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# Monodromy groups of supersingular abelian surfaces over $Q_p$

Moqing Chen

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Moqing Chen. Monodromy groups of supersingular abelian surfaces over $Q_p$. 2025. hal-05391509

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Distributed under a Creative Commons Attribution 4.0 International License
MONODROMY GROUPS OF SUPERSINGULAR ABELIAN SURFACES OVER $\mathbb{Q}_p$

MOQING CHEN

ABSTRACT. For primes $p \ge 7$, we give a parametrization of the filtered $\varphi$-modules attached to the $p$-adic Tate modules of abelian surfaces over $\mathbb{Q}_p$ with supersingular good reduction. We use this classification to determine the neutral components of the monodromy groups of the associated $p$-adic representations up to $\bar{\mathbb{Q}}_p$-isomorphism. Furthermore, we analyze the $p$-adic distribution of these groups in the moduli space of filtered $\varphi$-modules. In particular, we prove that the neutral components are generically isomorphic to $\mathrm{GL}_2 \times_{\det} \mathrm{GL}_2$.

CONTENTS

1. Introduction 2

1.1. Main classification result 2

1.2. A moduli space description 3

1.3. Distribution of Wintenberger types 4

1.4. Strategy of proof 5

1.5. Relation to previous work and broader motivation 5

Outline of the article 6

Acknowledgement 6

2. Preliminaries 7

2.1. Crystalline representations and filtered $\varphi$-modules 7

2.2. Hodge decomposition for admissible filtered $\varphi$-modules 8

2.3. Filtered $\varphi$-modules arising from geometry 8

2.4. Tools for computing monodromy groups 9

3. Filtered $\varphi$-modules arising from supersingular abelian surfaces over $\mathbb{Q}_p$ 10

3.1. Supersingular $p$-Weil polynomials of degree 4 10

3.2. Linear-algebraic constraints 10

3.3. The case $\chi_{\varphi}(X) = (X^2 \pm p)^2$ 12

3.4. The case $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$ 12

3.5. Proof of Theorem 3.10 13

3.6. Construction of certain skew forms 18

4. $p$-adic monodromy groups of supersingular abelian surfaces over $\mathbb{Q}_p$ 19

Date: December 1, 2025.
4.1. The case $\chi_{\varphi}(X) = (X^2 \pm p)^2$ 19

4.2. The case $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$ 19

5. A moduli space description 24

5.1. Parametrization by $\mathbb{P}^2(\mathbb{Q}_p)$ 25

5.2. The GIT viewpoint 29

5.3. Distribution of monodromy groups 31

References 32

# 1. INTRODUCTION

## 1.1. Main classification result.
In this paper we study the monodromy groups of the p-adic crystalline representations arising as the p-adic Tate modules of abelian surfaces over $\mathbb{Q}_p$ with supersingular good reduction. Let $\text{AbSurf}_{\mathbb{Q}_p}^{\text{ss}}$ be the full subcategory of the category of abelian varieties over $\mathbb{Q}_p$ consisting of such abelian surfaces. Via Fontaine's functor $D_{\text{cris}}$, we work in the category $\mathbf{MF}_{\mathbb{Q}_p}^{\text{wa}}$ of admissible filtered $\varphi$-modules:

$$ \text{AbSurf}_{\mathbb{Q}_p}^{\text{ss}} \xrightarrow{V_p} \text{Rep}_{\mathbb{Q}_p}^{\text{cris}}(\Gamma_{\mathbb{Q}_p}) \xrightarrow{D_{\text{cris}}} \mathbf{MF}_{\mathbb{Q}_p}^{\text{wa}}. $$

Denote the essential image of the functor $D_{\text{cris}} \circ V_p$ by $\mathbf{C}$. We give an explicit classification of objects in $\mathbf{C}$ and their associated monodromy groups.

**Theorem A** (Theorem 3.7, Theorem 3.10). Let $p \ge 7$. The objects in $\mathbf{C}$ are precisely the filtered $\varphi$-modules $D_{\epsilon'}^{\text{prod}}$, $D_{\epsilon'}^{\epsilon, \text{iso}}$, $D_{a'}^{\epsilon, \nu}$ or $D_{(a,b)}^{\epsilon, \mu}$ constructed in Definitions 3.5 and 3.8, with parameters satisfying the explicit arithmetic conditions given in Theorem 3.10.

Based on this classification, we determine the p-adic algebraic monodromy groups of supersingular abelian surfaces over $\mathbb{Q}_p$. Denote by $H_D$ the monodromy group of $D \in \mathbf{C}$ and by $H_D^\circ$ the neutral connected component.

**Theorem B** (Theorem 4.6). Fix a prime $p \ge 7$. For $\epsilon' \in \{\pm 1\}$, $\epsilon \in \{0, \pm 1\}$ and $a', a, b \in \mathbb{Q}_p$ with $ab \ne -1$, we have the following isomorphisms of algebraic groups over $\bar{\mathbb{Q}}_p$.

(1) $H_{D_{\epsilon'}^{\text{prod}}}^{\circ} \simeq \mathbf{G}_m^2,$

(2) $H_{D_{a'}^{\epsilon, \nu}}^{\circ} \simeq \begin{cases} \mathbf{G}_m^3 & \text{if } a' = \epsilon = 0 \\ \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2 & \text{otherwise,} \end{cases}$

(3) $H_{D_{(a,b)}^{\epsilon, \mu}}^{\circ} \simeq \begin{cases} \mathbf{G}_a^2 \times_g \mathbf{G}_m^2 & \text{or } \mathbf{G}_m^2 \text{ if } c(a,b) = -\epsilon p \\ \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2 & \text{otherwise,} \end{cases}$

(4) $H_{D_{\epsilon'}^{\epsilon, \text{iso}}}^{\circ} \simeq \mathbf{GL}_2$

where $c(a, b) = -\frac{a^2+\epsilon p+b^2 p^2}{ab+1}$ and $g(s, t) = \begin{pmatrix} s & 0 \\ 0 & t \end{pmatrix}$. In particular, for any object $A$ in $\text{AbSurf}_{\mathbb{Q}_p}^{\text{ss}}$, the neutral connected component $G_{V_p(A)}^{\circ}$ of the p-adic algebraic monodromy group associated with $A$ is $\bar{\mathbb{Q}}_p$-isomorphic to one of the following:

$$ \mathbf{G}_m^2, \mathbf{G}_m^3, \mathbf{G}_a^2 \times_g \mathbf{G}_m^2, \mathbf{GL}_2, \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2. $$
From this, one can easily see which of the objects in $\mathbb{C}$ are semisimple, as semisimplicity corresponds precisely to the reductiveness of the associated monodromy groups. Adding some direct computations, we see that $D \in \mathbb{C}$ is non-semisimple if and only if $3|p-1$ and $D \simeq D_{(-\zeta_3 p, 1)}^{1,\mu}$ for any primitive cube root of unity $\zeta_3 \in \mathbb{Q}_p^\times$.

**1.2. A moduli space description.** The families occurring in Theorem A are parametrized by "algebraic" subvarieties in a certain coarse moduli space of filtered $\varphi$-modules. Fix $\epsilon \in \{0, \pm 1\}$. We focus on the objects $D \in \mathbb{C}$ such that $\chi_{\varphi_D}(X) = X^4 + \epsilon p X^2 + p^2$, where $\varphi_D$ denotes the crystalline Frobenius of $D$. We denote the set of the isomorphism classes of these objects by $M^{\text{wa}}$. We show that $M^{\text{wa}}$ is in natural bijection (up to finitely many points) with the $\mathbb{Q}_p$-points of a projective GIT quotient of an open subvariety $Y^{\text{def}}$ of $\text{Gr}(2, 4)$.

**Theorem C (Theorem 5.7).** There exists a commutative diagram

<figure><img src="image_3.png" /></figure>

where

(1) $c$ is algebraic and becomes a projective GIT quotient after base change to $\bar{\mathbb{Q}}_p$;

(2) if the polynomial $X^2 + \epsilon p X + p^2$ has no root in $\mathbb{Q}_p$, the map $\bar{c}$ is injective;

(3) if the polynomial $X^2 + \epsilon p X + p^2$ splits over $\mathbb{Q}_p$, the map $\bar{c}$ is injective outside $\bar{c}^{-1}([-\epsilon p : 1])$ and $\bar{c}^{-1}([-\epsilon p : 1])$ consists of three points.

For a more concrete study, we construct a closed immersion $\mathbb{P}^2 \to Y^{\text{def}}$ such that the induced map $\pi : \mathbb{P}^2(\mathbb{Q}_p) \to M^{\text{wa}}$ is surjective.

<figure><img src="image_4.png" /></figure>

This approach clarifies the structure of the map $\bar{c}$. When $X^2 + \epsilon p X + p^2$ has two different roots $\mu_1, \mu_2$ over $\mathbb{Q}_p$, the preimage $(c \circ \iota)^{-1}([-\epsilon p : 1])$ consists of two projective lines $l_1, l_2$ in $\mathbb{P}^2(\mathbb{Q}_p)$ intersecting at a point $o$, and the preimages of the three points in $\bar{c}^{-1}([-\epsilon p : 1])$ under $\pi$ are $l_1 \setminus \{o\}, l_2 \setminus \{o\}$ and $\{o\}$. In particular, we see that under the quotient topology, the point $\pi(o)$ generalizes to $\pi(x_i)$ where $x_i \in l_i \setminus \{o\}, i = 1, 2$.

Moreover, we have natural lifts of the families $D_{\epsilon, \text{iso}}^{\epsilon, \text{iso}}$, $D_{\alpha', \nu}^{\epsilon, \nu}$ and $D_{(a,b)}^{\epsilon, \mu}$ (which are defined on $M^{\text{wa}}$) along $\pi$. We construct points $P_{\pm 1} \in \mathbb{P}^2(\mathbb{Q}_p)$ and morphisms

$$A^1 \xrightarrow{\nu} \mathbb{P}^2 \quad \text{and} \quad A^2 \setminus \{ab = -1\} \xrightarrow{\mu} \mathbb{P}^2$$
such that $[D_{\pm 1}^{\epsilon, iso}] = \pi(P_{\pm 1})$ and the diagrams

<figure><img src="image_5.png" /></figure>

commute. Here, $[D] \in \mathcal{M}^{\text{wa}}$ denotes the isomorphism classes of $D \in \mathbf{MF}_{\mathbb{Q}_p}^{\text{wa}}$.

In terms of our moduli space description, we have the following corollary of Theorem B.

**Theorem D (Theorem 5.13).** For each point $P \in \mathcal{M}^{\text{wa}}$, let $H_{D_P}$ denote the monodromy group of any filtered $\varphi$-module representing the isomorphism class corresponding to $P$. Then, we have $H_{D_P}^{\circ} \simeq \bar{\mathbb{Q}}_p \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$ for all except finitely many $P \in \mathcal{M}^{\text{wa}}$. The exceptions are

$$ H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \begin{cases} \mathbf{GL}_2 & \bar{c}(P) = [\pm 2p: 1] \\ \mathbf{G}_m^2 \text{ or } \mathbf{G}_a^2 \times \mathbf{G}_m^2 & \bar{c}(P) = [-\epsilon p: 1] \\ \mathbf{G}_m^3 & \epsilon = 0 \text{ and } \bar{c}(P) = [1: 0] \end{cases} $$

where $g(s, t) = \begin{pmatrix} s & 0 \\ 0 & t \end{pmatrix}$. When $\bar{c}(P) = [-\epsilon p: 1]$, in the context of Theorem C, the case $H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \mathbf{G}_m^2$ occurs when $P = \pi(o)$, and $H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \mathbf{G}_a^2 \times_g \mathbf{G}_m^2$ when $P = \pi(x_i)$ where $x_i \in l_i \setminus \{o\}$, $i = 1, 2$.

The distribution of monodromy group can be visualized when pulled back to $\mathbb{P}^2(\mathbb{Q}_p)$ (based on Remark 5.9). For example, when $\epsilon = 0$ we have the following description:

<figure><img src="image_6.png" /></figure>

1.3. **Distribution of Wintenberger types.** In the proof, we use a refinement of Hodge decomposition in the proof,

$$ D = \bigoplus_{\xi \in X_{per}} D_{\xi} $$

which was introduced in [Win84]. Here, $X_{per}$ is the abelian group of periodic functions from $\mathbb{Z}$ to itself, and we call the subset consisting of the $\xi$ that appear in this decomposition the *Wintenberger type* of $D$, denoted $X_D$. We study the distribution of $X_{D_P}$ for $P \in \mathcal{M}^{\text{wa}}$, where $D_P \in \mathbf{MF}_{\mathbb{Q}_p}^{\text{wa}}$ represents the isomorphism class associated to $P$.
**Theorem E (Corollary 5.8).** Let $P \in \mathcal{M}^{\text{wa}}$. The Wintenberger type of $D_P$ satisfies

$$X_{D_P} = \begin{cases} (A) & v_p(\bar{c}(P)) \le 0 \\ (B) & v_p(\bar{c}(P)) \ge 1 \end{cases}$$

where (A) and (B) are given in Lemma 3.4 and conventionally, we set $v_p([a : 1]) := v_p(a)$ and $v_p([1:0]) := -\infty$.

1.4. **Strategy of proof.** Throughout this subsection, we fix an unramified finite extension $K$ of $\mathbb{Q}_p$. The structure of any admissible filtered $\varphi$-module over $K$ is determined by its

crystalline Frobenius $\varphi$ and Hodge cocharacter<fnref n="1" /> $\mu : \mathbf{G}_m \to \mathbf{GL}_{4,K}$.

When $K = \mathbb{Q}_p$, the crystalline Frobenius is linear and hence determined by its characteristic polynomial, which must be a supersingular $p$-Weil polynomial of degree 4. We prove that, for $p \ge 7$, such polynomials are precisely

$$(X^2 \pm p)^2 \quad \text{and} \quad X^4 + \epsilon p X^2 + p^2, \epsilon \in \{0, \pm 1\}.$$

Moreover, the conjugacy class of the Hodge cocharacter is fixed, as the $p$-adic Tate module of any abelian surface has Hodge numbers $h^{1,0} = h^{0,1} = 2$. There are technical constraints on the relative position of $\varphi$ and $\mu$ inside $\mathbf{GL}_4(\mathbb{Q}_p)$ [Win84, Introduction], which translate into the arithmetic conditions in Theorem A. Finally, we complete the proof of Theorem A by verifying the conditions required in Volkov’s characterization of filtered $\varphi$-modules arising from abelian varieties with good reduction over $\mathbb{Q}_p$ [Vol05, Corollary 5.9].

In the proof of Theorem B, one of the main ingredients is a density result due to Pink [Pin98, Proposition 2.5], which asserts that for any admissible filtered $\varphi$-module $D \in \mathbf{MF}_K^{\text{wa}}$, the conjugations of $\mu(K^\times)$ by $\varphi^i$ together with $\varphi^{[K:\mathbb{Q}_p]}$ generates a Zariski dense subgroup in $H_D$. We also use the algebraic Lie correspondence [Mil17, Corollary 10.17] in our computations.

Theorem C is established by proving that the isomorphism class of $D_{a'}^{\epsilon, \nu}$ (resp. $D_{(a,b)}^{\epsilon, \mu}$) is determined by $c(\iota(\nu(a')))$ (resp. $c(\iota(\mu(a,b)))$), except for the case $c = [-\epsilon p: 1]$, which requires a special discussion.

<figure><img src="image_7.png" /></figure>

Theorem D and E are direct corollaries of Theorem B and C.

1.5. **Relation to previous work and broader motivation.** Our study provides examples of filtered $\varphi$-modules arising from $p$-adic geometry, which is inspired by and generalizes the work of Volkov [Vol23]. In [Vol23] §2 and §3, Volkov constructed two filtered $\varphi$-modules and claimed that they are both non-semisimple, which is equivalent to saying that the associated monodromy groups are both non-reductive. We can show that these two filtered $\varphi$-modules are isomorphic to $D_{-1}^{\text{prod}}$ and $D_{(-\zeta_3^{-1} p, 1)}^{1, \mu}$ respectively. The neutral components of the monodromy groups of these two filtered $\varphi$-modules are $\mathbf{G}_m^2$ and $\mathbf{G}_a^2 \times_g \mathbf{G}_m^2$

<footnote n="1">¹The cocharacter associated to the Hodge decomposition [Win84, Theorem 3.1.2].</footnote>
respectively. The second group is non-reductive, reproving Proposition 3.1 (b) in [Vol23], while the first
is reductive, in contrast with Proposition 2.1 (b) in [Vol23]<fnref n="2" />.

Theorem B lies in the general framework of algebraic monodromy groups arising from geometry, defined
as the Zariski closure of the Galois action on étale cohomology groups. When the base field is a number
field, these groups are deeply studied. For instance, they satisfy certain $\ell$-independence properties
(see [Ser00, §5], [LP92]) and are large for low-dimensional abelian varieties with constraints on their
endomorphism rings (see [Ser97, IV 2.2], [Ser00, §5]).

The case where the base field is a mixed characteristic *p*-adic field and $\ell = p$ is more delicate. The central
case concerns the monodromy group of the *p*-adic Tate module of an abelian variety with good reduction
over *K*. Pink [Pin98, Proposition 2.9] proved that when the reduction is ordinary, this *p*-adic algebraic
monodromy group is always solvable. When the reduction is supersingular, Serre [Ser97] showed that
for elliptic curves, the *p*-adic monodromy group is either $\text{GL}_2$ or a non-split Cartan subgroup of $\text{GL}_2$.
Theorem B may be viewed as an extension of Serre's result.

Another motivation for studying the distribution of *p*-adic monodromy groups is the theory of Hecke
operators on Shimura varieties, since monodromy groups are stable under the Hecke action. For instance,
the determination of local monodromy groups of *F*-isocrystals was one of the main ingredients for the
resolution of Chai–Oort Hecke orbit conjecture in positive characteristic [DvH22].

Yu Fu recently formulated a *p*-adic nowhere density conjecture [Fu25, Conjecture 1.3] for Hecke orbits
of positive-dimensional subvarieties in Hodge type Shimura varieties. To provide evidence for this
conjecture, she proves the *p*-adic density of the "big monodromy" locus in the formal neighborhood
of certain points $x \in \text{Sh}(\overline{\mathbb{F}}_p)$ [Fu25, Theorem 1.5]. In our work, we study instead the variation of
monodromy groups on the coarse moduli space $\mathcal{M}^{\text{wa}}$ of admissible filtered $\varphi$-modules with isomorphic
$\varphi$-action. These two spaces are related by Grothendieck–Messing type period maps.

**Outline of the article.** In §2 we recall the necessary background on filtered $\varphi$-modules and the Tannakian formalism.

In §3 we establish the classification Theorem A.

In §4 we compute the associated monodromy groups and prove Theorem B.

In §5 we study the space $\mathcal{M}^{\text{wa}}$, and we prove Theorem C–E.

**Acknowledgement.** I am grateful to my advisor Marco d'Addezio, who suggested that I study *p*-adic
local monodromy groups coming from geometry. I also thank him for helpful discussions on this topic
and for the many constructive suggestions he gave on the draft of this article. I thank Maja Volkov, who
kindly answered my questions on filtered *ϕ*-modules arising from abelian surfaces in her work. I have
also greatly benefited from discussions with Yu Fu, who is likewise interested in *p*-adic monodromy
groups arising from deformations of supersingular abelian varieties. This work was supported in part by
the Agence Nationale de la Recherche (ANR) under the project ANR-25-CE40-7869-01 (pDefi), and by
China Scholarship Council (CSC) program (File No. 202306340018).

<footnote n="2">²In fact, we found a mistake in her proof of Proposition 2.1. Using the notation in this proof, she claimed that the nontrivial $\varphi$-stable subspaces of $D$ are precisely the $D_i$. However, since $\varphi^2 = -p$ in this case, for any nonzero vector $v \in D$, the 2-dimensional subspace of $D$ spanned by $v$ and $\varphi(v)$ is $\varphi$-stable. Therefore, $D_1$ actually has a complement subobject in $D$. Indeed, setting $z := y_1 + x_2$, the subspace of $D$ spanned by $z$ and $\varphi(z)$ is such a complement. A similar argument implies that every subobject of $D$ has a complement and $D$ is semisimple.</footnote>
## 2. PRELIMINARIES

Let $K$ be a finite extension of $\mathbb{Q}_p$. We denote by $\Gamma_K$ the absolute Galois group of $K$. Let $K_0$ be the maximal unramified subextension inside $K$, we have a Frobenius lift $\sigma \in \text{Gal}(K_0/\mathbb{Q}_p)$.

**2.1. Crystalline representations and filtered $\varphi$-modules.** For a proper smooth variety $X$ over $K$ and $n \ge 0$, the $p$-adic étale cohomology groups $H_{\bar{\ell}t}^n(X_{\bar{K}}, \mathbb{Q}_p)$ give rise to Galois representations of $\Gamma_K$. Unlike the $\ell$-adic case, these representations are typically highly ramified. Nevertheless, they are controlled by Fontaine's period ring formalism (see [BC09, §5]): if $X$ has good reduction, its $p$-adic cohomology groups are crystalline representations; in the case of semistable reduction, one obtains semistable representations; and in general one always obtains de Rham representations. For our purposes, we will restrict attention to crystalline representations, which form a subcategory $\text{Rep}_{\mathbb{Q}_p}^{\text{cris}}(\Gamma_K)$ of the category $\text{Rep}_{\mathbb{Q}_p}(\Gamma_K)$ of finite-dimensional continuous $\mathbb{Q}_p$-representations of $\Gamma_K$ [BC09, §5, §9].

A central aspect of classical $p$-adic Hodge theory is the passage from categories of $p$-adic representations of $\Gamma_K$ to corresponding categories of semilinear algebraic objects. As an important example, the category of crystalline representations corresponds to the category of filtered $\varphi$-modules defined as follows.

**Definition 2.1 ([Fon79]).** A filtered $\varphi$-module over $K$ is a triple $(D, \varphi, \text{Fil}_\bullet)$ where

(1) $D$ is a finite-dimensional vector space over $K_0$,

(2) $\varphi$ is a $\sigma$-semilinear automorphism of $D$,

(3) $\text{Fil}_\bullet$ is a decreasing exhaustive and separated filtration on $D \otimes_{K_0} K$.

A morphism between two filtered $\varphi$-modules over $K$ is a $K_0$-linear map which commutes with the $\varphi$-actions and respects the filtration after tensoring $K$. We denote by $\text{MF}_K$ the category of filtered $\varphi$-modules over $K$.

The category $\text{MF}_K$ is not abelian. However, we can define a full subcategory $\text{MF}_K^{\text{wa}}$ of $\text{MF}_K$ which is abelian by introducing some additional properties relating the $\varphi$-action and the filtration called admissibility [BC09, §8], which requires the existence of a $\mathbb{Z}_p$-lattice $M \subset D$ such that $\sum_{i \in \mathbb{Z}} \frac{1}{p^i} (\text{Fil}_i D \cap M) = M$. The objects in $\text{MF}_K^{\text{wa}}$ are called admissible filtered $\varphi$-modules and $M$ is called an adapted lattice.

Based on works of many, Fontaine and Colmez [CF00] proved an equivalence of category from $\text{MF}_K^{\text{wa}}$ to the category $\text{Rep}_{\mathbb{Q}_p}^{\text{cris}}(\Gamma_K)$ of crystalline $\mathbb{Q}_p$-representations of $\Gamma_K$:

$$ \text{MF}_K^{\text{wa}} \xleftarrow{V_{\text{cris}}} \text{Rep}_{\mathbb{Q}_p}^{\text{cris}}(\Gamma_K) . $$

Moreover, the natural forgetful functors $\text{MF}_K^{\text{wa}} \to \text{Vec}_K$ (resp. $\text{Rep}_{\mathbb{Q}_p}^{\text{cris}} \to \text{Vec}_{\mathbb{Q}_p}$) makes $\text{MF}_K^{\text{wa}}$ (resp. $\text{Rep}_{\mathbb{Q}_p}^{\text{cris}}$) into a Tannakian category (resp. a neutral Tannakian category) over $\mathbb{Q}_p$ in the sense of [DM12], and the functors $D_{\text{cris}}$, $V_{\text{cris}}$ are compatible with the $\otimes$-structures. For an object $D \in \text{MF}_K^{\text{wa}}$, let $V = V_{\text{cris}}(D)$ and $\langle D \rangle^{\otimes}$ (resp. $\langle V \rangle^{\otimes}$) be the Tannakian subcategory of $\text{MF}_K^{\text{wa}}$ (resp. of $\text{Rep}_{\mathbb{Q}_p}^{\text{cris}}$) generated by $D$ (resp. by $V$). By Tannakian formalism [DM12], the automorphism groups $\text{Aut}^{\otimes}(\omega|_D)$ and $\text{Aut}^{\otimes}(\eta|_V)$ are representable by algebraic subgroups $H_D$ of $\text{GL}_D$ and $H_V$ of $\text{GL}_V$ respectively, which are called the monodromy groups of $D$ and of $V$. Moreover, $H_D$ and $H_V$ are of inner forms of each other (see [DM12, Theorem 3.2]), which are in particular isomorphic over $\bar{\mathbb{Q}}_p$. On the other hand, the algebraic monodromy group of a crystalline representation $V$, defined as the Zariski closure of the
image of $\rho_V$ inside $\mathbf{GL}_V$, is isomorphic to $H_V$, which hence can be computed from the corresponding
filtered $\varphi$-module.

2.2. **Hodge decomposition for admissible filtered $\varphi$-modules.** Suppose throughout this subsection that the extension $K/\mathbb{Q}_p$ is unramified. Wintenberger [Win84] proved that any object $D \in \mathbf{MF}_K^{\text{wa}}$ enjoys a natural splitting of its filtration, called the *Hodge decomposition* as an analogue of the Hodge decomposition in complex geometry, which is a functorial decomposition $D = \bigoplus_{i \in \mathbb{Z}} D_i$ such that for all $i \in \mathbb{Z}$, $\text{Fil}_i D = \bigoplus_{i' \ge i} D_{i'}$. Via Tannakian formalism, there arises a *Hodge cocharacter* $\mu : \mathbf{G}_{m,K} \to \mathbf{GL}_D$ which factors through the monodromy group $H_D$ of $D$. Moreover, the splitting satisfies and is determined by the following conditions.

**Theorem 2.2 ([Win84, Introduction and Theorem 3.1.2])**. Let $D = (D, \varphi, \text{Fil}_\bullet) \in \mathbf{MF}_K^{\text{wa}}$ and $X_{\text{per}} = \{\xi : \mathbb{Z} \to \mathbb{Z} \text{ periodic}\}$. There exists a unique pair $((D_i)_{i \in \mathbb{Z}}, u)$ where $D_i$ are $K$-linear subspaces of $D$, $u \in \mathbf{GL}(D)$ such that

(a) $D = \bigoplus_{i \in \mathbb{Z}} D_i$ and for all $i \in \mathbb{Z}$, $\text{Fil}_i D = \bigoplus_{i' \ge i} D_{i'}$,

(b) $(u - \text{Id}_D) \bigoplus_{i' \le i} D_{i'} \subset \bigoplus_{i' \le i-1} D_{i'}$,

(c) $D = \bigoplus_{\xi \in X_{\text{per}}} D_{\xi}$, where $D_{\xi} = \{x \in D | (f^{\text{el}})^j(x) \in D_{\xi(j)}\}$ with $f^{\text{el}} = u^{-1} \circ \varphi \circ \mu(p^{-1})$,

(d) For any adapted lattice $M$ of $D$, let $M_{\xi} = M \cap D_{\xi}$, $k$ be the residue field of $K$, $\bar{M} = M/pM$ and $\bar{M}_{\xi} = M_{\xi}/pM_{\xi}$.

(i) We have $M = \bigoplus_{\xi \in X} M_{\xi}$, $f^{\text{el}}(M) = M$ and $u(M) = M$.

(ii) Let $\bar{f}^{\text{el}}$ (resp. $\bar{u}$) be the residue of $f^{\text{el}}$ (resp. of $u$) on $\bar{M}$. Then, there exists a descending chain $0 = L_0 \subset \dots \subset L_r = \bar{M}$ of $\bar{f}^{\text{el}}$-stable $k$-linear subspaces such that for $1 \le r' \le r$, $L_{r'} = \bigoplus_{\xi \in X_{\text{per}}} (L_{r'} \cap \bar{M}_{\xi})$ and $(\bar{u} - \text{Id}_{\bar{M}}) L_{r'} \subset L_{r'-1}$.

The periodic functions $\xi \in X_{\text{per}}$ that appear in the decomposition $D = \bigoplus_{\xi \in X_{\text{per}}} D_{\xi}$ is clearly an
invariant of $D$. For simplicity, we take the following definition.

**Definition 2.3.** Let $D = (D, \varphi, \text{Fil}_\bullet) \in \mathbf{MF}_K^{\text{wa}}$. The set $X_D := \{\xi \in X_{\text{per}} | D_\xi \neq 0\}$ is called the Wintenberger type of $D$.

2.3. **Filtered $\varphi$-modules arising from geometry.** For a proper and smooth variety $X$ over $K$ with good reduction, an important invariant is the characteristic polynomial $h_{X,n}(T) \in \mathbb{Q}_\ell[T]$ of a mod-$p$ Frobenius lift acting on $H_{\ell t}^n(X_{\bar{K}}, \mathbb{Q}_\ell)$, which is independent of $\ell \neq p$. By Weil conjectures, this polynomial is defined over $\mathbb{Z}$ and indeed is a $q^n$-Weil polynomial where $q$ is the cardinality of the residue field of $K$.

**Definition 2.4.** Let $q$ be a power of $p$. A $q$-Weil number (resp. supersingular $q$-Weil number) is an algebraic integer $\alpha$ such that all of its embeddings in $\mathbb{C}$ admit absolute value $\sqrt{q}$ (resp. $\alpha$ is a product of $\sqrt{q}$ and a complex root of unity). A monic polynomial in $\mathbb{Z}[T]$ is called a $q$-Weil polynomial (resp. supersingular $q$-Weil polynomial) if all its roots in $\mathbb{Q}$ are $q$-Weil numbers (resp. supersingular $q$-Weil numbers) and its valuation at $T^2-q$ is even. Let $\mathcal{P}_{\text{Weil},q}$ (resp. $\mathcal{P}^{\text{ss}}_{\text{Weil},q}$) denote the set of all $q$-Weil polynomials (resp. supersingular $q$-Weil polynomials).

As a consequence of the Riemann Hypothesis, $h_{X,n}$ can also be recovered from $D_{\text{cris}}(H_{\ell t}^n(X_{\bar{K}}, \mathbb{Q}_p))$, the
filtered $\varphi$-module associated with the $p$-adic cohomology. If we denote by $\varphi$ the crystalline Frobenius of
this filtered $\varphi$-module, the characteristic polynomial of $\varphi^{[K_0:\mathbb{Q}_p]}$ is exactly $h_{X,n}(T)$.
Conversely, when $K = \mathbb{Q}_p$, Volkov [Vol05] gives a series of conditions which classify the admissible filtered $\varphi$-modules arising from abelian varieties with good reduction over $\mathbb{Q}_p$.

**Theorem 2.5** ([Vol05, Corollary 5.9]). Let $D \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$. There is an abelian variety $A/\mathbb{Q}_p$ with good reduction such that $D = D_{\text{cris}}(V_p(A))$ if and only if $D$ satisfies the following conditions.

(1) the Hodge numbers satisfy $h^0(D) = h^1(D) = \frac{1}{2} \dim D$,

(2) the $\varphi$-action on $D$ is semisimple and $\chi_{\varphi}$ is a $p$-Weil polynomial,

(3) there exists a nondegenerate skew form on $D$ under which $\varphi$ is a $p$-similitude and $\text{Fil}_1 D$ is totally isotropic.

**2.4. Tools for computing monodromy groups.** For $K/\mathbb{Q}_p$ unramified, any object $D \in \text{MF}_K^{\text{wa}}$ is determined by its crystalline Frobenius $\varphi$ together with the Hodge decomposition. By Tannakian theory [DM12, Proposition 2.21 (a)], this could be illustrated by the following.

**Proposition 2.6** ([Pin98, Proposition 2.5]). Let $D = (D, \varphi, \text{Fil}_\bullet) \in \text{MF}_K^{\text{wa}}$ and $\mu$ be the Hodge cocharacter. For $i \in \mathbb{Z}$, let $\mu_i : \text{G}_{m,K} \to H_D$ be the unique cocharacter such that for all $t \in K^\times$, $\mu_i(\sigma^i(t)) = \varphi^i \circ \mu(t) \circ \varphi^{-i}$. The subgroup of $H_D$ generated by $\varphi^m$ and the image of $\mu_i$ for all $i \in \mathbb{Z}$ is Zariski dense in $H_D$.

The following lemma allows us to reformulate Proposition 2.6 in terms of generation of Lie algebras.

**Lemma 2.7** ([Mil17, Corollary 10.17]). Let $H_1, \dots, H_n$ be smooth algebraic subgroups of a connected algebraic group $G$. If the Lie algebras of the subgroups $H_i$ generate the Lie algebra of $G$, then the subgroups $H_i$ generate $G$ as an algebraic group.

The following technical lemma will be used repeatedly in the later computation.

**Lemma 2.8.** Let $K$ be a field with $\text{char } K \neq 2$ and $I_2 \in \text{GL}_2(K)$ be the identity matrix.

(i) For $A, B, C, D \in \text{Mat}_{2 \times 2}(K)$, the Lie subalgebra of $\text{gl}_4(K)$ generated by the block matrices
$$ \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} A & B \\ C & D \end{pmatrix} \text{ contains} $$

$$ \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}, \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \text{ and } \begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}. $$

(ii) For $\alpha_{i,j}, \beta_{i,j} \in K$, the Lie subalgebra of $\text{gl}_4(K)$ generated by

$$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2} & 0 & 0 \\ \alpha_{2,1} & \alpha_{2,2} & 0 & 0 \\ 0 & 0 & \beta_{1,1} & \beta_{1,2} \\ 0 & 0 & \beta_{2,1} & \beta_{2,2} \end{pmatrix} $$

contains

$$ \begin{pmatrix} \alpha_{1,1} & 0 & 0 & 0 \\ 0 & \alpha_{2,2} & 0 & 0 \\ 0 & 0 & \beta_{1,1} & 0 \\ 0 & 0 & 0 & \beta_{2,2} \end{pmatrix}, \begin{pmatrix} 0 & \alpha_{1,2} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \beta_{1,2} \\ 0 & 0 & 0 & 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ \alpha_{2,1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \beta_{2,1} & 0 \end{pmatrix}. $$

*Proof.* By the definition of $\text{gl}_4(K)$, we have

$$ [(\begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} A & B \\ C & D \end{pmatrix})] = (\begin{pmatrix} 0 & B \\ -C & 0 \end{pmatrix}), [(\begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & B \\ -C & 0 \end{pmatrix})] = (\begin{pmatrix} 0 & B \\ C & 0 \end{pmatrix}). $$

It's clear that the $K$-linear subspace of $\text{gl}_4(K)$ spanned by these matrices and $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$ contains

$$ \begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix}, \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \text{ and } \begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}. $$

The assertion (i) follows. The assertion (ii) is a direct corollary of (i). $\square$
### 3. FILTERED $\varphi$-MODULES ARISING FROM SUPERSINGULAR ABELIAN SURFACES OVER $\mathbb{Q}_p$

For any abelian variety $A$ over $\mathbb{Q}_p$ with good reduction, the filtered $\varphi$-module $D_{\text{cris}}(V_p(A)) \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ associated with $A$ carries rich information. The aim of this section is to classify for $p \ge 7$ all the filtered $\varphi$-modules associated with supersingular abelian surfaces over $\mathbb{Q}_p$, which form a full subcategory $\mathbf{C}$ of $\text{MF}_{\mathbb{Q}_p}^{\text{wa}}$.

**3.1. Supersingular $p$-Weil polynomials of degree 4.** For $D = (D, \varphi, \text{Fil}_\bullet) \in \mathbf{C}$, the crystalline Frobenius $\varphi$ acts linearly on $D$, with characteristic polynomial $\chi_\varphi \in \mathscr{P}_{\text{Weil},p}^{\text{ss}}$. The next proposition proves the rigidity of the forms of supersingular $p$-Weil polynomials of degree 4.

**Proposition 3.1.** Let $p \ge 7$ be a prime number, then all of the supersingular $p$-Weil polynomials of degree 4 are

$$ (X^2 \pm p)^2 \quad \text{and} \quad X^4 + \epsilon p X^2 + p^2 \quad \text{where } \epsilon \in \{0, \pm 1\}. \qquad (3.1) $$

*Proof.* It suffices to classify supersingular $p$-Weil numbers of degree$^3$ smaller than 4. Let $p \ge 7$ be a prime number, $m$ a positive integer and $\zeta_m$ a fixed primitive $m$-th root of unity. Suppose that $[\mathbb{Q}(\zeta_m \sqrt{p}) : \mathbb{Q}] \le 4$. We have immediately

$$ [\mathbb{Q}(\zeta_m) : \mathbb{Q}] = \frac{[\mathbb{Q}(\zeta_m, \sqrt{p}) : \mathbb{Q}]}{[\mathbb{Q}(\zeta_m, \sqrt{p}) : \mathbb{Q}(\zeta_m)]} = \frac{[\mathbb{Q}(\zeta_m, \sqrt{p}) : \mathbb{Q}(\zeta_m \sqrt{p})][\mathbb{Q}(\zeta_m \sqrt{p}) : \mathbb{Q}]}{[\mathbb{Q}(\zeta_m, \sqrt{p}) : \mathbb{Q}(\zeta_m)]} $$

Since $\mathbb{Q}(\zeta_m, \sqrt{p}) = \mathbb{Q}(\zeta_m \sqrt{p}, \sqrt{p})$, the numerator of the right hand side is bounded by 8. Note that the left hand side is equal to the Euler function $\varphi(m)$.

Suppose that $\varphi(m) = 8$ and hence $m \in \{15, 16, 20, 24, 30\}$. In this case, any $p \ge 7$ is unramified in $\mathbb{Q}(\zeta_m)$. Therefore, the denominator of the right hand side $[\mathbb{Q}(\zeta_m, \sqrt{p}) : \mathbb{Q}(\zeta_m)]$ equals 2 and the left hand side is less than or equal to 4, contradiction.

It follows that $\varphi(m) \le 4$ and hence $m \in \{1, 2, 3, 4, 5, 6, 8, 10, 12\}$. Note that $\zeta_5 \sqrt{p}$ and $\zeta_{10} \sqrt{p}$ both have degree 8. For each of the rest $m$, we fix a $m$-th root of unity $\zeta_m$. It follows that the conjugacy classes of these $\zeta_m \sqrt{p}$ are

$$ \{\pm \sqrt{p}\}, \{\pm \zeta_4 \sqrt{p}\}, \{\pm \zeta_3 \sqrt{p}, \pm \bar{\zeta}_3 \sqrt{p}\}, \{\pm \zeta_8 \sqrt{p}, \pm \bar{\zeta}_8 \sqrt{p}\}, \{\pm \zeta_{12} \sqrt{p}, \pm \bar{\zeta}_{12} \sqrt{p}\}, $$

which are corresponding to the polynomials in (3.1). $\square$

<figure><img src="image_8.png" /></figure>

**3.2. Linear-algebraic constraints.** The classification of filtered $\varphi$-modules arising from supersingular abelian surfaces over $\mathbb{Q}_p$ is reduced to the analysis of the linear-algebraic conditions, as stated in the following lemma, which is a direct corollary of Theorem 2.5.

**Lemma 3.2.** An admissible filtered $\varphi$-module $D \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ belongs to $\mathbf{C}$ if and only if $D$ satisfies the following conditions.

S1 dim $D = 4$ and the Hodge numbers satisfy $h^0(D) = h^1(D) = 2$,

S2 the $\varphi$-action on $D$ is semisimple and $\chi_\varphi$ is a supersingular $p$-Weil polynomial,

S3 there exists a nondegenerate skew form on $D$ under which $\varphi$ is a $p$-similitude and $\text{Fil}_1 D$ is totally isotropic.

<footnote>$^3$The degree of an algebraic number $\alpha \in \bar{\mathbb{Q}}$ is the number $[\mathbb{Q}(\alpha) : \mathbb{Q}] \in \mathbb{Z}$.</footnote>
**Lemma 3.3.** Let $D$ be any filtered $\varphi$-module over $\mathbb{Q}_p$ (not necessarily admissible) satisfies **S1** and **S2**. Then, $D$ is admissible if and only if $\mathrm{Fil}_1 D$ is not stable under $\varphi$.

*Proof.* Let $D' \subset D$ be a proper sub-$\varphi$-module of $D$. We have $\dim D' = 2$ since it corresponds to a certain $\mathbb{Q}_p$-factor of a supersingular $p$-Weil polynomial. Each eigenvalue of $\varphi$ in $\bar{\mathbb{Q}}_p$ has valuation $\frac{1}{2}$, so we have $t_N(D') = 1$. On the other hand,

$$t_H(D') = \sum_{i \in \mathbb{Z}} i \dim_{\mathbb{Q}_p} (D' \cap \mathrm{Fil}_i D / D' \cap \mathrm{Fil}_{i+1} D) \stackrel{\mathrm{S1}}{=} \dim_{\mathbb{Q}_p} (D' \cap \mathrm{Fil}_1 D).$$

Therefore, $t_H(D') \le t_N(D')$ if and only if $\mathrm{Fil}_1 D \ne D'$. $D$ is admissible if and only if this holds for any such $D'$, that is, $\mathrm{Fil}_1 D$ itself is not $\varphi$-stable. $\square$

The Hodge decomposition of $D \in \mathbb{C}$ is rigid. Theorem 2.2 implies that there exists a decomposition of $\mathbb{Q}_p$-linear subspaces $D = D_1 \oplus D_0$ and a $u \in \mathrm{GL}(D)$ such that

(a) $D_1 = \mathrm{Fil}_1 D$,

(b) $(u - \mathrm{Id}_D)D_0 = 0$ and $(u - \mathrm{Id}_D)D_1 \subset D_0$,

(c) $D = \bigoplus_{\xi \in X} D_\xi$ where $D_\xi = \{x \in D | (f^{el})^j(x) \in D_{\xi(j)}\}$ with $f^{el} = u^{-1} \circ \varphi \circ \mu(p^{-1})$.

Moreover, the data $D_0$ and $u$ are unique if they satisfy the condition (d) in Theorem 2.2. We focus on the decomposition $D = \bigoplus_{\xi \in X_D} D_\xi$ first.

**Notation.** For $r \in \mathbb{Z}_{\ge 1}$ and $a_0, \dots, a_{r-1} \in \mathbb{Z}$, the symbol $[a_0 a_1 \cdots a_{r-1}]$ stands for a function $\xi \in X_{per}$ of period dividing $r$ such that $\xi(k) = a_k$ for $0 \le k \le r-1$.

**Lemma 3.4.** Suppose that $D \in \mathrm{MF}_{\mathbb{Q}_p}^{wa}$ satisfies the equivalent **S1** and **S2**. The Wintenberger type $X_D := \{\xi \in X_{per} | D_\xi \ne 0\}$ of $D$ is one of the followings.

(A). $\{[0011], [0110], [1100], [1001]\}$,

(B). $\{[01], [10]\}$,

(C). $\{[0], [1]\}$,

(D). $\{[001], [010], [100], [1]\}$.

*Proof.* For $\xi \in X_{per}$ and $k \in \mathbb{Z}$, denote by $P(\xi)$ the period of $\xi$ and $\xi^k(j) = \xi(j + k)$. It follows that $\dim D_\xi = \dim D_{\xi^k}$ for any $k \in \mathbb{Z}$. Moreover, by definition for $i = 1, 2$ we have $D_i = \bigoplus_{\xi \in X_D} D_\xi$ and

$$\# \{\xi \in X_D | \xi(0) = i\} = \dim D_i = 2.$$

Note that $\dim D_{[0]} \ne 1$. Otherwise for any $0 \ne x \in D_{[0]} \subset D_0$, we have $f^{el}(x)$ proportional to $x$, and

$$\varphi(x) = u(f^{el}(\mu(p)(x))) = u(\underbrace{f^{el}(px)}_{\in D_0}) = p f^{el}(x).$$

It follows that $x$ is an eigenvector of $\varphi$, contradictory to the fact that $\varphi$ has no root over $\mathbb{Q}_p$. We argue by cases, depending on $P_{\max}(D) := \max_{\xi \in X_D} P(\xi) \le 4$.

* If $P_{\max}(D) = 4$, clearly $X_D = (A)$.

* If $P_{\max}(D) = 3$, there exists $\xi \in X_D$ such that $P(\xi) = \dim D_\xi = 1$. It follows that $\xi = [1]$ and $X_D = (D)$.
* If $P_{\max}(D) = 2$, note that the only period functions with value in $\{0, 1\}$ are $\xi = [01]$ and $\xi^1 = [10]$. If $\dim D_{\xi} = \dim D_{\xi^1} = 1$, then $[0] \in X_D$ and $D_{[0]} = 1$ for dimensional reason, contradiction. Therefore, $\dim D_{\xi} = \dim D_{\xi^1} = 2$ and $X_D = (B)$.

* If $P_{\max}(D) = 1$, it's clear that $X_D = (C)$.

□

### 3.3. The case $\chi_{\varphi}(X) = (X^2 \pm p)^2$.

**Definition 3.5.** For $\epsilon' \in \{\pm 1\}$, we denote by $D_{\epsilon'}^{\text{prod}} \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ the filtered $\varphi$-modules satisfying **S1** determined by the following data

$$ (\text{prod}) \ D_{\epsilon'}^{\text{prod}} = (\mathbb{Q}_p^4, \varphi_{\epsilon'}^{\text{prod}}, \text{Fil}_{\bullet}), \ \text{Mat}_{\mathcal{B}}(\varphi_{\epsilon'}^{\text{prod}}) = \begin{pmatrix} 0 & 0 & \epsilon'p & 0 \\ 0 & 0 & 0 & \epsilon'p \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}, \ \text{Fil}_1 D_{\epsilon'}^{\text{prod}} = \text{span}(e_1, e_2). $$

**Remark 3.6.** The filtered $\varphi$-modules $D_{\epsilon'}^{\text{prod}}$ are products of two isomorphic subobjects of dimension 2.

**Theorem 3.7.** For $\epsilon' \in \{\pm 1\}$, $D_{\epsilon'}^{\text{prod}}$ represents the unique isomorphism class of admissible filtered $\varphi$-modules over $\mathbb{Q}_p$ satisfying **S1**, **S2** with $\chi_{\varphi}(X) = (X^2 - \epsilon'p)^2$. In this case, we have $X_{D_{\epsilon'}^{\text{prod}}} = (B)$ in Lemma 3.4.

*Proof.* Let $D$ be an admissible filtered $\varphi$-modules over $\mathbb{Q}_p$ satisfying **S1**, **S2** with $\chi_{\varphi}(X) = (X^2 - \epsilon'p)^2$. We claim that $\text{Fil}_1 D \cap \varphi(\text{Fil}_1 D) = 0$. By contradiction, suppose that for some nonzero $x \in \text{Fil}_1 D$ we have $\varphi(x) \in \text{Fil}_1 D$. Then $\varphi(x)$ is not proportional to $x$, since otherwise $x$ is an eigenvector of $\varphi$ on $D$, contradictory to the fact that $\chi_{\varphi}(X) = (X^2 - \epsilon'p)^2$ has no root over $\mathbb{Q}_p$. Therefore, $\text{Fil}_1 D$ is spanned by $x$ and $\varphi(x)$. Using that $\varphi^2(x) = \mp x$, we have immediately $\tilde{D} := \text{Fil}_1 D$ is a $\mathbb{Q}_p[\varphi]$-submodule of $D$. On the other hand, we have $t_N(\tilde{D}) = 1$ by the supersingularity of $\chi_{\varphi}(X)$. Therefore, we have

$$ t_H(\tilde{D}) = \sum_{i \in \mathbb{Z}} i \dim_{\mathbb{Q}_p}(\text{Fil}_i D \cap \tilde{D}) = 2 > 1 = t_N(\tilde{D}), $$

contradictory to the admissibility of $D$. Let $(x, y)$ be a basis of $\text{Fil}_1 D$. Since $\text{Fil}_1 D \cap \varphi(\text{Fil}_1 D) = 0$, the $\mathbb{Q}_p$-linear subspaces

$$ D' := \mathbb{Q}_p x \oplus \mathbb{Q}_p \varphi(x) \quad \text{and} \quad D'' := \mathbb{Q}_p y \oplus \mathbb{Q}_p \varphi(y) $$

define two isomorphic subobjects of $D \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ and $D = D' \oplus D''$. Since the Hodge decomposition is given functorially, it suffices to study the filtered $\varphi$-module $D'$, which lives in the unique isomorphism class of filtered $\varphi$-modules over $\mathbb{Q}_p$ with $\chi_{\varphi}(X) = X^2 - \epsilon'p$. Let $D' = D'_1 \oplus D'_0$ be the Hodge decomposition of $D'$ and $u'$ be the automorphism as in Theorem 2.2. As in the proof of Lemma 3.4, we have $\dim D'_{[0]} \neq 1$. It follows that $X_{D'} = (B)$, $D'_1 = D'_{[10]}$ and $D'_0 = D'_{[01]}$. Therefore, we have $X_D = (B)$ as well and $D$ is isomorphic to the direct sum of 2 copies of $D'$, and one can check that it is exactly $D_{\epsilon'}^{\text{prod}}$. □

### 3.4. The case $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$
Fix a parameter $\epsilon \in \{0, \pm 1\}$. In contrast to the case $\chi_{\varphi}(X) = (X^2 \pm p)^2$, there are infinitely many isomorphism classes of admissible filtered $\varphi$-modules with $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$. We can construct concretely families of such filtered $\varphi$-modules.

**Definition 3.8.** For $\epsilon' \in \{\pm 1\}$ and $a, b, a' \in \mathbb{Q}_p$ such that $ab \neq -1$, we denote by

$$ D_{\epsilon', \text{iso}}^{\epsilon, \nu}, \ D_{a', \epsilon, \nu}^{\epsilon, \mu}, \ D_{(a,b)}^{\epsilon, \mu} \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}} $$
the filtered $\varphi$-modules satisfying **S1** determined by the following data

$$ (iso) \quad D_{\epsilon'}^{\epsilon, \text{iso}} = (\mathbb{Q}_p^4, \varphi_{\epsilon'}^{\epsilon, \text{iso}}, \text{Fil}_{\bullet}), \quad \text{Mat}_{\mathcal{B}}(\varphi_{\epsilon'}^{\epsilon, \text{iso}}) = \begin{pmatrix} 0 & 0 & \epsilon'p & 0 \\ 0 & 0 & 0 & \epsilon'p \\ 1 & 0 & 0 & -(\epsilon+2\epsilon')p \\ 0 & 1 & 1 & 0 \end{pmatrix}, \quad \text{Fil}_1 D_{\epsilon'}^{\epsilon, \text{iso}} = \text{span}(e_1, e_2), $$

$$ (\nu) \quad D_{a'}^{\epsilon, \nu} = (\mathbb{Q}_p^4, \varphi_{a'}^{\epsilon, \nu}, \text{Fil}_{\bullet}), \quad \text{Mat}_{\mathcal{B}}(\varphi_{a'}^{\epsilon, \nu}) = \begin{pmatrix} 0 & 0 & 0 & 1 \\ -p^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & -a' \\ a' - \epsilon p & 0 & 1 & 0 \end{pmatrix}, \quad \text{Fil}_1 D_{a'}^{\epsilon, \nu} = \text{span}(e_1, e_2), $$

$$ (\mu) \quad D_{(a,b)}^{\epsilon, \mu} = (\mathbb{Q}_p^4, \varphi_{(a,b)}^{\epsilon, \mu}, \text{Fil}_{\bullet}), \quad \text{Mat}_{\mathcal{B}}(\varphi_{(a,b)}^{\epsilon, \mu}) = \begin{pmatrix} 0 & 0 & 0 & -p^2 \\ 0 & 0 & 1 & \frac{a^2+\epsilon p+b^2 p^2}{ab+1} \\ 1 & 0 & a & \frac{a^3+\epsilon a p-b p^2}{ab+1} \\ 0 & 1 & b & -a \end{pmatrix}, \quad \text{Fil}_1 D_{(a,b)}^{\epsilon, \mu} = \text{span}(e_1, e_2), $$

where $\mathcal{B} = (e_i)_{1\le i\le 4}$ is the canonical basis of $\mathbb{Q}_p^4$.

**Remark 3.9.** The filtered $\varphi$-modules $D_{\epsilon'}^{\epsilon, \text{iso}}$ should be thought of as "isolated" objects with respect to the natural families under consideration (see Proposition 5.5).

**Theorem 3.10.** Let $p \ge 7$ and $\epsilon \in \{0, \pm 1\}$. Let $D = (D, \varphi, \text{Fil}_{\bullet}) \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ be an admissible filtered $\varphi$-module with $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$, the following are equivalent.

* $D \in \mathbb{C}$;

* $D$ satisfies **S1**;

* $D$ is isomorphic to $D_{\epsilon'}^{\epsilon, \text{iso}}$, $D_{a'}^{\epsilon, \nu}$ or $D_{(a,b)}^{\epsilon, \mu}$ for some parameters $\epsilon'$, $a, b, a'$ satisfy the following conditions:

<table><thead><tr><th>&epsilon;'</th><th>&epsilon;' &isin; {&plusmn;1}</th></tr></thead><tbody><tr><td>a'</td><td>a' &isin; &epsilon;p + p<sup>2</sup>&#8477;<sub>p</sub></td></tr><tr><td>a, b</td><td>v<sub>p</sub>(a)&ge;1, v<sub>p</sub>(b)&ge;0 or<br>v<sub>p</sub>(a)=v<sub>p</sub>(b)+1</td></tr></tbody></table>

Moreover, for each $D$ above, the Hodge decomposition is $D = D_1 \oplus D_2$ where $D_1 = \text{span}(e_1, e_2)$ and $D_2 = \text{span}(e_3, e_4)$

**3.5. Proof of Theorem 3.10.** The rest of the section is devoted to proving Theorem 3.10. It suffices to classify admissible filtered $\varphi$-modules over $\mathbb{Q}_p$ satisfying **S1** with $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$. For simplicity, we adopt the following definition.

**Definition 3.11.** For $D \in \text{MF}_{\mathbb{Q}_p}^{\text{wa}}$ satisfying **S1**, a $\mathbb{Q}_p$-basis $(e_1, e_2, e_3, e_4)$ of $D$ is called *adapted* if $D_1 = \text{span}(e_1, e_2)$ and $D_0 = \text{span}(e_3, e_4)$, where $D = D_0 \oplus D_1$ is the Hodge decomposition.

Using this terminology, for the filtered $\varphi$-module structures defined on $\mathbb{Q}_p^4$ listed in Theorem 3.10, we will see that the canonical basis of $\mathbb{Q}_p^4$ is always adapted.

**Lemma 3.12.** Let $D$ be an admissible filtered $\varphi$-modules over $\mathbb{Q}_p$ satisfying Condition **S1** with $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$ for $\epsilon \in \{0, \pm 1\}$, then $X_D \neq (C)$ or $(D)$ in Lemma 3.4.

*Proof.* By contradiction, suppose $X_D = (C)$. It follows that $D_0 = D_{[0]}$ so $u\varphi(D_0) = D_0$. Take an adapted basis $\mathcal{B}$ of $D$. Under this basis, the matrix of $u$, $f^{el}$ and $\mu(p)$ have the following forms, where $I_2 = (\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix})$, $S, M_1, M_2 \in \text{Mat}_{2 \times 2}(\mathbb{Q}_p)$.

$$ \text{Mat}_{\mathcal{B}}(u) = \begin{pmatrix} I_2 & 0 \\ S & I_2 \end{pmatrix}, \quad \text{Mat}_{\mathcal{B}}(f^{el}) = \begin{pmatrix} M_1 & 0 \\ 0 & M_0 \end{pmatrix} \quad \text{and} \quad \text{Mat}_{\mathcal{B}}(\mu(p)) = \begin{pmatrix} pI_2 & 0 \\ 0 & I_2 \end{pmatrix}. $$

In this case, we have $\chi_{f^{el}} = \chi_{\varphi}$ and hence $f^{el}$ has no eigenvector in $D$. It follows that for any nonzero $x \in D_1$, if $\varphi(x) = y+z$ where $y \in D_1$ and $z \in D_0$, then $y$ is not proportional to $x$. For the same reason,
$\varphi(z) \in D_0$ is not proportional to $z$. So $\mathcal{B} := (x, y, z, \varphi(z))$ is an adapted basis of $D$, under which the matrix of $\varphi$ has the form

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & a & 0 & 0 \\ 1 & b & 0 & 0 \\ 1 & c & 0 & e \\ 0 & d & 1 & f \end{pmatrix} $$

with $a, b, c, d, e, f \in \mathbb{Q}_p$. A direct calculation shows

$$ \chi_{\varphi}(X) = X^4 - (b+f)X^3 + (bf-a-e)X^2 + (af+be)X + ae = X^4 + \epsilon p X^2 + p^2. $$

Comparing the coefficients, we have

$$ b+f=0, \quad bf-a-e=\epsilon p, \quad af+be=0 \quad \text{and} \quad ae=p^2. $$

If $b \neq 0$, we have $a = e = \pm p$ and $b^2 = -(a + e + \epsilon p) \in \{\pm p, \pm 2p, \pm 3p\}$, which is impossible for $p \ge 5$. Therefore, we have $b = f = 0$ and $a + e = -\epsilon p$. In other words, $\{a, e\}$ is the set of roots of $X^2 + \epsilon pX + p^2$. In particular, we have $v_p(a) = 1$. Let $M$ be an adapted lattice of $D$. By the definition of adapted lattices and Theorem 2.2 (d), we have $\mathrm{pr}_1 \circ \varphi_p(M \cap D_1) \subset D_1$. However, $\det(\mathrm{pr}_1 \circ \varphi_p|_{D_1}) = -\frac{a}{p^2} \notin \mathbb{Z}_p$, contradiction. $X_D \neq (C)$.

Suppose that $X_D = (D)$. Take $x, y \in D$ such that $D_{[1]} = \mathbb{Q}_p x$ and $D_{[100]} = \mathbb{Q}_p y$. One can check that $\mathcal{B} = (x, y, f^{el}(y), (f^{el})^2(y))$ is an adapted basis of $D$. Under this basis, the matrix of $u, f^{el}$ and $\mu(p)$ have the following forms, where $t \in \mathbb{Q}_p^\times$ and $a, b, c, d \in \mathbb{Q}_p$.

$$ \mathrm{Mat}_{\mathcal{B}}(u) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & b & 1 & 0 \\ c & d & 0 & 1 \end{pmatrix}, \quad \mathrm{Mat}_{\mathcal{B}}(f^{el}) = \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{t} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mathrm{Mat}_{\mathcal{B}}(\mu(p)) = \begin{pmatrix} p & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

We deduce from $\varphi = u \circ f^{el} \circ \mu(p)$ that

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} pt & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{t} \\ apt & p & 0 & \frac{b}{t} \\ cpt & 0 & 1 & \frac{d}{t} \end{pmatrix} $$

and by taking the characteristic polynomial, we have

$$ \chi_{\varphi}(X) = X^4 - \left(\frac{d}{t} + pt\right)X^3 + \left(pd - \frac{b}{t}\right)X^2 + \left(pb - \frac{p}{t}\right)X + p^2 = X^4 + \epsilon p X^2 + p^2. $$

Comparing the coefficients, we have $d = -pt^2$, $b = \frac{1}{t}$ and $pd - \frac{b}{t} = \epsilon p$. It follows that $p^2t^2 + \frac{1}{t^2} + \epsilon p = 0$ and hence $pt^2$ is a root of $X^2 + \epsilon pX + 1$. Note that for $\epsilon \in \{0, \pm 1\}$, roots of $X^2 + \epsilon pX + 1$ are roots of unity, whereas $2 \nmid v_p(pt^2)$, contradiction. $\square$

The following lemma is rather technical and will be used in the subsequent proofs.

**Lemma 3.13.** Let $a, b \in \mathbb{Q}_p$ with $ab \neq -1$. Then, the followings are equivalent.

(1) $c = -\frac{a^2+\epsilon p+b^2p^2}{ab+1} \in p\mathbb{Z}_p$.

(2) $v_p(a) \ge 1$ and $v_p(b) \ge 0$, or $v_p(a) = v_p(b) + 1$.

In both cases, there exists a $\mathbb{Z}_p$-lattice $N$ in $\mathbb{Q}_p^2$ satisfying the following conditions.

(i) $\begin{pmatrix} 0 & -1 \\ 1 & \frac{c}{p} \end{pmatrix} N = N,$

(ii) $\begin{pmatrix} -pb & a \\ a+bc & pb \end{pmatrix} N \subset N.$

*Proof.* We divided $\mathbb{Z}^2$ into four regions $D_1, D_2, D_3$ and $L$ as following:

$$ D_1 = \{x \le y, x \le 0\}, \quad D_2 = \{x \ge y + 2, y \le -1\}, \quad D_3 = \{x \ge 1, y \ge 0\}, \quad L = \{x - y = 1, x \le 0\}. $$
This is shown in the following figure.

<figure><img src="image_9.png" /></figure>

Then, (2) is equivalent to $(v_p(a), v_p(b)) \in L \sqcup D_3$. We verify that (1) implies (2) by contradiction. If $(v_p(a), v_p(b)) \in D_1$, we have $v_p(c) \le v_p(a^2) - \min\{v_p(ab), 1\} \le 0$. Similarly, if $(v_p(a), v_p(b)) \in D_2$, we have $v_p(c) \le v_p(b^2p^2) - \min\{v_p(ab), 1\} \le 0$.

Conversely, if $(v_p(a), v_p(b)) \in L$, we have $v_p(c) \ge v_p(a^2) - v_p(ab) = 1$. If $(v_p(a), v_p(b)) \in D_3$, we have $v_p(c) = \min\{v_p(a^2), v_p(b^2p^2), 1\} \ge 1$.

It remains to prove the existence of $N$. Suppose that $(v_p(a), v_p(b)) \in L$. Let $(e_1, e_2)$ be the canonical basis of $\mathbb{Q}_p^2$. Take $u = \frac{bp}{a} \in \mathbb{Z}_p^\times$ and $n = -v_p(a) \ge 1$. Consider $N = \mathbb{Z}_p e_1 \oplus \mathbb{Z}_p \frac{e_1+ue_2}{p^n}$, we claim that conditions (i), (ii) hold. In fact, under the basis $(e_1, \frac{1}{p^n}(e_1 + ue_2))$, conditions (i) and (ii) becomes

$$ \left( \begin{array}{c} -u^{-1} - \frac{1}{p^n}(u^{-1}+u+\frac{c}{p}) \\ \frac{p^n}{u} - u^{-1}(1+\frac{cu}{p}) \end{array} \right) \quad \text{and} \quad \left( \begin{array}{c} -\frac{a+bc}{u} - bp \quad \frac{au-(a+bc)u^{-1}-2bp}{p^n} \\ \frac{p^n(a+bc)}{u} + bp \end{array} \right) \in \mathbf{GL}_2(\mathbb{Z}_p). $$

Note that

$$ \frac{c}{p} = - \frac{a^2 + \epsilon p + b^2 p^2}{p(ab + 1)} = - \frac{a^2(1 + u^2) + \epsilon p}{a^2 u + p}. $$

We deduce that $v_p(u^{-1} + u + \frac{c}{p}) = v_p(-\frac{c+\epsilon p}{a^2 u}) \ge 2n$ and thereby $-\frac{1}{p^n}(u^{-1} + u + \frac{c}{p}) \in \mathbb{Z}_p$. On the other hand, we can verify the equality $a + bc = -\frac{c+\epsilon p+p^2b^2}{a}$. Using this equality repeatedly, we have immediately

$$ v_p \left( \frac{p^n(a + bc)}{u} \right) \ge n + v_p(p^2b^2) - v_p(a) = 0 $$

and

$$ -\frac{a+bc}{u} - bp = \frac{c+\epsilon p}{au} \in \mathbb{Z}_p, \quad \frac{au-(a+bc)u^{-1}-2bp}{p^n} = \frac{c+\epsilon p}{ap^n u} \in \mathbb{Z}_p. $$

Suppose that $(v_p(a), v_p(b)) \in D_3$. Let $(e_1, e_2)$ be the canonical basis of $\mathbb{Q}_p^2$. We can verify directly that conditions (i), (ii) hold for the standard lattice $N = \mathbb{Z}_p e_1 \oplus \mathbb{Z}_p e_2$. This completes the proof. $\square$

We now discuss the classification according to the cases $X_D = (A)$ or $(B)$. The following proposition is the main part of Theorem 3.10.

**Proposition 3.14.** Let $p \ge 7$ be a prime and $\epsilon \in \{0, \pm 1\}$. For $D \in \mathbf{MF}_{\mathbb{Q}_p}^{\text{wa}}$ satisfying S1 with $\chi_\varphi(X) = X^4 + \epsilon p X^2 + p^2$, exactly one of the following holds.

(1) $X_D = (A)$, and $D \simeq D_a^{\epsilon, \nu}$ where $a \in \epsilon p + p^2 \mathbb{Z}_p$,

(2) $X_D = (B)$, and $D \simeq D_{(a,b)}^{\epsilon, \mu}$ where $a, b \in \mathbb{Q}_p$ satisfy one of the followings:

(i) $v_p(a) \ge 1$ and $v_p(b) \ge 0$,

(ii) $v_p(a) = v_p(b) + 1$.
$$ (3) \quad X_D = (B), \text{ and } D \simeq D_{\epsilon'}^{\epsilon, \text{iso}} \text{ where } \epsilon' = \pm 1. $$

*Proof.* First, suppose that $X_D = (B)$. We have

$$ u^{-1}\varphi(D_1) = u^{-1}\varphi(p^{-1}D_1) = u^{-1}\circ\varphi\circ\mu(p^{-1})(D_1) = f^{el}(D_1) = D_0. $$

Therefore, we have $\varphi(D_1) = D_0$ by the unipotence of $u$. If there exists a nonzero $x \in D_1$ such that $y := u\varphi^2(x)$ is not proportional to $x$, we deduce immediately that $\mathcal{B} = (x, y, \varphi(x), \varphi(y))$ is an adapted basis of $D$. The matrix of $\varphi$ under $\mathcal{B}$ has the form

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -p^2 \\ 0 & 0 & 1 & c \\ 1 & 0 & a & d \\ 0 & 1 & b & e \end{pmatrix} $$

with $a, b, c, d, e, f \in \mathbb{Q}_p$. A direct calculation shows

$$ \chi_{\varphi}(X) = X^4 - (a+e)X^3 + (ae-bd-c)X^2 + (ac-d+bp^2)X + p^2 = X^4 + \epsilon p X^2 + p^2. $$

Comparing the coefficients, we have

$$ a + e = 0, \quad ae - bd - c = \epsilon p \quad \text{and} \quad ac - d + bp^2 = 0. $$

We deduce that $-c(ab + 1) = a^2 + \epsilon p + b^2 p^2$. If $ab = -1$ and $a^2 + \epsilon p + b^2 p^2 = 0$, we have $a^2 - \epsilon abp + b^2 p^2 = 0$. Since in this case $b \neq 0$, we have $-\frac{a}{bp}$ is a primitive 3rd root of unity. It follows that $\frac{a^2}{p} = \frac{a}{bp} \cdot ab$ has normalized valuation 0, which is impossible for $a \in \mathbb{Q}_p$. Therefore, $ab \neq -1$ and

$$ c = -\frac{a^2 + \epsilon p + b^2 p^2}{ab + 1}, \quad d = -\frac{a^3 + \epsilon ap - bp^2}{ab + 1} \quad \text{and} \quad e = -a. $$

Therefore, we have $D \simeq D_{(a,b)}^{\epsilon, \mu}$. Denote $S = \begin{pmatrix} -b & a \\ \frac{a+bc}{p^2} & b \end{pmatrix}$ and $M = \begin{pmatrix} 0 & -p^2 \\ 1 & c \end{pmatrix}$. We have $\mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & M \\ I_2 & SM \end{pmatrix}$ and it follows that

$$ \mathrm{Mat}_{\mathcal{B}}(u) = \begin{pmatrix} I_2 & 0 \\ S & I_2 \end{pmatrix}, \quad \mathrm{Mat}_{\mathcal{B}}(f^{el}) = \begin{pmatrix} 0 & M \\ \frac{1}{p}I_2 & 0 \end{pmatrix} \quad \text{and} \quad \mathrm{Mat}_{\mathcal{B}}(\mu(p)) = \begin{pmatrix} pI_2 & 0 \\ 0 & I_2 \end{pmatrix}. $$

Let $H$ be an adapted lattice of $D$ and $H_i = H \cap D_i$ for $i = 0, 1$. By Theorem 2.2 (d), we have $H = H_1 \oplus H_0$. For $i = 0, 1$, we denote $\mathrm{pr}_i \mathcal{B} = \mathcal{B} \cap D_i$. From now on, we identify $D_1$ and $D_0$ with lattices in $\mathbb{Q}_p^2$ canonically by identifying the bases $\mathrm{pr}_i \mathcal{B}$ with the canonical basis of $\mathbb{Q}_p^2$. Under this identification, the definition of an adapted lattice and Theorem 2.2 (d) imply the containment of $\mathbb{Z}_p$-submodules of $\mathbb{Q}_p^2$

$$ \frac{1}{p}H_1 \subset H_0, \quad MH_0 \subset H_1 \quad \text{and} \quad SH_1 \subset H_0. $$

From the first two inclusions, we deduce that $\frac{1}{p}MH_1 \subset H_1$. Note that $\det(\frac{1}{p}M) = 1$, so the first two inclusions are indeed equalities. The third inclusion turns into $pSH_1 \subset H_1$, which implies that $c + \epsilon p = \det(pS) \in \mathbb{Z}_p$. Moreover, if $c + \epsilon p$ is invertible in $\mathbb{Z}_p$, we have $(u - \mathrm{Id})H_1 = H_0$ in $D$. Using the notations in Theorem 2.2 (d), on $\bar{H}$ we have $(\bar{u} - \mathrm{Id}_{\bar{H}})\bar{H} = \bar{H}_0$ hence $\bar{H}_0 \subset L_{r-1}$. Since $L_{r-1}$ is $\bar{f}^{el}$-stable, we have $\bar{H}_1 = \bar{f}^{el}(\bar{H}_0) \subset L_{r-1}$. It follows that $L_{r-1} = \bar{H}$, contradiction. In conclusion, we have $c + \epsilon p \in p\mathbb{Z}_p$, hence $c \in p\mathbb{Z}_p$. The condition (d) of Theorem 2.2 can be summarized to the existence of a lattice $N$ in $\mathbb{Q}_p^2$ satisfying the following conditions.

(i) $\begin{pmatrix} 0 & -1 \\ 1 & c \end{pmatrix} N = N,$

(ii) $\begin{pmatrix} -pb & a \\ a+bc & pb \end{pmatrix} N \subset N,$
(iii) on $N/pN$, the image of the $\mathbb{F}_p$-endomorphism defined by $\begin{pmatrix} -pb & a \\ a+bc & pb \end{pmatrix}$ is stable under the action of $\begin{pmatrix} 0 & -1 \\ 1 & c_p \end{pmatrix}$.

Note that condition (iii) always holds. Indeed, denote the endomorphisms of $N/pN$ defined by the matrices $\begin{pmatrix} 0 & -1 \\ 1 & c_p \end{pmatrix}$ and $\begin{pmatrix} -pb & a \\ a+bc & pb \end{pmatrix}$ by $\bar{M}$ and $\bar{S}$ respectively, we have $\bar{M} \circ \bar{S} + \bar{S} \circ \bar{M} = c_p \bar{S}$. Apply both sides on $N/pN$ and note that $\bar{M}(N/pN) = N/pN$, we have $\bar{M}(\bar{S}(N/pN)) \subset \bar{S}(N/pN)$ as claimed. Finally, Lemma 3.13 shows that the canonical basis of $\mathbb{Q}_p^4$ is adapted for the module $D_{(a,b)}^{\epsilon, \mu}$ if and only if $v_p(a) \ge 1$ and $v_p(b) \ge 0$, or $v_p(a) = v_p(b) + 1$.

Suppose $X_D = (B)$ and for all $x \in D_1$, $u\varphi^2(x)$ is proportional to $x$. It follows that there exists a fix $\epsilon' \in \{\pm 1\}$ such that $u\varphi^2(x) = \epsilon'px$. Let $(x, y)$ be a basis for $D_1$. We deduce that $\mathcal{B} = (x, y, \varphi(x), \varphi(y))$ is an adapted basis of $D$ and under which the matrix of $\varphi$ has the form

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & \epsilon'p & 0 \\ 0 & 0 & 0 & \epsilon'p \\ 1 & 0 & a & c \\ 0 & 1 & b & d \end{pmatrix} $$

for some $a, b, c, d \in \mathbb{Q}_p$. A direct calculation shows

$$ \chi_{\varphi}(X) = X^4 - (a+d)X^3 + (ad-bc-2\epsilon'p)X^2 + (a+d)\epsilon'pX + p^2 = X^4 + \epsilon pX + p^2. $$

Comparing the coefficients, we get $a+d=0$ and $ad-bc = (\epsilon + 2\epsilon')p$. If $b=0$, we have $a^2 = -(\epsilon + 2\epsilon')p \in \{\pm p, \pm 2p, \pm 3p\}$, which is impossible for $p \ge 5$. As before, we deduce that

$$ \mathrm{Mat}_{\mathrm{pr}_1\mathcal{B}, \mathrm{pr}_0\mathcal{B}}(u|_{D_0}) = \frac{\epsilon'}{p} \begin{pmatrix} a & -\frac{a^2+(\epsilon+2\epsilon')p}{b} \\ b & -a \end{pmatrix} $$

and that for any adapted lattice $H \subset D$, after identifying $D_1$ and $D_0$, we have $H_0 = \frac{1}{p}H_1$ and

$$ (\epsilon + 2\epsilon')p = \det \begin{pmatrix} a & -\frac{a^2+(\epsilon+2\epsilon')p}{b} \\ b & -a \end{pmatrix} \in p\mathbb{Z}_p. $$

Therefore, we have $(\bar{u} - \mathrm{Id}_{\bar{H}})\bar{H}_1 \subsetneq \bar{H}_0$ is a 1-demensional subspace. For the sake of Theorem 2.2 (d), take a nonzero element $t \in (\bar{u} - \mathrm{Id}_{\bar{H}})\bar{H}_1$, let $r=2$ and $L_1 = \mathrm{span}(t, f^{el}(t))$. Since $\begin{pmatrix} a & -\frac{a^2+(\epsilon+2\epsilon')p}{b} \\ b & -a \end{pmatrix}^2 = -(\epsilon + 2\epsilon')p$, the chain $0 = L_0 \subsetneq L_1 \subsetneq L_2 = \bar{H}$ satisfies the requirements.

Finally, we should verify that $D \simeq D_{\epsilon'}^{\epsilon, \mathrm{iso}}$. We can take $P \in \mathbf{GL}_2(\mathbb{Q}_p)$ such that

$$ P \begin{pmatrix} a & -\frac{a^2+(\epsilon+2\epsilon')p}{b} \\ b & -a \end{pmatrix} P^{-1} = \begin{pmatrix} 0 & -(\epsilon+2\epsilon')p \\ 1 & 0 \end{pmatrix}, $$

since these two matrices have the same characteristic polynomial which is separable. It follows that, under the canonical bases of $\mathbb{Q}_p^4$ as the underlying vector space of both $D$ and $D_{\epsilon'}^{\epsilon, \mathrm{iso}}$, the matrix $\begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} \in \mathbf{GL}_4(\mathbb{Q}_p)$ defines an isomorphism between these two modules, and our assertion follows.

Suppose $X_D = (A)$. It's clear that $D$ is decomposed into 4 subspaces of dimension 1.

$$ D = \underbrace{D_{[1100]} \oplus D_{[1001]}}_{D_1} \oplus \underbrace{D_{[0011]} \oplus D_{[0110]}}_{D_0}. $$

Take a nonzero $x \in D_{[1100]}$, one can check that $\mathcal{B} = (x, f^{el}(x), (f^{el})^2(x), (f^{el})^3(x))$ is an adapted basis of $D$. Under this basis, the matrix of $u, f^{el}$ and $\mu(p)$ have the following forms, where $a, b, c, d \in \mathbb{Q}_p$.

$$ \mathrm{Mat}_{\mathcal{B}}(u) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & b & 1 & 0 \\ c & d & 0 & 1 \end{pmatrix}, \quad \mathrm{Mat}_{\mathcal{B}}(f^{el}) = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mathrm{Mat}_{\mathcal{B}}(\mu(p)) = \begin{pmatrix} p & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$
We deduce from $\varphi = u \circ f^{el} \circ \mu(p)$ that

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -1 \\ p & 0 & 0 & 0 \\ bp & p & 0 & -a \\ dp & 0 & 1 & -c \end{pmatrix} \quad \text{and} \quad \chi_{\varphi}(X) = X^4 + cX^3 + (pd + a)X^2 + pbX + p^2 = X^4 + \epsilon p X^2 + p^2. $$

Comparing the coefficients, we have

$$ b = c = 0 \quad \text{and} \quad dp = a - \epsilon p \quad \text{and} \quad \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -1 \\ p & 0 & 0 & 0 \\ 0 & p & 0 & -a \\ a - \epsilon p & 0 & 1 & 0 \end{pmatrix}. $$

It follows that $D \simeq D_a^{\epsilon, \nu}$. Theorem 2.2 (d) requires that any adapted lattice $H \subset D$ has a $\mathbb{Z}_p$-basis $\tilde{\mathcal{B}}$ in which each element is proportional to one of elements in $\mathcal{B} = ((f^{el})^j(x))_{0 \le j \le 3}$. We deduce from the definition of adapted lattice, namely $H = \varphi(H) + \frac{1}{p}\varphi(\mathrm{Fil}_1 H)$, that $\tilde{\mathcal{B}} = (\alpha_j \cdot (f^{el})^j(x))_{0 \le j \le 3}$ for $\alpha_j \in \mathbb{Q}_p^\times$ such that $v_p(\alpha_i) = v_p(\alpha_j)$ for any $0 \le i, j \le 3$. Without loss of generality, we can thereby suppose that $\alpha_i = \alpha_j$ for any $0 \le i, j \le 3$. The matrices of $f^{el}$ and $u$ under $\tilde{\mathcal{B}}$ are

$$ \mathrm{Mat}_{\tilde{\mathcal{B}}}(f^{el}) = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mathrm{Mat}_{\tilde{\mathcal{B}}}(u) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & 0 & 1 & 0 \\ 0 & \frac{a-\epsilon p}{p} & 0 & 1 \end{pmatrix}. $$

We claim that $\bar{u} = \mathrm{Id}_{\tilde{H}}$. Indeed, we deduce from $\bar{u}(H) = H$ that the numbers $\frac{a-\epsilon p}{p}$, $a$ belong to $\mathbb{Z}_p$, which implies $a \in p\mathbb{Z}_p$. Therefore, the image of $\bar{u} - \mathrm{Id}_{\tilde{H}}$ is contained in the 1-dimensional linear subspace of $\tilde{H}$ generated by $\bar{\varphi}^2(x)$. Theorem 2.2 (d) requires that this image is contained in an $L_{r-1} \subset \tilde{H}$ which is stable under $f^{el}$. However, there is no proper $f^{el}$-submodule of $\tilde{H}$ containing $\bar{\varphi}^2(x)$, and our assertion follows. We deduce immediately that $\frac{a-\epsilon p}{p} \in p\mathbb{Z}_p$. $\square$

**3.6. Construction of certain skew forms.** By Lemma 3.2, to verify that $D_{\epsilon'}^{prod}$ and the filtered $\varphi$-modules listed in Theorem 3.10 belong to $\mathcal{C}$, it suffices to verify the S3 for them.

On $D_{\epsilon'}^{prod}$, take a $\mathbb{Q}_p$-basis $(x, y)$ of $\mathrm{Fil}_1 D$ and consider the skew form $\delta : D_{\epsilon'}^{prod} \times D_{\epsilon'}^{prod} \to \mathbb{Q}_p$ whose matrix under the basis $\mathcal{B} := (x, y, \varphi(x), \varphi(y))$ is

$$ \mathrm{Mat}_{\mathcal{B}}(\delta) = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & -\epsilon' & 0 \\ 0 & \epsilon' & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}. $$

It's clear that $\delta$ is nondegenerate, and one can check that under $\delta$, $\varphi$ is a $p$-similitude and $\mathrm{Fil}_1 D$ is totally isotropic.

For the case $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$ for $\epsilon \in \{0, \pm 1\}$, we have the following Proposition, which completes the proof of Theorem 3.10.

**Proposition 3.15.** Let $p \ge 7$ be a prime and $\epsilon \in \{0, \pm 1\}$. Let $D \in \mathbf{MF}_{\mathbb{Q}_p}^{wa}$ be a filtered $\varphi$-modules satisfying Condition S1 with $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$. Then $D$ satisfies S2 and S3, which thereby is arising from an abelian surface over $\mathbb{Q}_p$.

*Proof.* Note that the polynomial $\chi_{\varphi}(X) = X^4 + \epsilon p X^2 + p^2$ is separable. Therefore, $D$ satisfies S2. To verify that $D$ satisfies S3, take a $\mathbb{Q}_p$-basis $(x, y)$ of $\mathrm{Fil}_1 D$ and $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{Q}_p$ such that

$$ y \equiv \sum_{1 \le j \le 3} \alpha_j \varphi^j(x) \mod \mathbb{Q}_p x. $$
Take $x_1, x_2, x_3 \in \mathbb{Q}_p$ such that

$$ \begin{cases} (\alpha_1 - (\epsilon - 1)p\alpha_2)x_1 + \alpha_2 x_2 = 0 \\ (x_1, x_2) \neq (0, 0) \\ x_3 = (1 - \epsilon)px_1 \end{cases} $$

Consider the skew form $\delta : D \times D \to \mathbb{Q}_p$ whose matrix under the basis $\mathcal{B} := (\varphi^i(x))_{0 \le i \le 3}$ is

$$ \mathrm{Mat}_{\mathcal{B}}(\delta) = \begin{pmatrix} 0 & x_1 & x_2 & x_3 \\ -x_1 & 0 & px_1 & px_2 \\ -x_2 & -px_1 & 0 & p^2 x_1 \\ -x_3 & -px_2 & -p^2 x_1 & 0 \end{pmatrix}. $$

It follows that

$$ \det(\mathrm{Mat}_{\mathcal{B}}(\delta)) = (p^2 x_1^2 - px_2^2 + px_1 x_3)^2 \stackrel{x_3 = (1-\epsilon)px_1}{=} p^2((2-\epsilon)px_1^2 - x_2^2)^2. $$

The equation $(2 - \epsilon)px_1^2 = x_2^2$ has no solution for any $(0, 0) \neq (x_1, x_2) \in \mathbb{Q}_p^2$ since $2 \nmid v_p((2 - \epsilon)px_1^2)$. It follows that $\delta$ is nondegenerate. One can check that under $\delta$, the endomorphism $\varphi$ is a $p$-similitude and the subspace $\mathrm{Fil}_1 D$ is totally isotropic. $\square$

## 4. $p$-ADIC MONODROMY GROUPS OF SUPERSINGULAR ABELIAN SURFACES OVER $\mathbb{Q}_p$

After establishing Theorem 3.7 and Theorem 3.10, we classify all filtered $\varphi$-modules $D$ arising from supersingular abelian surfaces over $\mathbb{Q}_p$. We proceed to compute the monodromy groups $H_D$ associated with these filtered $\varphi$-modules.

A key advantage of our classification is that the Hodge cocharacter is simply $\mu(t) = \mathrm{diag}(t, t, 1, 1) \in \mathrm{GL}(D)$, while $\varphi_D$ is given by a concrete matrix. This allows us to compute $H_D$ via Proposition 2.6. Moreover, the supersingularity implies that $\varphi_D$ has finite order in $\mathrm{PGL}(D)$. Therefore, the neutral component $H_D^\circ$ is generated by the family of tori $\{\varphi_D^i \circ \mu(t) \circ \varphi_D^{-i}\}_{i \in \mathbb{Z}}$ and scalars.

**4.1. The case** $\chi_\varphi(X) = (X^2 \pm p)^2$. When $D \simeq D_{\pm 1}^{mult}$, we can compute the monodromy groups directly.

**Proposition 4.1.** For $\epsilon' = \pm 1$, we have $H_{D_{\epsilon'}^{mult}} \simeq \mathbb{G}_{m, \mathbb{Q}_p}^2$.

*Proof.* Under the canonical adapted basis $\mathcal{B}$ of $D_{\pm 1}^{mult}$, $\varphi$ and the Hodge cocharacter $\mu$ becomes

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & \epsilon'p & 0 \\ 0 & 0 & 0 & \epsilon'p \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \quad \text{and} \quad \mu : t \mapsto \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

We can thereby verify that

$$ \{\varphi^i \circ \mu(t) \circ \varphi^{-i}\}_{i \in \mathbb{Z}} = \{\mathrm{diag}(t, t, 1, 1), \mathrm{diag}(1, 1, t, t)\}_{t \in \mathbb{Q}_p^\times}, $$

and the closed subgroup of $\mathrm{GL}_D = \mathrm{GL}_4$ generated by these tori together with scalars is clearly $\mathbb{G}_m^2$. $\square$

**4.2. The case** $\chi_\varphi(X) = X^4 + \epsilon p X^2 + p^2$. To compute the monodromy groups of $D_a^{\epsilon, \nu}$, $D_{(a,b)}^{\epsilon, \mu}$ and $D_{\epsilon'}^{\epsilon, \mathrm{iso}}$, we need to work on the Lie algebras side. Invoking Lemma 2.7, we can compute the corresponding Lie algebras in a straightforward manner.
**Proposition 4.2.** Let $D = D_{a'}^{\epsilon, \nu}$. If $a' = \epsilon = 0$, we have $H_D^\circ \simeq G_m^3$. Otherwise, we have

$$ H_D^\circ \simeq \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2 := \left\{ \begin{pmatrix} g & 0 \\ 0 & g' \end{pmatrix} \middle| g, g' \in \mathbf{GL}_2(\mathbb{Q}_p), \det(g) = \det(g') \right\}. $$

*Proof.* Under the canonical adapted basis $\mathcal{B} = (e_i)_{1\le i\le 4}$ of $D_{a'}^{\epsilon, \nu}$, $\varphi$ and the Hodge cocharacter $\mu$ becomes

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & 1 \\ -p^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & -a \\ a-\epsilon p & 0 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mu : t \mapsto \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

where $a \in \epsilon p + p^2\mathbb{Z}_p$. Denote $\mathcal{B}' = (e_1, e_3, e_2, e_4)$, we have

$$ \mathrm{Mat}_{\mathcal{B}'}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -a \\ -p^2 & 0 & 0 & 0 \\ a-\epsilon p & 1 & 0 & 0 \end{pmatrix} \quad \text{and} \quad \mu' : t \mapsto \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & t & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

By block matrix computation, the family of tori $\mathcal{F} = \{\mathrm{Mat}_{\mathcal{B}'}(\varphi)^i \circ \mu'(t) \circ \mathrm{Mat}_{\mathcal{B}'}(\varphi)^{-i}\}_{i \in \mathbb{Z}} \subset \mathbf{GL}_4$ is contained in $\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$. By Lemma 2.7, to verify that the tori in $\mathcal{F}$ generate $\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$, it suffices to show that the Lie algebra $\mathcal{Lie}(\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2)$ is generated by the Lie algebras of these tori, which are

$$ \mathrm{span}_{i \in \mathbb{Z}} \{ \mathrm{Mat}_{\mathcal{B}'}(\varphi)^i d\mu'(1) \mathrm{Mat}_{\mathcal{B}'}(\varphi)^{-i} \}. $$

Substuting $d\mu'(1) = \mathrm{diag}(1, 0, 1, 0)$, a direct calculation shows that

$$ \mathrm{Mat}_{\mathcal{B}'}(\varphi)^i d\mu'(1) \mathrm{Mat}_{\mathcal{B}'}(\varphi)^{-i} = \left\{ \begin{array}{ll} \begin{pmatrix} 0 & 0 & 0 & 0 \\ a & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -\frac{a+\epsilon p}{p^2} & 0 \end{pmatrix} & i=1 \\ \begin{pmatrix} -\frac{a^2+a\epsilon p}{p^2} & -\frac{a+\epsilon p}{p^2} & 0 & 0 \\ \frac{a^3+ap^2-a^2\epsilon p}{p^2} & \frac{a^2+p^2-a\epsilon p}{p^2} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\frac{\epsilon}{p} \end{pmatrix} & i=2 \\ \begin{pmatrix} \frac{p-\epsilon a}{p} & -\frac{\epsilon}{p} & 0 & 0 \\ -\frac{ap+\epsilon a^2}{p} & \frac{\epsilon a}{p} & 0 & 0 \\ 0 & 0 & \frac{p-\epsilon a}{p} & a-\epsilon p \\ 0 & 0 & -\frac{a}{p^2} & \frac{\epsilon a}{p} \end{pmatrix} & i=3 \\ \begin{pmatrix} \frac{a\epsilon p-a^2}{p^2} & -\frac{a}{p^2} & 0 & 0 \\ \frac{a^3-2a^2\epsilon p+2ap^2-\epsilon p^3}{p^2} & \frac{a^2-a\epsilon p+p^2}{p^2} & 0 & 0 \\ 0 & 0 & 0 & \epsilon p \\ 0 & 0 & 0 & 1 \end{pmatrix} & i=4 \\ \begin{pmatrix} 1 & 0 & 0 & 0 \\ \epsilon p-a & 0 & 0 & 0 \\ 0 & 0 & 0 & a \\ 0 & 0 & 0 & 1 \end{pmatrix} & i=5. \end{array} \right. $$

We can deduce from Lemma 2.8 (ii) that the first (resp. the last) matrix together with $d\mu'(1)$ generates $\mathrm{diag}(0, 1, 1, 0)$ (resp. $\mathrm{diag}(1, 0, 0, 1)$) and the Lie algebras of the tori in $\mathcal{F}$ generate

$$ \mathrm{span} \{d\mu'(1), \mathrm{diag}(0, 1, 1, 0), \mathrm{diag}(1, 0, 0, 1)\} = \{\mathrm{diag}(t_1, t_2, t_3, t_4) \in \mathfrak{gl}_4(\mathbb{Q}_p) | t_1 + t_2 = t_3 + t_4\}. $$

Using Lemma 2.8 (ii) again, these Lie algebras generate

$$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{a+\epsilon p}{p^2} & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ \frac{a^3+ap^2-a^2\epsilon p}{p^2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{\epsilon}{p} & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ -\frac{a+\epsilon p}{p} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{a}{p^2} & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -\frac{\epsilon}{p} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a-\epsilon p \\ 0 & 0 & 0 & 0 \end{pmatrix}, $$

and

$$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ \frac{a^3-2a^2\epsilon p+2ap^2-\epsilon p^3}{p^2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -\frac{a}{p^2} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \epsilon p \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 0 & 0 \\ \epsilon p-a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}. $$
For any fixed $a \in \mathbb{Q}_p$ and $\epsilon \in \{0, \pm 1\}$, if $(a, \epsilon) \neq (0, 0)$, we can verify directly that the $\mathbb{Q}_p$-subspace spanned by the matrices above is

$$ \left\{ \begin{pmatrix} 0 & \alpha_{1,2} & 0 & 0 \\ \alpha_{2,1} & 0 & 0 & 0 \\ 0 & 0 & 0 & \beta_{1,2} \\ 0 & 0 & \beta_{2,1} & 0 \end{pmatrix} \in \mathfrak{gl}_4(\mathbb{Q}_p) \middle| \alpha_{i,j}, \beta_{i,j} \in \mathbb{Q}_p \right\}. $$

It follows that the Lie algebras of the tori in $\mathcal{F}$ generate

$$ \left\{ \begin{pmatrix} g & 0 \\ 0 & g' \end{pmatrix} \middle| g, g' \in \mathfrak{gl}_2(\mathbb{Q}_p), \text{tr}(g) = \text{tr}(g') \right\} = \mathcal{L}\text{ie}(\mathbf{GL}_2 \times_{\det} \mathbf{GL}_2) $$

and we have $H_D^\circ = \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$. If $a = \epsilon = 0$, the matrices above are all zero. It follows that $H_D^\circ = \mathbf{G}_m^3$. $\square$

**Proposition 4.3.** For any $a, b \in \mathbb{Q}_p$ such that $ab \neq -1$, let $D = D_{(a,b)}^{\epsilon, \mu}$ and $c = -\frac{a^2+\epsilon p+b^2 p^2}{ab+1}$. Denote $S = \begin{pmatrix} -b & a \\ \frac{a+bc}{p^2} & b \end{pmatrix}$ and $M = \begin{pmatrix} 0 & -p^2 \\ 1 & c \end{pmatrix}$.

(i) If $c \neq -\epsilon p$ and $p \ge 5$,

$$ H_D^\circ = \left\{ \begin{pmatrix} \alpha_{1,1} I_2 + \alpha'_{1,1} M & \alpha_{1,2} S + \alpha'_{1,2} M S \\ \alpha_{2,1} S + \alpha'_{2,1} S M & \alpha_{2,2} I_2 + \alpha'_{2,2} M \end{pmatrix} \middle| \begin{array}{l} \alpha_{i,j}, \alpha'_{i,j} \in \mathbb{Q}_p, \det \neq 0 \\ \alpha_{1,1} \alpha'_{2,2} - \alpha_{2,2} \alpha'_{1,1} = -\frac{\epsilon p+c}{p^2} (\alpha_{1,2} \alpha'_{2,1} - \alpha_{2,1} \alpha'_{1,2}) \end{array} \right\}. $$

In this case, we have $H_D^\circ \simeq \mathbb{Q}_p \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$.

(ii) If $c = -\epsilon p$, we have

$$ H_D^\circ = \left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2} S \\ \alpha_{2,1} S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p, \det \neq 0 \right\}. $$

In this case, either $H_D^\circ \simeq \mathbf{G}_a^2 \times_g \mathbf{G}_m^2$ with $g(s, t) = \begin{pmatrix} s & 0 \\ 0 & t \end{pmatrix}$, which is non-reductive, or $H_D^\circ \simeq \mathbf{G}_m^2$.

*Proof.* Under the canonical adapted basis $\mathcal{B} = (e_i)_{1 \le i \le 4}$ of $D_{(a,b)}^{\epsilon, \mu}$, the matrices of $\varphi$ and the Hodge cocharacter are

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -p^2 \\ 0 & 0 & 1 & \frac{a^2+\epsilon p+b^2 p^2}{ab+1} \\ 1 & 0 & a & \frac{a^3+\epsilon a p-b p^2}{ab+1} \\ 0 & 1 & b & -a \end{pmatrix} \quad \text{and} \quad \mu : t \mapsto \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

By Proposition 2.6, $H_D^\circ$ is generated by the family of tori

$$ \mathcal{F} = \{ \mathrm{Mat}_{\mathcal{B}}(\varphi)^i \circ \mu(t) \circ \mathrm{Mat}_{\mathcal{B}}(\varphi)^{-i} \}_{i \in \mathbb{Z}} \subset \mathbf{GL}_4. $$

Denote $S = \begin{pmatrix} -b & a \\ \frac{a+bc}{p^2} & b \end{pmatrix}$, $M = \begin{pmatrix} 0 & -p^2 \\ 1 & c \end{pmatrix} \in \mathrm{Mat}_{2 \times 2}(\mathbb{Q}_p)$. The matrix $\mathrm{Mat}_{\mathcal{B}}(\varphi)$ becomes

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -p^2 \\ 0 & 0 & 1 & \frac{a^2+\epsilon p+b^2 p^2}{ab+1} \\ 1 & 0 & a & \frac{a^3+\epsilon a p-b p^2}{ab+1} \\ 0 & 1 & b & -a \end{pmatrix} = \begin{pmatrix} 0 & M \\ I_2 & SM \end{pmatrix} $$

and we can verify that $\begin{pmatrix} 0 & M \\ I_2 & SM \end{pmatrix}^{-1} = \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}$. Therefore, each torus in the family $\mathcal{F}$ becomes

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi)^i \circ \mu(t) \circ \mathrm{Mat}_{\mathcal{B}}(\varphi)^{-i} = \begin{pmatrix} 0 & M \\ I_2 & SM \end{pmatrix}^i \begin{pmatrix} tI_2 & 0 \\ 0 & I_2 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}^i $$

whose Lie algebra is $\begin{pmatrix} 0 & M \\ I_2 & SM \end{pmatrix}^i \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}^i$. We can verify that the matrices $S, M \in \mathrm{Mat}_{2 \times 2}(\mathbb{Q}_p)$ satisfy relations $S^2 = -\frac{\epsilon p+c}{p^2} I_2$, $M^2 - cM + p^2 I_2 = 0$ and $MS + SM = cS$. Moreover, we can use these relations to show that

$$ H := \left\{ \begin{pmatrix} \alpha_{1,1} I_2 + \alpha'_{1,1} M & \alpha_{1,2} S + \alpha'_{1,2} M S \\ \alpha_{2,1} S + \alpha'_{2,1} S M & \alpha_{2,2} I_2 + \alpha'_{2,2} M \end{pmatrix} \middle| \begin{array}{l} \alpha_{i,j}, \alpha'_{i,j} \in \mathbb{Q}_p, \det \neq 0 \\ \alpha_{1,1} \alpha'_{2,2} - \alpha_{2,2} \alpha'_{1,1} = -\frac{\epsilon p+c}{p^2} (\alpha_{1,2} \alpha'_{2,1} - \alpha_{2,1} \alpha'_{1,2}) \end{array} \right\} $$
is a connected algebraic subgroup of GL<sub>4</sub>(Q<sub>p</sub>) whose Lie algebra is

$$
\mathcal{L}ie(H) = \left\{ \begin{pmatrix} \alpha_{1,1} I_2 + \alpha'_{1,1} M & \alpha_{1,2} S + \alpha'_{1,2} M S \\ \alpha_{2,1} S + \alpha'_{2,1} S M & \alpha_{2,2} I_2 + \alpha'_{2,2} M \end{pmatrix} \middle| \alpha_{i,j}, \alpha'_{i,j} \in \mathbb{Q}_p, \alpha'_{1,1} = \alpha'_{2,2} \right\}.
$$

After the above discussion, we now prove that $H$ is generated by the tori in $\mathcal{F}$. When $c \neq -\epsilon p$, by Lemma 2.7, it suffices to show that $\mathcal{L}ie(H)$ is generated by the family $\left\{ \begin{pmatrix} 0 & M \\ I_2 & 0 \end{pmatrix}^i \begin{pmatrix} -S & I_2 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} S & 0 \\ M^{-1} & 0 \end{pmatrix}^i \right\}_{i \in \mathbb{Z}}$. Clearly, $\mathcal{L}ie(H)$ contains $\begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}$. One can use the relations between $S$ and $M$ to show that $\mathcal{L}ie(H)$ is stable under the conjugation by $\begin{pmatrix} 0 & M \\ I_2 & S \end{pmatrix}$. Therefore, the Lie subalgebra generated by the toric Lie subalgebras above is contained in $\mathcal{L}ie(H)$. Inversely, we have

$$
\begin{pmatrix} 0 & M \\ I_2 & S \end{pmatrix}^i \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}^i =
\begin{cases}
\begin{pmatrix} 0 & 0 \\ -S & I_2 \end{pmatrix} & i=1 \\
\begin{pmatrix} I_2 - \frac{\epsilon p+c}{p^2} M & -MS \\ -\frac{\epsilon p+c}{p^2} SM+S - \frac{\epsilon p+c}{p^2} (M-cI_2) & -\frac{\epsilon p+c}{p} I_2 - (\epsilon p+c)S+MS \\ -S - \frac{1}{p} SM & (1+\frac{\epsilon p+c}{p})I_2 \end{pmatrix} & i=2 \\
\begin{pmatrix} \frac{\epsilon p+c}{p} I_2 & -(\epsilon p+c)S+MS \\ -S - \frac{1}{p} SM & (1+\frac{\epsilon p+c}{p})I_2 \end{pmatrix} & i=3.
\end{cases}
$$

Using Lemma 2.8, the Lie algebras of the tori in $\mathcal{F}$ generate

$$
\begin{pmatrix} 0 & 0 \\ 0 & I_2 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ -S & 0 \end{pmatrix}, \begin{pmatrix} I_2 - \frac{\epsilon p+c}{p^2} M & 0 \\ 0 & -\frac{\epsilon p+c}{p^2} (M-cI_2) \end{pmatrix}, \begin{pmatrix} 0 & -MS \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} -\frac{\epsilon p+c}{p^2} SM+S & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & -(\epsilon p+c)S+MS \\ 0 & 0 \end{pmatrix}.
$$

It's not hard to verify that these matrices span $\mathcal{L}ie(H)$ over $\mathbb{Q}_p$. Therefore, the tori in $\mathcal{F}$ generate $H$.

Finally, we prove that $H \simeq_{\mathbb{Q}_p} \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2$. Note that over $\mathbb{Q}_p$, the matrix $M$ is conjugated to
$\tilde{M} = \begin{pmatrix} \mu_1 & 0 \\ 0 & \mu_2 \end{pmatrix}$ with $\mu_1 \neq \mu_2 \in \mathbb{Q}_p^\times$. Indeed, $M$ is of rank 2 with characteristic polynomial $X^2 - cX + p^2$.
By Jordan decomposition, if $M$ is not of this form, we have $c = \pm 2p$. It follows that $-\frac{a+\epsilon p+b^2 p^2}{ab+1} = \pm 2p$
and $(a \pm bp)^2 = -(\epsilon \pm 2)p \in \{\pm p, \pm 3p\}$, impossible when $p \ge 5$. On the other hand, using the relations
$S^2 = -\frac{\epsilon p+c}{p^2} I_2$ and $MS + SM = cS$, we can show that the same conjugation makes $S$ conjugate to
$\tilde{S} = \begin{pmatrix} 0 & \lambda_2 \\ \lambda_1 & 0 \end{pmatrix}$ with $\lambda_1, \lambda_2 \in \mathbb{Q}_p^\times$. It follows that $H$ is conjugated to the group

$$
\left\{ \begin{pmatrix} \alpha_{1,1} I_2 + \alpha'_{1,1} \tilde{M} & \alpha_{1,2} \tilde{S} + \alpha'_{1,2} \tilde{M} \tilde{S} \\ \alpha_{2,1} \tilde{S} + \alpha'_{2,1} \tilde{M} \tilde{S} & \alpha_{2,2} I_2 + \alpha'_{2,2} \tilde{M} \end{pmatrix} \left| \begin{array}{l} \alpha_{i,j}, \alpha'_{i,j} \in \mathbb{Q}_p, \det \neq 0 \\ \alpha_{1,1} \alpha'_{2,2} - \alpha_{2,2} \alpha'_{1,1} = \\ -\frac{\epsilon p+c}{p^2} (\alpha_{1,2} \alpha'_{2,1} - \alpha_{2,1} \alpha'_{1,2}) \end{array} \right. \right\} = \left\{ \begin{pmatrix} \beta_{1,1} & 0 & 0 & \beta_{1,2} \\ 0 & \beta'_{2,2} & \beta'_{2,1} & 0 \\ 0 & \beta'_{1,2} & \beta'_{1,1} & 0 \\ \beta_{2,1} & 0 & 0 & \beta_{2,2} \end{pmatrix} \left| \begin{array}{l} \beta_{i,j}, \beta'_{i,j} \in \mathbb{Q}_p, \det \neq 0 \\ \beta_{1,1} \beta_{2,2} - \beta_{1,2} \beta_{2,1} = \\ \beta'_{1,1} \beta_{2,2} - \beta'_{1,2} \beta_{2,1} \end{array} \right. \right\}
$$

which, after switching the second and the fourth vectors in the basis, becomes

$$
\left\{ \begin{pmatrix} \beta_{1,1} & \beta_{1,2} & 0 & 0 \\ \beta_{2,1} & \beta_{2,2} & 0 & 0 \\ 0 & 0 & \beta'_{1,1} & \beta'_{1,2} \\ 0 & 0 & \beta'_{2,1} & \beta'_{2,2} \end{pmatrix} \middle| \begin{array}{l} \beta_{i,j}, \beta'_{i,j} \in \bar{\mathbb{Q}}_p, \det \neq 0 \\ \beta_{1,1} \beta_{2,2} - \beta_{1,2} \beta_{2,1} = \\ \beta'_{1,1} \beta_{2,2} - \beta'_{1,2} \beta_{2,1} \end{array} \right\} = \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2.
$$

It remains to show that the tori in $\mathcal{F}$ generate $\left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}S \\ \alpha_{2,1}S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p, \det \neq 0 \right\}$ when $c = -\epsilon p$.
By Lemma 2.7, it suffices to show that the toric subalgebras $\left\{ \begin{pmatrix} 0 & M \\ I_2 & 0 \end{pmatrix}^i \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}^i \right\}_{i \in \mathbb{Z}}$ generate
$\left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}S \\ \alpha_{2,1}S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p \right\}$ as a Lie subalgebra. Since the latter contains $\begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}$ and is stable under the
conjugation by $\begin{pmatrix} 0 & M \\ I_2 & S \end{pmatrix}$, it contains the former. A direct calculation shows that

$$
\begin{pmatrix} 0 & M \\ I_2 & S \end{pmatrix} \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ -S & I_2 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 0 & M \\ I_2 & S \end{pmatrix}^2 \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} -S & I_2 \\ M^{-1} & 0 \end{pmatrix}^2 = \begin{pmatrix} I_2 & -MS \\ S & 0 \end{pmatrix}.
$$

The calculation above implies the generation of Lie algebra and hence that of algebra groups. Indeed,
when $ab \neq 0$, we have $a + bc = -\frac{c+\epsilon p+p^2 b^2}{a} = -\frac{p^2 b^2}{a}$ and

$$
MS = \left( \frac{b^2 p^2}{a} - bp^2}{\frac{b^3 p^2}{a^2} - \frac{b^2 p^2}{a}} \right) = - \frac{p^2 b}{a} S.
$$
Our assertion follows from Lemma 2.8 and the calculation above. In this case, we have $H_D^\circ \simeq G_a^2 \times_g G_m^2$ and the subgroup $\{ (\begin{pmatrix} 1 & \alpha_S \\ 0 & 1 \end{pmatrix}) | \alpha \in \mathbb{Q}_p \}$ of $H_D^\circ$ is nontrivial, normal and unipotent. Therefore, $H_D^\circ$ is not reductive. Finally, when $ab = 0$, we deduce from $c = -\epsilon p$ that $a^2 + b^2p^2 = 0$ and hence $a = b = 0$. It follows that $S = 0$ and we have clearly $H_D^\circ = G_{m, \mathbb{Q}_p}^2$. $\square$

**Exemple 4.4.** Volkov [Vol23, §3] gave an example of a nonsemisimple filtered $\varphi$-module arising from supersingular abelian surface: let $p \equiv 1 \pmod 3$ be a prime and $\zeta_3 \in \mathbb{Q}_p$ be a primitive third root of unity, consider a 4-dimensional $\mathbb{Q}_p$-vector space $D$ with a $\mathbb{Q}_p$-basis $\mathcal{B} = (x_1, y_1, x_2, y_2)$ equipped with a filtered $\varphi$-module structure given by

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & \zeta_3 p & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \zeta_3^{-1} p \\ 0 & 0 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mathrm{Fil}_0 D = D, \mathrm{Fil}_1 D = \mathbb{Q}_p x_1 \oplus \mathbb{Q}_p (y_1 + x_2), \mathrm{Fil}_2 D = 0. $$

We can show that $D \simeq D_{(-\zeta_3^{-1}p, 1)}^{1, \mu}$ and in this case, $c := -\frac{a^2+\epsilon p+b^2p^2}{ab+1} = -\epsilon p$. Theorem 4.3 implies that $H_D^\circ \simeq G_a^2 \times_g G_m^2$ which is non-reductive, which reproves that $D$ is non-semisimple [Vol23, Proposition 3.1(b)].

**Proposition 4.5.** *Denote $D = D_{\epsilon'}^{\epsilon, \text{iso}}$, we have*

$$ H_D^\circ = \left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2} S \\ \alpha_{2,1} S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p, \det \neq 0 \right\} \quad \text{where } S = \begin{pmatrix} 0 & -(\epsilon+2\epsilon')p \\ 1 & 0 \end{pmatrix}. $$

In this case, we have $H_{D_{\epsilon'}^{\epsilon, \text{iso}}} \simeq \bar{\mathbb{Q}}_p \mathbf{GL}_2$.

*Proof.* Under the canonical adapted basis $\mathcal{B} = (e_i)_{1 \le i \le 4}$ of $D_{\epsilon'}^{\epsilon, \text{iso}}$, the matrices of $\varphi$ and the Hodge cocharacter are

$$ \mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & \epsilon'p & 0 \\ 0 & 0 & 0 & \epsilon'p \\ 1 & 0 & 0 & -(\epsilon+2\epsilon')p \\ 0 & 1 & 1 & 0 \end{pmatrix} \quad \text{and} \quad \mu : t \mapsto \begin{pmatrix} t & 0 & 0 & 0 \\ 0 & t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

As in the statement, denote $S = \begin{pmatrix} 0 & -(\epsilon+2\epsilon')p \\ 1 & 0 \end{pmatrix}$. Note that $S^2 = -(\epsilon + 2\epsilon')p$. By Proposition 2.6, $H_D^\circ$ is generated by the family of tori

$$ \mathcal{F} = \{ \mathrm{Mat}_{\mathcal{B}}(\varphi)^i \circ \mu(t) \circ \mathrm{Mat}_{\mathcal{B}}(\varphi)^{-i} \}_{i \in \mathbb{Z}} = \left\{ \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^i \begin{pmatrix} tI_2 & 0 \\ 0 & I_2 \end{pmatrix} \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^{-i} \right\}_{i \in \mathbb{Z}}. $$

Using Lemma 2.7, it suffices to show that $\left\{ \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^i \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^{-i} \right\}_{i \in \mathbb{Z}}$ generates $\left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}S \\ \alpha_{2,1}S & \alpha_{2,2} \end{pmatrix} \right\}_{\alpha_{i,j} \in \mathbb{Q}_p}$ as a Lie subalgebra of $\mathfrak{gl}_4(\mathbb{Q}_p)$. A direct calculation shows that

$$ \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix} \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^{-1} = \begin{pmatrix} 0 & 0 \\ -\frac{\epsilon'}{p} S & I_2 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^2 \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & \epsilon'p \\ I_2 & S \end{pmatrix}^{-2} = \begin{pmatrix} \left(1 - \frac{\epsilon+2\epsilon'}{a}\right)I_2 & -S \\ \frac{a-(\epsilon+2\epsilon')}{a^2p}S & \frac{\epsilon+2\epsilon'}{a}I_2 \end{pmatrix}. $$

By Lemma 2.8 we have $\begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & I_2 \end{pmatrix} \in \mathrm{Lie}(H_D^\circ)$. Therefore, $\mathrm{Lie}(H_D^\circ)$ contains $\mathrm{diag}(\alpha, \alpha, \beta, \beta)$ for any $\alpha, \beta \in \mathbb{Q}_p$. Using Lemma 2.8 again, we have

$$ \begin{pmatrix} 0 & 0 \\ -\frac{\epsilon'}{p} S & 0 \end{pmatrix}, \begin{pmatrix} 0 & -S \\ \frac{a-(\epsilon+2\epsilon')}{a^2p} S & 0 \end{pmatrix} \in \mathrm{Lie}(H_D^\circ). $$

It follows that $\mathrm{Lie}(H_D^\circ)$ contains and thereby equals to $\left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}S \\ \alpha_{2,1}S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p \right\}$. Therefore, we have

$$ H_D^\circ = \left\{ \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}S \\ \alpha_{2,1}S & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p, \det \neq 0 \right\}. $$
as claimed. To prove that $H_D^\circ$ is isomorphic to $GL_2$ over $\bar{\mathbb{Q}}_p$, note that the matrix $S$ is conjugated to $(\begin{smallmatrix} \sigma_1 & 0 \\ 0 & \sigma_2 \end{smallmatrix})$ for some $\sigma_1, \sigma_2 \in \bar{\mathbb{Q}}_p^\times$. It follows that

$$H_{D, \bar{\mathbb{Q}}_p}^{\circ} \simeq \left\{ \begin{pmatrix} \alpha_{1,1} & 0 & \alpha_{1,2}\sigma_1 & 0 \\ 0 & \alpha_{1,1} & 0 & \alpha_{1,2}\sigma_2 \\ \alpha_{2,1}\sigma_1 & 0 & \alpha_{2,2} & 0 \\ 0 & \alpha_{2,1}\sigma_2 & 0 & \alpha_{2,2} \end{pmatrix} \middle| \alpha_{i,j} \in \mathbb{Q}_p, \det \neq 0 \right\}.$$

The map

$$\begin{pmatrix} \alpha_{1,1} & 0 & \alpha_{1,2}\sigma_1 & 0 \\ 0 & \alpha_{1,1} & 0 & \alpha_{1,2}\sigma_2 \\ \alpha_{2,1}\sigma_1 & 0 & \alpha_{2,2} & 0 \\ 0 & \alpha_{2,1}\sigma_2 & 0 & \alpha_{2,2} \end{pmatrix} \mapsto \begin{pmatrix} \alpha_{1,1} & \alpha_{1,2}\sigma_1 \\ \alpha_{2,1}\sigma_1 & \alpha_{2,2} \end{pmatrix}$$

defines an isomorphism between $H_{D, \bar{\mathbb{Q}}_p}^{\circ}$ and $GL_{2, \bar{\mathbb{Q}}_p}$.

<figure><img src="image_10.png" /></figure>

Combining all the cases above, we have the following conclusion.

**Theorem 4.6.** Fix a prime $p \ge 7$, we have the following isomorphisms over $\bar{\mathbb{Q}}_p$.

(1) $H_{D_{\epsilon'}^{\text{prod}}} \simeq \mathbf{G}_m^2,$

(2) $H_{D_{a'}^{\epsilon, \nu}} \simeq \begin{cases} \mathbf{G}_m^3 & \text{if } a' = \epsilon = 0 \\ \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2 & \text{otherwise,} \end{cases}$

(3) $H_{D_{(a,b)}^{\epsilon, \mu}} \simeq \begin{cases} \mathbf{G}_a^2 \rtimes_g \mathbf{G}_m^2 & \text{or } \mathbf{G}_m^2 \text{ if } c(a,b) = -\epsilon p \\ \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2 & \text{otherwise,} \end{cases}$

(4) $H_{D_{\epsilon'}^{\epsilon, \text{iso}}} \simeq \mathbf{GL}_2$

where $c(a,b) = -\frac{a^2+\epsilon p+b^2p^2}{ab+1}$ and $g(s,t) = (\begin{smallmatrix} s & 0 \\ 0 & t \end{smallmatrix})$. In particular, the neutral connected component $G_{V_p(A), \bar{\mathbb{Q}}_p}^\circ$ of the p-adic monodromy group associated with a supersingular abelian surface A over $\mathbb{Q}_p$ is isomorphic to one of the following.

$$\mathbf{G}_m^2, \mathbf{G}_m^3, \mathbf{G}_a^2 \rtimes_g \mathbf{G}_m^2, \mathbf{GL}_2, \mathbf{GL}_2 \times_{\det} \mathbf{GL}_2.$$

5. A MODULI SPACE DESCRIPTION

Fix a prime $p \ge 7$ and a parameter $\epsilon \in \{0, \pm 1\}$. In this section, we study the set $M^{\text{wa}}$ of isomorphism classes of objects in $\mathbb{C}$ with Weil polynomial $\chi_\varphi(X) = X^4 + \epsilon p X^2 + p^2$. By Proposition 3.15, they are precisely admissible filtered $\varphi$-modules over $\mathbb{Q}_p$ satisfying **S1** with $\chi_\varphi(X) = X^4 + \epsilon p X^2 + p^2$.

A main tool to describe $M^{\text{wa}}$ is the Grassmannian $Y = \text{Gr}(2, 4)$. Indeed, for any such $D$, we can take a certain $\mathbb{Q}_p$-basis $\mathcal{B}$ of $D$ such that

$$\mathrm{Mat}_{\mathcal{B}}(\varphi) = \begin{pmatrix} 0 & 0 & 0 & -p^2 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -\epsilon p \\ 0 & 0 & 1 & 0 \end{pmatrix}. \qquad (5.1)$$

Since $D$ satisfies **S1**, the only indetermined data is the choice of $\text{Fil}_1 D \subset \mathbb{Q}_p^4$, described by $Y(\mathbb{Q}_p)$.

**Definition 5.1.** We denote by $D_P$ the filtered $\varphi$-module represented by $P \in Y(\mathbb{Q}_p)$, and by $Y(\mathbb{Q}_p)^{\text{wa}}$ the subset of $Y(\mathbb{Q}_p)$ consisting of the points representing admissible filtered $\varphi$-modules. By Lemma 3.3, $Y(\mathbb{Q}_p)^{\text{wa}}$ is the complement of the $\varphi$-stable points in $Y(\mathbb{Q}_p)$.

Note that $\mathbf{GL}_4$ acts algebraically on $Y$, under which $Y(\mathbb{Q}_p)^{\text{wa}}$ is stable. After identifying $\varphi$ with the matrix (5.1), we have the following observation.
**Proposition 5.2.** Denote $Z_{\varphi}(\mathbb{Q}_p) = \{g \in \text{GL}_4(\mathbb{Q}_p) | g\varphi = \varphi g\}$. The quotient set $Y(\mathbb{Q}_p)^{\text{wa}}/Z_{\varphi}(\mathbb{Q}_p)$ is identified with $M^{\text{wa}}$ via $[P] \mapsto D_P$.

**Notation.** Let $K$ be any field and $P \in Y(K)$. Denote by $V_P \subset K^4$ the linear subspace defined by $P$. Denote by $U(K) \subset Y(K)$ the open subvariety of $Y$ defined by the condition

$$K^4 = V_P \oplus \varphi(V_P).$$

For $P \in U(K)$, we denote by $\text{pr}_P$ the projection from $K^4$ onto $V_P$ along $\varphi(V_P)$.

**Definition 5.3.** Fix the canonical embedding $i : \mathbb{A}^1 \to \mathbb{P}^1$, $a \mapsto [a : 1]$. We denote by $c$ the morphism from $U$ to $\mathbb{P}^1$ defined by $P \mapsto i(\text{tr}(\text{pr}_P \circ \varphi^2|_{V_P}))$. We denote by $Y^{\text{def}}$ the domain of definition of $c$ (the maximal locus where $c$ can be extended), which is an open subvariety of $Y$.

**5.1. Parametrization by $\mathbb{P}^2(\mathbb{Q}_p)$**. Let $K$ be any field, we can decompose $K^4$ as

$$K^4 = Ke_1 \oplus \bigoplus_{2 \le i \le 4} Ke_i,$$

where $(e_i)_{1 \le i \le 4}$ is the canonical basis of $K^4$. By considering only the 2-dimensional subspaces of $K^4$ containing $e_1$, we can define a closed immersion

$$\iota : \mathbb{P}^2 = \text{Gr}(1, 3) \hookrightarrow \text{Gr}(2, 4) = Y.$$

Moreover, we have $\iota(\mathbb{P}^2(\mathbb{Q}_p)) \subset Y(\mathbb{Q}_p)^{\text{wa}}$ (since there is no 2-dimensional $\varphi$-stable subspace of $\mathbb{Q}_p^4$ containing $e_1$).

**Proposition 5.4.** The composition $\pi : \mathbb{P}^2(\mathbb{Q}_p) \xrightarrow{\iota} Y(\mathbb{Q}_p)^{\text{wa}} \to M^{\text{wa}}$ is surjective.

*Proof.* Take $P \in Y(\mathbb{Q}_p)^{\text{wa}}$ and $0 \ne x \in \text{Fil}_1 D_P$. Since $D_P$ is weakly admissible, $\text{Fil}_1 D_P$ is not $\varphi$-stable. It follows that $\{\varphi^j(x)\}_{0 \le j \le 3}$ is a basis of $D_P$. Take $y = \sum_{0 \le j \le 3} c_j \varphi^j(x)$ such that $\text{Fil}_1 D_P = \mathbb{Q}_p x \oplus \mathbb{Q}_p y$. Let $Q = \iota([c_1 : c_2 : c_3])$. We have an isomorphism of filtered $\varphi$-modules $D_Q \simeq D_P$ via taking the canonical basis of $V = \mathbb{Q}_p^4$ to the basis $\{\varphi^j(x)\}_{0 \le j \le 3}$ of $D_P$. Therefore, we have $\pi(Q) = \pi(P)$ and $\pi$ is surjective. $\square$

For convenience, for $P \in \mathbb{P}^2(\mathbb{Q}_p)$, we denote by $D_P$ the filtered $\varphi$-module $D_{\iota(P)}$.

**Proposition 5.5.** Consider the points $P_{\pm} = [(\epsilon \pm 1)p : 0 : 1] \in \mathbb{P}^2(\mathbb{Q}_p)$ and morphisms

$$\mathbb{A}^1 \xrightarrow{\nu} \mathbb{P}^2 \quad \text{and} \quad \mathbb{A}^2 \setminus \{xy = -1\} \xrightarrow{\mu} \mathbb{P}^2$$

defined by

$$a' \xrightarrow{\nu} [a' : 0 : 1] \quad \text{and} \quad (a, b) \xrightarrow{\mu} \left[ -\frac{a + \epsilon p ab^2 - b^3 p^2}{ab + 1} : 1 : -b \right].$$

For $\epsilon' \in \{\pm 1\}$ and $a, b, a' \in \mathbb{Q}_p$ such that $ab \ne -1$, we have

$$D_{P_{\epsilon'}} \simeq D_{\epsilon', \text{iso}}^\epsilon, \quad D_{\nu(a')} \simeq D_{a', \nu}^\epsilon \quad \text{and} \quad D_{\mu(a,b)} \simeq D_{(a,b)}^{\epsilon, \mu}.$$

*Proof.* Recall that for $P = [t_1 : t_2 : t_3] \in \mathbb{P}^2(\mathbb{Q}_p)$, the isomorphism class of $D_P$ is determined by the condition that

$$\text{there exists } x \in D_P \text{ such that } \text{Fil}_1 D_P = \text{span} \left( x, \sum_{i=1}^3 t_i \varphi^i(x) \right). \qquad (5.2)$$
The canonical adapted basis of $D_{a'}^{\epsilon, \nu}$ has the form $(y, x, \varphi(x), \varphi^2(x))$ such that $\varphi^3(x) = y - a'\varphi(x)$. It follows that $\mathrm{Fil}_1 D_{a'}^{\epsilon, \nu}$ is generated by $x$ and $a'\varphi(x) + \varphi^3(x)$, and the assertion $D_{\nu(a')} \simeq D_{a'}^{\epsilon, \nu}$ follows from (5.2).

Similarly, the canonical adapted basis of $D_{\epsilon'}^{\epsilon, \text{iso}}$ has the form $(x, y, \varphi(x), \varphi(y))$ such that

$$ \begin{cases} \varphi^2(x) = \epsilon'px + \varphi(y) \\ \varphi^2(y) = \epsilon'py - (\epsilon + 2\epsilon')p\varphi(x). \end{cases} $$

It follows that

$$ \varphi^3(x) = \epsilon'p\varphi(x) + \varphi^2(y) = \epsilon'py - (\epsilon + \epsilon')p\varphi(x). $$

Therefore, $\mathrm{Fil}_1 D_{a'}^{\epsilon, \nu}$ is generated by $x$ and $(\epsilon + \epsilon')p\varphi(x) + \varphi^3(x)$ and our assertion $D_{P_{\epsilon'}} \simeq D_{\epsilon'}^{\epsilon, \text{iso}}$ follows from (5.2).

Finally, the canonical adapted basis of $D_{(a,b)}^{\epsilon, \mu}$ has the form $(x, y, \varphi(x), \varphi(y))$ such that

$$ \left\{ \begin{aligned} \varphi^2(x) &= y + a\varphi(x) + b\varphi(y) \\ \varphi^2(y) &= p^2x + cy + d\varphi(x) - a\varphi(y) \end{aligned} \right. \qquad (5.3) $$

where $c = -\frac{a^2+\epsilon p+b^2p^2}{ab+1}$ and $d = ac + bp^2$. Since $\mathrm{Fil}_1 D_{(a,b)}^{\epsilon, \mu} = \mathrm{span}(x, y)$, it suffices to compute the coordinates of $y$ under the basis $(\varphi^i(x))_{0 \le i \le 3}$. Using (5.3) repeatedly, we have

$$ \begin{aligned} \varphi^3(x) = \varphi(y) + a\varphi^2(x) + b\varphi^2(y) &= bp^2x + bcy + bd\varphi(x) + a\varphi^2(x) + (1 - ab)\varphi(y) \\ &= bp^2x + (a + bc - b^{-1})y + (bd - ab^{-1} + a^2)\varphi(x) + b^{-1}\varphi^2(x). \end{aligned} $$

A direct computation shows that $bd - ab^{-1} + a^2 = -\frac{a+\epsilon pab^2-b^3p^2}{ab+1}$. Therefore, we have

$$ (a + bc - b^{-1})y \equiv \frac{a + \epsilon pab^2 - b^3p^2}{ab + 1} \varphi(x) + \varphi^2(x) - b\varphi^3(x) \mod x. $$

We deduce that $D_{\mu(a,b)} \simeq D_{(a,b)}^{\epsilon, \mu}$ from (5.2). $\square$

<figure><img src="image_11.png" /></figure>

The following lemma is technical, and we will prove it at the end of next subsection.

**Lemma 5.6.** We have $Y^{\mathrm{def}}(\mathbb{Q}_p) = Y(\mathbb{Q}_p)^{\mathrm{wa}}$.

The first main theorem of this section is the following.

**Theorem 5.7.** The map (of sets) $c : Y(\mathbb{Q}_p)^{\mathrm{wa}} = Y^{\mathrm{def}}(\mathbb{Q}_p) \to \mathbb{P}^1(\mathbb{Q}_p)$ factors through $\mathcal{M}^{\mathrm{wa}}$. Equivalently, there exists a unique map (of sets) $\mathrm{barc} : \mathcal{M}^{\mathrm{wa}} \to \mathbb{P}^1(\mathbb{Q}_p)$ such that the diagram

<figure><img src="image_12.png" /></figure>

commutes. Moreover, $\bar{c}$ is injective outside $\bar{c}^{-1}(i(-\epsilon p))$ (where $i$ is the embedding in Definition 5.3).
**Proof.** By Proposition 5.4, $M^{\text{wa}}$ is a quotient set of $\mathbb{P}^2(\mathbb{Q}_p)$ under $\pi$. Two points $P, P' \in \mathbb{P}^2(\mathbb{Q}_p)$ are in the same fiber of $\pi$ if and only if $D_P \simeq D_{P'}$ as filtered $\varphi$-modules.

<figure><img src="image_13.png" /></figure>

Denote $\{\infty\} = \mathbb{P}^2(\mathbb{Q}_p)\backslash\text{im } i$. We can verify that $c \circ i$ is given by

$$[x : y : z] \mapsto -\frac{x^2 - \epsilon p x z + p^2 z^2}{\epsilon p z^2 + y^2 - xz} - \epsilon p. \qquad (5.4)$$

Consider the decomposition

$$\mathbb{P}^2(\mathbb{Q}_p) = U_\infty \sqcup U_\mu \sqcup U_\nu \sqcup \{P_\pm\} \qquad (5.5)$$

where the subsets $U_\infty$, $U_\mu$ and $U_\nu$ are defined as

$$\begin{align*}
U_\infty &= \{P \in \mathbb{P}^2(\mathbb{Q}_p) | c(\iota(P)) = \infty\}, \\
U_\mu &= \{[x : y : z] \in \mathbb{P}^2(\mathbb{Q}_p) | y \neq 0\} \backslash U_\infty, \\
U_\nu &= \{[a : 0 : 1] | a \in \mathbb{Q}_p \setminus \{\epsilon p, (\epsilon \pm 1)p\}\}.
\end{align*}$$

If $c(\iota(P)) = \infty$ for some $P = [x : y : z] \in \mathbb{P}^2(\mathbb{Q}_p)$, we claim that $\text{Fil}_1(D_P) \cap \varphi(\text{Fil}_1(D_P)) \neq 0$. Indeed, let $v := (xy, xz, yz, z^2) \in \text{Fil}_1(D_P) \subset \mathbb{Q}_p^4$, we have

$$\varphi(v) = (-z^2, xy, xz - \epsilon p z^2, yz) \in \mathbb{Q}_p^4.$$

Since $c(\iota(P)) = \infty$, by (5.4) we have $xz - \epsilon p z^2 = y^2$ and it's clear that $0 \neq \varphi(v) \in \text{Fil}_1(D_P)$ and our assertion follows. Invoking Theorem 3.10, $D = D_{\epsilon p}^{\epsilon, \nu}$ represents the only isomorphism class such that $\text{Fil}_1(D) \cap \varphi(\text{Fil}_1(D)) \neq 0$. Therefore, $D_P$ is isomorphic to $D_{\epsilon p}^{\epsilon, \nu}$.

Let $P, P' \in \mathbb{P}^2(\mathbb{Q}_p)\backslash U_\infty$ such that $D_P \simeq D_{P'}$. Then, $c(P) = c(P')$ by the intrinsic Definition 5.3. Therefore, the map $c$ factors through $\pi$, which gives the desired map $\bar{c} : M^{\text{wa}} \to \mathbb{P}^1(\mathbb{Q}_p)$.

To verify that $\bar{c}$ is injective outside $\bar{c}^{-1}(i(-\epsilon p))$, we analyse its behavior on each part of (5.5) separately. Denote by $V = \{(a, b) \in \mathbb{Q}_p^2 | ab \neq -1\} \subset \mathbb{Q}_p^2$. We can verify directly that $\mu$ gives an isomorphism between $V$ and $U_\mu$ and that $c(\iota(\mu(a, b))) = -\frac{a^2+\epsilon p+b^2p^2}{ab+1}$, which coincides with the notation $c$ in Theorem 3.10. For $(a_1, b_1), (a_2, a_2) \in V$, denote $S_i = \begin{pmatrix} -b_i & a_i \\ a_i+b_i c_i & b_i \end{pmatrix}$ and $M_i = \begin{pmatrix} 0 & -p^2 \\ 1 & c_i \end{pmatrix} \in \text{Mat}_{2 \times 2}(\mathbb{Q}_p)$ where $c_i = c(a_i, b_i)$. We have the relations

$$M_i S_i + S_i M_i = c_i S_i \quad \text{and} \quad (S_i)^2 = -\frac{\epsilon p + c_i}{p^2} I_2. \qquad (5.6)$$

If $c_1 = c_2 \neq -\epsilon p$, we claim that $D_{\mu(a_1, b_1)} \simeq D_{\mu(a_2, b_2)}$. Indeed, in this case we have $M_1 = M_2$. By (5.6), when $M_1 = M_2$ is diagonalized over $\bar{\mathbb{Q}}_p$, the same conjugation makes $S_i$ into $\tilde{S}_i := \begin{pmatrix} 0 & \alpha_i \\ \beta_i & 0 \end{pmatrix}$ for some $\alpha_i, \beta_i \in \bar{\mathbb{Q}}_p^\times$, $i = 1, 2$. We have $\alpha_1\beta_1 = \alpha_2\beta_2 = \frac{\epsilon p+c(a,b)}{p^2} \neq 0$ by our assumption. It follows that $\tilde{S}_1$ is conjugated to $\tilde{S}_2$ by $\begin{pmatrix} \alpha_1 & 0 \\ 0 & \beta_2 \end{pmatrix}$, which commutes with $\tilde{M}$. Therefore, we have $S_1 = H S_2 H^{-1}$ for some $H \in Z(M_1) \subset \text{GL}_2(\mathbb{Q}_p)$. Then the matrices $\text{Mat}_B(\varphi_{D_{(a_i, b_i)}^{\epsilon, \mu}}) = \begin{pmatrix} 0 & M_i \\ I_2 & S_i M_i \end{pmatrix}$ ($i = 1, 2$) are conjugate via $\begin{pmatrix} H & 0 \\ 0 & H \end{pmatrix}$, which proves that $D_{(a_1, b_1)}^{\epsilon, \mu} \simeq D_{(a_2, b_2)}^{\epsilon, \mu}$.
If $c_1 = c_2 = -\epsilon p$. We can conjugate simultaneously the matrices $M_1 = M_2$, $S_1$ and $S_2$ into $\tilde{M} = (\begin{pmatrix} \mu_1 & 0 \\ 0 & \mu_2 \end{pmatrix})$, $\tilde{S}_1 = (\begin{pmatrix} 0 & \lambda_2 \\ \lambda_1 & 0 \end{pmatrix})$ and $\tilde{S}_2 = (\begin{pmatrix} 0 & \lambda'_2 \\ \lambda'_1 & 0 \end{pmatrix})$ where $\mu_i$ are roots of $X^2 + \epsilon p X + p^2$ and $\lambda_i, \lambda'_i \in \mathbb{Q}_p$. By assumption, $\lambda_1 \lambda_2 = \lambda'_1 \lambda'_2 = 0$. In this case, the isomorphism class of $D_{(a_i,b_i)}^{\epsilon, \mu}$ depends on the $Z(\tilde{M})$-conjugation class of $\tilde{S}_i$. Since $Z(\tilde{M})$ is the canonical maximal torus in $GL_2$, there are three such classes, represented by $(\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix})$, $(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix})$ and $(\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix})$. Using (5.4), the three conjugation classes above are respectively corresponding to the cases

$$ \frac{x}{pz} = -\mu_1, \quad \frac{x}{pz} = -\mu_2 \quad \text{and} \quad x = z = 0 \qquad (5.7) $$

which gives the three preimages under $\bar{c}$ of $i(-\epsilon p)$.

For any $P \in U_\nu$, we claim that there exists $P' \in U_\mu$ such that $\pi(P) = \pi(P')$. Note that for $P = [x:0:z]$, $c(P) \neq \infty$ if and only if $z \neq 0$ and $x \neq \epsilon p z$. The function $a \mapsto c([a:0:1])$ is clearly identified with $a \mapsto a - \epsilon p + \frac{p^2}{a-\epsilon p}$. It suffices to prove that, for any $a' \in \mathbb{Q}_p \setminus \{\epsilon p, (\epsilon \pm 1)p\}$, there exists $(a,b) \in U'$ such that $D_{a'}^{\epsilon,\nu} \simeq D_{(a,b)}^{\epsilon,\mu}$. Let $(e_i)_{1\le i \le 4}$ be the canonical adapted basis of $D_{a'}^{\epsilon,\nu}$. Denote $B' = (e_2, e_1, \varphi(e_2), \varphi(e_1))$, the matrix of $\varphi_{D_{a'}^{\epsilon,\nu}}$ under $B'$ is

$$ \mathrm{Mat}_{B'}(\varphi_{D_{a'}^{\epsilon,\nu}}) = \begin{pmatrix} 0 & 0 & a'-\epsilon p & 0 \\ 0 & 0 & 0 & \frac{p^2}{a'-\epsilon p} \\ 1 & 0 & 0 & \frac{1}{a'-\epsilon p} \\ 0 & 1 & -p^2-a'^2+\epsilon a'p & 0 \end{pmatrix}. $$

Denote $H = (\begin{pmatrix} 1 & a'-\epsilon p \\ 1 & -p^2 \end{pmatrix})$. By the assumption that $a' \in \mathbb{Q}_p \setminus \{\epsilon p, (\epsilon \pm 1)p\}$, the matrix $H$ is well-defined and invertible. The conjugation of $\mathrm{Mat}_{B'}(\varphi_{D_{a'}^{\epsilon,\nu}})$ by $(\begin{pmatrix} H & 0 \\ 0 & H \end{pmatrix})$ has the form of $\varphi_{D_{(a,b)}^{\epsilon,\mu}}$ given in Definition 3.8, which gives an isomorphism between $D_{a'}^{\epsilon,\nu}$ and certain $D_{(a,b)}^{\epsilon,\mu}$ and verifies our assertion.

Finally, for $\epsilon' = \pm 1$, $c^{-1}(2\epsilon'p)$ is the single point $P_\pm = [(\epsilon \pm 1)p : 0 : 1]$, corresponding to the only isomorphism class $D_{\epsilon'}^{\epsilon, \mathrm{iso}}$. This completes our proof. $\square$

**Corollary 5.8.** Let $P \in Y(\mathbb{Q}_p)^{\mathrm{wa}}$. The Wintenberger type of $D_P$ satisfies

$$ X_{D_P} = \begin{cases} (A) & v_p(\bar{c}(P)) \le 0 \\ (B) & v_p(\bar{c}(P)) \ge 1. \end{cases} $$

*Proof.* Using Theorem 5.7 and Proposition 5.5, we can compute the value of $\bar{c}$ corresponding to the three cases in Proposition 3.14.

In the above proof we see that the function $a \mapsto c([a:0:1])$ is identified with $a \mapsto a - \epsilon p + \frac{p^2}{a-\epsilon p}$. The case (1) in Proposition 3.14 implies that $a - \epsilon p \in p^2\mathbb{Z}_p$, which yields $v_p(\bar{c}) \le 0$. In the proof of Proposition 3.14 (2), we have shown that $c(a,b) \in p\mathbb{Z}_p$, which implies that $v_p(\bar{c}) \ge 1$ in this case. Finally, case (3) corresponds to $\bar{c} = \pm 2p$, which completes our proof. $\square$

**Remark 5.9.** For illustrating the decomposition (5.5), we represent $\mathbb{P}^2(\mathbb{Q}_p)$ as the disjoint union of

$$ \mathbb{A}^2(\mathbb{Q}_p) = \{[x:y:z] \in \mathbb{P}^2(\mathbb{Q}_p) \mid y \neq 0\} \quad \text{and} \quad \mathbb{P}^1(\mathbb{Q}_p) = \{[x:0:z] \in \mathbb{P}^2(\mathbb{Q}_p)\} $$

in the following diagram, where the open disk corresponds to $\mathbb{A}^2(\mathbb{Q}_p)$ and half of its boundary corresponds to $\mathbb{P}^1(\mathbb{Q}_p)$. This is merely a topological model (recall that $\mathbb{P}^2(\mathbb{R})$ can be obtained from the closed disk $\mathbb{D}^2$ by identifying antipodal points on the boundary), intended to make the subsequent results more transparent.
<figure><img src="image_14.png" /></figure>

**Remark 5.10.** In the prove, we can see that if $X^2 + \epsilon pX + p^2$ has no solution in $\mathbb{Q}_p$, the set $c^{-1}(-\epsilon p)$ is a single point $o = [0:1:0]$ and thereby $\bar{c}$ is injective on $M^{\text{wa}}$. On the other hand, if $\mu_i \in \mathbb{Q}_p$ ($i = 1, 2$) are the two different solutions of $X^2 + \epsilon pX + p^2$, then $c^{-1}(-\epsilon p)$ is consisting of two projective lines $l_1, l_2$ in $\mathbb{P}^2(\mathbb{Q}_p)$ which pass $o$ and are of slopes $-\mu_1 p, -\mu_2 p$ respectively. Moreover, $l_1 \setminus \{o\}, l_2 \setminus \{o\}$ and $\{o\}$ belong to three different $Z_\varphi(\mathbb{Q}_p)$-orbits respectively. Therefore, under the quotient topology of $M^{\text{wa}}$, the point $\pi(o)$ generalizes to $\pi(l_1)$ and $\pi(l_2)$.

**5.2. The GIT viewpoint.** In this subsection, we fix an algebraically closed field $K$. Our goal is to prove that the morphism $\bar{c}$ becomes a projective GIT quotient after base change to $K$. We can suppose that $\varphi$ is diagonal

$$ \varphi = \begin{pmatrix} \lambda_1 & 0 & 0 & 0 \\ 0 & \lambda_2 & 0 & 0 \\ 0 & 0 & \lambda_3 & 0 \\ 0 & 0 & 0 & \lambda_4 \end{pmatrix} \in \mathbf{GL}_4(K) $$

where $(\lambda_i)_{1\le i\le 4} \in (K^\times)^4$ are pairwise distinct nonzero elements. Denote by $G$ the algebraic group such that

$$ G(R) = \{g \in \mathbf{SL}_4(R) \mid g\varphi = \varphi g\} $$

for any commutative ring $R$. It's clear that $G \simeq \mathbf{G}_m^3$ is a subtorus of the canonical maximal subtorus in $\mathbf{GL}_4$, which inherits the canonical action of $\mathbf{GL}_4$ on $Y$.

Generally, the machinery of GIT asks for a nice (very ample) linearisation of the action, which is equivalent to a $G$-equivariant projective embedding of $X$. In our case, we should consider the Plücker embedding

$$ Y = \text{Gr}(2, 4) \to \mathbb{P}^5 = \text{Proj} K[x_{ij}]_{1\le i<j\le 4} \qquad (5.8) $$

identify $Y$ with the closed subvariety of $\mathbb{P}^5$ defined by the Plücker relation

$$ x_{12}x_{34} - x_{13}x_{24} + x_{14}x_{23} = 0. $$

Equivalently, (5.8) induces an isomorphism $Y \simeq \text{Proj} R(Y)$ where

$$ R(Y) = K[x_{ij}]_{1\le i<j\le 4}/(x_{12}x_{34} - x_{13}x_{24} + x_{14}x_{23}). $$

The embedding (5.8) becomes $G$-equivariant under the homomorphism $G \to \mathbf{GL}(\bigoplus_{1\le i<j\le 4} K x_{ij})$ defined by $\text{diag}(a_1, a_2, a_3, a_4)x_{ij} = a_i a_j x_{ij}$. It turns out that in this case, we can determine the invariant subalgebra.

**Proposition 5.11.** The canonical morphism $K[x, y] \to R(Y)$ defined by $x \mapsto x_{12}x_{34}$ and $y \mapsto x_{14}x_{23}$ induces an isomorphism $K[x, y] \simeq R(Y)^G$, where $R(Y)^G$ is the $G$-invariant subalgebra.
Proof. Denote $S = K[x_{12}x_{34}, x_{14}x_{23}] \subset R(Y)$ and $\underline{d} = (d_{ij})_{1\le i<j\le 4}$. Let

$$D = \{ \underline{d} \in (\mathbb{Z}_{\ge 0})^6 | d_{12}d_{34} = d_{13}d_{24} = d_{14}d_{23} = 0 \}.$$

It follows that $S \subset R(Y)^G$ and that $R(Y)$ is a free $S$-module with basis

$$\left\{ \prod_{1 \le i < j \le 4} x_{ij}^{d_{ij}} | \underline{d} \in D \right\}.$$

The monomial $f = \prod_{1 \le i < j \le 4} x_{ij}^{d_{ij}}$ is a weight vector of weight $\alpha(f) = \prod_{1 \le i < j \le 4} (\alpha_i \alpha_j)^{d_{ij}}$, where $\alpha_i$ is the cocharacter of $G$ defined by

$$\alpha_i(\mathrm{diag}(a_1, a_2, a_3, a_4)) = a_i.$$

Therefore, $f$ is $G$-stable if and only if $\alpha(f)$ is trivial. Consider the map

$$\Phi : (\mathbb{Z}_{\ge 0})^6 \to (\mathbb{Z}_{\ge 0})^4, \quad \underline{d} \mapsto (\Phi_k(\underline{d}))_{k \in [1,4]}, \quad \text{where } \Phi_k(\underline{d}) = \sum_{i=1}^{k-1} d_{ik} + \sum_{j=k+1}^{4} d_{kj}.$$

For $n \in \mathbb{Z}_{\ge 0}$, denote

$$E_n = \{ \underline{d} \in (\mathbb{Z}_{\ge 0})^6 | \Phi_k(\underline{d}) = n, \forall k \in [1, 4] \}.$$

Then, $\alpha(f)$ is trivial if and only if $\underline{d} \in \bigsqcup_{n \ge 0} E_n$.

To verify that $R(Y)^G = S$, it suffices to show that $D \cap E_n = \emptyset$ for any $n \ge 1$. Denote

$$U = \{ \underline{d} \in (\mathbb{Z}_{\ge 0})^6 | \Phi_k(\underline{d}) > 0, \forall k \in [1, 4] \}$$

and for $k \in [1, 4]$,

$$V_k = \{ \underline{d} \in (\mathbb{Z}_{\ge 0})^6 | \Phi_k(\underline{d}) > \Phi_m(\underline{d}), \forall 0 \le m \le 4, m \ne k \}.$$

For any $n \ge 1$, by definition we have $E_n \subset U$ and $E_n \cap V_k = \emptyset$. We claim that $D \cap U \subset \bigsqcup_{k \in [1,4]} V_k$, which thereby implies that $D \cap E_n = \emptyset$.

Denote by $\Sigma$ the set consisting of the subsets of 2 elements in $[1, 4]$ and for $A = \{i < j\} \in \Sigma$, let $d_A = d_{ij}$. For any $\underline{d} \in D \cap U$, the set $\Sigma(\underline{d}) = \{A \in \Sigma | d_A \ne 0\}$ satisfies

* $A \in \Sigma(\underline{d})$ implies that $A^c \notin \Sigma(\underline{d})$

* $\bigcup_{A \in \Sigma(\underline{d})} A = [1, 4]$

which implies that $\#\Sigma(\underline{d}) = 3$ and hence $\#\bigcap_{A \in \Sigma(\underline{d})} A = 1$. It follows that $\bigcap_{A \in \Sigma(\underline{d})} A = \{k\}$ for a certain $k \in [1, 4]$, which implies that $\underline{d} \in V_k$ as claimed. $\square$

In particular, we see that the ideal $R(Y)^G_+$ is generated by $x_{12}x_{34}, x_{14}x_{23}$. By definition, the semistable set $Y^{ss}$ is given by the complement to the closed subscheme of $Y$ defined by $R(Y)^G_+$, and the GIT quotient of this action is given by $Y^{ss} \to Y/G := \mathrm{Proj} R(Y)^G \simeq \mathbb{P}^1$. This is a special case of Gelfand-MacPherson Correspondence (see [Kap92, Theorem 2.4.7]). Taking $K = \bar{\mathbb{Q}}_p$, we have the following application.

**Proposition 5.12.** There exist nonzero $G$-invariant sections $s_1, s_2 \in \Gamma(Y_{\bar{\mathbb{Q}}_p}, \mathcal{O}_{Y_{\bar{\mathbb{Q}}_p}}(2))^G$ such that $\Gamma(Y_{\bar{\mathbb{Q}}_p}, \mathcal{O}_{Y_{\bar{\mathbb{Q}}_p}}(2))^G = \bar{\mathbb{Q}}_p s_1 \oplus \bar{\mathbb{Q}}_p s_2$ and $\bar{c} = [s_1 : s_2]$. In particular, we have $Y_{\bar{\mathbb{Q}}_p}^{\mathrm{def}} = Y_{\bar{\mathbb{Q}}_p}^{ss}$ both being the complement to the common vanishing locus of $s_1$ and $s_2$. The rational map $\bar{c} : Y_{\bar{\mathbb{Q}}_p} \dashrightarrow \mathbb{P}_{\bar{\mathbb{Q}}_p}^1$ is the projective GIT quotient with respect to $\mathcal{O}_{Y_{\bar{\mathbb{Q}}_p}}(1)$.
**Proof.** Over $\bar{\mathbb{Q}}_p$, we can diagonalize the matrix of $\varphi$ into $\mathrm{diag}(\lambda_1, \lambda_2, \lambda_3, \lambda_4)$ and use the Plücker embedding (5.8). Compute by definition, for $[\underline{x}] \in V(x_{12}x_{34} - x_{13}x_{24} + x_{14}x_{23}) \subset \mathbb{P}^5(\bar{\mathbb{Q}}_p)$, we have

$$ \bar{c}([\underline{x}]) = \frac{(\lambda_1\lambda_3 + \lambda_2\lambda_4)(\lambda_1 - \lambda_3)(\lambda_2 - \lambda_4)x_{14}x_{23} - (\lambda_1\lambda_4 + \lambda_2\lambda_3)(\lambda_1 - \lambda_4)(\lambda_2 - \lambda_3)x_{13}x_{24}}{-(\lambda_1 - \lambda_3)(\lambda_2 - \lambda_4)x_{14}x_{23} + (\lambda_1 - \lambda_4)(\lambda_2 - \lambda_3)x_{13}x_{24}}. $$

Note that by Proposition 5.11, $x_{14}x_{23}$ and $x_{13}x_{24}$ consist of a $\bar{\mathbb{Q}}_p$-basis of $\Gamma(Y_{\bar{\mathbb{Q}}_p}, \mathcal{O}_{Y_{\bar{\mathbb{Q}}_p}}(2))^G$. It follows that $\bar{c}$ is induced by $G$-invariant sections. Moreover,

$$ \left| \begin{array}{cc} (\lambda_1\lambda_3 + \lambda_2\lambda_4)(\lambda_1 - \lambda_3)(\lambda_2 - \lambda_4) & -(\lambda_1\lambda_4 + \lambda_2\lambda_3)(\lambda_1 - \lambda_4)(\lambda_2 - \lambda_3) \\ -(\lambda_1 - \lambda_3)(\lambda_2 - \lambda_4) & (\lambda_1 - \lambda_4)(\lambda_2 - \lambda_3) \end{array} \right| = \prod_{1 \le i < j \le 4} (\lambda_i - \lambda_j) \ne 0. $$

Therefore, the morphism $\bar{c}$ differs by an automorphism of $\mathbb{P}^1$ from the projective GIT quotient obtained from Proposition 5.11. $\square$

**Proof of Lemma 5.6.** From the proof of Proposition 5.12, we see that for any $\bar{\mathbb{Q}}_p$-algebra $K$,

$$ \begin{aligned} Y^{\mathrm{def}}(K) &= Y(K) \setminus \{[\underline{x}] \in Y(K) | x_{12}x_{34} = x_{13}x_{24} = 0\} \\ &= \bigcup_{1 \le k \le 4} \underbrace{\{P \in Y(K) | V_P \subset \bigoplus_{i \ne k} K e_i\}}_{Z_k(K)} \cup \bigcup_{1 \le k \le 4} \underbrace{\{P \in Y(K) | e_k \subset V_P\}}_{W_k(K)}. \end{aligned} $$

Note that this decomposition is defined over $\mathbb{Q}_p$. Let $P \in Y(\mathbb{Q}_p)$ and denote by $V_P \subset \mathbb{Q}_p^4$ the 2-dimensional subspace defined by $P$. If $P \in Z_k(\mathbb{Q}_p)$, we have $\dim \sum_{i \ge 0} \varphi^i(V_P) \le 3$. Note that $\sum_{i \ge 0} \varphi^i(V_P)$ is $\varphi$-stable and we deduce that $\dim \sum_{i \ge 0} \varphi^i(V_P) = 2$ since the polynomial $\chi_\varphi$ has no factor of degree 3 over $\mathbb{Q}_p$. It follows that $V_P$ is $\varphi$-stable. If $P \in W_k(\mathbb{Q}_p)$, $V_P \otimes \mathbb{Q}_p$ contains an eigenvector $v$ of $\varphi$. It follows that $V_P \otimes \bar{\mathbb{Q}}_p$ is spanned by $\{\sigma(v)\}_{\sigma \in \Gamma_{\bar{\mathbb{Q}}_p}}$, which implies that $V_P$ is $\varphi$-stable.

By definition, $Y(\mathbb{Q}_p)^{\mathrm{wa}}$ is the complement of the $\varphi$-stable points, and Lemma 5.6 follows. $\square$

**5.3. Distribution of monodromy groups.** Invoking the results in Section 4, we can determine the neutral components of the algebraic monodromy groups associated with the isomorphism class $D_P$ for any $P \in \mathcal{M}^{\mathrm{wa}}$. Combining Theorem 4.6, Theorem 5.7 and Remark 5.10, we have the following result.

**Theorem 5.13.** We have $H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \mathbf{GL}_2 \times_{\mathrm{det}} \mathbf{GL}_2$ for all except finitely many $P \in \mathcal{M}^{\mathrm{wa}}$. The exceptions are

$$ H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \begin{cases} \mathbf{GL}_2 & \bar{c}(P) = i(\pm 2p) \\ \mathbf{G}_m^2 \text{ or } \mathbf{G}_a^2 \times_g \mathbf{G}_m^2 & \bar{c}(P) = i(-\epsilon p) \\ \mathbf{G}_m^3 & \epsilon = 0 \text{ and } \bar{c}(P) = \infty \end{cases} $$

where $g(s, t) = (s \ 0 \ t)$. When $\bar{c}(P) = i(-\epsilon p)$, in the context of Remark 5.10, the case $H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \mathbf{G}_m^3$ occurs when $P = \pi(o)$, and $H_{D_P}^{\circ} \simeq_{\bar{\mathbb{Q}}_p} \mathbf{G}_a^2 \times \mathbf{G}_m^2$ when $P = \pi(x_i)$ where $x_i \in l_i \setminus \{o\}$ for $i = 1, 2$.

**Proof.** This result can be viewed as a geometric presentation of Theorem 4.6, and it follows directly from that theorem in combination with Proposition 5.5. $\square$

We can also consider the distribution of algebraic monodromy groups of the filtered $\varphi$-modules associated with the points on $\mathbb{P}^2(\mathbb{Q}_p)$. The distribution for the case $\epsilon = 0$ is illustrated in the following diagram.
<figure><img src="image_15.png" /></figure>

Distribution of monodromy groups on $\mathbb{P}^2(\mathbb{Q}_p)$ when $\epsilon = 0$

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