# Minimisation of 2-coverings of genus 2 Jacobians

Tom Fisher  University of Cambridge, DPMMS, Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WB, UK<EMAIL_ADDRESS>and
Mengzhen Liu  Harvard University, Department of Mathematics, Science Center
Room 325, 1 Oxford Street, Cambridge, MA 02138, USA<EMAIL_ADDRESS>
(Date: 12th September 2023)

###### Abstract.

An important problem in computational arithmetic geometry is to find changes
of coordinates to simplify a system of polynomial equations with rational
coefficients. This is tackled by a combination of two techniques, called
minimisation and reduction. We give an algorithm for minimising certain pairs
of quadratic forms, subject to the constraint that the first quadratic form is
fixed. This has applications to 2-descent on the Jacobian of a genus $2$
curve.

##  1\. Introduction

###  1.1. Models for $2$-coverings

We work over a field $K$ with $\operatorname{char}(K)\not=2$. Let $C$ be a
smooth curve of genus $2$ with equation
$y^{2}=f(x)=f_{6}x^{6}+f_{5}x^{5}+\ldots+f_{1}x+f_{0}$ where $f\in K[x]$ is a
polynomial of degree $6$. We fix throughout the polynomial

$G=z_{12}z_{34}-z_{13}z_{24}+z_{23}z_{14}.$

The following two definitions are based on those in [FY, Section 2.4].

######  Definition 1.1.

A model (for a $2$-covering of the Jacobian of $C$) is a pair $(\lambda,H)$
where $\lambda\in K^{\times}$ and $H\in
K[z_{12},z_{13},z_{23},z_{14},z_{24},z_{34}]$ is a quadratic form satisfying

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

where $\mathbf{G}$ and $\mathbf{H}$ are the matrices of second partial
derivatives of $G$ and $H$.

We identify the space of column vectors of length $6$ and the space of
$4\times 4$ alternating matrices via the map

$A:z=\begin{pmatrix}z_{12}\\\ z_{13}\\\ z_{23}\\\ z_{14}\\\ z_{24}\\\
z_{34}\end{pmatrix}\mapsto\begin{pmatrix}0&z_{12}&z_{13}&z_{14}\\\
&0&z_{23}&z_{24}\\\
\lx@intercol\hfil\hbox{\multirowsetup\,$-$}\hfil\lx@intercol&0&z_{34}\\\
&&&0\end{pmatrix}$

so that $G(z)$ is the Pfaffian of $A(z)$. Then each $4\times 4$ matrix $P$
uniquely determines a $6\times 6$ matrix $\wedge^{2}P$ such that

$PA(z)P^{T}=A((\wedge^{2}P)z)$

for all column vectors $z$. For $F\in K[x_{1},\ldots,x_{n}]$ and
$M\in\operatorname{GL}_{n}(K)$ we write $F\circ M$ for the polynomial
satisfying $(F\circ M)(x)=F(Mx)$ for all columns vectors $x$. The Pfaffian
$\operatorname{Pf}(A)$ of an alternating matrix $A$ has the properties that
$\operatorname{Pf}(A)^{2}=\det(A)$ and $\operatorname{Pf}(PAP^{T})=(\det
P)\operatorname{Pf}(A)$. The latter tells us that $G\circ\wedge^{2}P=(\det
P)G$. It is also not hard to show that $\det(\wedge^{2}P)=(\det P)^{3}$.

######  Definition 1.2.

Two models are $K$-equivalent if they are in the same orbit for the action of
$K^{\times}\times\operatorname{PGL}_{4}(K)$ via

$(c,P):(\lambda,H)\mapsto\left(c\lambda,\frac{c}{\det
P}H\circ\wedge^{2}P\right).$

It may be checked using the above observations that this is a well defined
(right) group action on the space of models (for a fixed choice of genus $2$
curve $C$).

######  Example 1.3.

Let $C/\mathbb{Q}$ be the genus $2$ curve given by $y^{2}=f(x)$ where

$f(x)=-28x^{6}+84x^{5}-323x^{4}+506x^{3}-471x^{2}+232x-60.$

One of the elements of the 2-Selmer group of $\operatorname{Jac}C$ is
represented by the model

$\displaystyle(\lambda_{1}$
$\displaystyle,H_{1})=(42336,\,\,25128z_{12}^{2}+24480z_{12}z_{13}+14031z_{12}z_{23}+15408z_{12}z_{14}$
$\displaystyle+13959z_{12}z_{24}+25407z_{12}z_{34}+2232z_{13}^{2}-16407z_{13}z_{23}+4464z_{13}z_{14}$
$\displaystyle-22815z_{13}z_{24}+1161z_{13}z_{34}+2329z_{23}^{2}+15282z_{23}z_{14}+7687z_{23}z_{24}$
$\displaystyle-19547z_{23}z_{34}-2304z_{14}^{2}-17838z_{14}z_{24}-22590z_{14}z_{34}-134z_{24}^{2}$
$\displaystyle+41978z_{24}z_{34}-99584z_{34}^{2}).$

Applying the transformation $(c,P)$ with $c=1/3024$ and

(1) $P=\begin{pmatrix}2&-19&2&5\\\ 4&4&-31&38\\\ 2&2&37&40\\\
-7&-7&-14&7\end{pmatrix}$

gives the $\mathbb{Q}$-equivalent model

$\displaystyle(\lambda_{2}$
$\displaystyle,H_{2})=(14,\,\,z_{12}z_{23}+2z_{12}z_{14}-z_{12}z_{24}+8z_{12}z_{34}-7z_{13}^{2}-13z_{13}z_{23}$
$\displaystyle-12z_{13}z_{14}-15z_{13}z_{24}-20z_{13}z_{34}-5z_{23}^{2}-2z_{23}z_{14}-25z_{23}z_{24}$
$\displaystyle-59z_{23}z_{34}-4z_{14}^{2}-14z_{14}z_{24}-18z_{14}z_{34}+17z_{24}^{2}-37z_{24}z_{34}-11z_{34}^{2}).$

###  1.2. Relation to previous work

The change of coordinates (1) was found by a combination of two techniques,
called minimisation and reduction. Minimisation seeks to remove prime factors
from a suitably defined invariant (usually the discriminant). The prototype
example is using Tate’s algorithm to compute a minimal Weierstrass equation
for an elliptic curve. Reduction seeks to a make a final unimodular
substitution so that the coefficients are as small as possible. The prototype
example is the reduction algorithm for positive definite binary quadratic
forms.

Algorithms for minimising and reducing 2-, 3-, 4- and 5-coverings of elliptic
curves are given by Cremona, Fisher and Stoll [CFS], and Fisher [F], building
on earlier work of Birch and Swinnerton-Dyer [BSD] for $2$-coverings.
Algorithms for minimising some other representations associated to genus $1$
curves are given by Fisher and Radicevic [FR]. A general framework for
minimising hypersurfaces is described by Kollar [K], and this has been refined
by Elsenhans and Stoll [ES]; in particular they give practical algorithms for
plane curves (of arbitrary degree) and for cubic surfaces. Algorithms for
minimising Weierstrass equations for general hyperelliptic curves are given by
Q. Liu [L].

In this paper we give an algorithm for minimising $2$-coverings of genus $2$
Jacobians. These are represented by pairs of quadratic forms (see Definition
1.1) where the first quadratic form is fixed. We only consider minimisation
and not reduction, since the latter is already treated in [FY, Remark 4.3].

Our minimisation algorithm plays a key role in the work of the first author
and Jiali Yan [FY] on computing the Cassels-Tate pairing on the $2$-Selmer
group of a genus $2$ Jacobian. Indeed the method presented in loc. cit. for
computing the Cassels-Tate pairing relies on being able to find rational
points on certain twisted Kummer surfaces. Minimising and reducing our
representatives for the $2$-Selmer group elements simplifies the equations for
these surfaces, and so makes it more likely that we will be able to find such
rational points.

Earlier works on minimisation (see in particular [CFS]) considered both
minimisation theorems (i.e., general bounds on the minimal discriminant) and
minimisation algorithms (i.e., practical methods for finding a minimal model
equivalent to a given one). For $2$-coverings of hyperelliptic Jacobians, some
minimisation theorems have already been proved; see the papers of Bhargava and
Gross [BG, Section 8], and Shankar and Wang [SW, Section 2.4]. We will not
revisit these results, as our focus is on the minimisation algorithms.

######  Remark 1.4.

As noted in [CF, Lemma 17.1.1], [FH, Section 19.1] and [FY, Section 2.4] the
quadratic form $G=z_{12}z_{34}-z_{13}z_{24}+z_{23}z_{14}$ has two algebraic
families of $3$-dimensional isotropic subspaces. Moreover, the transformations
considered in Definition 1.2 do not describe the full projective orthogonal
group of $G$, but only the index $2$ subgroup that preserves (rather than
swaps over) these two algebraic families. Restricting attention to this index
$2$ subgroup (when defining equivalence) makes no difference to the
minimisation problem (see Remark 3.2), but as explained in [FY, Sections 2.4
and 2.5] it is important in the context of $2$-descent, since it means we can
distinguish between elements of the $2$-Selmer group with the same image in
the fake $2$-Selmer group.

Some Magma [BCP] code accompanying this article, including an implementation
of our algorithm, will be made available from the first author’s website.

### Acknowledgements

This work originated as a summer project carried out by the second author and
supervised by the first author. We thank the Research in the CMS Programme for
their support.

##  2\. Statement of the algorithm

We keep the notation of Section 1.1, but now let $K$ be a field with discrete
valuation $v:K^{\times}\to\mathbb{Z}$, valuation ring ${\mathcal{O}_{K}}$,
uniformiser $\pi$, and residue field $k$. If $F$ is a polynomial with
coefficients in $K$ then we write $v(F)$ for the minimum of the valuations of
its coefficients.

######  Definition 2.1.

A model $(\lambda,H)$ is integral if $v(H)\geqslant 0$. It is minimal if
$v(\lambda)$ is minimal among all $K$-equivalent integral models.

Using the action of $K^{\times}$ (see Definition 1.2) to clear denominators it
is clear that any model is $K$-equivalent to an integral model. By Definition
1.1 we have $v(\lambda)\geqslant(v(f_{6})-v(f_{i}))/(6-i)$ for all
$i=0,1,\ldots,5$. We cannot have $f_{0}=\ldots=f_{5}=0$ since $C$ is a smooth
curve of genus $2$. Therefore $v(\lambda)$ is bounded below, and minimal
models exist.

It also follows from Definition 1.1 that if
$v(f_{6})=v(\operatorname{disc}f)=0$ then any integral model $(\lambda,H)$ has
$v(\lambda)\geqslant 0$. Therefore, in global applications, minimality is
automatic at all but a finite set of primes, which we may determine by
factoring.

Returning to the local situation, there is an evident recursive algorithm for
computing minimal models if we can solve the following problem.

Minimisation problem. Given an integral quadratic form
$H\in{\mathcal{O}_{K}}[z_{12},\ldots,z_{34}]$ determine whether there exists
$P\in\operatorname{PGL}_{4}(K)$ such that

$v\left(\frac{1}{\det P}H\circ\wedge^{2}P\right)>0$

and find such a matrix $P$ if it exists.

Our solution to this problem (see Algorithm 2.4) is an iterative procedure
that computes the required transformation as a composition of simpler
transformations. These simpler transformations are either given by a matrix in
$\operatorname{GL}_{4}({\mathcal{O}_{K}})$, in which case we call the
transformation an integral change of coordinates, or given by one of the
following operations, corresponding to
$P=\operatorname{Diag}(1,1,1,\pi),\operatorname{Diag}(1,1,\pi,\pi)$ or
$\operatorname{Diag}(1,\pi,\pi,\pi)$.

######  Definition 2.2.

We define the following three operations on quadratic forms $H$:

  * •

Operation 1. Replace $H$ by $\frac{1}{\pi}H(z_{12},z_{13},z_{23},\pi
z_{14},\pi z_{24},\pi z_{34})$,

  * •

Operation 2. Replace $H$ by $H(\pi^{-1}z_{12},z_{13},z_{23},z_{14},z_{24},\pi
z_{34})$,

  * •

Operation 3. Replace $H$ by $\frac{1}{\pi}H(z_{12},z_{13},\pi
z_{23},z_{14},\pi z_{24},\pi z_{34})$,

The following algorithm suggests some transformations that we might try
applying to $H$. In applications $W\subset k^{6}$ will be a subspace
determined by the reduction of $H$ mod $\pi$. We write
$e_{12},e_{13},e_{23},e_{14},e_{24},e_{34}$ for the standard basis of $k^{6}$,
and identify the dual basis with $z_{12},z_{13},z_{23},z_{14},z_{24},z_{34}$.

######  Algorithm 2.3.

(Subalgorithm to suggest some transformations.) We take as input an integral
quadratic form $H\in{\mathcal{O}_{K}}[z_{12},\ldots,z_{34}]$ and a vector
space $W\subset k^{6}$ that is isotropic for $G$. When we make an integral
change of coordinates we apply the same transformation (or rather its
reduction mod $\pi$) to $W$. The output is either one or two transformations
$P\in\operatorname{PGL}_{4}(K)$.

  * •

If $\dim W=1$ then make an integral change of coordinates so that $W=\langle
e_{12}\rangle$. Then apply Operation 2.

  * •

If $\dim W=2$ then make an integral change of coordinates so that $W=\langle
e_{12},e_{13}\rangle$. Then apply either Operation 1 or Operation 3.

  * •

If $\dim W=3$ then either make an integral change of coordinates so that
$W=\langle e_{12},e_{13},e_{23}\rangle$ and apply Operation 1, or make an
integral change of coordinates so that $W=\langle e_{12},e_{13},e_{14}\rangle$
and apply Operation 3.

We write ${\overline{H}}\in k[z_{12},\ldots,z_{34}]$ for the reduction of $H$
mod $\pi$. If $\operatorname{char}(k)\not=2$ then the rank and kernel of
${\overline{H}}$ are defined as the rank and kernel of the corresponding
$6\times 6$ symmetric matrix. If $\operatorname{char}(k)=2$ then we assume
that $k$ is perfect, so that

${\overline{H}}=\frac{\partial{\overline{H}}}{\partial
z_{12}}=\ldots=\frac{\partial{\overline{H}}}{\partial z_{34}}=0$

defines a $k$-vector space, which we call $\ker{\overline{H}}$. We then define

$\operatorname{rank}{\overline{H}}=6-\dim\ker{\overline{H}}.$

We continue to write $G$ for the reduction of $G$ mod $\pi$, as it should
always be clear from the context which of these we mean.

######  Algorithm 2.4.

(Minimisation algorithm.) We take as input an integral quadratic form
$H\in{\mathcal{O}_{K}}[z_{12},\ldots,z_{34}]$. The output is TRUE/FALSE
according as whether there exists $P\in\operatorname{PGL}_{4}(K)$ such that

$v\left(\frac{1}{\det P}H\circ\wedge^{2}P\right)>0.$

  * Step 1.

Compute $r=\operatorname{rank}{\overline{H}}$. If $r=0$ then return TRUE.

  * Step 2.

If $r=1$ then try making an integral change of coordinates so that
${\overline{H}}=z_{34}^{2}$. If the reductions of $G$ and
$\pi^{-1}H(z_{12},\ldots,z_{24},0)$ mod $\pi$ have a common $3$-dimensional
isotropic subspace $W\subset\ker{\overline{H}}$, then (since running Algorithm
2.3 on any such subspace $W$ gives $v(H)>0$) return TRUE.

  * Step 3.

If $r=2$ then try running Algorithm 2.3 on each codimension $1$ subspace
$W\subset\ker{\overline{H}}$ that is isotropic for $G$. If one of the
suggested transformations gives $v(H)>0$ then return TRUE.

  * Step 4.

If $r\in\\{1,2\\}$ and ${\overline{H}}$ factors as a product of linear forms
defined over $k$, say ${\overline{H}}=\ell_{1}\ell_{2}$, then for each $i=1,2$
try making an integral change of coordinates so that $\ell_{i}=z_{34}$ and
then apply Operation 2. If at least one of these transformations gives
$v(H)\geqslant 0$ then select one with $\operatorname{rank}{\overline{H}}$ as
small as possible and go to Step 1.

  * Step 5.

If $r\in\\{2,3,4,5\\}$ then try running Algorithm 2.3 on
$W=\ker{\overline{H}}$ if this subspace is isotropic for $G$, and otherwise on
each codimension $1$ subspace $W\subset\ker{\overline{H}}$ that is isotropic
for $G$. If at least one of the suggested transformations gives $v(H)\geqslant
0$ then select one with $\operatorname{rank}{\overline{H}}$ as small as
possible and go to Step 1.

  * Step 6.

If this step is reached, or if after visiting Step 1 the first time and
returning to it a further 4 times we still do not have $v(H)>0$, then return
FALSE.

There is no difficulty in modifying the algorithm so that when it returns TRUE
the corresponding transformation $P\in\operatorname{PGL}_{4}(K)$ is also
returned. In Section 3 we give further details of the implementation, in
particular explaining how we make the integral changes of coordinates, and
giving further details of Step 2. In Sections 4 and 5 we prove that Algorithm
2.4 is correct.

##  3\. Remarks on implementation

In Algorithms 2.3 and 2.4 we are asked to try making various integral changes
of coordinates. It is important to realise that we are restricted to
considering matrices of the form $\wedge^{2}P$ for
$P\in\operatorname{GL}_{4}({\mathcal{O}_{K}})$, and not general elements of
$\operatorname{GL}_{6}({\mathcal{O}_{K}})$. Therefore some care is required
both in determining whether a suitable transformation exists, and in finding
one when it does.

Since the natural map
$\operatorname{GL}_{4}({\mathcal{O}_{K}})\to\operatorname{GL}_{4}(k)$ is
surjective, we may concentrate on the mod $\pi$ situation here. Notice however
that in the global application with $K=\mathbb{Q}$ and $v=v_{p}$ it is better
to use the surjectivity of
$\operatorname{SL}_{4}(\mathbb{Z})\to\operatorname{SL}_{4}(\mathbb{Z}/p\mathbb{Z})$,
so that minimisation at $p$ does not interfere with minimisation at other
primes.

Let $k^{4}$ have basis $e_{1},\ldots,e_{4}$. We identify
$\wedge^{2}k^{4}=k^{6}$ via $e_{i}\wedge e_{j}\mapsto e_{ij}$. Each linear
subspace $W\subset k^{6}$ determines a linear subspace $V_{0}\subset k^{4}$
given by

$V_{0}=\\{v\in k^{4}\mid v\wedge w=0\text{ for all }w\in W\\}$

where $\wedge$ is the natural map
$k^{4}\times\wedge^{2}k^{4}\to\wedge^{3}k^{4}$. Let $V_{1}$ be the analogue of
$V_{0}$ when $W$ is replaced by its orthogonal complement with respect to $G$.

######  Lemma 3.1.

Let $W\subset k^{6}$ be a subspace, and let $P\in\operatorname{GL}_{4}(k)$.

  1. (i)

If $\dim W=1$ then $\wedge^{2}P$ sends $W$ to $\langle e_{12}\rangle$ if and
only if $P$ sends $V_{0}$ to $\langle e_{1},e_{2}\rangle$.

  2. (ii)

If $\dim W=2$ or $3$ then $\wedge^{2}P$ sends $W$ to a subspace of $\langle
e_{12},e_{13},e_{14}\rangle$ if and only if $P$ sends $V_{0}$ to $\langle
e_{1}\rangle$.

  3. (iii)

If $\dim W=5$ then $\wedge^{2}P$ sends $W$ to $\langle
e_{12},e_{13},e_{14},e_{23},e_{24}\rangle$ if and only if $P$ sends $V_{1}$ to
$\langle e_{1},e_{2}\rangle$.

###### Proof.

In (i) we have $W=\langle e_{12}\rangle$ if and only if $V_{0}=\langle
e_{1},e_{2}\rangle$, and in (ii) we have $W\subset\langle
e_{12},e_{13},e_{14}\rangle$ if and only if $V_{0}=\langle e_{1}\rangle$.
Since the definition of $V_{0}$ in terms of $W$ behaves well under all changes
of coordinates this proves (i) and (ii). As noted in Section 1.1, all
transformations of the form $\wedge^{2}P$ preserve $G$ (up to a scalar
multiple). Therefore (iii) follows from (i) on replacing $W$ by its orthogonal
complement with respect to $G$. ∎

######  Remark 3.2.

Let $\mathbf{G}$ be the matrix of second partial derivatives of $G$, i.e., the
$6\times 6$ matrix with entries $1,-1,1,1,-1,1$ on the antidiagonal. A direct
calculation shows that for any $4\times 4$ matrix $P$ we have

$\wedge^{2}({\operatorname{adj}}(P)^{T})=(\det
P)\mathbf{G}(\wedge^{2}P)\mathbf{G}.$

Letting $\operatorname{PGL}_{4}$ act on the space of quadratic forms via
$P:H\mapsto\frac{1}{\det P}H\circ\wedge^{2}P$, this tells us that applying $P$
to a quadratic form $H(z_{12},z_{13},z_{23},z_{14},z_{24},z_{34})$ has the
same effect as applying $P^{-T}$ to its dual quadratic form which we define to
be $H(z_{34},-z_{24},z_{14},z_{23},-z_{13},z_{12})$. We note that the
substitution used to replace $H$ by its dual swaps over the two families of
isotropic subspaces in Remark 1.4.

We find the changes of coordinates in Algorithm 2.3 by using Lemma 3.1(i) and
(ii), and the analogue of (ii) after passing to the dual as in Remark 3.2. We
find the changes of coordinates in Steps 2 and 4 of Algorithm 2.4 using Lemma
3.1(iii).

######  Remark 3.3.

In Step 2 of Algorithm 2.4 we must find if possible a 3-dimensional subspace
$W\subset\langle e_{12},e_{13},e_{14},e_{23},e_{24}\rangle$ that is isotropic
for both $G$ and ${\overline{H}}_{1}$ where

$H_{1}(z_{12},\ldots,z_{24})=\pi^{-1}H(z_{12},\ldots,z_{24},0).$

To be isotropic for $G$ we need that $\langle e_{12}\rangle\subset W$. So such
a subspace $W$ can only exist if $\overline{H}_{1}(1,0,\ldots,0)=0$. We assume
that this is the case and write

$\overline{H}_{1}(z_{12},\ldots,z_{24})=z_{12}h_{1}(z_{13},z_{23},z_{14},z_{24})+h_{2}(z_{13},z_{23},z_{14},z_{24})$

where $h_{i}$ is a homogeneous polynomial of degree $i$. Our problem reduces
to that of finding a line contained in

$\\{z_{13}z_{24}-z_{23}z_{14}=h_{1}=h_{2}=0\\}\subset\mathbb{P}^{3}.$

The well known description of the lines on
$\\{z_{13}z_{24}-z_{23}z_{14}=0\\}\subset\mathbb{P}^{3}$ suggests that we
substitute
$(z_{13},z_{23},z_{14},z_{24})=(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2})$
into $h_{1}$ and $h_{2}$, take the GCD, and factor into irreducibles. The
lines of interest now correspond to linear factors of the form $\alpha
x_{1}+\beta x_{2}$ or $\gamma y_{1}+\delta y_{2}$.

######  Remark 3.4.

In Steps 3 and 5 of Algorithm 2.4, when $\ker{\overline{H}}$ is not itself
isotropic for $G$, we must find all codimension $1$ subspaces of
$\ker{\overline{H}}$ that are isotropic for $G$. Since the restriction of $G$
to $\ker{\overline{H}}$ is a non-zero quadratic form, it can have at most two
linear factors. There are therefore at most two codimension $1$ subspaces we
need to consider. In particular, the number of times that Algorithm 2.4
applies one of the operations in Definition 2.2 is uniformly bounded.

##  4\. Weights and Admissibility

Let $H\in{\mathcal{O}_{K}}[u_{0},\ldots,u_{5}]$ be an integral quadratic form
and suppose that there exists $P\in\operatorname{GL}_{4}(K)$ such that

$v\left(\frac{1}{\det P}H\circ\wedge^{2}P\right)>0.$

Then $P$ is equivalent to a matrix in Smith normal form, say

$P=U{\rm Diag}(\pi^{w_{1}},\pi^{w_{2}},\pi^{w_{3}},\pi^{w_{4}})V$

for some $U,V\in\operatorname{GL}_{4}({\mathcal{O}_{K}})$ and
$w_{1},w_{2},w_{3},w_{4}\in\mathbb{Z}$. We say that the weight
$w=(w_{1},w_{2},w_{3},w_{4})$ is admissible for $H$. It is clear that
permuting the entries of $w$, or adding the same integer to all entries, has
no effect on admissibility.

######  Definition 4.1.

The weight $w=(w_{1},w_{2},w_{3},w_{4})$ dominates the weight
$w^{\prime}=(w^{\prime}_{1},w^{\prime}_{2},w^{\prime}_{3},w^{\prime}_{4})$ if

(2) $\displaystyle\max(1+w_{1}+w_{2}+w_{3}+w_{4}-w_{i}-w_{j}-w_{k}-w_{l},0)$
$\displaystyle\hskip
50.00008pt\geqslant\max(1+w^{\prime}_{1}+w^{\prime}_{2}+w^{\prime}_{3}+w^{\prime}_{4}-w^{\prime}_{i}-w^{\prime}_{j}-w^{\prime}_{k}-w^{\prime}_{l},0)$

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

This definition is motivated by the fact that if $w$ dominates $w^{\prime}$
and $w$ is admissible for $H$ then $w^{\prime}$ is admissible for $H$. Our
next lemma shows that (for the purpose of proving that Algorithm 2.4 is
correct) it suffices to consider finitely many (in fact 12) weights.

######  Lemma 4.2.

Every weight $w=(0,a,b,c)\in\mathbb{Z}^{4}$ with $0\leqslant a\leqslant
b\leqslant c$ dominates one of the following weights

$\displaystyle(0,0,0,0),\,\,(0,0,0,1),\,\,(0,1,1,1),\,\,(0,0,1,1),\,\,(0,0,1,2),\,\,(0,1,2,2),$
$\displaystyle(0,1,1,2),\,\,(0,1,1,3),\,\,(0,2,2,3),\,\,(0,1,2,3),\,\,(0,1,2,4),\,\,(0,2,3,4).$

###### Proof.

We list the pairs $(i,j)$ and $(k,l)$ in Definition 4.1 in the order $(1,2)$,
$(1,3)$, $(2,3)$, $(1,4)$, $(2,4)$, $(3,4)$. Taking $w=(0,a,b,c)$, the left
hand side of (2) is $\max(\xi,0)$ where $\xi$ runs over the entries of the
following symmetric matrix.

$\begin{bmatrix}1+b+c-a&1+c&1+c-a&1+b&1+b-a&1\\\
1+c&1+a+c-b&1+c-b&1+a&1&1+a-b\\\ 1+c-a&1+c-b&1+c-a-b&1&1-a&1-b\\\
1+b&1+a&1&1+a+b-c&1+b-c&1+a-c\\\ 1+b-a&1&1-a&1+b-c&1+b-a-c&1-c\\\
1&1+a-b&1-b&1+a-c&1-c&1+a-b-c\end{bmatrix}$

We divide into $8$ cases according as to which of the inequalities $0\leqslant
a\leqslant b\leqslant c$ are equalities. In fact we make the following more
precise claims.

  * •

If $0=a=b=c$ then $w=(0,0,0,0)$.

  * •

If $0=a=b<c$ then $w$ dominates $(0,0,0,1)$.

  * •

If $0=a<b=c$ then $w$ dominates $(0,0,1,1)$.

  * •

If $0=a<b<c$ then $w$ dominates $(0,0,1,2)$.

  * •

If $0<a=b=c$ then $w$ dominates $(0,1,1,1)$.

  * •

If $0<a=b<c$ then $w$ dominates $(0,1,1,3)$, $(0,1,1,2)$ or $(0,2,2,3)$.

  * •

If $0<a<b=c$ then $w$ dominates $(0,1,2,2)$.

  * •

If $0<a<b<c$ then $w$ dominates $(0,1,2,4)$, $(0,1,2,3)$ or $(0,2,3,4)$.

In each case where we list three possibilities, we further claim that these
correspond to the subcases $a+b<c$, $a+b=c$ and $a+b>c$ (in that order).

Since the proofs are very similar, we give details in just one case. So
suppose that $0<a<b<c$ and $a+b=c$. Then we have $a\geqslant 1$, $b\geqslant
2$, $c\geqslant 3$, $b-a\geqslant 1$, $c-a\geqslant 2$ and $c-b\geqslant 1$.
Listing the pairs $(i,j)$ and $(k,l)$ in the same order as before, the left
hand side of (2) is at least

$\begin{bmatrix}5&4&3&3&2&1\\\ 4&3&2&2&1&0\\\ 3&2&1&1&0&0\\\ 3&2&1&1&0&0\\\
2&1&0&0&0&0\\\ 1&0&0&0&0&0\end{bmatrix}$

with equality if $(a,b,c)=(1,2,3)$. Therefore $w$ dominates $(0,1,2,3)$. ∎

Our next remark further reduces the number of weights we must consider.

######  Remark 4.3.

It is clear from Remark 3.2 that if $w\in\mathbb{Z}^{4}$ is admissible for $H$
then $-w$ is admissible for the dual of $H$. We say that the weights $w$ and
$-w$ (or any weights equivalent to these, in the sense of permuting the
entries, or adding the same integer to all entries) are dual. The list of $12$
weights in Lemma 4.2 consists of $4$ dual pairs $(0,a,b,c)$ and
$(0,c-b,c-a,c)$ with $a+b\not=c$, and $4$ self-dual weights $(0,a,b,a+b)$.

##  5\. Completion of the proof

In this section we complete the proof that Algorithm 2.4 is correct.

We first note that if $H$ and $H^{\prime}$ are related by an integral change
of coordinates, and the algorithm works for $H$ then it works for
$H^{\prime}$. This is because before applying Operations 1, 2 or 3 we always
make an integral change of coordinates that, by Lemma 3.1, is unique up to an
element of $\operatorname{GL}_{4}({\mathcal{O}_{K}})$ whose reduction mod
$\pi$ preserves a suitable subspace of $k^{4}$. The following elementary lemma
then shows that the transformed quadratic forms are again related by an
integral change of coordinates.

######  Lemma 5.1.

Let $\alpha=\operatorname{Diag}(I_{r},\pi I_{4-r})$ and
$P\in\operatorname{GL}_{4}({\mathcal{O}_{K}})$. Then
$P\in\alpha\operatorname{GL}_{4}({\mathcal{O}_{K}})\alpha^{-1}$ if and only if
the reduction of $P$ mod $\pi$ preserves the subspace $\langle
e_{1},\dots,e_{r}\rangle$.

###### Proof.

This is [CFS, Lemma 4.1]. ∎

Let $H\in{\mathcal{O}_{K}}[z_{12},\ldots,z_{34}]$ be a quadratic form. If
there exists $P\in\operatorname{PGL}_{4}(K)$ such that

(3) $v\left(\frac{1}{\det P}H\circ\wedge^{2}P\right)>0,$

then, as explained in Section 4, one of the $12$ weights in Lemma 4.2 is
admissible for $H$. Since the analysis for dual weights (see Remark 4.3) is
essentially identical, we only need to consider one weight from each dual
pair. It therefore suffices to consider the 8 weights listed in the table
below.

In the case of weight $(w_{1},\ldots,w_{4})$ we may suppose, by an integral
change of coordinates, that (3) holds with
$P=\operatorname{Diag}(\pi^{w_{1}},\ldots,\pi^{w_{4}})$. This implies certain
lower bounds on the valuations of the coefficients of $H$. To specify these
(in a way that is valid even when $\operatorname{char}(k)=2$), we relabel the
variables $z_{12},z_{13},z_{23},z_{14},z_{24},z_{34}$ as $z_{1},\ldots,z_{6}$
and write $H=\sum_{i\leqslant j}H_{ij}z_{i}z_{j}$. We also put
$H_{ji}=H_{ij}$. Then the lower bounds on the $v(H_{ij})$ are as recorded in
the table.

Case 1: $(0,0,0,0)$ |  Case 2: $(0,0,0,1)$ |  Case 3: $(0,0,1,1)$ |  Case 4: $(0,1,1,2)$
---|---|---|---
$\begin{bmatrix}1&1&1&1&1&1\\\ 1&1&1&1&1&1\\\ 1&1&1&1&1&1\\\ 1&1&1&1&1&1\\\ 1&1&1&1&1&1\\\ 1&1&1&1&1&1\end{bmatrix}$ | $\begin{bmatrix}2&2&2&1&1&1\\\ 2&2&2&1&1&1\\\ 2&2&2&1&1&1\\\ 1&1&1&0&0&0\\\ 1&1&1&0&0&0\\\ 1&1&1&0&0&0\end{bmatrix}$ | $\begin{bmatrix}3&2&2&2&2&1\\\ 2&1&1&1&1&0\\\ 2&1&1&1&1&0\\\ 2&1&1&1&1&0\\\ 2&1&1&1&1&0\\\ 1&0&0&0&0&0\end{bmatrix}$ | $\begin{bmatrix}3&3&2&2&1&1\\\ 3&3&2&2&1&1\\\ 2&2&1&1&0&0\\\ 2&2&1&1&0&0\\\ 1&1&0&0&0&0\\\ 1&1&0&0&0&0\end{bmatrix}$
$r=0$ | $r=1,2,3$ | $r=1,2$ | $r=1,2,3,4$
Case 5: $(0,0,1,2)$ |  Case 6: $(0,1,1,3)$ |  Case 7: $(0,1,2,3)$ |  Case 8: $(0,1,2,4)$
$\begin{bmatrix}4&3&3&2&2&1\\\ 3&2&2&1&1&0\\\ 3&2&2&1&1&0\\\ 2&1&1&0&0&0\\\ 2&1&1&0&0&0\\\ 1&0&0&0&0&0\end{bmatrix}$ | $\begin{bmatrix}4&4&3&2&1&1\\\ 4&4&3&2&1&1\\\ 3&3&2&1&0&0\\\ 2&2&1&0&0&0\\\ 1&1&0&0&0&0\\\ 1&1&0&0&0&0\end{bmatrix}$ | $\begin{bmatrix}5&4&3&3&2&1\\\ 4&3&2&2&1&0\\\ 3&2&1&1&0&0\\\ 3&2&1&1&0&0\\\ 2&1&0&0&0&0\\\ 1&0&0&0&0&0\end{bmatrix}$ | $\begin{bmatrix}6&5&4&3&2&1\\\ 5&4&3&2&1&0\\\ 4&3&2&1&0&0\\\ 3&2&1&0&0&0\\\ 2&1&0&0&0&0\\\ 1&0&0&0&0&0\end{bmatrix}$
$r=3,4$ | $r=3,4$ | $r=3,4$ | $r=5$

In our analysis of each case, we will assume we are not in an earlier case.
The possibilities for $r=\operatorname{rank}{\overline{H}}$ will be justified
below, but are recorded in the table for convenience. We complete the proof
that Algorithm 2.4 is correct by going through the $8$ cases. In fact we show
that if the cases are grouped as

Case 1 | Case 2 | Case 4 | Case 5 | Case 7
---|---|---|---|---
Case 3 | Case 6 | Case 8

then at each iteration of the algorithm we move at least one column to the
left. Therefore, if after visiting Step 1 the first time and returning to it a
further 4 times we still do not have $v(H)>0$ then the algorithm is correct to
return FALSE.

### Case 1:

$w=(0,0,0,0)$. In this case we already have $v(H)>0$, so $r=0$ and we are done
by Step 1.

### Case 2:

$w=(0,0,0,1)$. We see from the table that $\langle
e_{12},e_{13},e_{23}\rangle\subset\ker{\overline{H}}$ and so $r\leqslant 3$.
We cannot have $r=0$, otherwise we would be in Case 1. If $r=1$ then we are
done by Step 2. If $r=2$ then we are done by Step 3. If $r=3$ then Step 5
directly applies Operation 1. (By “directly” we mean that there is no
preliminary integral change of coordinates.) Since this gives $v(H)>0$ we are
in Case 1 on the next iteration.

### Case 3:

$w=(0,0,1,1)$. We see from the table that ${\overline{H}}=\ell z_{34}$ for
some linear form $\ell$. One of the transformations considered in Step 4 is to
directly apply Operation 2. Since this gives $v(H)>0$ we are in Case 1 on the
next iteration.

### Case 4:

$w=(0,1,1,2)$. We see from the table that $\langle
e_{12},e_{13}\rangle\subset\ker{\overline{H}}$ and so $r\leqslant 4$. If $r=4$
then Step 5 directly applies Operation 1 or Operation 3. Then on the next
iteration either $(0,0,0,1)$ or $(0,1,1,1)$ is admissible, which means we are
in Case 2 or its dual. If $r\leqslant 3$ then by applying a block diagonal
element of $\operatorname{GL}_{4}({\mathcal{O}_{K}})$ with blocks of sizes
$1$, $2$ and $1$, we may suppose that $H_{35}\equiv H_{45}\equiv 0\pmod{\pi}$.
If $r=3$ then $\ker{\overline{H}}=\langle
e_{12},e_{13},ae_{23}+be_{14}\rangle$ for some $a,b\in k$. If $a=0$ or $b=0$
then $\ker{\overline{H}}$ is isotropic for $G$. Otherwise $\langle
e_{12},e_{13}\rangle$ is the unique codimension 1 isotropic subspace. Either
way, Step 5 directly applies Operation 1 or Operation 3, and we are done as
before.

We now suppose that $r\leqslant 2$ and divide into the following cases.

  * •

Suppose that $H_{36}\not\equiv 0\pmod{\pi}$ and $H_{46}\not\equiv
0\pmod{\pi}$. Since $r\leqslant 2$ we have ${\overline{H}}=\ell z_{34}$ for
some linear form $\ell$. Since there is no integral change of coordinates
taking $\ell$ to $z_{34}$ the only possible outcome of Step 4 is to directly
apply Operation 2. This brings us to Case 3.

  * •

Suppose that $H_{36}\equiv 0\pmod{\pi}$ and $H_{46}\not\equiv 0\pmod{\pi}$.
Then $v(H_{33})=1$, otherwise we would be in Case 2. We again have
${\overline{H}}=\ell z_{34}$ for some linear form $\ell$. Although there does
now exist an integral change of coordinates taking $\ell$ to $z_{34}$,
following this up with Operation 2 does not preserve that $v(H)\geqslant 0$.
So again the only possible outcome of Step 4 is to directly apply Operation 2.
This brings us to Case 3.

  * •

Suppose that $H_{36}\not\equiv 0\pmod{\pi}$ and $H_{46}\equiv 0\pmod{\pi}$.
This is essentially the same as the previous case by duality.

  * •

Suppose that $H_{36}\equiv H_{46}\equiv 0\pmod{\pi}$. Then ${\overline{H}}$ is
a quadratic form in $z_{24}$ and $z_{34}$ only. If this factors over $k$ then
either of the transformations in Step 4 brings us to Case 3. Otherwise we
proceed to Step 5 which directly applies Operation 1 or Operation 3. As
before, this brings us to Case 2 or its dual.

### Case 5:

$w=(0,0,1,2)$. Applying a block diagonal element of
$\operatorname{GL}_{4}({\mathcal{O}_{K}})$ with blocks of sizes $2$, $1$ and
$1$, we may suppose that $H_{26}\equiv 0\pmod{\pi}$. Then $H_{36}\not\equiv
0\pmod{\pi}$ (otherwise we would be in Case 2) and $H_{44},H_{45},H_{55}$
cannot all vanish mod $\pi$ (otherwise we would be in Case 3). Therefore
$\langle e_{12},e_{13}\rangle\subset\ker{\overline{H}}$ and $r=3$ or $4$. The
only $3$-dimensional isotropic subspaces for $G$ that contain $\langle
e_{12},e_{13}\rangle$ are $\langle e_{12},e_{13},e_{23}\rangle$ and $\langle
e_{12},e_{13},e_{14}\rangle$. Therefore one of the transformations considered
in Step 5 is to directly apply Operation 1 or Operation 3 (the latter only
being a possibility if $H_{44}\equiv 0\pmod{\pi}$). It follows that at the
next iteration we have $r\leqslant 2$, and so are in Case 4 or earlier.

### Case 6:

$w=(0,1,1,3)$. Applying a block diagonal element of
$\operatorname{GL}_{4}({\mathcal{O}_{K}})$ with blocks of sizes $1$, $2$ and
$1$, we may suppose that $H_{15}\equiv 0\pmod{\pi^{2}}$. We have
$H_{44}\not\equiv 0\pmod{\pi}$ (otherwise we would be in Case 4) and
$H_{35}\not\equiv 0\pmod{\pi}$ (otherwise we would be in Case 5). Therefore
$\langle e_{12},e_{13}\rangle\subset\ker{\overline{H}}$ and $r=3$ or $4$.
Exactly as in Case 5 we find that at the next iteration we have $r\leqslant
2$, and so are in Case 4 or earlier.

### Case 7:

$w=(0,1,2,3)$. We have $H_{26}\not\equiv 0\pmod{\pi}$ (otherwise we would be
in Case 4), and $H_{35},H_{45},H_{55}$ cannot all vanish mod $\pi$ (otherwise
we would be in Case 3). Therefore $r=3$ or $4$, and $\langle
e_{12}\rangle\subset\ker{\overline{H}}\subset\langle
e_{12},e_{13},e_{23},e_{14}\rangle$.

If $r=4$ then $\ker{\overline{H}}=\langle
e_{12},ae_{13}+be_{23}+ce_{14}\rangle$ for some $a,b,c\in k$. If $b,c\not=0$
then $\langle e_{12}\rangle$ is the unique codimension $1$ subspace of
$\ker{\overline{H}}$ that is isotropic for $G$. Therefore, Step 5 directly
applies Operation 2, which brings us to Case 4. If $b=0$ then $c\not=0$, and
by applying a block diagonal element of
$\operatorname{GL}_{4}({\mathcal{O}_{K}})$ with blocks of sizes $1$, $1$ and
$2$, we may suppose that $a=0$. Then the $3$-dimensional isotropic subspaces
for $G$ containing $\ker{\overline{H}}=\langle e_{12},e_{14}\rangle$ are
$\langle e_{12},e_{13},e_{14}\rangle$ and $\langle
e_{12},e_{14},e_{24}\rangle$. Step 5 applies either
$\operatorname{Diag}(1,\pi,\pi,\pi)$ or $\operatorname{Diag}(1,1,\pi,1)$
bringing us to Case 5 or Case 6. The case $c=0$ is similar by duality.

If $r=3$ then $\ker{\overline{H}}=\langle
e_{12},e_{23}+ae_{13},e_{14}+be_{13}\rangle$ for some $a,b\in k$. By applying
a block diagonal element of $\operatorname{GL}_{4}({\mathcal{O}_{K}})$ with
blocks of sizes $2$ and $2$, we may suppose that $a=b=0$. Then $H_{35}\equiv
H_{36}\equiv H_{45}\equiv H_{46}\equiv 0\pmod{\pi}$ and $H_{55}\not\equiv
0\pmod{\pi}$. The codimension $1$ subspaces of $\ker{\overline{H}}=\langle
e_{12},e_{23},e_{14}\rangle$ that are isotropic for $G$ are $\langle
e_{12},e_{23}\rangle$ and $\langle e_{12},e_{14}\rangle$. The $3$-dimensional
isotropic subspaces for $G$ containing one of these spaces are

$\langle e_{12},e_{13},e_{23}\rangle,\,\,\,\langle
e_{12},e_{13},e_{14}\rangle,\,\,\,\langle
e_{12},e_{23},e_{24}\rangle,\,\,\,\langle e_{12},e_{14},e_{24}\rangle.$

The first two of these correspond to directly applying Operation 1 or
Operation 3, which brings us to Case 5 or its dual. The last two correspond to
transformations which fail to preserve that $v(H)\geqslant 0$, and so cannot
be selected by Step 5.

### Case 8:

$w=(0,1,2,4)$. We have $H_{35}\not\equiv 0\pmod{\pi}$ (otherwise we would be
in Case 5), $H_{26}\not\equiv 0\pmod{\pi}$ (otherwise we would be in Case 6),
and $H_{44}\not\equiv 0\pmod{\pi}$ (otherwise we would be in Case 7).
Therefore $r=5$ and $\ker{\overline{H}}=\langle e_{12}\rangle$. Step 5
directly applies Operation 2 which brings us to Case 6.

######  Example 5.2.

We give three examples where Algorithm 2.4 takes the maximum of $4$ iterations
to give $v(H)>0$. The first two examples start in Case 7, with
$\operatorname{rank}{\overline{H}}=3$ or $4$, and the final one starts in Case
8. In the first two examples there are two choices on the first iteration. We
made an arbitrary choice in each case, but in fact with the other choices the
algorithm would still have taken $4$ iterations.

Let $K=\mathbb{Q}$ and $v=v_{p}$ for any choice of prime number $p$. An arrow
labelled $(w_{1},\ldots,w_{4})$ indicates that we replace $H$ by
$\frac{1}{\det P}H\circ\wedge^{2}P$ where
$P=\operatorname{Diag}(p^{w_{1}},\ldots,p^{w_{4}})$.

$\displaystyle
p^{5}z_{12}^{2}+z_{13}z_{34}+pz_{23}^{2}+pz_{14}^{2}+z_{24}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,0,1)}}{{\longrightarrow}}p^{4}z_{12}^{2}+z_{13}z_{34}+z_{23}^{2}+p^{2}z_{14}^{2}+pz_{24}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,0)}}{{\longrightarrow}}p^{3}z_{12}^{2}+pz_{13}z_{34}+pz_{23}^{2}+pz_{14}^{2}+z_{24}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,1,0,1)}}{{\longrightarrow}}p^{3}z_{12}^{2}+z_{13}z_{34}+pz_{23}^{2}+pz_{14}^{2}+p^{2}z_{24}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,1)}}{{\longrightarrow}}p(z_{12}^{2}+z_{13}z_{34}+z_{23}^{2}+z_{14}^{2}+pz_{24}^{2}).$
$\displaystyle p^{5}z_{12}^{2}+z_{13}z_{34}+pz_{23}^{2}+z_{14}z_{24}$
$\displaystyle\stackrel{{\scriptstyle(0,0,0,1)}}{{\longrightarrow}}p^{4}z_{12}^{2}+z_{13}z_{34}+z_{23}^{2}+pz_{14}z_{24}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,0)}}{{\longrightarrow}}p^{3}z_{12}^{2}+pz_{13}z_{34}+pz_{23}^{2}+z_{14}z_{24}$
$\displaystyle\stackrel{{\scriptstyle(0,1,0,1)}}{{\longrightarrow}}p^{3}z_{12}^{2}+z_{13}z_{34}+pz_{23}^{2}+pz_{14}z_{24}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,1)}}{{\longrightarrow}}p(z_{12}^{2}+z_{13}z_{34}+z_{23}^{2}+z_{14}z_{24}).$
$\displaystyle p^{6}z_{12}^{2}+z_{13}z_{34}+z_{23}z_{24}+z_{14}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,1)}}{{\longrightarrow}}p^{4}z_{12}^{2}+pz_{13}z_{34}+z_{23}z_{24}+z_{14}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,0,1)}}{{\longrightarrow}}p^{3}z_{12}^{2}+pz_{13}z_{34}+z_{23}z_{24}+pz_{14}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,1,0,1)}}{{\longrightarrow}}p^{3}z_{12}^{2}+z_{13}z_{34}+pz_{23}z_{24}+pz_{14}^{2}$
$\displaystyle\stackrel{{\scriptstyle(0,0,1,1)}}{{\longrightarrow}}p(z_{12}^{2}+z_{13}z_{34}+z_{23}z_{24}+z_{14}^{2}).$

## References

  * [BG] M. Bhargava and B.H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and $L$-functions, D. Prasad, C. S. Rajan, A. Sankaranarayanan and J. Sengupta (eds.), 23–91, Tata Institute of Fundamental Research, Stud. Math. 22, Mumbai, 2013.
  * [BSD] B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves. I, J. reine angew. Math. 212 (1963), 7–25.
  * [BCP] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comb. 24, 235-265 (1997), http://magma.maths.usyd.edu.au/magma/
  * [CF] J.W.S. Cassels and E.V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, 230, Cambridge University Press, Cambridge, 1996.
  * [CFS] J.E. Cremona, T.A. Fisher, and M. Stoll, Minimisation and reduction of 2-,3- and 4-coverings of elliptic curves, Algebra & Number Theory 4 (2010), no. 6, 763–820.
  * [ES] A.-S. Elsenhans and M. Stoll, Minimization of hypersurfaces, preprint, 2021,
https://arxiv.org/abs/2110.04625

  * [F] T.A. Fisher, Minimisation and reduction of 5-coverings of elliptic curves, Algebra & Number Theory 7 (2013), no. 5, 1179–1205.
  * [FR] T.A. Fisher and L. Radičević, Some minimisation algorithms in arithmetic invariant theory, J. Théor. Nombres Bordeaux 30 (2018), no. 3, 801–828.
  * [FY] T.A. Fisher and J. Yan, Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian, preprint, 2023, https://arxiv.org/abs/2306.06011
  * [FH] W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.
  * [K] J. Kollár, Polynomials with integral coefficients, equivalent to a given polynomial, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 17–27.
  * [L] Q. Liu, Computing minimal Weierstrass equations, preprint, 2022,
https://arxiv.org/abs/2209.00469

  * [SW] A. Shankar and X. Wang, Rational points on hyperelliptic curves having a marked non-Weierstrass point, Compos. Math. 154 (2018), no. 1, 188–222.