# Distributed Optimization with Consensus Constraint for Multi-Robot Semantic
Octree Mapping††thanks:

Arash Asgharivaskasi  Nikolay Atanasov The authors are with the Department of
Electrical and Computer Engineering, University of California San Diego, CA
92093, USA<EMAIL_ADDRESS>gratefully acknowledge support
from ARL DCIST CRA W911NF-17-2-0181 and NSF FRR CAREER 2045945.

###### Abstract

This work develops a distributed optimization algorithm for multi-robot 3-D
semantic mapping using streaming range and visual observations and single-hop
communication. Our approach relies on gradient-based optimization of the
observation log-likelihood of each robot subject to a map consensus constraint
to build a common multi-class map of the environment. This formulation leads
to closed-form updates which resemble Bayes rule with one-hop prior averaging.
To reduce the amount of information exchanged among the robots, we utilize an
octree data structure that compresses the multi-class map distribution using
adaptive-resolution.

##  I Introduction

Dense semantically-rich environment representations, such as multi-class
volumetric maps [1], can be built online by mobile robots thanks to the
availability of on-board GPU-accelerated segmentation models. Distributed 3-D
mapping requires designing optimization algorithms that take into account the
communication and computation limits of multi-robot systems. Among the recent
works in distributed mapping for robotics, Corah et al. [2] proposed a
distributed Gaussian mixture model (GMM) in which a GMM representation is
communicated over a network of robots and is utilized by each robot to build a
local occupancy grid map for planning. The distributed estimation work by
Paritosh et al. [3] uses mirror descent optimization of an information-
theoretic cost function to obtain the probability density function (PDF) of
unknown continuous variables by communicating local estimates among the
agents. The metric-semantic mapping approach in Kimera-multi [4] builds a
dense multi-class mesh representation from visual-inertial observations and
local trajectory communication used to estimate a global reference frame in a
decentralized manner.

In this work, we propose a multi-class mapping algorithm for a team of mobile
robots. Our main contribution is an online distributed multi-robot semantic
octree mapping method, where each node in the octree maintains a categorical
distribution over various object classes, and is updated using local range and
category observations. We approach the multi-robot mapping problem using
distributed gradient optimization of the observation log-likelihood, and show
that every iteration contains a consensus step where neighbors average their
map distributions and a Bayesian update step where new local observations are
incorporated into the map distribution by each robot. Through a simulated
experiment, we show that our algorithm leads to globally consistent Bayesian
multi-class mapping and the octree data structure helps to achieve memory and
communication efficiency.

##  II Problem Statement

Figure 1: Semantically annotated point cloud obtained from a range and
category observation $z_{t}^{i}$. Each object category is shown with a unique
color.

Consider a network of robots represented by an undirected graph ${\cal
G}({\cal V},{\cal E})$ where ${\cal V}$ denotes the set of robots and ${\cal
E}\subseteq{\cal V}\times{\cal V}$ encodes the existence of a communication
link between each pair of robots, assumed to be constant throughout the task.
The robots are navigating in an environment consisting of a collection of
disjoint sets ${\cal S}_{k}\subset\mathbb{R}^{3}$, each associated with a
semantic category $c\in{\cal C}:=\left\\{0,1,\ldots,C\right\\}$. Let ${\cal
S}_{0}$ denote free space, while each ${\cal S}_{c}$ for $c>0$ represents a
different category, such as building, vegetation, terrain. Each robot $i$ is
equipped with a sensor that provides a stream of ray-based observations
$z^{i}_{t}\in\mathbb{R}\times{\cal C}$ at time $t$. Each observation contains
information about the distance to and semantic category of the observed object
along the ray. See Fig. 1 for an example. Such information may be obtained by
processing the measurements of an RGBD camera or a LiDAR with a semantic
segmentation algorithm.

The robots aim to construct a common multi-class map $\mathbf{m}$ of the
environment by utilizing their semantically-annotated local range measurements
as well as communication with their teammates. We model the map $\mathbf{m}$
as a grid of independent cells, where each individual cell $m$ is labeled with
a category in ${\cal C}$. To model measurement noise, we consider a PDF
$p(z^{i}_{t}\mid\mathbf{m})$ for each observation. The goal is to compute the
map estimate that maximizes the sum of expected log-likelihood of data, i.e.
local observations $\\{z_{\tau}^{i}\\}_{\tau=1}^{t}$ for each robot $i\in{\cal
V}$:

$\max_{p\in{\cal P}}\sum_{i=1}^{N}\mathbb{E}_{\mathbf{m}\sim
p}\big{[}\sum_{\tau=1}^{t}\log{p(z_{\tau}^{i}|\mathbf{m})}\big{]},$

where $N=|{\cal V}|$ and ${\cal P}$ is the space of all probability mass
functions (PMF) over the set of possible maps. To remove the constraint
$p\in{\cal P}$, we utilize a multi-class log-odds ratio representation of the
categorical distribution [1]:

$\mathbf{h}:=\begin{bmatrix}\log\frac{p(m=0)}{p(m=0)}&\cdots&\log\frac{p(m=C)}{p(m=0)}\end{bmatrix}^{\top}\in\mathbb{R}^{C+1}.$

A probability mass function and its log-odds representation have a one-to-one
correspondence through the softmax function
$\sigma:\mathbb{R}^{C+1}\mapsto\mathbb{R}^{C+1}$:

$p(m=c)=\sigma_{c+1}(\mathbf{h}):=\frac{\mathbf{e}_{c+1}^{\top}\exp(\mathbf{h})}{\mathbf{1}^{\top}\exp(\mathbf{h})},$

where $\mathbf{e}_{c}$ is the standard basis vector with $c$-th element equal
to $1$ and $0$ elsewhere, $\mathbf{1}$ is the vector with all elements equal
to $1$, and $\exp(\cdot)$ is applied element-wise to the vector $\mathbf{h}$.
Applying Bayes rule to the observation model $p(z^{i}_{\tau}|\mathbf{m})$ and
using the map cell independence assumption makes it possible to solve the
optimization for individual cells instead of maintaining the joint
distribution over all cells. Thus, the estimation objective can then be
written as an unconstrained optimization for each map cell:

$\displaystyle\max_{\mathbf{h}\in\mathbb{R}^{C+1}}\sum_{i=1}^{N}\sum_{m=0}^{C}\sigma_{m+1}(\mathbf{h})\log{\frac{q_{i}(m)}{\sigma_{m+1}(\mathbf{h})}},$

where $q_{i}(m)=\sqrt[t]{\prod_{\tau=1}^{t}p(m|z_{\tau}^{i})}$ is the temporal
geometric average of the inverse observation model. In order to enable
distributed optimization of the objective, we introduce a constraint that
requires the robots to agree on a common map estimate via only one-hop
communication as follows.

######  Problem.

Let ${\cal G}({\cal V},{\cal E})$ be a network of $N$ robots, where each robot
$i\in{\cal V}$ collects observations $\\{z_{\tau}^{i}\\}_{\tau=1}^{t}$ and
maintains its own local estimate of the map cell distribution
$\mathbf{h}_{i}$. Estimate the globally consistent common map such that:

$\displaystyle\max_{\mathbf{h}_{1:N}\in\mathbb{R}^{(C+1)\times N}}$
$\displaystyle\sum_{i=1}^{N}\sum_{m=0}^{C}\sigma_{m+1}(\mathbf{h}_{i})\log{\frac{q_{i}(m)}{\sigma_{m+1}(\mathbf{h}_{i})}},$
s.t. $\displaystyle\sum_{\\{i,j\\}\in{\cal
E}}A_{ij}\|\mathbf{h}_{j}-\mathbf{h}_{i}\|_{2}^{2}=0,$ (1)

where $A_{ij}$ is the $ij$-th entry of the weighted adjacency matrix of ${\cal
G}$ with $\sum_{j}A_{ij}=1$.

##  III Distributed Semantic Mapping

The optimization objective in (1) has a specific structure as the sum of local
objective functions over all robots, subject to a consensus constraint among
all $\mathbf{h}_{i}$, $i\in{\cal V}$. We develop an iterative gradient-based
method inspired by the general distributed optimization framework presented in
[5]. The underlying idea is to interleave local gradient updates for the
individual objective function and for the consensus constraint at each robot.
Alg. 1 presents the update steps for each map cell in order to solve (1) in a
distributed manner. We initialize the log-odds $\mathbf{h}^{(0)}_{i}$ with
prior $\mathbf{h}_{0}$ for each robot $i$. We continuously apply the update
steps until a maximum number of iterations is reached, or the update norm is
smaller than a threshold. The update step in line 7 guides the local log-odds
towards satisfaction of the consensus constraint, which only requires single-
hop communication between neighboring robots $j\in{\cal N}_{i}$. Line 12
incorporates the local observations
$\log{\mathbf{q}_{i}}=[\log{q_{i}(m)}]_{m=0}^{C}$ via
$\boldsymbol{\Delta}_{i}$, where $\odot$ is the element-wise multiplication.
This step is only local to each robot $i$ and does not require communication.
Note that both update steps 7 and 14 resemble the log-odds equivalent of the
Bayes rule for updating multi-class probabilities (see (8) in [1]). Alg. 1
results in a globally consistent multi-class probabilistic grid map of the
environment that is shared amongst all robots in ${\cal V}$. Proofs of
convergence to consensus and optimality with respect to the objective function
(1) will be shown in future work.

The distributed semantic mapping algorithm we developed assumes a regular grid
representation of the environment. To reduce the storage and communication
requirements, we may utilize a semantic octree data structure which provides a
lossless compression of the original 3-D multi-class map. In this case, the
update rules in Alg. 1 should be applied to all leaf nodes in the semantic
octree map of each robot $i$. Please refer to Alg. 3 in [1] for the octree
equivalents of the update steps in 7 and 14.

Algorithm 1 Distributed Semantic Mapping

1:Local observations $\\{z_{\tau}^{i}\\}_{\tau=1}^{t}$ for each robot
$i\in{\cal V}$

2:$\mathbf{h}^{(0)}_{i}=\mathbf{h}_{0}\quad\forall i\in{\cal V}$
$\triangleright$ Initialize all log-odds to $\mathbf{h}_{0}$

3:$k=0$

4:loop

5: for each robot $i\in{\cal V}$ do

6: $\triangleright$ Enforce consensus with step size $\epsilon$:

7:
$\tilde{\mathbf{h}}^{(k+1)}_{i}=\mathbf{h}^{(k)}_{i}+2\epsilon\sum_{j\in{\cal
N}_{i}}A_{ij}(\mathbf{h}^{(k)}_{j}-\mathbf{h}^{(k)}_{i})$

8: $\triangleright$ Gradient computation:

9:
$\boldsymbol{\Delta}_{i}=\tilde{\mathbf{h}}^{(k+1)}_{i}-\log{\mathbf{q}_{i}}$

10:
$\boldsymbol{\alpha}^{(k+1)}_{i}=(\exp(\tilde{\mathbf{h}}^{(k+1)}_{i})^{\top}\boldsymbol{\Delta}_{i})\mathbf{1}$

11:
$\boldsymbol{\beta}^{(k+1)}_{i}=(\exp(\tilde{\mathbf{h}}^{(k+1)}_{i})^{\top}\mathbf{1})\boldsymbol{\Delta}_{i}$

12:
$\mathbf{g}^{(k+1)}_{i}=(\boldsymbol{\alpha}^{(k+1)}_{i}-\boldsymbol{\beta}^{(k+1)}_{i})\odot\frac{\exp(\tilde{\mathbf{h}}^{(k+1)}_{i})}{(\exp(\tilde{\mathbf{h}}^{(k+1)}_{i})^{\top}\mathbf{1})^{2}}$

13: $\triangleright$ Apply gradient with step size $\gamma^{(k+1)}$:

14:
$\mathbf{h}^{(k+1)}_{i}=\tilde{\mathbf{h}}^{(k+1)}_{i}+\gamma^{(k+1)}\mathbf{g}^{(k+1)}_{i}$

15: $h^{(k+1)}_{i,1}=0$ $\triangleright$
$h^{(k+1)}_{i,1}=\log\frac{p(m=0)}{p(m=0)}=0$

16: $k=k+1$

17:return $\mathbf{h}_{i}^{(k)}\quad\forall i\in{\cal V}$

##  IV Evaluation

(a) Multi-robot exploration begins

(b) Exploration at $t=100\text{s}$

(c) Semantic octree map after $300$ s of exploration

(d) Photo-realistic Unity simulation environment

Figure 2: Time lapse of multi-robot autonomous exploration and semantic octree
mapping in the environment shown in (d). Different colors represent different
semantic categories (building, dirt road, grass, etc.). Local semantic maps
are overlaid with each other such that the transparency of each cell directly
relates to the deviation between the local map estimates.

We tested our distributed semantic octree mapping algorithm in a 3-D photo-
realistic Unity simulation environment, resembling an outdoor village area
with various object classes, such as grass, dirt road, building, car, etc. Six
ClearPath Husky robots each equipped with an RGB-D camera explore the unknown
environment using frontier-based exploration [6]. The RGB sensor measurements
are used to perform pixel classification, which combined with the depth input,
are provided as local observations to build semantic octree maps. We assume
known robot poses and fully connected communication graph ${\cal G}$
throughout the experiment. Each robot publishes its local map every $1$s, and
upon arrival of a map, one iteration of Alg. 1 is executed. Fig. 2 shows
several snapshots of the multi-robot semantic octree mapping, as well as the
simulation environment. Fig. 3 shows the total distance in local map
estimates, i.e. $\sum_{\\{i,j\\}\in{\cal
E}}\|\mathbf{h}_{j}-\mathbf{h}_{i}\|_{2}^{2}$, summed over all semantic octree
map leaf nodes. Note that we did not account for the communication strength
coefficients $A_{ij}\leq 1$ since doing so would underestimate the actual
divergence in the local map estimates. As the figure shows, our distributed
mapping algorithm leads to reduction of the total distance in local map
estimates. Regarding communication requirements of our distributed semantic
octree mapping, Fig. 4 illustrates the size of the communicated local semantic
octree maps, with comparison to running the same distributed mapping
experiment using uniform resolution grid maps. Our semantic octree map shows
more than four times saving in communication bandwidth, empirically validating
the benefit of our map representation for distributed multi-robot
applications.

Figure 3: Time evolution of the total distance in local map estimates.
Iteration of Alg. 1 is executed in each robot every one second, upon arrival
of a local map estimate from a neighboring robot. Figure 4: Packet size used
for local map communication for uniform resolution grid map (blue) and
semantic octree map (red). The solid lines and the dashed lines show the
average over all robots and one standard deviation from the average,
respectively.

In this work, we presented distributed construction of semantic octree maps of
unknown environments using a team of robots. The global map can be accessed
via query from any of the robots, eliminating the need for a central
estimation unit. Such common map can be utilized by robots for decentralized
planning. In a future work, we use our multi-robot mapping algorithm alongside
a distributed planning method in order to achieve multi-robot exploration.

## References

  * [1] A. Asgharivaskasi and N. Atanasov, “Semantic octree mapping and Shannon mutual information computation for robot exploration,” _IEEE Transactions on Robotics (TRO)_ , 2023.
  * [2] M. Corah, C. O’Meadhra, K. Goel, and N. Michael, “Communication-efficient planning and mapping for multi-robot exploration in large environments,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 2, pp. 1715–1721, 2019\.
  * [3] P. Paritosh, N. Atanasov, and S. Martinez, “Distributed bayesian estimation of continuous variables over time-varying directed networks,” _IEEE Control Systems Letters_ , vol. 6, pp. 2545–2550, 2022.
  * [4] Y. Tian, Y. Chang, F. Herrera Arias, C. Nieto-Granda, J. P. How, and L. Carlone, “Kimera-multi: Robust, distributed, dense metric-semantic slam for multi-robot systems,” _IEEE Transactions on Robotics_ , vol. 38, no. 4, pp. 2022–2038, 2022.
  * [5] S. M. Shah, “Distributed optimization on Riemannian manifolds for multi-agent networks,” _arXiv preprint arXiv:1711.11196_ , 2017.
  * [6] B. Yamauchi, “A frontier-based approach for autonomous exploration,” in _IEEE International Symposium on Computational Intelligence in Robotics and Automation_ , 1997, pp. 146–151.


# Genus two curves with everywhere good reduction over quadratic fields

Andrzej Dąbrowski  Institute of Mathematics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Mohammad Sadek  Faculty of Engineering and
Natural Sciences, Sabancı University, Tuzla, İstanbul, 34956 Turkey
<EMAIL_ADDRESS>
###### Abstract.

We address the question of existence of absolutely simple abelian varieties of
dimension 2 with everywhere good reduction over quadratic fields. The emphasis
will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic
number field and $C$ is a genus $2$ curve with everywhere good reduction over
$K$. We provide the first infinite sequence of pairs $(K,C)$ where $K$ is a
real (complex) quadratic field and $C$ has everywhere good reduction over $K$.
Moreover, we show that the Jacobian of $C$ is an absolutely simple abelian
variety.

††footnotetext:  Keywords: genus two curves, everywhere good reduction, Igusa
invariants, quadratic fields 2010 Mathematics Subject Classification: 11G30,
14H25

##  1\. Introduction

It is due to Fontaine, [11] (see also Abrashkin [1]), that there are no
abelian varieties with everwhere good reduction over any of the number fields
${\mathbb{Q}}$, ${\mathbb{Q}}(\sqrt{-1})$, ${\mathbb{Q}}(\sqrt{-3})$, or
${\mathbb{Q}}(\sqrt{5})$. The first example of an elliptic curve with
everywhere good reduction over a quadratic field was given by Tate over the
field $\mathbb{Q}(\sqrt{29})$, namely, the elliptic curve described by the
Weierstrass equation $y^{2}+xy+\left({5+\sqrt{29}\over 2}\right)^{2}y=x^{3}$.
Much work has been directed since then to giving examples of elliptic curves
with everwhere good reduction over number fields with due attention to
quadratic number fields, see [13, 26, 30, 33] and references therein. In [7],
an algorithm was given to find all elliptic curves over a number field with
good reduction outside a given finite set of primes.

Shafarevich conjectured [27] that for each number field $K$, finite set of
places $S$, and an integer $g\geq 2$, there are only finitely many
$K$-isomorphism classes of curves of genus $g$ over $K$ with good reduction
outside $S$. The proof was sketched by him in the hyperelliptic case; for
details see [23, 24]. According to Fontaine’s result, if $K={\mathbb{Q}}$ then
$S$ must contain at least one prime so that such curves exist. The papers [8,
20, 25] treated genus two curves over ${\mathbb{Q}}$ with bad reduction at
exactly one prime.

Fixing integers $g,k\geq 2$, one may pose the following question: does there
exist a genus $g$ curve with everywhere good reduction over some number field
$K$ of degree $k$? Note that if a curve $C$ has everywhere good reduction over
some number field, then the jacobian of $C$ also has such a property, but the
converse is not true in general. In [2], examples of genus 2 curves, hence
abelian varieties of dimension 2, with everywhere good reduction over quartic
fields were given. However, the Jacobian varieties of these curves are not
absolutely simple. The first examples of genus two curves with everywhere good
reduction over real quadratic fields were constructed in [9, 10]. This was
approached by combining the computation of Hecke eigenvalues of certain
Hilbert modular forms, using new rational models of Hilbert modular surfaces,
together with assuming Eichler-Shimura conjecture for Hilbert modular forms.
Up to the knowledge of the authors, the only example of a genus 2 curve with
everywhere good reduction over a complex quadratic field can be found in [3,
Theorem 5.2].

In this article we produce genus two curves with everywhere good reduction
over real quadratic fields without the current restrictions on the
discriminant or the class number of the field. We also display a $1$-parameter
family of pairs of genus 2 curves and real (complex) quadratic fields such
that the curve has everywhere good reduction over the quadratic field. Finally
we prove the existence of genus two curves with everywhere good reduction over
a quadratic field such that its Jacobian has trivial endomorphisms, hence the
Jacobian is an absolutely simple abelian variety.

Our approach makes use of Mestre’s algorithm to construct genus 2 curves from
their Igusa invariants. Starting from rational invariants that satisfy certain
local properties, we obtain genus two curves over the rational field. The only
nontrivial automorphism of these curves is the hyperelliptic involution.
Moreover, the minimal discriminants of these curves are of prescribed
valuations at every rational bad prime of the curve. After a quadratic base
change we prove that these curves attain everywhere good reduction. Mestre’s
algorithm can be adjusted to produce genus 2 curves with different
automorphism groups, [6]. This enables us to produce infinitely many such
curves with everywhere good reduction over corresponding quadratic fields.

##  2\. Preliminaries on genus two curves

Let $K$ be a perfect field. Let $C$ be a (projective smooth) curve of genus
two defined by a Weierstrass equation over $K$ of the form

(1) $\displaystyle E:\;y^{2}+Q(x)y=P(x),\qquad P(x),Q(x)\in K[x],\;\deg
Q(x)\leq 3,\;\deg P(x)\leq 6.$

Given a Weierstrass equation of the form
$y^{2}=f(x)=a_{0}x^{6}+a_{1}x^{5}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{2}+a_{5}x+a_{6}$,
Igusa associated to $f(x)$ the invariants $J_{2}$, $J_{4}$, $J_{6}$,
$J_{10}\in{\mathbb{Z}}[1/2,a_{0},a_{2},\cdots,a_{6}].$ In fact,
$J_{10}=2^{-12}\cdot\operatorname{Discriminant}(f)$ if $a_{0}\neq 0$, and
$J_{10}=2^{-12}a_{1}^{2}\cdot\operatorname{Discriminant}(f)$ otherwise. For
the full description of these invariants, see [12] when $a_{0}=0$, and [16,
§2] for the general case. We define Igusa invariants,$J_{2i}(E)$, $i=1,2,3,5$,
of the Weierstrass equation $E$ to be the ones associated to
$P(x)+Q(x)^{2}/4$.

Another set of invariants, Igusa-Clebsch invariants,
$I_{2},I_{4},I_{6},I_{10}$ may be associated to the Weierstrass equation $E$.
The Igusa invariants are linked to the Igusa-Clebsch invariants as follows:

$\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle I_{2}/8,$ $\displaystyle
J_{4}$ $\displaystyle=$ $\displaystyle(4J_{2}^{2}-I_{4})/96,$ $\displaystyle
J_{6}$ $\displaystyle=$ $\displaystyle(8J_{2}^{3}-160J_{2}J_{4}-I_{6})/576,$
$\displaystyle J_{8}$ $\displaystyle=$
$\displaystyle(J_{2}J_{6}-J_{4}^{2})/4,$ (2) $\displaystyle J_{10}$
$\displaystyle=$ $\displaystyle I_{10}/4096.$

If $E^{\prime}:y^{\prime 2}+S(x^{\prime})y^{\prime}=T(x^{\prime})$ is another
Weierstrass equation describing $C$, then there exist $a,b,c,d\in K$, $e\in
K^{\times}$, $H[x]\in K[x]$ such that

(3) $\displaystyle x^{\prime}=\frac{ax+b}{cx+d},\qquad
y^{\prime}=\frac{ey+H(x)}{(cx+d)^{3}},\qquad\textrm{ where }ad-bc\neq 0.$

In this case, we say that $E$ and $E^{\prime}$ are $K$-equivalent. Two
$K$-equivalent Weierstrass equations describe isomorphic genus 2 curves over
$K$.

If $\operatorname{char}K\neq 2$, then one may assume that $Q(x)=0$. Further,
one can see that

(4) $\displaystyle J_{10}(E^{\prime})=J_{10}(E)e^{-20}(ad-bc)^{30}.$

In fact, two Weierstrass equations $E$ and $E^{\prime}$ describe isomorphic
genus two curves over $\overline{K}$ if and only if there is a
$\lambda\in\overline{K}^{\times}$ such that
$J_{2i}(E^{\prime})=\lambda^{2i}J_{2i}(E)$.

We now assume that $K$ is a discrete valuation field with ring of integers
${\mathcal{O}_{K}}$, and normalized discrete valuation $\nu$. If $C$ is a
genus two curve defined over $K$ by a Weierstrass equation
$E:y^{2}+Q(x)y=P(x)$, then $E$ is said to be integral if
$P(x),Q(x)\in{\mathcal{O}_{K}}[x]$. An integral Weierstrass equation $E$ is
minimal over $K$ if $\nu(J_{10}(E))\leq\nu(J_{10}(E^{\prime}))$ for any other
Weierstrass equation $E^{\prime}$ describing $C$ over $K$. More precisely,
$\nu(J_{10}(E^{\prime}))=\nu(J_{10}(E))+10m$, where $m\geq 0$ is given by the
formula in (4).

The curve $C$ is said to have good reduction over $K$ if there exists an
integral Weierstrass equation $E$ describing $C$ over $K$ such that
$\nu(J_{10}(E))=0$. It is said to be of potential good reduction over $K$ if
there exists a finite extension $K\subset K^{\prime}$ such that $C$ attains
good reduction over $K^{\prime}$. The following criterion for potential good
reduction is due to Igusa, [12].

######  Theorem 2.1.

Let $C$ be a genus two curve defined over $K$ by the Weierstrass equation
$E:y^{2}+Q(x)y=P(x)$. The curve $C$ has potential good reduction over $K$ if
and only if $J_{2i}(E)^{5}/J_{10}(E)^{i}\in{\mathcal{O}_{K}}$ for every $i\leq
5$.

Assuming that $K$ is a number field with ring of integers ${\mathcal{O}_{K}}$.
A Weierstrass equation $E:y^{2}+Q(x)y=P(x)$ describing a genus two curve $C$
defined over $K$ is said to be integral if $P(x),Q(x)\in{\mathcal{O}_{K}}[x]$;
whereas an integral Weierstrass equation $E$ is said to be minimal at a prime
$\mathfrak{p}$ of $K$ if it is minimal over $K_{\mathfrak{p}}$, the completion
of $K$ at $\mathfrak{p}$. The equation $E$ is said to be globally minimal if
it is minimal at every prime $\mathfrak{p}$ of $K$. If $K$ has class number 1,
then one can always describe $C$ by a globally minimal Weierstrass equation,
see [17, Proposition 2].

Similarly, $C$ has good reduction at a prime $\mathfrak{p}$ of $K$ if it has
good reduction over $K_{\mathfrak{p}}$, and $C$ is said to have everywhere
good reduction if $C$ has good reduction at every prime $\mathfrak{p}$ of $K$.

Finally, one knows that if a genus two curve has good reduction at a prime
$\mathfrak{p}$ of $K$, then its Jacobian has good reduction at $\mathfrak{p}$.

##  3\. Mestre’s algorithm

Let $K$ be a number field. If $E$ is a Weierstrass equation describing a genus
2 curve such that the Igusa invariants $J_{2}(E),J_{4}(E),J_{6}(E)$ and
$J_{10}(E)$ are $K$-rational, then this does not mean necessarily that $C$ can
be described by a Weierstrass equation over $K$. In [21], Mestre was able to
give a necessary and sufficient condition for $C$ to be defined by a
Weierstrass equation over $K$ when the only nontrivial element in the
automorphism group of $C$ is the hyperelliptic involution. For genus 2 curves
with different automorphism groups, it was shown in [6] that if the Igusa
invariants of $E$ are $K$-rational, then $E$ is $\overline{K}$-equivalent to a
Weierstrass equation defined over $K$.

In what follows, we explain the construction of Mestre and Cardona-Quer. The
description of the construction is algorithmic and it is implemented in MAGMA
[5] and Sage [29]. Given a quadruple $(J_{2},J_{4},J_{6},J_{10})\in K^{4}$,
$J_{10}\neq 0$, we want to decide whether there is a Weierstrass equation
defined over $K$ whose Igusa invariants are given by this quadruple.
Equivalently, one may consider the following problem. If $F$ is a field with
characteristic $\neq 2,3,5$, we set

$\mathcal{M}_{2}(\overline{F})=\\{(J_{2},J_{4},J_{6},J_{10})\in\overline{F}^{4}:J_{10}\neq
0\\}/F^{\times}$

where the action of $F^{\times}$ is defined by a weighted scaling of the form

$a\cdot(J_{2},J_{4},J_{6},J_{10})=(a^{2}J_{2},a^{4}J_{4},a^{6}J_{6},a^{10}J_{10}).$

We write
$(J_{2}^{\prime},J_{4}^{\prime},J_{6}^{\prime},J_{10}^{\prime})\sim_{F}(J_{2},J_{4},J_{6},J_{10})$
if
$(J_{2}^{\prime},J_{4}^{\prime},J_{6}^{\prime},J_{10}^{\prime})=a\cdot(J_{2},J_{4},J_{6},J_{10})$
for some $a\in F^{\times}$. A moduli point $J\in\mathcal{M}_{2}(\overline{F})$
is defined over $F$ if it is the equivalence class of a quadruple
$(J_{2},J_{4},J_{6},J_{10})$ with all $J_{2i}\in F$, $i=1,2,3,5$.

Let $C$ be a curve of genus 2 defined over $\overline{K}$ by a Weierstrass
equation $E$. We say that $F\subset\overline{K}$ is a field of moduli for $C$
if the equivalence class of $(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))$ in
$\mathcal{M}_{2}(\overline{K})$ is defined over $F$. We say that
$F\subset\overline{K}$ is a field of definition for $C$ if there exists a
Weierstrass equation $E^{\prime}$ defined over $F$ which is
$\overline{K}$-equivalent to $E$. The problem now boils down to: given that
$K$ is a field of moduli for $C$, is $K$ a field of definition for $C$?

Let $C$ be a genus 2 curve such that $K$ is a field of moduli for $C$. One may
construct homogeneous ternary forms $Q_{J},T_{J}\in K[u,v,w]$ of degrees 2 and
3 respectively. The conic $M_{J}$ defined by $Q_{J}=0$ has a $K$-rational
point if $C$ has a nontrivial automorphism other than the hyperelliptic
involution, see [6]. Whereas, the conic defined by $M_{J}:Q_{J}=0$ may not
have a $K$-rational point if the only nontrivial automorphism of $C$ is the
hyperelliptic involution. In the latter case, if $M_{J}(K)=\emptyset$, then
there exists a quadratic extension $K\subset K^{\prime}$ such that
$M_{J}(K^{\prime})\neq\emptyset$, see [21]. The existence of a rational point
over $L\in\\{K,K^{\prime}\\}$ gives rise to a parametrization
$(u,v,w)=(u(x),v(x),w(x))$ where $u,v,w\in L[x]$ are polynomials of degree 2.
Now we get the Weierstrass equation $y^{2}=T_{J}(u(x),v(x),w(x))$ describing
$C$ where $T_{J}(u(x),v(x),w(x))\in L[x]$ is a degree 6 polynomial. The
following theorem summarizes the above discussion.

######  Theorem 3.1.

Let $E$ be a Weierstrass equation describing a genus 2 curve $C$. Assume that
$J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E)\in K$. Let $J$ be the class of the
quadruple of Igusa invariants $(J_{2}(E),J_{4}(E),\\\ J_{6}(E),J_{10}(E))$ of
$E$ in $\mathcal{M}_{2}(\overline{K})$.

  * (1)

(Mestre, [21]) If ${\operatorname{Aut}}(C)\cong{\mathbb{Z}}_{2}$, then $K$ is
a field of definition for $C$ if and only if $M_{J}(K)\neq\emptyset$.

  * (2)

(Cardona-Quer, [6]) If ${\operatorname{Aut}}(C)\not\cong{\mathbb{Z}}_{2}$,
then $K$ is always a field of definition for $C$.

One remarks that when ${\operatorname{Aut}}(C)\cong{\mathbb{Z}}_{2}$, if
$M_{J}(K)=\emptyset$, there are many quadratic extensions $K\subset
K^{\prime}$ over which $M_{J}(K^{\prime})\neq\emptyset$. We set two of the
coordinates $u,v,w$ to be fixed and solve for the third coordinate. This gives
a rational point on the conic over a quadratic extension $K\subset
K^{\prime}$.

One disadvantage about Mestre’s algorithm is that the coefficients of the
sextic $T_{J}(u(x),v(x),w(x))$ are large numbers. This is why we will spend
the next section discussing minimization of genus two curves.

##  4\. Minimization of genus two curves

In this section, $K$ is a complete discrete valuation field with ring of
integers ${\mathcal{O}_{K}}$, maximal ideal $\mathfrak{m}$, residue field
$k={\mathcal{O}_{K}}/\mathfrak{m}$, a uniformiser $\pi$, and normalized
discrete valuation $\nu$. We assume that $\operatorname{char}k>5$ throughout
this section. Moreover, we will be mainly concerned with genus two curves
described by Weierstrass equations $E$ defined over $K$ with potential good
reduction over $K$. In other words,
$J_{2i}^{5}(E)/J_{10}^{i}(E)\in{\mathcal{O}_{K}}$ for every $i\leq 5$, see
Theorem 2.1.

Let $(j_{2},j_{4},j_{6},j_{10})\in K^{4}$ where $j_{10}\neq 0$. Let $C$ be a
genus 2 curve defined by a Weierstrass equation $E$ over $K$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))=\lambda\cdot(j_{2},j_{4},j_{6},j_{10})$
for some $\lambda\in K^{\times}.$ Assume, moreover, that $C$ has potential
good reduction over $K$. In what follows, we discuss the possibilities for
$\nu(J_{10}(E))$ if $E$ is a minimal Weierstrass equation over $K$.

######  Proposition 4.1.

Let $E_{1}$ and $E_{2}$ be two minimal Weierstrass equations over $K$
describing two genus 2 curves with potential good reduction over $K$. Assume
moreover that $E_{1}$ and $E_{2}$ are $\overline{K}$-equivalent. Then

$|\nu(J_{10}(E_{1}))-\nu(J_{10}(E_{2}))|\in\\{0,5,10,15,20\\}.$

Proof: We set $\overline{\nu}$ to be an extension of $\nu$ to $\overline{K}$.
One knows that
$(J_{2}(E_{2}),J_{4}(E_{2}),J_{6}(E_{2}),J_{10}(E_{2}))=(\lambda^{2}J_{2}(E_{1}),\lambda^{4}J_{4}(E_{1}),\lambda^{6}J_{6}(E_{1}),\lambda^{10}J_{10}(E_{1}))$
for some $\lambda\in\overline{K}^{\times}$. Therefore, $[K(\lambda):K]\leq 2$
as $E_{2}$ is integral over $K$ and so
$\lambda^{2}J_{2}(E_{1})\in{\mathcal{O}_{K}}$. Since both curves have
potential good reduction over $K$ and $\operatorname{char}k>5$, this implies
that $\nu(J_{10}(E_{1})),\nu(J_{10}(E_{2}))\leq 20$, see [17, Remarque 13]. It
follows that
$\nu(J_{10}(E_{2}))=10\overline{\nu}(\lambda)+\nu(J_{10}(E_{1}))\leq 20$.

The facts that both $E$, $E^{\prime}$ are minimal, hence integral, and
$J_{2}(E^{\prime})=\lambda^{2}J_{2}(E)\in{\mathcal{O}_{K}}$, imply that
$\lambda^{2}\in K^{\times}$. Therefore,
$10\overline{\nu}(\lambda)\in\\{0,5,10,15,20\\}$. $\Box$

An integral Weierstrass equation $E:y^{2}+Q(x)y=P(x)$ over $K$ will be called
reduced over $K$ if $4P(x)+Q(x)^{2}\neq 0$ in $k[x]$, otherwise, it is called
non-reduced over $K$.

######  Remark 4.2.

It is easy to see that if $E$ is an integral Weierstrass equation describing a
genus 2 curve $C$ over $K$, and $E$ is non-reduced over $K$, then the
transformation $x\mapsto x$, $y\mapsto\pi^{1/2}y$ yields an integral
Weierstrass equation $E^{\prime}$ over $K$ that describes a genus 2 curve
isomorphic to $C$ over $K(\pi^{1/2})$, and
$\nu(J_{10}(E^{\prime}))=\nu(J_{10}(E))-10$.

We also remark that if $E$ is a minimal Weierstrass equation over $K$ with
$\nu(J_{10}(E))\leq 9$, then $E$ is reduced over $K$, see [17, Proposition 7].

######  Remark 4.3.

Let $E:y^{2}+Q(x)y=P(x)$ be integral over $K$ and
$4P(x)+Q(x)^{2}=\sum_{i=0}^{6}a_{i}x^{i}$. If

$\nu(a_{0})\geq 7,\;\nu(a_{1})\geq 5,\;\nu(a_{2})\geq 3,\;\nu(a_{3})\geq 1,$

then $E$ is $K$-equivalent to a non-reduced Weierstrass equation $E^{\prime}$
via the transformation $x\mapsto\pi^{2}x$, $y\mapsto\pi^{3}y$. Moreover,
$\nu(J_{10}(E^{\prime}))=\nu(J_{10}(E))$.

We recall that if $E$ is a minimal Weierstrass equation over $K$ describing a
genus 2 curve $C$ with $J_{2}(E),J_{4}(E),J_{6}(E)\in{\mathcal{O}_{K}}$ and
$J_{10}(E)\in\mathcal{O}_{K}^{\times}$, then the curve $C$ has potential good
reduction over $K$ since $J_{2i}^{5}(E)/J_{10}^{i}(E)\in{\mathcal{O}_{K}}$ for
every $i$, see Theorem 2.1. Therefore, there exists a minimal finite extension
$F$ of $K$ over which $C$ attains good reduction. In other words, $F$ is
contained in any other such field. Now we state the main result of this
section.

######  Theorem 4.4.

Let $J_{2},J_{4},J_{6},J_{10}$ be such that $J_{i}\in{\mathcal{O}_{K}}$,
$i=2,4,6$, and $J_{10}\in\mathcal{O}_{K}^{\times}$. Let $E$ be a minimal
Weierstrass equation over $K$ describing a genus 2 curve $C$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))\sim_{\overline{K}}(J_{2},J_{4},J_{6},J_{10})$.
Let $F$ be the minimal field extension over which $C$ attains good reduction
and $[F:K]>1$.

If $E$ is not $F$-equivalent to a non-reduced Weierstrass equation
$E^{\prime}$ over $F$ with $\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$,
then $\nu(J_{10}(E))\in\\{5,10,15\\}.$ Moreoer, one has

  * a)

If $\nu(J_{10}(E))=5$, then $[F:K]=6$.

  * b)

If $\nu(J_{10}(E))=10$, then $[F:K]=3$.

  * c)

If $\nu(J_{10}(E))=15$, then $[F:K]=2$.

Proof: One sees that
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))=(\lambda^{2}J_{2},\lambda^{4}J_{4},\lambda^{6}J_{6},\lambda^{10}J_{10})$
for some $\lambda\in\overline{K}^{\times}$. Now since $E$ is minimal hence
integral, it follows that $\lambda^{2}J_{2}\in{\mathcal{O}_{K}}$. Therefore,
$\lambda^{2}\in K$. Since
$\nu(J_{10}(E))=5\nu(\lambda^{2})+\nu(J_{10})=5\nu(\lambda^{2})\leq 20$, [17,
Remarque 13], it follows that $\nu(J_{10}(E))\in\\{0,5,10,15,20\\}$. We notice
that $\nu(J_{10}(E))\neq 0$ as $[F:K]>1$.

Since $C$ has potential good reduction over $K$, and $\operatorname{char}k>5$,
it follows that the minimal field $F$ over which $C$ attains good reduction is
a totally and tamely ramified extension of $K$, hence cyclic, with
$n_{F}:=[F:K]\in\\{2,3,4,5,6,8,10\\}$, see [15, Corollaire 4.1]. We set
$\pi_{F}$ to be a uniformiser in the ring of integers $\mathcal{O}_{F}$ of
$F$, and we assume without loss of generality that $\pi_{F}^{n_{F}}$ is either
$\pi$ or $-\pi$. We set $\nu_{F}$ to be the corresponding discrete valuation
on $F$.

One knows that since $E:y^{2}+Q(x)=P(x)$ is not minimal over $F$, this yields
that either $E$ is non-reduced over $K$ or
$4P(x)+Q(x)^{2}=\sum_{i=0}^{6}a_{i}x^{i}\in{\mathcal{O}_{K}}[x]$ has a root
$x_{0}$ of multiplicity at least $4$ in $k$, see for example [22, Lemma 5.3].

Since $E$ is reduced by assumption, then $4P(x)+Q(x)^{2}$ has a root $x_{0}$
of multiplicity at least 4. We use the translation map $x-t$ where $t$ is a
lift of $x_{0}$ in ${\mathcal{O}_{K}}$. This map does not change
$\nu(J_{10}(E))$, and it allows us to assume that $\nu(a_{i})\geq 1$ when
$i=0,1,2,3$, as now $0$ is the root of $4P(x)+Q(x)^{2}$ in $k[x]$ that has
multiplicity at least $4$. Now we may apply the transformation
$x\mapsto\pi_{F}^{r}x$, $y\mapsto\pi_{F}^{s}y$, for some $r,s\geq 1$, to
obtain the minimal Weierstrass equation $E^{\prime}$ over $F$ with
$\nu_{F}(J_{10}(E^{\prime}))=0$, see for example [22, Lemma 11.2]. It follows
that $n_{F}\nu(J_{10}(E))=20s-30r$, see (3) and (4). In the following table,
we run through the possibilities for $r$ and $s$ when
$n_{F}\in\\{2,3,4,5,6,8,10\\}$. The entries of the table contain the solutions
$(r,s)$ for the linear equation $n_{F}\nu(J_{10}(E))=20s-30r$ where $t\geq 0$
is an integer.

$\nu(J_{10}(E))$ $n_{F}$ | 2 | 3 | 4 | 5 | 6 | 8 | 10
---|---|---|---|---|---|---|---
5 | $(1+2t,2+3t)$ | - | $(2+2t,4+3t)$ | - | $(1+2t,3+3t)$ | $(2+2t,5+3t)$ | $(1+2t,4+3t)$
10 | $(2+2t,4+3t)$ | $(1+2t,3+3t)$ | $(2+2t,5+3t)$ | $(1+2t,4+3t)$ | $(2+2t,6+3t)$ | $(2+2t,7+3t)$ | $(2+2t,8+3t)$
15 | $(1+2t,3+3t)$ | - | $(2+2t,6+3t)$ | - | $(1+2t,6+3t)$ | $(2+2t,9+3t)$ | $(1+2t,9+3t)$
20 | $(2+2t,5+3t)$ | $(2+2t,6+3t)$ | $(2+2t,7+3t)$ | $(2+2t,8+3t)$ | $(2+2t,9+3t)$ | $(2+2t,11+3t)$ | $(2+2t,13+3t)$

One remarks that $E^{\prime}$ is given by $y^{2}+Q_{1}(x)y=P_{1}(x)$, thus

$4P_{1}(x)+Q_{1}(x)^{2}=\sum_{i=0}^{6}\pi_{F}^{ir-2s}a_{i}x^{i}\in\mathcal{O}_{F}[x].$

Since $E^{\prime}$ describes a curve with good reduction over $F$, it follows
that $4P_{1}(x)+Q_{1}(x)^{2}$ is separable of degree at least $5$ in
$k_{F}[x]$. Therefore, if $5r-2s>0$, then this means that the latter
polynomial is of degree at most $4$ in $k_{F}[x]$. Also, we note that if
$5r-2s<0$, then the integrality of $E^{\prime}$ implies
$\nu(a_{5})>\lfloor(2s-5r)/n_{F}\rfloor$; similarly, if $6r-2s<0$, then
$\nu(a_{6})>\lfloor(2s-6r)/n_{F}\rfloor$. Thus, if $6r-2s<0$, then
$4P_{1}(x)+Q_{1}(x)^{2}$ is again of degree at most $4$ in $k_{F}[x]$. This
leaves us with the following possibilities:

$\nu(J_{10}(E))$ $n_{F}$ | 2 | 3 | 4 | 5 | 6 | 8 | 10
---|---|---|---|---|---|---|---
5 | - | - | - | - | $(1,3)$ | $(2,5)$ | -
10 | - | $(1,3)$ | $(2,5)$ | - | $(2,6)$ |  $(2+2t,7+3t)$, $t=1$ |  $(2+2t,8+3t)$, $t=1$
15 | $(1,3)$ | - | - | - | $(1,6)$ |  $(2+2t,9+3t)$, $t=0,1,2$ |  $(1+2t,9+3t)$, $t=2,3$
20 | $(2,5)$ | $(2,6)$ |  $(2+2t,7+3t)$, $t=0,1$ | - |  $(2+2t,9+3t)$, $t=0,1,2$ |  $(2+2t,11+3t)$, $t=1,3$ |  $(2+2t,13+3t)$, $t=4$

We now consider the case $\nu(J_{10}(E))=5$. If $n_{F}=8$, the pair $(r,s)$
takes the value $(2,5)$. In the latter case, we have

$\nu_{F}(a_{0})\geq 16,\ \nu_{F}(a_{1})\geq 8,\ \nu_{F}(a_{2})\geq 8,\
\nu_{F}(a_{3})\geq 8,\ \nu_{F}(a_{4})\geq 8.$

Hence Remark 4.3 implies that $E$ is $F$-equivalent to a non-reduced
Weierstrass equation $E^{\prime}$ over $F$ with
$\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$.

We consider the case $\nu(J_{10}(E))=10$. If $n_{F}=4$, the pair $(r,s)$ takes
the value $(2,5)$. The integrality of $E$ over $K$ implies that

$\nu_{F}(a_{0})\geq 12,\ \nu_{F}(a_{1})\geq 8,\ \nu_{F}(a_{2})\geq 8,\
\nu_{F}(a_{3})\geq 4,\ \nu_{F}(a_{4})\geq 4.$

We conclude by Remark 4.3 that this pair does not allow $E$ to satisfy the
hypothesis of the theorem. For $n_{F}=6$ and the pair $(r,s)=(2,6)$, we
conclude similarly as then

$\nu_{F}(a_{0})\geq 12,\ \nu_{F}(a_{1})\geq 12,\ \nu_{F}(a_{2})\geq 12,\
\nu_{F}(a_{3})\geq 6,\ \nu_{F}(a_{4})\geq 6,\ \nu_{F}(a_{5})\geq 6.$

In a similar fashion, one has for $n_{F}=8$ and the pair $(r,s)=(4,10)$ the
following valuations

$\nu_{F}(a_{0})\geq 24,\ \nu_{F}(a_{1})\geq 16,\ \nu_{F}(a_{2})\geq 16,\
\nu_{F}(a_{3})\geq 8,\ \nu_{F}(a_{4})\geq 8,$

whereas for $n_{F}=10$ and the pair $(r,s)=(4,11)$, one has

$\nu_{F}(a_{0})\geq 30,\ \nu_{F}(a_{1})\geq 20,\ \nu_{F}(a_{2})\geq 20,\
\nu_{F}(a_{3})\geq 10,\ \nu_{F}(a_{4})\geq 10.$

A similar argument shows that when $\nu(J_{10}(E))=15$, then the only
admissible pair is $(1,3)$ when $n_{F}=2$, otherwise for $n_{F}\geq 6$ and a
correponding pair from the latter table will force $E$ to be $F$-equivalent to
a non-reduced Weierstrass equation $E^{\prime}$ over $F$ with
$\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$. As for $\nu(J_{10}(E))=20$,
any choice for $n_{F}$ and a corresponding pair of $(r,s)$ will again yield
that $E$ is $F$-equivalent to a non-reduced Weierstrass equation $E^{\prime}$
over $F$ with $\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$. $\Box$

In the proof of Theorem 4.4, we mentioned the minimization transformations for
$E$ over the minimal field $F$ to produce an integral equation $E^{\prime}$
over $F$ with $\nu_{F}(J_{10}(E^{\prime}))=0$. We list these transformations
in the remark below.

######  Remark 4.5.

Let $J_{2},J_{4},J_{6},J_{10}$ be such that $J_{i}\in{\mathcal{O}_{K}}$,
$i=2,4,6$, and $J_{10}\in\mathcal{O}_{K}^{\times}$. Let $E:y^{2}+Q(x)y=P(x)$
be a minimal Weierstrass equation over $K$ describing a genus 2 curve $C$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))\sim_{\overline{K}}(J_{2},J_{4},J_{6},J_{10})$.
Let $F$ be the minimal field extension over which $C$ attains good reduction
and $[F:K]>1$.

If $E$ is not $F$-equivalent to a non-reduced Weierstrass equation
$E^{\prime}$ over $F$ with $\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$,
then set $n_{F}=30/\nu(J_{10}(E))$ and $\pi_{F}=(\pm\pi)^{1/n_{F}}$. After a
translation, one may use the transformation
$(x,y)\mapsto(\pi_{F}x,\pi_{F}^{3}y)$ to obtain an integral Weierstrass
equation $E^{\prime}$ over $F$ with $\nu_{F}(J_{10}(E^{\prime}))=0$. One may
also see that $4P(x)+Q(x)^{2}$ has a root $x_{0}$ of multiplicity $6$ in
$k[x].$

######  Remark 4.6.

In Theorem 4.4, one may relax the condition that $E$ is not $F$-equivalent to
a non-reduced Weierstrass equation $E^{\prime}$ over $F$ with
$\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$ and replace it with $E$ is
not $K$-equivalent to a non-reduced Weierstrass equation $E^{\prime}$ over $K$
with $\nu(J_{10}(E^{\prime}))=\nu(J_{10}(E))$. This happens because in the
three cases of the theorem, one has $\nu_{F}(J_{10}(E))=30$, hence if $E$ is
$F$-equivalent to a non-reduced Weierstrass equation $E^{\prime}$ over $F$
with $\nu_{F}(J_{10}(E^{\prime}))=\nu_{F}(J_{10}(E))$, then one may replace
$E$ with an integral equation $E_{1}$ over $F$ with
$\nu_{F}(J_{10}(E_{1}))=20$. But since the curve $C$ attains good reduction
over $F$, then one may find $r$ and $s$ such that $20s-30r=20$, i.e.,
$(r,s)=(2+2t,4+3t)$ but as in the proof of the theorem, this implies that $C$
does not have good reduction over $F$.

######  Corollary 4.7.

Let $J_{2},J_{4},J_{6},J_{10}$ be such that $J_{i}\in{\mathcal{O}_{K}}$,
$i=2,4,6$, and $J_{10}\in\mathcal{O}_{K}^{\times}$. Let $E$ be a minimal
Weierstrass equation over $K$ describing a genus 2 curve $C$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))\sim_{\overline{K}}(J_{2},J_{4},J_{6},J_{10})$.

If $E:y^{2}+Q(x)y=P(x)$ is not $K$-equivalent to a non-reduced Weierstrass
equation $E^{\prime}$ over $F$ with $\nu(J_{10}(E^{\prime}))=\nu(J_{10}(E))$,
then the polynomial $4P(x)+Q(x)^{2}$ is irreducible over $K$.

Proof: Let $\mathcal{C}$ be a stable model of $C$ over the strict
henselisation of ${\mathcal{O}_{K}}$ in the sense of [15, Définition]. Let
$f:\mathcal{C}\to\mathcal{Z}:=\mathcal{C}/\langle\sigma\rangle$ where $\sigma$
is the extension of the hyperelliptic involution of $C$ to $\mathcal{C}$. We
set $\omega$ to be the point in the generic fiber of $\mathcal{Z}$
corresponding to $x=\infty$, and $\overline{\omega}$ its specialization in the
special fiber of $\mathcal{Z}$. Then $\overline{\omega}$ is said to be
ramified if $f$ is ramified at $\overline{\omega}$.

The fact that $n_{F}$ is either $6,3$, or $2$ when $\nu(J_{10}(E))=5,10$, or
$15$, respectively, implies that $\overline{\omega}$ is non-ramified, see [16,
Théorèm 1]. Thus $C\to{\mathbb{P}}_{K}^{1}$ is non-ramified at any rational
point $x$ in ${\mathbb{P}}^{1}_{K}$, see [16, Proposition 4.1.1], hence
$4P(x)+Q(x)^{2}$ is irreducible over $K$. $\Box$

##  5\. Globally minimal Weierstrass equations

In this section we show that the existence of a minimal Weierstrass equation
describing a genus 2 curve over a number field $K$ whose Igusa invariant
$J_{10}$ satisfies certain properties implies the existence of a genus two
curve with everywhere good reduction over a quadratic extension of $K$.

We assume throughout this section that $K$ is a number field with ring of
integers ${\mathcal{O}_{K}}$. If $\mathfrak{p}$ is a prime of $K$, then
$K_{\mathfrak{p}}$ is the completion of $K$ at $\mathfrak{p}$, with ring of
integers $\mathcal{O}_{K_{\mathfrak{p}}}$, maximal ideal
$\mathfrak{m_{\mathfrak{p}}}$, residue field $k_{\mathfrak{p}}$, a uniformiser
$\pi_{\mathfrak{p}}$ in ${\mathcal{O}_{K}}$, and normalized discrete valuation
$\nu_{\mathfrak{p}}$.

######  Theorem 5.1.

Let $K$ be a number field with class number $1$. Let
$(J_{2},J_{4},J_{6},J_{10})$ be such that $J_{i}\in{\mathcal{O}_{K}}$,
$i=2,4,6$, and $J_{10}\in\mathcal{O}_{K}^{\times}$. Let $E$ be a globally
minimal Weierstrass equation over $K$ describing a genus 2 curve $C$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))\sim_{\overline{K}}(J_{2},J_{4},J_{6},J_{10})$.
Assume that for every bad prime $\mathfrak{p}$ of $C$ in $K$, one has
$\gcd(\operatorname{char}k_{\mathfrak{p}},30)=1$, and moreover, one of the
following conditions is satisfied:

  * i)

either $\nu_{\mathfrak{p}}(J_{10}(E))=10$ and $E$ is non-reduced over
$K_{\mathfrak{p}}$,

  * ii)

or $\nu_{\mathfrak{p}}(J_{10}(E))=15$, and $E$ is not
$K_{\mathfrak{p}}$-equivalent to a non-reduced Weierstrass equation
$E^{\prime}$ over $K_{\mathfrak{p}}$ with
$\nu_{\mathfrak{p}}(J_{10}(E^{\prime}))=15$.

Then $C$ has everywhere good reduction over
$K\left((\pm\pi_{{\mathfrak{p}}_{1}}\ldots\pi_{{\mathfrak{p}}_{k}})^{1/2}\right)$
where $\mathfrak{p}_{i}$’s are the primes such that
$\nu_{\mathfrak{p}_{i}}(J_{10}(E))=15$.

Proof: In view of Remark 4.2, if $E:y^{2}+Q(x)y=P(x)$ is non-reduced over
$K_{\mathfrak{p}}$ and $\nu_{\mathfrak{p}}(J_{10}(E))=10$, then we apply the
transformation $x\mapsto x$ and $y\mapsto\pi_{\mathfrak{p}}^{1/2}y$. This
yields an integral Weierstrass equation $E^{\prime}$ over $K_{\mathfrak{p}}$
with $\nu_{\mathfrak{p}}(J_{10}(E^{\prime}))=0$.

So we may (and will) assume that evey bad prime of $C$ is of type ii). Now
that $\gcd(\operatorname{char}k_{\mathfrak{p}},30)=1$, Remark 4.5 asserts that
for every such prime $\mathfrak{p}$ one may use a translation $x\mapsto
x+x_{\mathfrak{p}}$ where $x_{\mathfrak{p}}$ is an integral lift of the
multiplicity-$6$ root of $4P(x)+Q(x)^{2}$ in $k_{\mathfrak{p}}$. A Chinese
Remainder Theorem argument shows that there exists $x_{0}\in{\mathcal{O}_{K}}$
such that $x_{0}$ is a lift of $x_{\mathfrak{p}}$ for every bad prime
$\mathfrak{p}$. So using the transformation $x\mapsto x+x_{0}$, which neither
changes $\nu_{\mathfrak{p}}(J_{10}(E))$ for any $\mathfrak{p}$ nor the
integrality of $E$ over $K$, will allow us to use the minimization
transformation $x\mapsto(\pm\pi_{\mathfrak{p}})^{1/2}x$,
$y\mapsto(\pm\pi_{\mathfrak{p}})^{3/2}y$ for every bad prime $\mathfrak{p}$ of
$C$ to obtain an integral Weierstrass equation $E_{\mathfrak{p}}$ over
$F_{\mathfrak{p}}:=K_{\mathfrak{p}}((\pm\pi_{\mathfrak{p}})^{1/2})$ with
$\nu_{F_{\mathfrak{p}}}(J_{10}(E_{\mathfrak{p}}))=0$, see Theorem 4.4.
Therefore, the choice of $\pi_{\mathfrak{p}}$ in ${\mathcal{O}_{K}}$ shows
that using the transformation

$x\mapsto(\pm\pi_{\mathfrak{p}_{1}}\pi_{\mathfrak{p}_{2}}\ldots\pi_{\mathfrak{p}_{k}})^{1/2}x,\qquad
y\mapsto(\pm\pi_{\mathfrak{p}_{1}}\pi_{\mathfrak{p}_{2}}\ldots\pi_{\mathfrak{p}_{k}})^{3/2}y$

will produce an integral Weierstrass equation $E^{\prime}$ over
$L=K\left((\pm\pi_{{\mathfrak{p}}_{1}}\ldots\pi_{{\mathfrak{p}}_{k}})^{1/2}\right)$
where ${\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{k}$ are the bad primes of $C$
with $\nu_{\mathfrak{p}}(J_{10}(E))=15$, and $E^{\prime}$ describes a genus
two curve over $L$ with everywhere good reduction. $\Box$

We recall that any Weierstrass equation $E:y^{2}+Q(x)y=P(x)$ satisfying the
assumptions of Theorem 5.1 must satsisfy that $4P(x)+Q(x)^{2}$ is irreducible
over $K$, see Corollary 4.7.

######  Remark 5.2.

The assumption that $K$ has class number $1$ in Theorem 5.1 above ensures the
existence of a globally minimal Weierstrass equation describing $C$ over $K$,
see [17, Remarque 6]. However, the existence of a globally minimal Weierstrass
equation for $C$ over
$K\left((\pm\pi_{{\mathfrak{p}}_{1}}\ldots\pi_{{\mathfrak{p}}_{k}})^{1/2}\right)$
is not guaranteed.

######  Remark 5.3.

In [14], it was shown that there is no elliptic curve defined over the field
of rational numbers that attains good reduction at every finite place under
quadratic base change. Theorem 5.1 shows that a genus two curve defined over
${\mathbb{Q}}$ may attain good reduction everywhere over a quadratic field
extension of ${\mathbb{Q}}$. In fact, all genus 2 curves constructed in the
coming sections are defined over ${\mathbb{Q}}$ and they assume everywhere
good reduction over a quadratic field extension.

##  6\. Genus two curves with everywhere good reduction over quadratic fields

In this section we describe the algorithm to compute genus 2 curves with
everywhere good reduction over a quadratic extension of ${\mathbb{Q}}$. Then
we list some of the Weierstrass equations $E$ provided by the algorithm. It is
worth mentioning that in this section we will be producing such equations with
$J_{2}(E)\neq 0$, whereas we will leave the case of $J_{2}(E)=0$ for §7.

######  Algorithm 6.1.

Input: A quadruple $J=(J_{2},J_{4},J_{6},\pm 1)$ where $J_{i}\in{\mathbb{Z}}$.

Output:

  * i)

A quadratic field extension $K$ of ${\mathbb{Q}}$.

  * ii)

A genus two curve $C$ with everywhere good reduction over $K$ described by an
integral Weierstrass equation $E$ over $K$ and
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))=(J_{2},J_{4},J_{6},\pm 1)$. Moreover,
$E$ is globally minimal except possibly at the primes of $K$ lying above the
prime $2$.

The steps are as follows:

  * 1)

If either $J_{2}=0$; or $J_{2}\neq 0$ and $M_{J}({\mathbb{Q}})\neq\emptyset$,
see §3, then there exists a genus two curve $C$ defined by a globally minimal
Weierstrass equation $E:y^{2}+Q(x)y=P(x)$ over ${\mathbb{Q}}$ with
$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))\sim_{\overline{{\mathbb{Q}}}}(J_{2},J_{4},J_{6},\pm
1)$. In other words,

$(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))=(\lambda^{2}J_{2},\lambda^{4}J_{4},\lambda^{6}J_{6},\pm\lambda^{10})\in{\mathbb{Q}}^{4}$

for some $\lambda\in\overline{{\mathbb{Q}}}$ with
$[{\mathbb{Q}}(\lambda):{\mathbb{Q}}]\leq 2$.

  * 2)

We list the curve if for every prime $p\mid J_{10}(E)$ one either has
$\nu_{p}(J_{10}(E))=10$ and $E$ is non-reduced over ${\mathbb{Q}}_{p}$; or if
$\gcd(J_{10}(E),30)=1$, $\nu_{p}(J_{10}(E))=15$ and $E$ is not
${\mathbb{Q}}_{p}$-equivalent to a non-reduced Weierstrass equation $E_{p}$
over ${\mathbb{Q}}_{p}$ with $\nu_{p}(J_{10}(E^{\prime}))=15$.

  * 3)

For the curves in 2):

    * i)

For every prime $p$ with $\nu_{p}(J_{10}(E))=10$ since $E$ is non-reduced over
${\mathbb{Q}}_{p}$, we consider the Weierstrass equation
$y^{2}=(4P(x)+Q(x)^{2})/p$.

    * ii)

For every prime $p$ with $\nu_{p}(J_{10}(E))=15$, the polynomial
$4P(x)+Q(x)^{2}$ has a root $x_{p}$ of multiplicity $6$ in
$\mathbb{F}_{p}[x]$. Let $x_{0}\in{\mathbb{Z}}$ be a lift of $x_{p}$ for every
$p$. Then use the transformation $x\mapsto x+x_{0}$.

  * 4)

If $p_{1},\ldots,p_{n}$ is the list of primes in 3-ii), then use the
transformation

$x\mapsto(\pm p_{1}\ldots p_{n})^{1/2}x,\qquad y\mapsto(\pm p_{1}\ldots
p_{n})^{3/2}y.$

One obtains an integral Weierstrass equation over $K={\mathbb{Q}}(\sqrt{\pm
p_{1}\cdots p_{n}})$, where $p_{1},\cdots,p_{n}$ are the prime divisors of
$J_{10}(E)$ with $\nu_{p}(J_{10}(E))=15$. Moreover, the latter Weierstrass
equation has everywhere good reduction over $K$, see Theorem 5.1.

######  Example 6.2.

To illustrate the algorithm, we consider the quadruple
$J=(J_{2},J_{4},J_{6},J_{10})=(1,1,1,1)$. Using Sage one knows that
$M_{J}({\mathbb{Q}})\neq\emptyset$ and a hyperelliptic curve $C$ corresponding
to this moduli point is given. Meanwhile, one may use MAGMA to obtain a
minimal Weierstrass equation $E$ describing $C$ over $\mathbb{Q}$:

$\displaystyle y^{2}$ $\displaystyle+$
$\displaystyle(-x^{3}+2x^{2}-3x-1)\;y=319942251280530334443517978446668563494310791\;x^{6}$
$\displaystyle-$ $\displaystyle
1348554939815014300021284658314762636324173354\;x^{5}+2368396716413597071671039091048166857266661725\;x^{4}$
$\displaystyle-$ $\displaystyle
2218395544527845496581055490986877026155157575\;x^{3}+1168815723018377789171495695634957245626408185\;x^{2}$
$\displaystyle-$ $\displaystyle
328436816477826084131551647893678647088163154\;x+38454430231425908654742571540402070766182706.$

The discriminant $J_{10}(E)=5^{10}\cdot 17^{15}\cdot 193^{15}\cdot 1193^{15}$.
In fact, $E$ is non-reduced at $5$, so using the minimization transformations
in the algorithm described above, we obtain a genus $2$ curve with everywhere
good reduction over the quadratic field $\mathbb{Q}(\sqrt{17\cdot 193\cdot
1993})$. A minimal Weierstrass equation $E^{+}_{\min}$ describing the curve is
given as follows: We set $a=\sqrt{17\cdot 193\cdot 1993}$, and remark that
$D=17\cdot 193\cdot 1993\equiv 1$ mod $4$.

$\displaystyle E^{+}_{\min}:\quad y^{2}+(-x^{3}+2x^{2}-3x-1)\;y$
$\displaystyle=63988450256106066888703595689333712698862158\;x^{6}$
$\displaystyle+(1+14905246457204355953359713280206751857219456\cdot a)\;x^{5}$
$\displaystyle+5662551478017146195184525794000839333887206015356\;x^{4}$
$\displaystyle+\frac{1}{2}(5+586228886401597062058919441470649503237809367473\cdot
a)\;x^{3}$
$\displaystyle+33406538328776427426641190939693462447792426559697895\;x^{2}$
$\displaystyle+\frac{1}{2}(-3+1037547005393506172459895771934774485127403691693305\cdot
a)\;x$
$\displaystyle+13138915763144011510044623243960742638241073415065183915$

In a similar fashion, the Weierstrass equation $E$ yields a genus $2$ curve
with everywhere good reduction over the quadratic field
$\mathbb{Q}(\sqrt{-17\cdot 193\cdot 1993})$. Setting $a=\sqrt{-17\cdot
193\cdot 1193}$, a minimal Weierstrass equation $E^{-}_{\min}$ describing the
curve is given as follows:

$\displaystyle E^{-}_{\min}:\quad y^{2}+(-x^{3}+2x^{2}-3x-a)\;y$
$\displaystyle=63988450256106066888703595689333712698862158\;x^{6}$
$\displaystyle+(1-14905246457204355953359713280206751857219456\cdot a)\;x^{5}$
$\displaystyle-5662551478017146195184525794000839333887206015361\;x^{4}$
$\displaystyle+(3+293114443200798531029459720735324751618904683736\cdot
a)\;x^{3}$
$\displaystyle+(33406538328776427426641190939693462447792426559697894+a)\;x^{2}$
$\displaystyle-518773502696753086229947885967387242563701845846654\cdot a\;x$
$\displaystyle-13138915763144011510044623243960742638241073415064205357.$

In the Appendix, we list more globally minimal Weierstrass equations
$E:y^{2}+Q(x)y=P(x)$ defined over a quadratic field $K=\mathbb{Q}(\sqrt{D})$
that describe genus two curves with everywhere good reduction over $K$.

##  7\. An infinite family of genus two curves with everywhere good reduction

In this section we consider integral Weierstrass equations $E$ with
$J_{2}(E)=0$. In [6], Mestre’s algorithm was generalized to include genus 2
curves with automorphism group different from ${\mathbb{Z}}_{2}$. These
include genus two curves described by Weierstrass equations with the Igusa
invariant $J_{2}=0$.

In [19, Theorem 3.11], it was shown that if either $J_{2}=J_{4}=0$, or
$J_{2}=J_{6}=0$, then the genus 2 curves that correspond to the moduli points
$(J_{2},J_{4},J_{6},J_{10})$ in $\mathcal{M}_{2}(\overline{{\mathbb{Q}}})$ are
described by $1$-parameter families of Weierstrass equations. In the following
theorem we consider the $1$-parameter family corresponding to $J_{2}=J_{4}=0$,
and we show that for infintely many rational values of the parameter these
Weierstrass equations describe genus 2 curves with everywhere good reduction
over real (complex) quadratic extensions of ${\mathbb{Q}}$.

######  Theorem 7.1.

Let $E_{j(t)}$ be the Weierstrass equation given as follows

$\displaystyle y^{2}-x^{2}y$
$\displaystyle=(40960000000000-14580000000j^{5}+531441j^{10})\;x^{6}-(86400000000j^{2}-39366000j^{7})\;x^{5}$
$\displaystyle-(1/4+48600000j^{4}+177147j^{9}/4)\;x^{4}+(320000000j+291600j^{6})\;x^{3}-1350000j^{3}\;x^{2}$
$\displaystyle+(1600000+243j^{5})\;x-500j^{2}$

where $j(t)=20t-3$, $t$ is an integer. Then one has that $E_{j(t)}$ describes
a genus 2 curve with everywhere good reduction over
$K={\mathbb{Q}}\left(\sqrt{\pm(-3200000+729j(t)^{5})}\right)$, for every
integer $t$.

Proof: We take $J_{10}=\pm 1$ in the parametric Weierstrass equation with
Igusa invariants $J_{2}=J_{4}=0$ described in [19, Theorem 3.11] (iv). We then
minimize the Weierstrass equation at $3,5$ and the primes dividing $j$. One
then obtains the Weierstrass equation

$\displaystyle E^{\prime}_{j}:y^{2}$
$\displaystyle=40960000000000-14580000000j^{5}+531441j^{10}+(-172800000000j^{2}+78732000j^{7})\,x$
$\displaystyle-(194400000j^{4}+177147j^{9})\,x^{2}+(2560000000j+2332800j^{6})\,x^{3}-21600000j^{3}x^{4}$
$\displaystyle+(51200000+7776j^{5})\,x^{5}-32000j^{2}x^{6},$

whose discriminant $J_{10}(E^{\prime}_{j})=-2^{30}(-3200000+729j^{5})^{15}$.
For any integer value of $j$, it is clear that $E^{\prime}_{j}$ is reduced
over ${\mathbb{Q}}_{p}$ for any prime $p>5$.

Taking $j\equiv-3$ mod $20$ assures that the primes $3$ and $5$ do not divide
$J_{10}(E^{\prime}_{j})$, and moreover $2\nmid(-3200000+729j^{5})$. Setting
$j=20t-3$, we use the transformation $x\mapsto 1/2x$,
$y\mapsto(8y-4x^{2})/(2x)$ to obtain the equation

$\displaystyle E_{j(t)}:\;y^{2}-x^{2}y$
$\displaystyle=(40960000000000-14580000000j^{5}+531441j^{10})\;x^{6}-(86400000000j^{2}-39366000j^{7})\;x^{5}$
$\displaystyle-(1/4+48600000j^{4}+177147j^{9}/4)\;x^{4}+(320000000j+291600j^{6})\;x^{3}-1350000j^{3}\;x^{2}$
$\displaystyle+(1600000+243j^{5})\;x-500j^{2},$

whose discriminant $J_{10}(E_{j(t)})$ is
$D(t):=-(-3200000+729(20t-3)^{5})^{15}$, hence $\gcd(J_{10}(E_{j(t)}),60)=1$
for any integer value of $t$.

One recalls that if $p^{k}$, $k\geq 2$, divides $D(t)$ for some integer $t$,
then $E_{j(t)}$ is not minimal over ${\mathbb{Q}}_{p}$, since otherwise the
corresponding curve does not have potential good reduction over
${\mathbb{Q}}_{p}$, [17, Remarque 13]. Our choice of $j(t)$ implies that any
prime divisor of $D(t)$ does not divide the leading coefficient of the sextic
polynomial. Now in accordance with Corollary 4.7, one has that if
$E_{j(t)}$does not describe a curve with good reduction over
${\mathbb{Q}}_{p}$, then the minimal field over which good reduction is
attained is a quadratic extension of ${\mathbb{Q}}_{p}$, [16, Théorème 1 (a)].
Therefore, in view of Theorem 4.4, we may assume without loss of generality
that $\nu_{p}(J_{10}(E_{j(t)}))$ is either $0$ or $15$. It follows that for
every integer value of $t$, $E_{j(t)}$ satisfies the assumptions of Theorem
5.1 for all prime divisors of $J_{10}(E_{j(t)})$, hence $E_{j(t)}$ describes a
curve with everywhere good reduction over
${\mathbb{Q}}\left(\sqrt{\pm(-3200000+729j(t)^{5})}\right)$. $\Box$

######  Corollary 7.2.

Let $J_{j(t)}$, $t\in{\mathbb{Z}}$, be the Jacobian of the genus two curve
described by $E_{j(t)}$ in Theorem 7.1. Then
${\operatorname{End}}(J_{j(t)})={\mathbb{Z}}$. In particular, $J_{j(t)}$ is an
absolutely simple abelian variety.

Proof: One may check using MAGMA that if we set $t=1$ in
$E_{j(t)}:y^{2}+Q_{t}(x)y=P_{t}(x)$, the Galois group of he polynomial
$4P_{1}(x)+Q_{1}(x)^{2}$ is the symmetric group on six elements; $S_{6}$.
Therefore, the polynomial $4P_{t}(x)+Q_{t}(x)^{2}$ must have generic Galois
group $S_{6}$. According to [32, Theorem 2.1], for any curve $C$ described by
an equation of the form $y^{2}=f(x)$, $n=\deg f\geq 5$, over a field $K$ with
characteristic $0$, if $f$ has Galois group $S_{n}$ or $A_{n}$ then
${\operatorname{End}}(\operatorname{Jac}(C))=\mathbb{Z}$. $\Box$

##  8\. Final remarks and comments

(i) Let us mention the following "Open Question" stated in [10, p. 2].

Find a real quadratic field $F$, and an abelian surface $A$ defined over $F$
such that $A$ has trivial conductor with
${\operatorname{End}}_{F}(A)=\mathbb{Z}$.

We gave an answer to this question in Corollary 7.2 using Jacobian varieties
of genus two curves described by Weierstrass equations $E$ for which
$J_{2}(E)=J_{4}(E)=0$, hence the automorphism group of the curves contain a
nontrivial automorphism different from the hyperelliptic involution. Below we
give an answer to the same question using the Jacobian of a genus two curve
whose automorphism group is isomorphic to ${\mathbb{Z}}_{2}$, in particular it
is described by a Weierstrass equation $E$ with $J_{2}(E)\neq 0$.

Consider the curve $C$ described by the Weierstrass equation $E:y^{2}=f(x)$
over the field $K=\mathbb{Q}(\sqrt{31\cdot 47\cdot 107\cdot 269})$, where

$\displaystyle f(x)$
$\displaystyle=1/4\cdot(66861242074781803311769723545637855624989525807865019\
-212180300372212133803172192824208735245404632954\cdot a\;x$
$\displaystyle+11765761679797475148212972485522051834951241828993\;x^{2}-8297325990561138463673721463315177816731190\cdot
a\;x^{3}$
$\displaystyle+138030288410295455310019117974278390625714666\;x^{4}-29202077961801148532323213140299077108\cdot
a\;x^{5}$ $\displaystyle+107953688207841230963812605490049186473\;x^{6})$

and $a=\sqrt{31\cdot 47\cdot 107\cdot 269}$. The Weierstrass equation $E$ is
globally minimal except at the primes lying above $2$. Moreover, $E$ has Igusa
invariants $(J_{2}(E),J_{4}(E),J_{6}(E),J_{10}(E))=(1,8,-4,1)$. Using MAGMA ,
we may check that $C$ has everywhere good reduction over $K$ and the
polynomial $f(x)$ has the symmetric group $S_{6}$ as its Galois group. Now, in
view of [32, Theorem 2.1], we see that
${\operatorname{End}}(\operatorname{Jac}(C))=\mathbb{Z}$.

(ii) Dembélé and Kumar in [9] restricted the fields over which everywhere good
reduction is attained to be real quadratic fields with narrow class number
$=1$. We have examples of curves with everywhere good reduction over
$K=\mathbb{Q}(\sqrt{D})$ and whose narrow class number $>1$. One such example
is curve $y^{2}=f(x)$ over $K=\mathbb{Q}(\sqrt{3244322821})$ with
$(J_{2},J_{4},J_{6},J_{10})=(1,6,20,1)$ where

$\displaystyle f(x)$
$\displaystyle=1/4\cdot(56603909596961676599859889534368406320800421600598321259089511276547396625161821435597444$
$\displaystyle-659996593966528308096994020555497202477042400305477225967492795853797578598449848\cdot
a\;x$
$\displaystyle+10402788757353302209521712779904492235857842873743815936622007573635398892236426913\;x^{2}$
$\displaystyle-26954577662737621340558324771176695069369536668994228423683609591814811006\cdot
a\;x^{3}$
$\displaystyle+127456465699588438544545940903163923096316718284638517203001087373472362051\;x^{4}$
$\displaystyle-99075409972806731283591723191139320859584248134216074085707760902\cdot
a\;x^{5}$
$\displaystyle+104107664637185788902792490762217692317930964923924351034101286541\;x^{6})$

and $a=\sqrt{3244322821}$. One easily checks that $h(3244322821)=3$, hence the
narrow class number is $>1$ (it is a multiple of the usual class number; more
precisely it differs by a power of $2$).

(iii) In [9, Table 7], a genus 2 curve $C$ with everywhere good reduction over
${\mathbb{Q}}(\sqrt{461})$ is given. The following example describes a genus 2
curve with everywhere good reduction over ${\mathbb{Q}}(\sqrt{461})$. The
Jacobian of this curve is isogenous to the Jacobian of $C$. For
$(J_{2},J_{4},J_{6},J_{10})=(-36,-18,-4,1)$, our algorithm gives the following
minimal Weierstrass equation $y^{2}+Q(x)y=P(x)$ over
${\mathbb{Q}}(\sqrt{461})$ where

$\displaystyle Q(x)$ $\displaystyle=1,$ $\displaystyle P(x)$
$\displaystyle=12130483888208271296798313727552769926465009660-598631135278559067562986005319927574362349047\cdot
a\;x$
$\displaystyle+5674533451428904589539319523651961900875388818\;x^{2}-62229859794374632967785456230016040735670239\cdot
a\;x^{3}$
$\displaystyle+176966449088848750932408731229660847277811408\;x^{4}-582211599034682482959237037970428723757123\cdot
a\;x^{5}$ $\displaystyle+367925978657389359077234026398703090110814\;x^{6}$

and $a=\sqrt{461}$.

We moreover obtain a curve with everywhere good reduction over
${\mathbb{Q}}(\sqrt{-461})$ described by the Weierstrass equation
$y^{2}+Q^{-}(x)y=P^{-}(x)$ where

$\displaystyle Q^{-}(x)$ $\displaystyle=a,$ $\displaystyle P^{-}(x)$
$\displaystyle=-12130483888208271296798313727552769926465009545+598631135278559067562986005319927574362349047\cdot
a\;x$
$\displaystyle+5674533451428904589539319523651961900875388818\;x^{2}-62229859794374632967785456230016040735670239\cdot
a\;x^{3}$
$\displaystyle-176966449088848750932408731229660847277811408\;x^{4}+582211599034682482959237037970428723757123\cdot
a\;x^{5}$ $\displaystyle+367925978657389359077234026398703090110814\;x^{6}$

and $a=\sqrt{-461}$.

## Appendix

We fix $J_{2}=J_{10}=\pm 1$. We consider $J_{4}$, $1\leq J_{4}\leq 6$, and run
the algorithm over the following values of $J_{6}$: $1\leq J_{6}\leq 50$. We
were able to obtain the following examples of globaly minimal Weierstrass
equations $y^{2}+Q(x)y=P(x)$ describing genus 2 curves with everywhere good
reduction over a real/complex quadratic field. In what follows
$K_{D}={\mathbb{Q}}(\sqrt{D})$, and $a_{D}=\sqrt{D}$.

For $(J_{2},J_{4},J_{6},J_{10})=(1,3,41,1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=283\cdot 983\cdot 292541$ with

$\displaystyle Q(x)$ $\displaystyle=-x^{3}-x^{2}-2\;x-1$ $\displaystyle P(x)$
$\displaystyle=14493149125598449395929549754038474084310380241491931823193074055994900474640051957369500281928261566976890$
$\displaystyle-(1+1665438444662855865052620278000056560398794643738231308609985628460383079986338377382156048929621\cdot
a_{D})\;x$
$\displaystyle+6489479481704636494295796125228374586785347202783533368433878328443971354643739735858926097025003\;x^{2}$
$\displaystyle-\frac{1}{2}(3+331431032874618735976084441445190757259603113874300807585890963582001508777869309138483\cdot
a_{D})\;x^{3}$
$\displaystyle+193716095692334168253009328934954019939268327342175262407259127392365275308074763764493\;x^{4}$
$\displaystyle-\frac{1}{2}(1+2968043547749256931085921692531919016034618196595381301553295464182002510035\cdot
a_{D})\;x^{5}$
$\displaystyle+385505245693700062768252022867858762437647346167922980067063605138542968539\;x^{6}.$

For $(J_{2},J_{4},J_{6},J_{10})=(-1,3,-41,-1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=-283\cdot 983\cdot 292541$ with

$\displaystyle Q(x)$ $\displaystyle=-x^{3}-a_{D}\;x^{2}-2\;x-a_{D}$
$\displaystyle P(x)$
$\displaystyle=-14493149125598449395929549754038474084310380241491931823193074055994900474640051957369500281928241221554828$
$\displaystyle+1665438444662855865052620278000056560398794643738231308609985628460383079986338377382156048929620\cdot
a_{D}\;x$
$\displaystyle+6489479481704636494295796125228374586785347202783533368433878328443971354643739735858966787869128\;x^{2}$
$\displaystyle-165715516437309367988042220722595378629801556937150403792945481791000754388934654569243\cdot
a_{D}\;x^{3}$
$\displaystyle-193716095692334168253009328934954019939268327342175262407259127392365275308054418342433\;x^{4}$
$\displaystyle+1484021773874628465542960846265959508017309098297690650776647732091001255017\cdot
a_{D}\;x^{5}$
$\displaystyle+385505245693700062768252022867858762437647346167922980067063605138542968539\;x^{6}.$

For $(J_{2},J_{4},J_{6},J_{10})=(1,4,4,1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=26404561$ with

$\displaystyle Q(x)$ $\displaystyle=-x^{3}-x-1$ $\displaystyle P(x)$
$\displaystyle=3980068297155832423457786128424974929592052330976661679652318339872233$
$\displaystyle-\frac{1}{2}(1+2615260210718839277044005360804018934630474400152774950170473767\cdot
a_{D})\;x$
$\displaystyle+4726579929822021780213733516738163820518783843378859759003500803\;x^{2}$
$\displaystyle-\frac{1}{2}(1+690174444833251297883378495386018560180595136487572674839\cdot
a_{D})\;x^{3}$
$\displaystyle+374207277614825042495998202528157743572394326529883851355\;x^{4}$
$\displaystyle-8196253017457146326678417525184726280926437083474\cdot
a_{D}\;x^{5}$
$\displaystyle+1975086830915274164097017785890796271744368850030\;x^{6}.$

For $(J_{2},J_{4},J_{6},J_{10})=(-1,4,-4,-1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=-26404561$ with

$\displaystyle Q(x)$ $\displaystyle=-x^{3}-x-a_{D}$ $\displaystyle P(x)$
$\displaystyle=-3980068297155832423457786128424974929592052330976661679652318333271093$
$\displaystyle+1307630105359419638522002680402009467315237200076387475085236883\cdot
a_{D}\;x$
$\displaystyle+4726579929822021780213733516738163820518783843378859759003500803\;x^{2}$
$\displaystyle-345087222416625648941689247693009280090297568243786337420\cdot
a_{D}\;x^{3}$
$\displaystyle-374207277614825042495998202528157743572394326529883851356\;x^{4}$
$\displaystyle+8196253017457146326678417525184726280926437083474\cdot
a_{D}\;x^{5}$
$\displaystyle+1975086830915274164097017785890796271744368850030\;x^{6}.$

For $(J_{2},J_{4},J_{6},J_{10})=(-1,6,40,-1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=22639\cdot 3787361$ with

$\displaystyle Q(x)$ $\displaystyle=(-x^{3}+a_{D}\cdot x^{2}-2\;x-a_{D})$
$\displaystyle P(x)$
$\displaystyle=-20102088440135614030470128986946346563555781945937611141050719826788916139991909714976694933$
$\displaystyle-4617292887655574165961428503504867769633166770554856646846593036443400760381748059\cdot
a_{D}\;x$
$\displaystyle-37889281434253382202455676997526996644934137545830281733194190489890863460961881935\;x^{2}$
$\displaystyle-1933971633574852387913620699071895736642779234966827401936437145373079117\cdot
a_{D}\;x^{3}$
$\displaystyle-4761023220312967350217124372495946438441280174280887209278673812202315630\;x^{4}$
$\displaystyle-72904659362732357860282616825152037719939877323542568131715001\;a_{D}\;x^{5}$
$\displaystyle-39883473190506942475701006247163415226409169450474757057166290\;x^{6}.$

For $(J_{2},J_{4},J_{6},J_{10})=(1,6,-40,1)$, we find the following minimal
Weierstrass equation describing a genus 2 curve with everywhere good reduction
over $K_{D}$, $D=-22639\cdot 3787361$ with

$\displaystyle Q(x)$ $\displaystyle=-x^{3}+x^{2}-2\;x-1$ $\displaystyle P(x)$
$\displaystyle=20102088440135614030470128986946346563555781945937611141050719826788916139991909693541178513$
$\displaystyle+(-1+4617292887655574165961428503504867769633166770554856646846593036443400760381748058\cdot
a_{D}\;)x$
$\displaystyle-37889281434253382202455676997526996644934137545830281733194190489890863503832914774\;x^{2}$
$\displaystyle+\frac{1}{2}(1-3867943267149704775827241398143791473285558469933654803872874290746158235\cdot
a_{D})\;x^{3}$
$\displaystyle+4761023220312967350217124372495946438441280174280887209278673790766799208\;x^{4}$
$\displaystyle+\frac{1}{2}(1+145809318725464715720565233650304075439879754647085136263430003\cdot
a_{D})\;x^{5}$
$\displaystyle-39883473190506942475701006247163415226409169450474757057166290\;x^{6}.$

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