\varepsilon}(k_{v})$ is any local point avoiding
the zeros and poles of $g$.

By [Se, Chapter XIV, Section 2, Proposition 5] the element
$c_{v}\in\operatorname{Br}(k_{v})$ is represented by the quaternion algebra
$(a,g(P_{v}))$. In particular its local invariant is $0$ or $1/2$ according as
the Hilbert symbol $(a,g(P_{v}))_{v}$ is $1$ or $-1$. By abuse of notation, we
let the Hilbert symbol take values in ${\mathbb{F}}_{2}$. The Cassels-Tate
pairing is then given by summing the Hilbert symbols $(a,g(P_{v}))_{v}$ as
claimed. $\Box$

###  3.3. Remnants of the group law

In this section we explain how the rational function $g$ in Theorem 3.5 can be
computed by specialising a certain $(2,2,2)$-form. In later sections we
explain how to compute this $(2,2,2)$-form.

The Kummer surface ${\mathcal{K}}={\mathcal{J}}/[-1]$ retains some remnants of
the group law on ${\mathcal{J}}$. As usual we embed
${\mathcal{K}}\subset{\mathbb{P}}^{3}$ with coordinates $x_{1},\ldots,x_{4}$.
Flynn [F2] computed biquadratic forms $\Phi_{ij}$ such that for all
$P,Q\in{\mathcal{J}}$ we have

(44) $\displaystyle\Phi_{ij}(x_{1}(P),$
$\displaystyle\ldots,x_{4}(P);x_{1}(Q),\ldots,x_{4}(Q))$
$\displaystyle\propto\big{(}x_{i}(P+Q)x_{j}(P-Q)+x_{i}(P-Q)x_{j}(P+Q)\big{)}$

where the constant of proportionality is independent of $1\leqslant
i,j\leqslant 4$. We package these biquadratic forms as a $(2,2,2)$-form on
${\mathbb{P}}^{3}\times{\mathbb{P}}^{3}\times({\mathbb{P}}^{3})^{\vee}$.
Explicitly, we put ${\mathbf{x}}=(x_{1},\ldots,x_{4})$,
${\mathbf{y}}=(y_{1},\ldots,y_{4})$, ${\mathbf{z}}=(z_{1},\ldots,z_{4})$ and
define

$\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=\sum_{i,j=1}^{4}\Phi_{ij}({\mathbf{x}},{\mathbf{y}})z_{i}z_{j}.$

Since the $\Phi_{ij}$ are symmetric, i.e.,
$\Phi_{ij}({\mathbf{x}},{\mathbf{y}})=\Phi_{ij}({\mathbf{y}},{\mathbf{x}})$,
we have
$\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=\Phi({\mathbf{y}},{\mathbf{x}},{\mathbf{z}})$.
If $P,Q\in{\mathcal{J}}$ then by (44) we have

(45) $\displaystyle\Phi(x_{1}(P),\ldots,$ $\displaystyle
x_{4}(P);x_{1}(Q),\ldots,x_{4}(Q);c_{1},c_{2},c_{3},c_{4})$
$\displaystyle\propto\big{(}\textstyle\sum_{i=1}^{4}c_{i}x_{i}(P+Q)\big{)}\big{(}\textstyle\sum_{j=1}^{4}c_{j}x_{j}(P-Q)\big{)}.$

where the constant of proportionality is independent of $c_{1},\ldots,c_{4}$.
We note for later use the immediate consequence that for fixed
$Q\in{\mathcal{J}}$ and $(c_{1}:\ldots:c_{4})\in({\mathbb{P}}^{3})^{\vee}$ the
left hand side of (45) is a non-zero function of $P$.

Just as the $(2,2,2)$-form $\Phi$ corresponds to the group law on
${\mathcal{J}}$, there is a twisted $(2,2,2)$-form $\Phi_{{\varepsilon},\eta}$
corresponding to the twisted group law

$\mu:{\mathcal{J}}_{\varepsilon}\times{\mathcal{J}}_{\eta}\to{\mathcal{J}}_{{\varepsilon}+\eta}$

defined in Lemma 3.7. The analogue of (45) is that if
$P\in{\mathcal{J}}_{\varepsilon}$ and $Q\in{\mathcal{J}}_{\eta}$ then

(46) $\displaystyle\Phi_{{\varepsilon},\eta}(x_{1}(P),\ldots,x_{4}$
$\displaystyle(P);x_{1}(Q),\ldots,x_{4}(Q);c_{1},c_{2},c_{3},c_{4})$
$\displaystyle\propto\big{(}\textstyle\sum_{i=1}^{4}c_{i}x_{i}(\mu(P,Q))\big{)}\big{(}\textstyle\sum_{j=1}^{4}c_{j}x_{j}(\mu(P,\iota_{\eta}(Q)))\big{)}.$

The next theorem computes the rational function $g$ in Theorem 3.5.

######  Theorem 3.9.

Let $Q\in{\mathcal{J}}_{\eta}$ and let
$\phi=\mu(-,Q):{\mathcal{J}}_{\varepsilon}\to{\mathcal{J}}_{{\varepsilon}+\eta}$.
Let $H_{\varepsilon}$ and $H_{{\varepsilon}+\eta}$ be the divisors on
${\mathcal{J}}_{\varepsilon}$ and ${\mathcal{J}}_{{\varepsilon}+\eta}$
obtained by pulling back the hyperplane sections $\\{x_{1}=0\\}$ and
$\\{c_{1}x_{1}+\ldots+c_{4}x_{4}=0\\}$ on
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$
and
${\mathcal{J}}_{{\varepsilon}+\eta}\to{\mathcal{K}}_{{\varepsilon}+\eta}\subset{\mathbb{P}}^{3}$.
Then

$g=\Phi_{{\varepsilon},\eta}(x_{1},\ldots,x_{4};x_{1}(Q),\ldots,x_{4}(Q);c_{1},c_{2},c_{3},c_{4})/x_{1}^{2}$

is a rational function on ${\mathcal{J}}_{\varepsilon}$ with divisor

(47)
$\operatorname{div}(g)=\phi^{*}H_{{\varepsilon}+\eta}+\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})-2H_{\varepsilon}.$

###### Proof.

Let $P\in{\mathcal{J}}_{\varepsilon}$ and
$P^{\prime}=\phi(P)=\mu(P,Q)\in{\mathcal{J}}_{{\varepsilon}+\eta}$. If
$c_{1}x_{1}(P^{\prime})+\ldots+c_{4}x_{4}(P^{\prime})=0$ then by (46) we have
$g(P)=0$. Therefore $g$ vanishes on $\phi^{*}H_{{\varepsilon}+\eta}$. Since
$g$ factors via the double cover
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}$ it also vanishes
on $\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})$.

By the observation following (45) we know that $g$ does not vanish identically
on ${\mathcal{J}}_{\varepsilon}$. Assuming that
$\phi^{*}H_{{\varepsilon}+\eta}$ and
$\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})$ are distinct and
irreducible it follows that

(48)
$\operatorname{div}(g)=\phi^{*}H_{{\varepsilon}+\eta}+\iota_{\varepsilon}^{*}(\phi^{*}H_{{\varepsilon}+\eta})+E-2H_{\varepsilon}$

for some effective divisor $E$. By considering the quotient of $g$ and the
rational function of the same name in Theorem 3.5(ii), it follows that $E=0$.

This completes the proof under the assumption in the last paragraph. We showed
in [Y1, Section 5.2.3] that the assumption is satisfied for generic choices of
$Q\in{\mathcal{J}}$ and $(c_{1}:\ldots:c_{4})\in({\mathbb{P}}^{3})^{\vee}$,
and from this we were able to deduce that the theorem holds in general. ∎

In [Y1] the second author computed the twisted $(2,2,2)$-form
$\Phi_{{\varepsilon},\eta}$ from the untwisted $(2,2,2)$-form $\Phi$ by
directly making the changes of coordinates $\psi_{\varepsilon}$, $\psi_{\eta}$
and $\psi_{{\varepsilon}+\eta}$ as defined in (36). Since these $4\times 4$
matrices are typically defined over (different) degree $16$ number fields,
this method is rather slow. Our improved method for computing the twisted
Kummer surfaces (see Section 2.6) also means we do not have these matrices to
hand, although it would be possible to compute them in hindsight by matching
up the nodes on ${\mathcal{K}}$ and ${\mathcal{K}}_{\varepsilon}$.

In Section 3.6 we give a formula, analogous to that in [Fi2], that enables us
to compute the twisted $(2,2,2)$-form $\Phi_{{\varepsilon},\eta}$ without
working in field extensions of such large degree.

###  3.4. The Heisenberg group

In this section we describe some of the geometry behind our method for
computing the $(2,2,2)$-forms.

######  Definition 3.10.

The standard level $2$ Heisenberg group is the subgroup
$H_{2}\subset\operatorname{GL}_{4}$ generated by the matrices

$\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{pmatrix},\,\,\,\,\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix},\,\,\,\,\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix},\,\,\,\,\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}.$

It is a non-abelian group of order $2^{5}$ and sits in an exact sequence

$0\to\\{\pm 1\\}\to H_{2}\to\overline{H}_{2}\to 0,$

where $\overline{H}_{2}\cong({\mathbb{Z}}/2{\mathbb{Z}})^{4}$ is the image of
$H_{2}$ in $\operatorname{PGL}_{4}$.

######  Lemma 3.11.

Let ${\mathcal{K}}\subset{\mathbb{P}}^{3}$ be a Kummer surface defined over an
algebraically closed field of characteristic not $2$. We may change
coordinates on ${\mathbb{P}}^{3}$ so that

  1. (i)

the isomorphisms ${\mathcal{K}}\to{\mathcal{K}}$ corresponding to a
$(2,4)$-partition of the Weierstrass points (equivalently to translation by
some $0\not=T\in{\mathcal{J}}[2]$) are given by the non-trivial elements of
$\overline{H}_{2}$, and

  2. (ii)

the isomorphisms ${\mathcal{K}}\to{\mathcal{K}}^{\vee}$ corresponding to a
$(1,5)$, respectively $(3,3)$, partition of the Weierstrass points are given
by the skew-symmetric, respectively symmetric, matrices in $\overline{H}_{2}$.

###### Proof.

The existence of a change of coordinates satisfying (i) is a special case of
[BL, Example 6.7.4]. Since all the matrices are orthogonal, the conclusion in
(i) for ${\mathcal{K}}$ also holds for the dual Kummer ${\mathcal{K}}^{\vee}$.
The claims in (ii) follow by Remark 2.1 and the fact that $\overline{H}_{2}$
is its own centraliser inside $\operatorname{PGL}_{4}$. ∎

######  Lemma 3.12.

Let ${\mathcal{K}}\subset{\mathbb{P}}^{3}$ be a Kummer surface defined over a
field of characteristic not $2$. Let $B_{1},\ldots,B_{10}$ be $4\times 4$
symmetric matrices whose images in $\operatorname{PGL}_{4}$ correspond to the
$(3,3)$-partitions of the Weierstrass points. If we define quadratic forms by
$Q_{i}(x_{1},\ldots,x_{4})=\frac{1}{2}{\mathbf{x}}^{T}B_{i}{\mathbf{x}}$ and
$Q_{i}^{*}(y_{1},\ldots,y_{4})=\frac{1}{2}{\mathbf{y}}^{T}B_{i}^{-1}{\mathbf{y}}$
then

(49)
$\sum_{i=1}^{10}Q_{i}(x_{1},\ldots,x_{4})Q_{i}^{*}(y_{1},\ldots,y_{4})=\left(\sum_{j=1}^{4}x_{j}y_{j}\right)^{2}.$

###### Proof.

The conclusions of the lemma are unchanged if we extend our field, rescale the
matrices $B_{i}$, or make a change of coordinates on ${\mathbb{P}}^{3}$. By
Lemma 3.11 we may therefore assume that the $B_{i}$ are (lifts to
$\operatorname{GL}_{4}$ of) the symmetric matrices in $\overline{H}_{2}$. The
left hand side of (49) is now

$\displaystyle\frac{1}{4}\sum_{u,v\in\\{\pm 1\\}}(x_{1}^{2}+ux_{2}^{2}$
$\displaystyle+vx_{3}^{2}+uvx_{4}^{2})(y_{1}^{2}+uy_{2}^{2}+vy_{3}^{2}+uvy_{4}^{2})$
$\displaystyle+\sum_{u\in\\{\pm
1\\}}(x_{1}x_{2}+ux_{3}x_{4})(y_{1}y_{2}+uy_{3}y_{4})$
$\displaystyle+\sum_{u\in\\{\pm
1\\}}(x_{1}x_{3}+ux_{2}x_{4})(y_{1}y_{3}+uy_{2}y_{4})$
$\displaystyle+\sum_{u\in\\{\pm
1\\}}(x_{1}x_{4}+ux_{2}x_{3})(y_{1}y_{4}+uy_{2}y_{3}).$

Expanding this out gives the right hand side of (49). ∎

######  Lemma 3.13.

The space of quartic forms invariant under the natural action of the standard
level $2$ Heisenberg group $H_{2}$ is $5$-dimensional, spanned by

$\displaystyle x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4},\quad
x_{1}^{2}x_{2}^{2}+x_{3}^{2}x_{4}^{2},\quad
x_{1}^{2}x_{3}^{2}+x_{2}^{2}x_{4}^{2},\quad
x_{1}^{2}x_{4}^{2}+x_{2}^{2}x_{3}^{2},\quad x_{1}x_{2}x_{3}x_{4}.$

Up to scalars, there are exactly $10$ quartic forms in this space that factor
as the square of a quadratic form. These are the squares of the quadratic
forms corresponding to the symmetric matrices in $\overline{H}_{2}$.

###### Proof.

The lemma may be proved either by a brute force calculation, or by decomposing
the 2nd and 4th symmetric powers of the standard representation of $H_{2}$
into irreducible representations. ∎

###  3.5. A formula for the untwisted $(2,2,2)$-form

As usual let ${\mathcal{C}}$ be the genus $2$ curve with equation
$y^{2}=f(x)$, where the coefficients of $f$ are labelled as in (3). We recall
that Flynn [F2] computed the biquadratic forms
$\Phi_{ij}({\mathbf{x}},{\mathbf{y}})$, with coefficients in
${\mathbb{Z}}[f_{0},\ldots,f_{6}]$, and in Section 3.3 we packaged these as a
$(2,2,2)$-form

(50)
$\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=\sum_{i,j=1}^{4}\Phi_{ij}({\mathbf{x}},{\mathbf{y}})z_{i}z_{j}.$

A generic calculation shows that

(51)
$\Phi({\mathbf{x}};{\mathbf{x}};z_{1},z_{2},z_{3},0)=2(z_{2}^{2}-z_{1}z_{3})F({\mathbf{x}})$

where $F$ is the equation (5) for ${\mathcal{K}}\subset{\mathbb{P}}^{3}$.

In this section we give a new formula for the $(2,2,2)$-form $\Phi$, and in
the next section we modify it to give a formula for the twisted $(2,2,2)$-form
$\Phi_{{\varepsilon},\eta}$. Both formulae involve working with the étale
algebra of degree $10$, say $L_{10}$, corresponding to the $(3,3)$-partitions
of the Weierstrass points.

######  Remark 3.14.

As explained in [St, Lemma 5.3], after possibly replacing $f(x)$ by $f(x+n)$
for some $n\in{\mathbb{Z}}$, we can take $L_{10}$ to be defined by the degree
$10$ polynomial specified in [CF, page 56]. If $L_{10}$ is a field and $f$ has
roots $\theta_{1},\ldots,\theta_{6}$ then this is the minimal polynomial of
$\theta_{1}\theta_{2}\theta_{3}+\theta_{4}\theta_{5}\theta_{6}$.

Supposing that $f$ factors as the product of two cubics, say,

(52)
$f(x)=f_{6}(x^{3}+r_{2}x^{2}+r_{1}x+r_{0})(x^{3}+s_{2}x^{2}+s_{1}x+s_{0}),$

we define a column vector

(53)
$\lambda=(\lambda_{1},\ldots,\lambda_{4})^{T}=(f_{6}(r_{0}s_{2}+r_{2}s_{0}),f_{6}(r_{0}+s_{0}),f_{6}(r_{1}+s_{1}),1)^{T}$

and a symmetric matrix

$A=f_{6}\begin{pmatrix}r_{2}-s_{2}&-r_{1}+s_{1}&r_{0}-s_{0}\\\
-r_{1}+s_{1}&r_{0}+r_{1}s_{2}-r_{2}s_{1}-s_{0}&-r_{0}s_{2}+r_{2}s_{0}\\\
r_{0}-s_{0}&-r_{0}s_{2}+r_{2}s_{0}&r_{0}s_{1}-r_{1}s_{0}\end{pmatrix}.$

######  Lemma 3.15.

If $f$ factors as in (52) as the product of two cubic polynomials, then the
$(2,2,2)$-form $\Phi$ has the following properties.

  1. (i)

$\Phi({\mathbf{x}},{\mathbf{y}},\lambda)=({\mathbf{x}}^{T}B{\mathbf{y}})^{2}$
where

$B=\lambda\lambda^{T}+\begin{pmatrix}\operatorname{Adj}A&0\\\
0&0\end{pmatrix}.$

  2. (ii)

If we put $b_{ii}=B_{ii}$ and $b_{ij}=2B_{ij}$ for $i\not=j$ then

$b_{ij}b_{kl}=\sum_{m\leqslant n}{\rm
coeff}(\Phi_{mn}|x_{i}x_{j}y_{k}y_{l})b_{mn}$

for all $1\leqslant i,j,k,l\leqslant 4$.

###### Proof.

We checked this by computer algebra, using Flynn’s formulae for the
biquadratic forms $\Phi_{ij}({\mathbf{x}},{\mathbf{y}})$ specialised to $f$ of
the form (52). ∎

######  Remark 3.16.

The $b_{ij}$ are unchanged when we swap over the two cubic factors in (52),
and are therefore elements of $L_{10}$. In fact they form a basis. Lemma
3.15(iii) then says that the $10^{3}$ coefficients of the $(2,2,2)$-form
$\Phi$ are the structure constants for the degree $10$ étale algebra $L_{10}$
with respect to this basis.

######  Proposition 3.17.

Let $\lambda_{1},\lambda_{2},\lambda_{3}\in L_{10}$ be given by (53). Then the
quartic form

$P(x_{1},\ldots,x_{4})=2(\lambda_{2}^{2}-\lambda_{1}\lambda_{3})F({\mathbf{x}})+2\sum_{i=1}^{3}\lambda_{i}\Phi_{i4}({\mathbf{x}},{\mathbf{x}})+\Phi_{44}({\mathbf{x}},{\mathbf{x}})$

factors as $P=Q^{2}$ where $Q\in L_{10}[x_{1},x_{2},x_{3},x_{4}]$ is a
quadratic form. If we scale $Q$ so that the coefficient of $x_{4}^{2}$ is $1$
and write $Q^{*}$ for the dual quadratic form (in the sense of Lemma 3.12)
then the $(2,2,2)$-form is given by

$\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=\operatorname{Tr}_{L_{10}/k}(Q({\mathbf{x}})Q({\mathbf{y}})Q^{*}({\mathbf{z}})).$

###### Proof.

By (50) and (51) we have
$\Phi({\mathbf{x}},{\mathbf{x}},\lambda)=P(x_{1},\ldots,x_{4})$. By Lemma
3.15(i) this factors as $P=Q^{2}$ where $Q({\mathbf{x}})=\sum_{i\leqslant
j}b_{ij}x_{i}x_{j}$ and the coefficient of $x_{4}^{2}$ is $1$. By Lemma
3.15(ii) we have

(54) $Q({\mathbf{x}})Q({\mathbf{y}})=\left(\sum_{i\leqslant
j}b_{ij}x_{i}x_{j}\right)\left(\sum_{k\leqslant
l}b_{kl}y_{k}y_{l}\right)=\sum_{m\leqslant
n}\Phi_{mn}({\mathbf{x}},{\mathbf{y}})b_{mn}.$

On the other hand by Lemma 3.12 we have

(55)
$\operatorname{Tr}_{L_{10}/k}(b_{mn}Q^{*}({\mathbf{z}}))=\left\\{\begin{array}[]{ll}z_{m}z_{n}&\text{
if }m=n,\\\ 2z_{m}z_{n}&\text{ if }m\not=n.\end{array}\right.$

Combining (54) and (55) gives

$\operatorname{Tr}_{L_{10}/k}(Q({\mathbf{x}})Q({\mathbf{y}})Q^{*}({\mathbf{z}}))=\sum_{m,n=1}^{4}\Phi_{mn}({\mathbf{x}},{\mathbf{y}})z_{m}z_{n}=\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}}).\qed$

######  Remark 3.18.

As noted in [CF, page 24], the doubling map on the Kummer surface
${\mathcal{K}}=\\{F=0\\}\subset{\mathbb{P}}^{3}$ is given by

$(x_{1}:x_{2}:x_{3}:x_{4})\mapsto(\Phi_{14}({\mathbf{x}},{\mathbf{x}}):\Phi_{24}({\mathbf{x}},{\mathbf{x}}):\Phi_{34}({\mathbf{x}},{\mathbf{x}}):\tfrac{1}{2}\Phi_{44}({\mathbf{x}},{\mathbf{x}})).$

The quartic form $P$ in Proposition 3.17 is a linear combination of the
Heisenberg invariant quartics
$F,\Phi_{14}({\mathbf{x}},{\mathbf{x}}),\ldots,\Phi_{44}({\mathbf{x}},{\mathbf{x}})$
that factors as the square of a quadratic form. By Lemma 3.13 this quadratic
form $Q$ corresponds to a $(3,3)$-partition of the Weierstrass points. In
particular, an alternative way to compute $Q$ is as the quadratic form defined
by the symmetric matrix (14).

###  3.6. A formula for the twisted $(2,2,2)$-form

In the last section we gave a formula for the untwisted $(2,2,2)$-form $\Phi$.
We now modify this to compute the twisted $(2,2,2)$-form
$\Phi_{{\varepsilon},\eta}$.

######  Proposition 3.19.

Let $\lambda_{1},\ldots,\lambda_{4}\in L_{10}$ be given by (53). Let
${\varepsilon}\in S^{(2)}({\mathcal{J}}/k)$ be represented by a model $(G,H)$,
and let $F_{0},\ldots,F_{4}\in k[x_{1},\ldots,x_{4}]$ be the associated
quartic forms, as defined in Section 2.6. Then the quartic form

(56)
$P_{\varepsilon}(x_{1},\ldots,x_{4})=(\lambda_{2}^{2}-\lambda_{1}\lambda_{3})F_{0}(x_{1},\ldots,x_{4})+\sum_{i=1}^{4}\lambda_{i}F_{i}(x_{1},\ldots,x_{4})$

factors as

(57)
$P_{\varepsilon}(x_{1},\ldots,x_{4})=\alpha_{\varepsilon}\,Q_{\varepsilon}(x_{1},\ldots,x_{4})^{2}$

where $\alpha_{\varepsilon}\in L_{10}^{\times}$ and $Q_{\varepsilon}\in
L_{10}[x_{1},\ldots,x_{4}]$ is a quadratic form.

###### Proof.

For the proof we are free to extend our field $k$. Since the $F_{i}$ are
covariants we may then reduce to the case where $H$ is as given in Section
2.2. In this case we find that $F_{0},\ldots,F_{4}$ are (up to an overall
scaling) the quartics

$2F,\,\,2\Phi_{14}({\mathbf{x}},{\mathbf{x}}),\,\,2\Phi_{24}({\mathbf{x}},{\mathbf{x}}),\,\,2\Phi_{34}({\mathbf{x}},{\mathbf{x}}),\,\,\Phi_{44}({\mathbf{x}},{\mathbf{x}}).$

We are done by the first part of Proposition 3.17. ∎

######  Theorem 3.20.

Let ${\varepsilon},\eta\in S^{(2)}({\mathcal{J}}/k)$. The twisted
$(2,2,2)$-form is given by

$\Phi_{{\varepsilon},\eta}({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=\operatorname{Tr}_{L_{10}/k}(\mu
Q_{\varepsilon}({\mathbf{x}})Q_{\eta}({\mathbf{y}})Q_{{\varepsilon}+\eta}^{*}({\mathbf{z}}))$

for some $s\in k^{\times}$ and $\mu\in L_{10}^{\times}$ satisfying
$\alpha_{\varepsilon}\alpha_{\eta}/\alpha_{{\varepsilon}+\eta}=s\mu^{2}$.

###### Proof.

Let $\psi_{\varepsilon}\in\operatorname{GL}_{4}(\overline{k})$ be the matrix
in (36) scaled so that the quartic forms in Propositions 3.17 and 3.19 are
related by $P\circ\psi_{\varepsilon}=P_{\varepsilon}$. The corresponding
quadratic forms are then related by
$Q\circ\psi_{\varepsilon}=\sqrt{\alpha_{\varepsilon}}\,Q_{\varepsilon}$. We
have the same relations with ${\varepsilon}$ replaced by $\eta$ or
${\varepsilon}+\eta$.

We write $\operatorname{Tr}$ for the trace from
$\overline{L}_{10}=L_{10}\otimes_{k}\overline{k}$ to $\overline{k}$. First by
definition and then using Proposition 3.17, the twisted $(2,2,2)$-form
$\Phi_{{\varepsilon},\eta}$ is a scalar multiple of

$\displaystyle\Phi(\psi_{\varepsilon}\,{\mathbf{x}},\psi_{\eta}\,{\mathbf{y}},\psi_{{\varepsilon}+\eta}^{-T}\,{\mathbf{z}})$
$\displaystyle=\operatorname{Tr}\big{(}(Q\circ\psi_{\varepsilon})({\mathbf{x}})\,(Q\circ\psi_{\eta})({\mathbf{y}})\,(Q\circ\psi_{{\varepsilon}+\eta})^{*}({\mathbf{z}})\big{)}$
$\displaystyle=\operatorname{Tr}\big{(}\sqrt{\alpha_{\varepsilon}\alpha_{\eta}/\alpha_{{\varepsilon}+\eta}}\,Q_{\varepsilon}({\mathbf{x}})\,Q_{\eta}({\mathbf{y}})\,Q_{{\varepsilon}+\eta}^{*}({\mathbf{z}})\big{)}.$

Viewing the trace as a sum over conjugates, this last formula writes
$\Phi_{{\varepsilon},\eta}$ as a linear combination of 10 $(2,2,2)$-forms. It
may be checked using Lemma 3.11 that these 10 forms are linearly independent.
Since $\Phi_{{\varepsilon},\eta}$ is defined over $k$ this forces
$\sqrt{\alpha_{\varepsilon}\alpha_{\eta}/\alpha_{{\varepsilon}+\eta}}=s_{0}\mu$
for some $s_{0}\in\overline{k}^{\times}$ and $\mu\in L_{10}^{\times}$.
Squaring both sides then shows that $s_{0}^{2}\in k$. ∎

######  Remark 3.21.

Theorem 3.20 is not always sufficient to compute the twisted $(2,2,2)$-form
since the equation
$\alpha_{\varepsilon}\alpha_{\eta}/\alpha_{{\varepsilon}+\eta}=s\mu^{2}$ does
not determine $s$ and $\mu$ uniquely. This is not a problem if for example
$\operatorname{Gal}(f)=S_{6}$. In this case $L_{10}$ is a field, and it has no
quadratic subfields. It follows that $\mu$ is uniquely determined up to
multiplication by an element of $k^{\times}$. This is sufficient for computing
the twisted $(2,2,2)$-form, again up to multiplication by an element of
$k^{\times}$, but as explained in Remark 3.6(ii) this scaling is not
important.

In order to handle arbitrary Galois actions on the Weierstrass points it
remains to compute $\mu$ in a more systematic way. Our answer depends on the
scaling of the quadratic forms $Q_{\varepsilon}$, $Q_{\eta}$ and
$Q_{{\varepsilon}+\eta}$, and indeed such a dependence is necessary since in
(57) we are free to multiply $Q_{\varepsilon}$ by any element of
$\mu_{2}(L_{10})$ without changing $\alpha_{\varepsilon}$.

Let $W$ be the set of Weierstrass points, and $\Gamma$ the set of
$(3,3)$-partitions of $W$. Let
$X=\\{x\in\operatorname{Map}(W,\mu_{2}):\prod_{\theta\in W}x(\theta)=1\\}$.
There is a commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{{\mathcal{J}}[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}(\Gamma,\mu_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}(\Gamma,\mu_{2})/\mu_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the middle vertical map is given by
$\iota(x)(\gamma_{1}|\gamma_{2})=\prod_{\theta\in\gamma_{1}}x(\theta).$ Taking
Galois cohomology gives another commutative diagram with exact rows

(58)
$\textstyle{k^{\times}/(k^{\times})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(k,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{H^{1}(k,{\mathcal{J}}[2])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup
c}$$\textstyle{\operatorname{Br}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{\times}/(k^{\times})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{10}^{\times}/(L_{10}^{\times})^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(k,\mu_{2}(\overline{L}_{10})/\mu_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Br}(k)}$

We identify

$H^{1}(k,X)\cong\frac{\\{(\xi,m)\in L^{\times}\times
k^{\times}|N_{L/k}(\xi)=m^{2}\\}}{\\{(\nu^{2},N_{L/k}(\nu))|\nu\in
L^{\times}\\}}.$

so that the top row (where the first map is now $r\mapsto(r,r^{3})$) gives us
the isomorphism (23).

######  Definition 3.22.

For $(\xi,m)\in L^{\times}\times k^{\times}$ with $N_{L/k}(\xi)=m^{2}$, let
$w(\xi,m)\in L_{10}=\operatorname{Map}_{k}(\Gamma,\overline{k})$ be given by

$(\theta_{1}\theta_{2}\theta_{3}|\theta_{4}\theta_{5}\theta_{6})\mapsto\xi(\theta_{1})\xi(\theta_{2})\xi(\theta_{3})+\xi(\theta_{4})\xi(\theta_{5})\xi(\theta_{6})+2m$

and likewise under all permutations of $\theta_{1},\ldots,\theta_{6}$.

######  Lemma 3.23.

The map $\iota_{*}:H^{1}(k,X)\to L_{10}^{\times}/(L_{10}^{\times})^{2}$ sends
$(\xi,m)\mapsto w(\xi,m)$, provided that $w(\xi,m)$ is a unit.

###### Proof.

Let $(\xi,m)\in L^{\times}\times k^{\times}$ with $N_{L/k}(\xi)=m^{2}$, and
let $w=w(\xi,m)$. We pick square roots $\sqrt{\xi(\theta)}$ in such a way that
$\prod_{\theta\in W}\sqrt{\xi(\theta)}=m$. We can then take
$\sqrt{\xi}\in\operatorname{Map}(W,\overline{k})=L\otimes_{k}\overline{k}$ to
be given by $\theta\mapsto\sqrt{\xi(\theta)}$, and
$\sqrt{w}\in\operatorname{Map}(\Gamma,\overline{k})=L_{10}\otimes_{k}\overline{k}$
to be given by

$(\gamma_{1}|\gamma_{2})\mapsto\prod_{\theta\in\gamma_{1}}\sqrt{\xi(\theta)}+\prod_{\theta\in\gamma_{2}}\sqrt{\xi(\theta)}$

We find that $\sigma\sqrt{w}/\sqrt{w}=\iota(\sigma\sqrt{\xi}/\sqrt{\xi})$ for
all $\sigma\in\operatorname{Gal}(\overline{k}/k)$, and from this the lemma
follows. ∎

######  Proposition 3.24.

Let ${\varepsilon}\in S^{(2)}({\mathcal{J}}/k)$ be represented by a model
$(G,H)$. Let ${\mathbf{G}}$ and ${\mathbf{H}}$ be the $6\times 6$ symmetric
matrices corresponding to $G$ and $H$, and let $S$ be the quadratic form
defined by ${\mathbf{H}}{\mathbf{G}}^{-1}{\mathbf{H}}$. Let $Z$ and $W$ be the
$4\times 4$ skew symmetric matrices defined in (19), and let $Y$ be the
analogue of $W$ where $H$ is replaced by $S$. Let

$A_{ij}(u_{0},\ldots,u_{5})=(W^{*}ZY^{*})_{ij}+(W^{*}ZY^{*})_{ji}.$

Then the forms $\Xi$ and $M$ in Lemma 2.11 satisfy

(59) $-f_{6}w(\Xi,M)=\alpha_{\varepsilon}\left(\sum c_{ij}A_{ij}\right)^{2}$

where $\alpha_{\varepsilon}$ and $Q_{\varepsilon}=\sum c_{ij}x_{i}x_{j}$ are
as given in Proposition 3.19.

###### Proof.

Since $P_{\varepsilon}\in L_{10}[x_{1},\ldots,x_{4}]$ is a quartic form, we
may write

$P_{\varepsilon}(x_{1},\ldots,x_{4})=\widetilde{P}_{{\varepsilon}}(x_{1}^{2},x_{1}x_{2},\ldots,x_{4}^{2})$

where $\widetilde{P}_{{\varepsilon}}$ is a quadratic form in $10$ variables.
This does not determine $\widetilde{P}_{{\varepsilon}}$ uniquely, but in view
of (57) there is a canonical choice, namely the one for which the associated
symmetric bilinear form sends

$(x_{1}^{2},x_{1}x_{2},\ldots,x_{4}^{2};y_{1}^{2},y_{1}y_{2},\ldots,y_{4}^{2})\mapsto\alpha_{\varepsilon}Q_{\varepsilon}(x_{1},\ldots,x_{4})Q_{\varepsilon}(y_{1},\ldots,y_{4}).$

The proof of (59) is now reduced to showing that

$-f_{6}w(\Xi,M)=\widetilde{P}_{{\varepsilon}}(A_{11},A_{12},\ldots,A_{44}).$

Each side is a form of degree $6$ in $u_{0},\ldots,u_{5}$ whose coefficients
have degree $15$ in the coefficients $h_{ij}$ of the model. We checked this by
computer algebra for the model $(G,H)$ in Section 2.2. The general case
follows by carefully keeping track of the effect of a change of coordinates. ∎

We solve for $s$ and $\mu$ in Theorem 3.20 as follows. First we pick
$(\xi_{\varepsilon},m_{\varepsilon})$ by specialising the forms $(\Xi,M)$ in
Lemma 2.11. Specialising (59) at the same point gives
$-f_{6}w(\xi_{\varepsilon},m_{\varepsilon})=\alpha_{\varepsilon}\chi_{\varepsilon}^{2}$
for some explicit $\chi_{\varepsilon}\in L_{10}$, which is a unit if we
specialised at a sufficiently general point. We repeat everything with
${\varepsilon}$ replaced by $\eta$ or ${\varepsilon}+\eta$. By (23) we have

(60)
$(\xi_{\varepsilon}\xi_{\eta}\xi_{{\varepsilon}+\eta},m_{\varepsilon}m_{\eta}m_{{\varepsilon}+\eta})=(r\nu^{2},r^{3}N_{L/k}(\nu)),$

for some $r\in k^{\times}$ and $\nu\in L^{\times}$. We then use the next lemma
to solve for $s$ and $\mu$ satisfying
$\alpha_{\varepsilon}\alpha_{\eta}/\alpha_{{\varepsilon}+\eta}=s\mu^{2}$.

######  Lemma 3.25.

Let $(\xi_{i},m_{i})\in L^{\times}\times k^{\times}$ with
$N_{L/k}(\xi_{i})=m_{i}$ for $i=1,2,3$. If

$(\xi_{1}\xi_{2}\xi_{3},m_{1}m_{2}m_{3})=(r\nu^{2},r^{3}N_{L/k}(\nu))$

then

$\prod_{i=1}^{3}w(\xi_{i},m_{i})=s\mu^{2}$

for some $s\in k$ and $\mu\in L_{10}$. Moreover we may take $s=r^{3}$ and

(61)
$\mu(\gamma_{1}|\gamma_{2})=\left(\prod_{\theta\in\gamma_{1}}\nu(\theta)\right)\prod_{i=1}^{3}\left(1+\frac{m_{i}}{\prod_{\theta\in\gamma_{1}}\xi_{i}(\theta)}\right).$

###### Proof.

If the $w(\xi_{i},m_{i})$ are units, then the existence of $s$ and $\mu$
follows from Lemma 3.23 and the commutative diagram (58). To prove the result
in general and to give explicit formulae we proceed as follows. By Definition
3.22 we have

$w(\xi_{i},m_{i})(\gamma_{1}|\gamma_{2})=\left(\prod_{\theta\in\gamma_{1}}\xi_{i}(\theta)\right)\left(1+\frac{m_{i}}{\prod_{\theta\in\gamma_{1}}\xi_{i}(\theta)}\right)^{2}.$

It follows that $\prod_{i=1}^{3}w(\xi_{i},m_{i})=s\mu^{2}$ where $s$ and $\mu$
are as given in the statement of the lemma. A calculation then shows that (61)
is unchanged when we swap $\gamma_{1}\leftrightarrow\gamma_{2}$ and so defines
an element of $L_{10}$. ∎

We have now reduced the problem of solving for $s$ and $\mu$ in Theorem 3.20
to that of solving for $r$ and $\nu$ in (60). This still does not have a
unique solution, but our choices for $\nu$ are more limited than they were for
$\mu$. Indeed $\nu$ is unique up to multiplying by elements of $k^{\times}$
(which do not matter) and elements of

$\frac{\\{\nu\in L^{\times}|\nu^{2}\in
k^{\times},N_{L/k}(\nu)=\nu^{6}\\}}{k^{\times}}\cong{\mathcal{J}}[2](k).$

This leaves us with $\\#{\mathcal{J}}[2](k)$ choices. Since this is the same
as the number of choices we had in defining the twisted group law (see Remark
3.8), and so in defining the twisted $(2,2,2)$-form, all of these choices
work.

##  4\. Implementation and Examples

Let ${\mathcal{C}}$ be a genus $2$ curve defined over a number field $k$, and
let ${\mathcal{J}}$ be its Jacobian. We have written a program in Magma [BCP]
for computing the Cassels-Tate pairing

$\langle~{},~{}\rangle_{{\operatorname{CT}}}:S^{(2)}({\mathcal{J}}/k)\times
S^{(2)}({\mathcal{J}}/k)\to{\mathbb{F}}_{2}$

in the case $k={\mathbb{Q}}$. We have also worked out some first examples when
$k$ is a small quadratic field. The main steps are as follows.

  1. (i)

We compute the fake $2$-Selmer group $S_{\rm fake}^{(2)}({\mathcal{J}}/k)$
using the Magma function TwoSelmerGroup. This calls programs of Stoll (when
$k={\mathbb{Q}}$) and Bruin (when $k\not={\mathbb{Q}}$). Our insistence that
${\mathcal{C}}$ has an equation $y^{2}=f(x)$ where $f$ has degree $6$ can slow
these programs down, so in the presence of a rational Weierstrass point it may
be better to work with a degree $5$ model and then convert the fake $2$-Selmer
group elements back to a degree $6$ model.

  2. (ii)

We compute the $2$-Selmer group $S^{(2)}({\mathcal{J}}/k)$ from the fake
Selmer group, representing its elements as pairs $(\xi,m)$ where
$N_{L/k}(\xi)=m^{2}$ as in (23). Although this is simply a matter of
extracting square roots of the norms, we must be careful to keep track of the
group structure.

  3. (iii)

We convert each $2$-Selmer group element $(\xi,m)$ to a model $(G,H)$ as
described in Section 2.4. This involves solving for a $3$-dimensional
isotropic subspace of a quadratic form in $6$ variables. When $k={\mathbb{Q}}$
we use the existing function IsotropicSubspace in Magma [BCP]. This uses
techniques of minimisation and reduction based on a paper of Simon [Si]. When
$k$ is a small quadratic field we find that similar ideas work reasonably well
in practice.

  4. (iv)

We use the action of $\operatorname{PGL}_{4}(k)$ on the space of models to
simplify our list of models. This is achieved by a combination of minimisation
and reduction. The algorithm for minimisation will be described in joint work
of the first author and Mengzhen (August) Liu [FL], based on a summer project
in 2022. See Remark 4.3 below for details of the method we used for reduction.

  5. (v)

We select elements ${\varepsilon},\eta\in S^{(2)}({\mathcal{J}}/k)$ and aim to
compute $\langle{\varepsilon},\eta\rangle_{{\operatorname{CT}}}$. If the value
of the pairing can be inferred from earlier values computed, or from the
properties of the pairing listed at the start of Section 3 (specifically, it
is symmetric, bilinear and satisfies Lemma 3.1), then there is no need for the
following steps (other than for testing purposes). The deficient places may be
computed using the algorithm in [St, Section 7], as implemented in the Magma
function IsDeficient.

  6. (vi)

Starting from our models representing ${\varepsilon}$, $\eta$ and
${\varepsilon}+\eta$ we use (18) to compute equations for the twisted Kummer
surfaces ${\mathcal{K}}_{\varepsilon}$, ${\mathcal{K}}_{\eta}$ and
${\mathcal{K}}_{{\varepsilon}+\eta}$. As described in Section 2.6, computing
$2\times 2$ minors instead of $3\times 3$ minors (and scaling by $-f_{6}$)
gives us pushout forms for ${\mathcal{K}}_{\varepsilon}$ and
${\mathcal{K}}_{\eta}$, i.e., quartic forms that when set equal to a square
define the double covers
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}$ and
${\mathcal{J}}_{\eta}\to{\mathcal{K}}_{\eta}$. Since we need them in Step
(viii) below, we also compute the covariants describing the covering maps
${\mathcal{K}}_{\varepsilon}\to{\mathcal{K}}$,
${\mathcal{K}}_{\eta}\to{\mathcal{K}}$ and
${\mathcal{K}}_{{\varepsilon}+\eta}\to{\mathcal{K}}$.

  7. (vii)

We search for $k$-points on ${\mathcal{K}}_{\eta}\subset{\mathbb{P}}^{3}$.
When $k={\mathbb{Q}}$ we use the function PointSearch in Magma, slightly
modified to allow for the fact that Kummer surfaces are always singular. When
$k\not={\mathbb{Q}}$ we took an ad hoc approach (intersecting with random
lines) which could doubtless be improved.

  8. (viii)

Assuming we found a rational point $P\in{\mathcal{K}}_{\eta}(k)$, we compute
$a\in k^{\times}/(k^{\times})^{2}$ by evaluating the pushout form describing
${\mathcal{J}}_{\eta}\to{\mathcal{K}}_{\eta}$ at this point $P$. We then
compute the rational function $g$ on ${\mathcal{K}}_{\varepsilon}$ using
Theorems 3.9 and 3.20. There is no need to compute the $(2,2,2)$-form in full,
since we immediately specialise the second and third sets of variables. These
are specialised to the coordinates of
$P\in{\mathcal{K}}_{\eta}\subset{\mathbb{P}}^{3}$ and an arbitrary point on
$({\mathbb{P}}^{3})^{\vee}$. For the latter we take
$(c_{1},c_{2},c_{3},c_{4})=(1,0,0,0)$.

  9. (ix)

We compute $\langle{\varepsilon},\eta\rangle_{{\operatorname{CT}}}$ using
Theorem 3.5(iii). We are careful to pick local points on
${\mathcal{K}}_{\varepsilon}$ that lift to local points on
${\mathcal{J}}_{\varepsilon}$. The sum over all places is reduced to a finite
sum as explained in Section 4.1 below.

  10. (x)

We repeat Steps (v) to (ix) until we have computed the matrix of the Cassels-
Tate pairing $\langle~{},~{}\rangle_{{\operatorname{CT}}}$ relative to a basis
for $S^{(2)}({\mathcal{J}}/k)$ as an ${\mathbb{F}}_{2}$-vector space.

######  Remark 4.1.

For our methods to work, we do not need to find rational points on all of the
twisted Kummer surfaces. Accordingly, in Step (vii) we use a relatively small
search bound, and only return to search again if we do not succeed with other
choices of $\eta$. For this reason it is recommended to give our program the
full $2$-Selmer group as input, and not some subgroup or just a pair of
elements.

######  Remark 4.2.

The main obstacles to generalising to number fields of larger degree are the
need to carry out minimisation and reduction in Steps (iii) and (iv), and the
point search in Step (vii). We found it particularly helpful if the ring of
integers of $k$ is a Euclidean domain since then our code for minimisation
over ${\mathbb{Q}}$ (which calls the Magma function SmithForm) could be
adapted relatively easily.

######  Remark 4.3.

For simplicity we take $k={\mathbb{Q}}$. In the reduction part of Step (iv) we
are looking for a matrix in $\operatorname{GL}_{4}({\mathbb{Z}})$ representing
a change of coordinates that simplifies our equation for the twisted Kummer
surface ${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$. The same matrix,
acting as described in Definition 2.7(ii), may then be used to simplify the
model $(G,H)$. To find this matrix in $\operatorname{GL}_{4}({\mathbb{Z}})$ we
use the natural generalisation of [CFS, Corollary 6.3], namely we embed
${\overline{\mathbb{Q}}}\subset{\mathbb{C}}$ and perform lattice reduction
with respect to the Gram matrix

$\sum_{T\in{\mathcal{J}}[2]}\frac{M_{T}^{\dagger}M_{T}\,\,}{\,\,|\det
M_{T}\,|^{1/2}}$

where the matrices $M_{T}$ were computed in Remark 2.12, and the superscript
$\dagger$ denotes complex conjugate transpose. An alternative method would be
to run Magma’s ReduceCluster on the set of nodes of
${\mathcal{K}}_{\varepsilon}$, but the need to numerically solve for the nodes
makes this method less efficient.

Both of these methods are specific to curves of genus $2$. A method that works
for $2$-coverings of hyperelliptic Jacobians in general is currently being
developed by Jack Thorne [T2].

Presented with the output of a long and complicated program, it is natural to
wonder whether the answers computed are correct. We used the methods in
Section 2.5 to check that the models computed in Step (iii) do indeed
correspond to the Selmer elements claimed. For computing the pairing itself
the following checks are available. The first was the most important for
catching bugs.

  * •

In applying Theorem 3.5(iii) we may compute many $k_{v}$-points on
${\mathcal{K}}_{\varepsilon}$ and check that each makes the same local
contribution to the pairing.

  * •

In Step (viii) we may make different choices of $P$, of
$(c_{1},\ldots,c_{4})$, and of the overall scaling of $g$, and check that the
pairing does not change.

  * •

We may check that the pairing as computed is symmetric, bilinear and satisfies
Lemma 3.1.

  * •

We may check that the image of the map ${\mathcal{J}}(k)/2{\mathcal{J}}(k)\to
S^{(2)}({\mathcal{J}}/k)$ is contained in the kernel of the pairing.

  * •

We may check that our answers are consistent with Remark 3.2.

In Section 4.1 we explain how the sum over all places in our formula for the
pairing reduces to a finite sum. In Section 4.2 we give some examples over
$k={\mathbb{Q}}$ where our methods improve the upper bound for the rank of
${\mathcal{J}}({\mathbb{Q}})$ coming from $2$-descent by $2$ or more. In
Section 4.3 we explain how our methods compare to the visibility method
employed by Bruin and Stoll, and how the methods can be combined. Finally in
Section 4.4 we explain how we used our methods to unconditionally determine
the rank of every genus $2$ Jacobian in the LMFDB.

###  4.1. Controlling the bad primes

We work over a number field $k$ with ring of integers ${{\mathcal{O}}_{k}}$.
Let ${\mathcal{C}}$ be a genus $2$ curve with equation
$y^{2}=f(x)=f_{6}x^{6}+f_{5}x^{5}+\ldots+f_{1}x+f_{0}$ where the $f_{i}$ are
in ${{\mathcal{O}}_{k}}$. Let $S_{0}$ be a finite set of places of $k$
containing all the infinite places and all the primes dividing $2f_{6}\,{\rm
disc}(f)$.

######  Theorem 4.4.

Let $v\not\in S_{0}$ be a place of $k$. Suppose that

  1. (i)

${\mathcal{K}}_{\varepsilon}$ arises from a model $(G,H)$ (see Definition 2.7)
and $H$ has $v$-adically integral coefficients.

  2. (ii)

$a\in k^{\times}$ is a $v$-adic unit.

  3. (iii)

$g=\gamma(x_{1},\ldots,x_{4})/x_{1}^{2}$ where $\gamma\in
k[x_{1},\ldots,x_{4}]$ is a quadratic form, with all coefficients $v$-adically
integral and at least one a $v$-adic unit.

Then the local contribution at $v$ in Theorem 3.5(iii) is trivial.

###### Proof.

Replacing the local field $k_{v}$ by an unramified extension of odd degree we
may assume that the order of the residue field ($q$ say) is arbitrarily large.
Under the assumptions in the statement of the theorem, the construction of
${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$ and the double cover
${\mathcal{J}}_{\varepsilon}\to{\mathcal{K}}_{\varepsilon}$ from the model
$(G,H)$ carries over to the residue field. By the Lang-Weil estimates there
are $\Omega(q^{2})$ smooth points on the reduction of
${\mathcal{K}}_{\varepsilon}$ mod $v$ that lift to ${\mathbb{F}}_{q}$-points
on the reduction of ${\mathcal{J}}_{\varepsilon}$ mod $v$. The reduction of
$\gamma$ mod $v$ cannot be identically zero on the surface (it is a quadratic
form, whereas the surface has degree $4$) and so vanishes at $O(q)$ of these
points. Since $q$ is large there are some points left over. Hensel lifting any
one of these (and noting that for $v\nmid 2\infty$ the Hilbert symbol of two
$v$-adic units is trivial) shows that the local contribution at $v$ is
trivial. ∎

###  4.2. First examples over ${\mathbb{Q}}$

Our first example was chosen as a genus $2$ curve with equation $y^{2}=f(x)$
where $f$ has small integer coefficients and Galois group $S_{6}$, and a
$2$-descent does not appear to give a sharp upper bound for the rank of the
Jacobian.

######  Example 4.5.

Let ${\mathcal{C}}$ be the genus $2$ curve with equation

$y^{2}=-3x^{6}+3x-15.$

Let $L={\mathbb{Q}}[\theta]={\mathbb{Q}}[x]/(x^{6}-x+5)$. We represent
elements of the $2$-Selmer group $S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ as
pairs $(\xi,m)\in L^{\times}\times{\mathbb{Q}}^{\times}$ where
$N_{L/{\mathbb{Q}}}(\xi)=m^{2}$. Magma computes that
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{3}$ is
generated by

$\displaystyle c$ $\displaystyle=(1,-1),$ $\displaystyle{\varepsilon}$
$\displaystyle=(\theta^{5}-3\theta^{4}-2\theta^{3}+5\theta^{2}+4\theta-9,1),$
$\displaystyle\eta$
$\displaystyle=(\theta^{5}-2\theta^{4}+2\theta^{3}-\theta^{2}+1,1).$

We find that the Selmer group elements ${\varepsilon}$, $\eta$,
${\varepsilon}+\eta$ are represented by models $(G,\frac{1}{2}H_{1})$,
$(G,\frac{1}{2}H_{2})$, $(G,\frac{1}{2}H_{3})$ where
$G=u_{0}u_{5}+u_{1}u_{4}+u_{2}u_{3}$ and

$\displaystyle H_{1}$
$\displaystyle=u_{0}u_{1}-2u_{0}u_{2}-u_{1}u_{2}+u_{1}u_{3}-2u_{1}u_{5}-2u_{2}u_{5}-2u_{3}u_{4}-2u_{4}u_{5}+2u_{5}^{2},$
$\displaystyle H_{2}$
$\displaystyle=u_{0}u_{2}-2u_{0}u_{3}-2u_{1}^{2}-u_{1}u_{2}+u_{2}^{2}-u_{2}u_{4}+3u_{2}u_{5}+2u_{3}^{2}+2u_{3}u_{5}-4u_{4}u_{5},$
$\displaystyle H_{3}$
$\displaystyle=-2u_{0}u_{3}+u_{1}u_{2}+u_{1}u_{4}-2u_{1}u_{5}-u_{2}u_{3}+2u_{2}u_{4}-u_{3}u_{4}+4u_{4}u_{5}+2u_{5}^{2}.$

The identity element is represented by $(G,H)$ as given in Section 2.2, and
the remaining four elements of $S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ are given
by reversing the order of the variables $u_{0},u_{1},\ldots,u_{5}$. (As noted
in Remark 2.10 this corresponds to adding $c$.) It may be checked using the
formulae in Section 2.5 that these models do indeed correspond to the pairs
$(\xi,m)$ we started with.

The procedure in Section 2.6 shows that
${\mathcal{K}}_{\varepsilon}\subset{\mathbb{P}}^{3}$ has equation
$F_{\varepsilon}=0$ where

$\displaystyle F_{{\varepsilon}}$
$\displaystyle=\,2x_{1}^{3}x_{2}+4x_{1}^{3}x_{3}+4x_{1}^{3}x_{4}-18x_{1}^{2}x_{2}^{2}+22x_{1}^{2}x_{2}x_{3}+2x_{1}^{2}x_{2}x_{4}-8x_{1}^{2}x_{3}^{2}$
$\displaystyle-4x_{1}^{2}x_{3}x_{4}-2x_{1}^{2}x_{4}^{2}+6x_{1}x_{2}^{3}+6x_{1}x_{2}^{2}x_{3}+7x_{1}x_{2}^{2}x_{4}-6x_{1}x_{2}x_{3}^{2}$
$\displaystyle-24x_{1}x_{2}x_{3}x_{4}-6x_{1}x_{2}x_{4}^{2}+4x_{1}x_{3}^{3}+6x_{1}x_{4}^{3}-x_{2}^{4}+3x_{2}^{3}x_{3}+3x_{2}^{3}x_{4}$
$\displaystyle-3x_{2}^{2}x_{3}x_{4}-5x_{2}^{2}x_{4}^{2}+2x_{2}x_{3}^{3}-2x_{2}x_{3}^{2}x_{4}+2x_{2}x_{4}^{3}-2x_{3}^{2}x_{4}^{2}+2x_{3}x_{4}^{3}+x_{4}^{4}.$

We likewise find equations for ${\mathcal{K}}_{\eta}\subset{\mathbb{P}}^{3}$
and ${\mathcal{K}}_{{\varepsilon}+\eta}\subset{\mathbb{P}}^{3}$. On
${\mathcal{K}}_{\eta}$ we find the rational point $P=(1:0:-1:-1)$ whose
inverse image in ${\mathcal{J}}_{\eta}$ is defined over
${\mathbb{Q}}(\sqrt{-3})$. Since $\operatorname{Gal}(f)=S_{6}$, computing the
twisted $(2,2,2)$-form is simplified as described in Remark 3.21. Taking
$(c_{1},c_{2},c_{3},c_{4})=(1,0,0,0)$ in Theorem 3.9 we find that
$g=\gamma(x_{1},\ldots,x_{4})/x_{1}^{2}$ where
$\gamma=12x_{1}x_{2}-2x_{2}^{2}+6x_{2}x_{3}+3x_{2}x_{4}$. According to Theorem
3.5 the Cassels-Tate pairing is given by

$\langle{\varepsilon},\eta\rangle_{\operatorname{CT}}=\sum_{v}(-3,\gamma(P_{v}))_{v}$

where for each place $v$ of ${\mathbb{Q}}$ we choose a local point
$P_{v}\in{\mathcal{K}}_{\varepsilon}({\mathbb{Q}}_{v})$ that lifts to
${\mathcal{J}}_{\varepsilon}({\mathbb{Q}}_{v})$ and satisfies
$\gamma(P_{v})\not=0$. As before, our convention is that the Hilbert symbol
$(~{},~{})_{v}$ takes values in ${\mathbb{F}}_{2}$.

By Theorem 4.4 the only places that could contribute to the pairing are the
primes dividing the discriminant $-2^{8}\cdot 3^{10}\cdot 5^{6}\cdot 7\cdot
31\cdot 43$, together with $\infty$. For each of these places $v$ we specify a
local point $P_{v}\in{\mathcal{K}}_{\varepsilon}({\mathbb{Q}}_{v})$ whose
first three coordinates are exact, and whose last coordinate is either exact
or given to sufficient precision to determine the point uniquely. It is
important to note that we have chosen local points that lift to
${\mathcal{J}}_{\varepsilon}({\mathbb{Q}}_{v})$. The third column lists
$\gamma(P_{v})$ as an element of
${\mathbb{Q}}_{v}^{\times}/({\mathbb{Q}}_{v}^{\times})^{2}$.

$\begin{array}[]{c|c|c|c}\text{ place }v&\text{ local point $P_{v}$ on
${\mathcal{K}}_{\varepsilon}$ }&\gamma(P_{v})&(-3,\gamma(P_{v}))_{v}\\\
\hline\cr 2&(3:4:1:2^{2}+O(2^{3}))&-2&1\\\ 3&(1:-1:0:-3^{2}+O(3^{3}))&1&0\\\
5&(0:1:1:1)&2&0\\\ 7&(0:1:0:1)&1&0\\\ 31&(0:1:0:1)&1&0\\\ 43&(0:1:0:1)&1&0\\\
\infty&(0:1:1:1)&1&0\end{array}$

Adding up the entries in the right hand column shows that
$\langle{\varepsilon},\eta\rangle_{{\operatorname{CT}}}\not=0$.

Since ${\mathcal{C}}$ has an even number of deficient places (these are $3$
and $\infty$) we know by Lemma 3.1 that the Cassels-Tate pairing on
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ has even rank. Since we have shown that
this pairing is non-zero, this improves the upper bound
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})\leqslant 3$ coming from
$2$-descent to ${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})\leqslant
1$.

Further calculations show that the pairing is given by

$\begin{array}[]{c|ccc}\langle~{},~{}\rangle_{{\operatorname{CT}}}&c&{\varepsilon}&\eta\\\
\hline\cr c&0&1&1\\\ {\varepsilon}&1&1&1\\\ \eta&1&1&1\end{array}$

Since the kernel of the pairing is $1$-dimensional, spanned by
${\varepsilon}+\eta$, this suggests we should look for rational points on
${\mathcal{J}}_{{\varepsilon}+\eta}$. We find that the point $(1:-2:-2:0)$ on
${\mathcal{K}}_{{\varepsilon}+\eta}$ lifts to a ${\mathbb{Q}}$-point on
${\mathcal{J}}_{{\varepsilon}+\eta}$. Using the formulae for the covering map
in Section 2.6, this point maps to the point $(124:238:199:3607)$ on
${\mathcal{K}}$, which in turn lifts to a point of infinite order in
${\mathcal{J}}({\mathbb{Q}})$, represented by a pair of points on
${\mathcal{C}}$ with $x$-coordinates satisfying $124x^{2}-238x+199=0$. It
follows that ${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=1$ and
$\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$. Our
calculations further show that the Cassels-Tate pairing on $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2]$ is non-degenerate, and hence $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2^{\infty}]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$.

Next we give an example where the rank of ${\mathcal{J}}({\mathbb{Q}})$ had
previously only been computed conditional on the Birch Swinnerton-Dyer
conjecture. We discuss further such examples in Sections 4.3 and 4.4.

######  Example 4.6.

Let ${\mathcal{C}}$ be the genus $2$ curve with equation

$y^{2}=-3x^{6}-4x^{5}-10x^{4}-51x^{2}+80x-28,$

and LMFDB label 49471.a.49471.1. Since the right hand side factors as the
product of two cubics, the canonical element of the $2$-Selmer group is
trivial. It follows by Lemma 3.1 that the Cassels-Tate pairing on
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$ is
alternating. A calculation similar to that in the previous example shows that
the pairing is non-zero. Therefore
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=0$ and $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2^{\infty}]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$.

Our next example is taken from a paper of Logan and van Luijk [LvL].

######  Example 4.7.

Let ${\mathcal{C}}$ be the genus $2$ curve with equation

$y^{2}=-6(x^{2}+1)(x^{2}-2x-1)(x^{2}+x-1).$

We have ${\mathcal{J}}({\mathbb{Q}})[2]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$.
We find that
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{4}$ is
generated by

$\displaystyle c$ $\displaystyle=(1,-1),$ $\displaystyle{\varepsilon}$
$\displaystyle=(\theta^{4}-\theta^{3}-\theta^{2}-2\theta-2,18),$
$\displaystyle\eta$ $\displaystyle=(5\theta^{3}-14\theta^{2}+1,900),$
$\displaystyle\nu$ $\displaystyle=(\theta^{4}+\theta^{3}+5\theta^{2}-3,270),$

and the Cassels-Tate pairing on $S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ is given
by

$\begin{array}[]{c|cccc}\langle~{},~{}\rangle_{{\operatorname{CT}}}&c&{\varepsilon}&\eta&\nu\\\
\hline\cr c&0&1&0&0\\\ {\varepsilon}&1&1&0&0\\\ \eta&0&0&0&0\\\
\nu&0&0&0&0\end{array}$

It follows that ${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=0$ and
$\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2^{\infty}]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$.

We were able to compute the pairing in this example despite the result of
Logan and van Luijk [LvL, Proposition 3.33] that
${\mathcal{K}}_{{\varepsilon}}({\mathbb{Q}})=\emptyset$. This illustrates the
point we made in Remark 4.1. In fact, in this example we find that of the $15$
twisted Kummer surfaces (associated to the non-zero elements of
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$) all except
${\mathcal{K}}_{{\varepsilon}}$ and
${\mathcal{K}}_{{\varepsilon}+c}\cong({\mathcal{K}}_{\varepsilon})^{\vee}$
have rational points of small height.

We end this section with an example where the pairing has higher rank.

######  Example 4.8.

Let ${\mathcal{C}}$ be the genus $2$ curve with equation

$y^{2}=f(x)=3x^{6}+10x^{5}-16x^{4}+18x^{3}+23x^{2}-54x-17.$

We have $\operatorname{Gal}(f)=S_{6}$ and so the Jacobian ${\mathcal{J}}$ has
no rational $2$-torsion. We find that
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{5}$ is
generated by

$\displaystyle c$ $\displaystyle=(1,-1),$ $\displaystyle{\varepsilon}$
$\displaystyle=(3\theta^{2}-8\theta+10,2041),$ $\displaystyle\eta$
$\displaystyle=(2\theta^{2}-4\theta+15,16657/3),$ $\displaystyle\nu$
$\displaystyle=(6\theta^{3}+14\theta^{2}-16\theta-31,37503),$
$\displaystyle\phi$
$\displaystyle=(2\theta^{4}+9\theta^{3}-3\theta^{2}-5\theta+1,2925),$

and the Cassels-Tate pairing on $S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ is given
by the rank $3$ matrix

$\begin{array}[]{c|ccccc}\langle~{},~{}\rangle_{{\operatorname{CT}}}&c&{\varepsilon}&\eta&\nu&\phi\\\
\hline\cr c&1&0&0&0&1\\\ {\varepsilon}&0&0&1&1&0\\\ \eta&0&1&0&0&1\\\
\nu&0&1&0&0&1\\\ \phi&1&0&1&1&1\\\ \end{array}$

The points $(11:-98:78:-2217)$ and $(44:72:124:-245)$ on the Kummer surface
${\mathcal{K}}$ lift to independent points of infinite order in
${\mathcal{J}}({\mathbb{Q}})$. It follows that
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=2$ and $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2^{\infty}]\cong({\mathbb{Z}}/2{\mathbb{Z}})^{3}$.

###  4.3. Examples from an experiment of Bruin and Stoll

Bruin and Stoll conducted an experiment [BS] where they attempted to determine
the existence of rational points on all genus $2$ curves
$y^{2}=f_{6}x^{6}+f_{5}x^{5}+\ldots+f_{1}x+f_{0}$ where the $f_{i}$ are
integers with $|f_{i}|\leqslant 3$. They reduced this to the determination of
the ranks of $47$ genus $2$ Jacobians, of which $36$ are expected to have rank
$0$, a further $10$ are expected to have rank $1$, and one is expected to have
rank $2$.

We start by considering one of the cases where the rank is expected to be $1$.

######  Example 4.9.

Let ${\mathcal{C}}$ be the genus $2$ curve with equation

$y^{2}=-x^{6}+2x^{5}+3x^{4}+2x^{3}-x-3.$

Magma computes that
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{3}$ is
generated by

$\displaystyle c$ $\displaystyle=(1,-1),$ $\displaystyle{\varepsilon}$
$\displaystyle=(\theta^{2}+\theta+1,4),$ $\displaystyle\eta$
$\displaystyle=(\theta^{3}-4\theta^{2}+3\theta-1,8).$

We find that the Selmer group elements ${\varepsilon}$, $\eta$,
${\varepsilon}+\eta$ are represented by the models $(G,\frac{1}{2}H_{1})$,
$(G,\frac{1}{2}H_{2})$, $(G,\frac{1}{2}H_{3})$, where
$G=u_{0}u_{5}+u_{1}u_{4}+u_{2}u_{3}$ and

$\displaystyle H_{1}$
$\displaystyle=u_{0}u_{2}-u_{0}u_{4}+2u_{1}u_{3}-2u_{1}u_{5}+2u_{2}^{2}+2u_{2}u_{3}+4u_{3}^{2}-10u_{3}u_{5}-2u_{4}u_{5},$
$\displaystyle H_{2}$
$\displaystyle=2u_{0}^{2}+2u_{0}u_{1}+2u_{0}u_{4}+4u_{1}u_{5}+2u_{2}^{2}+2u_{2}u_{3}+4u_{2}u_{5}-u_{3}u_{4}+3u_{3}u_{5},$
$\displaystyle H_{3}$
$\displaystyle=u_{0}u_{4}+2u_{1}u_{2}-2u_{1}u_{3}+4u_{1}u_{4}-2u_{2}u_{3}+8u_{3}u_{5}-u_{4}^{2}-6u_{4}u_{5}-2u_{5}^{2}.$

Using these models we compute that the Cassels-Tate pairing is given by

$\begin{array}[]{c|ccc}\langle~{},~{}\rangle_{{\operatorname{CT}}}&c&{\varepsilon}&\eta\\\
\hline\cr c&0&0&0\\\ {\varepsilon}&0&0&1\\\ \eta&0&1&0\end{array}$

Since the point $(9536:-5312:5008:53113)$ on ${\mathcal{K}}$ lifts to a point
of infinite order in ${\mathcal{J}}({\mathbb{Q}})$ it follows that
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=1$.

######  Remark 4.10.

Let $d$ be a squarefree integer. We write ${\mathcal{C}}_{d}$ for the
quadratic twist of ${\mathcal{C}}$ by $d$, and ${\mathcal{J}}_{d}$ for its
Jacobian. It is well known that

(62)
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}}(\sqrt{d}))={\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})+{\operatorname{rank}}\,{\mathcal{J}}_{d}({\mathbb{Q}}).$

Bruin (see [BF], [BS]) observed that the upper bound for the rank of
${\mathcal{J}}({\mathbb{Q}})$ coming from $2$-descent can sometimes be
improved by carrying out a $2$-descent over ${\mathbb{Q}}(\sqrt{d})$. Indeed,
this gives an upper bound for the rank of
${\mathcal{J}}({\mathbb{Q}}(\sqrt{d}))$ which, when combined with (62) and a
lower bound for the rank of ${\mathcal{J}}_{d}({\mathbb{Q}})$, gives an upper
bound for the rank of ${\mathcal{J}}({\mathbb{Q}})$. When this method succeeds
it is because there are elements in $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})[2]$ that are in the kernel of the restriction
map

$\mbox{\wncyr Sh}({\mathcal{J}}/{\mathbb{Q}})\to\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}}(\sqrt{d})).$

One way to think about this kernel is as the elements of $\mbox{\wncyr
Sh}({\mathcal{J}}/{\mathbb{Q}})$ that are visible (in the sense defined by
Mazur) in the restriction of scalars of ${\mathcal{J}}$. Accordingly Bruin
calls his method visibility.

Bruin and Stoll [BS] were already able to show that
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=1$ in Example 4.9 by
taking $d=-1$ in Remark 4.10. In fact this is just one of $5$ curves on their
list for which the same thing happens, that is, by computing the Cassels-Tate
pairing we are able to prove that
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=1$, but Bruin and Stoll
already proved this using visibility. Likewise the conclusion
${\operatorname{rank}}\,{\mathcal{J}}({\mathbb{Q}})=1$ in Example 4.5 can be
proved by taking $d=-3$ in Remark 4.10. However our method has the advantage
that we only had to carry out a $2$-descent over ${\mathbb{Q}}$ instead of
${\mathbb{Q}}(\sqrt{d})$, and so only had to verify a class group calculation
in a degree 6 rather than a degree 12 number field.

In the remaining $42$ examples we find that the Cassels-Tate pairing on
$S^{(2)}({\mathcal{J}}/{\mathbb{Q}})$ is identically zero. This is consistent
with the calculations of Bruin and Stoll, who showed in each case that
$\mbox{\wncyr Sh}({\mathcal{J}}/{\mathbb{Q}})$ has analytic order
(approximately) $16$, and so is expected to be isomorphic to
$({\mathbb{Z}}/4{\mathbb{Z}})^{2}$. Visibility was not sufficient for Bruin
and Stoll to compute the ranks in these cases, except in $4$ cases where they
obtained results conditional on GRH. However by combining visibility with
computing the Cassels-Tate pairing over a quadratic field
${\mathbb{Q}}(\sqrt{d})$, we are able to prove that in all but $4$ of the $42$
examples the Jacobian does have the expected rank. For this we took