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# Dynamic Appearance Modeling of Clothed 3D Human Avatars
using a Single Camera

Hansol Lee1 Junuk Cha 1 Yunhoe Ku 1 Jae Shin Yoon2* Seungryul Baek1* 1UNIST
2Adobe Research

###### Abstract

The appearance of a human in clothing is driven not only by the pose but also
by its temporal context, i.e., motion. However, such context has been largely
neglected by existing monocular human modeling methods whose neural networks
often struggle to learn a video of a person with large dynamics due to the
motion ambiguity, i.e., there exist numerous geometric configurations of
clothes that are dependent on the context of motion even for the same pose. In
this paper, we introduce a method for high-quality modeling of clothed 3D
human avatars using a video of a person with dynamic movements. The main
challenge comes from the lack of 3D ground truth data of geometry and its
temporal correspondences. We address this challenge by introducing a novel
compositional human modeling framework that takes advantage of both explicit
and implicit human modeling. For explicit modeling, a neural network learns to
generate point-wise shape residuals and appearance features of a 3D body model
by comparing its 2D rendering results and the original images. This explicit
model allows for the reconstruction of discriminative 3D motion features from
UV space by encoding their temporal correspondences. For implicit modeling, an
implicit network combines the appearance and 3D motion features to decode
high-fidelity clothed 3D human avatars with motion-dependent geometry and
texture. The experiments show that our method can generate a large variation
of secondary motion in a physically plausible way.

Figure 1: The clothed 3D human avatars with motion-variant appearance and its
intermediate representation. Using a 3D model mesh, we first estimate explicit
pose-variant geometry. We then reconstruct implicit motion-dependent clothed
3D human avatars conditioned by the temporal surface correspondences from the
explicit geometry. The color coding in the pose-variant geometry represents a
visual interpretation of our temporal modeling results, demonstrating 3D
displacement over time.

††Co-corresponding authors$*$.

##  1 Introduction

The modeling of 3D clothed human avatars with a realistic secondary motion
enables various applications such as physically plausible film-making, gaming,
augmented reality (AR), virtual reality (VR), and e-commerce industries.
Existing methods [47, 3, 4, 17, 6] have studied how to model appearance and
geometry of a human using a single camera for the widespread of such self-
avatars in those applications. However, many of them have largely ignored the
importance of motion-dependent appearance modeling for the realistic animation
of a clothed 3D human avatar. In this paper, we model dynamic humans, a
clothed 3D human avatar with high-fidelity motion-dependent geometry and
texture, using a single camera.

Modeling motion-dependent appearance is challenging due to the lack of 3D
ground truth data. In particular, we are missing three types of data: 1) 3D
body pose that aligns with images to encode the pose-to-appearance
relationship; 2) 3D surface to learn to reconstruct high-quality 3D geometry;
and 3) temporal surface correspondences to model motion-dependent geometry.
While the recent advance in 3D pose prediction allows us to have accurate 3D
pose, obtaining high-quality 3D temporal correspondences and geometry is still
challenging, which prevents us from modeling the motion-dependent appearance
of a high-fidelity clothed 3D human avatar.

Existing monocular 3D human modeling has addressed these challenges in two
ways: explicit and implicit modeling. For explicit approach, previous works
[47, 3, 4, 5] have mainly focused on the reconstruction of high-fidelity
static geometry, e.g., the surface for a A-posed person, by predicting the
person-specific residual geometry of an explicit 3D body model, e.g., SMPL
[23] which provides topology-invariant mesh surface. While such explicit
representation provides explicit correspondences over different body poses,
this representation significantly limits the capability of the residual
geometry to describe its local details due to the nature of the fixed topology
of the 3D body model.

The implicit approaches [43, 32, 42, 31, 15, 30, 38] effectively overcome the
limitation of explicit methods by utilizing a number of variations of implicit
representations such as occupancy [25], signed/unsigned distance fields [27,
7] or radiance fields [26] where they are more flexible to describe a complex
and challenging shape since they are free from a fixed topology. However,
these implicit models inhibit the establishment of the temporal
correspondences of the reconstructed 3D surface.

In this paper, we use the complementary properties of the explicit and
implicit representations by newly designing a compositional human modeling
framework. For explicit modeling, we perform joint modeling of 3D body pose
and pose-dependent geometry of humans by predicting explicit geometric offsets
between the clothed humans (observed from a video) and unclothed parametric 3D
body model (i.e., SMPL [23]). This explicit model is topology-invariant to
body poses so that it enables the joint training of the appearance and 3D
surface motion features in the canonical UV coordinates. The modeled
appearance and motion features are conditioned in two implicit functions
dedicated to decoding motion-dependent texture and geometry, respectively. We
highlight that securing well-aligned explicit geometry is important to
establish accurate temporal surface correspondences, which allows the implicit
models to reconstruct the high-fidelity dynamic human avatars with plausible
local details.

The experiments show that our method can generate motion-dependent avatars
with plausible clothes movements even for unseen pose sequences. We compare
our method with state-of-the-art avatar reconstruction methods and it
outperforms other methods in generating motion-dependent avatars with
plausible clothes movements, thereby highlighting the effectiveness of our
compositional framework.

Our contribution can be summarized as follows:

  * •

We introduce a novel compositional framework that can take advantage of both
implicit and explicit representation to model the high-fidelity clothed 3D
human avatar with motion-dependent appearance (texture and geometry).

  * •

We demonstrate that conditioning motion information on implicit functions
enables more effective learning for the video with a person with large
dynamics and capturing the subtle variations of motion-dependent appearance,
leading to physically plausible human avatar animation.

  * •

We demonstrate that our method is capable of generating more realistic
renderings of dynamic human appearance, particularly for large motions,
compared to other methods. Moreover, our approach exhibits strong
generalizability to unseen poses, indicating its effectiveness in handling
variations in pose and motion.

Figure 2: The system overview. We propose a compositional framework that
combines explicit and implicit representations of clothed 3D human avatars.
First, we reconstruct explicit geometry by predicting the surface residuals
from a 3D body mesh model. Using their temporal correspondences, we extract
high-dimensional motion and appearance features, which are conditioned on an
implicit network to reconstruct the clothed 3D human avatar with motion-
dependent geometry and appearance.

##  2 Related works

3D Clothed Human Reconstruction. Existing methods [18, 21, 8] utilize
parametric 3D body models (e.g., SMPL and SMPL-X [23, 28]) to represent 3D
human pose and shape. However, these models are fundamentally limited in
expressing fine details such as clothing since they were designed specifically
for minimally clothed humans. To overcome this limitation, other methods [2,
1, 4, 22, 49, 40, 41] estimate 3D offsets on top of body mesh vertices. While
these methods add some details, their performance on loose clothing is limited
due to the constrained 3D mesh topologies. Researchers have proposed various
implicit representation techniques for human reconstruction and modeling to
address these limitations, including signed distance fields [36], occupancy
fields [32, 15, 42, 31, 30], and NeRF-based methods [43, 38]. ICON [42]
leverages local features to model the high-frequency details of the clothed
humans’ local geometry. ECON [41] improves upon ICON’s limited generalization
ability for various clothing topologies, particularly loose clothing, by
completing the full 3D shape using front and back normals and surface
information extracted from RGB images. HumanNeRF [38] reconstructs clothed
avatars as a free-viewpoint mixture of affine fields. Despite these advances,
such approaches often suffer from temporal inconsistency in shape
reconstruction because of the lack of temporal surface correspondences,
leading to physically implausible 3D animations.

Animatable Human Avatars. Some works [43, 17] proposed methods for the co-
training of the SDF and NeRF network constraining neural radiance field by
implicit human body model. However, they require a large corpus of posed human
scans and can not generalize well to the poses beyond the distribution of the
training set. SCANimate [33] introduced the end-to-end trainable framework to
build a high-quality parametric clothed human model from raw scans by encoding
locally pose-aware implicit surface representation to model pose-dependent
clothing deformation. However, it requires massive 3D ground truth data, which
is not possible to obtain using a single camera. Animatable NeRF [29] designed
a novel transformation field between view and canonical space to better
memorize the seen poses. Zheng  _et al_. [48] proposed a method to use a set
of structured local radiance fields attached to a human body template. HF-
Avatar [47] proposed coarse-to-fine explicit representation using a dynamic
surface network that refines the unclothed body shape by estimating clothed
shape offsets. Anim-NeRF [6] proposed a geometry-guided deformable NeRF for
explicit pose-guided deformation that enables explicit control of a human
avatar. While those existing methods have shown promising results, they often
struggle to learn a dynamic avatar with large movements due to the absence of
explicit conditioning temporal analysis to model motion-dependent appearance.

##  3 Method

Our method aims to generate high-fidelity clothed 3D human avatars with
motion-dependent appearance from a single camera. It consists of two main
components: explicit pose-dependent appearance modeling (Sec. 3.1) and
implicit motion-dependent appearance modeling (Sec. 3.2). The explicit
modeling component generates pose-dependent geometry and appearance from a 3D
body model whose temporal information is used to extract motion features. The
implicit modeling component decodes the motion features to reconstruct motion-
dependent appearance and geometry. Figure 2 provides an overview of our
pipeline.

Figure 3: (Left) Explicit pose-dependent appearance modeling with 3D
performance capture. It predicts the pose-dependent geometry
$\mathbf{G}_{\text{p}}$ and appearance feature $\mathbf{A}_{\text{p}}$ on
single image. Implicit motion-dependent appearance modeling. (Right) It
predicts the motion-conditioned clothed human mesh $\mathbf{G}_{\text{M}}$
under the target image pose. The global and local motion features are
extracted from the previous $n$ frame’s explicit geometries.

###  3.1 Explicit Pose-dependent Appearance Modeling.

In this section, we model the person-specific 3D human avatar in clothing that
captures the pose-dependent appearance and shape:

$\displaystyle\centering\mathbf{G}_{\text{p}},\mathbf{A}_{\text{p}}\leftarrow
f(\mathbf{P}).\@add@centering$ (1)

$f$ is the modeling function that generates pose-dependent geometry
$\mathbf{G}_{\text{p}}\in\mathbb{R}^{n\times 3}$ and appearance feature
$\mathbf{A}_{\text{p}}\in\mathbb{R}^{u\times v\times 64}$, where $n$ is the
number of vertices, and $\mathbf{A}$ is the canonical UV appearance map whose
width and height are $u$ and $v$, respectively.
$\mathbf{P}\in\mathbb{R}^{n\times 3}$ is the vertices of 3D mesh model (i.e.,
SMPL [23]) parameterized by a human pose parameter p.

Inspired by existing human geometry processing [24], we design the modeling
function $f$ in a way that predicts the pose-dependent geometry in the
residual space for more effective generalization to unseen body poses as
follows:

$\displaystyle\centering\mathbf{G}_{\text{p}},\mathbf{A}_{\text{p}}\leftarrow
f(\mathbf{P}),\ \ \ \ \ \text{where}\ \ \ \
\mathbf{G}_{\text{p}}=\mathbf{P}+\Delta\mathbf{G}_{\text{p}}.\@add@centering$
(2)

$\Delta\mathbf{G}_{\text{p}}\in\mathbb{R}^{n\times 3}$ is the pose corrective
residual.

We train the explicit modeling function $f$ using the following objectives:

$\displaystyle
L_{\text{f}}=\omega_{\text{mask}}L_{\text{mask}}+\omega_{\text{normal}}L_{\text{normal}}+\omega_{\text{lap}}L_{\text{lap}}$
$\displaystyle+\omega_{\text{rgb}}L_{\text{rgb}}+\omega_{\text{vgg}}L_{\text{vgg}}+\omega_{\text{GAN}}L_{\text{GAN}},$
(3)

where mask loss $L_{\text{mask}}$ encourages the silhouette of the
reconstructed 3D mesh to be aligned with the ground truth mask by measuring
their difference: $L_{\text{mask}}=\|\mathbf{M}_{\rm p}-\mathbf{M}_{\rm gt}\|$
where $\mathbf{M}_{\rm p}$ is the rendered 2D mask from the reconstructed
geometry $\mathbf{G}_{\rm p}$, and $\mathbf{M}_{\rm gt}$ is the ground-truth
mask detected from the video using existing human segmentation method [10]. In
a similar way, the normal loss $L_{\text{normal}}$ penalizes the difference
between the rendered surface normal $\mathbf{N}_{\rm p}$ from $\mathbf{G}_{\rm
p}$ and the detected one $\mathbf{N}_{\rm gt}$ from the existing surface
normal detection method [16], i.e., $L_{\text{normal}}=\|\mathbf{N}_{\rm
p}-\mathbf{N}_{\rm gt}\|$, which leads to the reconstruction of the detailed
geometry. The Laplacian smoothing loss $L_{\text{lap}}$ regularizes the
distribution of the predicted offsets by minimizing the second-order
derivatives of the 3D surface similar to the approach from Grassal et al.
[11]. RGB loss $L_{\text{rgb}}$ measures the L1 difference between the
rendered RGB image and the ground truth RGB image, i.e.,
$L_{\text{rgb}}=\|\mathbf{I}_{\rm p}-\mathbf{I}_{\rm gt}\|$, where
$\mathbf{I}_{\rm p}$ is the rendered image and $\mathbf{I}_{\rm gt}$ is the
ground truth image from the video. Additionally, we use VGG loss
$L_{\text{vgg}}$ [34] to consider the content and style of the rendered image.
Finally, we apply the unconditional adversarial loss $L_{\text{GAN}}$ [9] for
producing realistic human texture where the rendered image is used as a fake
and the ground truth as real.
The overview of the network for this section is shown on the left side of Fig.
3, and please also refer to the supplementary material for the network
details.

Implementation Details. In this section, our network inputs are SMPL pose
parameters and outputs are pose-dependent human specific geometry offsets and
appearance features. We first estimate the SMPL pose parameters p and camera
pose $\mathbf{C}$ from a single video using a two-step approach. In the first
step, we obtain the initial pose estimate using an existing 3D pose estimation
method, e.g., VIBE [20]. In the second step, we optimize the initial pose
based on the video cue by minimizing the difference of the 3D mesh surface
with the detected surface (from an existing human surface detection, i.e.,
DensePose [14]) in the image space. From an RGB image, we obtain the 3D mesh
$\mathbf{P}$, its corresponding $\rm p$, and $\mathbf{C}$. We use the VIBE
[20] model for initial $\mathbf{P}$ and $\mathbf{C}$ estimates, and DensePose
for obtaining surface correspondences with the 3D body model in the image
space. We fix the shape of the 3D body model $\mathbf{P}$ with the mean shape.
To better express mesh deformation, we subdivide each triangular face of the
body mesh into four using edge subdivision. We up-sample the template body
mesh vertices from 6,890 to 27,554.

We enable the explicit appearance modeling function $f$, using two networks
for geometry and appearance.

Given an unclothed 3D body mesh model [23], our approach is designed to
generate pose-dependent geometric offset and appearance to reconstruct a
person-specific avatar by following [12].

The geometry network learns to predict the pose-dependent geometry offset. In
practice, it consists of both static and dynamic offset, i.e.,
$\Delta\mathbf{G}_{\text{p}}=\Delta\mathbf{G}^{\text{\rm
s}}_{\text{p}}+\Delta\mathbf{G}^{\rm d}_{\text{p}}$. While the static offset
is a subject-specific offset across all frames, the dynamic offset is a pose-
specific offset unique to each frame. The static offset
$\Delta\mathbf{G}^{\text{\rm s}}_{\text{p}}$ is obtained by feeding a zero-
value vector into the network. This vector has the exact dimensions as the
SMPL pose parameters. The output serves as the static offset, establishing the
subject’s base shape. For the dynamic offset $\Delta\mathbf{G}^{\rm
d}_{\text{p}}$, the network uses SMPL pose parameters to generate pose-variant
offsets. Static and dynamic offsets are combined during the training process
to generate a comprehensive pose-dependent geometry offset. This architecture
enables the model to capture the subject’s inherent, pose-invariant geometry
while also facilitating pose-variant adjustments.

The appearance networks is composed of two parts: appearance feature
extraction and appearance texture rendering. We employ a U-Net architecture
for appearance feature extraction. The input to this U-Net is the geometry
represented in UV coordinates, denoted as
$\mathcal{G}_{\text{p}}=u^{-1}(\mathbf{G}_{\text{p}})$. The output is a pose-
dependent appearance feature $\mathbf{A}_{\text{p}}$ in the UV coordinates. We
warp these appearance features to the image space to render the texture using
a final CNN layer.

###  3.2 Implicit Motion-dependent Appearance Modeling.

In this section, we model an implicit avatar with a motion-dependent
appearance by encoding the temporal context of the explicit pose-dependent
geometry $\mathbf{G}_{\text{p}}$ and appearance feature
$\mathbf{A}_{\text{p}}$ modeled in Sec 3.1:

$\displaystyle\centering\mathbf{o},\mathbf{c}\leftarrow
g(\\{\mathbf{G}_{\text{p}^{t}},\mathbf{G}_{\text{p}^{t-1}},\cdots,\mathbf{G}_{\text{p}^{t-n}}\\},\mathbf{A}_{\text{p}^{t}};\mathbf{X}),\@add@centering$
(4)

where $g$ is the implicit function designed with multi-layer perceptrons
(MLPs) which take the sequence of 3D pose-dependent geometry
$\\{\mathbf{G}_{\text{p}^{t}},\mathbf{G}_{\text{p}^{t-1}},...,\mathbf{G}_{\text{p}^{t-n}}\\}$,
explicit appearance feature $\mathbf{A}_{\text{p}^{t}}$ at the target pose,
and the sampled 3D point $\mathbf{X}$; and outputs implicit avatar with the
motion-dependent appearance in the form of occupancy $\mathbf{o}\in[0,1]$ and
color $\mathbf{c}\in\mathbb{R}^{3}$.

We further extract the high-dimensional motion features using an encoder
$E_{\rm m}$ which allows encoding more discriminative and robust motion
context [45]:

$\displaystyle\mathbf{o},\mathbf{c}\leftarrow g(\\{\mathcal{M}_{t}^{\rm
G},\mathcal{M}_{t}^{\rm L}\\},\mathbf{A}_{\text{p}^{t}};\mathbf{X})$
$\displaystyle=g(E_{\rm
m}\\{\mathbf{G}_{\text{p}^{t}},\mathbf{G}_{\text{p}^{t-1}},\cdots,\mathbf{G}_{\text{p}^{t-n}}\\},\mathbf{A}_{\text{p}^{t}};\mathbf{X}).$
(5)

In practice, the motion encoder $E_{\rm m}$ extracts the high-dimensional
features in the UV coordinates, which result in the high-dimensional global
motion features $\mathcal{M}_{t}^{\rm G}\in\mathbb{R}^{1\times 64}$ that
guides the motion with respect to the entire body; and the high-dimensional
local motion features $\mathcal{M}_{t}^{\rm L}\in\mathbb{R}^{u\times v\times
64}$ that guides the motion of the local surface, e.g., secondary clothing
movement as shown in Fig. 3-(right).

We train implicit modeling function $g$ using the following objectives:

$\displaystyle
L_{\text{g}}=\omega_{\text{IOU}}L_{\text{IOU}}+\omega_{\text{color}}L_{\text{color}}+\omega_{\text{norm}}L_{\text{norm}}+\omega_{\text{eik}}L_{\text{eik}},$
(6)

where the IOU loss $L_{\text{IOU}}$ encourages the alignment between the
estimated segmentation mask and the ground-truth mask. It is computed as the
Intersection Over Union:
$L_{\text{IOU}}=1-\text{IOU}(\mathbf{M}_{\text{M}},\mathbf{M}_{\rm gt})$,
where $\mathbf{M}_{\text{M}}$ is the rendered 2D mask from geometry
$\mathbf{G}_{\text{M}}$ using a differentiable renderer, as described by [39].

The color loss $L_{{\text{color}}}$ measures the L1 distance between the
rendered color $\hat{\mathbf{c}}_{\mathbf{p}}$ at pixel $\mathbf{p}$,
generated by the implicit differential renderer, and the ground-truth color
$\mathbf{c}^{\text{gt}}_{\mathbf{p}}$ at pixel $\mathbf{p}$ from the input
image:
$L_{\text{color}}=(1/|\text{P}|)\cdot\sum_{\mathbf{p}\in\text{P}}|\hat{\mathbf{c}}_{\mathbf{p}}-\mathbf{c}^{\text{gt}}_{\mathbf{p}}|$.
This loss helps to refine the geometry and texture of the implicit
representation. We employ the Implicit Differentiable Renderer (IDR) [44],
which enables the efficient and differentiable rendering of an implicit model.

Similarly, the normal loss $L_{\text{norm}}$ measures the discrepancy between
the predicted rendered normal $\hat{\mathbf{n}}_{\mathbf{p}}$ at pixel
$\mathbf{p}$ and the pseudo ground-truth normal
$\mathbf{n}^{\text{gt}}_{\mathbf{p}}$ at pixel $\mathbf{p}$ using the normal
map $\mathbf{N}_{\rm gt}$. It can be formulated as follows:
$L_{\text{norm}}=(1/|\text{P}|)\cdot\sum_{\mathbf{p}\in{\text{P}}}\|\hat{\mathbf{n}}_{\text{p}}-\mathbf{n}^{\text{gt}}_{\mathbf{p}}\|_{2}.$

Lastly, the eikonal loss $L_{\text{eik}}$ is a regularization term inspired by
IGR [13] that enforces the property of signed distance functions, where the
L2-norm of the gradient of sampling points on the surface should be 1. This is
achieved by computing the L2-norm of the gradient of the sampling points on
the surface and penalizing deviations from 1 using a squared loss.
The overview of the network for this section is shown on the right side of
Fig. 3.

Implementation Details. Our network inputs are pose-variant explicit geometry
information and appearance features from Sec 3.1 and outputs are motion-
dependent implicit geometry and rendered appearance.

To handle modality mismatch between the 3D mesh and 2D inputs to the neural
network, in practice, we encode the geometry in the UV coordinates,
$\mathcal{G}_{\text{p}}=u^{-1}(\mathbf{G}_{\text{p}})$, where the inverse UV
mapping function $u^{-1}$ converts the 3D geometry $\mathbf{G}_{\text{p}}$
into a 2D representation in the UV domain
$u^{-1}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$. Since the UV coordinate is
pose-invariant representation, we represent the motion by simply concatenating
temporally consecutive $\mathcal{G}_{\text{p}}$:
$\bigoplus(\mathcal{G}_{\text{p}^{t}},\mathcal{G}_{\text{p}^{t-1}},\cdots,\mathcal{G}_{\text{p}^{t-n}})$.
Utilizing this, we extract global motion features and local motion features
for implicit motion-dependent appearance modeling with a U-Net structure:

$\displaystyle E_{\rm
m}:\bigoplus(\mathcal{G}_{\text{p}^{t}},\mathcal{G}_{\text{p}^{t-1}},\cdots,\mathcal{G}_{\text{p}^{t-n}})\in\mathbb{R}^{\text{T}\times
128\times 128\times 3}\;$ $\displaystyle\rightarrow\;\mathcal{M}_{t}^{\rm
G}\in\mathbb{R}^{1\times 64},\;\mathcal{M}_{t}^{\rm L}\in\mathbb{R}^{128\times
128\times 64}$ (7)

where the global motion feature $\mathcal{M}_{t}^{\rm G}$ is from the latent
space of the network, and the local motion features $\mathcal{M}_{t}^{\rm L}$
is from the final output from the decoder. We assign the extracted motion
features to the closest 3D points sampled for the implicit motion-dependent
avatar.

For the texture network, we modify the IDR [44]. It receives the zero
isosurface’s point, its normal, view direction, and its global geometry
feature and outputs the point’s color along the view direction. Furthermore,
we modify this network to render our implicit geometry with high-quality
texture. Our modified network receives appearance feature
$\mathbf{A}_{\text{p}^{t}}$ estimated from Section 3.1 as additional input. We
feed $\mathbf{A}_{{\text{p}}}$ to each point after UV unwrapping changing the
shape from $128\times 128\times 64$ to $27,554\times 64$, and perform the same
local sampling. Using this texture network, we generate the final image
$\mathbf{I}_{\text{M}}$. Please refer to the supplementary document for the
network details.

Figure 4: We compare our method to several baselines (HF-Avatar [47], Anim-
NeRF [6]) on various sequences(Youtube1-2, MPI and Custom1). For each example,
we show the ground truth (GT) target appearances, the rendered appearances by
each method, and the reconstructed geometries from sampled two frames.

| Custom 1 | MPI | Youtube1 | Youtube2
---|---|---|---|---
| SSIM$\uparrow$ | LPIPS $\downarrow$ | tOF$\downarrow$ | SSIM $\uparrow$ | LPIPS$\downarrow$ | tOF$\downarrow$ | SSIM $\uparrow$ | LPIPS$\downarrow$ | tOF $\downarrow$ | SSIM $\uparrow$ | LPIPS$\downarrow$ | tOF $\downarrow$
HF-Avatar [16] | 0.865 | 0.115 | 0.7021 | 0.878 | 0.132 | 0.4861 | 0.836 | 0.207 | 1.1018 | 0.8690 | 0.182 | 0.7921
Anim-NeRF [6] | 0.888 | 0.109 | 0.6088 | 0.867 | 0.106 | 0.5082 | 0.859 | 0.101 | 0.8573 | 0.8782 | 0.149 | 0.6130
Ours | 0.911 | 0.074 | 0.5031 | 0.882 | 0.077 | 0.4220 | 0.872 | 0.086 | 0.7877 | 0.9122 | 0.078 | 0.4176

Table 1: Quantitative Results. We compared our method to state-of-the-arts (HF-Avatar [47], Anim-NeRF [6] on four subjects. We evaluated on three metrics: SSIM $(\uparrow$), LPIPS ($\downarrow$), and tOF ($\downarrow$). Our method outperforms state-of-the-arts for all subjects. Figure 5: More qualititative results of our method tested on People-Snapshot [3] dataset. This is reconstruction results for ground truth images. Sequential rows depict our method’s geometry results, rendered appearance results, and ground truth images (GT).  Figure 6: The ablation study on each module of our method: (column1) Stage 1: only with the explicit pose-dependent appearance model, (column2) without Stage1: utilizing temporal motion features extracted from unclothed human model meshes with the implicit human representation, (column3) Ours. | Custom 1
---|---
| SSIM$\uparrow$ | LPIPS $\downarrow$ | tOF $\downarrow$
Stage 1 (explicit model) | 0.923 | 0.092 | 0.5034
w/o Stage 1 (only SMPL) | 0.904 | 0.081 | 0.5240
Ours | 0.911 | 0.074 | 0.5031
Table 2: Ablation of each stage. We evaluated its module on Subject 1. Each
module shows the significant performance improvement on three metrics: SSIM
$(\uparrow$), LPIPS ($\downarrow$), and tOF ($\downarrow$). Figure 7: Our
method can generate motion avatars with random SMPL sequences. We utilize SMPL
parameters from publicly available AIST [35] datasets to obtain random
parameters for unseen poses, and leverage our previously trained MPI dataset
to generate new motion-dependent avatars.

##  4 Experiments

Datasets. We perform experiments using a number of video sequences of people
in loose clothing with dynamic movements such as dancing and jumping captured
from a previous work [45], which are including two dance videos from YouTube,
and a video sequence from prior work [19]. We use l.5k frames from each video
for training and measure the performance on unseen sequences with 200 frames.
We use pose and camera parameters extracted from [20] for the test sequences.
We also use People-Snapshot dataset [3] for the qualitative result. This
dataset provides videos of subjects rotating in the A-pose. Unlike other
datasets, the People-Snapshot dataset is characterized by small motion
dynamics.

Metrics. We evaluate the quality of rendered images from the reconstructed 3D
avatars using three metrics: 1) Structure Similarity (SSIM)[37] assesses
luminance, contrast, and structure quality; higher SSIM indicates greater
similarity to the ground truth. 2) Learned Perceptual Image Patch Similarity
(LPIPS)[46] compares generated images with ground truth in feature space using
a deep neural network, determining perceptual similarity. 3) Temporal Optical
Flow (tOF) measures the difference in optical flow between synthesized and
ground-truth videos; lower tOF scores suggest more physically plausible
appearances relative to real videos.

Baselines. In our experiments, we compare our method to three state-of-the-art
animatable modeling approaches using a single camera. HF-Avatar [47] refines
the unclothed body shape of an explicit 3D model by estimating clothing shape
offsets. Anim-NeRF [6] employs an implicit Neural Radiance Field to model 3D
avatars and a skinning function for pose adaptation. For HF-Avatar, we provide
initial SMPL pose, shape, and camera parameters extracted from [20] due to
failures in our evaluation dataset with significant motion dynamics.

Results. We qualitatively and quantitatively evaluate our dynamic human
modeling approach against baseline methods. Table 1 shows our method
outperforms baselines in image quality and temporal plausibility. Fig 4
demonstrates our method’s ability to generate high-quality clothed human
avatars conditioned on motion. HF-Avatar struggles with rotating motions and
over-smoothed results due to topology constraints. Anim-Nerf shows significant
noise around clothing with large dynamics, failing to learn motion-dependent
appearance. Our method excels in generating high-quality clothed human
avatars. Our method captures fine details in geometry and appearance,
accurately corresponds motion features with implicit geometry, and employs an
adapted texture network to generate high-quality textures, thus outperforming
baseline methods in image quality and temporal plausibility. We also
demonstrate our avatar modeling results including the geometry and appearance
on People-Snapshot [3] data in Fig. 5. By achieving high-quality results on
this low-motion dataset, we demonstrate the robust and consistent performance
of our approach under a variety of conditions.

Ablation Study. Our compositional human modeling pipeline consists of two
stages. Stage 1 estimates pose-dependent explicit 3D reconstruction of clothed
humans and UV appearance features. In stage 2, implicit networks refine
geometry conditioned on motion features on the 3D-clothed surface. Fig. 6 and
Table. 2 show our full method outperforms ablation baselines in terms of LPIPS
and tOF. The 3D human model generated with explicit representation only
(Stage1) is constrained by SMPL topologies, limiting fine-grained surface
geometry. Using unclothed SMPL body models without Stage 1 (w/o Stage 1)
restricts fine-grained 3D point sampling, leading to drops in geometric
quality. However, Stage 1 performs better in SSIM measurement as it heavily
relies on generative neural network functions without considering geometric
quality impact.

Application. Our method can generate 3D avatar animation from random SMPL
sequences. Fig. 7 shows an example of its application. We show that our method
is highly generalizable to unseen poses. Please refer to the supplementary
video for the full animation.

##  5 Conclusion

In this paper, we introduce a method to model a dynamic clothed 3D human
avatar with a plausible motion-dependent appearance using a monocular video.
To achieve this, we propose a novel compositional framework that can take the
benefits from both explicit and implicit approaches. An explicit network
predicts the UV texture features and topology-invariant geometry residuals,
and an implicit function constructs high-fidelity local geometry and
appearance conditioned on the explicit surface motion. We evaluate our method
on multiple video sequences with large motion dyanmics, and our method shows
the best performance compared to previous methods.

Limitation. Since our training method is completely free from 3D ground truth,
it often includes reconstruction noise on geometry. Our rendering model shows
a weak performance (blur artifacts) to learn the appearance with fast motion.
Our method highly depends on initial body mesh alignment performance, i.e.,
the errors from monocular performance tracking propagate to the human avatar
modeling pipeline.

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# Abelian varieties not isogenous to Jacobians over global fields

Ananth N. Shankar and Jacob Tsimerman

###### Abstract.

We prove the existence of abelian varieties not isogenous to Jacobians over
characterstic $p$ function fields. Our methods involve studying the action of
degree $p$ Hecke operators on hypersymmetric points, as well as their effect
on the formal neighborhoods using Serre Tate co-ordinates. We moreover use our
methods to provide another proof over number fields, as well as proving a
version of this result over finite fields.

##  1\. Introduction

This paper concerns the following question: _Given an algebraically closed
field $k$ and $g\geq 4$, Does there exist an Abelian variety over $k$ of
dimension $g$ which is not isogenous to the Jacobian of a stable curve?_. The
history of the question, as far as we know, is that it was first asked by
Nicholas Katz and Frans Oort in the setting of $k=\overline{\mathbb{Q}}$, and
then posed more generally by Ching-Li Chai and Oort in [4]. We mention that an
observation of Poonen (unpublished, but credited by Chai and Oort) which
helped clarify the situation, was to formulate the question more generally for
an arbitrary subvariety of $\mathcal{A}_{g}$ in place of the Torelli locus. Of
course, one can also generalize from $\mathcal{A}_{g}$ to an arbitrary Shimura
variety.

In the case of $k=\overline{\mathbb{Q}}$ the question has been answered in the
affirmative: Chai-Oort[4] proved it conditional on the Andre-Oort conjecture
for $\mathcal{A}_{g}$, and then the second-named author made their proof
unconditional [13]. Since then the Andre-Oort conjecture itself has been
proven [14] in this case, removing the need for [13]. Recently, Masser-Zannier
[11] proved a stronger result, producing Abelian varieties satisfying the
condition which are defined over low-degree fields, and in fact showing that
‘most’ such Abelian varieties satisfy the condition, in a precise sense.

We now turn our attention to the case of finite characteristic, about which
much less is known. Over $\overline{\mathbb{F}}_{p}$, the authors[12] have a
previous work in which they make conjectures based on arithmetic statistics
which suggests that the statement should not have affirmative answer for all
$g\geq 4$. In fact, [12] suggests that if $D\subset\mathcal{A}_{g}$ is a
generically ordinary divisor defined over $\overline{\mathbb{F}}_{p}$ (with
$g>1$), then every $x\in\mathcal{A}_{g}(\overline{\mathbb{F}}_{p})$ should be
isogenous to some $y\in D(\overline{\mathbb{F}}_{p})$.

In the way of unconditional results, there is only a result of Chai-Oort[4,
§4] which deals with the analogous situation of a curve inside $X(1)^{2}$
(which produces non-ordinary points) and a result of the authors[12, Thm 4.1]
for a specific hypersurface in $X(1)^{N}$ for $N\geq 270$, using additive
combinatorics. Relatedly, work of the first author, Asvin G. and Q.He [9]
shows that if $C_{1},C_{2}$ are two generically ordinary curves contained in
suitable mod $p$ Hilbert modular surfaces, then there are infinitely many
closed points on $C_{1}$ isogenous to some point on $C_{2}$.

The most analogous setting to number fields is function fields in one
variable, and thus the natural analogue to $\overline{\mathbb{Q}}$ is
$\overline{\mathbb{F}_{p}(T)}$. The main goal of this paper is to deal with
precisely that case.

Our main theorem is as follows:

######  Theorem 1.1.

Let $k$ be a finite field, let $F=\overline{k(t)}$, and $g>1$. Let
$D\subset\mathcal{A}_{g}/F$ be a divisor. There exists an $F$-valued point of
$\mathcal{A}_{g}$ which is not isogenous to any $F$-points of $D$.

As a stepping stone to our main theorem, we first prove the following:

######  Theorem 1.2.

Let $k$ be a finite field, and $g>1$. Let $D\subset\mathcal{A}_{g}/k$ be a
divisor. Let $F=\overline{k(t)}$. There exists an $F$-valued point of
$\mathcal{A}_{g}$ which is not isogenous to any $F$-points of $D$.

The heuristics offered in [12] suggest that Theorem 1.2 is strongest possible
theorem that is true in positive characteristic; namely, that
$\overline{k(t)}$ is the smallest algebraically closed field over which one
can expect the existence of an abelian variety not isogenous to any point on
$D$.

Of course, one may formulate the above theorem instead in terms of curves
contained in $\mathcal{A}_{g}$ by using the familiar translation between
curves and function fields.

###  1.1. Other results

Theorem 1.2 yields yet another proof of the number field version of Theorem
1.2 (namely, the existence of abelian varieties over number fields not
isogenous to Jacobians). Our method also yields the following result over
finite fields:

######  Theorem 1.3.

Let $D\subset\mathcal{A}_{g}$ denote a divisor over
$\overline{\mathbb{F}}_{p}$. Then, there exists an ordinary
$x\in\mathcal{A}_{g}(\overline{\mathbb{F}}_{p})$ such that $T(x)\not\subset D$
for every prime-to-$p$ Hecke correspondence $T$.

Theorem 1.3 acts as a substitute over $\overline{\mathbb{F}}_{p}$, because as
stated before we expect that every ordinary
$x\in\mathcal{A}_{g}(\overline{\mathbb{F}}_{p})$ is isogenous to some point in
$D(\overline{\mathbb{F}}_{p})$.

We remark also that Theorem 1.3 also highlights one of the
differences/difficulties in handling the $\overline{\mathbb{F}}_{p}$ case: The
Galois orbits are just much smaller, so that $T(x)$ cannot be a single orbit,
unlike the case of number fields or function fields.

###  1.2. Ideas of Proofs

####  1.2.1. Proof of Theorem 1.2

Fix a divisor $D\subset(\mathcal{A}_{g})_{\overline{k}}$. Our first
observation is that in the function field case one can impose very strong
‘local’ conditions111It is unclear to us whether one can impose analogously
strong local conditions over number fields. on our $F$-point, which rule out
being contained in $D$. Thinking of $F$-valued point as curves
$C\subset\mathcal{A}_{g}$, this amounts to making a curve $C$ which is highly
singular at a given point, such that for example its formal neighborhood
contains much of the formal neighborhood of $\mathcal{A}_{g}$. There are some
difficulties with implementing this strategy, namely:

  1. (1)

While the local structure of prime-to-$p$ Hecke operators is very well
understood, $p$-power Hecke operators behave poorly in characteristic $p$.
These are not etale (or even finite) and strongly distort the local structure.

  2. (2)

It is a-priori unclear how to insist that every _irreducible_ Hecke translate
of $C$ also has this property. A-priori, different branches of $C$ through a
point $x$ could separate upon applying a Hecke operator.

We overcome the second difficulty by insisting that the curve has maximal
monodromy by using Lefschetz theorems in conjunction with Bertini-style
theorems due to Poonen and Charles-Poonen. The first difficulty is the main
one, and our idea in overcoming it is to use _hypersymmetric_ abelian
varieties (see Definition Definition for the definition). The $p$-power Hecke
orbit of a hypersymmetric point consists of points defined over the same field
as the original point (similar in spirit to the prime-to-$p$ Hecke orbit of a
supersingular point). In particular, the orbit of a hypersymmetric point under
$p$-power isogenies is finite.

This prevents ’escape to infinity’ and allows us to impose certain local
conditions on this whole finite set at once, which we can then control. To
impose our local conditions, we use the billinear structure on the tangent
space (and in fact the entire formal neighborhood) inherited from the Serre-
Tate co-ordinates to isolate certain favored directions which behave stably
under $p$-power isogenies.

####  1.2.2. Proof of Theorem 1.1

It turns out that the hardest case of this theorem is Theorem 1.2. Our
approach is essentially to notice that $\overline{k}(T)$ contains many
distinct subfields - in fact, copies of itself: $\overline{k}(P(T))$. One can
show that if $D$ is defined over some function field $E$ of $F$, then we may
find a smaller subfield $E^{\prime}\subset E$ such that the intersection of
$D$ with its own Galois conjugates over $E^{\prime}$ is defined over
$\overline{k}$. But if $C$ is a curve defined over $E^{\prime}$, then
$C\subset D$ implies that $C$ is contained in the intersection of $D$ with all
its galois conjugates over $E^{\prime}$. One must be a bit careful, and we
work with curves instead of function fields, but this idea can essentially be
pushed through to reduce to Theorem 1.2.

####  1.2.3. Proofs of Additional Results

This same idea can be carried out (with a lot less difficulty, as we no longer
face any charcteristic $p$ issues) to find a curve $C\subset\mathcal{A}_{g}$
none of whose Hecke translates are contained in $D$, where $C$ is now a curve
defined over a _Number Field_. We use the existence of such a curve, along
with a "big monodromy" result due to Zywina, to provide an intersection-
theoretic proof of the existence of Number Field-valued points of
$\mathcal{A}_{g}$ that are not isogenous to any point of $D$. This method also
goes through to prove Theorem 1.3.

###  1.3. Structure of Paper

In section 2 we show how to construct Abelian varieties over curves with
prescribed local and monodromy conditions. In section 3 we review the theory
of Serre-Tate coordinates, introduce the notion of primitive Serre-Tate
directions and study how they behave under isogenies. This is the technical
heart of the paper. We then go on to prove Theorem 1.2, and then upgrade it to
Theorem 1.1. Finally, in Section 4 we give a couple of other results: we
explain how our methods can give a rather short proof of the same result over
number fields (although relying on a strong Monodromy theorem of Zywina) and
we prove Theorem 1.3.

###  1.4. Acknowledgements

The first author is grateful to Bjorn Poonen for useful conversations. The
second author is grateful to Frans Oort for introducing him to the problem and
many useful discussions. We are also grateful to Daniel Litt for providing us
with a reference for tame Lefschetz theorems.

##  2\. Constructing families of Abelian varieties with specified local
conditions

In what follows, for an irreducible curve $C$ when referring to the _geometric
monodromy_ of $C$, we mean $\pi_{1}(C_{\eta})$ wehere $\eta$ is the generic
point.

We fix a prime222The purpose of $r$-torsion is simply to rigidify the moduli
problem. $r>6p$ which is distinct from the characteristic $p$. We fix a smooth
compactification of $\overline{\mathcal{A}}_{g}[r]$, with boundary divisor
$D$. Note that over $\mathcal{A}_{g}[r]$ we have the $n$-torsion local systems
$\mathcal{L}_{n}$ for $n$ prime to $p$, as well as the $p^{m}$-torsion local
systems $\mathcal{L}_{p^{m}}$ over the ordinary locus
$\mathcal{A}_{g}^{\operatorname{ord}}[r]$, defined as the dual of the
connected component of the $p^{m}$-torsion of the universal Abelian scheme.

We shall need the following result:

######  Theorem 2.1.

Fix $g>1$, and an auxiliary prime $r>6p$. For any finite set of
$\operatorname{Spec}\overline{\mathbb{F}}_{p}[[t]]$ points $x_{1},\dots,x_{n}$
whose differential is injective in $\mathcal{A}^{\operatorname{ord}}_{g}[r]$,
and positive integers $m_{1},\dots,m_{n}$ there exists a curve
$C\subset\mathcal{A}^{\operatorname{ord}}_{g}[r]$ such that

  1. (1)

$C$ admits $\operatorname{Spec}\overline{\mathbb{F}}_{p}[[t]]$ points
specializing to all of the $x_{i}$ mod the $t^{m_{i}}$ and

  2. (2)

The geometric monodromy of $C$ surjects onto
$\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$

where

$\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell})):=\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell})/\mathbb{Z}^{\times}_{\ell}$

The idea for the proof is twofold: For the prime-to-$p$ part we use tame
Lefschetz theorems (see [6, Theorem 7.3] and [7]), and for the $p$-part we
simply pick enough $\mathbb{F}_{q}$ points whose Frobenius images generate.

######  Lemma 2.2.

For a positive integer $n$ prime to $p$, the $n$-torsion local system
$\mathcal{L}_{n}$ over $\mathcal{A}_{g}[r]$ is tamely ramified at $D$.

###### Proof.

The image of inertia at the boundary in
$\operatorname{GSp}_{2g}(\mathbb{Z}/n\mathbb{Z})$ is well-known to be
unipotent, and therefore has order prime to $p$.

∎

Next, let $k$ be a finite field over which the union of all the $x_{i}$ is
defined. We pick finitely many points in $\mathcal{A}_{g}[r](k)$ whose
Frobenius images contain every conjugacy class in
$\operatorname{GL}_{g}(\mathbb{Z}/p^{2}\mathbb{Z})$. Call the union of all
those $R$. It follows that every curve defined over $k$ containing $R$ as a
smooth divisor has fundamental group surjecting onto
$\operatorname{GL}_{g}(\mathbb{Z}/p^{2}\mathbb{Z})$ and hence onto all of
$\operatorname{GL}_{g}(\mathbb{Z}_{p})$. We moreover claim that such a curve
$C$ has _geometric_ fundamental group surjecting onto
$\operatorname{GL}_{g}(\mathbb{Z}/p^{2}\mathbb{Z})$. Indeed, pick a point
$Q\in R\subset C$ such that the Frobenius at $Q$ maps trivially to
$\operatorname{GL}_{g}(\mathbb{Z}/p^{2}\mathbb{Z})$. Then from the sequence

$1\rightarrow\pi_{1}(C_{\overline{k}},\overline{Q})\rightarrow\pi_{1}(C_{k},\overline{Q})\rightarrow\pi_{1}(k)\rightarrow
1$

the claim follows.

Next, as we will be applying Goursat’s lemma to join the $p$-part and prime-
to-$p$ part of the Monodromy, we shall need to consider quotients of
$\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))$ which can occur as
quotients of $\operatorname{GL}_{g}(\mathbb{Z}_{p})$. Since the Chevalley
groups are simple (with finitely many possible exceptions) most factors don’t
contribute, and so it is easy to see that all such quotients factor through a
single finite quotient group $H$ of
$\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))$ and $H^{\prime}$ of
$\operatorname{GL}_{g}(\mathbb{Z}_{p})$.

Now, by possibly increasing $k$ we increase $R$ to contain points in
$\mathcal{A}^{\operatorname{ord}}_{g}[r](k)$ whose Frobenius image contain
every conjugacy class in $H\times H^{\prime}$. As above, every curve defined
over $k$ and containing $R$ as a smooth divisor has geometric fundamental
group surjecting onto $H\times H^{\prime}$. We moreover insist that $R$ is
distinct from the closed points in the image of the $x_{i}$. We are now ready
to complete the proof.

###### Proof.

of Theorem 2.1: First, consider a series of blowups at points of
$\mathcal{A}_{g}[r]$ to obtain a smooth variety $B$ in which all the $x_{i}$
seperate. Now we apply Poonens result [15, Thm 1.3] to obtain a smooth curve
$C_{0}$ which is obtained from $B$ by repeatedly intersecting with the ample
class, such that $C_{0}$ is smooth, intersects $D$ transversally, is defined
over $k$, and contains the lifts of the $x_{i}$ mod $t^{m_{i}}$. Note that by
[5, Thm 1.1] we may insist that $C_{0}$ is moreover irreducible. We will
define $C$ to be the image of $C_{0}$, so this already implies that condition
(1) is satisfied.

Next, we will moreover ensure (using the same result) that $C_{0}$ contains
the points above $R$ as smooth divisors, which will imply that the geometric
monodromy of $C_{0}$ surjects onto $\operatorname{GL}_{g}(\mathbb{Z}_{p})$.
Separately, the Lefschetz theorem proven by Esnault-Kindler together with
lemma 2.2 implies that the fundamental group of $C_{0}$ surjects onto
$\Gamma_{g}(\mathbb{Z}_{r})\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))$.

Finally, Goursat’s lemma together with the fact that the geometric monodromy
of $C_{0}$ surjects onto $H\times H^{\prime}$ implies that the fundamental
group of $C_{0}$ surjects onto all of
$\Gamma_{g}(\mathbb{Z}_{r})\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$
as desired.

∎

##  3\. Hecke translates in characteristic $p$

###  3.1. Background on Serre-Tate coordinates

We will briefly recall Serre-Tate coordinates and describe the local action of
$p$-power hecke operators in terms of these coordinates. For a thorough
treatment of Serre-Tate coordinates, see [10], or [3].

Let $\mathscr{G}$ denote an ordinary $p$-divisible group over an algebraically
closed field $k$ with dimension $g$ and height $2g$. We have that
$\mathscr{G}\simeq\mathscr{G}^{\operatorname{mult}}\times\mathscr{G}^{\operatorname{et}}$,
where $\mathscr{G}^{\operatorname{mult}}\simeq\mu_{p^{\infty}}^{g}$ and
$\mathscr{G}^{\operatorname{et}}\simeq(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{g}$.
The $p$-divisible groups $\mathscr{G}^{\operatorname{mult}}$ and
$\mathscr{G}^{\operatorname{et}}$ are rigid, and the functoriality of the
connected-étalè exact sequence implies that deformations of of $\mathscr{G}$
to an Artin local $k$-algebra $R$ are in bijection with extensions of
$\mathscr{G}^{\operatorname{et}}\times\operatorname{Spec}R$ by
$\mathscr{G}^{\operatorname{mult}}\times\operatorname{Spec}R$. In particular,
the deformation space of $\mathscr{G}$ has a natural group structure.

Let $T_{p}(\mathscr{G})$ denote the Tate-module of
$\mathscr{G}^{\operatorname{et}}$, and let $T_{p}(\mathscr{G}^{\vee})$ denote
the Tate-module of $(\mathscr{G}^{\operatorname{mult}})^{\vee}$, where ∨
denotes taking the Cartier dual. Then, the formal deformation space of
$\mathscr{G}$, which we have already seen has the structure of a group,
actually has the structure of a formal torus, and is canonically isomorphic to
$\operatorname{Hom}(T_{p}(\mathscr{G})\otimes
T_{p}(\mathscr{G}^{\vee}),\hat{\mathbb{G}}_{m})$. It will later also be
convenient to use the canonical identification of this torus with
$(T_{p}(\mathscr{G})^{*}\otimes
T_{p}(\mathscr{G}^{\vee})^{*})\otimes\hat{\mathbb{G}}_{m}$. Here, ∗ denotes
taking the linear dual of a $\mathbb{Z}_{p}$-module.

Specifically, let $R$ denote any Artin local $k$-algebra, with maximal ideal
$\mathfrak{m}$. Then, the deformations of $\mathscr{G}$ to $R$ are in
bijection with bilinear maps $q:T_{p}(\mathscr{G})\times
T_{p}(\mathscr{G}^{\vee})\rightarrow 1+\mathfrak{m}$. Therefore, a
$\mathbb{Z}_{p}$-basis $e_{i}=\\{e_{1}\ldots e_{g}\\}$ of $T_{p}(\mathscr{G})$
and $m_{j}=\\{m_{1}\ldots m_{g}\\}$ of $T_{p}(\mathscr{G}^{\vee})$ yield
coordinates $Q=\\{q_{ij}\\}$ on the deformation space, i.e. the characteristic
$p$ deformation space is isomorphic to $\operatorname{Spf}k[[q_{ij}-1]]$, and
an $R$-valued point of the deformation space corresponds to a choice of
$g^{2}$ elements in $1+\mathfrak{m}$, which is the same data as a bilinear map
$q:T_{p}(\mathscr{G})\times T_{p}(\mathscr{G}^{\vee})\rightarrow
1+\mathfrak{m}$. Changing the coordinates $e_{i}$ by
$A\in\operatorname{GL}_{g}(\mathbb{Z}_{p})$ and $m_{j}$ by
$B\in\operatorname{GL}_{g}(\mathbb{Z}_{p})$ yields a new set of coordinates
$Q^{\prime}=\\{q^{\prime}_{ij}\\}$, with $Q^{\prime}=A^{-1}QB^{\textrm{T}}$.
The same analysis holds when $R$ is a complete local $k$-algebra. Here we
think of $Q$ as an element of $M_{g}((1+m)^{\times})$, which inherits the
structure of a $M_{g}(\mathbb{Z}_{p})$ bi-module from the
$\mathbb{Z}_{p}$-module structure of $(1+m)^{\times}$.

Let $\mathscr{G}^{\prime}$ denote another $p$-divisible group and let
$\phi:\mathscr{G}\rightarrow\mathscr{G}^{\prime}$ be an isogeny. We pick bases
$\\{e^{\prime}_{i},m^{\prime}_{j}\\}$ for
$T_{p}(\mathscr{G}^{\prime}),T_{p}(\mathscr{G}^{\prime\vee})$, and let $X$ and
$Y$ denote the matrices (necessarily with non-zero determinant) of
$\phi:T_{p}(\mathscr{G})\rightarrow T_{p}(\mathscr{G}^{\prime})$ and
$\phi^{\vee}:T_{p}(\mathscr{G}^{\prime\vee})\rightarrow
T_{p}(\mathscr{G}^{\vee})$ in these coordinates. Let
$\tilde{\mathscr{G}}/k[[t]]$ deform $\mathscr{G}$, and let $Q=\\{q_{ij}(t)\\}$
denote its Serre-Tate coordinates. The following result is well known (see for
example [10, Theorem 2.1] or [3, Proposition 2.23] ).

######  Proposition 3.1.

Notation as above, and consider the matrix $Q^{\prime}=X^{-1}QY$. Note that
a-priori the entries of $Q^{\prime}$ are valued only in
$k((t^{\frac{1}{p^{\infty}}}))$.

  1. (1)

Suppose that the entries of $Q^{\prime}$ are valued in $1+tk[[t]]$. Then,
$\phi$ lifts uniquely to an isogeny
$\tilde{\mathscr{G}}\rightarrow\tilde{\mathscr{G}}^{\prime}$ over $k[[t]]$
(where $\tilde{\mathscr{G}}^{\prime}$ is necessarily a deformation of
$\mathscr{G}^{\prime}$) and the Serre-Tate coordinates (with respect to
$\\{e^{\prime}_{i},m^{\prime}_{j}\\}$) of $\tilde{\mathscr{G}}^{\prime}$ are
$Q^{\prime}$.

  2. (2)

If the entries of $Q^{\prime}$ aren’t valued in $1+tk[[t]]$ (eg. if some entry
looks like $(1+t)^{\frac{1}{p}}$), then $\phi$ doesn’t lift to an isogeny over
$k[[t]]$ with source $\tilde{\mathscr{G}}$. However, let
$n=n(\tilde{\mathscr{G}},\phi)$ denote the smallest positive integer such that
the entries of $Q^{\prime}(t^{p^{n}})$ are valued in $1+tk[[t]]$. Then, $\phi$
lifts uniquely to an isogeny
$\tilde{\mathscr{G}}\times_{\operatorname{Spf}k[[t]]}\operatorname{Spf}k[[s]]\rightarrow\tilde{\mathscr{G}}^{\prime}$,
where the map $\operatorname{Spf}k[[s]]\rightarrow\operatorname{Spf}k[[t]]$ is
given by $t\mapsto s^{p^{n}}$, and $\tilde{\mathscr{G}}^{\prime}$ is a
deformation of $\mathscr{G}^{\prime}$ to $k[[s]])$. The Serre-Tate coordinates
of
$\tilde{\mathscr{G}}\times_{\operatorname{Spf}k[[t]]}\operatorname{Spf}k[[s]]$
are $Q(t)=Q(s^{p^{n}})$, and of $\tilde{\mathscr{G}}^{\prime}$ are
$Q^{\prime}(s)=X^{-1}Q(s^{p^{n}})Y$.

In either case, we will let $\tilde{\phi}$ denote the isogeny that lifts
$\phi$, which is already defined over $\operatorname{Spf}k[[t]]$ in the first
case, but which is only defined over
$\operatorname{Spf}k[[s]]=\operatorname{Spf}k[[t^{\frac{1}{p^{n}}}]]$ in the
second case.

Suppose now that $\mathscr{G}$ was equipped with a principal polarization,
$\lambda$. This yields canonical isomorphisms $T_{p}(\mathscr{G})\rightarrow
T_{p}(\mathscr{G}^{\vee})$ and $T_{p}(\mathscr{G})^{*}\rightarrow
T_{p}(\mathscr{G}^{\vee})^{*}$, both of which we will also denote by
$\lambda$. Suppose that we have chosen coordinates such that
$m_{i}=\lambda(e_{i})$ (which yield dual bases $\mu_{i}=\lambda(\epsilon_{i})$
of $T_{p}(\mathscr{G})^{*},T_{p}(\mathscr{G}^{\vee})^{*}$). Then, $\lambda$
lifts to a deformation of $\mathscr{G}$ precisely when $q_{ij}=q_{ji}$ ([3,
Corollary 2.2.4]). The deformation space of $(\mathscr{G},\lambda)$ is
$\operatorname{Hom}^{\operatorname{Sym}}(T_{p}(\mathscr{G})\otimes
T_{p}(\mathscr{G}^{\vee}),\hat{\mathbb{G}}_{m})\subset\operatorname{Hom}(T_{p}(\mathscr{G})\otimes
T_{p}(\mathscr{G}^{\vee}),\hat{\mathbb{G}}_{m})$, the set of all symmetric
maps. This space is canonically isomorphic to
$\big{(}T_{p}(\mathscr{G})^{*}\otimes
T_{p}(\mathscr{G}^{\vee})^{*}\big{)}^{\operatorname{Sym}}\otimes\hat{\mathbb{G}}_{m}$,
where $(T_{p}(\mathscr{G})^{*}\otimes
T_{p}(\mathscr{G}^{\vee})^{*})^{\operatorname{Sym}}$ is the
$\mathbb{Z}_{p}$-span of
$\\{\epsilon_{i}\otimes\mu_{j}+\epsilon_{j}\otimes\mu_{i},\epsilon_{i}\otimes\mu_{i}\\}$.
In coordinates, this space equals
$\operatorname{Spf}k[[q_{ij}]]/(\\{q_{ij}-q_{ji}\\})$.

###  3.2. Description of $p$-power Hecke operators on $\mathcal{A}_{g}\mod p$

We now briefly describe $p$-power Hecke operators on
${\mathcal{A}_{g}}^{\operatorname{ord}}\mod p$. For further details, the
interested reader may consult [2, Chapter VII Section 4].

Let $(\mathscr{G},\lambda),(\mathscr{G}^{\prime},\lambda^{\prime})$ denote
ordinary principally polarized $p$-divisible groups as above, and let
$\phi\rightarrow\mathscr{G}^{\prime}$ denote an isogeny such that
$\phi^{*}(\lambda_{2})=p^{n}\lambda_{1}$. Suppose that we choose bases for
$T_{p}(\mathscr{G}),T_{p}(\mathscr{G}^{\vee}),T_{p}(\mathscr{G}^{\prime})$ and
$T_{p}(\mathscr{G}^{\prime,\vee})$ which are compatible with
$\lambda,\lambda^{\prime}$. The choice of bases induce isomorphisms
$\mathscr{G},\
\mathscr{G}^{\prime}\rightarrow(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{g}\times(\mu_{p^{\infty}})^{g}$.
The isogeny $\phi$ acts by multiplication by a matrix $X\in
M_{g}(\mathbb{Z}_{p})$ on the first summand, and by $Y\in
M_{g}(\mathbb{Z}_{p})$ on the second summand, and the condition
$\phi^{*}\lambda^{\prime}=p^{n}\lambda$ is equivalent to $X\cdot
Y^{T}=p^{n}\operatorname{Id}$. We say that the matrix of $\phi$ is $(A,B)$.

To that end, let $(n:n_{1},n_{2},\ldots n_{g})$ denote a sequence of integers
such that $n\geq n_{1}\geq n_{2}\ldots\geq n_{g}\geq 0$. A $p$-power Hecke
operator $\tau$ is said to have type $(n:n_{1},n_{2},\ldots n_{g})$ if it
parameterizes all ordinary isogenies
$\\{\phi:\mathscr{G}\rightarrow\mathscr{G}^{\prime}\\}$ such that the matrix
$\phi$ is $(A,B)$ where $AB^{T}=p^{n}\operatorname{Id}$ and

$A\in\operatorname{GL}_{g}(\mathbb{Z}_{p})\left[\begin{array}[]{cccc}p^{n_{1}}&&&\\\
&p^{n_{2}}&&\\\ &&\ddots&\\\ &&&p^{n_{g}}\\\
\end{array}\right]\operatorname{GL}_{g}(\mathbb{Z}_{p}).$

We note that the Hecke correspondence $\tau$ is Frobenius precisely when it
has type $(1:0,0,\ldots 0)$ and is Verschibung precisely when it has type
$(1:1,1,\ldots 1)$. Further, $\tau$ factors through Frobenius precisely when
$n>n_{1}$, and factors through Verschibung when $n_{g}>0$. The Hecke
correspondence $\tau$ induces a closed subvariety
$\mathcal{A}_{g}^{\operatorname{ord}}[\tau]\subset\mathcal{A}_{g}^{\operatorname{ord}}\times\mathcal{A}_{g}^{\operatorname{ord}}$,
with the two canonical projections $\Pr_{1}(\tau),\Pr_{2}(\tau)$ to
$\mathcal{A}_{g}$. The maps $\Pr_{i}(\tau)$ are usually inseparable.

Now, let $C\subset\mathcal{A}_{g}$ denote a reduced generically ordinary
curve, and let $C^{\operatorname{ord}}$ be the intersection of $C$ with the
ordinary locus of $\mathcal{A}_{g}$. We define
$\tau(C^{\operatorname{ord}})\subset\mathcal{A}_{g}^{\operatorname{ord}}$ to
be the unique reduced subscheme corresponding to
$\Pr(\tau_{1})_{*}\Pr(\tau)_{2}^{*}C^{\operatorname{ord}}$, and we define
$\tau(C)$ to be the Zariski closure of $\tau(C^{\operatorname{ord}})$ in
$\mathcal{A}_{g}$. We make the following observation that will be used below:
Let $\tau$ denote a Hecke correspondence which doesn’t factor through
Frobenius, and suppose the Serre-Tate coordinates of some branch of the
completion of $C$ at an ordinary point are $Q=q_{ij}$, where none of the
$q_{ij}$ are pth powers. Then the Serre-Tate coordinates of $\tau$ applied to
this branch are calculated using the recipe outlined in Proposition 3.1 (2),
i.e. the Serre-Tate coordinates of the various branches equal
$A^{-1}Q^{p^{n}}B^{T}$, where $A$ is as in 3.2 and $B$ satisfies
$AB^{T}=p^{n}\operatorname{Id}$.

###  3.3. $p$-power Hecke correspondence on Serre-Tate coordinates

We will now describe the action of $p$-power Hecke correspondences locally in
terms of Serre-Tate coordinates. For a point $x\in\mathcal{A}_{g}$, we will
denote the associated principally polarized abelian variety by $A_{x}$ and its
associated $p$-divisible group by either $A_{x}[p^{\infty}]$ or
$\mathscr{G}_{x}$. We note that the formal neighbourhood
$\hat{\mathcal{A}_{g}}^{x}$ of $\mathcal{A}_{g}$ at an ordinary point $A$ is
canonically isomorphic to the polarized deformation space of $A_{x}$ (which,
by the Serre-Tate lifting theorem, is the same as the polarized deformation
space of $\mathscr{G}_{x}$). Therefore, this space canonically has the
structure of a formal torus, and in coordinates, is isomorphic to
$\operatorname{Hom}^{\operatorname{Sym}}(T_{p}(\mathscr{G}_{x})\otimes
T_{p}(\mathscr{G}_{x}^{\vee}),\hat{\mathbb{G}}_{m})$.

######  Definition.

Let $x\in\mathcal{A}_{g}$ denote an ordinary point, and let $\ell\subset
T_{x}\mathcal{A}_{g}$ denote a one-dimensional subspace. We say that $\ell$ is
a _Serre-Tate_ direction if it equals the tangent space of a rank-1 formal
subtorus of $\hat{\mathcal{A}_{g}}^{x}$, i.e. $L\otimes\hat{\mathbb{G}}_{m}$
where $L\subset(T_{p}(\mathscr{G})^{*}\otimes
T_{p}(\mathscr{G}^{\vee})^{*})^{\operatorname{Sym}}$ is a saturated rank-1
$\mathbb{Z}_{p}$-submodule. We say that $\ell$ is a _Primitive Serre-Tate_
direction if in addition $L$ is spanned by a primitive tensor in
$(T_{p}(\mathscr{G})^{*}\otimes
T_{p}(\mathscr{G}^{\vee})^{*})^{\operatorname{Sym}}$.

Analogously, we define $T_{x}^{k}\mathcal{A}_{g}$ to be the $k$’th order
neighborhood of $x$ in $\mathcal{A}_{g}$. We define a primitive kth order
Serre-Tate direction to be the base change of a rank-1 formal subtorus of
$\hat{\mathcal{A}_{g}}^{x}$ to this neighborhood.

We note that for any positive $k$, there are only finitely many kth order
Serre-Tate directions.

######  Definition.

An ordinary abelian variety $A/\mathbb{F}_{q}$ is said to be hypersymmetric if
$A$ is isogenous to $E^{g}$, where $E/\mathbb{F}_{q}$ is an ordinary Elliptic
curve.

Let $C\subset\mathcal{A}_{g}$ denote a generically ordinary reduced
irreducible curve, that has the following properties:

  1. (1)

C passes through every $\mathbb{F}_{q}$-rational ordinary hypersymmetric point
$x$.

  2. (2)

For any $x\in\mathcal{A}_{g}(\mathbb{F}_{q})$ with $A_{x}$ as above, $C$ has
nodal singularities at $x$, and for every kth order primitive Serre-Tate
direction $\ell_{k}$ at $x$, there exists a formally smooth local branch
$C_{\ell_{k},x}$ of $C$ at $x$ whose tangent space equals $\ell_{k}$.

We have the following key theorem:

######  Theorem 3.2.

Let $\tau$ denote any $p$-power Hecke correspondence, and let $C$ be as above.
Then $\tau(C)$ contains every ordinary $x\in\mathcal{A}_{g}(\mathbb{F}_{q})$
which is hypersymmetric. Further, $\tau(C)$ contains every kth order primitive
Serre-Tate direction in $T_{x}^{k}\mathcal{A}_{g}$.

######  Lemma 3.3.

Let $x\in\mathcal{A}_{g}(\mathbb{F}_{q})$ be ordinary and hypersymmetric.
Then, $\tau(x)\in\mathcal{A}_{g}(\mathbb{F}_{q})$.

###### Proof.

Every subgroup of $A_{x}[p^{\infty}]$ has the form
$G_{\operatorname{et}}\times G_{\operatorname{mult}}$, where
$G_{\operatorname{et}}\subset A_{x}[p^{\infty}]_{\operatorname{et}}$, and
$G_{\operatorname{mult}}\subset A_{x}[p^{\infty}]_{\operatorname{mult}}$.
Since $A_{x}$ is hypersymmetric, the Galois representation on
$T_{p}(A_{x}[p^{\infty}]_{\operatorname{et}})$ acts through a scalar (because
this statement is trivially true for $E$). It follows that every subgroup of
$A_{x}[p^{\infty}]$ is defined over $\mathbb{F}_{q}$, whence the result
follows. ∎

###### Proof of Theorem 3.2.

Let $x\in\mathcal{A}_{g}(\mathbb{F}_{q})$ denote some fixed hypersymmetric
point, and let $\ell_{k}$ denote a kth order primitive Serre-Tate direction,
and let $C_{x,\ell_{k}}=\operatorname{Spf}\mathbb{F}[[t]]$ denote the formally
smooth local branch of $C$ at $x$ that realizes $\ell_{k}$. We fix
polarization-compatible bases $\\{e_{i}\\}$ of $T_{p}(\mathscr{G}_{x})$ and
$\\{m_{j}\\}$, with dual bases $\\{\epsilon_{i}\\}$ and $\\{\mu_{j}\\}$, such
that $\ell_{k}$ is realized by the sub-torus corresponding to the span of
$\epsilon_{1}\otimes\mu_{1}$.

Let $y\in\tau^{\vee}(x)$, such that the associated isogeny
$\phi^{\vee}:A_{x}\rightarrow A_{y}$ is obtained by the quotienting $A$ by the
diagonal subgroup with type $(n:n^{\prime}_{1},\ldots n^{\prime}_{g})$. By
Lemma 3.3, $y\in\mathcal{A}_{g}(\mathbb{F}_{q})$, and therefore our hypothesis
implies that $C$ passes through $y$, and there exists a formally smooth branch
of $C$ through $y$ realizing any fixed kth order primitive Serre-Tate
direction. Let $e^{\prime}_{i}\in T_{p}(\mathscr{G}_{y})$ be defined as =
$e^{\prime}_{i}=\frac{1}{p^{n^{\prime}_{i}}}\phi^{\vee}(e_{i})$, and let
$m^{\prime}_{j}\in T_{p}(\mathscr{G}_{y}^{\vee})$ be defined analogously –
$\\{e^{\prime}_{i}\\}$ and $\\{m^{\prime}_{j}\\}$ are bases of $T_{p}(A_{y})$
and $T_{p}(A_{y}^{\vee})$ respectively. Let $\\{\epsilon^{\prime}_{i}\\}$ and
$\\{\mu^{\prime}_{j}\\}$ denote dual bases. Consider the branch
$C_{y,\ell^{\prime}_{k}}$, where $\ell^{\prime}_{k}$ is realized by
$\epsilon_{1}^{\prime}\otimes\mu_{1}^{\prime}$. The Serre-Tate coordinates of
$C_{y,\ell^{\prime}_{k}}$ are given by functions $q^{\prime}_{ij}(t)$ where
$q^{\prime}_{11}(t)\equiv 1+t\mod t^{k+1}$ and $q^{\prime}_{ij}(t)\equiv 1\mod
t^{k+1}$.

Let $\phi:A_{y}\rightarrow A_{x}$ denote the dual of $\phi^{\vee}$. We now
lift $\phi$ to an isogeny $\tilde{\phi}$ whose source is the abelian scheme
$A_{C_{y,\ell^{\prime}_{k}}}$. By Proposition 3.1, the Serre-Tate coordinates
of $\tilde{\phi}(A_{C_{y,\ell^{\prime}_{k}}})$ are given by
$Q(s)=X^{-1}Q^{\prime}(s^{p^{n}})Y$, where $X$ and $Y$ are diagonal matrices
with ith entry $p^{n_{i}}$ and $p^{n-n_{i}}$ respectively. It follows that the
kth order tangent direction of $\tilde{\phi}(A_{C_{y,\ell^{\prime}_{k}}})$ is
equal to $\ell_{k}$. As $x$, $\ell_{k}$ and $\tau$ were arbitrary, the result
follows. ∎

###  3.4. Zariski-Density of rank-1 formal branches

######  Proposition 3.4.

Let $x$ be a hypersymmetric point, and suppose $D\subset\mathcal{A}_{g}$ is a
variety such that $T_{x}D$ contains all rank-1 primitive subtori. Then
$D=\mathcal{A}_{g}$.

###### Proof.

We will in fact prove the stronger claim that all rank-1 formal tori are dense
in $\operatorname{Sym}^{2g}(\mathbb{Z}_{p})\otimes\hat{\mathbb{G}_{m}}$.
Letting $q_{ij}$ be the co-ordinates as before, we consider the element
$v\in\mathbb{Z}_{p}^{g}$ such that $v_{i}=p^{N^{i}}$ where $N$ will be chosen
to be a large integer. Then the corresponding primitive rank 1 torus has co-
ordinates $Q_{ij}(t)=t^{p^{N^{i}+N^{j}}}$. Now let $f\in k[[q_{ij}]]$ be a
non-zero power series which vanishes on all such primitive rank 1 tori. There
are finitely many monomials in $f$ which are dominated by all the rest. Note
that $\displaystyle\prod_{i,j}q_{ij}^{r_{ij}}$ evaluates at $\vec{Q}(t)$ to
$t^{M_{ij}}$ where $M_{ij}=\displaystyle\sum_{i,j}r_{ij}p^{N^{i}+N^{j}}$. For
large enough $N$ these exponents are all distinct, which means there is a
single term of $f(\vec{Q}(t))$ of smallest degree, which is a contradiction.

∎

###  3.5. Algebraicity of pure Serre-Tate directions

As in [1, Prop 5.4], for we may consider the completion along the diagonal in
$D\times D$ as a formal scheme $D^{\wedge}$ over $D$, and we may also view
this as sitting inside the pull back $i^{*}\mathcal{A}_{g}^{\wedge}$. Now for
any integer $m>0$ we may also take the corresponding $n$’th order
infinitesimal subscheme $D_{m}$ sitting inside $i^{*}{\mathcal{A}_{g}}_{,n}$.
Now, if we go up to a level cover
${\mathcal{A}_{g}}^{\operatorname{ord}}_{p^{n}}$ where the $p$-adic tate
module has been trivialized mod $p^{n}$, then the infinitesimal scheme
$(\mathcal{A}_{g})_{n}$ can be trivialized as
$\operatorname{Sym}^{2}(\mathbb{Z}/p^{m}\mathbb{Z})^{g}\otimes{\mathbb{G}_{m}}_{n}$
and there is a closed subscheme consisting of the pure Serre-Tate directions.
Hence the same is true in $\mathcal{A}_{g}^{\operatorname{ord}}$. Thus,
pulling back, to $D$ we see that there is a closed subvariety $R_{n}\subset
i^{*}{\mathcal{A}_{g}}_{,n}$ whose fibers at closed point consist of the
unions of all pure Serre-Tate directions to order $n$. Moreover, $R_{n}$ is
etale-locally a product, thus its restriction to any closed subvariety of $D$
is irreducible. It follows that there is a Zariski-closed subscheme
$D_{st,n}$of $D$ consisting of all the points at which $D$ contains all the
pure Serre-tate direcitons to order $n$.

###  3.6. Proof of Theorem 1.2

Suppose that $D\subsetneq\mathcal{A}_{g}$ is a subvariety, and let
$x\in\mathcal{A}_{g}(\overline{k})$ be a hypersymmetric point. If $D$ does not
contain any point isogenous to $x$ the result follows simply by picking a
curve $C$ passing through $x$. Thus, at the cost of increasing the finite
field $k$, we assume $D$ contains a $k$\- point isogenous to $x$. By Lemma 3.4
it follows that $D$ does not contain all formal subtori passing through $x$.
It follows that there is some positive integer $M$ such that $T^{M}_{x}D$ does
not contain all $M$’th order pure Serre-Tate directions. Thus, by the
discussion in the previous subsection, $x\in D_{st,M}$. It follows by
Noetherianity that the tower $D_{st,M}$ stabilizes, and thus for which the
above statement is true for all hypersymmetric points.

We now pick a prime $r>6p$ and increase $k$ so that the pre-images of $x$ in
$\mathcal{A}_{g}[r]$ are defined over $k$. We now use Theorem 2.1 to product a
curve $C\subset\mathcal{A}_{g}[r]$ which

  1. (1)

For every hypersymmetric point isogenous to $x$, and every $M$’th order pure
Serre-Tate direction $\eta$ through $x$, $C$ admits a $\overline{k}[[t]]$
point specializing to $\eta$, and

  2. (2)

The geometric monodromy of $C$ surjects onto
$\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$.

By condition 2 above, every Hecke translate $TC$ of $C$ is irreducible. Since
the isogenous curves to $C$ correspond to irreducible components of its Hecke
translates,it suffices to show that $TC\not\subset D$ for every hecke operator
$T$. We factor $T=T_{e}T_{p}$ for $T_{e}$ a prime-to-$p$ Hecke correspondence
and $T_{p}$ a $p$-power Hecke correspondence.

By Theorem 3.2, the curve $T_{p}C$ also satisfies condition 1 above. Now since
$TC$ is irreducible, it follows that for each point $y\in T_{e}x$ the formal
neigborhood of $TC=T_{e}(T_{p}C)$ at $y$ contains the image of the formal
neigborhood of $T_{p}C$ at $x$. Sine $T_{e}$ is etale, it follows that $TC$
satisfies condition 1 above at $y$. Therefore, $TC$ cannot be contained in
$D.$

The result about function field valued points follows immediately by
considering the generic point of $C$.

$\vskip 6.0pt plus 2.0pt minus 2.0pt\square$

###  3.7. Proof of Theorem 1.1

We now suppose that $D\subsetneq(\mathcal{A}_{g})_{F}$ is a subvariety.
Identifying $k(T)$ with the function field of $\mathbf{P}^{1}_{l}$ we may
realize $D$ as the generic point of a (strict) subvariety
$\mathcal{D}\subset(\mathcal{A}_{g}\times\mathbf{P}^{1})_{k}$, no generic
points of which are contained in a fiber. We let
$E\subset(\mathcal{A}_{g})_{\overline{k}}$ be the constant part $D$, so that
$E_{F}\subset D$ and $E$ is the maximum such subvariety333One may construct
$E$ by intersecting all base changes of $D$ along automorphisms of $F$ over
$\overline{k}$.. By increasing $k$ we assume $E$ is defined over $k$. We are
looking for a curve $C\subset(\mathcal{A}_{g}\times\mathbf{P}^{1})_{k}$ such
that no Hecke translate of $C$ is contained inside $\mathcal{D}$.

As a first step, we pick $N$ sufficiently large so that $A_{N}$ is not a sub-
quotient of $\prod_{\ell\neq
p}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$,
and let $\phi:\mathbf{P}^{1}\rightarrow\mathbf{P}^{1}$ be a degree $N$ map
whose Galois group is $A_{N}$. We shall need the following two lemmas:

######  Lemma 3.5.

Let $K\subset L$ be a finite separable extension of fields, and let $G$ be the
Galois group of the normal closure of $L$ over $K$. Suppose that $H$ is a
profinite group with no irreducible constituent in common with $G$, and let
$\phi:\operatorname{Gal}_{K}\rightarrow H$ be a group homomorphism. Then
$\phi(\operatorname{Gal}_{L})=\phi(\operatorname{Gal}_{K})$.

###### Proof.

By increasing $L$ we may as well assume $L/K$ is Galois. Then
$\operatorname{Gal}_{L}\subset\operatorname{Gal}_{K}$ is normal, and so we get
a map $G\rightarrow\phi(\operatorname{Gal}_{K})/\phi(\operatorname{Gal}_{L})$,
which must be trivial by our assumptions. The statement follows. ∎

######  Lemma 3.6.

Let $\mathcal{D}\subset(\mathcal{A}_{g}\times\mathbf{P}^{1})_{k}$ be as above.
There is an integer $M$ such that for any $M$ distinct points
$x_{1},\dots,x_{M}$ of $\mathbf{P}^{1}(\overline{k})$, we have
$\displaystyle\bigcap_{i=1}^{m}\mathcal{D}_{x_{i}}=E$.

###### Proof.

We will base change to an uncountable field $k$, and prove the statement there
(from which the original statement clearly follows).

For any $M$, consider the set $S_{M}$ of points $x_{1}$ for which one can find
distinct $x_{2},\dots,x_{M}$ such that
$\displaystyle\bigcap_{i=1}^{m}\mathcal{D}_{x_{i}}\neq E$. This is clearly a
constructible set, so is either finite or co-finite. Moreover the set $S_{M}$
is clearly descending (perhaps non-strictly) with $M$. It follows that if
$S_{M}$ is not eventually empty, then $\bigcap_{M}S_{M}\neq\emptyset.$ Thus we
may find a point $x_{1}$ which is contained in this intersection.

We may now iterate with $x_{2}$, etc.., to produce a countably infinite
sequence of points $x_{i}$ such that
$\displaystyle\bigcap_{i=1}^{m}\mathcal{D}_{x_{i}}\neq E$ for any positive
integer $m$. By Noetherianity this intersectio eventually stabilizes to a
subvariety $E^{\prime}$ properly containing $E$. This $E^{\prime}$ is
contained in infinitely many fibers of $\mathcal{D}$ and hence in all of them.
Thus $E^{\prime}_{F}\subset\mathcal{D}$ contradicting the definition of $E$.

∎

We now construct our curve $C$ as follows. Fix a prime $r>6p$. Following
Theorem 1.2 we next construct an irreducible curve
$C_{0}\subset\mathcal{A}_{g}[r]$ such that

  1. (1)

No Hecke translate of $C_{0}$ is contained in $E$.

  2. (2)

The geometric monodromy of $C\cap\mathcal{A}_{g}^{\operatorname{ord}}[r]$
surjects onto $\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$,
where $r>6p$ is a prime as in the proof of Theorem 1.2.

We do this the same way as in the proof of Theorem 1.1. Next, we pick a map
$\psi:C_{0}\rightarrow\mathbf{P}^{1}$ arbitrarily, and let $C_{1}$ be the
graph of this map in $\mathcal{A}_{g}[r]\times\mathbf{P}^{1}$. Finally, we let
$C$ be the base change of this map under
$\phi\times\operatorname{Id}_{\mathcal{A}_{g}[r]}$, for $N>\deg\psi$. It
follows that $C$ is irreducible since $A_{N}$ does not admit a non-trivial
homomorphism to $S_{\deg\psi}$,

By lemma 3.6, the monodromy group of $C$ surjects onto
$\Gamma_{g}(\mathbb{Z}_{r})\times\prod_{\ell\neq
p,r}\mathbf{P}(\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell}))\times\operatorname{GL}_{g}(\mathbb{Z}_{p})$.
It follows that for each Hecke operator $T$, that $TC$ is irreducible. Suppose
that $TC\subset\mathcal{D}$.

For a generic point $s\in\mathbf{P}^{1}$, the fiber $\phi^{-1}(s)$ consists of
$N$ points $x_{1},\dots,x_{N}$, and so it follows from theorem 3.6 that

$(TC)_{x_{j}}=\bigcap_{i=1}^{m}(TC)_{x_{i}}\subset\bigcap_{i=1}^{m}\mathcal{D}_{x_{i}}=E.$

It follows that $TC\subset E$. Since $E$ is constant, by projecting to
$\mathcal{A}_{g}$, this in fact means that $TC_{0}\subset E$ which is a
contradiction.

##  4\. Additional results

###  4.1. The case of number fields

Let $H\subset\mathcal{A}_{g}$ denote the Hodge line bundle. For any proper
curve $C\subset\mathcal{A}_{g}$, we define $(C.H)$ to denote the degree of
this line bundle pulled back to $C$.

######  Lemma 4.1.

Let $T$ denote a Hecke operator on $\mathcal{A}_{g}$. Then, for any proper
curve $C\subset\mathcal{A}_{g}$ we have $\deg T(C)=\deg(T)\cdot\deg(C)$.

###### Proof.

This follows directly from the fact that the two different pull-backs of $H$
to the graph of the Hecke correspondence are equal. ∎

######  Definition.

Let $X$ be some variety defined over a field $L$, and let $K/L$ denote a
finite extension. We say that a point $x\in X(K)$ is _primitive_ if $K$ is the
smallest field of definition of $x$.

######  Lemma 4.2.

Let $C\subset\mathcal{A}_{g}$ denote a curve defined over $\mathbb{Q}$ with
maximal monodromy. There exists a constant $c\in\mathbb{Z}$ such that for any
positive integer $n\in\mathbb{Z}$, there exists a number field $K/\mathbb{Q}$
which is Galois and has degree $\geq n$, and a primitive point $x\in C(K)$
such that the monodromy of $A_{x}$ has index bounded by $c$ in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$.

###### Proof.

Let $\pi:C\rightarrow\mathbb{P}^{1}$ denote a finite map of degree $d$, and
consider $B=\operatorname{Res}_{C/\mathbb{P}^{1}}(A)$, which is an abelian
scheme over an open subset $U\subset\mathbb{P}^{1}$ of dimension $gd$. Work of
Zywina [16, Theorem 1.1] implies that for any number field $L$, there exists a
constant $c_{L}$ such that for a density 1 set of points $y\in U(L)$, the
monodromy of $B_{y}$ has index bounded by $c_{L}$ in the monodromy of $B_{U}$.
In Section 1.2 of _loc. cit._ , Zywina explicitly describes the constant
$c_{L}$. Indeed, if $L/\mathbb{Q}$ were a Galois extension which has the
property that the monodromy of $B_{U_{L}}$ were the same as the monodromy of
$B_{U_{\mathbb{Q}}}$, then the constant $c_{L}$ equals $c_{\mathbb{Q}}$. It is
easy to see that there exist fields $L$ with this property with
$[L:\mathbb{Q}]$ arbitrarily large; for example, by choosing $L$ so that
$\operatorname{Gal}(L/\mathbb{Q})=G(\mathbb{Z}/\ell^{n}\mathbb{Z})$, where
$G/\mathbb{Z}_{\ell}$ is some exceptional simple group and $\ell$ is any
prime.

We now use the above discussion to deduce the existence of number fields $K$
with arbitrarily large degree and points $x\in C(K)$ with monodromy as
claimed. Let $y\in U(L)$ satisfy the conclusions of the previous paragraph. By
Hilbert irreducibility, we may also assume that that
$\pi^{-1}(y)=\\{x_{1}\ldots x_{d}\\}$, where $x_{i}\in C(K_{i})$ where
$K_{i}/L$ are conjugate degree d extensions of $L$ and the points $x_{i}$ are
conjugate to each other. We have that $\prod_{i=1}^{d}A_{x_{i}}$ descends to
$L$ and equals $B_{y}$, and that the monodromy of
$(\prod_{i=1}^{d}A_{x_{i}})_{K_{1}}$ equals the intersections of the
monodromies of $B_{C}$ and $B_{y}$ inside the monodromy of $B_{U}$. It then
follows that the monodromy of $(\prod_{i=1}^{d}A_{x_{i}})_{K_{1}}$ has index
bounded by $c=c_{\mathbb{Q}}$ inside the monodromy of $B_{C}$, and projecting
onto the first factor yields that the monodromy of $A_{x_{1}}$ has index
bounded by $c$ in the monodromy of $A_{C}$, which is equal to
$\operatorname{GSp}(\hat{\mathbb{Z}})$ by our assumption on the curve $C$.

∎

######  Theorem 4.3.

Let $V\subsetneq\mathcal{A}_{g}$ denote a closed subvariety. Then, there
exists an abelian variety over a number field which is not isogenous to any
point of $V$.

###### Proof.

Without loss of generality, we suppose that $V$ is a divisor, and by replacing
$V$ by the union of its Galois conjugates, we may assume that $V$ is defined
over $\mathbb{Q}$. We may proceed as in the positive characteristic setting to
find a curve $C\subset\mathcal{A}_{g}$ having maximal monodromy which
satisfies $T(C)\not\subset V$ for every Hecke correspondence $T$. In fact, by
using the characteristic-zero Lefschetz theorem detailed in [8, Theorem 1.2]
for smooth quasi-projective varieties, we may find a proper curve
$C\subset\mathcal{A}_{g}$ defined over $\mathbb{Q}$ with maximal monodromy,
and which satisfies such that $T(C)\not\subset V$ for every Hecke
correspondence $T$.

We first note that for a proper curve $C\subset\mathcal{A}_{g}$, we have

(4.1.1) $(T(C)\cdot V)=\deg(T)(C\cdot V).$

This is true when $V=H$ by Lemma 4.1. In general, we note that
$\mathcal{A}_{g}$ has Picard group $\mathbb{Z}$, and hence $(C^{\prime}\cdot
V)=n_{V}(C^{\prime}\cdot H)$ for any proper curve $C\subset\mathcal{A}_{g}$,
where the divisors $[H],[V]$ satisfy $n_{V}[H]=V$ in
$\textrm{Pic}(\mathcal{A}_{g})$. By Lemma 4.2, there exists a Galois extension
$K/Q$ of degree $\geq n$ and a primitive point $x\in C_{\mathbb{Q}}$ whose
monodromy has index bounded by $c$ in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$, where we now choose $n>c(C\cdot
V).$ For any fixed Hecke correspondence, we have that $T(x)$ is stabilized by
$\operatorname{Gal}(\overline{K}/K)$, and the monodromy assumptions on $A_{x}$
implies that each orbit has at least $\frac{\deg(T)}{c}$ elements. The same
statement holds for $\sigma(x)$ with
$\sigma\in\operatorname{Gal}(K/\mathbb{Q})$, and further, we have that the
sets $T(x)$ and $T(\sigma(x))$ are in the same
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ orbit.

Suppose now that $T(x)$ intersected $V$ non-trivially for some Hecke operator
$T$. As we have assumed $V$ is defined over $\mathbb{Q}$, it follows that
$T(x)\cap V$ is stable under
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

The previous paragraph implies that $T(\sigma(x))\cap V$ has size
$\geq\frac{\deg(T)}{c}$ for every $\sigma\in\operatorname{Gal}(K/\mathbb{Q})$.
Further, even if the same point $y$ is contained444This can happen if
$\sigma_{1}(x)$ is isogenous to $\sigma_{2}(x)$. in $T(\sigma_{1}(x))\cap V$
and $T(\sigma_{2}(x))\cap V$, this point contributes twice to the intersection
$T(C)\cdot V$, because this implies that $T(C)$ must be singular at $y$, with
branches corresponding to the branch of $C$ through $\sigma_{1}(x)$ and
through $\sigma_{2}(x)$. It therefore follows that

(4.1.2) $T(C)\cdot V\geq\frac{\deg(T)}{c}\cdot n.$

But our initial choice of $n$ was such that $n>c\cdot N_{V}\cdot\deg(C)$, and
so Equations (4.1.1) and (4.1.2) cannot both hold simultaneously. It therefore
follows that $T(x)$ must be disjoint from $V$ for every Hecke correspondence
$T$, and the result follows.

∎

###  4.2. An analogue over $\overline{\mathbb{F}}_{p}$.

We will now prove that for $D\subset\mathcal{A}_{g}$ a divisor, there exists
an ordinary $x\in\mathcal{A}_{g}(\overline{\mathbb{F}}_{p})$ such that
$T(x)\not\subset D$ for every prime-to-$p$ Hecke operator $T$.

###### Proof of Theorem 1.3.

By the main result of this paper, there exists a generically ordinary
$C\subset\mathcal{A}_{g}$ that satisfies $T(C)\not\subset\mathcal{A}_{g}$ for
every Hecke operator $T$. An identical intersection-theoretic argument as in
the proof of Theorem 4.3 yields that for a point $x\in
C(\overline{\mathbb{F}}_{p})$, whose minimal field of definition is
$\mathbb{F}_{q}$ where $q$ is a large enough power of $p$, $T(x)\not\subset
D$. The result follows.

∎

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