# Abelian surfaces with supersingular good reduction and non semisimple Tate
module

Maja Volkov  Université de Mons, Département de Mathématique, Place du Parc
20, B-7000 Mons, Belgium<EMAIL_ADDRESS>
###### Abstract.

We show the existence of abelian surfaces $\mathcal{A}$ over
${\mathbb{Q}}_{p}$ having good reduction with supersingular special fibre
whose associated $p$-adic Galois module $V_{p}(\mathcal{A})$ is not
semisimple.

2000 Mathematics Subject Classification: 11G10, 14K15, 14G20.

Keywords: Abelian varieties, local fields, Galois representations.

###### Contents

  1. 1 The general method
  2. 2 A lift of the twofold product of a supersingular elliptic curve
  3. 3 A lift of a simple supersingular abelian surface

## Introduction

Fix a prime number $p$ and an algebraic closure $\overline{\mathbb{Q}}_{p}$ of
${\mathbb{Q}}_{p}$. Write
$G=\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/{\mathbb{Q}}_{p})$ for the
absolute Galois group of ${\mathbb{Q}}_{p}$. For a $d$-dimensional abelian
variety ${\mathcal{A}}$ over ${\mathbb{Q}}_{p}$ let ${\mathcal{A}}[p^{n}]$ be
the group of $p^{n}$-torsion points with values in $\overline{\mathbb{Q}}_{p}$
and

$V_{p}({\mathcal{A}})\underset{\text{\tiny
def}}{=}\,{\mathbb{Q}}_{p}\otimes_{\mathbb{Z}_{p}}\varprojlim_{n\geq
1}{\mathcal{A}}[p^{n}].$

This is a $2d$-dimensional ${\mathbb{Q}}_{p}$-vector space on which $G$ acts
linearly and continuously. We want to consider the following problem: find
abelian varieties ${\mathcal{A}}$ over ${\mathbb{Q}}_{p}$ having good
reduction with supersingular special fibre and such that the Galois module
$V_{p}({\mathcal{A}})$ is not semisimple. In this paper we show the existence
of two such varieties with nonisogenous special fibres for the least dimension
possible, namely for $d=2$. In fact our procedure easily generalises to any
$d\geq 2$, however we stick to surfaces as they furnish low-dimensional hence
simple to describe representations.

The existence of such surfaces follows from the characterisation of $p$-adic
representations of $G$ arising from abelian varieties with (tame) potential
good reduction obtained in [Vo], and indeed provides an example of application
of this result. In order to explicitely describe our objects we use Fontaine’s
contravariant functor establishing an equivalence between crystalline $p$-adic
representations of $G$ and admissible filtered $\varphi$-modules. In section 1
we briefly review this theory as well as the characterisation in [Vo] (Theorem
1.2), and outline the general strategy. In sections 2 and 3 we construct two
filtered $\varphi$-modules arising from abelian surfaces over
${\mathbb{Q}}_{p}$ with good reduction that enjoy the required properties
(Propositions 2.1 and 3.1).

##  1\. The general method

Recall from [Fo2] that the objects $D$ in the category ${\bf
MF}_{{\mathbb{Q}}_{p}}(\varphi)$ of filtered $\varphi$-modules are finite
dimensional ${\mathbb{Q}}_{p}$-vector spaces together with a Frobenius map
$\varphi\in\operatorname{Aut}_{{\mathbb{Q}}_{p}}(D)$ and a decreasing
filtration $\operatorname{Fil}=(\operatorname{Fil}^{i}D)_{i\in\mathbb{Z}}$ on
$D$ by subspaces such that $\operatorname{Fil}^{i}D=D$ for $i\ll 0$ and
$\operatorname{Fil}^{i}D=0$ for $i\gg 0$, and the morphisms are
${\mathbb{Q}}_{p}$-linear maps commuting with $\varphi$ and preserving the
filtration. The dual of $(D,\operatorname{Fil})$ is the
${\mathbb{Q}}_{p}$-linear dual $D^{*}$ with $\varphi_{D^{*}}=\varphi^{*\,-1}$
and $\operatorname{Fil}^{i}D^{*}$ consists of linear forms on $D$ vanishing on
$\operatorname{Fil}^{j}D$ for all $j>-i$. The Tate twist $D\\{-1\\}$ of
$(D,\operatorname{Fil})$ is $D$ as a ${\mathbb{Q}}_{p}$-vector space with
$\varphi_{D\\{-1\\}}=p\varphi$ and
$\operatorname{Fil}^{i}D\\{-1\\}=\operatorname{Fil}^{i-1}D$. The filtration
$\operatorname{Fil}$ has Hodge-Tate type $(0,1)$ if
$\operatorname{Fil}^{i}D=D$ for $i\leq 0$, $\operatorname{Fil}^{i}D=0$ for
$i\geq 2$, and $\operatorname{Fil}^{1}D$ is a nontrivial subspace. The full
subcategory ${\bf MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$ of
${\bf MF}_{{\mathbb{Q}}_{p}}(\varphi)$ consists of objects
$(D,\operatorname{Fil})$ satisfying a property relating the Frobenius with the
filtration, called admissibility and defined as follows. For a
$\varphi$-stable sub-${\mathbb{Q}}_{p}$-vector space $D^{\prime}$ of $D$
consider the Hodge and Newton invariants

$t_{H}(D^{\prime})\underset{\text{\tiny
def}}{=}\,\sum_{i\in\mathbb{Z}}i\dim_{{\mathbb{Q}}_{p}}\\!\bigl{(}D^{\prime}\\!\cap\operatorname{Fil}^{i}\\!D\big{/}D^{\prime}\\!\cap\operatorname{Fil}^{i+1}\\!D\bigr{)}\quad\text{
and }\quad t_{N}(D^{\prime})\underset{\text{\tiny
def}}{=}\,v_{p}(\det\varphi_{\\!\mid D^{\prime}})$

where $v_{p}$ is the normalised $p$-adic valuation on ${\mathbb{Q}}_{p}$. Then
$(D,\operatorname{Fil})$ is admissible if

  * (i)

$t_{H}(D)=t_{N}(D)$

  * (ii)

$t_{H}(D^{\prime})\leq t_{N}(D^{\prime})$ for any
sub-${\mathbb{Q}}_{p}[\varphi]$-module $D^{\prime}$ of $D$.

A sub-${\mathbb{Q}}_{p}[\varphi]$-module $D^{\prime}$ endowed with the induced
filtration
$\operatorname{Fil}^{i}D^{\prime}=D^{\prime}\cap\operatorname{Fil}^{i}D$ is a
subobject of $(D,\operatorname{Fil})$ in ${\bf
MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$ if and only if
$t_{H}(D^{\prime})=t_{N}(D^{\prime})$.

Let $B_{\text{\tiny cris}}$ be the ring of $p$-adic periods constructed in
[Fo1] and for a $p$-adic representation $V$ of $G$ put

$\operatorname{{\bf D}^{*}_{\text{\tiny cris}}}(V)\underset{\text{\tiny
def}}{=}\,\operatorname{Hom}_{{\mathbb{Q}}_{p}[G]}(V,B_{\text{\tiny cris}}).$

We always have $\dim_{{\mathbb{Q}}_{p}}\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V)\leq\dim_{{\mathbb{Q}}_{p}}V$ and $V$ is said to be crystalline when
equality holds. The functor $V\mapsto\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V)$ establishes an anti-equivalence between the category of
crystalline $p$-adic representations of $G$ and ${\bf
MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$, a quasi-inverse being
$\operatorname{{\bf V}^{*}_{\text{\tiny
cris}}}(D,\operatorname{Fil})=\operatorname{Hom}_{\varphi,\operatorname{Fil}}(D,B_{\text{\tiny
cris}})$ ([Co-Fo]). These categories are well-suited to our problem since for
an abelian variety ${\mathcal{A}}$ over ${\mathbb{Q}}_{p}$ the $G$-module
$V_{p}({\mathcal{A}})$ is crystalline if and only if ${\mathcal{A}}$ has good
reduction ([Co-Io] Thm.4.7 or [Br] Cor.5.3.4.).

A $p$-Weil number is an algebraic integer such that all its conjugates have
absolute value $\sqrt{p}$ in $\mathbb{C}$. Call a monic polynomial in
$\mathbb{Z}[X]$ a $p$-Weil polynomial if all its roots in
$\overline{\mathbb{Q}}\mspace{1.0mu}$ are $p$-Weil numbers and its valuation
at $X^{2}-p$ is even. Consider the following conditions on a filtered
$\varphi$-module $(D,\operatorname{Fil})$ over ${\mathbb{Q}}_{p}$:

  * (1)

$\varphi$ acts semisimply and $\operatorname{P_{char}}(\varphi)$ is a $p$-Weil
polynomial

  * (2)

the filtration has Hodge-Tate type $(0,1)$

  * (3)

there exists a nondegenerate skew form on $D$ under which $\varphi$ is a
$p$-similitude and $\operatorname{Fil}^{1}D$ is totally isotropic.

Recall that $\varphi$ is a $p$-similitude under a bilinear form $\beta$ if
$\beta(\varphi x,\varphi y)=p\beta(x,y)$ for all $x,y\in D$ and
$\operatorname{Fil}^{1}D$ is totally isotropic if $\beta(x,y)=0$ for all
$x,y\in\operatorname{Fil}^{1}D$. The map sending
$\delta\in\operatorname{Isom}_{{\mathbb{Q}}_{p}}(D^{*},D)$ to
$\beta:(x,y)\mapsto\delta^{-1}(x)(y)$ identifies the antisymmetric
isomorphisms of filtered $\varphi$-modules from $D^{*}\\{-1\\}$ to $D$ with
the forms satisfying (3). A ${\mathbb{Q}}_{p}$-linear map
$\delta:D^{*}\rightarrow D$ is an antisymmetric morphism in ${\bf
MF}_{{\mathbb{Q}}_{p}}(\varphi)$ if $\delta^{*}=-\delta$ (under the canonical
isomorphism ${D^{*}}^{*}\simeq D$), $\varphi\delta=p\delta\varphi^{*\,-1}$,
and $\delta(\operatorname{Fil}^{1}D)^{\perp}\subseteq\operatorname{Fil}^{1}D$.

######  Remark 1.1.

Let $\operatorname{Hom}_{\varphi}^{\text{a}}(D^{*}\\{-1\\},D)$ be the
${\mathbb{Q}}_{p}$-vector space of antisymmetric $\varphi$-module morphisms
from $D^{*}\\{-1\\}$ to $D$ and pick any
$\delta\in\operatorname{Isom}_{\varphi}^{\text{a}}(D^{*}\\{-1\\},D)$. Then
$\alpha^{{\dagger}}=\delta\alpha^{*}\delta^{-1}$ defines an involution
${\dagger}$ on $\operatorname{End}_{\varphi}(D)$ and the map
$\alpha\mapsto\alpha\delta$ establishes an isomorphim
$\operatorname{End}_{\varphi}(D)^{{\dagger}}\xrightarrow{\sim}\operatorname{Hom}_{\varphi}^{\text{a}}(D^{*}\\{-1\\},D)$
where $\operatorname{End}_{\varphi}(D)^{{\dagger}}$ is the subspace of
elements fixed by ${\dagger}$.

######  Theorem 1.2 ([Vo] Corollary 5.9).

Let $V$ be a crystalline $p$-adic representation of $G$. The following are
equivalent:

  * (i)

there is an abelian variety $\mathcal{A}$ over ${\mathbb{Q}}_{p}$ such that
$V\simeq V_{p}(\mathcal{A})$

  * (ii)

$\operatorname{{\bf D}}^{*}_{\text{\em\tiny cris}}(V)$ satisfies conditions
$(1)$, $(2)$ and $(3)$.

Note that the restriction $p\neq 2$ in [Vo] Theorem 5.7 and its Corollary 5.9
is unnecessary as Kisin shows that a crystalline representation with Hodge-
Tate weights in $\\{0,1\\}$ arises from a $p$-divisible group unrestrictidly
on the prime $p$ ([Ki] Thm.0.3).

Let $\mathcal{A}$ be an abelian variety over ${\mathbb{Q}}_{p}$ having good
reduction and $(D,\operatorname{Fil})=\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V_{p}(\mathcal{A}))$. The $\varphi$-module $D$ satisfies (1) by the
Weil conjectures for abelian varieties over $\mathbb{F}_{p}$. Tate’s theorem
on endomorphisms of the latter (see [Wa-Mi] II) shows that the isomorphism
class of the $\varphi$-module $D$, given by semisimplicity by
$\operatorname{P_{char}}(\varphi)$, determines the isogeny class of the
special fibre of $\mathcal{A}$ over ${\mathbb{F}}_{\\!p}$. Any polarisation on
${\mathcal{A}}$ induces a form on $D$ satisfying (3) and the filtration
satisfies (2) by the Hodge decomposition for $p$-divisible groups and (3).

Conversely let $V$ be a crystalline $p$-adic representation of $G$ such that
$\operatorname{{\bf D}^{*}_{\text{\tiny cris}}}(V)$ satisfies (1), (2), (3).
From (1) the Honda-Tate theory ([Ho-Ta]) furnishes an abelian variety $A$ over
$\mathbb{F}_{p}$ with the right Frobenius. From (2) Kisin’s result [Ki]
furnishes a $p$-divisible group over ${\mathbb{Z}}_{p}$ lifting $A(p)$. The
Serre-Tate theory of liftings then produces a formal abelian scheme
$\mathcal{A}$ over ${\mathbb{Z}}_{p}$ with special fibre isogenous to $A$.
Finally (3) furnishes a polarisation on $\mathcal{A}$ which ensures by
Grothendieck’s theorem on algebraisation of formal schemes ([Gr] 5.4.5) that
$\mathcal{A}$ is a true abelian scheme. The proof of Theorem 5.7 in [Vo]
details this construction.

Thus we want to construct an admissible filtered $\varphi$-module
$(D,\operatorname{Fil})$ over ${\mathbb{Q}}_{p}$ satisfying conditions (1),
(2), (3) of theorem 1.2 and such that

  * (a)

$\operatorname{P_{char}}(\varphi)$ is a supersingular $p$-Weil polynomial

  * (b)

$(D,\operatorname{Fil})$ is not semisimple.

Recall that a $p$-Weil polynomial is supersingular if its roots are of the
form $\zeta\sqrt{p}$ with $\zeta\in\overline{\mathbb{Q}}\mspace{1.0mu}$ a root
of unity, and that an abelian variety $A$ over ${\mathbb{F}}_{\\!p}$ is
supersingular if and only if the characteristic polynomial of its Frobenius is
supersingular. Regarding (a) in section 2 we take
$\operatorname{P_{char}}(\varphi)(X)=(X^{2}+p)^{2}$ which is the
characteristic polynomial of the Frobenius of the product of a supersingular
elliptic curve $E$ over ${\mathbb{F}}_{\\!p}$ with itself. In section 3 we
take $\operatorname{P_{char}}(\varphi)(X)=X^{4}+pX^{2}+p^{2}$ which is the
characteristic polynomial of the Frobenius of a simple supersingular abelian
surface over ${\mathbb{F}}_{\\!p}$.

Regarding (b) we assume $p\equiv 1\bmod 3\mathbb{Z}$ in section 3. In each
(a)-case we find a subobject $D_{1}$ of $(D,\operatorname{Fil})$ in ${\bf
MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$ and a quotient object
$D_{2}$ (endowed with the quotient filtration
$\operatorname{Fil}^{i}D_{2}=\operatorname{Fil}^{i}D\bmod D_{1}$) such that
the sequence

(s)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$incl$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$proj$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

is exact and $D_{2}$ is not a subobject. Thus (s) does not split and therefore
$(D,\operatorname{Fil})$ is not semisimple. Of course when
$(D,\operatorname{Fil})\simeq\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V_{p}(\mathcal{A}))$ this means that there is a nonsplit short exact
sequence of $G$-modules

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p}(\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

with $V_{i}\simeq\operatorname{{\bf V}^{*}_{\text{\tiny cris}}}(D_{i})$ for
$i=1,2$, and it follows that $V_{p}(\mathcal{A})$ is not a semisimple
$G$-module.

##  2\. A lift of the twofold product of a supersingular elliptic curve

Consider the filtered $\varphi$-module $(D,\operatorname{Fil})$ over
${\mathbb{Q}}_{p}$ defined as follows. There is a ${\mathbb{Q}}_{p}$-basis
$\mathcal{B}=(x_{1},y_{1},x_{2},y_{2})$ for $D$ so that

$D={\mathbb{Q}}_{p}x_{1}\oplus{\mathbb{Q}}_{p}y_{1}\oplus{\mathbb{Q}}_{p}x_{2}\oplus{\mathbb{Q}}_{p}y_{2}$

is a 4-dimensional ${\mathbb{Q}}_{p}$-vector space. The matrix of $\varphi$
over $\mathcal{B}$ is

$\operatorname{Mat}_{\mathcal{B}}(\varphi)=\,\begin{pmatrix}0&-p&0&0\\\
1&0&0&0\\\ 0&0&0&-p\\\ 0&0&1&0\end{pmatrix}\in\,GL_{4}({\mathbb{Q}}_{p})$

and the filtration is given by

$\operatorname{Fil}^{0}D=D,\quad\operatorname{Fil}^{1}D=\,{\mathbb{Q}}_{p}x_{1}\oplus{\mathbb{Q}}_{p}(y_{1}+x_{2}),\quad\operatorname{Fil}^{2}D=0.$

######  Proposition 2.1.

There is an abelian surface $\mathcal{A}$ over ${\mathbb{Q}}_{p}$ such that
$(D,\operatorname{Fil})\simeq\operatorname{{\bf D}}^{*}_{\text{\em\tiny
cris}}(V_{p}(\mathcal{A}))$. Further

  * (a)

$\mathcal{A}$ has good reduction with special fibre isogenous to the product
of two supersingular elliptic curves over ${\mathbb{F}}_{\\!p}$

  * (b)

the $G$-module $V_{p}(\mathcal{A})$ is not semisimple.

###### Proof.

The filtration has Hodge-Tate type $(0,1)$ with
$\dim\operatorname{Fil}^{1}D=2$ and $\det\varphi=p^{2}$ hence
$t_{H}(D)=2=t_{N}(D)$. Since
$\operatorname{P_{char}}(\varphi)(X)=(X^{2}+p)^{2}$ the nontrivial
$\varphi$-stable subspaces of $D$ are the
$D_{i}={\mathbb{Q}}_{p}x_{i}\oplus{\mathbb{Q}}_{p}y_{i}$ for $i=1,2$ both
having Newton invariant $t_{N}(D_{i})=1$. However
$D_{1}\cap\operatorname{Fil}^{1}D={\mathbb{Q}}_{p}x_{1}$ whereas
$D_{2}\cap\operatorname{Fil}^{1}D=0$, so $t_{H}(D_{1})=1$ and
$t_{H}(D_{2})=0$. Therefore $(D,\operatorname{Fil})$ is admissible, $D_{1}$ is
a subobject, $D_{2}$ is a quotient that is not a subobject, the short exact
sequence (s) does not split and $(D,\operatorname{Fil})$ is not semisimple.

The action of $\varphi$ is semisimple and
$\operatorname{P_{char}}(\varphi)=\operatorname{P_{char}}(\text{Fr}_{E})^{2}$
where $E$ is a supersingular elliptic curve over ${\mathbb{F}}_{\\!p}$ with
$\operatorname{P_{char}}(\text{Fr}_{E})(X)=X^{2}+p$. Thus
$(D,\operatorname{Fil})$ satisfies condition (1) of theorem 1.2 as well as
condition (a) of section 1 and it obviously satisfies (2). It remains to check
condition (3) that is to find a
$\delta\in\operatorname{Isom}_{{\mathbb{Q}}_{p}}(D^{*},D)$ satisfying
$\delta^{*}=-\delta$, $\varphi\delta=p\delta\varphi^{*-1}$, and
$\delta(\operatorname{Fil}^{1}D)^{\perp}=\operatorname{Fil}^{1}D$. Let
$\mathcal{B}^{*}=(x_{1}^{*},y_{1}^{*},x_{2}^{*},y_{2}^{*})$ be the dual basis
of $\mathcal{B}$ for $D^{*}$ where $z^{*}$ is the linear form on $D$ sending
$z\in D$ to $1$ and vanishing on all vectors noncolinear to $z$. The matrix of
$p\varphi^{*\,-1}$ over $\mathcal{B}^{*}$ is

$p\operatorname{Mat}_{\mathcal{B}}(\varphi^{-1})^{t}=\,\begin{pmatrix}0&-1&0&0\\\
p&0&0&0\\\ 0&0&0&-1\\\ 0&0&p&0\end{pmatrix}$

where $M^{t}$ is the transpose of $M$ and

$(\operatorname{Fil}^{1}D)^{\perp}=\,{\mathbb{Q}}_{p}y_{2}^{*}\oplus{\mathbb{Q}}_{p}(y_{1}^{*}-x_{2}^{*}).$

Let $\delta:D^{*}\rightarrow D$ be the ${\mathbb{Q}}_{p}$-linear morphism with
matrix over the bases $\mathcal{B}^{*}$ and $\mathcal{B}$

$\operatorname{Mat}_{\mathcal{B}^{*}\\!\\!,\mathcal{B}}(\delta)=\,\begin{pmatrix}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{pmatrix}.$

Then $\delta$ is invertible and satisfies the relations $\delta^{*}=-\delta$
and $\varphi\delta=p\delta\varphi^{*\,-1}$. Further
$\delta(\operatorname{Fil}^{1}D)^{\perp}=\delta\bigl{(}{\mathbb{Q}}_{p}y_{2}^{*}\oplus{\mathbb{Q}}_{p}(y_{1}^{*}-x_{2}^{*})\bigr{)}={\mathbb{Q}}_{p}x_{1}\oplus{\mathbb{Q}}_{p}(y_{1}+x_{2})=\operatorname{Fil}^{1}D$.
∎

######  Remark 2.2.

Any 2-dimensional object satisfying conditions (1) and (2) of theorem 1.2 also
satisfies condition (3). Hence theorem 1.2 applied to the admissible filtered
$\varphi$-modules $(D_{1},\operatorname{Fil}^{i}D\cap D_{1})$ and
$(D_{2},\operatorname{Fil}^{i}D\bmod D_{1})$ shows the existence of elliptic
schemes $\mathcal{E}_{i}$ over ${\mathbb{Z}}_{p}$ such that
$D_{i}\simeq\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V_{p}(\mathcal{E}_{i}))$ for $i=1,2$. The special fibres of the
$\mathcal{E}_{i}$ are ${\mathbb{F}}_{\\!p}$-isogenous to $E$. Thus we obtain a
nonsplit exact sequence of $G$-modules

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V_{p}(\mathcal{E}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
87.75644pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
87.75644pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V_{p}(\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
143.90732pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
143.90732pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V_{p}(\mathcal{E}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
202.16376pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
202.16376pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

By Tate’s full faithfulness theorem [Ta] the $G$-module $V_{p}(\mathcal{A})$
determines the $p$-divisible group $\mathcal{A}(p)$ over ${\mathbb{Z}}_{p}$ up
to isogeny, therefore $\mathcal{A}(p)$ is not ${\mathbb{Z}}_{p}$-isogenous to
$\mathcal{E}_{1}(p)\times\mathcal{E}_{2}(p)$.

######  Remark 2.3.

The same construction works starting with the square of any supersingular
$p$-Weil polynomial of degree two (when $p\geq 5$ there is only $X^{2}+p$ but
when $p=2$ or $3$ there are also the $X^{2}\pm pX+p$). However it fails when
dealing with the product of two distinct such. Indeed let
$\alpha_{1}\neq\alpha_{2}\in p{\mathbb{Z}}_{p}$ and $D$ be a semisimple
4-dimensional $\varphi$-module with
$\operatorname{P_{char}}(\varphi)(X)=(X^{2}+\alpha_{1}X+p)(X^{2}+\alpha_{2}X+p)$.
Then $D=D_{1}\oplus D_{2}$ with
$D_{i}=\operatorname{Ker}(\varphi^{2}+\alpha_{i}\varphi+p)$, which are the
nontrivial $\varphi$-stable subspaces of $D$, and $t_{N}(D_{i})=1$. Since
$\alpha_{1}\neq\alpha_{2}$ one checks that any ${\mathbb{Q}}_{p}$-linear
$\delta:D^{*}\rightarrow D$ satisfying $\delta^{*}=-\delta$ and
$\varphi\delta=p\delta\varphi^{*\,-1}$ sends $D_{2}^{\perp}$ into $D_{1}$ and
$D_{1}^{\perp}$ into $D_{2}$. Endowing $D$ with an admissible Hodge-Tate
$(0,1)$ filtration such that (s) does not split amounts to picking a
$2$-dimensional subspace $\operatorname{Fil}^{1}D$ such that $\dim
D_{1}\cap\operatorname{Fil}^{1}D=1$ and $\dim
D_{2}\cap\operatorname{Fil}^{1}D=0$ (or vice versa) ; then $\dim
D_{1}\cap\delta(\operatorname{Fil}^{1}D)^{\perp}=0$ and $\dim
D_{2}\cap\delta(\operatorname{Fil}^{1}D)^{\perp}=1$, therefore
$\delta(\operatorname{Fil}^{1}D)^{\perp}\neq\operatorname{Fil}^{1}D$. This
shows that the $p$-adic Tate modules of abelian schemes over
${\mathbb{Z}}_{p}$ with special fibre ${\mathbb{F}}_{\\!p}$-isogenous to the
product of two nonisogenous supersingular elliptic curves are semisimple.

######  Remark 2.4.

One constructs in a similar fashion for each integer $n\geq 2$ a lift of the
$n$-fold product of a supersingular elliptic curve over ${\mathbb{F}}_{\\!p}$
with nonsemisimple $p$-adic Tate module.

##  3\. A lift of a simple supersingular abelian surface

In this section we assume $p\equiv 1\bmod 3\mathbb{Z}$ which is equivalent to
$\zeta_{3}\in{\mathbb{Q}}_{p}$ where $\zeta_{3}$ is a primitive 3rd root of
unity. Consider the filtered $\varphi$-module $(D,\operatorname{Fil})$ defined
as follows. There is a ${\mathbb{Q}}_{p}$-basis
$\mathcal{B}=(x_{1},y_{1},x_{2},y_{2})$ for $D$ so that

$D={\mathbb{Q}}_{p}x_{1}\oplus{\mathbb{Q}}_{p}y_{1}\oplus{\mathbb{Q}}_{p}x_{2}\oplus{\mathbb{Q}}_{p}y_{2}$

is a 4-dimensional ${\mathbb{Q}}_{p}$-vector space. The matrix of $\varphi$
over $\mathcal{B}$ is

$\operatorname{Mat}_{\mathcal{B}}(\varphi)=\,\begin{pmatrix}0&\zeta_{3}p&0&0\\\
1&0&0&0\\\ 0&0&0&\zeta_{3}^{-1}p\\\
0&0&1&0\end{pmatrix}\in\,GL_{4}({\mathbb{Q}}_{p})$

and the filtration is given by

$\operatorname{Fil}^{0}D=D,\quad\operatorname{Fil}^{1}D={\mathbb{Q}}_{p}x_{1}\oplus{\mathbb{Q}}_{p}(y_{1}+x_{2}),\quad\operatorname{Fil}^{2}D=0.$

######  Proposition 3.1.

There is an abelian surface $\mathcal{A}$ over ${\mathbb{Q}}_{p}$ such that
$(D,\operatorname{Fil})\simeq\operatorname{{\bf D}}^{*}_{\text{\em\tiny
cris}}(V_{p}(\mathcal{A}))$. Further

  * (a)

$\mathcal{A}$ has good reduction with special fibre isogenous to a
supersingular simple abelian surface over ${\mathbb{F}}_{\\!p}$

  * (b)

the $G$-module $V_{p}(\mathcal{A})$ is not semisimple.

###### Proof.

Just as in the proof of proposition 2.1 we have $t_{H}(D)=2=t_{N}(D)$. Since

$\operatorname{P_{char}}(\varphi)(X)=X^{4}+pX^{2}+p^{2}=(X^{2}-\zeta_{3}p)(X^{2}-\zeta_{3}^{-1}p)$

the nontrivial sub-${\mathbb{Q}}_{p}[\varphi]$-modules of $D$ are the
$D_{i}={\mathbb{Q}}_{p}x_{i}\oplus{\mathbb{Q}}_{p}y_{i}$ for $i=1,2$ both
having Newton invariant $t_{N}(D_{i})=1$, and Hodge invariants
$t_{H}(D_{1})=1$, $t_{H}(D_{2})=0$. Again we obtain a nonsplit exact sequence
(s) in ${\bf MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$ and
$(D,\operatorname{Fil})$ is not semisimple.

The action of $\varphi$ is semisimple and
$\operatorname{P_{char}}(\varphi)=\operatorname{P_{char}}(\text{Fr}_{A})$
where $A$ is a supersingular simple abelian surface over ${\mathbb{F}}_{\\!p}$
with $\operatorname{P_{char}}(\text{Fr}_{A})(X)=X^{4}+pX^{2}+p^{2}$. Thus
$(D,\operatorname{Fil})$ satisfies condition (1) of theorem 1.2 as well as
condition (a) of section 1. It obviously satisfies (2) and it remains to check
(3). Let $\mathcal{B}^{*}=(x_{1}^{*},y_{1}^{*},x_{2}^{*},y_{2}^{*})$ be the
dual basis of $\mathcal{B}$ for $D^{*}$. Again
$(\operatorname{Fil}^{1}D)^{\perp}={\mathbb{Q}}_{p}y_{2}^{*}\oplus{\mathbb{Q}}_{p}(y_{1}^{*}-x_{2}^{*})$
and the matrix of $p\varphi^{*\,-1}$ over $\mathcal{B}^{*}$ is

$p\operatorname{Mat}_{\mathcal{B}}(\varphi^{-1})^{t}=\,\begin{pmatrix}0&\zeta_{3}^{-1}&0&0\\\
p&0&0&0\\\ 0&0&0&\zeta_{3}\\\ 0&0&p&0\end{pmatrix}.$

Let $\delta:D^{*}\rightarrow D$ be the ${\mathbb{Q}}_{p}$-linear morphism with
matrix over the bases $\mathcal{B}^{*}$ and $\mathcal{B}$

$\operatorname{Mat}_{\mathcal{B}^{*}\\!\\!,\mathcal{B}}(\delta)=\,\begin{pmatrix}0&0&0&\zeta_{3}\\\
0&0&1&0\\\ 0&-1&0&0\\\ -\zeta_{3}&0&0&0\end{pmatrix}.$

As in the proof of proposition 2.1 one checks that $\delta$ is invertible,
satisfies $\delta^{*}=-\delta$, $\varphi\delta=p\delta\varphi^{*\,-1}$, and
that $\delta(\operatorname{Fil}^{1}D)^{\perp}=\operatorname{Fil}^{1}D$. ∎

######  Remark 3.2.

The objects $(D_{1},\operatorname{Fil}^{i}D\cap D_{1})$ and
$(D_{2},\operatorname{Fil}^{i}D\bmod D_{1})$ in ${\bf
MF}_{{\mathbb{Q}}_{p}}^{\operatorname{ad}}(\varphi)$ do not arise from
elliptic schemes over ${\mathbb{Z}}_{p}$, however [Ki] Thm.0.3 shows the
existence of $p$-divisible groups $\mathcal{G}_{i}$ over ${\mathbb{Z}}_{p}$
such that $D_{i}\simeq\operatorname{{\bf D}^{*}_{\text{\tiny
cris}}}(V_{p}(\mathcal{G}_{i}))$. The special fibre of $\mathcal{A}(p)$ is
${\mathbb{F}}_{\\!p}$-isogenous to the product of the special fibres of the
$\mathcal{G}_{i}$, themselves being nonisogenous. Thus we obtain a nonsplit
exact sequence of $G$-modules

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p}(\mathcal{G}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p}(\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p}(\mathcal{G}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

and Tate’s full faithfulness theorem shows that $\mathcal{A}(p)$ is not
${\mathbb{Z}}_{p}$-isogenous to $\mathcal{G}_{1}\times\mathcal{G}_{2}$.

######  Remark 3.3.

Starting with $X^{4}-pX^{2}+p^{2}$ when $p\equiv 1\bmod 3\mathbb{Z}$ and
$X^{4}+p^{2}$ when $p\equiv 1\bmod 4\mathbb{Z}$ one obtains alike
nonsemisimple $4$-dimensional supersingular representations (just replace
$\zeta_{3}$ by $\zeta_{6}$ or $\zeta_{4}$). More generally the

$p^{d}\Phi_{n}\Bigl{(}\frac{X^{2}}{p}\Bigr{)}=\prod_{i\in(\mathbb{Z}/n\mathbb{Z})^{\times}}(X^{2}-\zeta_{n}^{i}p)\qquad\text{
with }\,d=\\#\bigl{(}\mathbb{Z}/n\mathbb{Z}\bigr{)}^{\times}\geq 2$

where $\Phi_{n}$ is the $n$th cyclotomic polynomial are supersingular $p$-Weil
polynomials leading when $p\equiv 1\bmod n\mathbb{Z}$ to similar higher-
dimensional constructions.

## References

  * [Br] C. Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés, Annals of Math. 152 (2000), 489-549.
  * [Co-Io] R. Coleman and A. Iovita, The Frobenius and Monodromy operators for Curves and Abelian Varieties, Duke Math. J. 97 (1999), 171-215.
  * [Co-Fo] P. Colmez et J.-M. Fontaine, Construction des représentations $p$-adiques semi-stables, Invent. math. 140, 1 (2000), 1-43.
  * [Fo1] J.-M. Fontaine, Le corps des périodes $p$-adiques, in Périodes $p$-adiques, Astérisque 223, Soc. Math. de France (1994).
  * [Fo2] J.-M. Fontaine, Représentations $p$-adiques semi-stables, in Périodes $p$-adiques, Astérisque 223, Soc. Math. de France (1994).
  * [Gr] A. Grothendieck, EGA III, Inst. Hautes Études Sci. Publ. Math. 11 (1961).
  * [Ho-Ta] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki 352 (1968), 15p.
  * [Ki] M. Kisin, Crystalline representations and $F$-crystals, in Algebraic Geometry and Number Theory In Honor of Vladimir Drinfeld’s 50th Birthday, Progress in Mathematics 253 (2006), 459-496.
  * [Ta] J. Tate, $p$-Divisible groups over local fields, in Proceedings of a Conference on Local Fields, Driebergen 1966, Springer-Verlag (1967), 158-183.
  * [Vo] M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over ${\mathbb{Q}}_{p}$, Math. Ann. 331 (2005), no. 4, 889–923.
  * [Wa-Mi] W.C. Waterhouse and J.S. Milne, Abelian Varieties over Finite Fields, in AMS Proceedings of Symposia in Pure Mathematics XX (1971), 53-64.