# Constructing all Genus 2 Curves with Supersingular Jacobian

Andreas Pieper  Andreas Pieper, Institut für Algebra und Zahlentheorie,
Universität Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany  andreas.pieper@uni-
ulm.de

###### Abstract.

L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow\mathbb{P}^{1}$
of genus 2 curves with supersingular Jacobian. In this paper we first classify
the reducible fibers of a Moret-Bailly family using linear algebra over a
quaternion algebra. The main result is an algorithm that exploits properties
of two reducible fibers to compute a hyperelliptic model for any irreducible
fiber of a Moret-Bailly family.

###### Contents

  1. 1 Hermitian Forms over Quaternion Algebras
  2. 2 Affine Group Schemes
  3. 3 Theta Groups
    1. 3.1 Definition
    2. 3.2 Theta Groups and Pullback
    3. 3.3 Representations of the Theta group
  4. 4 The Construction of Moret-Bailly
    1. 4.1 Notation
    2. 4.2 Results of Moret-Bailly
    3. 4.3 Reducible Fibers I: Classification of reducible fibers
    4. 4.4 Reducible Fibers II: Role of 2-torsion
    5. 4.5 Reducible Fibers III: Theta groups
    6. 4.6 Construction of irreducible fibers
    7. 4.7 Examples

## Introduction

Let $k$ be an algebraically closed field of characteristic $p>0$. Recall that
an Abelian variety $A/k$ is called supersingular if it is isogenous to a
product of supersingular elliptic curves. The supersingular locus in the
moduli space of $g$-dimensional principally polarized Abelian varieties has
been intensively studied. K.-Z. Li and F. Oort [7] prove that the
supersingular locus is birational to a variety explicitly described in terms
of semi-linear algebra (Dieudonné theory). This allows one to determine the
number of irreducible components of the supersingular locus and shows that it
is equidimensional of dimension $\lfloor\frac{g^{2}}{4}\rfloor$.

If $g=2$ this result says more precisely that the supersingular locus is the
union of rational curves, as was proven earlier by T. Katsura and Oort in [3].
They prove that every supersingular principally polarized Abelian variety is a
fiber of one of the families $\mathcal{A}\rightarrow\mathbb{P}^{1}$
constructed by Moret-Bailly in [9]. Therefore the supersingular locus is the
union of the images of the $\mathbb{P}^{1}$s from Moret-Bailly’s construction.

This construction works as follows for $p>2$ (the case $p=2$ is similar but
slightly different): Indeed fix a supersingular elliptic curve $E/k$. Then the
kernel of geometric Frobenius is $E[F]\cong\alpha_{p}$. Since
$\operatorname{Hom}(\alpha_{p},\alpha_{p}^{2})=k^{2}$, the subgroup schemes
isomorphic to $\alpha_{p}$ in $E^{2}$ are parametrized by $\mathbb{P}^{1}$.
Moret-Bailly studies the Abelian scheme $\mathcal{A}\rightarrow\mathbb{P}^{1}$
obtained as a quotient of $E^{2}_{\mathbb{P}^{1}}$ by a certain finite flat
group scheme. (Here and in the following for two $k$-schemes $S$ and $T$ the
notation $S_{T}$ is an abbreviation for $S\times_{k}T$). This puts all the
possible ways of taking a quotient of $E^{2}$ by $\alpha_{p}$ into a family of
Abelian surfaces over $\mathbb{P}^{1}$.

Furthermore Moret-Bailly explains how to equip $\mathcal{A}$ with a principal
polarization. For that purpose we need a polarization $\eta$ on $E^{2}$ with
$\ker(\eta)=E^{2}[F]$. Then Moret-Bailly proves that $\eta_{\mathbb{P}^{1}}$
descends to a principal polarization $\lambda$ on $\mathcal{A}$. He shows that
there exists a relative divisor $\mathfrak{C}\subset\mathcal{A}$ inducing
$\lambda$, a “theta divisor”. The fibers of
$\mathfrak{C}\rightarrow\mathbb{P}^{1}$ are either smooth curves of genus 2 or
two elliptic curves intersecting transversely. In this way we get finitely
many families of curves of genus 2 corresponding to the equivalence classes of
$\eta$. (Here the equivalence relation is conjugation by $\text{Aut}(E^{2})$.
See [3, Theorem 5.7] for a discussion of the cardinality of the equivalence
classes.) The Jacobians of those curves are supersingular as they are
isogenous to $E^{2}$.

The goal of the present paper is to make this theory effective by giving an
algorithm that computes all the non-singular fibers of a Moret-Bailly family
$\mathfrak{C}\rightarrow\mathbb{P}^{1}$. For that purpose we need to be able
to compute quotients by $\alpha_{p}$. In the paper [8] D. Lubicz and D. Robert
spread the idea that separable isogenies between higher dimensional Abelian
varieties should be computed using Mumford’s theory of theta groups. This
point of view can be extended to inseparable isogenies.

Mumford defines for an ample line bundle $\mathcal{L}$ on an Abelian variety
$A$, a group scheme $\mathcal{G}(\mathcal{L})$ with center $\mathbb{G}_{m}$
(the so-called theta group). Furthermore $\mathcal{G}(\mathcal{L})$ acts
irreducibly on $\operatorname{H}^{0}(A,\mathcal{L})$ such that the center acts
by multiplication. But he also shows that there exists only one representation
of $\mathcal{G}(\mathcal{L})$ with this property.

In the separable case, that is $\text{char}(k)\nmid\deg\mathcal{L}$, Mumford
proves that $\mathcal{G}(\mathcal{L})$ is isomorphic to an explicit group
$\mathcal{G}(\delta)$, a so-called Heisenberg group. He also writes down an
explicit irreducible representation $V(\delta)$ of $\mathcal{G}(\delta)$ such
that the center acts by multiplication. From the uniqueness of the
representation one gets an isomorphism
$\operatorname{H}^{0}(A,\mathcal{L})\cong V(\delta)$. That is a very important
result and can be used to prove theorems in a purely algebraic fashion that
were classically proven over $\mathbb{C}$ via the analytic theory of theta
functions.

Unfortunately the structure theorem for theta groups fails in the inseparable
case. However in our case we are lucky as the theta group we are using is
nevertheless a Heisenberg group
$\mathbb{G}_{m}\times\alpha_{p}\times\alpha_{p}$ with group law as explained
in Definition 3.12.

To construct such an isomorphism with a Heisenberg group it turns out that we
need two fibers of $\mathfrak{C}\rightarrow\mathbb{P}^{1}$, one for each copy
of $\alpha_{p}$. Then we exploit that there are plenty ($5p-5>2$) of fibers
that are as simple as possible, namely the reducible fibers. For that purpose
we develop a classification of the reducible fibers in terms of linear algebra
over the quaternion algebra $B=\operatorname{End}(E)\otimes\mathbb{Q}$.

Using two reducible fibers also has the advantage that we can construct a
rational function $g$ on $E^{2}$ from them. This rational function will be
interpreted as a section of a line bundle on $E^{2}$, via the correspondence
between divisors and line bundles. Then we can construct further rational
functions on $E^{2}$ by translating $g$ using the group structure on $E^{2}$.
In the end the desired curve can be obtained from the zero divisor of such a
translate of $g$. We refer to the main body of the text for the details.

The outline of the paper is as follows: In chapter 1 we prove some results
about Hermitian lattices over maximal orders in quaternion algebras. These are
analogous to similar statements for Hermitian lattices over the ring of
integers of a number field and the proofs work the same way. In chapter 2 we
recall some facts about group schemes related to $\alpha_{p}$. In chapter 3 we
introduce theta groups, which are the cornerstone of our method. The theory is
usually formulated in terms of line bundles. Unfortunately we need to prove
some technical compatibilities when we translate from line bundles to
divisors.

Chapter 4 discusses the main theorems. First we classify the reducible fibers
of a Moret-Bailly family and give an algorithm that computes them. Then we
bootstrap ourselves and use two reducible fibers to compute all the
irreducible fibers.
Acknowledgements: The material presented here is part of the author’s PhD
thesis written under the supervision of S. Wewers. I want to thank him for
reading the manuscript carefully and suggesting various improvements. The
author wishes to thank the anonymous referees for valuable comments. Thanks go
to I. Bouw, B. Dina, J. Hanselman, and J. Sijsling for valuable discussions.

##  1\. Hermitian Forms over Quaternion Algebras

In this section let $R$ be a Dedekind ring with field of fractions $K$ and $B$
a quaternion algebra over $K$, i.e., a central simple $K$-algebra with
$[B:K]=4$. Denote by $\overline{(\cdot)}$ the standard involution on $B$. Let
further $\mathsf{O}$ be a maximal $R$-order of $B$. Later we will only
consider the case, where $R=\mathbb{Z}$ or a localization $\mathbb{Z}_{(l)}$
and $B$ is the unique quaternion algebra over $\mathbb{Q}$ ramified only at
$(p,\infty)$. However as it does not lead to additional difficulties we will
present all the statements in most general form. The goal of this section is
to prove the Lemma 1.5 about Hermitian forms over $\mathsf{O}$. It says that
for a perfect hermitian pairing $(\cdot,\cdot)$ on an $\mathsf{O}$-module $M$
and a primitive vector $v\in M$ there exists a $w\in M$ with $(v,w)=1$. This
lemma is analogous to the similar statement over the ring of integers of a
number field and the proof works the same way.

We first need some definitions concerning $\mathsf{O}$-modules. We will freely
use the notions of projective modules (see [6, Chapter 1.2]) and invertible
fractional ideals (see [2, p. 220]).

######  Definition 1.1.

Let $M$ be a finitely generated projective right $\mathsf{O}$-module. An
element $v\in M$ is called a _primitive vector_ if the following two
conditions are satisfied:

  * •

$\operatorname{Ann}_{\mathsf{O}}(v)=0$

  * •

If $v=v^{\prime}a$ for some $v^{\prime}\in M,a\in\mathsf{O}$, then
$a\in\mathsf{O}^{\times}$.

Notice that the first condition follows from the second if $B$ is a division
algebra. We are now ready to discuss Hermitian forms.

######  Definition 1.2.

A _Hermitian $\mathsf{O}$-lattice_ is a finitely generated projective right
$\mathsf{O}$-module $M$ together with a pairing $(\cdot,\cdot):M\times
M\longrightarrow\mathsf{O}$ such that for all $v,v_{1},v_{2}\in
M,\lambda_{1},\lambda_{2}\in\mathsf{O}$

  * i)

$(\cdot,\cdot)$ is linear in the second component:

$(v,v_{1}\lambda_{1}+v_{2}\lambda_{2})=(v,v_{1})\lambda_{1}+(v,v_{2})\lambda_{2}\,.$

  * ii)

$(\cdot,\cdot)$ is semilinear in the first component:

$(v_{1}\lambda_{1}+v_{2}\lambda_{2},v)=\overline{\lambda_{1}}(v_{1},v)+\overline{\lambda_{2}}(v_{2},v)\,.$

  * iii)

$(\cdot,\cdot)$ is hermitian symmetric:
$(v_{1},v_{2})=\overline{(v_{2},v_{1})}\,.$

A pairing satisfying i) and ii) is called a sesquilinear pairing.

######  Lemma 1.3.

Let $M$ be a projective right $\mathsf{O}$-module. Then

$M^{\dagger}=\left\\{\phi:M\longrightarrow\mathsf{O}|\,\phi(v_{1}\lambda_{1}+v_{2}\lambda_{2})=\overline{\lambda_{1}}\phi(v_{1})+\overline{\lambda_{2}}\phi(v_{2})\right\\}$

is naturally a projective right $\mathsf{O}$-module. To give a sesquilinear
pairing on $M$ is equivalent to giving a linear map $M\longrightarrow
M^{\dagger}$.

###### Proof.

Clear. ∎

######  Definition 1.4.

We say that a sesquilinear pairing on $M$ is _non-degenerate_ (resp.
_perfect_) if the induced map $M\longrightarrow M^{\dagger}$ is injective
(resp. bijective).

######  Lemma 1.5.

Assume now that every invertible fractional left $\mathsf{O}$-ideal is free.
Let $M$ be a projective right $\mathsf{O}$-module such that $M\otimes_{R}K$ is
a free $B$-module. Let further $(\cdot,\cdot)$ be a perfect sesquilinear
pairing on $M$. Let $v\in M$ be a primitive vector. Then there exists a $w\in
M$ such that $(w,v)=1$.

###### Proof.

Let $\phi$ be the image of $v$ under the map $M\longrightarrow M^{\dagger}$
induced by the pairing $(\cdot,\cdot)$. Denote by $I\subseteq\mathsf{O}$ the
image of $\phi$. Then $I$ is naturally a left ideal of $\mathsf{O}$ (notice
that the anti-linearity of $\phi$ interchanges left and right.). We claim that
$KI=B$. Indeed $KI$ is a left ideal of $B$. By Wedderburn’s theorem $B$ is
either a division algebra or a $2\times 2$-matrix algebra over $K$. If $B$ is
a division algebra there is nothing to show. Assume now that $B=M_{2,2}(K)$ is
a matrix algebra over $K$. Suppose by way of contradiction that $KI\neq B$. It
is well known and can be shown using elementary linear algebra that every
proper left ideal of a matrix algebra $M_{n,n}(K)$ over a field is of the form
$M_{n,n}(K)A$ where $\operatorname{rk}A<n$. Therefore there is a matrix $A\in
M_{2,2}(K)$ with $\operatorname{rk}A<2$ such that $KI=BA$. Choose a basis
$e_{1},\ldots,e_{d}$ of the free $B$-module $M\otimes_{R}K$. Consider the
$B$-antilinear map $\phi_{K}:M\otimes K\longrightarrow B$ obtained by
extension of scalars. Since $\operatorname{im}(\phi_{K})=KI$, there exist
$r_{1},\ldots,r_{d}\in B$ such that $\phi_{K}(e_{i})=r_{i}A$. This implies
that $\phi_{K}$ is the map $M\otimes K\longrightarrow
B,\,\sum_{i=1}^{d}e_{i}\lambda_{i}\mapsto\overline{\lambda_{i}}r_{i}A$. Hence
there is a $\phi^{\prime}\in M^{\dagger}\otimes K=(M\otimes K)^{\dagger}$ such
that $\phi=\phi^{\prime}A$. On the other hand clearly $M\otimes
K\longrightarrow M^{\dagger}\otimes K$ is an isomorphism. Therefore there is
an $v^{\prime}\in M\otimes K$ such that $v=v^{\prime}A$. But since $A$ is not
of full rank there exists an $r\in B\setminus\\{0\\}$ such that $Ar=0$. By
multiplying with an element in $R$ we can assume that $r\in\mathsf{O}$. Then
one has $vr=0$ contradicting the primitivity of $v$. We thus conclude that
$KI=B$.

By [2, Theorem 7] this implies that $I$ is an invertible left ideal. Since we
assumed that all invertible left ideals are principal, there exists an
$a\in\mathsf{O}$ such that $I=\mathsf{O}a$. As $KI=B$ the element $a$ cannot
be a zero-divisor. Therefore $N(a)=a\overline{a}\neq 0$. Now the map
$\phi^{\prime}:M\longrightarrow\mathsf{O},\,w\mapsto\phi(w)\frac{\overline{a}}{N(a)}$
is well-defined and gives an element in $M^{\dagger}$ satisfying
$\phi=\phi^{\prime}a$. Because the pairing $(\cdot,\cdot)$ is perfect, the map
$M\longrightarrow M^{\dagger}$ is an isomorphism. Denote by $v^{\prime}$ the
pre-image of $\phi^{\prime}$. Then $v=v^{\prime}a$ and by the primitivity of
$v$ we conclude that $a\in\mathsf{O}^{\times}$. Therefore $I=\mathsf{O}$ and
hence by the definition of $I$ there exists an element $w\in M$ with
$(w,v)=1$. ∎

##  2\. Affine Group Schemes

In this chapter we recall some well-known facts about the group scheme
$\alpha_{p}$ and related group schemes we will use. Throughout this chapter
$k$ will be a field of characteristic $p>0$ and $S$ a scheme over $k$.

######  Definition 2.1.

An _$\alpha$ -group scheme_ of rank $r$ over $S$ is a finite flat group scheme
$G$ over $S$, such that there exists an fppf-covering $T\rightarrow S$ such
that $G\times_{S}T\cong\alpha_{p,T}^{r}$ as $T$-group schemes.

######  Theorem 2.2.

The functor

$\left\\{\begin{tabular}[]{@{}l@{}}$\alpha$-group schemes\\\ over
$S$\end{tabular}\right\\}\longrightarrow\left\\{\begin{tabular}[]{@{}l@{}}Locally
free sheaves\\\ of finite rank on $S$\end{tabular}\right\\}$

that sends $G$ to $\text{\emph{Lie}}(G)^{\vee}$ is a rank preserving anti-
equivalence of categories.

###### Proof.

Indeed we claim that the inverse functor

$\left\\{\begin{tabular}[]{@{}l@{}}Locally free sheaves\\\ of finite rank on
$S$\end{tabular}\right\\}\rightarrow\left\\{\begin{tabular}[]{@{}l@{}}$\alpha$-group
schemes\\\ over $S$\end{tabular}\right\\}$

is defined as follows: For a locally free sheaf of finite rank $\mathcal{F}$
on $S$ we define the following coherent sheaf of $\mathcal{O}_{S}$-Hopf
algebras $\mathcal{A}=\oplus_{i=0}^{p-1}\text{Sym}^{i}(\mathcal{F})$. The
multiplication on $\mathcal{A}$ is given by
$\text{Sym}^{i}(\mathcal{F})\otimes\text{Sym}^{j}(\mathcal{F})\rightarrow\text{Sym}^{i+j}(\mathcal{F})$
if $i+j<p$ and
$\text{Sym}^{i}(\mathcal{F})\otimes\text{Sym}^{j}(\mathcal{F})\rightarrow 0$
if $i+j\geqslant p$. The comultiplication is defined as follows: One can
easily show that there is a unique algebra map
$c:\mathcal{A}\rightarrow\mathcal{A}\otimes_{\mathcal{O}_{S}}\mathcal{A}$ such
that $c_{|\mathcal{F}}$ equals
$(\operatorname{id},\operatorname{id}):\mathcal{F}\rightarrow(\mathcal{F}\otimes\mathcal{O}_{S})\oplus(\mathcal{O}_{S}\otimes\mathcal{F})\subset\mathcal{A}\otimes\mathcal{A}$.
The counit is the map $\mathcal{A}\rightarrow\mathcal{O}_{S}$ that projects
onto the first summand. Then we define
$\alpha(\mathcal{F})=\underline{\text{Spec}}_{S}(\mathcal{A})$. This defines
an $\alpha$-group scheme of rank $\text{rk}(\mathcal{F})$.

It is clear that both maps are contravariant functors. [1, exposé VIIA,
Theorem 7.4] implies that $\alpha(\cdot)$ is an anti-equivalence with inverse
$\text{Lie}(\cdot)^{\vee}$. ∎

Notice that the previous theorem implies that $\alpha$-group schemes are
isomorphic to $\alpha_{p}^{r}$ locally for the Zariski topology (instead of
fppf).

######  Definition 2.3.

For a locally free sheaf $\mathcal{F}$ we define the corresponding
$\alpha$-group scheme $\alpha(\mathcal{F})$ as in the previous proof.

The previous theorem can be used to compute an isomorphism $G\cong\alpha_{p}$
for a group scheme which is abstractly isomorphic to $\alpha_{p}$ but no
isomorphism is given a priori.

######  Lemma 2.4.

Let $G$ be a finite group scheme over a field $k$ which is isomorphic to
$\alpha_{p}$. Suppose the Hopf-Algebra of $G$ is given as $A_{G}=k[t]/(t^{p})$
(but the comultiplication does not have to be $t\mapsto t\otimes 1+1\otimes
t$).

Let $\omega=\sum_{i=0}^{p-1}a_{i}t^{i}dt$ be a non-zero invariant differential
form. Then there is a unique isomorphism

$\iota:G\longrightarrow\alpha_{p}=\operatorname{Spec}k[x]/(x^{p})$

with $\iota^{*}(dx)=\omega$. It is given by

$x\mapsto\sum_{i=0}^{p-2}\frac{a_{i}t^{i+1}}{i+1}\,,$

i.e. the logarithm of $G$ which is defined as the integral of $\omega$.

Furthermore $a_{p-1}=0$.

###### Proof.

The existence and uniqueness of $\iota$ follows from Theorem 2.2. The explicit
formula is an elementary computation. ∎

We will also need a fact about extensions of group schemes.

######  Theorem 2.5.

Let $G$ be a finite commutative group scheme over $k$ and

$0\longrightarrow\mathbb{G}_{m}\longrightarrow E\longrightarrow
G\longrightarrow 0$

an extension in the category of commutative group schemes. Suppose $k$ is
perfect and $G$ is unipotent. Then the extension is uniquely split.

###### Proof.

See [1, exposé XVII, Théorème 6.1.1(C)]. ∎

######  Remark 2.6.

Notice that over a non-perfect field there is a non-split extension of
$\alpha_{p}$ by $\mathbb{G}_{m}$.

##  3\. Theta Groups

In this section we are going to recall the facts about theta groups needed for
chapter 4. Let $A/k$ be an Abelian variety and $\mathcal{L}$ a line bundle on
$A$. The theta group of $\mathcal{L}$ is a group that naturally encodes the
geometric interplay of $\mathcal{L}$ with the group structure of $A$. The
theta group satisfies nice properties under pullback. Therefore it is a useful
theoretical as well as practical tool for studying a situation where a
polarization is descended along an isogeny as in Moret-Bailly’s construction.
For this reason Moret-Bailly also uses theta groups in [9].

The theory of theta groups is due to Mumford and was developed in the case
where $\text{char}(k)\nmid\deg(\mathcal{L})$ in [10]. The general case is
explained in a letter from Mumford to T. Sekiguchi published in [12]. Our
exposition follows [14]. Notice that Sekiguchi’s definition of a Heisenberg
group differs from the one used here.

###  3.1. Definition

Let $A/k$ be an Abelian variety and $\mathcal{L}$ a line bundle. The line
bundle induces a homomorphism $\lambda_{\mathcal{L}}:A\longrightarrow
A^{\vee},\,x\mapsto\mathcal{L}\otimes t^{*}_{x}\mathcal{L}^{-1}$. We denote
the kernel of $\lambda_{\mathcal{L}}$ by $K(\mathcal{L})$. By [11, p. 57] the
line bundle $\mathcal{L}$ is ample if and only if $\lambda$ is an isogeny and
$\operatorname{H}^{0}(A,\mathcal{L})\neq 0$.

######  Definition 3.1.

Let $A/k$ be an Abelian variety and $\mathcal{L}$ a line bundle. Then we
define the _theta group of $\mathcal{L}$_ via the functor of points

$\displaystyle\mathcal{G}(\mathcal{L}):\\{\text{Sch}/k\\}\longrightarrow\\{\text{Groups}\\}$
$\displaystyle T\mapsto\Big{\\{}(x,\phi)|x\in A(T)\textrm{ and }$
$\displaystyle\phi:\mathcal{L}_{T}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}t_{x}^{*}\mathcal{L}_{T}$
$\displaystyle\text{ an isomorphism of line bundles on }A\times T\Big{\\}}\,.$

The group structure is defined by
$(x_{1},\phi_{1})\cdot(x_{2},\phi_{2})=(x_{1}+x_{2},t_{x_{2}}^{*}\phi_{1}\circ\phi_{2})$.

######  Lemma 3.2.

The functor $\mathcal{G}(\mathcal{L})$ is represented by a group scheme over
$k$. There is an exact sequence of group schemes

$0\longrightarrow\mathbb{G}_{m}\longrightarrow\mathcal{G}(\mathcal{L})\longrightarrow
K(\mathcal{L})\longrightarrow 0$

where the map $\mathcal{G}(\mathcal{L})\rightarrow K(\mathcal{L})$ is given by
$(x,\phi)\mapsto x$.

###### Proof.

See [14, Lemma 8.2]. ∎

######  Definition/Proposition 3.3.

If $\mathcal{L}$ is ample, then $\mathbb{G}_{m}$ is equal to the center of
$\mathcal{G}(\mathcal{L})$. Therefore the commutator map will factor through a
non-degenerate alternating pairing

$e^{\mathcal{L}}:K(\mathcal{L})\times K(\mathcal{L})\rightarrow\mathbb{G}_{m}$

called the _commutator pairing_.

###### Proof.

See [14, Corollary 8.20]. ∎

###  3.2. Theta Groups and Pullback

######  Proposition 3.4.

Let $A$ be an Abelian variety over a field $k$ and $\mathcal{L}$ be a line
bundle on $A$. Let $\mathcal{H}\subset K(\mathcal{L})$ be a closed subgroup
scheme. Denote the quotient map $A\rightarrow B=A/\mathcal{H}$ by $\pi$. Then
there is a natural bijection

$\left\\{\begin{tabular}[]{@{}l@{}}\emph{line bundles}\\\ $\mathcal{L}_{B}$
\emph{on} $B$ \emph{such}\\\ \emph{that}
$\pi^{*}\mathcal{L}_{B}\cong\mathcal{L}$\end{tabular}\right\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\left\\{\begin{tabular}[]{@{}l@{}}\emph{splittings
of the}\\\ \emph{exact sequence in}\\\ \emph{Lemma \ref{ThetaExt} over}
$\mathcal{H}\,.$\end{tabular}\right\\}\,.$

###### Proof.

See [14, Theorem 8.10]. ∎

######  Corollary 3.5.

Suppose further that $\mathcal{L}$ is ample and $K(\mathcal{L})$ is unipotent.
Then a line bundle $\mathcal{L}_{B}$ on $B$ with
$\pi^{*}\mathcal{L}_{B}\cong\mathcal{L}$ exists if and only if
$\mathcal{H}\subset K(\mathcal{L})$ is isotropic under the commutator pairing
on $K(\mathcal{L})$. Furthermore $\mathcal{L}_{B}$ is then unique.

###### Proof.

Follows easily from Theorem 2.5. ∎

######  Definition 3.6.

A level subgroup $\tilde{\mathcal{H}}\subset\mathcal{G}(\mathcal{L})$ is a
subgroup scheme such that $\mathbb{G}_{m}\cap\tilde{\mathcal{H}}=\\{1\\}$.

A level subgroup will map isomorphically onto its image under
$\mathcal{G}(\mathcal{L})\rightarrow K(\mathcal{L})$. Let us denote the image
by $\mathcal{H}$. By Proposition 3.4 the level subgroup $\tilde{\mathcal{H}}$
will determine a descended line bundle on $A/\mathcal{H}$.

######  Definition/Proposition 3.7.

A subgroup $\tilde{\mathcal{H}}\subset\mathcal{G}(\mathcal{L})$ is called a
maximal level subgroup if it is a level subgroup satisfying one (and hence
both) of the following two equivalent properties:

  * i)

$\sharp(\tilde{\mathcal{H}})^{2}=\sharp(K(\mathcal{L}))$. Here and in the rest
of the article $\sharp(\cdot)$ denotes the rank of a finite group scheme.

  * ii)

Denote by $\mathcal{H}$ the image of $\tilde{\mathcal{H}}$ under
$\mathcal{G}(\mathcal{L})\rightarrow K(\mathcal{L})$. Then
$\mathcal{H}^{\perp}=\mathcal{H}$. The orthogonal complement should be taken
with respect to the commutator pairing on $K(\mathcal{L})$.

If furthermore $k$ is algebraically closed then i) and ii) are equivalent to

  * iii)

$\tilde{\mathcal{H}}$ is maximal in the set of level subgroups ordered by
inclusion.

###### Proof.

See [14, Lemma 8.22]. ∎

For our applications it will be useful to translate the Proposition 3.4 into
the language of divisors. This leads to the following definition:

######  Definition 3.8.

Let $A/k$ be an Abelian variety and $\mathcal{H}\subset A$ be a subgroup
scheme. A closed subscheme $Z\subset A$ is called $\mathcal{H}$-invariant if
the scheme theoretic preimages of $Z$ under the two maps

$\text{pr}_{2}:\mathcal{H}\times A\rightarrow A,\,m:\mathcal{H}\times
A\rightarrow A,\,(h,a)\mapsto h+a$

agree.

For the proof of the next proposition we need a technical definition taken
from [11, p. 104]. The reader who does not want to study this proof can skip
the definition. It will not be used in other parts of the paper.

######  Definition 3.9.

Let $A/k$ be an Abelian variety, $\mathcal{H}\subset A$ be a subgroup scheme
and $\mathcal{F}$ a coherent sheaf on $A$. A _lift of the action_
$m:\mathcal{H}\times A\rightarrow A$ to $\mathcal{F}$ is an isomorphism
$\lambda:\text{pr}_{2}^{*}(\mathcal{F})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}m^{*}(\mathcal{F})$
of sheaves on $\mathcal{H}\times A$ such that the following diagram of sheaves
on $\mathcal{H}\times\mathcal{H}\times A$ commutes:

${\text{pr}_{3}^{*}(\mathcal{F})}$${\xi^{*}(\mathcal{F})}$${\eta^{*}(\mathcal{F})}$$\scriptstyle{\operatorname{pr}_{23}^{*}(\lambda)}$$\scriptstyle{(m_{12}\times\operatorname{id}_{A})^{*}(\lambda)}$$\scriptstyle{(\operatorname{id}_{\mathcal{H}}\times
m_{23})^{*}(\lambda)}$

where
$\xi=\operatorname{pr}_{23}\circ\,m,\,\eta=(m_{12}\times\operatorname{id}_{A})\circ
m=(\operatorname{id}_{\mathcal{H}}\times m_{23})\circ m$.

We can now state the translation of Proposition 3.4 into the language of
divisors.

######  Proposition 3.10.

Let $A/k$ be an Abelian variety and $\mathcal{H}\subset A$ be a finite
subgroup scheme. Suppose an effective divisor $D\subset A$ is
$\mathcal{H}$-invariant. Denote by $\mathcal{L}=\mathcal{O}(D)$ and
$\pi:A\rightarrow B=A/\mathcal{H}$ the natural quotient map.

  * •

$\mathcal{H}\subset K(\mathcal{L})$ and there is a natural homomorphism
$\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ splitting the exact sequence
in Lemma 3.2 over $\mathcal{H}$.

  * •

$\mathcal{M}=\mathcal{O}(\pi(D))$ is a line bundle satisfying
$\mathcal{L}\cong\pi^{*}\mathcal{M}$. Under the bijection in Proposition 3.4
this corresponds to the splitting
$\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ above.

###### Proof.

This is well-known and similar ideas are contained in [10]. As we could not
find a proof in the literature we will indicate it here. The reader can skip
the proof as it is quite technical and the techniques will not be used in the
rest of the paper. Indeed consider the two maps
$\text{pr}_{2},m:\mathcal{H}\times A\rightarrow A$. Then both maps are flat:
Indeed flatness of $\text{pr}_{2}$ is clear and flatness of $m$ follows from
the commutative diagram

${\mathcal{H}\times A}$${\mathcal{H}\times
A}$${A}$$\scriptstyle{(h,a)\mapsto(h,h+a)}$$\scriptstyle{\sim}$$\scriptstyle{m}$$\scriptstyle{\text{pr}_{2}}$

Flatness implies that the natural map
$\text{pr}_{2}^{*}\mathcal{O}(D)\rightarrow\mathcal{O}(\text{pr}_{2}^{-1}(D))$
is an isomorphism and similarly
$m^{*}\mathcal{O}(D)\cong\mathcal{O}(m^{-1}(D))$. Because we assumed $D$ to be
$\mathcal{H}$-invariant, one has $\text{pr}_{2}^{-1}(D)=m^{-1}(D)$. Therefore
we obtain an isomorphism
$\lambda:\text{pr}_{2}^{*}\mathcal{O}(D)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}m^{*}\mathcal{O}(D)$.
By the definition of the theta group, this isomorphism determines a map of
schemes $s:\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$. We check that $s$
is a group homomorphism. Indeed this means that the two maps $s\circ
m_{\mathcal{H}},\,m_{\mathcal{G}(\mathcal{L})\circ(s\times
s)}:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ are
equal. Considered as an element in
$\mathcal{G}(\mathcal{L})(\mathcal{H}\times\mathcal{H})$ the first map
corresponds to the isomorphism
$(m_{1,2}\circ\operatorname{id}_{A})^{*}\lambda$. The other map is the product
of the elements $s\circ\operatorname{pr}_{1},\,s\circ\operatorname{pr}_{2}$ in
$\mathcal{G}(\mathcal{L})(\mathcal{H}\times\mathcal{H})$. By the definition of
the group law on the theta group this product corresponds to the isomorphism
$(\operatorname{id}_{\mathcal{H}}\times
m_{23})^{*}(\lambda)\circ\operatorname{pr}_{23}^{*}(\lambda)$. We are
therefore reduced to proving that the following diagram of sheaves on
$\mathcal{H}\times\mathcal{H}\times A$ commutes:

${\text{pr}_{3}^{*}(\mathcal{O}(D))}$${\xi^{*}(\mathcal{O}(D))}$${\eta^{*}(\mathcal{O}(D))}$$\scriptstyle{\operatorname{pr}_{23}^{*}(\lambda)}$$\scriptstyle{(m_{12}\times\operatorname{id}_{A})^{*}(\lambda)}$$\scriptstyle{(\operatorname{id}_{\mathcal{H}}\times
m_{23})^{*}(\lambda)}$

where $\xi=\operatorname{pr}_{23}\circ
m,\,\eta=(m_{12}\times\operatorname{id}_{A})\circ
m=(\operatorname{id}_{\mathcal{H}}\times m_{23})\circ m$. Indeed this is true
because $\operatorname{pr}_{3}^{-1}(D)=\xi^{-1}(D)=\eta^{-1}(D)$ and all the
maps are induced from the equality of these divisors. Therefore $s$ must be a
group homomorphism.

It is clear that the composition
$\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})\rightarrow
K(\mathcal{L})\subset A$ is the natural inclusion. Thus $\mathcal{H}\subset
K(\mathcal{L})$ and $\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ is a
splitting of the exact sequence in Lemma 3.2 over $\mathcal{H}$.

We will now prove the second part of the proposition. It is clear that we can
instead consider $\mathcal{L}=\mathcal{O}(-D)$. In the first part of the proof
we constructed a group homomorphism
$\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ splitting the exact sequence
in Lemma 3.2 over $\mathcal{H}$. Then Proposition 3.4 gives a line bundle
$\mathcal{M}$ on $B$ such that $\mathcal{L}\cong\pi^{*}\mathcal{M}$. We have
to show that $\mathcal{M}=\mathcal{O}(-\pi(D))$. Indeed by the proof of
Proposition 3.4 (for which we refer to [14, Theorem 8.10]) the group
homomorphism $\mathcal{H}\rightarrow\mathcal{G}(\mathcal{L})$ determines a
lift of the multiplication action of $\mathcal{H}$ to $\mathcal{L}$. It is not
difficult to check that this lift of the action is given by the isomorphism
$\lambda$ from above. Furthermore the natural inclusion
$\iota:\mathcal{O}(-D)\rightarrow\mathcal{O}_{A}$ is equivariant for this
action, since this is equivalent to the commutativity of

${\operatorname{pr}_{2}^{*}\mathcal{O}(-D)}$${m^{*}\mathcal{O}(-D)}$${\operatorname{pr}_{2}^{*}\mathcal{O}_{A}}$${m^{*}\mathcal{O}_{A}}$$\scriptstyle{\lambda}$$\scriptstyle{\operatorname{pr}_{2}^{*}(\iota)}$$\scriptstyle{m^{*}(\iota)}$

where we use the equality $\mathcal{O}_{\mathcal{H}\times
A}=\operatorname{pr}_{2}^{*}\mathcal{O}_{A}=m^{*}\mathcal{O}_{A}$. Now by [11,
p. 104, Theorem 1] $\mathcal{M}=\pi_{*}(\mathcal{L})^{\mathcal{H}}$ where
$\pi_{*}(\mathcal{L})^{\mathcal{H}}\subseteq\pi_{*}(\mathcal{L})$ is the
subsheaf of $\mathcal{H}$-invariant sections as defined in the proof of the
quoted theorem. But since the inclusion
$\mathcal{O}(-D)\subset\mathcal{O}_{A}$ is $\mathcal{H}$-equivariant we have

$\mathcal{M}=\pi_{*}(\mathcal{L})^{\mathcal{H}}=\pi_{*}(\mathcal{O}(-D))\cap\pi_{*}(\mathcal{O}_{A})^{\mathcal{H}}=\pi_{*}(\mathcal{O}(-D))\cap\mathcal{O}_{B}\,.$

Because $\pi$ is finite and thus affine, the latter sheaf equals the coherent
sheaf of ideals whose vanishing set is the closed set $\pi(D)$ (this can be
checked on affine opens). We conclude $\mathcal{M}=\mathcal{O}(-\pi(D))$. This
finishes the proof of the proposition. ∎

######  Proposition 3.11.

Let $A$ be an Abelian variety over an algebraically closed or finite field
$k$. Let $\mathcal{L}$ be a line bundle on $A$ such that $K(\mathcal{L})$
contains a non-zero Abelian subvariety $B\subseteq A$. Denote by
$\pi:A\longrightarrow A/B$ the quotient map. Then there is a line bundle
$\mathcal{L}_{0}$ on $A$ algebraically equivalent to $\mathcal{O}_{A}$ and a
line bundle $\mathcal{L}_{A/B}$ on $A/B$ such that
$\mathcal{L}\cong\mathcal{L}_{0}\otimes\pi^{*}\mathcal{L}_{A/B}$

###### Proof.

The case where $k$ is algebraically closed is given as an exercise in [14,
11.3]. As there is not proof given there we shall include it here: Consider
the restriction $\mathcal{L}_{|B}$. Since $B\subseteq K(\mathcal{L})$, the
line bundle $\mathcal{L}_{|B}$ is algebraically equivalent to
$\mathcal{O}_{B}$. Therefore it gives an element in
$B^{\vee}(k)=\operatorname{Pic}^{0}(B)(k)$. Consider the exact sequence

$0\longrightarrow(A/B)^{\vee}\longrightarrow A^{\vee}\longrightarrow
B^{\vee}\longrightarrow 0\,.$

Since $k$ is finite or algebraically closed the sequence

$0\longrightarrow(A/B)^{\vee}(k)\longrightarrow A^{\vee}(k)\longrightarrow
B^{\vee}(k)\longrightarrow 0$

is exact by Lang’s theorem or Hilbert’s Nullstellensatz respectively.
Therefore there is a line bundle $\mathcal{L}_{0}\in A^{\vee}(k)$ such that
$(\mathcal{L}_{0}^{-1}\otimes\mathcal{L})_{|B}\cong\mathcal{O}_{B}$. It
remains to show that $\mathcal{L}_{0}^{-1}\otimes\mathcal{L}$ descends to
$A/B$. Indeed after replacing $\mathcal{L}$ with
$\mathcal{L}_{0}^{-1}\otimes\mathcal{L}$ we will now assume that
$\mathcal{L}_{|B}\cong\mathcal{O}_{B}$. Consider the exact sequence

$0\longrightarrow\mathbb{G}_{m}\longrightarrow\mathcal{G}(\mathcal{L})\longrightarrow
K(\mathcal{L})\longrightarrow 0$

and push it out along $B\subseteq K(\mathcal{L})$:

$0\longrightarrow\mathbb{G}_{m}\longrightarrow\mathcal{G}(\mathcal{L})_{|B}\longrightarrow
B\longrightarrow 0\,.$

The resulting exact sequence is the sequence of
$\mathcal{G}(\mathcal{L}_{|B})$ on $B$. This is the extension corresponding to
$\mathcal{L}_{|B}$ under the identification
$B^{\vee}(k)=\operatorname{Ext}^{1}(B,\mathbb{G}_{m})$. Therefore the exact
sequence splits. This implies that $B$ lifts to a level group in
$\mathcal{G}(\mathcal{L})$. We conclude that $\mathcal{L}$ descends to $A/B$.
∎

######  Definition 3.12.

Let $\mathcal{H}$ be a finite $k$-group scheme. We will define a group scheme
$\mathcal{G}(\mathcal{H})$, the so called Heisenberg group associated to
$\mathcal{H}$: As a scheme

$\mathcal{G}(\mathcal{H})=\mathbb{G}_{m}\times\mathcal{H}\times\mathcal{H}^{D}$

and the group structure is defined on $T$-valued points by

$(r,h,\chi)\cdot(r^{\prime},h^{\prime},\chi^{\prime})=(rr^{\prime}\chi(h^{\prime}),h+h^{\prime},\chi+\chi^{\prime})$

where $\mathcal{H}^{D}$ denotes the Cartier dual of $\mathcal{H}$. We use the
identification
$\mathcal{H}^{D}(T)=\operatorname{Hom}_{T}(\mathcal{H}_{T},\mathbb{G}_{m,T})$.

Notice that Heisenberg groups have very similar properties as the theta group
of an ample line bundle. However such a theta group will not always be
isomorphic to a Heisenberg group. On the other hand it will be isomorphic to a
Heisenberg group if the degree of $\mathcal{L}$ is not divisible by the
characteristic of the ground field. Another example will be the line bundle
appearing in the construction of Moret-Bailly although in this case
$\deg\mathcal{L}=p$. There is the following characterization for the theta
group being a Heisenberg group:

######  Lemma 3.13.

Let $\mathcal{L}$ be an ample line bundle on an Abelian variety $A$. Suppose
there are two maximal level subgroups
$\tilde{\mathcal{H}}_{1},\tilde{\mathcal{H}_{2}}\subset\mathcal{G}(\mathcal{L})$
with
$\tilde{\mathcal{H}}_{1}\cap\tilde{\mathcal{H}_{2}}=\\{1\\}$ (scheme-theoretic
intersection). Then $\tilde{\mathcal{H}}_{1}\cong\tilde{\mathcal{H}}_{2}^{D}$
and
$\mathcal{G}(\mathcal{L})\cong\mathcal{G}(\tilde{\mathcal{H}}_{1})$.

###### Proof.

Follows immediately from the proof of [14, Lemma 8.24]. The assumption that
$k$ is algebraically closed is not needed for the part we use. ∎

###  3.3. Representations of the Theta group

In this section we recall the important properties of the action of
$\mathcal{G}(\mathcal{L})$ on the global sections
$\operatorname{H}^{0}(A,\mathcal{L})$. First there is the following
definition:

######  Definition 3.14.

Let $\mathcal{L}$ be a line bundle on an Abelian variety $A/k$. We say that a
linear representation of $\mathcal{G}(\mathcal{L})$ has _weight_
$n\in\mathbb{Z}$ if the restriction to $\mathbb{G}_{m}$ is the character
$z\mapsto z^{n}$.

######  Theorem 3.15.

Let $\mathcal{L}$ be an ample line bundle on an Abelian variety $A/k$. Then
there is a natural absolutely irreducible weight 1 representation of the group
scheme $\mathcal{G}(\mathcal{L})$ on $\operatorname{H}^{0}(A,\mathcal{L})$.

###### Proof.

See [12, p. 710]. ∎

######  Theorem 3.16.

Let $\mathcal{L}$ be an ample line bundle on an Abelian variety $A/k$. Then
there is a unique absolutely irreducible weight 1 representation $V$ of
$\mathcal{G}(\mathcal{L})$. Its dimension is
$\dim(V)=\sqrt{\sharp(K(\mathcal{L}))}=\deg(\mathcal{L})$.

###### Proof.

The existence of such a representation follows from Theorem 3.15. The
uniqueness is proven in [14, theorem 8.32] in the case where $k$ is
algebraically closed. We claim that the general case can be deduced using
descent. 111Alternatively it is not difficult to see that the cited proof goes
through in the general case if we replace irreducible with absolutely
irreducible everywhere. Indeed suppose $V,W$ are two absolutely irreducible
$\mathcal{G}(\mathcal{L})$-modules of weight $1$. Then
$V\otimes\overline{k},W\otimes\overline{k}$ are irreducible
$\mathcal{G}(\mathcal{L})_{\overline{k}}$-modules of weight $1$. By the quoted
theorem in the case where the ground field is algebraically closed
$V\otimes\overline{k}$ and $W\otimes\overline{k}$ are isomorphic. The next
Lemma 3.17 implies that $V$ and $W$ are isomorphic as
$\mathcal{G}(\mathcal{L})$-modules. This proves the claim. ∎

######  Lemma 3.17.

Let $k$ be a field and $G$ an affine group scheme over $k$ and $V,W$ be two
irreducible $G$-modules. Suppose that $V\otimes\overline{k}$ and
$W\otimes\overline{k}$ are isomorphic as $G_{\overline{k}}$-modules. Then $V$
and $W$ are isomorphic as $G$-modules.

###### Proof.

Consider the $k$-vector space $\operatorname{Hom}_{G}(V,W)$ of $G$-equivariant
$k$-linear maps. By Schur’s lemma it suffices to show that
$\operatorname{Hom}_{G}(V,W)\neq\\{0\\}$.

Indeed $\operatorname{Hom}_{G}(V,W)$ equals the vector space of $k$-linear
maps $\varphi:V\rightarrow W$ such that the following diagram commutes:

${V}$${W}$${V\otimes A_{G}}$${W\otimes
A_{G}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi\otimes\operatorname{id}}$

where $A_{G}$ denotes the Hopf-algebra of $G$ and the vertical arrows are
induced from the action of $G$ on $V$ (resp. $W$).

This description of $\operatorname{Hom}_{G}(V,W)$ implies that for any field
extension $k^{\prime}/k$ one has

$\operatorname{Hom}_{G_{k^{\prime}}}(V\otimes k^{\prime},W\otimes
k^{\prime})=\operatorname{Hom}_{G}(V,W)\otimes k^{\prime}\,.$

Since by assumption $V\otimes\overline{k}$ and $W\otimes\overline{k}$ are
isomorphic as $G_{\overline{k}}$-modules we obtain
$\operatorname{Hom}_{G_{\overline{k}}}(V\otimes\overline{k},W\otimes\overline{k})\neq\\{0\\}$.
Therefore $\operatorname{Hom}_{G}(V,W)\neq\\{0\\}$ and the lemma follows. ∎

The representation of the theta group can be used to characterize the image of
the global sections under pullback.

######  Proposition 3.18.

Let $\pi:A\rightarrow B$ be a finite surjective map of Abelian varieties with
kernel $\mathcal{H}$. Let further $\mathcal{M}$ be an ample line bundle on
$B$. Then the image of

$\pi^{*}:\operatorname{H}^{0}(B,\mathcal{M})\rightarrow\operatorname{H}^{0}(A,\pi^{*}\mathcal{M})$

is $\operatorname{H}^{0}(A,\pi^{*}\mathcal{M})^{\tilde{\mathcal{H}}}$ where
$\tilde{\mathcal{H}}\subset\mathcal{G}(\pi^{*}\mathcal{M})$ is the level
subgroup over $\mathcal{H}$ coming from Proposition 3.4.

###### Proof.

Mumford uses this without proof in [10, §1, Theorem 4]. The result follows
from the proof of Proposition 3.4, which we do not discuss, and [11, p. 104,
Theorem 1]. ∎

We will also need a result about the action of the level group when we have a
situation as in Proposition 3.10.

######  Proposition 3.19.

Let $A/k$ be an Abelian variety and $\mathcal{H}\subset A$ be a finite
subgroup scheme. Suppose an ample effective divisor $D\subset A$ is
$\mathcal{H}$-invariant.

Then the action of the level group
$\tilde{\mathcal{H}}\subset\mathcal{G}(\mathcal{O}(D))$ coming from
Proposition 3.10 on $\operatorname{H}^{0}(A,\mathcal{O}(D))$ is given by the
translation action of $\mathcal{H}$ on the function field $K(A)$.

###### Proof.

From the proof of Proposition 3.10 we get a lift of the translation action of
$\mathcal{H}$ to the sheaf $\mathcal{O}(D)$. Furthermore we have seen there
that the inclusion $\mathcal{O}\rightarrow\mathcal{O}(D)$ is equivariant for
this action. By taking the stalk at the generic point of
$\mathcal{O}\rightarrow\mathcal{O}(D)$ we see that the action on global
sections induced by the action of $\mathcal{H}$ on the sheaf $\mathcal{O}(D)$
is given by translation of rational functions. On the other hand one can check
using the definitions that the action of $\mathcal{H}$ on global sections
induced from the action on the sheaf $\mathcal{O}(D)$ coincides with the
restriction of the representation of the theta group to the level group
$\tilde{\mathcal{H}}$. ∎

##  4\. The Construction of Moret-Bailly

Let $k$ be an algebraically closed field of characteristic $p>2$. Fix a
supersingular elliptic curve $E/k$ defined over $\mathbb{F}_{p}$ such that the
relative Frobenius satisfies $F^{2}+p=0$. All schemes in this section will be
$k$-schemes to allow ourselves to apply theorems for schemes over
algebraically closed fields. However as we see later everything will be
defined over $\mathbb{F}_{p^{2}}$.

###  4.1. Notation

We shall fix the following notation for the rest of the article.

  * •

$E$ is a supersingular elliptic curve defined over $\mathbb{F}_{p}$ such that
the relative Frobenius satisfies $F^{2}+p=0$.

  * •

$\mathsf{O}=\text{End}(E)$ and $B=\mathsf{O}\otimes\mathbb{Q}$. Notice that
the conditions on $E$ imply that all the endomorphism of $E$ are defined over
$\mathbb{F}_{p^{2}}$.

  * •

There is an additive bijection

$\gamma:\text{Mat}_{m,n}(\mathsf{O})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\text{Hom}(E^{n},E^{m})$
$\Phi=(\varphi_{ij})\mapsto\left(\psi:E^{n}\rightarrow
E^{m},\,(P_{1},\ldots,P_{n})\mapsto\left(\sum_{j=1}^{n}\varphi_{1j}(P_{j})),\ldots,\sum_{j=1}^{n}\varphi_{mj}(P_{j})\right)\right)\,.$

This bijection turns matrix multiplication into composition of maps. Notice
that we use the convention that maps should be composed from right to left.

  * •

$\mu:E^{2}\rightarrow(E^{2})^{\vee}$ denotes the natural product polarization.

  * •

The map

$f\mapsto(\mu\circ\gamma(f):E^{2}\rightarrow(E^{2})^{\vee})$

is a bijection from the set of positive definite hermitian matrices in
$M_{2,2}(\mathsf{O})$ to the set of polarizations on $E^{2}$.

  * •

Let $A$ be an Abelian variety. By an elliptic curve $E_{1}\subset A$ we mean a
closed subscheme which is a smooth curve of genus 1 with a distinguished
rational point, but not necessarily the origin. Thus $E_{1}$ does not have to
be an Abelian subvariety.

###  4.2. Results of Moret-Bailly

In this section we recall the construction of the families of Moret-Bailly.
All the results in this section are taken from [9]. Thus they are just stated
here without proof for the convenience of the reader and to fix the notation.

Consider the $\alpha$-group scheme $\alpha(\mathcal{O}(1))$ over
$\mathbb{P}^{1}$ (see Definition 2.3 for the notation). The map
$\mathcal{O}^{2}\rightarrow\mathcal{O}(1)$ induces a map of
$\mathbb{P}^{1}$-group schemes
$\alpha(\mathcal{O}(1))\rightarrow\alpha_{p,\mathbb{P}^{1}}^{2}$. After
choosing an isomorphism $E[F]\cong\alpha_{p}$ there is a composed map
$\iota:\alpha(\mathcal{O}(1))\rightarrow E^{2}_{\mathbb{P}^{1}}$. The map will
depend on the choice of the isomorphism $E[F]\cong\alpha_{p}$, but its image
will not. By [1, exposé 5 Théorème 4.1] the quotient
$E^{2}_{\mathbb{P}^{1}}/\iota(\alpha(\mathcal{O}(1)))$ exists. It is an
Abelian scheme over $\mathbb{P}^{1}$.

######  Definition 4.1.

We define $\mathcal{A}=E^{2}_{\mathbb{P}^{1}}/\iota(\alpha(\mathcal{O}(1)))$.

Moret-Bailly proved that one can put a principal polarization on
$\mathcal{A}$:

######  Lemma 4.2.

Let $\eta$ be a polarization on $E^{2}$ such that $\ker(\eta)=E^{2}[F]$. Then
$\eta_{\mathbb{P}^{1}}$ descends to a principal polarization $\lambda$ on the
Abelian scheme $\mathcal{A}\rightarrow\mathbb{P}^{1}$.

###### Proof.

For the proof we refer to [9, §1.4]. ∎

The next proposition explains the properties of the polarization $\lambda$ on
the Abelian scheme $\mathcal{A}\rightarrow\mathbb{P}^{1}$:

######  Theorem 4.3.

There exists a symmetric divisor $\mathfrak{C}$ on $\mathcal{A}$ that is
relatively ample and flat over $\mathbb{P}^{1}$ and induces the polarization
$\lambda$. Then we have the following properties:

  * •

The generic fiber of $\mathfrak{C}\rightarrow\mathbb{P}^{1}$ is a smooth
geometrically connected curve of genus 2.

  * •

$5p-5$ fibers of $\mathfrak{C}\rightarrow\mathbb{P}^{1}$ are of the form
$E_{1}\cup E_{2}$ where $E_{1},E_{2}$ are elliptic curves intersecting
transversely in a point contained in $\mathcal{A}[2]$. We are going to refer
to these fibers as the reducible fibers. By [3, formula 3.3] the reducible
fibers are fibers over $\mathbb{F}_{p^{2}}$-valued points of $\mathbb{P}^{1}$
(necessary but not sufficient condition).

Denote by $\mathfrak{D}$ the preimage of $\mathfrak{C}$ under
$E^{2}_{\mathbb{P}^{1}}\rightarrow\mathcal{A}$. Then $\mathfrak{D}$ is a
symmetric divisor on $E^{2}_{\mathbb{P}^{1}}$ that is relatively ample and
flat over $\mathbb{P}^{1}$ and induces the polarization
$\eta_{\mathbb{P}^{1}}$. Furthermore we have the following properties:

  * •

The generic fiber of $\mathfrak{D}\rightarrow\mathbb{P}^{1}$ is a singular
geometrically integral curve of geometric genus 2 and arithmetic genus $p+1$.

  * •

$5p-5$ fibers of $\mathfrak{D}\rightarrow\mathbb{P}^{1}$ are of the form
$E_{1}\cup E_{2}$ where $E_{1},E_{2}$ are elliptic curves intersecting with
multiplicity $p$ in one of $10$ points contained in $E^{2}[2]$.

###### Proof.

See [9, §2.1]. ∎

Notice that $\mathfrak{C}$ is not unique but only unique up to translating by
a $2$-torsion point.

###  4.3. Reducible Fibers I: Classification of reducible fibers

######  Proposition 4.4.

Let $f\in M_{2,2}(\mathsf{O})$ be a hermitian positive definite matrix
inducing a polarization $\eta$ on $E^{2}$ with kernel $E^{2}[F]$. Then there
exists an ample divisor $D=E_{1}+E_{2}$ inducing the polarization $\eta$ where
$E_{1},E_{2}$ are elliptic curves intersecting in the origin with multiplicity
$p$. To give such a decomposition is equivalent to writing $f=f_{1}+f_{2}$
where $f_{1},f_{2}\in M_{2,2}(\mathsf{O})$ are hermitian rank 1 matrices with
integral entries.

Furthermore if we are given $f_{1},f_{2}$ as above then
$E_{i}=\ker(\gamma(f_{i}))$. In particular the $E_{i}$ and $D$ are defined
over $\mathbb{F}_{p^{2}}$.

###### Proof.

The existence of an ample divisor $D$ of the form $D=E_{1}+E_{2}$ with above
properties follows from Theorem 4.3. We prove now that to give such a divisor
is equivalent to writing $f=f_{1}+f_{2}$ with $f_{1},f_{2}\in
M_{2,2}(\mathsf{O})$ hermitian of rank $1$. Indeed consider the map

$r:\text{NS}(E^{2})\longrightarrow\operatorname{End}(E^{2}),\,D\mapsto\mu^{-1}\circ\lambda_{\mathcal{O}(D)}$

where $\mu$ denotes the natural product polarization on $E^{2}$. By [11, p.
174, Theorem 2 and p. 214, Theorem 3] $r$ is an injective homomorphism and
$\text{im}(r)$ is the subgroup of endomorphisms fixed by the Rosati involution
induced by $\mu$. Under the identification
$\operatorname{End}(E^{2})=M_{2,2}(\mathsf{O})$ the Rosati involution is given
by the hermitian transpose. The desired bijection is defined by
$f_{i}=r(E_{i}),\,i=1,2$.

We prove that this bijection has the right properties. Indeed suppose we are
given an ample divisor $D=E_{1}+E_{2}$ inducing the polarization $\eta$. We
have to show that $f_{i}=r(E_{i})$ have rank 1 for $i=1,2$. To show this
notice first that clearly $E_{i}\subseteq\ker(\gamma(f_{i})),\,i=1,2$.
Furthermore by Poincaré’s reducibility theorem $E^{2}$ is isogenous to
$E_{1}\times E_{2}$ and therefore $E_{1},E_{2}$ must be supersingular.
Therefore there is an isogeny $E\longrightarrow E_{i}$. The composed map
$E\rightarrow E_{i}\hookrightarrow E^{2}$ corresponds to a
$v_{i}\in\mathsf{O}^{2}\setminus\\{0\\}$. Now $f_{i}v_{i}=0$ proving that
$\operatorname{rk}f_{i}\leqslant 1$. Since certainly $f_{i}\neq 0$ we deduce
$\operatorname{rk}f_{i}=1$.

Conversely suppose we are given $f_{1},f_{2}\in M_{2,2}(\mathsf{O})$ hermitian
of rank 1 such that $f=f_{1}+f_{2}$. Consider $D_{i}=r^{-1}(f_{i}),\,i=1,2$.
We have to show that the $D_{i}$ are algebraically equivalent to elliptic
curves with the required properties. Indeed as $\operatorname{rk}f_{i}=1$,
there exist $v_{i}\in\mathsf{O}^{2}\setminus\\{0\\}$ such that $f_{i}v_{i}=0$.
The $v_{i}$ correspond to maps $E\longrightarrow E^{2}$ whose image $E_{i}$ is
an elliptic curve satisfying $E_{i}\subset\ker(\gamma(f_{i}))$. It follows
from Proposition 3.11 that $D_{i}$ is algebraically equivalent to $n_{i}E_{i}$
for some $n_{i}\in\mathbb{Z}$. On the other hand
$D.D=2\deg(D)=2\sqrt{\deg\eta}=2p=2n_{1}n_{2}E_{1}.E_{2}$. Therefore

$(1,1,p),(1,p,1),(p,1,1),(-1,-1,p),(-1,-p,1),(-p,-1,1)$

are all the possibilities for $(n_{1},n_{2},E_{1}.E_{2})$. The last three
possibilities are excluded by ampleness of $D$. The 2nd and 3rd would imply
$E_{1}[p]\subseteq\ker(\gamma(f_{1}))\cap\ker(\gamma(f_{2}))\subseteq\ker(\eta)$
respectively $E_{2}[p]\subseteq\ker(\eta)$ contradicting
$\ker(\eta)=E^{2}[F]$. This shows that $D=E_{1}+E_{2}$ and $E_{1}.E_{2}=p$.

We will now prove that $E_{i}=\ker(\gamma(f_{i}))$. Indeed denote by
$\mathcal{L}_{i}=\mathcal{O}(E_{i})$ the line bundle corresponding to $E_{i}$.
Since $f_{i}=\mu^{-1}\circ\lambda_{\mathcal{L}_{i}}$, the claim is equivalent
to $K(\mathcal{L}_{i})=E_{i}$. Thus we will compute $K(\mathcal{L}_{i})$.
Indeed clearly $E_{i}\subset K(\mathcal{L}_{i})$ and $K(\mathcal{L}_{i})\neq
E^{2}$ because $E_{i}$ is not algebraically equivalent to $0$. Therefore
$G=K(\mathcal{L}_{i})/E_{i}$ is finite. On the other hand if
$\pi:E^{2}\rightarrow E^{2}/E_{i}$ denotes the natural quotient map, then
$\mathcal{L}_{i}=\pi^{*}\mathcal{O}(0)$. By Proposition 3.4 $E_{i}$ lifts to a
level subgroup in $\mathcal{G}(\mathcal{L}_{i})$. We compute the centralizer
$\mathfrak{C}(E_{i})$ in $\mathcal{G}(\mathcal{L}_{i})$. For this we note that
the commutator pairing factors through $E_{i}\times
G\rightarrow\mathbb{G}_{m}$. This induces a group homomorphism
$E_{i}\rightarrow G^{D}$. But $E_{i}$ is proper and geometrically connected
and $G^{D}$ is finite. Therefore $E_{i}\rightarrow G^{D}$ is trivial. We
conclude $\mathfrak{C}(E_{i})=\mathcal{G}(\mathcal{L}_{i})$. Using [14,
Proposition 8.15] one obtains
$\mathbb{G}_{m}=\mathcal{G}(\mathcal{O}(0_{E^{2}/E_{i}}))=\mathfrak{C}(E_{i})/E_{i}$.
This implies $G=0$ and we have proven $E_{i}=\ker(\gamma(f_{i}))$. ∎

######  Corollary 4.5.

In the notation of the previous proposition, the scheme theoretic intersection
$\mathcal{H}=E_{1}\cap E_{2}\subset E^{2}$ is a closed subgroup scheme
isomorphic to $\alpha_{p}$. The quotient $A=E^{2}/\mathcal{H}$ is isomorphic
to $E_{1}^{(p)}\times E_{2}^{(p)}$. Also $\eta$ descends along the isogeny
$E^{2}\rightarrow A$ to a principal polarization $\lambda$ on $A$. In
particular $(A,\,\lambda)$ is a fiber of the Moret-Bailly family constructed
from $E^{2},\eta$. Furthermore $\lambda$ is the natural product polarization
on $E_{1}^{(p)}\times E_{2}^{(p)}$.

###### Proof.

Choose a uniformizer $\mathfrak{z}\in\mathcal{O}_{E_{1},0}$. Since
$E_{1},E_{2}$ intersect with multiplicity $p$, the intersection
$\mathcal{H}=E_{1}\cap E_{2}$ is given by the image of the closed immersion
$\operatorname{Spec}(\mathcal{O}_{E_{1},0}/\mathfrak{z}^{p})\hookrightarrow
E_{1}$. Therefore $\mathcal{H}=E_{1}[F]$ as a closed subscheme. We conclude
that $\mathcal{H}$ is a closed subgroup scheme isomorphic to $\alpha_{p}$.

Consider the quotient $A=E^{2}/\mathcal{H}$ and denote the quotient map by
$\pi:E^{2}\rightarrow A$. Now because $\mathcal{H}\subset E_{1}$ and
$\mathcal{H}\subset E_{2}$ is a subgroup scheme, the divisor $D$ is
$\mathcal{H}$-invariant. Therefore by Proposition 3.10 $\mathcal{O}(D)$
descends along $\pi$ to the line bundle $\mathcal{O}(\pi(D))$. Since $D$
induces the polarization $\eta$, $\eta$ thus descends to a polarization
$\lambda$ induced by $\pi(D)$ on $A$. Computing degrees yields that $\lambda$
is a principal polarization. On the other hand
$\mathcal{O}(D)=\pi^{*}\mathcal{O}(\pi(D))$ implies $D=\pi^{-1}(\pi(D))$.
Therefore the map $D\rightarrow\pi(D)$ is a finite radicial map of degree $p$
being the pull-back of such a map. Thus $\pi(D)=E_{1}^{(p)}+E_{2}^{(p)}$ and
by a result of A. Weil $A\cong E_{1}^{(p)}\times E_{2}^{(p)}$ and $\lambda$ is
the natural product polarization. ∎

######  Corollary 4.6.

In the notation of the previous proposition there is a bijective map

$\left\\{\begin{tabular}[]{@{}l@{}}\emph{Unordered pairs} $f_{1},f_{2}\in
M_{2,2}(\mathsf{O})$\\\ \emph{hermitian} \emph{of rank 1}\\\ \emph{such that}
$f=f_{1}+f_{2}$\end{tabular}\right\\}\rightarrow\left\\{\begin{tabular}[]{@{}l@{}}\emph{reducible
fibers of}\\\ \emph{the Moret-Bailly family}\\\
$\mathfrak{C}\rightarrow\mathbb{P}^{1}$ \emph{constructed from}
$f$\end{tabular}\right\\}\,.$

###### Proof.

Proposition 4.4 and Corollary 4.5 give us such a map. We claim that the map is
injective. Indeed suppose that the two pairs $\\{f_{1},f_{2}\\}$ and
$\\{f_{1}^{\prime},f_{2}^{\prime}\\}$ map to the same element in the target.
Define $D$ to be the divisor $\ker(\gamma(f_{1}))+\ker(\gamma(f_{2}))$ on
$E^{2}$ and similarly $D^{\prime}=\ker(f_{1}^{\prime})+\ker(f_{2}^{\prime})$.
Define further $\mathcal{H}=\ker(\gamma(f_{1}))\cap\ker(\gamma(f_{2}))$ and
$\mathcal{H}^{\prime}=\ker(f_{1}^{\prime})\cap\ker(f_{2}^{\prime})$. Then one
has $\mathcal{H}=\mathcal{H}^{\prime}$ as subgroup schemes because
$\\{f_{1},f_{2}\\}$ and $\\{f_{1}^{\prime},f_{2}^{\prime}\\}$ map to the same
element in the target. Denote by $\pi:E^{2}\rightarrow E^{2}/\mathcal{H}$ the
quotient map. Then $\pi(D),\,\pi(D^{\prime})$ induce the same principal
polarization. Therefore there exists a $2$-torsion point $P$ such that
$\pi(D)=t_{P}(\pi(D^{\prime}))$. But the two components of $\pi(D)$ (resp.
$\pi(D^{\prime})$) intersect only at $0$. Therefore one must have $P=0$. This
implies $\pi(D)=\pi(D^{\prime})$. But by Proposition 3.10 one has
$D=\pi^{-1}(\pi(D))$ (resp. $D^{\prime}=\pi^{-1}(\pi(D)^{\prime})$) and thus
we conclude $D=D^{\prime}$. Proposition 4.4 gives
$\\{f_{1},f_{2}\\}=\\{f_{1}^{\prime},f_{2}^{\prime}\\}$ as unordered pairs.
This proves the injectivity. We prove the surjectivity. Indeed let
$x\in\mathbb{P}^{1}$ be a point such that $\mathfrak{C}_{x}$ is reducible.
Consider the divisor $\mathfrak{D}_{x}$ on $E^{2}$. By Theorem 4.3
$\mathfrak{D}_{x}$ is the union of two elliptic curves intersecting at a
$2$-torsion point $P\in E^{2}[2]$. Define $D=t_{P}(\mathfrak{D}_{x})$. Then
$D$ is an ample divisor inducing the polarization $\eta$ and the union of two
elliptic curves intersecting at the origin. Proposition 4.4 gives the desired
preimage $\\{f_{1},f_{2}\\}$ under our map. ∎

We further investigate these decompositions $f=f_{1}+f_{2}$ from the
proposition.

######  Proposition 4.7.

Let $f\in M_{2,2}(\mathsf{O})$ be a hermitian positive definite matrix
inducing a polarization $\eta$ on $E^{2}$ with kernel $E^{2}[F]$. Consider the
hermitian lattice $L=(\mathsf{O}^{2},f)$. To write $f=f_{1}+f_{2}$ where
$f_{1},f_{2}\in M_{2,2}(\mathsf{O})$ are hermitian rank 1 matrices is
equivalent to giving two sublattices $I_{1},I_{2}\subseteq L$ with the
following properties i)-iii). If $I_{1},I_{2}$ satisfy i) and ii), then
condition iii) is equivalent to iii’):

  * i)

$I_{1},I_{2}$ are invertible right-$\mathsf{O}$-modules.

  * ii)

$I_{1}\perp I_{2}$ with respect to $f$.

  * iii)

$L/(I_{1}\oplus I_{2})\cong\mathsf{O}/F$.

  * iii’)

$I_{2}$ is primitive and for every $v\in I_{1}$ one has
$v^{\dagger}fv=p[I_{1}:v\mathsf{O}]$.

The bijection is defined as follows: Given $f_{1},f_{2}$ we define
$I_{i}=\ker(f_{i}),\,i=1,2$ considered as an $\mathsf{O}$-submodule of
$\mathsf{O}^{2}$.

Conversely given $I_{1},I_{2}$ there is a direct sum decomposition

$L\otimes B=(I_{1}\otimes B)\oplus(I_{2}\otimes B)\,.$

Denote by $P_{i}\in M_{2,2}(B)$ the matrices of the projection maps onto the
two factors respectively. Then one defines
$f_{1}=P_{2}^{\dagger}fP_{2},\,f_{2}=P_{1}^{\dagger}fP_{1}$.

###### Proof.

“$\Rightarrow$” Suppose we have a decomposition $f=f_{1}+f_{2}$ where
$f_{i}\in M_{2,2}(\mathsf{O})$ are hermitian rank 1 matrices with integral
entries. Then we define $I_{i}=\ker(f_{i})$ considered as an
$\mathsf{O}$-submodule of $\mathsf{O}^{2}$. It is clear that property i)
holds. For showing ii) we take $v_{1}\in I_{1},v_{2}\in I_{2}$ arbitrary. Then
$v_{1}^{\dagger}fv_{2}=v_{1}^{\dagger}f_{1}v_{2}+v_{1}^{\dagger}f_{2}v_{2}=v_{2}^{\dagger}f_{1}v_{1}=0$
using the definition of the $I_{i}$.

We turn to property iii): Indeed i) and ii) imply the direct sum decomposition

$L\otimes B=(I_{1}\otimes B)\oplus(I_{2}\otimes B)\,.$

Denote by $P_{i}\in M_{2,2}(B)$ the matrices of the projection maps onto the
two factors respectively. Then one has the identities
$f_{i}P_{i}=0,\,P_{1}+P_{2}=\mathds{1}_{2}$. From this we deduce
$f_{1}=f_{1}(P_{1}+P_{2})=f_{1}P_{2}$ and similarly $f_{2}=f_{2}P_{1}$. This
implies $fP_{2}=f_{1}P_{2}+f_{2}P_{2}=f_{1}$ and similarly $fP_{1}=f_{2}$. We
will now deduce iii) by considering the localization at all prime numbers $l$.
Indeed notice first that $I_{1}\subset L$ is a primitive submodule, i.e.,
$L/I_{i}$ is torsion free. This implies that also $I_{1,(l)}\subset L_{(l)}$
is primitive. Furthermore since $\mathbb{Z}_{(l)}$ is a DVR, $I_{1,(l)}$ is
free, generated by $v_{1}\in I_{1,(l)}$, say. Assume first that $l\neq p$.
Then since $\ker(\eta)=E^{2}[F]$ and $l\neq p$, the pairing on $L_{(l)}$ is
perfect. Thus by Lemma 1.5 there exists an $w\in L_{(l)}$ such that
$w^{\dagger}fv_{1}=1$. On the other hand $P_{1}$ is the orthogonal projection
onto the subspace generated by $v_{1}$ and thus by basic linear algebra
$P_{1}=\frac{v_{1}v_{1}^{\dagger}f}{v_{1}^{\dagger}fv_{1}}$. We now compute
$w^{\dagger}f_{2}w=w^{\dagger}fP_{1}w=\frac{w^{\dagger}fv_{1}v_{1}^{\dagger}fw}{v_{1}^{\dagger}fv_{1}}=\frac{1}{v_{1}^{\dagger}fv_{1}}\in\mathbb{Z}_{(l)}$.
This implies that $v_{1}^{\dagger}fv_{1}\in\mathbb{Z}_{(l)}^{\times}$.
Therefore $P_{1}\in M_{2,2}(\mathsf{O}_{(l)})$. Similarly one proves $P_{2}\in
M_{2,2}(\mathsf{O}_{(l)})$. This implies $L_{(l)}=I_{1,(l)}\oplus I_{2,(l)}$.
It remains to consider the case $l=p$. Then the pairing given by $f$ induces a
map $L\longrightarrow L^{\dagger}$ with image $L^{\dagger}F$. The same
argument as in Lemma 1.5 shows the existence of a $w\in L$ such that
$w^{\dagger}fv_{1}=F$. We now compute
$w^{\dagger}f_{2}w=w^{\dagger}fP_{1}w=\frac{w^{\dagger}fv_{1}v_{1}^{\dagger}fw}{v_{1}^{\dagger}fv_{1}}=\frac{F\overline{F}}{v_{1}^{\dagger}fv_{1}}=\frac{p}{v_{1}^{\dagger}fv_{1}}\in\mathbb{Z}_{(p)}$.
This implies that
$P_{1}=\frac{v_{1}v_{1}^{\dagger}f}{v_{1}^{\dagger}fv_{1}}\in\frac{F}{p}M_{2,2}(\mathsf{O}_{(p)})$
because $f\in FM_{2,2}(\mathsf{O})$. Similarly
$P_{2}\in\frac{F}{p}M_{2,2}(\mathsf{O}_{(p)})$. We conclude that
$L_{(p)}/(I_{1,(p)}\oplus I_{2,(p)})$ is killed by $F$. This implies that
$L_{(p)}/(I_{1,(p)}\oplus I_{2,(p)})$ is either $\mathsf{O}/F$ or
$(\mathsf{O}/F)^{2}$. However the second case is impossible because of the
primitivity of $I_{1,(p)}$. This proves iii).

“$\Leftarrow$”: Conversely suppose we are given $I_{1}$ and $I_{2}$ satisfying
i)-iii). There is a direct sum decomposition

$L\otimes B=(I_{1}\otimes B)\oplus(I_{2}\otimes B)\,.$

Denote by $P_{i}\in M_{2,2}(B)$ the matrices of the projection maps onto the
two factors respectively. Then one defines
$f_{1}=P_{2}^{\dagger}fP_{2},\,f_{2}=P_{1}^{\dagger}fP_{1}$. By construction
one has $f_{1}P_{1}=f_{2}P_{2}=0$. From these equalities and
$P_{1}+P_{2}=\mathds{1}_{2}$ one easily infers
$f=f_{1}+f_{2},\,f_{1}=fP_{2},\,f_{2}=fP_{1}$. It remains to show
$f_{1},f_{2}\in M_{2,2}(\mathsf{O})$. Indeed from ii) one gets
$P_{1},P_{2}\in\frac{F}{p}M_{2,2}(\mathsf{O})$. Hence one obtains
$f_{1}=fP_{2},\,f_{2}=fP_{1}\in M_{2,2}(\mathsf{O})$.

We show now that for $I_{1},I_{2}$ satisfying conditions i)-iii) property
iii’) holds true. Indeed we have already explained in this proof why $I_{2}$
must be primitive. We prove now the formula
$v^{\dagger}fv=p[I_{1}:v\mathsf{O}],\,\forall v\in I_{1}$ by comparing the
$l$-adic valuations of both sides for all prime numbers. Since
$I_{1,(l)},I_{2,(l)}$ are free, there exist $v_{1}\in I_{1,(l)},\,v_{2}\in
I_{2,(l)}$ such that
$I_{1,(l)}=v_{1}\mathsf{O}_{(l)},I_{2,(l)}=v_{2}\mathsf{O}_{(l)}$. Because
$I_{1}\perp I_{2}$, the matrix representing the hermitian form on
$I_{1,(l)}\oplus I_{2,(l)}$ induced by $f$ is of the form

$\begin{pmatrix}d_{1}&0\\\ 0&d_{2}\end{pmatrix}$

with $d_{1},d_{2}\in\mathbb{Z}_{(l)}$. Assume first that $l\neq p$. Then the
hermitian pairing on $I_{1,(l)}\oplus I_{2,(l)}$ is perfect an thus
$d_{1},d_{2}\in\mathbb{Z}_{(l)}$. This implies
$\nu_{l}(v^{\dagger}fv)=\nu_{l}([I_{1}:v\mathsf{O}]),\,\forall v\in I_{1}$.
Assume now that $l=p$. Then as $F|f$ we must have $p|d_{1},d_{2}$. On the
other hand the map $I_{1,(p)}\oplus I_{2,(p)}\longrightarrow
I_{1,(p)}^{\dagger}\oplus I_{2,(p)}^{\dagger}$ induced by $f$ has cokernel of
length $4$ over the local ring $\mathsf{O}_{(p)}$ with maximal ideal
$F\mathsf{O}_{(p)}=\mathsf{O}_{(p)}F$: This follows from $L/(I_{1}\oplus
I_{2})=\mathsf{O}/F$ and $\operatorname{im}(L\longrightarrow
L^{\dagger})=L^{\dagger}F$. This implies that
$\nu_{p}(d_{1}d_{2})=\nu_{p}(F^{4})=2$. Therefore
$\nu_{p}(d_{1})=\nu_{p}(d_{2})=1$. We conclude
$\nu_{p}(v^{\dagger}fv)=1+\nu_{p}([I_{1}:v\mathsf{O}]),\,\forall v\in I_{1}$.
Since the sign of $v^{\dagger}fv$ is positive we have proven
$v^{\dagger}fv=p[I_{1}:v\mathsf{O}],\,\forall v\in I_{1}$.

Next we prove that for $I_{1},I_{2}$ satisfying conditions i), ii), iii’)
property iii) holds true. Indeed given $I_{1},I_{2}$ we define $f_{1},f_{2}$
by the formulae above. We claim that $f_{1},f_{2}$ have integral coefficients.
This is proven by localizing at all prime numbers $l$. Using the same notation
as in the previous paragraph we have
$f_{1}=\frac{fv_{1}v_{1}^{\dagger}f}{v_{1}^{\dagger}fv_{1}}$. The matrix
coefficients of numerator have $l$-adic valuation $\geqslant\nu_{l}(p)$, while
the denominator has
$\nu_{l}(v_{1}^{\dagger}fv_{1})=\nu_{l}(d_{1})=\nu_{l}(p)$. This implies that
$f_{1}\in M_{2,2}(\mathsf{O})$. Therefore also $f_{2}=f-f_{1}\in
M_{2,2}(\mathsf{O})$. Furthermore by construction $I_{i}\subseteq\ker(f_{i})$.
Equality holds because all the lattices are primitive sublattices. Therefore
by the direction “$\Rightarrow$” we already proved, $I_{1},I_{2}$ satisfy
conditions i)-iii). ∎

######  Corollary 4.8.

Using the notation of the previous proposition, the isomorphism classes of the
elliptic curves $E_{i}$ in Proposition 4.4 are determined as follows: By a
result of Deuring there is a bijection between isomorphism classes of
invertible right-$\mathsf{O}$-modules and isomorphism classes of supersingular
elliptic curves over $\overline{\mathbb{F}_{p}}$. Then $E_{i}$ is the elliptic
curve corresponding to $I_{i}$ under this bijection.

In particular one has $E_{i}\cong E$ if and only if $I_{i}$ is free as a
$\mathsf{O}$-module generated by some vector $v_{i}\in\mathsf{O}^{2}$. In this
case the map $E_{i}\hookrightarrow E^{2}$ is the one given by the vector
$v_{i}$.

###### Proof.

We have to show that the bijection

$\left\\{\begin{tabular}[]{@{}l@{}}Invertible right\\\
$\mathsf{O}$-modules\end{tabular}\right\\}_{/\cong}\longrightarrow\left\\{\begin{tabular}[]{@{}l@{}}Supersingular
elliptic\\\ curves over $\mathbb{F}_{p^{2}}$\end{tabular}\right\\}_{/\cong}$

maps $I_{1}$ to the elliptic curve $E_{1}$. Indeed choose $v\in
I_{1}\setminus\\{0\\}$. Then $v\in\mathsf{O}^{2}$ determines a map
$\phi:E\rightarrow E^{2}$. Since $v\in I_{1}=\ker(f_{1})$, one has
$\operatorname{im}(\phi)\subseteq E_{1}$. Since $v\neq 0$ the map
$\phi:E\longrightarrow E_{1}$ is an isogeny. One checks that $\phi$ is the
isogeny constructed from the invertible left $\mathsf{O}$-ideal

$\\{a\in B|va\in I_{1}\\}^{-1}$

in the construction in [15, Section 42.2]. ∎

######  Example 4.9.

Consider for example $E/\mathbb{F}_{3}$ with Weierstraß equation
$y^{2}=x^{3}-x$. Choose a square root of -1 in $\mathbb{F}_{9}$ and denote it
by $i$. Then $E$ has the endomorphism $x\mapsto-x,\,y\mapsto iy$ defined over
$\mathbb{F}_{9}$ also denoted $i$ by abuse of notation. Therefore $E$ has CM
by the order $\mathbb{Z}[i]$. Since $(3)$ is inert in $\mathbb{Q}(i)$ the
curve $E$ is supersingular and

$\operatorname{End}(E)\otimes\mathbb{Q}=\mathbb{Q}[i,F]/(iF+Fi)$

is the endomorphism algebra. We can take

$f=\begin{pmatrix}3&(1+i)F\\\ -(1+i)F&3\end{pmatrix}\,.$

Then one decomposition as in the proposition is

$f_{1}=\begin{pmatrix}2&(1+i)F\\\
-(1+i)F&3\end{pmatrix},f_{2}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\,.$

One computes that $I_{1}=\begin{pmatrix}-F\\\ 1+i\end{pmatrix}\mathsf{O}$ and
$I_{2}=\begin{pmatrix}0\\\ 1\end{pmatrix}\mathsf{O}$. Hence

$P_{1}=\begin{pmatrix}1&0\\\
\frac{(1+i)F}{3}&0\end{pmatrix},\,P_{2}=\begin{pmatrix}0&0\\\
\frac{-(1+i)F}{3}&1\end{pmatrix}\,.$

One checks that the formulas
$f_{1}=P_{2}^{\dagger}fP_{2},\,f_{2}=P_{1}^{\dagger}fP_{1}$ are indeed
satisfied.

The following algorithm computes for a given $f$ and an invertible right
$\mathsf{O}$-module $I$ all the sublattices $I_{1}\oplus I_{2}$ with
properties i)-iii) of the last proposition with $I_{1}\cong I$. As it turns
out this computation reduces to a short vector problem.

1

input : An hermitian positive definite matrix $f\in M_{2,2}(\mathsf{O})$
inducing a polarization $\eta$ on $E^{2}$ with kernel $E^{2}[F]$. A fractional
ideal $I\subset B$ with right order $\mathsf{O}$.

output : All decompositions $f=f_{1}+f_{2}$ as in Proposition 4.4 such that
$\ker(f_{1})\cong I$

2 begin

3 Consider the hermitian lattice $(\mathsf{O}^{2},f)$. By forgetting the
$\mathsf{O}$-module structure we get a quadratic space over $\mathbb{Z}$.
Consider the sub-$\mathbb{Z}$-module $I^{-1}\oplus I^{-1}$ (notice that this
is not a right $\mathsf{O}$-module). Compute the set of short vectors
$S=\\{v\in I^{-1}\oplus I^{-1}|v^{\dagger}fv\leqslant p[I:\mathsf{O}]\\}$.

4 output $\\{(f_{1}=\frac{fvv^{\dagger}f}{v^{\dagger}fv},f_{2}=f-f_{1})|v\in
S\\}$

5 end

6

Algorithm 1 Compute decompositions

###### Proof.

We prove the correctness of the algorithm. First we claim that the length of
every vector in $I^{-1}\oplus I^{-1}$ is divisible by $p[I:\mathsf{O}]$.
Indeed it suffices to show this for every localization at a prime number $l$.
There exists an $a\in\mathsf{O}_{(l)}$ such that $I^{-1}=\mathsf{O}_{(l)}a$.
Therefore every element in $I^{-1}_{(l)}\oplus I^{-1}_{(l)}$ is of the form
$va$ for $v\in\mathsf{O}_{(l)}^{2}$. This implies that
$\nu_{l}((va)^{\dagger}f(va))=\nu_{l}(N(a)v^{\dagger}fv)\geqslant\nu_{l}(N(a))+\nu_{l}(p)=\nu_{l}([\mathsf{O}:I^{-1}])+\nu_{l}(p)=\nu_{l}([I:\mathsf{O}])+\nu_{l}(p)$.
Since $l$ was an arbitrary prime we conclude that the length of every vector
in $I^{-1}\oplus I^{-1}$ is divisible by $p[I:\mathsf{O}]$. This implies that
$S=\\{v\in I^{-1}\oplus I^{-1}|v^{\dagger}fv=p[I:\mathsf{O}]\\}$. In
particular $S$ will either be empty or the set of shortest vectors.

Let $v\in S$. Then we define $I_{1}=vI\subset\mathsf{O}^{2}$ and $I_{2}$ to be
the $f$-orthogonal complement of $I_{1}$. Then $I_{1},I_{2}$ are right
$\mathsf{O}$-submodules of $\mathsf{O}^{2}$ satisfying i), ii), iii’). By
Proposition 4.7 this corresponds to a decomposition $f=f_{1}+f_{2}$ where
$f_{1}$ is given by the formula $\frac{fvv^{\dagger}f}{v^{\dagger}fv}$.

Conversely suppose we are given a decomposition $f=f_{1}+f_{2}$ with
$I_{1}=\ker(f_{1})\cong I$. Then $I_{1}$ will satisfy property iii’) of
Proposition 4.7. This implies that a generator of the free rank 1
$\mathsf{O}_{L}(I)$-module222See [15, 10.2.6] for the definition of
$\mathsf{O}_{L}(I)$, the left order of $I$. $I_{1}I^{-1}$ occurs in $S$. This
proves that
$\\{(f_{1}=\frac{fvv^{\dagger}f}{v^{\dagger}fv},f_{2}=f-f_{1})|v\in S\\}$
gives all pairs as in Proposition 4.4 such that $\ker(f_{1})\cong I$. Notice
that this list will contain some duplicate elements. ∎

###  4.4. Reducible Fibers II: Role of 2-torsion

Using the same situation as in section 4.2 we want to further understand the
reducible fibers of $\mathfrak{D}\rightarrow\mathbb{P}^{1}$. Notice that this
is not fully answered by Proposition 4.4: If $D=E_{1}\cup E_{2}$ is a
reducible fiber of $\mathfrak{D}\rightarrow\mathbb{P}^{1}$, then $E_{1}$ and
$E_{2}$ do not necessarily intersect in the origin but rather some point in
$E^{2}[2]$. In this section we explain how to compute this $2$-torsion point.
First we need a definition:

######  Definition 4.10.

Let $A$ be an Abelian variety and $D$ a symmetric divisor on $A$. Define the
function $e^{D}_{*}:A[2]\rightarrow\\{\pm
1\\},\,P\mapsto(-1)^{m_{D}(P)-m_{D}(0)}$ where $m_{D}(P)$ denotes the
multiplicity of $D$ at $P$.

The function $e^{D}_{*}$ has the following properties:

######  Proposition 4.11.

Let $A$ be an Abelian variety and $D$ a symmetric divisor on $A$. Denote
$\mathcal{L}=\mathcal{O}(D)$.

  1. i)

$e^{D}_{*}$ depends only on $\mathcal{L}$.

  2. ii)

$D$ is rationally equivalent to $0$ if and only if $D$ is algebraically
equivalent to $0$ and $e^{D}_{*}(P)=1,\,\forall P\in A[2]$.

  3. iii)

$e^{D}_{*}$ is a quadratic function whose associated bilinear form is the
restriction of the commutator pairing on $K(\mathcal{L}^{2})$ to $A[2]$.

  4. iv)

$e^{D}_{*}$ is locally constant in families, i.e., if $S$ is a $k$-scheme and
$\mathfrak{D}\subset A\times_{k}S$ is a symmetric divisor flat over $S$, then
for any $P\in A[2](k)$ the function $|S|\rightarrow\\{\pm 1\\},\,x\mapsto
e^{\mathfrak{D}_{x}}_{*}(P)$ is locally constant.

###### Proof.

The first assertion is true because Mumford gives another definition for
$e^{D}_{*}$ in terms of $\mathcal{L}$ and shows in [10, §2, Proposition 2]
that it is equivalent to our definition. The second assertion follows from
[10, §2, First Properties, iv)]. For the third assertion see [10, Corollary 1,
p. 314]. For the fourth assertion notice that it follows easily from Mumford’s
alternative definition that $|S|\rightarrow\\{\pm 1\\},\,x\mapsto
e^{\mathfrak{D}_{x}}_{*}(P)$ is continuous in the Zariski topology. Here
$\\{\pm 1\\}$ is considered as a closed subset of $\mathbb{G}_{m}$. ∎

######  Corollary 4.12.

Two fibers of $\mathfrak{D}\rightarrow\mathbb{P}^{1}$ are rationally
equivalent divisors in $E^{2}$.

###### Proof.

Indeed by Theorem 4.3 they both induce the polarization $\eta$. Therefore they
are algebraically equivalent. Notice that by assumption $\mathfrak{D}$ is
symmetric and thus Proposition 4.11 applies. By parts ii) and iv) we conclude
that the fibers are rationally equivalent. ∎

######  Remark 4.13.

Although one could infer the corollary from the fact that the base of
$\mathfrak{D}\rightarrow\mathbb{P}^{1}$ is a rational curve, we prefer this
more involved argument because it generalizes to Li’s and Oort’s higher
dimensional theory.

######  Definition 4.14.

  * •

For $y\in\mathbb{P}^{1}(k)$ arbitrary we denote
$\mathcal{L}=\mathcal{O}(\mathfrak{D}_{y})$. By corollary 4.12 this line
bundle is independent of $y$. The letter $\mathcal{L}$ is reserved for this
line bundle for the rest of this paper.

  * •

Denote by

$e^{\mathfrak{D}}_{*}:E^{2}[2]\rightarrow\\{\pm 1\\}$

the function $e^{\mathfrak{D}}_{*}=e^{\mathfrak{D}_{y}}_{*}$ for any
$y\in|\mathbb{P}^{1}|$.

We are now ready to describe the formula for the $2$-torsion point that is the
intersection point of the components of a reducible fiber. Using the previous
proposition this boils down to linear algebra over $\mathbb{F}_{2}$.

Consider a divisor $D=E_{1}\cup E_{2}$ as in Proposition 4.4. By Corollary 4.5
there is a point $x\in\mathbb{P}^{1}(\mathbb{F}_{p^{2}})$ such that
$\mathcal{A}_{x}\cong E_{1}^{(p)}\times E_{2}^{(p)}$ with the product
polarization.

######  Lemma 4.15.

Using the notation preceding the lemma there is a unique point $P\in E^{2}[2]$
such that $t_{P}(D)=\mathfrak{D}_{x}$. Furthermore $P$ can be computed as
follows: Choose a basis $\mathfrak{B}$ of $E[2]^{2}$ as an
$\mathbb{F}_{2}$-vector space. Take upper triangular matrices
$\mathfrak{Q},Q\in M_{4,4}(\mathbb{F}_{2})$ such that the quadratic functions
$e^{\mathfrak{D}}_{*}$ and $e^{D}_{*}$ are given by
$e^{\mathfrak{D}}_{*}((v)_{\mathfrak{B}})=(-1)^{v^{t}\mathfrak{Q}v},\,e^{D}_{*}((v)_{\mathfrak{B}})=(-1)^{v^{t}Qv}$.
$\mathfrak{Q}+Q$ is a diagonal matrix and denote its diagonal by the vector
$d\in\mathbb{F}_{2}^{4}$. Then

$(Q+Q^{t})^{-1}d$

is the coordinate vector of the point $P$ in the basis $\mathfrak{B}$.

###### Proof.

We know that $\mathcal{A}_{x}\cong E_{1}^{(p)}\times E_{2}^{(p)}$ and the
polarization $\lambda_{x}$ is the product polarization by corollary 4.5.
Therefore $\lambda_{x}$ is induced by the symmetric divisor
$E_{1}^{(p)}\times\\{0\\}+\\{0\\}\times E_{2}^{(p)}$. By assumption
$\mathfrak{C}_{x}$ is another symmetric divisor inducing $\lambda_{x}$.
Because two symmetric divisors inducing the same polarization are translates
by a $2$-torsion point, there exists a $P\in A_{x}[2]$ such that
$\mathfrak{C}_{x}=t_{P}(E_{1}^{(p)}\times\\{0\\}+\\{0\\}\times E_{2}^{(p)})$.
Now $E^{2}[2]\cong A_{x}[2]$ canonically, so by abuse of notation denote the
preimage of $P$ under this isomorphism also by $P$. Then
$t_{P}(D)=\mathfrak{D}_{x}$.

This proves the existence of $P$. To prove uniqueness it suffices to establish
the formula. Indeed it is clear that
$e^{\mathfrak{D}}_{*}(\cdot)=e^{t_{P}(D)}(\cdot)=e^{D}_{*}(P+\cdot)$. We claim
that $P$ is uniquely characterized by this property. This is a question about
quadratic forms over $\mathbb{F}_{2}$. Define now
$L=\mathcal{O}(E_{1}+E_{2})$. Then $\mathcal{L},L$ are ample symmetric line
bundles inducing the polarization $\eta$. Thus $\mathcal{L},L$ are
algebraically equivalent and by part ii) of Proposition 4.11 one has
$\mathcal{L}^{2}\cong L$. Part iii) of Proposition 4.11 gives:

$e_{*}^{\mathfrak{D}}(P_{1}+P_{2})=e_{*}^{\mathfrak{D}}(P_{1})e_{*}^{\mathfrak{D}}(P_{2})e^{\mathcal{L}^{2}}(P_{1},P_{2})\,\forall
P_{1},P_{2}\in E^{2}[2]$

and similarly for $D$. Since $\mathcal{L}^{2}\cong L^{2}$, this implies that
the bilinear forms associated to the quadratic functions
$e_{*}^{\mathfrak{D}},e_{*}^{D}$ agree. From that one deduces
$\mathfrak{Q}+\mathfrak{Q}^{t}=Q+Q^{t}$ and thus $\mathfrak{Q}+Q$ is indeed a
diagonal matrix.

On the other hand $e^{\mathcal{L}^{2}}(\cdot,\cdot)_{|{E^{2}[2]}}$ is non-
degenerate because it is the restriction of a non-degenerate pairing to the
$2$-primary part. Therefore $Q+Q^{t}$ is invertible. We conclude that
$P=((Q+Q^{t})^{-1}d)_{\mathfrak{B}}$ is well-defined. An elementary
computation shows that $P$ is the unique point in $E^{2}[2]$ satisfying
$e^{\mathfrak{D}}_{*}(\cdot)=e^{D}_{*}(P+\cdot)$ ∎

###  4.5. Reducible Fibers III: Theta groups

In this section we continue our study of the reducible fibers of the map
$\mathfrak{D}\rightarrow\mathbb{P}^{1}$. For the main algorithm it will be
important to understand how two reducible fibers can be used to pin down an
isomorphism of $\mathcal{G}(\mathcal{L})$ with the Heisenberg group
$\mathcal{G}(\alpha_{p})$.

Suppose we are given two distinct reducible fibers
$\mathfrak{D}_{x},\mathfrak{D}_{x^{\prime}}$. Then Corollary 4.5 gives a
subgroup scheme $\mathcal{H}\subset E^{2}$ isomorphic to $\alpha_{p}$ such
that $\mathfrak{D}_{x}$ is $\mathcal{H}$-invariant. Similarly we get
$\mathcal{H}^{\prime}\subset E^{2}$ isomorphic to $\alpha_{p}$ such that
$\mathfrak{D}_{x^{\prime}}$ is $\mathcal{H}^{\prime}$-invariant. By
Proposition 3.10 $\mathcal{H}$ and $\mathcal{H}^{\prime}$ uniquely lift to
level subgroups $\tilde{\mathcal{H}}$ (resp. $\tilde{\mathcal{H}^{\prime}}$)
inside $\mathcal{G}(\mathcal{L})$.

######  Proposition 4.16.

Use the notation preceding the proposition. For any isomorphism
$\mathcal{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\alpha_{p}$ there is a
unique isomorphism
$\phi:\mathcal{G}(\mathcal{L})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{G}(\alpha_{p})$
such that
$\phi(\tilde{\mathcal{H}})=\\{1\\}\times\alpha_{p}\times\\{0\\},\phi(\tilde{\mathcal{H}^{\prime}})=\\{1\\}\times\\{0\\}\times\alpha_{p}$
and the diagram

${\tilde{\mathcal{H}}}$${\\{1\\}\times\alpha_{p}\times\\{0\\}}$${\mathcal{H}}$${\alpha_{p}}$$\scriptstyle{\phi_{|\tilde{\mathcal{H}}}}$$\scriptstyle{\sim}$

commutes.

###### Proof.

Immediate consequence of Lemma 3.13 and Corollary 4.5. ∎

######  Remark 4.17.

Notice that we do not fix an isomorphism
$\mathcal{H}^{\prime}\cong\alpha_{p}$, instead such an isomorphism is then
determined by $\phi$. In our thesis we will explain how this isomorphism can
be computed. One has to make the commutator pairing on
$\mathcal{G}(\mathcal{L})$ explicit. There are two distinct formulas for this
computation. The first involves certain translation invariant vector fields on
$E^{2}$. The second uses the action of Frobenius and certain endomorphisms of
$E$ on de Rham cohomology.

###  4.6. Construction of irreducible fibers

We drop now the assumption that $k$ is algebraically closed. Instead $k$ is
any field containing $\mathbb{F}_{p^{2}}$. In this section we will explain our
algorithm that constructs all the irreducible fibers of a Moret-Bailly family.
To define a Moret-Bailly family one needs an hermitian positive definite
matrix $f\in M_{2,2}(\mathsf{O})$ inducing a polarization $\eta$ on $E^{2}$
with kernel $E^{2}[F]$. The basic idea is to use the information obtained from
two reducible fibers of the family $\mathfrak{D}\rightarrow\mathbb{P}^{1}$. In
order to obtain two reducible fibers we need to decompositions
$f=f_{1}+f_{2}=f_{1}^{\prime}+f_{2}^{\prime}$ as in Proposition 4.4. The
fibers will be distinct if and only if the sets $\\{f_{1},f_{2}\\}$ and
$\\{f_{1}^{\prime},f_{2}^{\prime}\\}$ are distinct by Corollary 4.6.

Now given $\\{f_{1},f_{2}\\}$ (resp. $\\{f_{1}^{\prime},f_{2}^{\prime}\\}$) we
can compute the corresponding reducible fiber $D_{0}$ (resp. $D_{\infty}$).
From the divisor $D_{0}$ we can construct a subgroup scheme
$\mathcal{H}\cong\alpha_{p}$ of $E^{2}$ using Corollary 4.5. Then using Lemma
2.4 we can choose an isomorphism of Hopf algebras $A_{\mathcal{H}}\cong
k[z]/(z^{p})$ such that the comultiplication is given by $z\mapsto z\otimes
1+1\otimes z$. The next theorem explains how to compute the other fibers
$D_{x}$ for $x=[t:1]\in\mathbb{P}^{1}(k)\setminus\\{[1:0]\\}$. Indeed since
all the $D_{x}$ are rationally equivalent it suffices to determine a rational
function $g_{x}\in K(E^{2})$ such that $(g_{x})=D_{x}-D_{0}$. To start with
there exists a rational function $g\in K(E^{2})$ such that
$(g)=D_{\infty}-D_{0}$. We claim that we can compute $g_{x}$ from $g$:

######  Theorem 4.18.

In the notation preceding the Theorem let $g_{\mathcal{H}}\in
K(E^{2})\otimes_{k}A_{\mathcal{H}}$ denote the image of $g$ under the map
$K(E^{2})\rightarrow K(E^{2})\otimes_{k}A_{\mathcal{H}}$ describing the
translation action of $\mathcal{H}$ on $E^{2}$.
Then if one takes $g_{x}\in K(E^{2})$ to be the image of $g_{\mathcal{H}}$
under the $K(E^{2})$-linear map

$K(E^{2})\otimes A_{\mathcal{H}}\rightarrow K(E^{2})$ $\rho\otimes
z^{2i}\mapsto\frac{(2i)!(-t)^{i}\rho}{i!2^{i}},\,\rho\otimes z^{2i+1}\mapsto
0$

one has $(g_{x})=D_{x}-D_{0}$

###### Proof.

From the divisor $D_{\infty}$ we can construct a subgroup scheme
$\mathcal{H}^{\prime}\subset E^{2}$ using Corollary 4.5. Consider the line
bundle $\mathcal{L}=\mathcal{O}(D_{0})$. There exists a unique level subgroup
over $\mathcal{H}$ (resp. $\mathcal{H}^{\prime}$) in
$\mathcal{G}(\mathcal{L})$ that we will by abuse of notation denote using the
same letter. From Proposition 4.16 we get an isomorphism
$\mathcal{G}(\mathcal{L})\cong\mathcal{G}(\mathcal{H})=\mathbb{G}_{m}\times\mathcal{H}\times\mathcal{H}^{D}$
that maps $\mathcal{H}$ to $\\{1\\}\times\mathcal{H}\times\\{0\\}$ and
$\mathcal{H}^{\prime}$ maps to $\\{1\\}\times\\{0\\}\times\mathcal{H}^{D}$. By
Theorem 3.15 there is an absolutely irreducible weight 1 representation of
$\mathcal{G}(\mathcal{L})$ on $\operatorname{H}^{0}(E^{2},\mathcal{L})$. By
Theorem 3.16 (or Riemann-Roch) we have
$\dim_{k}(\operatorname{H}^{0}(E^{2},\mathcal{L}))=p$. On the other hand
$\mathcal{G}(\mathcal{H})$ also has a weight 1 representation on the
$p$-dimensional $k$-vector space $A_{\mathcal{H}^{D}}$ given by

$((c,h,\chi)f)(\chi_{0})=c\cdot\chi_{0}(h)\cdot f(\chi+\chi_{0})$

for any $k$-scheme $T$ and
$c\in\mathbb{G}_{m}(T),h\in\mathcal{H}(T),\chi\in\mathcal{H}^{D}(T),\chi_{0}\in\mathcal{H}^{D}(T),f\in
A_{\mathcal{H}^{D}}\otimes\operatorname{H}^{0}(T,\mathcal{O}_{T})=\operatorname{Hom}_{\text{Sch}}(\mathcal{H}^{D}_{T},\mathbb{A}^{1}_{T})$.
Then Theorem 3.16 implies that there exists an isomorphism of $k$-vector
spaces

$\psi:A_{\mathcal{H}^{D}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{H}^{0}(E^{2},\mathcal{L})$

that identifies the two representations. By Schur’s lemma $\psi$ is uniquely
determined up to a scalar.

We have $(A_{\mathcal{H}^{D}})^{\mathcal{H}^{D}}=k$. On the other hand we
claim that
$\operatorname{H}^{0}(E^{2},\mathcal{L})^{\mathcal{H}^{\prime}}=kg$. Indeed
$g$ gives an element in
$\operatorname{H}^{0}(E^{2},\mathcal{L})=\operatorname{H}^{0}(E^{2},\mathcal{O}(D_{0}))$.
Furthermore $g$ is $\mathcal{H}^{\prime}$-invariant by Proposition 3.10
because it maps to $1$ under the isomorphism
$\operatorname{H}^{0}(E^{2},\mathcal{O}(D_{0}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{H}^{0}(E^{2},\mathcal{O}(D_{\infty})),\,\rho\mapsto
g^{-1}\rho$. This proves the claim. Now the claim implies that we can without
loss assume $\psi(1)=g$. Consider now the action of $\mathcal{H}$ on
$A_{\mathcal{H}^{D}}$ given by restricting the representation of
$\mathcal{G}(\mathcal{H})$. This determines a map

$A_{\mathcal{H}^{D}}\rightarrow A_{\mathcal{H}^{D}}\otimes A_{\mathcal{H}}\,.$

It follows from the definition of the representation of
$\mathcal{G}(\mathcal{H})$ that this map is given by multiplying with the
element in $A_{\mathcal{H}^{D}}\otimes
A_{\mathcal{H}}=A_{\mathcal{H}}^{\vee}\otimes A_{\mathcal{H}}$ which comes
from Cartier duality. To make this explicit notice that we have chosen an
isomorphism $\mathcal{H}\cong\alpha_{p}$. The autoduality of $\alpha_{p}$
gives an isomorphism $\mathcal{H}^{D}\cong\alpha_{p}$. This gives a generator
of $A_{\mathcal{H}^{D}}$ that we will also denote by $z$ (this can not cause
any confusion). Then the map

$A_{\mathcal{H}^{D}}\rightarrow A_{\mathcal{H}^{D}}\otimes A_{\mathcal{H}}$

is given by multiplication with $\exp(z\otimes z)\in
A_{\mathcal{H}^{D}}\otimes A_{\mathcal{H}}$. In particular $1$ maps to
$\exp(z\otimes z)$. Here $\exp(\cdot)$ denotes the usual exponential series.
Notice that the expression $\exp(z\otimes z)$ is well-defined although
$\text{char}(k)=p$ since $z^{p}=0$.

On the other hand $\mathcal{H}$ also acts on
$\operatorname{H}^{0}(E^{2},\mathcal{L})$ by restricting the action of
$\mathcal{G}(\mathcal{L})$. This gives a map

$\operatorname{H}^{0}(E^{2},\mathcal{L})\rightarrow\operatorname{H}^{0}(E^{2},\mathcal{L})\otimes
A_{\mathcal{H}}\,.$

By Proposition 3.19 this action is given by the translation action on
$K(E^{2})$ using the identification

$\operatorname{H}^{0}(E^{2},\mathcal{L})=\operatorname{H}^{0}(E^{2},\mathcal{O}(D_{0}))=\\{\rho\in
K(E^{2})|(\rho)+D_{0}\text{ is effective}\\}\,.$

In particular $g$ maps to $g_{\mathcal{H}}$ under this map. Because the
isomorphism $\psi$ is equivariant, one has
$(\psi\otimes\operatorname{id}_{A_{\mathcal{H}}})(\exp(z\otimes
z))=g_{\mathcal{H}}$.

We will now focus on the point $x=[t:1]\in\mathbb{P}^{1}$. Indeed this point
determines a map
$\alpha_{p}\rightarrow\alpha_{p}\times\alpha_{p},\,s\mapsto(ts,s)$. Let us
denote by $\mathcal{H}_{x}$ the image of the composition
$\alpha_{p}\rightarrow\alpha_{p}\times\alpha_{p}\cong\mathcal{H}\times\mathcal{H}^{D}$.
By Lemma 4.2 and Proposition 3.4 there is a unique level subgroup in
$\mathcal{G}(\mathcal{H})$ above $\mathcal{H}_{x}$. We will denote by abuse of
notation this level subgroup by $\mathcal{H}_{x}$. Its image under the
isomorphism $\mathcal{G}(\mathcal{H})\cong\mathcal{G}(\mathcal{L})$ is also
denoted $\mathcal{H}_{x}$. Now the desired rational function $g_{x}$ will be a
generator of the 1-dimensional vector space
$\operatorname{H}^{0}(E^{2},\mathcal{L})^{\mathcal{H}_{x}}$ (see Proposition
3.18). On the other hand $(A_{\mathcal{H}^{D}})^{\mathcal{H}_{x}}$ is
generated by the element $\exp(\frac{-tz^{2}}{2})$, see [9, proof of
Proposition 1.5]. Therefore we can take $g_{x}=\psi(\exp(\frac{-tz^{2}}{2}))$.
The theorem follows because $\exp(\frac{-tz^{2}}{2})\in A_{\mathcal{H}^{D}}$
is the image of $\exp(z\otimes z)$ under the map $A_{\mathcal{H}^{D}}\otimes
A_{\mathcal{H}}\rightarrow A_{\mathcal{H}^{D}}$
$\rho\otimes z^{2i}\mapsto\frac{(2i)!(-t)^{i}\rho}{i!2^{i}},\,\rho\otimes
z^{2i+1}\mapsto 0$. ∎

######  Remark 4.19.

Notice that the coordinate system on $\mathbb{P}^{1}$ used in the previous
theorem is not the same as the coordinate system in section 4.1. For the
coordinate system in section 4.1 a point $[t_{0}:t_{1}]$ corresponds to the
embedding
$\alpha_{p}\stackrel{{\scriptstyle(t_{0},t_{1})}}{{\hookrightarrow}}\alpha_{p}^{2}\cong
E^{2}[F]$, where the last isomorphism comes from an isomorphism
$\alpha_{p}\cong E[F]$.

On the other hand for the coordinate system in the previous theorem a point
$[t_{0}:t_{1}]$ corresponds to the embedding
$\alpha_{p}\stackrel{{\scriptstyle(t_{0},t_{1})}}{{\hookrightarrow}}\alpha_{p}^{2}\cong\mathcal{H}\times\mathcal{H}^{\prime}=E^{2}[F]$,
where the isomorphism $\alpha_{p}\cong\mathcal{H}^{\prime}$ is the one induced
by Proposition 4.16 from the choice of an isomorphism
$\alpha_{p}\cong\mathcal{H}$.

With the first coordinate system one can directly write down the image of
$\alpha_{p}\hookrightarrow E^{2}$ as a closed subgroup scheme. This is not
possible with the second coordinate system. It is possible to compute the
change of coordinates matrix, see also Remark 4.17.

We are ready to explain the main algorithm.

input : An hermitian positive definite matrix $f\in M_{2,2}(\mathsf{O})$
inducing a polarization $\eta$ on $E^{2}$ with kernel $E^{2}[F]$. Two
decompositions $f=f_{1}+f_{2}=f_{1}^{\prime}+f_{2}^{\prime}$ as in Proposition
4.4 corresponding to distinct divisors. A point $x\in\mathbb{P}^{1}(k)$ such
that the fiber $\mathfrak{C}_{x}$ of the Moret-Bailly family
$\mathfrak{C}\rightarrow\mathbb{P}^{1}$ constructed from $(E^{2},\eta)$ is
irreducible.

output : The curve $\mathfrak{C}_{x}$

1 begin

2
$E_{i}\longleftarrow\ker(\gamma(f_{i})),\,E_{i}^{\prime}\longleftarrow\ker(\gamma(f_{i}^{\prime})),\,i=1,2$

3 $\mathfrak{D}_{0}\longleftarrow E_{1}+E_{2},\,D_{\infty}\longleftarrow
E_{1}^{\prime}+E_{2}^{\prime}$ as a divisor on $E^{2}$.

4 Compute the $2$-torsion point $P$, such that
$t_{P}(D_{\infty})\sim\mathfrak{D}_{0}$ using the formula of Lemma 4.15.

5 $\mathfrak{D}_{\infty}\longleftarrow t_{P}(D_{\infty})$.

6 Compute a rational function $g\in K(E^{2})$ such that
$(g)=\mathfrak{D}_{\infty}-\mathfrak{D}_{0}$.

7 Compute the rational function $g_{x}$ using the formula from Theorem 4.18.
The vanishing locus  $\mathcal{V}(g_{x})\subset E^{2}$ is the divisor
$\mathfrak{D}_{x}$.

8 Put $\mathfrak{C}_{x}$ equal to $\mathfrak{D}_{x}^{\text{norm},(p)}$, i.e.,
the Frobenius twist of the normalization of $\mathfrak{D}_{x}$.

9 Output: $\mathfrak{C}_{x}$

10 end

11

Algorithm 2 Irreducible fibers of a Moret-Bailly family

###### Proof.

We prove the correctness of the algorithm. Indeed Theorem 4.18 gives us a
rational function $g_{x}$ such that
$(g_{x})=\mathfrak{D}_{x}-\mathfrak{D}_{0}$. Therefore
$\mathfrak{D}_{x}=\mathcal{V}(g_{x})$. The map
$\mathfrak{D}_{x}\rightarrow\mathfrak{C}_{x}$ is purely inseparable of degree
$p$. By [13, Tag 0CCX] there is an isomorphism
$\mathfrak{C}_{x}\cong\mathfrak{D}_{x}^{\text{norm},(p)}$. ∎

######  Remark 4.20.

The algorithm constructs the divisor $\mathfrak{D}_{x}$ without making any
field extensions. Indeed notice that the condition $F^{2}+p=0$ implies that
all endomorphism and all the 2-torsion points of $E$ are defined over
$\mathbb{F}_{p^{2}}$ because the $p^{2}$-Frobenius acts trivially. It follows
that $\mathfrak{D}\subset E^{2}$ is defined over $\mathbb{F}_{p^{2}}$ as a
closed subscheme. This implies that $\mathfrak{C}$ and the map
$\mathfrak{C}\rightarrow\mathbb{P}^{1}$ are also defined over
$\mathbb{F}_{p^{2}}$. The algorithm correctly computes the fibers of this map
up to isomorphism over the ground field (not over its algebraic closure).

######  Remark 4.21.

Notice that steps 1-6 are independent of $x$. Thus after performing steps 1-6
we can quickly compute many fibers.

In fact we could theoretically even construct the generic fiber (or a model
for $\mathfrak{C}\rightarrow\mathbb{P}^{1}$) by applying the algorithm to the
generic point. However this is very expensive in practice because normalizing
the very singular curve $\mathfrak{D}_{x}$ will take too long over a function
field. By replacing step 8 with a technique that avoids normalization one can
remedy this (see the author’s PhD thesis for details).

###  4.7. Examples

We have implemented Algorithms 1 and 2 in MAGMA. The source code is available
on GitHub via
https://github.com/Andreas-Pieper/Supersingular. We present some examples that
we computed using these programs.

######  Example 4.22.

Consider $p=1601$ and $B=\left(\frac{-3,-p}{\mathbb{Q}}\right)=\langle
1,i,j,k\rangle_{\mathbb{Q}}$ with multiplication given by
$i^{2}=-3,j^{2}=-p,ij=k=-ji$ and the maximal order $\mathsf{O}=\Large\langle
1,\frac{1}{2}+\frac{1}{2}i,\frac{1}{2}+\frac{1}{6}i+\frac{1}{2}j+\frac{1}{6}k,-\frac{1}{2}+\frac{1}{6}i-\frac{1}{3}k\Large\rangle_{\mathbb{Z}}$
of $B$. This maximal order contains $j$, which generates a two-sided ideal of
reduced norm $p$, and thus there exists a supersingular elliptic curve
$E/\overline{\mathbb{F}}_{p}$ defined over $\mathbb{F}_{p}$ with endomorphism
algebra $\mathsf{O}$ (see [15, Lemma 42.4.1]). The geometric Frobenius will
satisfy $F^{2}+p=0$ because of the Hasse-Weil bound. Choose the matrix

$f=\begin{pmatrix}1601&40j\\\ -40j&1601\end{pmatrix}\in
M_{2,2}(\mathsf{O})\,.$

We apply Algorithm 1 to $f$ and all the right ideal classes of $\mathsf{O}$
and find $8000=5p-5$ decompositions $f=f_{1}+f_{2}$ as in Proposition 4.7. The
first few are:

$f_{1}\in\Big{\\{}\begin{pmatrix}1601&40j\\\
-40j&1600\end{pmatrix},\,\begin{pmatrix}1054&\frac{27}{2}-\frac{1}{6}i+\frac{53}{2}j-\frac{1}{6}k\\\
\frac{27}{2}+\frac{1}{6}i-\frac{53}{2}j+\frac{1}{6}k&1067\end{pmatrix}$
$\begin{pmatrix}1094&13+27j\\\
13-27j&1067\end{pmatrix},\,\begin{pmatrix}1054&\frac{27}{2}+\frac{1}{6}i+\frac{53}{2}j+\frac{1}{6}k\\\
\frac{27}{2}-\frac{1}{6}i-\frac{53}{2}j-\frac{1}{6}k&1067\end{pmatrix},\ldots\Big{\\}}\,.$

The computation took 4min 50s on a 3.10GHz dual Intel processor.

######  Example 4.23.

Consider $p=5$. The field extension $\mathbb{F}_{5^{2}}/\mathbb{F}_{5}$ is
generated by a primitive third root of unity $\zeta_{3}\in\mathbb{F}_{5^{2}}$.
We will take the supersingular elliptic curve
$E/\overline{\mathbb{F}}_{5}:y^{2}=x^{3}+1$. Its endomorphism algebra is a
maximal order in

$B=\mathbb{Q}[\zeta_{3},F]/(\zeta_{3}^{2}+\zeta_{3}+1,F^{2}+5,\zeta_{3}F-F\zeta_{3}^{2})$

where $\zeta_{3}$ is the endomorphism $E\rightarrow
E,\,(x,y)\mapsto(\zeta_{3}x,y)$ and $F$ is the geometric Frobenius.

We apply Algorithm 2 with input

$f=\begin{pmatrix}5&2F\\\ -2F&5\end{pmatrix}\in M_{2,2}(\mathsf{O})$
$f_{1}=\begin{pmatrix}4&2F\\\ -2F&5\end{pmatrix},\,f_{2}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\in M_{2,2}(\mathsf{O})$
$f_{1}^{\prime}=\begin{pmatrix}5&2F\\\
-2F&4\end{pmatrix},\,f_{2}^{\prime}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}\in
M_{2,2}(\mathsf{O})\,.$

And $x=[a:1]\in\mathbb{P}^{1}(\mathbb{F}_{5^{4}})$, where
$a\in\mathbb{F}_{5^{4}}$ is a root of the irreducible polynomial
$x^{4}+4x^{2}+4x+2\in\mathbb{F}_{5}[x]$. The resulting curve
$\mathfrak{C}_{x}$ is computed on a 3.10GHz dual Intel processor in 4.5s. We
get the hyperelliptic model

$\mathfrak{C}_{x}:y^{2}=(4a^{3}+3a^{2}+1)x^{5}+(4a^{3}+a^{2}+4a+3)x^{4}+(3a^{3}+3a^{2}+2a)x^{3}+(4a^{3}+4a^{2})x^{2}+$
$(4a^{3}+2a^{2}+3a+3)x+(3a^{3}+3a+3)\,.$

N.B.: The coefficients on the right are expressed in the
$\mathbb{F}_{5}$-basis $a^{3},a^{2},a,1$ of $\mathbb{F}_{5^{4}}$. Replacing
$a$ by another element of $\mathbb{F}_{5^{4}}$ does not give a curve with a
supersingular Jacobian in general.

######  Remark 4.24.

The matrices $f$ describing the polarization $\eta$ were found by hand in
these examples. In general for a fixed $p$ there are finitely many choices for
$f$ up to conjugation by $\text{Gl}_{2}(\mathsf{O})$. These are in bijection
with the set of isomorphism classes of quaternion hermitian lattices in a
fixed genus. There is an algorithm which computes a list of representatives
for the isomorphism classes. This uses the so-called method of neighboring
lattices. The method was developed by M. Kneser for the classification of
quadratic lattices over $\mathbb{Z}$ (see [4]).

The method generalizes to quaternion hermitian lattices by using the general
strong approximation theorem of M. Kneser [5] applied to the group of
isometries of a quaternion hermitian lattice. The quaternion hermitian case is
less technical because the latter group is simply-connected whereas
$\text{SO}_{n}$ is not.

## References

  * [1] Michel Demazure and Alexander Grothendieck.  Schémas en Groupes I (SGA 3, vol. I).  Springer, 1970.
  * [2] Irving Kaplansky.  Submodules of quaternion algebras.  Proc. London Math. Soc., (3):219–232, 1969.
  * [3] Toshiyuki Katsura and Frans Oort.  Families of Supersingular Abelian Surfaces.  Compositio Mathematica, 62:107–167, 1987.
  * [4] Martin Kneser.  Klassenzahlen definiter quadratischer Formen.  Archiv der Mathematik, 1957.
  * [5] Martin Kneser.  Strong approximation.  In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages 187–196. American Mathematical Society, 1966.
  * [6] Tsit-Yuen Lam.  Lectures on Modules and Rings.  Springer, 1999.
  * [7] Ke-Zheng Li and Frans Oort.  Moduli of Supersingular Abelian Varieties.  Springer Lecture Notes in Mathematics, 1998.
  * [8] David Lubicz and Damien Robert.  Computing isogenies between Abelian Varieties.  Compositio Mathematica, 148(5), 2012.
  * [9] Laurent Moret-Bailly.  Familles de courbes et de variétés abéliennes sur $\mathbb{P}^{1}$, II Exemples.  Astérisque, 86, 1981.
  * [10] David Mumford.  On the equations defining abelian varieties I.  Inventiones mathematicae, (1):287–354, 1966.
  * [11] David Mumford.  Abelian Varieties.  Hindustan book agency, 2nd edition, 1974.
  * [12] Tsutomu Sekiguchi.  On projective normality of abelian varieties II.  Journal of the Mathematical Society of Japan, 29(4):709 – 727, 1977\.
  * [13] The Stacks Project Authors.  Stacks Project.  https://stacks.math.columbia.edu, 2018.
  * [14] Gerard van der Geer and Ben Moonen.  Abelian Varieties (draft book).  https://www.math.ru.nl/~bmoonen/research.html#bookabvar.
  * [15] John Voight.  Quaternion algebras.  Springer, 2020.