# Compact Transformer Tracker with Correlative Masked Modeling

Zikai Song1, Run Luo1, Junqing Yu1, Yi-Ping Phoebe Chen2, Wei
Yang111footnotemark: 1
indicates co-corresponding author.

###### Abstract

Transformer framework has been showing superior performances in visual object
tracking for its great strength in information aggregation across the template
and search image with the well-known attention mechanism. Most recent advances
focus on exploring attention mechanism variants for better information
aggregation. We find these schemes are equivalent to or even just a subset of
the basic self-attention mechanism. In this paper, we prove that the vanilla
self-attention structure is sufficient for information aggregation, and
structural adaption is unnecessary. The key is not the attention structure,
but how to extract the discriminative feature for tracking and enhance the
communication between the target and search image. Based on this finding, we
adopt the basic vision transformer (ViT) architecture as our main tracker and
concatenate the template and search image for feature embedding. To guide the
encoder to capture the invariant feature for tracking, we attach a lightweight
correlative masked decoder which reconstructs the original template and search
image from the corresponding masked tokens. The correlative masked decoder
serves as a plugin for the compact transform tracker and is skipped in
inference. Our compact tracker uses the most simple structure which only
consists of a ViT backbone and a box head, and can run at 40 _fps_. Extensive
experiments show the proposed compact transform tracker outperforms existing
approaches, including advanced attention variants, and demonstrates the
sufficiency of self-attention in tracking tasks. Our method achieves state-of-
the-art performance on five challenging datasets, along with the VOT2020,
UAV123, LaSOT, TrackingNet, and GOT-10k benchmarks. Our project is available
at https://github.com/HUSTDML/CTTrack.

##  1 Introduction

Figure 1: Our compact transformer tracker adopts the simple ViT structure
(encoder) with the concatenation of the template and search image as input,
which essentially exploits the standard self-attention mechanism for
information aggregation. The encoded tokens pass through a box head to
estimate the result bounding box. And we develop a correlative masked decoder
reconstructing the original template and search pixels to enhance the
information aggregation, which is skipped during inference.

Visual Object Tracking is one of the fundamental tasks in computer vision with
applications ranging from human-computer interaction, surveillance, traffic
flow monitoring and etc. It aims to estimate the location, denoted as a
bounding box, of an arbitrary target object throughout the subsequent video
sequence. Deep Learning based trackers have achieved great success due to
their strong representation ability. Trackers (Bertinetto et al. 2016; Nam and
Han 2016; Li et al. 2018, 2019) derived from Convolutional Neural Networks
(CNN) (Krizhevsky, Sutskever, and Hinton 2012; Simonyan and Zisserman 2015; He
et al. 2016) produce tracking accuracy that beyond the comparison of
traditional approaches, especially the trackers built on Siamese network
(Bertinetto et al. 2016; Xu et al. 2020; Li et al. 2018, 2019; Voigtlaender et
al. 2020; Yu et al. 2020; Guo et al. 2021). The key of Siamese network
trackers is to produce the cross-correlation and measure the similarity
between the target template and search image. Nowadays, transformer-based
trackers (Chen et al. 2021; Wang et al. 2021; Yan et al. 2021; Shen et al.
2022; Song et al. 2022; Cui et al. 2022) have shown great strength by
introducing the attention mechanism (Vaswani et al. 2017) to enhance and fuse
the features of querying sample and tracked objects. Prevalent transformer
trackers (Chen et al. 2021; Yan et al. 2021; Cui et al. 2022) more or less
adapt the attention for aggregating information across the template and search
image.

We find that the advanced variants of attention mechanism in recent research,
including mix-attention (Cui et al. 2022) and cross-attention (Yu et al. 2020;
Chen et al. 2021), are equivalent or even just a subset of the packed self-
attention (i.e., standard self-attention with the concatenation of the
template and search image as input). Then the question is which parts of the
self-attention mechanism play an important role in visual object tracking? We
revisited the transformer tracking framework and find that the tracking
results are generated from tokens corresponding to the search image (search
tokens), while the tokens corresponding to the template (template tokens) are
always discarded in the last. The representational ability of search tokens
comes from two parts: the cross-information enhancement from the template
tokens and the self-information enhancement from the search tokens themselves.
In this paper, we prove that self-information enhancement in multi-image
attention plays a greater role than cross-information aggregation, though
cross-information aggregation is indispensable in visual object tracking but
not greatly beneficial.

Driven by this analysis, we propose a compact transformer tracker combined
with correlative masked modeling for the cross-information aggregation and
self-information reinforcement. As shown in Figure 1, our tracker adopts the
basic vision transformer as the main branch and applies a lightweight masked
decoder to enhance the implicit representation capability of the packed self-
attention. The correlative masked decoder, which is inspired by Masked Image
Modeling (He et al. 2022; Xie et al. 2022), reconstructs the both original
template and search pixels from the corresponding masked tokens, to guide the
encoder to capture the invariant feature for tracking. In addition, our
decoder can be plugged into other transformer trackers, which can effectively
improve the tracking performance without compromising speed. Applying our
correlative masked modeling strategy to the compact transformer tracker can
improve the AUC from 64.0% to 65.8% on the LaSOT (Fan et al. 2019) dataset.
Extensive comparison experiments on 5 challenging datasets including VOT2020
(Kristan et al. 2020), UAV123 (Mueller, Smith, and Ghanem 2016), LaSOT,
GOT-10k (Huang, Zhao, and Huang 2019), and TrackingNet (Muller et al. 2018)
exhibits the state-of-the-art performance, which further evidence the
correctness of our analysis regarding the self-attention in visual tracking.

To summarize, our main contributions include:

  1. 1.

We present a unified analyzing method for the attention mechanism and find
that the advanced variants of the attention mechanism are equivalent or even
just a subset of the self-attention. We also prove that self-information
enhancement in multi-image attention plays a greater role than cross-
information aggregation.

  2. 2.

We develop a compact transformer tracker with a correlative masked decoder,
which has a very simple structure and achieves state-of-the-art accuracy at a
high Frames-Per-Seconds (_fps_) tracking speed. The decoder reconstructs the
original template and search image from the corresponding masked tokens and
serves as a training plugin for the tracker. The experiment demonstrates that
our analysis regarding self-attention is correct.

##  2 Related Work

Traditional trackers. Traditional single object tracking algorithms can be
roughly summarized as Correlation Filter based trackers (CF), Deep Network
based trackers (DLN). CF-based trackers(Bolme et al. 2010; Henriques et al.
2015; Danelljan et al. 2016, 2017, 2019; Bhat et al. 2019) exploit the
convolution theorem and learn a filter in the Fourier domain that maps known
target images to the desired output. DLN-based trackers refer to algorithms
employing deep neural networks for the tracking process. Earlier approaches
(Nam and Han 2016; Pu et al. 2018) treat the tracking task as a classification
problem and exploit deep features for locating the target. Shortly afterwards
more trackers adopt the Siamese network (Bertinetto et al. 2016; Li et al.
2018, 2019) for its effectiveness in measuring similarity. The Siamese network
consists of two branches, one operates on the template and the other for the
search area.

Above all, these methods mainly consist of a backbone which extracts the
features of search image and template separately, a similarity measuring
module, and heads to predict the location and bounding box. Compared to our
framework, traditional trackers have too many modules and a very complex
design, we simply adapt a ViT backbone with a box head to get better tracking
results.

Transformer trackers. The ViT (Dosovitskiy et al. 2021) first introduces the
transformer to image recognition tasks and presents an impressive performance.
Ever since, transformer has been widely applied in image
classification(Dosovitskiy et al. 2021; Wu et al. 2021; Liu et al. 2021),
object detection(Carion et al. 2020; Li et al. 2022), visual object
tracking(Yan et al. 2021; Chen et al. 2021; Wang et al. 2021; Song et al.
2022; Shen et al. 2022; Cui et al. 2022) and etc. Transformer-based tracking
methods have become the mainstream tracking algorithms nowadays. TransT (Chen
et al. 2021) proposes a feature fusion network and employs an attention
mechanism to combine the features of the template and search region. STARK
(Yan et al. 2021) develops a spatial-temporal architecture based on the
encoder-decoder transformer. CSWinTT (Song et al. 2022) proposes a transformer
architecture with multi-scale cyclic shifting window attention for visual
tracking, elevating the attention from pixel level to window level. MixFormer
(Cui et al. 2022) constructs a compact tracking framework and designs a mixed
attention module that unifies the process of feature extraction and
information matching module.

Instead of designing a complex attention mechanism as in the previous tracking
approaches, we compare the essential differences of attention variants(such as
mix-attention and cross-attention) and find these attention variants are
equivalent or even just a subset of the packed self-attention. To verify the
capability of self-attention in information aggregation, we design a compact
transformer tracker using the most simple pipeline which only consists of a
ViT backbone and a box head, without any extra design including separate
modules of feature extraction and aggregation, and multi-layer feature
aggregation.

Masked image modeling (MIM). MIM masks an area of the original images and
predicts the missing pixels, which aims to enhance the representation of
models. Recently, MIM approaches((Chen et al. 2020; He et al. 2022; Xie et al.
2022; Wei et al. 2022; Bao, Dong, and Wei 2021)) are extended to the modern
vision transformers (Dosovitskiy et al. 2021; Liu et al. 2021). iGPT (Chen et
al. 2020) first proposes a transformer to predict unknown pixels from a
sequence of low-resolution pixels. BEiT (Bao, Dong, and Wei 2021) tokenizes
the images via an additional dVAE (Ramesh et al. 2021) network with a block-
wise masking strategy. SimMIM (Xie et al. 2022) find that a moderately large
masked patch size of the input image for pixel predictions makes a strong pre-
text task. MAE (He et al. 2022) develops an asymmetric encoder-decoder
architecture, the encoder operates on a small proportion of the visible
patches, and the decoder reconstructs the original pixels. MaskFeat (Wei et
al. 2022) reconstructs the feature descriptors such as HoG (Dalal and Triggs
2005) instead of pixels.

Our approach is inspired by the previous MIM method (Xie et al. 2022; He et
al. 2022), but we have to deal with two fundamental problems in the tracking
framework: (1) Visual tracking is a downstream vision task that generally does
not have the pre-train process to apply the MIM strategy. We develop a masked
decoder to leverage the search and the template tokens to predict the original
images, which is embedded as an attachment plugin in the training phase to
implement an end-to-end model. (2) MIM methods reconstructing the single image
do not fit the tracking framework which involves cross-aggregation of multiple
images. According to the properties of packed self-attention, we design a
self-decoder and a cross-decoder to reconstruct the original template and
search image from the corresponding masked tokens. As far as we know, we are
the first to artfully introduce the MIM into the visual tracking field to
improve the information aggregation capabilities.

Figure 2: Information streams in the attention mechanism. The four information
streams of Q-K-V are corresponding to the four parts in the attention map.
Variants of attention can be uniformly explained under this analytical
approach.

##  3 Approach

In this section, we introduce our compact transformer tracker with correlative
masked modeling in detail. Before proceed, we first present a analysis on the
key component of transformer tracker, and demonstrate that existing attention
variants are equivalent to the packed self-attention.

###  3.1 Revisiting Transformer Tracker

Transformer tracking framework. As described in ViT(Vaswani et al. 2017), the
query-key-value attention mechanism is applied with query $\mathbf{Q}$, key
$\mathbf{K}$, and value $\mathbf{V}$. The linear weights of $\mathbf{Q}$,
$\mathbf{K}$, $\mathbf{V}$ are $\mathbf{W}_{Q}$, $\mathbf{W}_{K}$,
$\mathbf{W}_{V}$ respectively. The attention (Attn) is computed as:

$\text{Attn}(\text{X})=\text{softmax}(\dfrac{\mathbf{X}\mathbf{W}_{Q}\cdot\mathbf{W}_{K}^{T}\mathbf{X}^{T}}{\sqrt{d_{k}}})\cdot\mathbf{X}\mathbf{W}_{V}$
(1)

where the X is the input token and the $d_{k}$ is the dimension of the key.
For a clearer description of the post-order steps, we apply an attention
calculation with the inputs of two different tokens, the token
$\mathbf{X}_{Q}$ computed with query and the token $\mathbf{X}_{K}V$ computed
with key and value. We modify the attention formula and define the attention
map (AMap) as:

$\begin{split}\text{Attn}(\mathbf{X}_{Q},\mathbf{X}_{KV})&=\text{AMap}(\mathbf{X}_{Q},\mathbf{X}_{KV})\cdot\mathbf{X}_{KV}\mathbf{W}_{V}\\\
\text{AMap}(\mathbf{X}_{Q},\mathbf{X}_{KV})&=\text{softmax}(\frac{\mathbf{X}_{Q}\mathbf{W}_{Q}\cdot\mathbf{W}_{K}^{T}\mathbf{X}_{KV}^{T}}{\sqrt{d}})\end{split}$
(2)

Our compact transformer tracker consists of two parts: a transformer backbone
for information aggregation and a box head for the bounding box estimation.
Give the template $z$ in the initial frame and a search image $s$. We obtain
the tokens $X_{t}\in\mathbb{R}^{L_{z}\times d}$ and
$X_{s}\in\mathbb{R}^{L_{s}\times d}$ respectively through patch embedding,
where $d$ represents the number of channels. The packed self-attention (PSelf-
Attn) in the tracking field is defined as the self-attention with the input of
the concatenation (Cat) of the template and the search image:

$\text{PSelf-
Attn}=\text{Attn}\Big{(}Cat(\mathbf{X}_{z},\mathbf{X}_{s}),Cat(\mathbf{X}_{z},\mathbf{X}_{s})\Big{)}$
(3)

(a) PSelf-Attn

(b) AMix-Attn

(c) Cross-Attn

Figure 3: Configurations of information stream in attention map of packed
self-attention (PSelf-Attn), asymmetric mix-attention(AMix-Attn) and cross-
attention (Cross-Attn).

Analysis on Attention. As shown in Figure 2, we divide the computation of
attention mechanism, which involves both template and search image, into four
information streams:

  1. (1)

self-information enhancement on template;

  2. (2)

cross-information aggregation on template;

  3. (3)

cross-information aggregation on search image;

  4. (4)

self-information enhancement on search image.

These four information streams are also reflected in the four parts of the
attention map (In Figure 2, the index of each part in the attention map
corresponds to the information stream). Based on this dissection, we can
conveniently compare the differences between existing attention, including
packed self-attention, mix-attention, and cross-attention.

The PSelf-Attn and the mix-attention(Cui et al. 2022) are essentially
equivalent, the mix-attention is calculated as:

$\begin{split}&\text{PSelf-Attn}==\text{Mix-Attn}=\\\
\text{Cat}\Big{(}\text{AMap}\big{(}\mathbf{X}_{z}&,Cat(\mathbf{X}_{z},\mathbf{X}_{s})\big{)},\text{AMap}\big{(}\mathbf{X}_{s},Cat(\mathbf{X}_{z},\mathbf{X}_{s})\big{)}\Big{)}\\\
\end{split}$ (4)

which is the same as Eqn. 3, and they include all four information streams
(the attention map is shown as Figure 3a).

By the same analysis, the asymmetric mix-attention (AMix-Attn)  contains three
information streams (#1, #3, #4 info stream), which is shown in the Figure 3b
and is calculated as follows:

$\begin{split}&\text{AMix-Attn}=\\\
\text{Cat}\Big{(}\text{AMap}\big{(}\mathbf{X}_{z},\mathbf{X}_{z}&\big{)},\text{AMap}\big{(}\mathbf{X}_{s},Cat(\mathbf{X}_{z},\mathbf{X}_{s})\big{)}\Big{)}\end{split}$
(5)

The cross-attention contains two information streams (#2,#3 info stream) for
cross information aggregation, which is shown in the Figure 3c and is
calculated as follows:

$\text{Cross-
Attn}=\text{Cat}\Big{(}\text{AMap}\big{(}\mathbf{X}_{z},\mathbf{X}_{s}\big{)},\text{AMap}\big{(}\mathbf{X}_{s},\mathbf{X}_{z}\big{)}\Big{)}$
(6)

Table 1: The effectiveness of information streams in the attention mechanism on the LaSOT dataset. The visualized four parts in the attention map (AMap) correspond to the four information streams at the matched location. # | AMap | No. Info Stream | AUC | Prec
---|---|---|---|---
1 | 2 | 3 | 4
1 |  | ✓ | ✓ | ✓ | ✓ | 61.7 | 64.2
2 |  | ✓ |  | ✓ | ✓ | 64.0 | 67.7
3 |  |  | ✓ | ✓ | ✓ | 60.6 | 63.7
4 |  | ✓ | ✓ | ✓ |  | 58.8 | 60.1
5 |  |  | ✓ | ✓ |  | 57.9 | 58.5

In order to fully verify the importance of each part of packed attention, it
is necessary to evaluate the impact of each information stream individually.
The key of visual object tracking is to find the target in the search image,
there must be a cross-information aggregation of the search image (#3 info
stream). The other information streams can be blocked out to verify their
performance.

Based on the above idea, we conduct detailed experiments and the result is
shown in Table 1. Removing cross-information aggregation of the template (#2
info stream) of self-attention can greatly improve tracking performance (the
AUC and Prec of Table 1 #2 are better than that of Table 1 #1), and the cross-
information aggregation of the template will introduce a lot of noise in
template features, which is not recommended in visual tracking. However,
removing self-information enhancement (#3 and #4 info stream) of self-
attention severely degrades the tracking performance (the AUC and Prec of
Table 1 #3 and #4 are worse than that of Table 1 #1). From the results we can
conclude that self-information enhancement in multi-image attention plays a
greater role than cross-information aggregation, the cross-information
aggregation is indispensable in tracking but not greatly beneficial.

###  3.2 Correlative Masked Modeling

According to the above analysis, the best tracking performance can be achieved
by adopting three information streams: self-information on the template(#1
info stream), cross-information on the search image (#3 info stream), and
self-information on the search image (#4 info stream). These three information
streams can be grouped into two categories: two self-information enhancements
and one cross-information aggregation. We designed a correlative masked
modeling method to enhance the information aggregation of our tracking
framework, as shown in Figure 1. The ViT backbone is an encoder, and the
correlative masked decoder reconstructs the original image (the template and
search image respectively) from randomly masked tokens to enhance the self-
information and reconstructs the template image from search tokens to improve
cross-information aggregation. In parallel with the masked decoder, the search
image tokens go through a box estimation head as in (Yan et al. 2021) to
generate the result bounding box.

Figure 4: The correlative masked decoders consists of a self-decoder and a
cross-decoder. The self-decoder reconstructs the two original images, template
and search image, from its corresponding masked tokens. The cross-decoder
reconstructs the template image from search tokens.

Decoder. The decoders in our framework consist of a self-decoder and a cross-
decoder, these two decoders have the same structure but do not share weights,
each one is composed of a series of transformer blocks similar to the MAE, and
the last layer of the decoder is a linear projection with output channels
equal to the number of pixels in a patch. As shown in Figure 4, the decoder
takes masked tokens as input and predicts the original image pixels
corresponding to the template token and the search image token, where the
template tokens are only self-reconstructed to the template image for
enhancing the #1 information stream, search tokens are used to crossly
reconstruct the template image (for #3 info stream) and self-reconstruct the
search image (for #4 info stream).

Masking and Reconstruction. The encoder embeds the concatenation set of
template tokens and search tokens. Then we split the encoded tokens into
template tokens and search tokens, crop the search tokens using Precise RoI
Pooling(Jiang et al. 2018) to the same size as the template tokens, and sample
a subset of them. We randomly sample tokens at a high masking ratio (75%). Our
decoder predicts the pixel values for each masked token, and the output of the
decoder is reshaped to form a reconstructed image. We use the mean squared
error (MSE) between the reconstructed and original images on masked tokens as
our loss function.

###  3.3 Training and Inference

Our decoder is only used in the training phase, while does not participate in
the inference phase, hence it doesn’t affect the tracking speed. During the
training phase, our tracker takes a triplet input consisting of one search
region and two templates similar to STARK(Yan et al. 2021). We randomly sample
multiple frames from sequences in the training set, select the first frame and
the second frame as templates, and the last frame as the search region. In the
target localization training, we train the whole network except the scoring
head in an end-to-end manner with the combination of $L1$ Loss, generalized
IoU loss (Rezatofighi et al. 2019), and decoder loss $L_{dec}$. The full loss
function is defined as follows:

$Loss=\lambda_{L1}L_{1}(B_{i},\hat{B}_{i})+\lambda_{g}L_{g}(B_{i},\hat{B}_{i})+\lambda_{dec}L_{dec}$
(7)

where $\lambda_{L1}=5.0$, $\lambda_{g}=2.0$ and $\lambda_{dec}=0.3$ are the
weighting factors of three losses, $\hat{B}_{i}$ is the estimated box of the
target and $B_{i}$ is the ground-truth bounding box. The decoder loss
$L_{dec}$ is defined as:

$L_{dec}=L_{2}(z,z_{p})+L_{2}(s,s_{p})+L_{2}(z,s_{p})$ (8)

where the $L_{2}$ is the MSE loss, $z$ and $s$ represent the original template
image and search image, $z_{p}$ and $s_{p}$ represent the predicting template
image and search image respectively.

In the inference phase, we use two templates of the same size as the input.
One of which is the initial template and fixed, the other is online updated
and always set to the latest tracking result with high confidence. We use a
score head to control the updating of the online template. Our score head
consists of the multilayer perceptron (MLP) that receives a class-
token(Dosovitskiy et al. 2021) as input and evaluates the accuracy of current
tracking results.

##  4 Experiments

###  4.1 Implementation Details

In order to effectively verify the correctness of our analysis, we design the
compact transformer tracker without any other extra attention mechanisms. The
only structures remaining are feature extraction and aggregation, and multi-
layer feature aggregation. The main tracker only consists of a ViT backbone
and a box estimation head, we test both ViT-Base and ViT-Large, and the ViT
parameters are initialized with MAE (He et al. 2022) pre-trained model. We
refer our Compact Transformer tracker as CTTrack-B (the backbone of ViT-Base)
and CTTrack-L (the backbone of ViT-Large) in this section.

We adopt CoCo(Lin et al. 2014), LaSOT(Fan et al. 2019), GOT-10k(Huang, Zhao,
and Huang 2019), and TrackingNet(Muller et al. 2018) as our training dataset
except the GOT-10k benchmark. The training samples are directly sampled from
the same sequence and we apply common data augmentation operations including
brightness jitter and horizontal flip. The size of the input template is
128$\times$128, the search region is $5^{2}$ times of the target box area and
further resized to 320$\times$320\. The decoder parameters are initialized
with Xavier Uniform. The AdamW optimizer (Loshchilov and Hutter 2018) is
employed with initial learning rate (lr) of 1e-4 with the layer-wise decay
0.75, and the lr decreases according to the cosine function with the final
decrease factor of 0.1. We adopt a warm-up lr with the 0.2 warm-up factor on
the first 5 epochs. We train our model on 4 Nvidia Tesla V100 GPUs for a total
of 500 epochs, each epoch uses $6\times 10^{4}$ images. The mini-batch size is
set to 128 images with each GPU hosting 32 images. Our approach is implemented
in Python 3.7 with PyTorch 1.7.

###  4.2 Ablation Study

We ablate our compact transformer tracker on several intriguing properties
using the challenging LaSOT dataset and report the Area Under the Curve (AUC)
and Precision (Prec) as the validation accuracy.

Table 2: Model size and speed using different backbones. Methods | Params(M) | FLOPs(G) | Speed(_fps_)
---|---|---|---
CTTrack-B | 93.8 | 48.1 | 40
CTTrack-L | 313.9 | 163.7 | 22

Backbone Comparison. Table 2 shows the comparison of the transformer backbones
between the ViT-Base and ViT-Large backbone. The CTTrack-B reaches a higher
tracking speed while the CTTrack-L exhibits a better performance.

Reconstruction Streams. Our decoder enforces three types of reconstruction
streams as shown in Figure 4. Table 3 exhibits different configurations of
reconstruction streams, through varied combinations of search tokens
reconstruct search image (s2s), template tokens reconstruct template image
(t2t) and search tokens reconstruct template image(s2t). The result is
consistent with the conclusion of our previous analysis that self-information
enhancement (#5) plays the most important role in transformer tracking,
compared to cross-information aggregation(#4). Besides, search image
information has more influence than the template information, the s2s (#2)
improves performance the most among all streams (#2, #3, #4), from 64.0 to
64.7 in AUC score. After adopting all three reconstruction streams, tracking
accuracy improved by an impressive AUC score of 1.8%, which validates the
effectiveness of our masked modeling decoders.

Table 3: Ablation Study for the reconstruction streams. s2s represents search tokens reconstruct search image, t2t denotes template tokens reconstruct template image and s2t means search tokens reconstruct template image. # |  | Recons Type |  | AUC | Prec
---|---|---|---|---|---
| s2s | t2t | s2t |
1 |  | - | - | - |  | 64.0 | 67.7
2 |  | ✓ | - | - |  | 64.7 | 69.1
3 |  | - | ✓ | - |  | 64.4 | 68.4
4 |  | - | - | ✓ |  | 64.4 | 68.6
5 |  | ✓ | ✓ | - |  | 65.1 | 69.9
6 |  | ✓ | ✓ | ✓ |  | 65.8 | 70.9

Masking ratio. When we conduct reconstruction streams, we randomly mask the
input tokens according to a pre-defined ratio. Table 4 shows the influence of
different masking ratios. We mask the encoded template token and search tokens
with a random sampling strategy at different masking rates. Similar to the
conclusion obtained by the MAE(He et al. 2022), the optimal ratios are
relatively high, and the accuracy increases steadily with the masking ratio
growing until reaching 75%, which produces the best tracking results.

Table 4: Comparison on masking ratio. Mask Ratio | 25% | 50% | 75% | 90%
---|---|---|---|---
AUC | 64.6 | 65.7 | 65.8 | 64.9
Prec | 69.0 | 70.7 | 70.9 | 69.5

Online Template Updating. We evaluate the effect of the online update strategy
in our method. The ablation study result is shown in Table 5, #1 represents
the performance without template updating. We can see that applying a fixed
interval to update the online template (#2) is ineffective as it greatly
reduces the quality of template and causes tracking drift. It can be seen in
#3, there is a 0.2% improvement in the AUC score after applying the scoring
head to evaluate the accuracy of current tracking results.

Table 5: Ablation for the online template updating component.Online denotes updating the template at a fixed update interval. Score represents the online template is only updated with high confident samples.  | Online | Score | AUC | Prec
---|---|---|---|---
CTTrack-B | - | - | 65.8 | 70.9
✓ | - | 64.9 | 69.9
✓ | ✓ | 66.0 | 71.1
Figure 5: Visualization of attention map which compares the difference between
training with correlative decoder (w) and training without correlative
decoder(w/o). S-to-S is self-information enhancement on search image, T-to-T
is self-information enhancement on template, S-to-T is cross-information
aggregation on search image.

Visualization of attention maps. We visualize attention maps in Figure5, our
tracker adopting the correlative decoder has a stronger discriminative
ability. The baseline transformer without a reconstruction decoder tends to
lose the target position, and the distractors in the background get suppressed
with the training by the correlative decoder.

###  4.3 Comparison with the SOTA

Table 6: Comparisons with previous state-of-the-art trackers on four challenge benchmarks. The red, green and blue indicate performances ranked at first, second, and third places. The tracker -GOT denotes only trained on the GOT-10k train split. Methods | UAV123 |  | LaSOT |  | TrackingNet |  | GOT-10k
---|---|---|---|---|---|---|---
AUC | P |  | AUC | PNorm | P |  | AUC | PNorm | P |  | AO | SR0.5 | SR0.75
CTTrack-L | 71.3 | 93.3 |  | 69.8 | 79.7 | 76.2 |  | 84.9 | 89.1 | 83.5 |  | 75.3 | 84.5 | 74.0
CTTrack-B | 68.8 | 89.5 |  | 67.8 | 77.8 | 74.0 |  | 82.5 | 87.1 | 80.3 |  | 73.5 | 83.5 | 70.6
CTTrack-L -GOT | - | - |  | - | - | - |  | - | - | - |  | 72.8 | 81.3 | 71.5
CTTrack-B -GOT | - | - |  | - | - | - |  | - | - | - |  | 71.3 | 80.7 | 70.3
MixFormer(Cui et al. 2022) | 69.5 | 91.0 |  | 70.1 | 79.9 | 76.3 |  | 83.9 | 88.9 | 83.1 |  | 70.7 | 80.0 | 67.8
CSWinTT(Song et al. 2022) | 70.5 | 90.3 |  | 66.2 | 75.2 | 70.9 |  | 81.9 | 86.7 | 79.5 |  | 69.4 | 78.9 | 65.4
UTT(Shen et al. 2022) | - | - |  | 64.6 | - | 67.2 |  | 79.7 | - | 77.0 |  | 67.2 | 76.3 | 60.5
STARK(Yan et al. 2021) | - | - |  | 67.1 | 77.0 | - |  | 82.0 | 86.9 | - |  | 68.8 | 78.1 | 64.1
TransT(Chen et al. 2021) | 68.1 | 87.6 |  | 64.9 | 73.8 | 69.0 |  | 81.4 | 86.7 | 80.3 |  | 67.1 | 76.8 | 60.9
TrDiMP(Wang et al. 2021) | 67.0 | 87.6 |  | 64.0 | 73.2 | 66.6 |  | 78.4 | 83.3 | 73.1 |  | 68.8 | 80.5 | 59.7
STMTrack(Fu et al. 2021) | 64.7 | - |  | 60.6 | 69.3 | 63.3 |  | 80.3 | 85.1 | 76.7 |  | 64.2 | 73.7 | 57.5
AutoMatch(Zhang et al. 2021) | 64.4 | 83.8 |  | 58.2 | 67.5 | 59.9 |  | 76.0 | 82.4 | 72.5 |  | 65.2 | 76.6 | 54.3
SiamGAT(Guo et al. 2021) | 64.6 | 84.3 |  | 53.9 | 63.3 | 53.0 |  | - | - | - |  | 62.7 | 74.3 | 48.8
KYS(Bhat et al. 2020) | - | - |  | 55.4 | 63.3 | 55.8 |  | 74.0 | 80.0 | 68.8 |  | 63.6 | 75.1 | 51.5
MAML(Wang et al. 2020) | - | - |  | 52.3 | - | 53.1 |  | 75.7 | 82.2 | 72.5 |  | - | - | -
SiamAttn(Yu et al. 2020) | 65.0 | 84.5 |  | 56.0 | 64.8 | - |  | 75.2 | 81.7 | - |  | - | - | -
SiamFC++(Xu et al. 2020) | 61.8 | 80.4 |  | 54.4 | 62.3 | 54.7 |  | 75.4 | 80.0 | 70.5 |  | 59.5 | 69.5 | 47.9
SiamRPN++(Li et al. 2019) | 64.2 | 84.0 |  | 49.6 | 56.9 | 49.1 |  | 73.3 | 80.0 | 69.4 |  | 51.7 | 61.6 | 32.5
DiMP(Bhat et al. 2019) | 64.2 | 84.9 |  | 57.7 | 66.4 | 57.9 |  | 74.0 | 80.1 | 68.7 |  | 61.1 | 71.7 | 49.2
ATOM(Danelljan et al. 2019) | 61.7 | 82.7 |  | 51.5 | 57.6 | 50.5 |  | 70.3 | 77.1 | 64.8 |  | 55.6 | 63.4 | 40.2
Table 7: Comparisons on VOT2020, where trackers only predict bounding boxes rather than masks. Methods | EAO$\uparrow$ | Accuracy$\uparrow$ | Robustness$\uparrow$
---|---|---|---
SiamFC | 0.179 | 0.418 | 0.502
ATOM | 0.271 | 0.462 | 0.734
DiMP | 0.274 | 0.457 | 0.740
UPDT | 0.278 | 0.465 | 0.755
TransT | 0.293 | 0.477 | 0.754
CSWinTT | 0.304 | 0.480 | 0.787
CTTrack-L | 0.287 | 0.453 | 0.787

We compare our compact tracker with the state-of-the-art trackers on
UAV123(Mueller, Smith, and Ghanem 2016), LaSOT(Fan et al. 2019),
TrackingNet(Muller et al. 2018), GOT-10k(Huang, Zhao, and Huang 2019), and
VOT2020(Kristan et al. 2020). For a fairer comparison, here we adopt relative
position biases in our ViT backbones, this addition improves AUC by around 1
point.

UAV123 gathers an application-specific collection of 123 sequences. It adopts
the AUC and Precision (P) as the evaluation metrics. As shown in Table 1, Our
CTTrack-L outperforms previous trackers and exhibits very competitive
performance (71.3% AUC) when compared to the previous best-performing tracker
CSWinTT (70.5% AUC).

LaSOT is a long-term dataset including 1400 sequences and distributed over 14
attributes, the testing subset of LaSOT contains 280 sequences. Methods are
ranked by the AUC, P, and Normalized Precision (PNorm). Our CTTrack-L achieves
the AUC (69.8%) and Prec (76.2%), which is an excellent result that
outperforms other methods only except the MixFormer. Our tracker has lower
performance than MixFormer on LaSOT because it contains long-term sequences
and large variations in content. ViT backbone is a plain and non-hierarchical
architecture that maintains feature maps at a certain scale, which may not be
able to well handle long-term tracking sequences with scale variations.

TrackingNet is a large-scale tracking dataset consisting of 511 sequences for
testing. The evaluation is performed on the online server. Table 1 shows that
CTTrack-L performs better quality and ranks first in AUC score at 84.9%. The
gain is 1.0% improvement when compared with the previous best results.

GOT-10k contains over 10k videos for training and 180 for testing. It forbids
the trackers to use external datasets for training. We follow this protocol by
retraining our trackers to only use the GOT10k train split. As in Table 1,
MixFormer and CSWinTT provide the best performance, with an AO score of 70.7%
and 69.4%. Our CTTrack-L has obtained an AO score of 72.8%, significantly
outperforming the best existing tracker by 2.1%.

VOT2020 benchmark contains 60 challenging videos. The performance is evaluated
using the expected average overlap (EAO), which takes both accuracy (A) and
robustness (R). Since our algorithm does not output a segmentation mask,
trackers that only predict bounding boxes are selected for comparisons to
ensure fairness. It can be seen from Table 7 that our CTTrack-L obtains an EAO
of 0.287.

##  5 Conclusion

In this work, we analyze the information stream in the attention mechanism in
depth. We prove that the vanilla self-attention structure is sufficient for
information aggregation, and employ the three information streams of the
packed self-attention in the transformer tracking framework. To enhance the
information representation, we design the correlative masked decoder
consisting of a self-decoder and a cross-decoder to reconstruct the original
pixels of both template and search image. Extensive experiments demonstrate
the effectiveness of our correlative masked modeling strategy and our compact
transformer tracker exhibits impressive performance over previous trackers. In
addition, our correlative masked decoder can be plugged into other transformer
trackers, which can effectively improve the tracking performance without
compromising speed. In the future, we plan to combine the feature pyramid or
convolution module for better performance on long-term tracking sequences.

## Acknowledgments

This work is supported by the national key research and development program of
China under Grant No.2020YFB1805601, National Natural Science Foundation of
China (NSFC No. 62272184), and CCF-Tencent Open Research Fund (CCF-Tencent
RAGR20220120). The computation is completed in the HPC Platform of Huazhong
University of Science and Technology.

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# Class number divisibility of $\mathbb{Q}(\sqrt{3p},\sqrt{m-21pn^{2}})$
constructed from elliptic curves of $2$-Selmer rank exactly $1$

Debopam Chakraborty  Department of Mathematics
BITS-Pilani, Hyderabad campus
Hyderabad, INDIA<EMAIL_ADDRESS>,  Vinodkumar Ghale
Department of Mathematics
BITS-Pilani, Hyderabad campus
Hyderabad, INDIA<EMAIL_ADDRESS>and  Md Imdadul Islam
Department of Mathematics
BITS-Pilani, Hyderabad campus
Hyderabad, INDIA<EMAIL_ADDRESS>
###### Key words and phrases:

Class number; Elliptic curve; Selmer group.

###### 2020 Mathematics Subject Classification:

Primary 11R29, 11G07; Secondary 11G05, 11R11

## Abstract

The class number divisibility problem for number fields is one of the
classical problems in algebraic number theory, which originated from Gauss’
class number conjectures. The relation between the points on an elliptic curve
and class number divisibility of a number field has been explored through the
works of various mathematicians. Here, we explicitly construct an unramified
abelian extension of a bi-quadratic field generated from points of a certain
type of elliptic curve. Moreover, showing the $2$-Selmer rank of the said
elliptic curve as $1$, we also construct an infinite family of bi-quadratic
fields of even class number.

##  1\. Introduction

The class group of a number field $K$ measures how much the ring of integers
$\mathcal{O}_{K}$ deviates from having unique factorization into irreducible
elements. The ideal class group of a number field is the quotient of the group
of fractional ideals of $K$ by the group of principal fractional ideals. The
ideal class group of a number field $K$ can be identified with the Galois
group of the maximal unramified abelian extension of $K$. Hence, an unramified
abelian extension plays an important role in the class number divisibility
problem for a number field.
Gauss was the first to formalize the definition of class groups for quadratic
number fields through the language of binary quadratic forms. He conjectured
that for $d<0$, the class number $h(d)$ of $\mathbb{Q}(\sqrt{d})$ tends to
infinity as $-d$ tends to infinity. Heilbronn proved this result in [6]. The
general Gauss’ class number problem for imaginary quadratic fields was solved
through the works of Goldfield [3] [4], Gross, and Zagier [5]. Several
mathematicians studied ideal class groups of a number field through different
tools that originate from the arithmetic of the elliptic curve. In [7] and
[8], T. Honda used elliptic curves to construct infinitely many quadratic
fields with class numbers divisible by $3$. A.Sato followed Honda’s work with
a geometric interpretation and extended the results to the cases for primes
$5$ and $7$ in [12]. In [14], R. Soleng gave a class number divisibility
condition for quadratic fields with ideal class groups isomorphic to the
torsion group of certain elliptic curves. Lemmermeyer [9] constructed an
unramified quadratic extension of cubic fields generated from points of
elliptic curves. A similar approach for bi-quadratic fields was taken by
Chakraborty et al. in [1]. In a fundamental work of Coates and Sujatha [2], it
was shown that the fine Selmer group, a subgroup of the classical Selmer
group, satisfies stronger local conditions and shows stronger finiteness
properties than the regular Selmer group. In [10], Kumar Murty et al. proved
the similarity between the growth of the $p$-rank of the Fine Selmer group and
the growth of the ideal class group of a number field.
In this work, we construct a bi-quadratic field from the points of an elliptic
curve and then simultaneously try to understand the $2$-part of the Selmer and
Shafarevich Tate group of the said curve and also the class number
divisibility problem of the bi-quadratic field constructed from the points of
that curve. The main result of this work is the following theorem, which
explicitly constructs an unramified abelian extension of a bi-quadratic field
generated from an elliptic curve of $2$-Selmer rank exactly 1. We note that
the elliptic curve here is distinctively different from the curve
$y^{2}=x^{3}+m$ that was used in both [9] and [1].

######  Theorem 1.1.

Let $E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$ be an elliptic curve where $p$ is a
prime such that either $p\equiv 17,113\pmod{120}$. Then the $2$-Selmer rank of
$E_{p}$ is exactly $1$. Moreover, if
$\big{(}\frac{r}{t^{2}},\frac{s}{t^{3}}\big{)}$ is an arbitrary point on
$E_{p}$, then there exists an unramified quadratic extension of the bi-
quadratic field $K=\mathbb{Q}(\sqrt{3p},\sqrt{r-21pt^{2}})$ whenever $t$ is
even and $\gcd(s,3p)=1$.

The following result deals with the case when $p$ is a prime of the form
$p\equiv 53,77\pmod{120}$. The result ensures the rank of $E_{p}$ to be zero.

######  Corollary 1.2.

Let $E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$ be an elliptic curve where $p$ is a
prime such that $p\equiv 53,77\pmod{120}$. Then the $2$-Selmer rank of $E_{p}$
is $0$.

The following result constructs an infinite family of bi-quadratic fields of
even class number generated from $E_{p}$ for a fixed prime $p\equiv
17,113\pmod{120}$.

######  Corollary 1.3.

If $P_{0}=\Big{(}\frac{r_{0}}{t_{0}^{2}},\frac{s_{0}}{t_{0}^{3}}\Big{)}$ is an
arbitrary point on $E_{p}$ such that $t_{0}$ is even and $\gcd(s_{0},3p)=1$,
then there exists an infinite family of the bi-quadratic fields
$K_{i}=\mathbb{Q}(\sqrt{3p},\sqrt{r_{i}-21pt_{i}^{2}})$ with even class
number, $i=0,1,2,...$ and
$P_{i}=2^{i}P_{0}=\Big{(}\frac{r_{i}}{t_{i}^{2}},\frac{s_{i}}{t_{i}^{3}}\Big{)}$.

##  2\. Unramified abelian extension

In this section, we construct an unramified quadratic extension of a bi-
quadratic field generated from the elliptic curve $E:y^{2}=(x-a)(x+b)(x-c)$
such that each of $a,b$, and $c$ are positive and $ab+bc=ac$. We start with
the following lemma, which introduces an element in the bi-quadratic field of
our interest.

######  Lemma 2.1.

Let $\big{(}\frac{r}{t^{2}},\frac{s}{t^{3}}\big{)}$ be an arbitrary point on
$E:y^{2}=(x-a)(x+b)(x-c)$ defined above. Let $K$ denotes the bi-quadratic
field $\mathbb{Q}(\sqrt{abc},\sqrt{r+(b-a-c)t^{2}})$. If
$\alpha=s+t^{3}\sqrt{abc}\in K$, then $\alpha\mathcal{O}_{K}=\mathfrak{a}^{2}$
for some ideal $\mathfrak{a}\subset\mathcal{O}_{K}$ whenever $\gcd(s,abc)=1$
and $t$ is even.

###### Proof.

We first notice that $P=\big{(}\frac{r}{t^{2}},\frac{s}{t^{3}}\big{)}\in
E(\mathbb{Q})$ implies $s^{2}=r^{3}+(b-a-c)r^{2}t^{2}+abct^{6}$. Assuming
$\bar{\alpha}=s-t^{3}\sqrt{abc}$, this in return shows that

$N_{K/\mathbb{Q}}(\alpha)=\alpha^{2}{\bar{\alpha}}^{2}=(s^{2}-abct^{6})^{2}=(r\sqrt{r+(b-a-c)}t^{2})^{4}.$

Without loss of generality, one can choose $\gcd(r,t)=\gcd(s,t)=1$. Hence, if
$t$ is even, both $r$ and $s$ must be odd. Now for any prime ideal
$\wp\subset\mathcal{O}_{K}$, if $\alpha,\bar{\alpha}\in\wp$, then
$\alpha+\bar{\alpha}=2s,\alpha\cdot\bar{\alpha}=r^{3}+(b-a-c)r^{2}t^{2}\in\wp$.
This implies that $2\not\in\wp$ as $r^{3}+(b-a-c)r^{2}t^{2}$ is odd. Now,
$s,r^{3}+(b-a-c)r^{2}t^{2}\in\wp$ implies that $abct^{6}\in\wp$ which is again
a contradiction as $\gcd(s,abc)=\gcd(s,t)=1$.
As $\gcd(\alpha,\bar{\alpha})=1$, we can now conclude that
$\alpha^{2}\mathcal{O}_{K}=\mathfrak{a}^{4}$ now for some ideal
$\mathfrak{a}\subset\mathcal{O}_{K}$. This concludes the proof. ∎

We conclude this section with the following result concerning the ramification
of primes in a quadratic extension of $K$.

######  Lemma 2.2.

Let $\big{(}\frac{r}{t^{2}},\frac{s}{t^{3}}\big{)}$ be an arbitrary point on
$E:y^{2}=(x-a)(x+b)(x-c)$ defined above. Let $K$ denotes the bi-quadratic
field $\mathbb{Q}(\sqrt{abc},\sqrt{r+(b-a-c)t^{2}})$. If
$\alpha=s+t^{3}\sqrt{abc}\in K$, then $K(\sqrt{\alpha})/K$ is an unramified
abelian extension if $t$ is even, $\gcd(s,abc)=1$, and $K$ is either imaginary
or $3$ ramifies in $K/\mathbb{Q}$.

###### Proof.

The extension $K(\sqrt{\alpha})/K$ is quadratic, hence abelian. For the
ramification of primes, we first notice that
$\alpha\mathcal{O}_{K}=\mathfrak{a}^{2}$ for an ideal $\mathfrak{a}$ implies
that there are no finite primes except possible primes above $2$ that can
ramify in $K(\sqrt{\alpha})/K$.
For the ramification of primes above $2$, we first notice that $t$ even
implies that $s$ is odd, hence $\alpha\equiv s\equiv 1,3\pmod{4}$. If $s\equiv
1\pmod{4}$, no primes above $2$ ramify in $K(\sqrt{\alpha})/K$. If
$\alpha\equiv s\equiv 3\pmod{4}$ and $K$ is imaginary, replacing $\alpha$ by
$-\alpha$ again make the extension $K(\sqrt{\alpha})/K$ unramified for all
primes including infinite primes. In case, $K$ is real and $\alpha\equiv
s\equiv 3\pmod{4}$, because $3$ ramifies in $K/\mathbb{Q}$, replacing $\alpha$
by $3\alpha$ make the extension $K(\sqrt{\alpha})/K$ unramified for all
primes. This concludes the proof. ∎

##  3\. The $2$-descent method

We first recall the $2$-descent method to find the $2$-Selmer group for the
elliptic curve

(3.1) $E_{p}:\;\;y^{2}=(x+6p)(x-9p)(x-18p),$

where $p$ is any prime. The discriminant of the elliptic curve $E_{p}$ can be
observed to be $16\cdot 2^{6}\cdot 3^{8}\cdot 5^{2}\cdot p^{6}$. Reducing
$E_{p}$ modulo $11$ and modulo $13$ one can immediately observe that
$|(\tilde{E}_{p})(\mathbb{F}_{11})|=12$ and
$|(\tilde{E}_{p})(\mathbb{F}_{13})|=8\text{ or }20$. Since $E[2]\subset
E_{tors}(\mathbb{Q})$ and $E_{tors}(\mathbb{Q})$ injects into both
$\tilde{E}(\mathbb{F}_{11})$ and $\tilde{E}(\mathbb{F}_{13})$ for any elliptic
curve $E$, one can see that
$(E_{p})_{tors}(\mathbb{Q})\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.
Let $S$ be the set consisting of all finite places at which $E_{p}$ has a bad
reduction, the infinite places, and the prime $2$, i.e.,
$S=\\{p,\,2,\,3,\,5,\,\infty\\}$. We define

(3.2) $\displaystyle\mathbb{Q}(S,2)$
$\displaystyle=\left\\{b\in\mathbb{Q}^{*}/(\mathbb{Q}^{*})^{2}:\text{ord}_{l}(b)\equiv
0~{}(\bmod\text{ }{2})~{}\text{for all primes}~{}l\not\in S\right\\}$
$\displaystyle=\left\\{\pm 1,\;\pm 2,\;\pm 3,\;\pm 5,\;\pm p,\;\pm 6,\;\pm
10,\;\pm 2p,\;\pm 15,\;\pm 3p,\;\pm 5p,\;\pm 30,\;\pm 6p,\;\pm 10p,\;\pm
15p,\;\pm 30p\right\\}.$

By the method of 2-descent (see [13], Proposition X.$1.4$), there exists an
injective homomorphism

$\phi:E_{p}(\mathbb{Q})/2E_{p}({\mathbb{Q}})\longrightarrow\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$

defined by

$\displaystyle\phi(x,y)=\begin{cases}(x+6p,x-9p)&\text{if}\text{
}x\neq-6p,9p,\\\ (10,-15p)&\text{if}\text{ }x=-6p,\\\
(15p,-15)&\text{if}\text{ }x=9p,\\\ (1,1)&\text{if}\text{ }x=\infty,\text{
i.e., if }(x,y)=O,\end{cases}$

where $O$ is the fixed base point (also identified as the point of infinity
$[0,1,0]$ in the projective plane) that acts as the identity element in the
group $E_{p}(\mathbb{Q})$. Moreover, if
$(b_{1},b_{2})\in\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$ is a pair that is not
in the image of one of the three points $O,(-6p,0),(9p,0)$, then
$(b_{1},b_{2})$ is the image of a point $P=(x,y)\in
E_{p}(\mathbb{Q})/2E_{p}(\mathbb{Q})$ if and only if the equations

(3.3) $\displaystyle b_{1}z_{1}^{2}-b_{2}z_{2}^{2}=15p,$ (3.4) $\displaystyle
b_{1}z_{1}^{2}-b_{1}b_{2}z_{3}^{2}=24p,$

have a solution
$(z_{1},z_{2},z_{3})\in\mathbb{Q}^{*}\times\mathbb{Q}^{*}\times\mathbb{Q}$.
These equations represent a finite set of smooth curves, each isomorphic to
$E_{p}$ over $\bar{\mathbb{Q}}$ together with a simply transitive algebraic
group action of $E_{p}$ on each of them over $\mathbb{Q}$. These smooth curves
are called homogeneous space of $E_{p}$ defined over $\mathbb{Q}$. Computing
$E_{p}(\mathbb{Q})/2E_{p}(\mathbb{Q})$ boils down to determining the existence
of $\mathbb{Q}$-rational points in these spaces. Furthermore, the general
failure of the Hasse-Minkowski principle motivates the definition of the
Selmer group, which studies the adelic points on the homogeneous spaces (see
Remark X$.1.2$, X$.4.9$, and Section $3$ of chapter X in [13]).
The image of $E_{p}(\mathbb{Q})/2E_{p}({\mathbb{Q}})$ under the $2$-descent
map is contained in a subgroup of $\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$ known
as the $2$-Selmer group $\mbox{Sel}_{2}(E_{p}/\mathbb{Q})$, which fits into an
exact sequence (see Chapter X, [13], Theorem X$.4.2$)

(3.5) $0\longrightarrow
E_{p}(\mathbb{Q})/2E_{p}(\mathbb{Q})\longrightarrow\mbox{Sel}_{2}(E_{p}/\mathbb{Q})\longrightarrow\Sha(E_{p}/\mathbb{Q})[2]\longrightarrow
0.$

The elements in $\mbox{Sel}_{2}(E/\mathbb{Q})$ for any elliptic curve $E$
correspond to the pairs $(b_{1},b_{2})\in\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$
such that the system of equations (3.3) and (3.4) has non-trivial local
solutions in $\mathbb{Q}_{l}$ at all primes $l$ of $\mathbb{Q}$ including
infinity. Note that
$\\#\,E_{p}(\mathbb{Q})/2E_{p}(\mathbb{Q})=2^{2+r(E_{p})}$. It is customary to
write $\\#\,\mbox{Sel}_{2}(E_{p}/\mathbb{Q})=2^{2+s(E_{p})}$, and refer to
$s(E_{p})$ as the $2$-Selmer rank. We have

(3.6) $0\leq r(E_{p})\leq s(E_{p}).$

From the inequality above as well as the exact sequence (3.5), one can say
that the $2$-Selmer group provides information about both $E_{p}(\mathbb{Q})$
and $\Sha(E_{p}/\mathbb{Q})[2]$.

##  4\. Local solutions of the homogeneous space

In this section, we examine the properties of the $l$-adic solutions for (3.3)
and (3.4) that are associated with the $2$-Selmer group. We use these
properties later to bound the size of the $2$-Selmer group. Any $l$-adic
number $a$ can be written as $a=l^{n}\cdot u\text{ where }n\in\mathbb{Z}$,
$u\in\mathbb{Z}_{l}^{*}$. Notice that the $l$-adic valuation $v_{l}(a)$ of $a$
is just $n$. We first prove the following result for all primes $l\neq 5$.

######  Lemma 4.1.

Suppose (3.3) and (3.4) have a solution
$(z_{1},z_{2},z_{3})\in\mathbb{Q}_{l}\times\mathbb{Q}_{l}\times\mathbb{Q}_{l}$
for any odd prime $l\neq 5$. If $v_{l}(z_{i})<0$ for any one
$i\in\\{1,2,3\\}$, then $v_{l}(z_{1})=v_{l}(z_{2})=v_{l}(z_{3})=-k<0$ for some
integer $k$.

###### Proof.

Let $z_{i}=l^{k_{i}}u_{i}$, where $k_{i}\in\mathbb{Z}$ and
$u_{i}\in\mathbb{Z}_{l}^{*}$ for $i\in\\{1,2,3\\}$. Then $v_{l}(z_{i})=k_{i}$
for all $i\in\\{1,2,3\\}$.
Suppose $k_{1}<0$. Then from (3.3) one can get that

$b_{1}u_{1}^{2}-b_{2}u_{2}^{2}l^{2(k_{2}-k_{1})}=15pl^{-2k_{1}}.$

If $k_{2}>k_{1}$, then $l^{2}$ must divide $b_{1}$, a contradiction as $b_{1}$
is square-free. Hence $k_{2}\leq k_{1}<0$. Now if $k_{2}<k_{1}<0$ then again
from (3.3) we get

$b_{1}u_{1}^{2}l^{2(k_{1}-k_{2})}-b_{2}u_{2}^{2}=15pl^{-2k_{2}},$

which implies $l^{2}$ must divide $b_{2}$, a contradiction again. Hence if
$k_{1}<0$, then we have $k_{1}=k_{2}=-k<0$ for some integer $k$. For
$k_{2}<0$, one similarly gets $k_{1}=k_{2}=-k<0$.
From (3.4), we have

$b_{1}u_{1}^{2}-b_{1}b_{2}u_{3}^{2}l^{2(k_{3}-k_{1})}=24pl^{-2k_{1}}.$

If $k_{1}<0$ and $k_{3}>k_{1}$, then $l^{2}$ must divide $b_{1}$, a
contradiction as before. Hence $k_{3}\leq k_{1}<0$ if $k_{1}<0$. For
$k_{3}<k_{1}<0$, we can rewrite the above equation as

(4.1) $b_{1}u_{1}^{2}l^{2(k_{1}-k_{3})}-b_{1}b_{2}u_{3}^{2}=24pl^{-2k_{3}},$

which implies $l^{2}$ must divide $b_{1}b_{2}$ i.e. $l$ divides both $b_{1}$
and $b_{2}$. If $l=2,3$ or $p$, then from (4.1) we arrive at the contradiction
that $l^{3}$ divides $b_{1}b_{2}$ whereas $b_{1}$ and $b_{2}$ are square-free.
Hence $k_{1}<0$ implies $k_{3}=k_{1}<0$. Together, we obtain
$k_{1}=k_{2}=k_{3}=-k<0$ for some integer $k$ if $k_{1}<0$.
Now if $k_{2}<0$ but $k_{1}\geq 0$, then from 3.3, we get

$b_{1}u_{1}^{2}l^{2(k_{1}-k_{2})}-b_{2}u_{2}^{2}=15pl^{-2k_{2}}\implies
b_{2}\equiv 0\pmod{l^{2}},$

a contradiction. Hence $k_{2}<0\implies k_{1}<0$ and we are back to the first
case which implies $k_{1}=k_{2}=k_{3}=-k$ for some integer $k\geq 0$.
Similarly if $k_{3}<0$ but $k_{1}\geq 0$, we get from 3.4, for $l\neq 5$,

$b_{1}u_{1}^{2}l^{2(k_{1}-k_{3})}-b_{1}b_{2}u_{3}^{2}=24pl^{-2k_{3}}\implies
l^{3}\text{ divides }b_{1}b_{2},\text{ a contradiction }.$

So again we observe that $k_{3}<0$ implies $k_{1}<0$ and hence
$k_{1}=k_{2}=k_{3}=-k$ for some positive integer $k$ again. This concludes the
proof. ∎

An immediate observation from the proof above is that the conclusion of Lemma
4.1 stays true even for the case $l=5$ as long as $b_{1}b_{2}\not\equiv
0\pmod{25}$. We conclude this section with the following lemma for the case
$l=5$.

######  Lemma 4.2.

Suppose $5$ divides $\gcd(b_{1},b_{2})$ and the corresponding equations (3.3)
and (3.4) have a solution
$(z_{1},z_{2},z_{3})\in\mathbb{Q}_{5}\times\mathbb{Q}_{5}\times\mathbb{Q}_{5}$.
If $v_{5}(z_{i})<0$ for any one $i\in\\{1,2\\}$, then
$v_{5}(z_{1})=v_{5}(z_{2})=v_{5}(z_{3})=-k<0$ for some integer $k$. If
$v_{5}(z_{3})<0$ but $v_{5}(z_{1})\neq v_{5}(z_{3})$, then $v_{5}(z_{3})=-1$
and $v_{5}(z_{1})\geq 0$.

###### Proof.

The first part of the statement follows similarly to the proof of Lemma 4.1.
Hence we only focus on the case $k_{3}=v_{5}(z_{3})<0$. Now if
$k_{1}=v_{5}(z_{1})\leq k_{3}<0$ then that will imply from (3.4),

$b_{1}u_{1}^{2}-b_{1}b_{2}u_{3}^{2}5^{2(k_{3}-k_{1})}=24\cdot p\cdot
5^{-2k_{1}}\implies b_{1}\equiv 0\pmod{25},$

a contradiction. Now if $k_{3}<k_{1}$, and $k_{3}\leq-2$, we again get from
(3.4),

$b_{1}u_{1}^{2}5^{2(k_{1}-k_{3})}-b_{1}b_{2}u_{3}^{2}=24\cdot p\cdot
5^{-2k_{3}}\implies b_{1}b_{2}\equiv 0\pmod{5^{3}},$

a contradiction. Hence $k_{3}<0$ implies $k_{3}=-1$ and $k_{1}>k_{3}$ i.e.
$k_{1}\geq 0$. Now the result follows. ∎

##  5\. Bounding the size of the $2$-Selmer group of $E_{p}$

In this section, we bound the size of the $2$-Selmer group of the elliptic
curve $E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$. The $2$-Selmer group
$\mbox{Sel}_{2}(E_{p}/\mathbb{Q})$ consists of those pairs $(b_{1},b_{2})$ in
$\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$ for which equations (3.3) and (3.4)
have an $l$-adic solution at every place $l$ of $\mathbb{Q}$. We now limit the
size of $\mbox{Sel}_{2}(E_{p}/\mathbb{Q})$ by ruling out local solutions for
specific pairs $(b_{1},b_{2})$ by exploiting the results of the previous
sections.

######  Lemma 5.1.

Let $(b_{1},b_{2})\in\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$. Then
$(i)$ The corresponding homogeneous space will have no $l$-adic solution for
the case $l=\infty$ if $b_{1}<0$.
$(ii)$ If $\gcd(b_{1},b_{2})\equiv 0\pmod{2}$, the corresponding homogeneous
space will not have $2$-adic solutions.

###### Proof.

$(i)$ The result follows from the observation that $b_{1}<0$ in equations
(3.3) and (3.4) either implies $15p<0$ when $b_{2}>0$ or $24p<0$ when
$b_{2}<0$.
$(ii)$ Suppose $\gcd(b_{1},b_{2})$ is even. If moreover, $v_{2}(z_{i})<0$ for
some $i\in\\{1,2,3\\}$, then from Lemma 4.1, we can say that $v_{2}(z_{i})=-k$
for some positive integer $k$ and for all $i\in\\{1,2,3\\}$. From (3.4) we can
now get

$b_{1}u_{1}^{2}-b_{1}b_{2}u_{3}^{2}=24\cdot p\cdot 2^{2k},$

where $z_{i}=u_{i}\cdot 2^{-k},u_{i}\in{\mathbb{Z}_{2}}^{*}$ for all
$i\in\\{1,2,3\\}$. This immediately implies that $b_{1}\equiv 0\pmod{4}$, a
contradiction. Hence $v_{2}(z_{i})\geq 0$ for all $i\in\\{1,2,3\\}$ which is
again a contradiction as then (3.3) implies $15p\equiv 0\pmod{2}$. Hence the
result follows. ∎

Let
$A=\\{(1,1),(10,-15p),(15p,-15),(6p,p)\\}\subset\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$
be the image of $(E_{p})_{tors}(\mathbb{Q})$ under the map $\phi$. We notice
the fact that it is enough to check the local solutions for homogeneous space
corresponding to only one $(b_{1},b_{2})$ for each coset in
$(\mathbb{Q}(S,2)\times\mathbb{Q}(S,2))/A$ as the existence or non-existence
of local solutions for one pair $(b_{1},b_{2})$ determines the same for all
four members of the coset. This observation leads us to the following lemma.

######  Lemma 5.2.

$(i)$ The homogeneous space corresponding to $(b_{1},b_{2})$ can not have
$l$-adic solution for all primes $l\leq\infty$ if $b_{2}$ is even.
$(ii)$ Let $\gcd(b_{1},b_{2})\equiv 0\pmod{3}$. Then
$(b_{1},b_{2})\equiv(c_{1},c_{2})\pmod{A}$ such that $\gcd(c_{1},c_{2})$ is
not divisible by either $3$ or $p$.
$(iii)$ Let $\gcd(b_{1},b_{2})\equiv 0\pmod{p}$. Then
$(b_{1},b_{2})\equiv(c_{1},c_{2})\pmod{A}$ such that $\gcd(c_{1},c_{2})$ is
not divisible by either $3$ or $p$.

Before proving Lemma 5.2, we first explain the importance of it in the
$2$-descent method here. Once proved, Lemma 5.2 will imply that without loss
of generality, we can assume $b_{2}$ is odd and $\gcd(b_{1},b_{2})$ is not
divisible by either $3$ or $p$ while looking for local solutions as long as
$(b_{1},b_{2})$ satisfies the conditions above.

###### Proof.

$(i)$ Let us suppose $b_{2}$ is even. Then from Lemma 5.1, we can deduce that
$b_{1}$ is odd. But then $(b_{1},b_{2})$ and $(6pb_{1},pb_{2})$ belong to the
same coset of $A$ and the homogeneous space corresponding to
$(6pb_{1},pb_{2})$ has no $2$-adic solution from Lemma 5.1. Hence the result
follows.
$(ii)$ We prove this by observing that modulo $A$, the coset of
$(b_{1},b_{2})$ consists of
$\\{(b_{1},b_{2}),(10b_{1},-15pb_{2}),(15pb_{1},-15b_{2}),(6pb_{1},pb_{2})\\}$.
The result is now evident from the fact that one of
$(15pb_{1},-15b_{2}),(6pb_{1},pb_{2})$ will always satisfy the condition of
the statement depending on whether $3$ or $p$ divides $b_{1},b_{2}$ or not.
$(iii)$ The proof is identical to the proof of the previous result. ∎

We are now left with $256$ points. As mentioned above, from now on we only
consider $(b_{1},b_{2})$ such that $b_{2}$ is odd and $\gcd(b_{1},b_{2})$ is
not divisible by either $3$ or $p$. This leaves us with $64$ possible pairs to
look into. The following lemma helps us omit some of those $64$ possibilities.

######  Lemma 5.3.

Let us assume $p$ is a prime such that $p\equiv 17,113\pmod{120}$. Then
$(i)$ The homogeneous space corresponding to $(b_{1},b_{2})$ has no $3$-adic
solution if $b_{1}=6,15,3p$ or $30p$.
$(ii)$ The homogeneous space corresponding to $(b_{1},b_{2})$ has no $3$-adic
solution if $b_{2}=3,15p,-15$ or $-3p$.
$(iii)$ The homogeneous space corresponding to $(b_{1},b_{2})$ has no $p$-adic
solution if $b_{1}=3p,5p,6p$ or $10p$.
$(iv)$ The homogeneous space corresponding to $(b_{1},b_{2})$ has no $p$-adic
solution if $b_{2}=\pm 3p,\pm 15p$.
$(v)$ Let $5$ divides $\gcd(b_{1},b_{2})$. Then the homogeneous space
corresponding to $(b_{1},b_{2})$ has no $5$-adic solution if
$\Big{(}\frac{b_{1}b_{2}/25}{5}\Big{)}=1$.

###### Proof.

We start the proof by first noticing the fact $p\equiv 17,113\pmod{120}$
implies $\big{(}\frac{2}{p}\big{)}=1$ and
$\big{(}\frac{3}{p}\big{)}=\big{(}\frac{5}{p}\big{)}=-1$. The idea behind each
proof is to manipulate this observation and (3.3).
$(i)$ From Lemma 4.1, we know that if $(z_{1},z_{2})$ is a solution of (3.3),
and $v_{3}(z_{i})<0$ for any $i\in\\{1,2\\}$, then
$v_{3}(z_{1})=v_{3}(z_{2})=-k$ for some positive integer $k$. This transform
(3.3) into $b_{1}u_{1}^{2}-b_{2}u_{2}^{2}=15\cdot p\cdot 3^{2k}$, where
$u_{i}$ denotes $3$-adic units. From here, it is evident that $b_{1}$ can not
be divisible by $3$ as $\gcd(b_{1},b_{2})\not\equiv 0\pmod{3}$. Hence
$b_{1}\equiv 0\pmod{3}\implies z_{2}\equiv 0\pmod{3}$. This in turn implies

$\frac{b_{1}}{3}z_{1}^{2}\equiv
5p\pmod{3}\implies\Big{(}\frac{b_{1/3}}{3}\Big{)}=\Big{(}\frac{2p}{3}\Big{)}=1.$

Hence, if $3$ divides $b_{1}$, then $b_{1}$ can only be one of $3,6p,30$ or
$15p$ as a necessary condition for the existence of $3$-adic solutions. Now
the result follows.
$(ii)$ In a very similar manner to the previous method, we notice from (3.3)
that if $b_{2}\equiv 0\pmod{3}$, then

$-\frac{b_{2}}{3}z_{2}^{2}\equiv
5p\pmod{3}\implies\Big{(}\frac{b_{2/3}}{3}\Big{)}=-\Big{(}\frac{2p}{3}\Big{)}=-1.$

Hence $b_{2}$ can only assume one of the values from $15,3p,-3$ or $-15p$ for
the existence of $3$-adic solutions. The result now follows immediately.
$(iii)$ Applying the same method used in the first part of this proof, we can
immediately see that if $p$ divides $b_{1}$, then $p$ must divide $z_{2}$ from
(3.3) and hence
$\Big{(}\frac{b_{1}/p}{p}\Big{)}=\Big{(}\frac{15}{p}\Big{)}=1\implies
b_{1}=p,2p,15p,30p$. This concludes the proof.
$(iv)$ Similar to the previous case, one can notice that $b_{2}\equiv
0\pmod{p}$ implies $\Big{(}\frac{-b_{2}}{p}\Big{)}$ is a quadratic residue
modulo $p$ i.e. $b_{2}=\pm p$ or $\pm 15p$. The result now follows
immediately.
$(v)$ Suppose $\gcd(b_{1},b_{2})\equiv 0\pmod{5}$. Then from Lemma 4.1 and,
Lemma 4.2, we can rewrite (3.4) as either
$b_{1}u_{1}^{2}-b_{1}b_{2}u_{3}^{2}=24\cdot p\cdot 5^{2k}$ or
$b_{1}z_{1}^{2}-\frac{b_{1}b_{2}}{25}u_{3}^{2}=24p$ where
$u_{i}\in\mathbb{Z}_{5}^{*},z_{i}\in\mathbb{Z}_{5}$ for $i\in\\{1,3\\}$ and
$k$ is a positive integer. The solution to the first equation implies
$b_{1}\equiv 0\pmod{5^{2}}$, a contradiction. Whereas a solution to the second
equation implies
$\Big{(}\frac{b_{1}b_{2}/25}{5}\Big{)}=\Big{(}\frac{-24p}{5}\Big{)}=\Big{(}\frac{p}{5}\Big{)}=-1$.
The result now follows immediately. ∎

Now, we are left with the following six cosets.

Table 1.  Distribution of points | Representative
---
of a coset of A
Points in the coset Of A |  | Representative
---
of a coset of A
Points in the coset Of A
$(1,15)$ | $(10,-p),(15p,-1),(6p,15p)$ | $(2p,15)$ | $(30,-1),(5p,-p),(3,15p)$
$(3,\pm p)$ | $(30,\mp 15),(5p,\mp 15p),(2p,\pm 1)$ | $(3,\pm 5p)$ | $(2p,\pm 5),(30,\mp 3),(5p,\mp 3p)$

We rule out the possibility of five of the six cosets mentioned above in the
following lemma.

######  Lemma 5.4.

Let us suppose that $p\equiv 17,113\pmod{120}$. Then the homogeneous space
corresponding to $(b_{1},b_{2})$ has no $5$-adic solution if
$(b_{1},b_{2})\in\\{(1,15),(2p,15),(3,\pm 5p)\\}$. Moreover, The homogeneous
space corresponding to $(3,-p)$ has no $3$-adic solution.

###### Proof.

The first part of the result follows from the following observation. If $5$
does not divide $b_{1}$ while dividing $b_{2}$, then from 3.3 one can get that
$z_{1}\equiv 0\pmod{5}$ and $z_{2}\not\equiv 0\pmod{5}$. This, in return
implies that $\big{(}\frac{b_{2}/5}{5}\big{)}=\big{(}\frac{3p}{5}\big{)}=1$, a
condition that rules out the cases $b_{2}=15,\pm 5p$ if $5$ does not divide
$b_{1}$. Hence the result follows.
For the case $(b_{1},b_{2})=(3,-p)$, we first observe that subtracting 3.4
from 3.3 yield into the equation $-3pz_{3}^{2}+pz_{2}^{2}=-9p$. Noticing the
fact that $\frac{z_{2}}{3}\in\mathbb{Z}_{3}^{*}$, one can reach to the
conclusion that $\big{(}\frac{z_{2}}{3}\big{)}^{2}\equiv-1\pmod{3}$, a
contradiction. Hence the result follows. ∎

##  6\. A homogeneous space with everywhere local solution

In this section, we prove that for the elliptic curve
$E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$ where $p\equiv 17,113\pmod{120}$, the
homogeneous space corresponding to $(3,p)$ has local solutions for all places
$l\leq\infty$. We prove the following lemma regarding that.

######  Lemma 6.1.

Equations (3.3) and (3.4) have a local solution in $\mathbb{Q}_{l}$ for every
prime $l$ for $(b_{1},b_{2})=(3,p)$.

###### Proof.

A twist of a smooth projective curve $C/\mathbb{Q}$ is a smooth curve
$C^{\prime}/\mathbb{Q}$ that is isomorphic to $C$ over $\bar{\mathbb{Q}}$ (see
[13], Section X$.2$). We use the following version of Hensel’s lemma to show
that for $(b_{1},b_{2})=(3,p)$, equations (3.3) and (3.4) have a local
solution in $\mathbb{Q}_{l}$ for every place $l$. Hensel’s lemma states that
if $f(x)$ is a polynomial with coefficients that are $l$-adic integers and
$f(x_{1})\equiv 0\pmod{l}$ for $x_{1}\in\mathbb{Z}$, then if
$f^{\prime}(x_{1})\not\equiv 0\pmod{l}$, there exists a $l$-adic integer $x$
with $x\equiv x_{1}\pmod{l}$ such that $f(x)=0$ (see [15], Theorem A$.2$).
Every homogeneous space of $E_{p}$ is a twist of $E_{p}$ (see [13],
Proposition $3.2$). First, we consider $l>5$, $l\neq p$. Suppose $C$ is the
homogeneous space given by (3.3) and (3.4) corresponding to the pair $(3,p)$.
Then $C$ is a twist of $E_{p}$, and in particular, it has genus $1$. By the
Hasse-Weil bound, we have

$\\#C(\mathbb{F}_{l})\;\geq\;1+l-2\sqrt{l}\;\geq\;2\qquad\mbox{ for
}l>5,\text{ }l\neq p.$

We can choose a solution
$(z_{1},z_{2},z_{3})\in\mathbb{F}_{l}\times\mathbb{F}_{l}\times\mathbb{F}_{l}$
such that not all three of them are zero modulo $l$. Now $z_{1}\equiv
z_{2}\equiv 0\pmod{l}$ implies $l^{2}$ divides $15p$, a contradiction.
Similarly, $z_{1}\equiv z_{3}\equiv 0\pmod{l}$ implies $24p\equiv
0\pmod{l^{2}}$, contradiction again. One can now suitably choose two of
$z_{1},z_{2}$ and $z_{3}$ to convert equations (3.3) and (3.4) into one single
equation of one variable with a simple root over $\mathbb{F}_{l}$. That common
solution can then be lifted to $\mathbb{Q}_{l}$ via Hensel’s lemma.
For $l=p$, one can notice that fixing $z_{1}=0$ turn equations (3.3) and (3.4)
with simple roots $z_{i}\not\equiv 0\pmod{p}$ for $i\in\\{2,3\\}$ and hence
can be lifted to a solution in $\mathbb{Z}_{p}$ via Hensel’s lemma.
For $l=3$, first, we fix $z_{2}=z_{3}=1$ to reduce equations (3.3) and (3.4)
to equations of only one variable. Now observing $z_{1}\not\equiv 0\pmod{3}$
is a simple root that Hensel’s lemma can lift, we conclude this case.
For the case $l=5$, notice that if $p\equiv 2\pmod{5}$, then fixing $z_{2}=1$
and $z_{3}=2$ gives the equation $3z_{1}^{2}\equiv 2\pmod{5}$ for which
$z_{1}\equiv 2\pmod{5}$ is a simple root and hence can be lifted by Hensel’s
lemma. Similarly, for the case $p\equiv 3\pmod{5}$, we can fix $z_{2}=1$ and
$z_{3}=2$ and then notice that $z_{1}\equiv 1\pmod{5}$ is then a simple root
and hence can be lifted. This concludes the proof for $l=5$.
Finally for the case $l=2$, choose $z_{2}=2$ and $z_{3}=1$. This turns (3.3)
and (3.4) into the following;

(6.1) $\displaystyle 3z_{1}^{2}-4=15p\implies 3z_{1}^{2}\equiv 3\pmod{8},$
(6.2) $\displaystyle 3z_{1}^{2}-3pz_{3}^{2}=24p\equiv 0\pmod{8}\implies
3z_{1}^{2}\equiv 3z_{3}^{2}\pmod{8}.$

In both the cases we now have $z_{1}^{2}\equiv 1\pmod{8}$ which by Hensel’s
lemma can be lifted to $\mathbb{Q}_{2}$. Hence proved. ∎

##  7\. Class number divisibility and $2$-Selmer rank

We are now in a position to prove Theorem 1.1. Identifying $a,b$ and $c$ in
Lemma 2.1 with $9p,6p$ and $18p$ respectively, we first notice that the
elliptic curve $E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$ mentioned in Theorem 1.1 is
same as the elliptic curve $E$ in Lemma 2.1.

###### Proof of Theorem 1.1.

Let $E_{p}:y^{2}=(x+6p)(x-9p)(x-18p)$ be the elliptic curve where $p\equiv
17,113\pmod{120}$. the first part of the statement is now evident as Lemma 6.1
proves that the homogeneous space corresponding to $(3,p)$ is the only one
with local solutions for all primes $l\leq\infty$. This proves that the
$2$-Selmer rank of $E_{p}$ is exactly one.
Now for the second part, we get that
$\mathbb{Q}(\sqrt{abc},\sqrt{r+(b-a-c)t^{2}})=\mathbb{Q}(\sqrt{3p},\sqrt{r-21pt^{2}})$.
We notice that $\gcd(s,3p)=1\implies\gcd(s,abc)=1$. Also $3$ ramifies in
$\mathbb{Q}(\sqrt{3p},\sqrt{r-21pt^{2}})$. Hence, from Lemma 2.2, the result
follows. ∎

###### Proof of Corollary 1.2.

We first notice that $p\equiv 53,77\pmod{120}\implies p\equiv 5\pmod{8}$,
which implies $\big{(}\frac{2}{p}\big{)}=-1$. From the $2$-descent method in
Section $5$, one can see Lemma 5.3.$(iii)$ is the only argument where we have
used the fact $\big{(}\frac{2}{p}\big{)}=1$. So, in this case, the homogeneous
spaces corresponding to each possible point except $(5p,\pm 5),(1,15),(1,5p)$
and $(3,p)$ in $\mathbb{Q}(S,2)\times\mathbb{Q}(S,2)$ are still ruled out from
having $l$-adic solutions for all primes $l\leq\infty$.
Firstly, for $(5p,\pm 5)$, we notice that $v_{p}(z_{1})<0$ is not possible,
and hence $v_{p}(z_{i})\geq 0$ for $i\in\\{1,2\\}$ from Lemma 4.1. This, in
turn implies that

$z_{2}\equiv 0\pmod{p}\implies z_{1}^{2}\equiv 3\pmod{p},\text{ a
contradiction}.$

In a very similar way, one can notice that local solutions for the homogeneous
space corresponding to $(1,15)$ implies $z_{1}^{2}\equiv-p\pmod{5}$ and to
$(1,15)$ implies $z_{1}^{2}\equiv-3\pmod{5}$, both contradictions.
For the homogeneous space corresponding to $(3,p)$, we first observe that for
$l=p$, $v_{p}(z_{1})<0$ is not a possibility from Lemma 4.1. Hence,
$v_{p}(z_{1}),v_{p}(z_{3})\geq 0$. But then from 3.3, we get that
$3z_{1}^{2}-3pz_{3}^{2}=24p$ which implies $z_{1}\equiv 0\pmod{p}$ and hence
$z_{3}^{2}\equiv-8\pmod{p}$, a contradiction as $\big{(}\frac{2}{p}\big{)}=-1$
now that $p\equiv 5\pmod{8}$. This proves that $(3,p)$ will not have any
$p$-adic solution and hence shows that the $2$-Selmer rank of $E_{p}$ is zero
if $p\equiv 53,77\pmod{120}$. This ends the proof of the result. ∎

We can now prove Corollary 1.3 by explicitly constructing an infinite family
of bi-quadratic fields with even class numbers from the points of $E_{p}$. The
Parity conjecture ensures $r(E_{p})=1$ for $p\equiv 17,113\pmod{120}$ as the
$2$-Selmer rank of $E_{p}$ is $1$. We, though, use a more straightforward
characterization for the existence of points of infinite order below.

###### Proof of Corollary 1.3.

We first notice that the $t_{0}$ even implies the Mordell-Weil rank of $E_{p}$
is at least $1$. If
$P_{0}=\big{(}\frac{r_{0}}{t_{0}^{2}},\frac{s_{0}}{t_{0}^{3}}\big{)}$ is a
point of infinite order in $E_{p}(\mathbb{Q})$ with $t_{0}$ even, one can
immediately observe from the duplication formula that

$x(2P)=\Big{(}\frac{r_{1}}{t_{1}^{2}}\Big{)}=\Big{(}\frac{r_{0}^{4}-8\cdot
972p^{3}\cdot r_{0}t_{0}^{6}+4\cdot 21p\cdot 972p^{3}\cdot
t_{0}^{8}}{4s_{0}^{2}t_{0}^{2}}\Big{)}.$

This implies $t_{1}$ is even as $r_{0}^{4}-8\cdot 972p^{3}\cdot
r_{0}t_{0}^{6}+4\cdot 21p\cdot 972p^{3}\cdot t_{0}^{8}$ is odd when $t_{0}$ is
even. A similar calculation involving duplication formula shows that
$\gcd(s_{0},3p)=1\implies\gcd(s_{1},3p)=1$ too. This is because if $s$ is
divisible by either $3$ or $p$, then so is $r$ and hence

$\gcd(s_{0},3p)=1\implies\gcd(r_{0},3p)=1\implies\gcd(r_{1},3p)=1\implies\gcd(s_{1},3p)=1.$

Also, $3$ ramifies in $K_{1}/\mathbb{Q}$. Hence the construction of an
unramified abelian extension for the bi-quadratic field
$K_{1}=\mathbb{Q}(\sqrt{3p},\sqrt{r_{1}-21pt_{1}^{2}})$ follows through Lemma
2.2. The proof now follows through an inductive argument. ∎

We conclude this section with a small table containing elliptic curves $E_{p}$
for primes $p\equiv 17,113\pmod{120}$ with their respective $2$-Selmer ranks
and bi-quadratic fields with corresponding class numbers in support of our
result. Each example below can generate an infinite family of bi-quadratic
fields with even class numbers by Corollary 1.3. Computations have been done
in Magma [11] software. While we define
$\\#\,\mbox{Sel}_{2}(E_{m,p}/\mathbb{Q})=2^{2+s(E_{m,p})}$, and refer to
$s(E_{m,p})$ as the $2$-Selmer rank in the following table, $2+s(E_{m,p})$
will denote the $2$-Selmer rank in Magma.

Table 2. $2$-Selmer rank and Class numbers | $p$
---
$s(E_{p})$ |  | $P=\big{(}\frac{r}{t^{2}},\frac{s}{t^{3}}\big{)}\in E_{p}(\mathbb{Q})$
---
$h(K)$
$17$ | $1$ | $\left(\dfrac{5257}{16},\;\dfrac{83581}{64}\right)$ | $2560$
$113$ | $1$ | $\left(\dfrac{103952278736903209}{48681040752400},\;\dfrac{6066145443591398465996627}{339656383916830232000}\right)$ | No information

We conclude by noting that the choice of $-6p,9p$ and $18p$ is solely to
ensure the ramification of the prime $3$ in the extension $K/\mathbb{Q}$ as
well as to ensure $ab+bc=ac$ as mentioned in Lemma 2.1. A different choice
satisfying the conditions of Section $2$ would still result in an unramified
abelian extension and hence even class number.

## Acknowledgement

The first author would like to acknowledge DST-SERB for providing the grant
through the Start-Up Research Grant (SRG/2020/001937) as well as BITS-Pilani,
Hyderabad Campus, for providing amenities. The second author would like to
acknowledge the fellowship (File No:09/1026(0029)/2019-EMR-I) and amenities
provided by the Council of Scientific and Industrial Research, India (CSIR)
and BITS-Pilani, Hyderabad. The third author would like to acknowledge the
fellowship (File No:09/1026(0036)/2020-EMR-I) and amenities provided by the
Council of Scientific and Industrial Research, India (CSIR) and BITS-Pilani,
Hyderabad.

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