#  Light-weighted CNN-Attention based architecture for Hand Gesture
Recognition via ElectroMyography

###### Abstract

Advancements in Biological Signal Processing (BSP) and Machine-Learning (ML)
models have paved the path for development of novel immersive Human-Machine
Interfaces (HMI). In this context, there has been a surge of significant
interest in Hand Gesture Recognition (HGR) utilizing Surface-Electromyogram
(sEMG) signals. This is due to its unique potential for decoding wearable data
to interpret human intent for immersion in Mixed Reality (MR) environments. To
achieve the highest possible accuracy, complicated and heavy-weighted Deep
Neural Networks (DNNs) are typically developed, which restricts their
practical application in low-power and resource-constrained wearable systems.
In this work, we propose a light-weighted hybrid architecture (HDCAM) based on
Convolutional Neural Network (CNN) and attention mechanism to effectively
extract local and global representations of the input. The proposed HDCAM
model with $58,441$ parameters reached a new state-of-the-art (SOTA)
performance with $82.91\%$ and $81.28\%$ accuracy on window sizes of $300$ ms
and $200$ ms for classifying $17$ hand gestures. The number of parameters to
train the proposed HDCAM architecture is $18.87\times$ less than its previous
SOTA counterpart.

Index Terms—  Attention Mechanism, Biological Signal Processing (BSP), Mixed
Reality (MR), surface Electromyogram.

##  1 Introduction

Surface Electromyogram (sEMG)-based Hand Gesture Recognition (HGR) is regarded
as a promising approach for a wide range of applications, including
myoelectric control prosthesis [1, 2, 3, 4], virtual reality technologies [5,
6], Human Computer Interactions (HCI) [7], and rehabilitative gaming systems
[8]. sEMG signals contain electrical activities of the muscle fibers that can
be employed to decode hand gestures and thereby enhance immersive HMI wearable
systems for immersion in Mixed Reality (MR) environments [9, 10].
Consequently, there has been a surge of interest in the development of Deep
Neural Networks (DNNs) and Machine Learning (ML) models to identify hand
gestures using sEMG signals. Generally speaking, sEMG datasets can be
collected based on “sparse multichannel sEMG” or “High-Density sEMG (HD-
sEMG)”. Despite advantages of HD-sEMG, its utilization leads to structural
complexity [11, 12], while adoption of sparse multichannel sEMG signals
requires fewer electrodes making it the common modality of choice for
incorporation into wearable devices. Therefore, development of DNNs based on
sparse sEMG signals has gained significant recent importance.

Despite extensive research in this area and the fact that academic researchers
achieve high classification accuracy in laboratory conditions, there is still
a gap between academic research in sEMG pattern recognition and commercialized
solutions [9]. In this context, one of the objectives for reducing the gap is
to focus on the development of DNN-based models that not only have high
recognition accuracy but also have minimal processing complexity, allowing
them to be embedded in low-power devices such as wearable controllers [1, 13].
Furthermore, the designed DNN-based models should be based on the minimum
number of electrodes while estimating the desired gestures within an
acceptable delay time [9, 14]. Consequently, in this paper, we develop the
novel Hierarchical Depth-wise Convolution along with the Attention Mechanism
(HDCAM) model for HGR based on sparse sEMG signals to fill this gap by meeting
criteria such as improving the accuracy and reducing the number of parameters.
The HDCAM is developed based on the Ninapro [15, 16] database, which is one of
the most well-known sparse multi-channel sEMG benchmark datasets.

Fig. 1: Comparing different variants of our proposed HDCAM model with SOTA
designs for an input window size of $300$ ms. The x-axis shows the number of
parameters and the y-axis displays the classification accuracy on the Ninapro
DB2 dataset. Our HDCAM shows a better compute versus accuracy trade-off
compared to recent approaches.

Using Convolutional Neural Networks (CNN) [17, 18, 19, 20] is a common
approach for hand movement classification, where sEMG signals are first
converted into images and then used as input for CNN-based architectures.
However, the nature of sEMG signals is sequential, and CNN architectures only
take into account the spatial features of the sEMG signals. Therefore, in
recent literature [21, 12, 13], authors proposed using recurrent-based
architectures such as Long Short Term Memory (LSTM) networks to exploit the
temporal features of sEMG signals. On the other hand, it is suggested [23, 22,
24] to use hybrid models (CNN-LSTM architecture) instead of using a single
model to capture the temporal and spatial characteristics of sEMG signals.
Although recent academic researchers are improving the performance by using
Recurrent Neural Networks (RNNs) or hybrid architectures, the sequence
modeling with recurrent-based architectures has several drawbacks such as
consuming high memory, lack of parallelism, and lack of stable gradient during
the training [25, 4]. It is demonstrated [26] that sequence modeling using
RNN-based models does not always outperform CNN-based designs. Specifically,
CNN architectures have several advantages over RNNs such as lower memory
requirements and faster training if designed properly [26]. Therefore, in the
recent literature [25, 4, 27, 28], the authors took advantage of $1$-D
Convolutions developed based on the dilated causal convolutions, where the
sequence of sEMG signals can be processed as a whole with lower memory
requirement during the training compared to RNNs. Convolution operation in
CNNs, however, has two main limitations, i.e., (i) it has a local receptive
field, which makes it incapable of modeling global context, and; (ii) their
learned weights remain stationary at inference time, therefore, they cannot
adapt to changes in input. Attention mechanism [29] can mitigate both of these
problems. Consequently, the authors in the recent research papers [31, 32, 1,
30, 2] used the attention mechanism combined with CNNs and/or RNNs to improve
the performance of sEMG-based HGR. The attention mechanism’s major
disadvantage is that it is often computationally intensive. Therefore, a
carefully engineered design is required to make attention-based models
computationally viable, especially for low-power devices.

In this paper, we develop the HDCAM architecture by effectively combining the
complementary advantages of CNNs and the attention mechanisms. Our proposed
architecture shows a favorable improvement in terms of parameter reduction and
accuracy compared to the state-of-the-art (SOTA) methods for sparse
multichannel sEMG-based hand gesture recognition (see Fig. 1). The
contributions of the HDCAM architecture can be summarized as follows:

  * •

Efficiently combining advantages of Attention- and CNN-based models and
reducing the number of parameters (i.e., computational burden).

  * •

Efficiently extracting local and global representations of the sEMG sequence
by coupling convolution and attention-based encoders.

  * •

Integration of Depth-wise convolution ($DwConv$) a hierarchical structure in
the proposed Hierarchical Depth-wise Convolution (HDConv) encoder, which not
only extracts a multi-scale local representation but increases the receptive
field in a single block.

The small version of the proposed HDCAM with $58,441$ parameters achieves new
SOTA $82.91\%$ top-1 classification accuracy on Ninapro DB2 dataset with
$18.87$ times less number of parameters compared to the previous SOTA approach
[13].

##  2 The Proposed HDCAM Architecture

Fig. 2: The proposed architecture: (a) The overall architecture of proposed
HDCAM model. (b) The HDConv Encoder, which uses Hierarchical Depth-wise
Convolution for multi-scale temporal feature mixing. (c) The MHSAtten Encoder
consists of a Multi-Head Self-Attention (MHA) module to encode the global
representation of the input feature maps.

The primary objective of this study is to build a lightweight hybrid
architecture that successfully combines advantages of Attention- and CNN-based
models for low-powered devices. In the proposed framework, a sliding window
strategy with window of size $\mathit{W}$ $\in\\{150,200,250,300\,ms\\}$ is
adapted to use the multi-variate temporal sEMG information, resulting in
dataset $\mathcal{D}{=}\\{(\bm{X}_{i},\mathrm{y}_{i})\\}_{i=1}^{N}$. More
specifically, $\mathrm{y}_{i}\in\mathbb{R}$ is the label assigned to the
$i^{\text{th}}$ segmented sequence $X_{i}\in\mathbb{R}^{L\times C}$. Here, $L$
is the length of the segmented sequential input corresponding to the number of
samples obtained at a frequency of $2$ kHz for a window of size $\mathit{W}$,
and $C$ denotes the number of channels in the input segment corresponding to
the number of input features/sensors. As illustrated in Fig. 2, the HDCAM
framework has a hybrid design based on CNN and the “Multi-Head Self-Attention
(MHA) mechanism” to reap the advantages of both methods for designing a
lightweight architecture.

(i) Overview of the Architecture: As shown in Fig 2(a), the overall HDCAM
architecture consists of four different stages, the first three for multi-
scale feature extraction and the last one for classification. HDCAM is made up
of two primary components, namely “Hierarchical Depth-wise Convolution
(HDConv)” encoder and “Multi-Head Self-Attention (MHSAtten)” encoder, where
the former and latter aim to model the local and global information in the
sequential input, respectively. Formally, for a given segmented sequential
input $X_{i}\in\mathbb{R}^{L\times C}$, HDCAM begins by patchifing $X_{i}$
with a $10\times 1$ strided convolution followed by a Layer Normalization
(LN). Patching mechanism helps reduce memory and computation requirements in
downstream layers resulting in ${L/10}\times C_{1}$ feature maps. Afterward,
local features are extracted using a HDConv encoder. Further processing of the
feature maps takes place in the second and third stages, which follow almost
the same architectural structure. Both of which start with the downsampling
layer followed by consecutive HDConv encoders for local feature extraction and
ended with MHSAtten block to encode the global representations of the input.
The downsampling layer consists of an LN followed by a $2\times 1$ strided
convolution, which reduces the sequential feature maps length by half and
increases the channels, resulting ${L/20}\times C_{2}$ and ${L/40}\times
C_{3}$ dimensional features for second and third stages, respectively. In the
final stage, a Global Average Pooling operation is used to reduce the feature
maps’ dimension to $1\times C_{3}$ followed by a Linear layer for
classification. Here, $C_{i}$ refers to number of channels in $i^{\text{th}}$
stage, for $i\in\\{1,2,3\\}$.

(ii) HDConv Encoder: As shown in Fig. 2(b), the proposed HDConv block combines
depth-wise convolution with a hierarchical structure to extract local features
at multi-scales. The proposed multi-scale feature extractor is inspired by the
Res2Net [33] module, which combines features with different resolutions.
Different from the Res2Net module, we omitted the first point-wise convolution
layer and added a $3\times 1$ depth-wise convolution to the first branch.
Also, the number of active branches in the hierarchical convolutional
structure is dynamic and varies depending on the stage. In HDConv module,
input feature maps of shape $L\times C$ is evenly splitted into $s$
subsets/scales, denoted by $\bm{x}_{i}$ of shape $L\times C/s$, where
$i\in\\{1,2,\dots,s\\}$. Then, $3\times 1$ depth-wise convolutions, denoted by
$DwConv_{i}$, is applied on each subset $\bm{x}_{i}$ after combining with the
previous branch output features, denoted by $\bm{y}_{i-1}$. Generally, we can
write the output features of each branch $\bm{y}_{i}$ as follows

$\bm{y}_{i}=\begin{cases}DwConv_{i}(\bm{x}_{i})&i=1\\\
DwConv_{i}(\bm{x}_{i}+\bm{y}_{i-1})&2\leq i\leq s\\\
\end{cases}\vspace{-0.1in}$ (1)

As shown in Fig 2(b) and Eq. (1), the hierarchical structure allows each
depth-wise convolution $DwConv_{i}$ receive the information from all previous
splits, {$\bm{x}_{j},j\leq i$}. The output feature maps of all branches are
concatenated and passed through an LN followed by point-wise convolution to
enrich the multi-scale local representation, and finally Gaussian Error Linear
Unit (GELU) activation is used for adding non-linearity to the model. For
information flow through the network hierarchy, residual connection is used in
HDConv encoder. The HDConv encoder can be represented as follows

$X_{out}=X_{in}+Linear_{GELU}(LN(HDwConv(X_{in}))),\vspace{-.1in}$ (2)

where $X_{in}$ and $X_{out}$ are the HDConv input and output feature maps,
both of shape $L\times C$, $Linear_{GELU}$ is point-wise convolution followed
by GELU non-linearity $LN$ is Layer Normalization, and $HDwConv$ is
hierarchical depth-wise convolution operation.

(iii) MHSAtten Encoder: In [29], the authors showed that the attention
mechanism allows a model to present global information in a given input
sequence. Furthermore, attention-based architectures [31, 32, 1, 30, 2] have
shown promising performance in the context of sEMG-based HGR by extracting
particular bits of information from the sequential nature of the sEMG signals.
However, most of these models are still heavy-weight to be used in resource-
constrained devices. Hence, in the proposed HDCAM architecture, we designed a
hybrid architecture that combines convolutions and attention mechanism
advantages. Specifically, due to spatial inductive biases in convolution
operation, the CNN-based encoder (HDConv) assists our hybrid model to learn
local representations with fewer parameters than solely attention-based
models. However, to effectively learn global representations, we also used an
attention-based encoder (MHSAtten). Since computation in the MHA has quadratic
relation to input size, we only used the MHSAtten encoder in the second and
third stages of the HDCAM to efficiently encode the global representation,
where the length of the sequential feature maps are $1/20$ and $1/40$ of the
original input of the network, respectively. The MHSAtten encoder can be
represented as follows

$X_{out}=Linear_{GELU}(LN(X_{in}+M\\!H\\!A(LN(X_{in}))))+X_{in}\vspace{-.075in}$
(3)

where $X_{in}$ and $X_{out}$ are the MHSAtten input and output feature maps,
both of shape $L\times C$, $Linear_{GELU}$ is point-wise convolution followed
by GELU non-linearity $LN$ is Layer Normalization, and $M\\!H\\!A$ is Multi-
Head Self-Attention mechanism. In $M\\!H\\!A$, the input feature maps $X_{in}$
of shape $L\times C$ are passed through a Linear projection to create Queries
$Q$ , i.e., a matrix with the same shape as the input feature maps. Then,
Queries $Q$ is evenly splitted into $h$ subsets, denoted by $\bm{q}_{i}$ of
shape $L\times C/h$, where $i\in\\{1,2,\dots,h\\}$ and $h$ is number of the
heads. In parallel, the same approach has been applied to construct Keys and
Values subsets, i.e, $\bm{k}_{i}$ and $\bm{v}_{i}$. Finally, on each head, the
attention block measures the pairwise similarity of each $\bm{q}_{i}$ and all
$\bm{k}_{h}$ to assign a weight to each $\bm{v}_{h}$. The entire operation is

$A_{h}=Softmax(\frac{\bm{q}_{h}\bm{k}_{h}^{T}}{\sqrt{d}})\bm{v}_{h},\vspace{-0.15in}$
(4)

where $\mathit{d}{=}C/h$ denotes the dimension of $\bm{k}_{h}$ and
$\bm{q}_{h}$ subsets. Then concatenation of attention feature maps of all
heads is projected to get the final attention maps of the MHA mechanism, i.e.,
$M\\!H\\!A(X_{in}){=}Linear(Concat(A_{1},A_{2},\dots,A_{h}))$.

##  3 Experiments and Results

3.1. Database

The proposed HDCAM is trained and tested using the second Ninapro dataset [16]
referred to as the DB2, which is the most commonly used sparse sEMG benchmark.
In the DB2 dataset, muscle electrical activities are measured by the Delsys
Trigno Wireless EMG system with $12$ electrodes at a frequency rate of $2$
kHz. The DB2 dataset consists of $50$ different hand movements (including
rest) recorded from $40$ healthy users. Each gesture is repeated six times by
each user, where each repetition lasts for $5$ seconds, followed by $3$
seconds of rest. For a fair comparison, as well as following the
recommendations of the database [16] and previous literature [30, 1, 13, 24],
we considered two repetitions (i.e., $2$ and $5$) for testing and the rest for
training. The DB2 dataset is presented in three sets of exercises (B, C, and
D). Following [1, 13, 24], the focus is on Exercise B which consists of $17$
hand movements.

Table 1: HDCAM Architecture variants. Description of the models’ layers with
respect to kernel size, and output channels, repeated $n$ times. We use a
hierarchical structure in HDConv Encoder to extract multi-scale local
features. Also, MHSAtten Encoder is used to extract global representations of
the feature maps.

| Layer | #Layers ($n$) | Kernel Size | Output Channels
---|---|---|---|---
| XXSmall | XSmall | Small
Stage 1 | Stem | $1$ | $10\times 1$ | $16$ | $24$ | $24$
HDConv Encoder | $1$ | $3\times 1$ | $16(s=2)$ | $24(s=3)$ | $24(s=3)$
Stage 2 | Downsampling | $1$ | $2\times 1$ | $24$ | $32$ | $32$
HDConv Encoder | $2$ | $3\times 1$ | $24(s=3)$ | $32(s=4)$ | $32(s=4)$
MHSAtten Encoder | $1$ | $-$ | $24(h=3)$ | $32(h=4)$ | $32(h=4)$
Stage 3 | Downsampling | $1$ | $2\times 1$ | $32$ | $48$ | $64$
HDConv Encoder | $4$ | $3\times 1$ | $32(s=4)$ | $48(s=4)$ | $64(s=4)$
MHSAtten Encoder | $1$ | $-$ | $32(h=4)$ | $48(h=4)$ | $64(h=4)$
| Global Avg. Pooling | $1$ | $-$ | $32$ | $48$ | $64$
| Linear | $1$ | $1$ | $17$ | $17$ | $17$
| Model Parameters |  |  | $20,689$ | $40,281$ | $58,441$

3.2. Results and Discussions

In this section, a comprehensive set of experiments is conducted to evaluate
the performance of the proposed HDCAM architecture. Table 1 represents the
sequence of the HDConv and MHSAtten encoders along with design information of
the extra-extra small (XXSmall), extra-small (XSmall), and small (Small)
versions of the model. As shown in Table 1, the type, number, and sequence of
the component blocks in the overall model architecture (illustrated in Fig. 2)
are maintained across all HDCAM architecture variants. The number of output
channels in each stage is what distinguishes the XXSmall, XSmall, and Small
models from one another. Since the number of active branches ($s$) and
attention heads ($h$) in the HDConv and MHSAtten encoders is proportional to
the number of output channels of the corresponding stage, we maintained the
fundamental rule for all model variants, which requires having at least eight
channels per-head/per-branch. Additionally, the maximum allowed number of
heads/branches is set to four. All models were trained using the Adam
optimizer at a learning rate of $0.0001$ with Cross-Entropy loss. Furthermore,
we chose a mini-batch size of $32$. For the pre-processing step, we employed
the $1^{\text{st}}$ order $1$ Hz low-pass Butterworth filter to smooth raw
sEMG signals as described in the recent literature [34, 17, 35, 16]. Moreover,
we normalized and scaled the sEMG signals by the $\mu$-law technique [3] with
the $\mu$ value of $256$. Table 2 shows the average recognition accuracy of
different variants over all subjects. In the following sections, we focus on
five experiments to evaluate our proposed HDCAM model:

The Model’s Dimension: This experiment analyzes the recognition accuracy of
the HDCAM by varying the number of channels in each stage, yielding XXSmall,
XSmall, and Small models. In this regard, Table 2 shows the results for all
variants of the proposed architecture for different window sizes. For the same
arrangement of component layers, it can be seen from Tables 1 and 2 that the
accuracy of the model is improved by increasing the dimensions of the stages.
More specifically, the dimension of the stage $3$ is the only difference
between the XSmall and Small architectures, resulting in more informative
high-level features in the Small model, which leads to better performance.
Comparing XXSmall versus two other variants, the dimension of all stages has
reduced leading to lower performance (Table 2). From Table 1, it can be
observed that there is a trade-off between the complexity of the model and the
accuracy.

The Effect of Window Size: Comparing outcomes in each column of Table 2 shows
that increasing window size ($W$) led to better performance for all model
variants. It is worth mentioning that $W$ is required to be under $300$ ms to
have a real-time response in peripheral human machine intelligence systems
[14]. According to this observation, the proposed HDCAM architecture is
capable of extracting/utilizing information from longer sequences of inputs.
Larger $W$ leads to longer sequential feature maps in the second and third
stages, which leads to more memory requirements in the attention mechanism. We
would like to emphasize that our proposed hybrid architecture is still far
superior to the sole attention-based approach since the sequence lengths in
the second and third stages are significantly decreased, as previously noted.

Table 2: Accuracy of HDCAM variants over different window sizes
($\mathit{W}$).

| Model ID | XXSmall | XSmall | Small
---|---|---|---|---
$\mathit{W}$=
150 ms |  Accuracy ($\%$) | $80.53$ | $81.51$ | $82.21$
STD ($\%$) | $6.9$ | $6.6$ | $6.7$
$\mathit{W}$=
200 ms |  Accuracy ($\%$) | $81.10$ | $81.77$ | $82.28$
STD ($\%$) | $6.8$ | $6.8$ | $6.6$
$\mathit{W}$=
250 ms |  Accuracy ($\%$) | $81.26$ | $82.17$ | $82.57$
STD ($\%$) | $6.8$ | $6.7$ | $6.6$
$\mathit{W}$=
300 ms |  Accuracy ($\%$) | $81.73$ | $82.61$ | $82.91$
STD ($\%$) | $6.7$ | $6.6$ | $6.5$

Comparison with State-of-the-Art (SOTA): In Table 3, HDCAM is compared with
recent SOTA recurrent (Dilated LSTM) [13], convolutional (CNN), hybrid LSTM-
CNN [24], and hybrid attention-CNN [1] models on Ninapro DB2 dataset [16].
Overall, our model demonstrates better accuracy versus the number of
parameters compared to other methods. As shown in Table 3, for the window size
of $200$ ms, all variants of the proposed model outperform other SOTA
approaches. For instance, our XXSmall model has $53.3$ times less parameter
than Dilated LSTM, but obtains a $2.1\%$ gain in the top-$1$ accuracy.
Compared to the best performing TC-HGR model (Model $4$), our XXSmall and
Small models improve the accuracy for $0.38\%$ and $1.56\%$ with $4.59$ and
$1.62$ times less number of parameters, respectively. Moreover, as shown in
Table 3, for the window size of $300$ ms, XSmall and Small variants of HDCAM
obtain $82.61\%$ and $82.91\%$ top-1 accuracy respectively, both surpassing
all previous SOTA methods with fewer number of parameters (see Fig. 1).
Dilation-based LSTM [13], as the previous SOTA model on DB2 dataset, reached
to $82.4\%$ top-1 accuracy with $1,102,801$ number of parameters, while our
XSmall model attains better accuracy ($82.61\%$) with only $40,281$
parameters, i.e., $27.38$ times fewer. It is worth noting that our Small model
achieved $82.91\%$ top-1 accuracy with $58,441$ parameters, reaching a new
SOTA performance that demonstrates the effectiveness and the generalization of
our design.

Effectiveness of the Multi-scales Local Representation: To extract multi-
scales local features, we integrated depth-wise convolution ($DwConv$) with a
hierarchical structure in the proposed HDConv encoder. The hierarchical
structure besides the multi-scale feature extraction increases the receptive
field in a single block. As shown in Table 4, replacing the “hierarchical”
$DwConv$ structure in HDConv with a standard $DwConv$ layer degrades the
accuracy in all variants of HDCAM, indicating its usefulness in our design. As
an example, the top-1 accuracy of the Small model decreased by $0.56\%$ in its
non-hierarchical variant.

Table 3: Performance comparison with SOTA models.

| Models |  $\bm{W=200}$ ms |  $\bm{W=300}$ ms
---|---|---|---
| Parameters$\downarrow$ | Accuracy$\uparrow$ ($\%$) | Parameters$\downarrow$ | Accuracy$\uparrow$ ($\%$)
Dilated
LSTM [13] | 4-layer 3rd Order Dilation | $1,102,801$ | $79.0$ | $1,102,801$ | $82.4$
4-layer 3rd Order Dilation (pure LSTM) | $\\_$ | $\\_$ | $466,944$ | $79.7$
LSTM$-$
CNN [24] | CNN | - | - | $\approx 1.4M$ | $77.30$
Hybrid LSTM-CNN | - | - | $\approx 1.1M$ | $81.96$
TC-HGR
[1] | Model 1 | $49,186$ | $80.29$ | 52,066 | $80.84$
Model 2 | $68,445$ | $80.63$ | $72,285$ | $81.59$
Model 3 | $69,076$ | $80.51$ | $67,651$ | $80.95$
Model 4 | $94,965$ | $80.72$ | $92,945$ | $81.65$
Ours
( HDCAM ) | XXSmall | 20,686 | $81.10$ | 20,686 | $81.73$
XSmall | $40,281$ | $81.77$ | $40,281$ | $82.61$
Small | $58,441$ | 82.28 | $58,441$ | 82.91

Table 4: Evaluating the effectiveness of the multi-scales local representation
extraction in the HDConv encoder. For Hierarchical models, Table 1 shows the
scale values ($s$) for each stage, otherwise, $s$ is equal to $1$ at all
stages.

HDConv Encoder | Accuracy ($\%$)
---|---
XXSmall | XSmall | Small
Hierarchical structure | $81.73$ | $82.61$ | $82.91$
Non-hierarchical structure | $81.27$ | $82.21$ | $82.35$

Importance of Using MHSAtten Encoders: To examine the importance of MHSAtten
encoder, we conducted two ablation studies using this encoder at different
stages of the network for $W{=}300$ ms. In Table 5, we kept the total number
of HDConv and MHSAtten encoders fixed to $[1,3,5]$ for experiment $1$ to $4$.
While in experiment $5$ to $8$, the number of HDConv encoder is set to
$[1,2,4]$ for all stages, and MHSAtten encoder is progressively added to the
end of stages. According to both experiments, adding the MHSAtten encoder
gradually in the last two stages increases accuracy($\checkmark$) and the
number of parameters($\bm{\times}$). In addition, adding a global MHSAtten
encoder to the first stage is not beneficial since the features in this stage
are not mature enough. Furthermore, we conducted another experiment to
investigate the impact of using the MHSAtten encoder at the beginning (after
downsampling) versus end of each stage on the HDCAM architecture. As shown in
Table 6, better performance is achieved by using the MHSAtten encoder as the
final block of the stages.

Table 5: Evaluating impact of MHSAtten encoder at different stages of the
network. The listed values show the number of the corresponding encoder in
stages $1$ to $3$ in order. Highlighted rows indicate the Small model.

ID: | Model Configuration | Accuracy$\uparrow$ ($\%$) | Parameters$\downarrow$
---|---|---|---
$\bm{1:}$ | $\text{{HD}Conv}=[1,3,5],\text{{MHS}Atten}=[0,0,0]$ | $81.56$ | $37,673$
$\bm{2:}$ | $\text{{HD}Conv}=[1,3,4],\text{{MHS}Atten}=[0,0,1]$ | $82.45$ | $54,249$
$\bm{3:}$ | $\text{{HD}Conv}=[1,2,4],\text{{MHS}Atten}=[0,1,1]$ | $82.91$ | $58,441$
$\bm{4:}$ | $\text{{HD}Conv}=[0,2,4],\text{{MHS}Atten}=[1,1,1]$ | $82,26$ | $60,817$
$\bm{5:}$ | $\text{{HD}Conv}=[1,2,4],\text{{MHS}Atten}=[0,0,0]$ | $81.93$ | $31,785$
$\bm{6:}$ | $\text{{HD}Conv}=[1,2,4],\text{{MHS}Atten}=[0,0,1]$ | $82.55$ | $52,969$
$\bm{7:}$ | $\text{{HD}Conv}=[1,2,4],\text{{MHS}Atten}=[0,1,1]$ | $82.91$ | $58,441$
$\bm{8:}$ | $\text{{HD}Conv}=[1,2,4],\text{{MHS}Atten}=[1,1,1]$ | $82.48$ | $61,585$

Table 6: Evaluating the impact of using MHSAtten encoder at the beginning vs.
end of each stage for the window size of $300$ ms.

MHSAtten Encoder | Accuracy ($\%$)
---|---
XXSmall | XSmall | Small
Fist block of Stage ($\text{{MHS}Atten}=[0,1,1]$) | $81.31$ | $82.37$ | $82.70$
Latest block of Stage ($\text{{MHS}Atten}=[0,1,1]$) | $81.73$ | $82.61$ | $82.91$

##  4 Conclusion

In this paper, a novel resource-efficient architecture, referred to as the
HDCAM, is developed for HGR from sparse multichannel sEMG signals. HDCAM is
developed by effectively combining the advantages of Attention-based and CNN-
based models for low-powered devices, which is an important step toward
incorporating DNN models into wearable for immersive HMI. To efficiently
extract local and global representations of the input sEMG sequence, HDCAM is
empowered with convolution and attention-based encoders, namely HDConv and
MHSAtten, respectively. Specifically, we showed that by proper design of
convolution-based architectures, we not only can extract a multi-scale local
representation but also can increase the receptive field in a single block. A
path for future study is to address the issue of misplacement or relocation of
sEMG electrodes/sensors, which is an active field of research. This may be
seen as both a constraint for the current study and a promising area for
future investigation, which is the subject of our continuing research.

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# An application of Birch-Tate formula to tame kernels of real quadratic
number fields

Li-Tong Deng and Yong-Xiong Li

###### Abstract.

Let $F$ be a real quadratic number field with discriminant $D$ and
${\mathcal{O}}_{F}$ the ring of integers in $F$. Let $\chi_{F}$ be the
Dirichlet character associated to $F/{\mathbb{Q}}$. Write $L(\chi_{F},s)$ for
the Dirichlet L-function of $\chi_{F}$. By an induction argument for
imprimitive Dirichlet L-values, we get several $2$-divisibility results on
$L(\chi_{F},-1)$ when $D$ has arbitrarily finitely many prime divisors. As an
application, by making use of the Birch-Tate formula for $F$, we determine the
$2$-primary part for the second $K$ group $K_{2}{\mathcal{O}}_{F}$. We also
give a new proof for an old theorem of Browkin and Schinzel.

###### Key words and phrases:

Birch-Tate formula, Tame kernels, Dirichlet L-functions

###### 2010 Mathematics Subject Classification:

11R70 (primary),11M06 11R42 (secondary)

##  1\. Introduction

Let $F$ be a real quadratic number field with the ring of integers
${\mathcal{O}}_{F}$. For a ring $R$, let $K_{2}R$ be the Milnor group of $R$.
The kernel of the surjective homomorphism

$K_{2}F\to\bigoplus_{\mathfrak{p}}({\mathcal{O}}_{F}/\mathfrak{p})^{\times},$

given by the tame symbols at all finite primes $\mathfrak{p}$ of $F$, is
called the tame kernel of $F$ and is known to be isomorphic to
$K_{2}{\mathcal{O}}_{F}$. A theorem of Garland [4] shows that
$K_{2}{\mathcal{O}}_{F}$ is a finite abelian group. For any prime $\ell$, we
write $K_{2}{\mathcal{O}}_{F}(\ell)$ the $\ell$-primary subgroup of
$K_{2}{\mathcal{O}}_{F}$. In this article, we shall determine an explicit
structure for $K_{2}{\mathcal{O}}_{F}(2)$ for several families of real
quadratic number fields $F$ where the discriminant of $F$ can have arbitrarily
finitely many prime factors.

For any odd prime $\ell$, we denote by $\left(\frac{\cdot}{\ell}\right)$ the
Legendre symbol. The main result in this article is given in the following
theorem.

######  Theorem 1.1.

Let $F={\mathbb{Q}}(\sqrt{D})$ be a real quadratic number field with
discriminant $D=4\ell_{1}\cdots\ell_{n}$ where $\ell_{i}$ are distinct primes
such that $\ell_{1}\equiv 3\ \mathrm{mod}\ 8$ and $\ell_{i}\equiv 5\
\mathrm{mod}\ 8$ for $2\leq i\leq n$. Assume that
$\left(\frac{\ell_{i}}{\ell_{j}}\right)=-1$ for $2\leq i<j\leq n$,
$\left(\frac{\ell_{1}}{\ell_{i}}\right)=1$ for $i\geq 3$,
$\left(\frac{\ell_{1}}{\ell_{2}}\right)=-1$ and $n$ is even. Then

$K_{2}{\mathcal{O}}_{F}(2)\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{n-1}\times({\mathbb{Z}}/2^{\delta}{\mathbb{Z}})$

where $\delta$ is an integer at least $3$.

We remark that for $n=2$, the above theorem is due to Qin [10] (see also [2]
for a lower bound). So the theorem can be viewed as a generalization of Qin’s
result to discriminant of $F$ with arbitrarily finitely many prime factors.
Some numerical data (included in section §4) suggest that $\delta$ is always
$3$. However, we currently see no way of proving it theoretically.

For a finite abelian group $\mathfrak{A}$ and any integer $n\geq 0$, we write
$r_{2^{n}}(\mathfrak{A})$ for the number of cyclic components of
$\mathfrak{A}$ of order divisible by $2^{n}$. The number
$r_{2^{n}}(\mathfrak{A})$ is called the $2^{n}$-rank of $\mathfrak{A}$.
Therefore Theorem 1.1 shows that $r_{8}(K_{2}{\mathcal{O}}_{F})$ is always
one. As far as we know, it is difficult to determine the $8$-rank for
$K_{2}{\mathcal{O}}_{F}$. In fact, our proof for Theorem 1.1 can not recover
the result of Qin [10] for $n=2$, in fact, the proof does depend on the result
of Browkin and Schinzel [2] showing that $r_{8}(K_{2}{\mathcal{O}}_{F})\geq 1$
when $n=2$. In addition to Theorem 1.1 for $r_{8}(K_{2}{\mathcal{O}}_{F})=1$,
we also get the following theorem for $r_{8}(K_{2}{\mathcal{O}}_{F})=0$, in
which we determine the $4$-rank for $K_{2}{\mathcal{O}}_{F}$.

######  Theorem 1.2.

Let $F={\mathbb{Q}}(\sqrt{D})$ with discriminant $D>0$.

  1. (1)

Assume that $D=\ell_{1}\cdots\ell_{n}$ where $\ell_{i}$ are distinct primes
such that $\ell_{i}\equiv 5\ \mathrm{mod}\ 8$ and
$\left(\frac{\ell_{i}}{\ell_{j}}\right)=-1$ for $1\leq i<j\leq n$. Assume
further that $n$ is odd. Then

$K_{2}{\mathcal{O}}_{F}(2)\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{n+1}.$

  2. (2)

Assume that $D=4\ell_{1}\cdots\ell_{n}$ satisfying the prime divisors
$\ell_{1}\equiv 3\ \mathrm{mod}\ 8$ and $\ell_{i}\equiv 5\ \mathrm{mod}\
8\,(2\leq i\leq n)$. Assume that $n$ is odd and
$\left(\frac{\ell_{i}}{\ell_{j}}\right)=-1$ for $1\leq i<j\leq n$. Then
$K_{2}{\mathcal{O}}_{F}(2)$ is isomorphic to
$({\mathbb{Z}}/2{\mathbb{Z}})^{n-1}\times({\mathbb{Z}}/4{\mathbb{Z}})$ (resp.
$({\mathbb{Z}}/2{\mathbb{Z}})^{2}$) if $n\geq 3$ (resp. $n=1$).

For $n=1$, the above theorem is due to Browkin and Schinzel [2, Corollary 1]
(see also [10]). Fortunately our proof for Theorem 1.2 recovers the case of
Browkin and Schinzel. We also remark that the Theorem 1.2 is essentially used
in the proof of Theorem 1.1.

Now, we say something about the main ideas in the proof for Theorems 1.1 and
1.2. Previously, the $2^{n}$-rank $r_{2^{n}}(K_{2}{\mathcal{O}}_{F})$ was
determined in an arithmetic way, i.e., using Hilbert symbols or solving
certain Diophantine Equations in integers (see [2],[8],[9],[10],[17], [3] and
[18]). It may be possible that the method in [10], [9] may show Theorems 1.1
and 1.2. However, we stress that our method showing Theorems 1.1 and 1.2 is
analytic. The main tool is the Birch-Tate formula (see Theorem 2.1), which
relates the order of $K_{2}{\mathcal{O}}_{F}$ to $L(\chi_{F},-1)$ (see
Corollary 2.3). Here, $\chi_{F}$ is the Dirichlet character associated to
$F/{\mathbb{Q}}$ and $L(\chi_{F},s)$ is the Dirichlet L-functions associated
to $\chi_{F}$. The Birch-Tate formula also shows that $L(\chi_{F},-1)$ is a
non-zero rational number. Thus if one can get a good estimate on the $2$-adic
valuation for $L(\chi_{F},-1)$, then one sees a rough picture of
$K_{2}{\mathcal{O}}_{F}(2)$. Although there are well-known special value
formulas for $L(\chi_{F},-1)$, it is very difficult to get enough information
on the $2$-adic valuation for $L(\chi_{F},-1)$ when the number of prime
divisors of $D$ becomes large. To get around this difficulty, we use an
induction method for imprimitive Dirichlet L-functions. This method is
inspired by Zhao who proved some cases for the Birch and Swinnerton-Dyer
conjecture for elliptic curves with complex multiplication. By using this
induction argument, we can get a more precise estimate on the $2$-adic
valuation for $L(\chi_{F},-1)$ when $D$ has an arbitrarily number of prime
factors. Then by combining a well-known method for determining the $2$-rank
and $4$-rank for ideal class groups and $K$ groups, we prove both Theorems. In
the last step, we use a relation (due to Feng and Yue [17]) between the
$4$-ranks of ideal class groups and $K$ groups.

In the final part of the introduction, we briefly outline the contents of the
paper. In next section §2, we will state the Birch-Tate formula and give a
precise relation between $K_{2}{\mathcal{O}}_{F}$ and $L(\chi_{F},-1)$. The
main part of the paper is §3, where we prove a special value formula for
imprimitive Dirichlet L-functions. Then by using the induction argument, we
get the required estimate on the $2$-adic valuation for the Dirichlet L-values
(see Theorem 3.8). In the final section §4, we summarise the method to compute
the $2$ (resp. $4$)-rank of both the ideal class groups and $K$ groups. We
then prove both theorems and give some numerical data in the last section.

Acknowledgement We would like to thank Professors C. Greither, P. A. Østvær,
H. Qin for helpful email correspondences, and thank Y. Kezuka for comments.

##  2\. Birch-Tate formula

Let ${\mathcal{K}}$ be a number field and ${\mathcal{O}}_{\mathcal{K}}$ the
ring of integers in ${\mathcal{K}}$. Let $\zeta_{\mathcal{K}}(s)$ be the
Dedekind zeta function of ${\mathcal{K}}$. For any ring $R$, we let $K_{2}R$
denote the Milnor group of $R$ (equivalently, the second K group of $R$). By a
theorem of Garland [4], $K_{2}{\mathcal{O}}_{\mathcal{K}}$ is a finite abelian
group. In this section, for a real quadratic number field $F$, we will state
the Birch-Tate formula, which gives a precise relation between the order of
$K_{2}{\mathcal{O}}_{F}$ and $\zeta_{F}(-1)$. Therefore, we can reduce the
study of the structure of $K_{2}{\mathcal{O}}_{F}$ to the zeta values.

Let $\mathcal{F}$ be a totally real number field. The original Birch-Tate
conjecture (see [13] for history and [7] for an update) predicts the following
equality

$\zeta_{\mathcal{F}}(-1)=\frac{\\#(K_{2}({\mathcal{O}}_{\mathcal{F}}))}{\\#(W_{2}(\mathcal{F}))},$

where $W_{2}(\mathcal{F})=H^{0}(\mathcal{F},{\mathbb{Q}}/{\mathbb{Z}}(2))$. By
Wiles’s theorem on Iwasawa main conjecture ([16], see also [5] for abelian
number fields) and a remark made by Kolster (see [14]), the above equality
holds up to a power of $2$. Furthermore, if $\mathcal{F}$ is an abelian
extension over ${\mathbb{Q}}$, the equality holds. Therefore, for a real
quadratic number field $F$, we have the following Birch-Tate formula.

######  Theorem 2.1.

Keep the above notations, then we have

(2.1) $\zeta_{F}(-1)=\frac{\\#(K_{2}({\mathcal{O}}_{F}))}{\\#(W_{2}(F))}.$

The term $w_{2}(F)=\\#(W_{2}(F))$ is easy to compute, we list the result in
the following lemma.

######  Lemma 2.2.

Write $F={\mathbb{Q}}(\sqrt{D})$ with $D$ the discriminant of $F$. Then

  1. (1)

If $D=5$, $w_{2}(F)=120$.

  2. (2)

If $D=8$, $w_{2}(F)=48$.

  3. (3)

If $D>8$, $w_{2}(F)=24$.

###### Proof.

This is a well-known results. We include a proof for readers’ convenience.
From [6], we have the formula

$w_{2}(F)=4\cdot\prod_{[\mathfrak{L}:F]=2}\frac{1}{2}\cdot\\#(\mu(\mathfrak{L}))$

where $\mathfrak{L}/F$ runs over all quadratic extensions of $F$, and
$\mu(\mathfrak{L})$ is the group of roots of unity in $\mathfrak{L}$. By
noting that a field extension over ${\mathbb{Q}}$ with degree $4$ can only
contain $4,5,6,8$-th roots of unity, the assertion in the lemma follows. ∎

When $F={\mathbb{Q}}(\sqrt{2})$ or ${\mathbb{Q}}(\sqrt{5})$, one can easily
determine the structure of $K_{2}{\mathcal{O}}_{F}$ (at least for the
$2$-primary part). From now on, we assume that $D>8$.

######  Corollary 2.3.

For $F={\mathbb{Q}}(\sqrt{D})$ satisfying $D>8$, we have

$\\#(K_{2}{\mathcal{O}}_{F})=24\cdot\zeta_{F}(-1).$

In particular, if we denote by $\chi_{F}$ the Dirichlet character associated
to $F/{\mathbb{Q}}$, then

$\\#(K_{2}{\mathcal{O}}_{F})=-2\cdot L(\chi_{F},-1).$

Here, $L(\chi_{F},s)$ is the Dirichlet L-series associated to $\chi_{F}$.

###### Proof.

Denote by $\zeta(s)$ the Riemann’s zeta function. The corollary follows by the
well-known formula $\zeta_{F}(s)=\zeta(s)L(s,\chi_{F})$ and
$\zeta(-1)=-\frac{1}{12}$. ∎

##  3\. A special value formula and induction method

We know from section §2 that one can reduce the study of
$K_{2}{\mathcal{O}}_{F}$ to $L(\chi_{F},-1)$. It is also well-known that
$L(\chi_{F},-1)$ is a nonzero rational number and there is a special value
formula for $L(\chi_{F},-1)$ in terms of the generalized Bernoulli numbers.
However, for any prime $\ell$, to determine the precise $\ell$-adic valuation
of $L(\chi_{F},-1)$ is very difficult (even for $\ell=2$). To overcome this
difficulty, we borrow some ideas of Zhao, who proved some cases for the Birch
and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication
(see [19]). The ideas consist of two main ingredients: one is a special value
formula for imprimitive L-values of Dirichlet characters at $s=-1$; the other
is an induction method for these imprimitive L-functions. By using Zhao’s
method, we can get a more precise estimate on the $2$-adic valuation of
$L(\chi_{F},-1)$ when $D$ has arbitrarily finitely many prime factors. This
key result will help us to determine the $2$-primary part of
$K_{2}{\mathcal{O}}_{F}$ in the next section.

Let us first recall some basic facts on Dirichlet L-functions and generalized
Bernoulli numbers. Let $\chi$ be a Dirichlet character with conductor
$f_{\chi}=f$. Let $g$ be any positive integer such that $f$ divides $g$. We
define $L^{(g)}(\chi,s)(s\in{\mathbb{C}})$ to be the analytic continuation of
the following power series

$\sum^{\infty}_{n=1,\,(n,g)=1}\frac{\chi(n)}{n^{s}},\qquad\textrm{where
}{\mathrm{Re}}(s)>1.$

This is an imprimitive Dirichlet L-series when $f\neq g$. When $f=g$,
$L^{(g)}(\chi,s)$ is a primitive L-series and we simply denote it by
$L(\chi,s)$.

We define the generalized Bernoulli number $B_{n,\chi,g}$ via the following
identity

(3.1)
$\sum^{g}_{z=1}\frac{\chi(z)te^{zt}}{e^{gt}-1}=\sum^{\infty}_{n=0}B_{n,\chi,g}\frac{t^{n}}{n!}.$

For each integer $n\geq 0$, we recall the definition of the Bernoulli
polynomial $B_{n}(X)$ in the following equality

(3.2) $\frac{te^{Xt}}{e^{t}-1}=\sum^{\infty}_{n=0}B_{n}(X)\frac{t^{n}}{n!}.$

We remark that the dependence of $B_{n,\chi,g}$ on $g$ in (3.1) can be ignored
by the following lemma.

######  Lemma 3.1.

For any integer $g$ such that $f\mid g$, the number $B_{n,\chi,g}$ is
independent of $g$. We denote this number by $B_{n,\chi}$. Then

$B_{n,\chi}=g^{n-1}\sum^{g}_{a=1}\chi(a)\cdot B_{n}\left(\frac{a}{g}\right).$

Furthermore, if $\chi$ is non-trivial and $\chi(-1)=1$, we have

(3.3) $B_{2,\chi}=\frac{1}{g}\sum^{g}_{a=1}\chi(a)a^{2}.$

###### Proof.

Set $y=\frac{g}{f}$ and make a change of variables

$z=b+cf,\qquad 1\leq b\leq f,\,0\leq c\leq y-1.$

Noting that $\chi$ has conductor $f$, we have

(3.4)
$\sum^{g}_{z=1}\chi(z)\cdot\frac{te^{zt}}{e^{gt}-1}=\sum^{f}_{b=1}\sum^{y-1}_{c=0}\chi(b)\cdot\frac{te^{t(b+cf)}}{e^{gt}-1}.$

Since the following identity

$\sum^{y-1}_{c=0}e^{tcf}=\frac{e^{tg}-1}{e^{tf}-1},$

the right hand side of (3.4) turns to be

$\sum^{f}_{b=1}\sum^{y-1}_{c=0}\chi(b)\cdot\frac{te^{t(b+cf)}}{e^{gt}-1}=\sum^{f}_{b=1}\chi(b)\cdot\frac{te^{bt}}{e^{ft}-1}.$

Therefore, we get

$\sum^{g}_{z=1}\chi(z)\cdot\frac{te^{zt}}{e^{gt}-1}=\sum^{f}_{b=1}\chi(b)\cdot\frac{te^{bt}}{e^{ft}-1}$

and the first assertion follows. Next, by formula (3.2), we get

$\sum^{\infty}_{n=0}g^{n-1}\sum^{g}_{a=1}\chi(a)B_{n}\left(\frac{a}{g}\right)\cdot\frac{t^{n}}{n!}=\sum^{g}_{a=1}\chi(a)\frac{te^{at}}{e^{gt}-1}.$

Comparing to (3.1), the first equality in the lemma follows. Finally, note
that $B_{2}(X)=X^{2}-X+\frac{1}{6}$, then formula (3.3) follows from the first
equality. Here, we use the fact that $\chi$ is non-trivial and $\chi(-1)=1$.
Therefore, the lemma follows. ∎

For any integer $n\geq 1$, the following special value formula for $L(\chi,s)$
at $s=1-n$ is well-known

(3.5) $L(\chi,1-n)=-\frac{B_{n,\chi}}{n}.$

One can find a detailed proof in [15, Theorem 4.2].

Next, we give a special value formula for imprimitive Dirichlet L-functions.
For this purpose, we introduce some notation. For a positive integer $M\equiv
0$ or $1\ \mathrm{mod}\ 4$, we put $\chi_{M}=\left(\frac{M}{\cdot}\right)$,
where $\left(\frac{M}{\cdot}\right)$ is the Jacobian symbol. Here, we make the
convention that $\left(\frac{M}{-1}\right)=1$ since $M>0$ (see also [12]). Let
$d_{i}(1\leq i\leq n)$ be discriminant of some real quadratic fields such that
the greatest common divisor of $d_{i},d_{j}(i\neq j)$ is $1$ or $4$. For any
$d=d_{i_{1}}\cdots d_{i_{s}}$ where $\\{i_{1},\cdots,i_{s}\\}$ is a non-empty
subset of $\\{1,\cdots,n\\}$, we let $d^{\prime}=D/d$.

######  Proposition 3.2.

Let $L^{(D)}(\chi_{d},s)$ be the imprimitive Dirichlet L-function associated
to $\chi_{d}$. Then

(3.6)
$L^{(D)}(\chi_{d},-1)=-\frac{1}{2D}\sum^{D}_{a=1}\chi_{Dd^{\prime}}(a)a^{2}.$

###### Proof.

By the Euler product of $L^{(D)}(\chi_{d},s)$ and that
$\chi_{d}(\ell)=\left(\frac{d}{\ell}\right)$ for any prime $\ell\nmid d$, we
get

$L^{(D)}(\chi_{d},s)=\prod_{\ell\nmid
d^{\prime}}\left(1-\left(\frac{d}{\ell}\right)\ell^{-s}\right)^{-1},\qquad\textrm{where}\quad{\mathrm{Re}}(s)>1.$

Since for these $\ell\nmid d^{\prime}$, we always have
$\chi_{Dd^{\prime}}(\ell)=\chi_{d}(\ell)$. Therefore, the above equality
implies that

$L^{(D)}(\chi_{d},s)=L(\chi_{Dd^{\prime}},s).$

Here, we should note that the conductor of $\chi_{Dd^{\prime}}$ is $D$. Now,
our formula (3.6) follows from the above equality, formulas (3.5) and (3.3).
Therefore, the Proposition follows. ∎

The next example are the very few cases where we can get a precise $2$-adic
valuation for $L(\chi_{F},-1)$ only using formula (3.6). We denote by
$v=v_{2}$ the normalized additive valuation on ${\mathbb{Q}}_{2}$ such that
$v(2)=1$.

######  Example 3.3.

Let $p\equiv 3\ \mathrm{mod}\ 8$ be a prime and $d=4p$. For
$F={\mathbb{Q}}(\sqrt{d})$, we have

$v(L(\chi_{d},-1))=1.$

Our motivation for this example comes from some arithmetic results by Browkin-
Schinzel in [2]. Of course, the discriminant $D_{F}=d$ of $F$ is greater than
$8$, thus, from Birch-Tate formula, we know that
$24\zeta_{F}(-1)\in{\mathbb{Z}}$. By $\zeta(-1)=\frac{1}{12}$, formula (3.6)
and

$\zeta_{F}(-1)=\zeta(-1)L(\chi_{d},-1),$

we see that $\frac{1}{d}\sum^{d}_{a=1}\chi_{d}(a)a^{2}$ is an integer. By
$\chi_{d}$ takes zero value on even integers and recall that $d=4p$, we get

(3.7)
$\frac{1}{d}\sum^{d}_{a=1}\chi(a)a^{2}=\frac{1}{4p}\sum^{2p}_{b=1}\chi_{d}(2b-1)\cdot(2b-1)^{2}.$

Since $\chi_{d}$ is nontrivial and $\chi_{d}(-1)=1$, one can easily check that

$\sum^{2p}_{b=1}\chi_{d}(2b-1)=\sum^{2p}_{b=1}\chi_{d}(2b-1)b=0.$

Thus, from formula (3.7), we get

(3.8)
$\frac{1}{d}\sum^{d}_{a=1}\chi(a)a^{2}=\frac{1}{p}\sum^{2p}_{b=1}\chi_{d}(2b-1)\cdot
b^{2}.$

Since $\chi_{d}(2b-1)=\left(\frac{p}{2b-1}\right)$, if we break up the second
sum in (3.8) according as $b=2c$ or $b=2c-1$ ($1\leq c\leq p$), by modulo $8$
and noting that

$(2c-1)^{2}\equiv 1\ \mathrm{mod}\ 8\quad\textrm{and
}\quad\sum^{p}_{c=1}\left(\frac{p}{4c-3}\right)=0,$

then we get

$\frac{1}{d}\sum^{d}_{a=1}\chi(a)a^{2}\equiv\frac{1}{p}\sum^{p}_{c=1}\left(\frac{p}{4c-1}\right)4c^{2}\
\mathrm{mod}\ 8.$

Now, the example will follow from the formula (3.6) and the following claim

$\sum^{p}_{c=1}\left(\frac{p}{4c-1}\right)c^{2}\equiv 1\ \mathrm{mod}\ 2.$

To show this claim, since the Jacobian symbol takes values in $\pm 1$, thus,
by modulo $2$, we have

$\sum^{p}_{c=1}\left(\frac{p}{4c-1}\right)c^{2}\equiv\sum^{p}_{c=1}c^{2}-\left(\frac{p+1}{4}\right)^{2}\
\mathrm{mod}\ 2.$

Then one can easily check the right hand side in the congruence is indeed an
odd integer, and our claim follows.

Recall that $D=d_{1}\cdots d_{n}$ with each $d_{i}$ is a discriminant of some
real quadratic fields satisfying $(d_{i},d_{j})=1$ or $4$ for $i\neq j$. We
remark that when $n$ is large, it is hardly to get a precise $2$-adic
valuation of $L(\chi_{D},-1)$, so now we introduce the induction method to get
around this problem. Let us define

$\mathcal{G}=\\{a\in{\mathbb{Z}}:1\leq a\leq D,\,\chi_{d_{i}}(a)=1\,\textrm{
for any }1\leq i\leq n\\}.$

The heart of the induction method is to take sum of all the imprimitive
L-values $L^{(D)}(\chi_{d},-1)$, where $d=d_{i_{1}}\cdots d_{i_{s}}$ such that
$\\{i_{1},\cdots,i_{s}\\}$ runs over all non-empty subset of
$\\{1,\cdots,n\\}$.

######  Proposition 3.4.

We have

(3.9) $\sum_{d=d_{i_{1}}\cdots
d_{i_{s}}}L^{(D)}(\chi_{d},-1)=-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}+\frac{D}{6}\varphi(D)+\frac{1}{12}\prod_{p\mid
D}(1-p).$

Here, the first sum is taken over all $d_{i_{1}}\cdots d_{i_{s}}$ such that
$\\{i_{1},\cdots,i_{s}\\}$ runs over all non-empty subset of
$\\{1,\cdots,n\\}$. The last product is taken over all prime divisors $p$ of
$D$. The function $\varphi(D)$ is the Euler function.

Before proving the Proposition, we need some elementary facts on Mobius
function $\mu$.

######  Lemma 3.5.

Assume $D$ as above and $D>1$, then we have

$\sum_{m\mid D}\mu(m)=0,\qquad\sum_{m\mid
D}\mu(m)\frac{D}{m}=\varphi(D),\qquad\textrm{and}\qquad\sum_{m\mid
D}\mu(m)m=\prod_{p\mid D}(1-p).$

###### Proof.

All are standard facts on Mobius functions, for a detailed proof, see [1]. ∎

Now, we come back to the proof of Proposition 3.4.

###### Proof of Proposition 3.4.

Writing $d=d_{i_{1}}\cdots d_{i_{s}}$ and adding up the formulas (3.6) for all
such $d\neq 1$, we get

$\sum_{d\neq
1}L^{(D)}(\chi_{d},-1)=-\frac{1}{2D}\sum^{D}_{a=1}\chi_{D}(a)a^{2}\sum_{d^{\prime}\neq
D}\left(\frac{d^{\prime}}{a}\right).$

Here, we recall that $d^{\prime}=\frac{D}{d}$ and
$\chi_{Dd^{\prime}}(a)=\chi_{D}(a)\left(\frac{d^{\prime}}{a}\right)$. By the
following identity

$\sum_{d^{\prime}\mid
D}\left(\frac{d^{\prime}}{a}\right)=\prod^{n}_{i=1}\left(1+\left(\frac{d_{i}}{a}\right)\right),$

we see that

$\sum_{d\neq
1}L^{(D)}(\chi_{d},-1)=-\frac{1}{2D}\sum^{D}_{a=1}\chi_{D}(a)a^{2}\left(\prod^{n}_{i=1}\left(1+\left(\frac{d_{i}}{a}\right)\right)-\left(\frac{D}{a}\right)\right).$

For the term on the right hand of the above formula, by separating the
bracket, we know the first term is non-zero if and only if
$a\in{\mathcal{G}}$. Noting that $\chi_{D}(a)=\left(\frac{D}{a}\right)$, then
we get

(3.10) $\sum_{d\neq 1}L^{(D)}(\chi_{d},-1)=-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}+\frac{1}{2D}\sum_{1\leq a\leq D,\,(a,D)=1}a^{2}.$

To handle the second sum on right hand side in (3.10), we invoke some relation
on arithmetic function on ${\mathbb{Z}}$. For two such arithmetic functions
$h_{1},h_{2}$, their Dirichlet product is given as follows

(3.11) $(h_{1}*h_{2})(n)=\sum_{m\mid n}h_{1}(m)h_{2}(n/m).$

For the two arithmetic functions $\mathfrak{h}_{1}(n)=\sum_{1\leq a\leq
n,\,(a,n)=1}\frac{a^{2}}{n^{2}}$ and $\mathfrak{h}(n)=\sum_{1\leq a\leq
n}\frac{a^{2}}{n^{2}}$, one can easily show that
$\mathfrak{h}_{1}=\mu*\mathfrak{h}$ (see also [1, Exercise 2.14]). Now, write
the second sum in right hand side of (3.10) as

$\frac{1}{2D}\sum_{1\leq a\leq D,\,(a,D)=1}a^{2}=\frac{D}{2}\sum_{1\leq a\leq
D,\,(a,D)=1}\frac{a^{2}}{D^{2}}.$

Applying formula (3.11) and Lemma 3.5, one can directly check the following
equality holds

$\frac{D}{2}\sum_{1\leq a\leq
D,\,(a,D)=1}\frac{a^{2}}{D^{2}}=\frac{1}{12}\sum_{m\mid
D}\mu(D/m)\cdot(m+1)(2m+1)\cdot\frac{D}{m}.$

By using Lemma 3.5 again, we see that

$\frac{1}{12}\sum_{m\mid
D}\mu(D/m)\cdot(m+1)(2m+1)\cdot\frac{D}{m}=\frac{D}{6}\varphi(D)+\frac{1}{12}\prod_{p\mid
D}(1-p).$

The Proposition now follows. ∎

In the final part of this section, we apply the formula (3.9) to some special
$d_{i}$’s to get some useful estimation on the $2$-adic valuation for
$L(\chi_{D},-1)$.

######  Proposition 3.6.

Let $D=d_{1}\cdots d_{n}$ be given as before. Assume that each
$d_{i}=p_{i}\equiv 5\ \mathrm{mod}\ 8$ are primes, then we have

$v(L(\chi_{D},-1))\geq n.$

Moreover, if we assume that $\left(\frac{p_{i}}{p_{j}}\right)=-1$ for all
$1\leq i<j\leq n$, then

  1. (1)

If $n$ is even, $v(L(\chi_{D},-1))\geq n+1$.

  2. (2)

If $n$ is odd, $v(L(\chi_{D},-1))=n$.

###### Proof.

Let us first prove the general lower bound in the Proposition. The main idea
is by induction on $n$ using formula (3.9). The case for $n=1$ is similar to
Example 3.3. We simply write $d_{1}=p$ and assume that $p>5$. Recall that

(3.12)
$L(\chi_{p},-1)=-\frac{1}{2p}\sum^{p}_{a=1}\left(\frac{p}{a}\right)a^{2}.$

For the sum $\sum^{p}_{a=1}\left(\frac{p}{a}\right)a^{2}$, separating it
according as $a=2b$ or $2b-1$ with $1\leq b\leq\frac{p-1}{2}$ and noting the
following fact (since $p\equiv 5\ \mathrm{mod}\ 8$)

$\sum^{\frac{p-1}{2}}_{b=1}\left(\frac{p}{2b-1}\right)=\frac{1}{2}\sum^{\frac{p-1}{2}}_{b=1}\left(\left(\frac{p}{2b-1}\right)+\left(\frac{p}{p-(2b-1)}\right)\right)=0,$

we get

$\sum^{p}_{a=1}\left(\frac{p}{a}\right)a^{2}=4\left(\sum^{\frac{p-1}{2}}_{b=1}\left(\frac{p}{2b}\right)b^{2}+\sum^{\frac{p-1}{2}}_{b=1}(b^{2}-b)\left(\frac{p}{2b-1}\right)\right)=:4\mathfrak{s}.$

We then claim that $\mathfrak{s}$ is odd. In fact, by modulo $2$ and recalling
that $p\equiv 5\ \mathrm{mod}\ 8$, we see that

$\mathfrak{s}\equiv\sum^{\frac{p-1}{2}}_{b=1}b\equiv\frac{(p+1)(p-1)}{8}\equiv
1\ \mathrm{mod}\ 2.$

Therefore, by formula (3.12), we have $v(L(\chi_{p},-1))=1$. The case for
$n=1$ then follows. Now, we let $n\geq 2$ and assume the proposition holds
when $d\neq D$, i.e. $v(L(\chi_{d},-1))\geq s$. From formula (3.9), we get the
following key formula in our induction argument

(3.13) $\sum_{d\neq 1}L^{(D)}(\chi_{d},-1)=-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}+\left(\frac{D}{6}+\frac{(-1)^{n}}{12}\right)\varphi(D).$

Here, the first sum is taken with all $d=d_{i_{1}}\cdots d_{i_{s}}$ dividing
$D$ such that $s>0$. For the term $\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}$, by the lemma below, it has $2$-adic valuation
$1$ or at least $2$ according as $n=2$ or $n\geq 4$. In either cases, we
always have

$v\left(-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}\right)\geq n.$

It is also easily to see that
$v\left(\left(\frac{D}{6}+\frac{(-1)^{n}}{12}\right)\varphi(D)\right)=2n-2$,
which is at least $n$ since $n\geq 2$. Next, we deal with each term
$L^{(D)}(\chi_{d},-1)$ in the left hand side in (3.13). Note the equality

$L^{(D)}(\chi_{d},-1)=\left(\prod_{p\nmid
d^{\prime}}(1-\chi_{d}(p)p)\right)\cdot L(\chi_{d},-1)\qquad d^{\prime}=D/d.$

The induction hypothesis shows that if $d\neq D$, $v(L(\chi_{d},-1))\geq s$.
Noting that $v(1-\chi_{d}(p)p)\geq 1$ for all $p\mid d^{\prime}$, we get

$v(L^{(D)}(\chi_{d},-1))=v\left(\left(\prod_{p\nmid
d^{\prime}}(1-\chi_{d}(p)p)\right)\right)+v(L(\chi_{d},-1))\geq n$

for all $d\neq D$. Finally, by formula (3.13), we must have

$v(L(\chi_{D},-1))\geq n,$

thus the first assertion of the proposition follows. For the next two special
cases, since the method are similar, we only give a detailed proof for (2). We
assume now that $\left(\frac{p_{i}}{p_{j}}\right)=-1$ for $i\neq j$ and $n$ is
odd. We still use induction on $n$, the case $n=1$ is done in the above. Write
$d=d_{i_{1}}\cdots d_{i_{s}}$. If $s$ is even, by $n$ is odd, we have $s+1\leq
n$. Then $L^{(D)}(\chi_{d},-1)=\left(\prod_{p\mid
d^{\prime}}(1-p)\right)L(\chi_{d},-1)$ has $2$-adic valuation at least
$2n-s\geq n+1$. Since $n\geq 3$, by the lemma below and a similar method as
above, each term in the right hand side in (3.13) has $2$-adic valuation at
least $n+1$. Now, for the terms $L^{(D)}(\chi_{d},-1)$ with $s<n$ odd, by
induction hypothesis, we have $v(L^{(D)}(\chi_{d},-1))=n$. Note that the
number of all such terms are odd, therefore, from (3.13), we must have
$v(L(\chi_{D},-1))=n$, (1) then follows. ∎

We now prove the following lemma used in the proof of Proposition 3.6.

######  Lemma 3.7.

Recall that $D=d_{1}\cdots d_{n}$ with $d_{i}=p_{i}\equiv 5\ \mathrm{mod}\ 8$
are primes. For $\mathfrak{s}_{1}=\sum_{1\leq a\leq
D,\,a\in\mathcal{G}}a^{2}$, we have

$\mathfrak{s}_{1}\equiv 2^{n-1}\mathfrak{t}\ \mathrm{mod}\ 4.$

where $\mathfrak{t}$ is some odd integer. In particular, when $n\geq 3$,
$\mathfrak{s}_{1}$ is always divisible by $4$.

###### Proof.

For $i=0,1$, we define $\mathfrak{G}_{i}$ to be the set
$\\{a\in\mathcal{G}:a\equiv i\ \mathrm{mod}\ 2\\}$. Since sending
$x\in\mathfrak{G}_{0}$ to $D-x\in\mathfrak{G}_{1}$ is a bijection and that it
is easy to show that
$\\#(\mathfrak{G}_{1}\cup\mathfrak{G}_{2})=\frac{\varphi(D)}{2^{n}}$, then we
have

$\\#(\mathfrak{G}_{1})=\frac{1}{2}\prod^{n}_{i=1}\frac{p_{i}-1}{2}.$

Now, the lemma follows from the following congruence

$\mathfrak{s}_{1}\equiv\\#(\mathfrak{G}_{1})\ \mathrm{mod}\ 4.$

∎

The following theorem is the main result in this section.

######  Theorem 3.8.

Let $D=d_{1}\cdots d_{n}$ satisfying $d_{1}=4p_{1}$ where $p_{1}\equiv 3\
\mathrm{mod}\ 8$ is a prime and $d_{i}=p_{i}\equiv 5\ \mathrm{mod}\ 8$ are
primes for $i>1$. Then we have

(3.14) $v(L(\chi_{D},-1))\geq n.$

Furthermore, if we assume that $\left(\frac{p_{i}}{p_{j}}\right)=-1$ for all
$2\leq i<j\leq n$.

  1. (1)

If $\left(\frac{p_{1}}{p_{j}}\right)=-1$ for $j>1$ and $n$ is odd, then
$v(L(\chi_{D},-1))=n$.

  2. (2)

Assume that one of the following conditions holds

    1. (2a)

$\left(\frac{p_{1}}{p_{j}}\right)=-1$ for $j\geq 2$ and $n$ is even.

    2. (2b)

$\left(\frac{p_{1}}{p_{j}}\right)=1$ for $j\geq 2$.

    3. (2c)

$\left(\frac{p_{1}}{p_{2}}\right)=-1$, $\left(\frac{p_{1}}{p_{j}}\right)=1$
for $j>2$ and $n$ is even.

Then $v(L(\chi_{D},-1))\geq n+1$.

Here, (2c) will be our concerned case to study the structure of the
$2$-primary part of $K_{2}{\mathcal{O}}_{F}$ in section §4. Before giving the
proof, we must say that the proof of (2c) involves all the before cases, i.e.,
Proposition 3.6, (1) and (2a) to (2b) in Proposition 3.8 in the induction
arguments and also a key arithmetic result of Browkin and Schinzel [2] (see
also [10]), which we state in the following lemma.

######  Lemma 3.9 (Browkin and Schinzel).

Let $D=d_{1}d_{2}$ satisfying $d_{1}=4p_{1}$ where $p_{1}\equiv 3\
\mathrm{mod}\ 8$ is a prime and $d_{2}=p_{2}\equiv 5\ \mathrm{mod}\ 8$ is a
prime. Then we have

$v(L(\chi_{D},-1))\geq 3.$

###### Proof.

Write $F={\mathbb{Q}}(\sqrt{D})$. Recall that $K_{2}{\mathcal{O}}_{F}(2)$
denotes the $2$-primary part of $K_{2}{\mathcal{O}}_{F}$. By [2, Corollary 2],
we have $\\#(K_{2}{\mathcal{O}}_{F}(2))\geq 16$. Then the lemma follows from
Corollary 2.3. ∎

As usual the formula (3.9) in Proposition 3.4 in this case becomes

(3.15) $\sum_{d\neq 1}L^{(D)}(\chi_{d},-1)=-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}+\left(\frac{D}{6}+\frac{(-1)^{n+1}}{24}\right)\varphi(D).$

By a more delicate analysis on elements in ${\mathcal{G}}$, we can get the
following key estimation for the $2$-adic valuation of the sum $\sum_{1\leq
a\leq D,\,a\in{\mathcal{G}}}a^{2}$.

######  Lemma 3.10.

Let $D$ be given as in Theorem 3.8. Put $\mathfrak{t}_{1}=\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}$. Then

$\mathfrak{t}_{1}\equiv 2^{n}\mathfrak{y}\ \mathrm{mod}\ 16.$

Here, $\mathfrak{y}$ is some odd integer. In particular, when $n\geq 4$, we
have $v(\mathfrak{t}_{1})\geq 4$.

###### Proof.

Note that $D$ is even. Thus any $a\in{\mathcal{G}}$ is an odd integer. For
$i=0,1$, we put

$\mathfrak{W}_{i}=\\{a\in{\mathcal{G}}:a^{2}\equiv 1+8i\ \mathrm{mod}\ 16\\}.$

One can easily see that $a\in{\mathcal{G}}$ if and only if
$D-a\in{\mathcal{G}}$. We claim further that the automorphism of
${\mathcal{G}}$ sending $a$ to $D-a$ maps $\mathfrak{W}_{0}$ onto
$\mathfrak{W}_{1}$. In fact, for $i=0,1$, note that $a\equiv\pm(1+2i)\
\mathrm{mod}\ 8$ if and only if $a^{2}\equiv 1+8i\ \mathrm{mod}\ 16$, i.e.,
$a\in\mathfrak{W}_{i}$. By a direct checking on the residue class for $D-a$,
the claim follows. From the claim, we know that
$\\#(\mathfrak{W}_{1})=\\#(\mathfrak{W}_{2})$. Therefore, we get

$\mathfrak{t}_{1}\equiv\\#(\mathfrak{W}_{0})+9\cdot\\#(\mathfrak{W}_{1})=10\cdot\\#(\mathfrak{W}_{0})\
\mathrm{mod}\ 16.$

Since ${\mathcal{G}}$ is a disjoint union of $\mathfrak{W}_{0}$ and
$\mathfrak{W}_{1}$ and that ${\mathcal{G}}$ has $\frac{\varphi(D)}{2^{n}}$
elements, then we have

$\mathfrak{t}_{1}\equiv 10\cdot\prod^{n}_{i=1}\left(\frac{p_{i}-1}{2}\right)\
\mathrm{mod}\ 16.$

Therefore, put
$\mathfrak{y}=\left(10\cdot\prod^{n}_{i=1}\left(\frac{p_{i}-1}{2}\right)\right)/2^{n}$,
the lemma follows. ∎

Now, we come back to the proof for Theorem 3.8.

###### Proof of Theorem 3.8.

We first show that inequality (3.14) holds. When $n=1$, it has been done in
Example 3.3. Using both the method and results in Proposition 3.6, one can
also show that (3.14) holds for $n=2,3$. So, we may assume that $n\geq 4$. The
two terms on the right hand side in (3.15) have $2$-adic valuations given as
follows

$v\left(\left(\frac{D}{6}+\frac{(-1)^{n+1}}{24}\right)\varphi(D)\right)=2n-3\geq
n+1,$

and

$v\left(-\frac{2^{n-1}}{D}\sum_{1\leq a\leq
D,\,a\in{\mathcal{G}}}a^{2}\right)\geq n+1.$

Here, we use the fact that $n\geq 4$ in the first inequality and Lemma 3.10 in
the second. For each term with $d\neq D$ on the left hand side in (3.15), by
Euler product for $L^{(D)}(\chi_{d},-1)$ and the following facts

$v(1-\chi_{d}(2)2)=0\quad\textrm{ and }\quad v(1-\chi_{d}(p)p)\geq 1$

for every odd prime $p\mid D$, noting also the induction hypothesis, we get

$v(L(\chi_{D},-1))\geq n.$

Therefore, the inequality (3.14) follows. Next, we prove (1). For $n=1$, it is
done in the above proof. One can also check directly that the case for $n=3$
is true. So we may assume that $n\geq 5$. Then each term on the right hand
side in (3.15) has $2$-adic valuation at least $n+1$. Thus, we only need to
deal with the terms $L^{(D)}(\chi_{d},-1)$ (with $d\neq D$) on the left hand
side of (3.15). Recall that $s$ is the positive integer such that
$d=d_{i_{1}}\cdots d_{i_{s}}$. By Proposition 3.6, formula (3.14) and the
induction hypothesis, we can show that $v(L^{(D)}(\chi_{d},-1))\geq n+1$ holds
for the following two cases: either $s$ is even or $s$ is odd and $d_{1}$ does
not divide $d$. For the remaining case where $s$ is odd and $d_{1}\mid d$,
since $\chi_{d}(p)=-1$ for every prime $p\mid d^{\prime}=D/d$, thus
$L^{(D)}(\chi_{d},-1)=\prod_{p\mid d^{\prime}}(1+p)\cdot L(\chi_{d},-1)$.
Since the induction hypothesis shows that $v(L(\chi_{d},-1))=s$, noting that
each $p\mid d^{\prime}$ satisfies $p\equiv 5\ \mathrm{mod}\ 8$, we must have
$v(L^{(D)}(\chi_{d},-1))=n$. Since $n\geq 5$, we see that the total number of
$d\neq D$ in this case is odd, therefore, from formula (3.15), we get
$v(L(\chi_{D},-1))=n$. Therefore, (1) follows. To prove (2), by using the same
proof for $n=1$ in (3.14), Lemma 3.9 and a direct computation, we can prove
that (2) holds for $n<4$. Therefore, we may assume that $n\geq 4$, then the
right hand side of (3.15) has $2$-adic valuation at least $n+1$. So, we only
need to deal with the terms $L^{(D)}(\chi_{d},-1)$ for $d\neq D$. Let us first
prove (2a). If $s$ is even, $v(L^{(D)}(\chi_{d},-1))\geq n+1$ follows from the
induction hypothesis. If $s$ is odd, from (1) and $\chi_{d}(p)=-1$, we see
that $v(L^{(D)}(\chi_{d},-1))=n$. Note that the number of all $d\neq D$ such
that $s$ is odd is an even integer, thus by (3.15) we must have
$v(L(\chi_{D},-1))\geq n+1$, then (2a) follows. For the case (2b), from the
induction hypothesis and Proposition 3.14, we can easily check that
$v(L^{(D)}(\chi_{d},-1))\geq n+1$ holds for the cases: either $d_{1}\mid d$ or
$d_{1}\nmid d$ and $s$ is even. For the remaining case when $d_{1}\nmid d$ and
$s$ is odd, note that $p_{1}\mid d^{\prime}$, and for each $p\mid d^{\prime}$,
$\chi_{d}(p)=1$ or $-1$ according as $p=p_{1}$ or $p\neq p_{1}$, we get

$v(L^{(D)}(\chi_{d},-1))=v((1-p_{1})\cdot\prod_{p\mid d^{\prime},\,p\neq
p_{1}}(1+p)L(\chi_{d},-1))=n.$

Here we use (2) in Proposition 3.6 in the last equality. Since the number of
all such $d$’s is even, thus the sum of all these $L^{(D)}(\chi_{d},-1)$’s has
$2$-adic valuation at least $n+1$. Therefore, from equality (3.15), we have
$v(L(\chi_{D},-1))\geq n+1$. Therefore, (2b) follows. Finally, we prove (2c).
Since $n$ is even, and from Lemma 3.9, we know that (2c) holds for $n=2$. Thus
we may assume that $n\geq 4$. So, the $2$-adic valuation on the right hand
side of (3.15) is at least $n+1$. To get the assertion in (2c), we only need
to study the terms $L^{(D)}(\chi_{d},-1)$ for $d\neq D$ on the left hand side
of (3.15). We divide the proof into the following four cases:

  1. Case (i)

If $d_{1}\mid d$ and $s$ is even. Then by either the induction hypothesis or
(2b), we get $v(L(\chi_{d},-1))\geq s+1$. Therefore, we must have
$v(L^{{D}}(\chi_{d},-1))\geq n+1$ for such $d$’s.

  2. Case(ii)

If $d_{1}\mid d$ and $s$ is odd. For every $p\mid d^{\prime}$, we see that
$\chi_{d}(p)=-1$ (resp. $1$) if $p_{2}\mid d^{\prime}$ and $p=p_{2}$ (resp.
$p\neq p_{2}$). Then $v\left(\prod_{p\mid d^{\prime}}(1-\chi_{d}(p)p)\right)$
is at least $2(n-s)-1$ (resp. $2(n-s)$) according as $p_{2}\mid d^{\prime}$
(resp. $p_{2}\nmid d^{\prime}$). If $p_{2}\mid d^{\prime}$, so $p_{2}\nmid d$,
by (2b), we have $v(L(\chi_{d},-1))\geq s+1$. Thus we get
$v(L^{(D)}(\chi_{d},-1))\geq 2n-s\geq n+1$ since $n$ is even. If $p_{2}\nmid
d^{\prime}$, so $p_{2}\mid d$, by the inequality (3.14), we have
$v(L(\chi_{d},-1))\geq s$. Thus we get $v(L^{(D)}(\chi_{d},-1))\geq 2n-s\geq
n+1$. Therefore, case (ii) follows.

  3. Case(iii)

If $d_{1}\nmid d$ and $s$ is even. From (1) in Proposition 3.6, we always have
$v(L(\chi_{d},-1))\geq s+1$. Thus we get $v(L^{(D)}(\chi_{d},-1))\geq n+1$.

  4. Case(iv)

If $d_{1}\nmid d$ and $s$ is odd. From (2) in Proposition 3.6, we have
$v(L(\chi_{d},-1))=s$. Since $p_{1}\mid d^{\prime}$, for every $p\mid
d^{\prime}$, we can easily check that $\chi_{d}(p)=-1$ or $1$ according as
$p_{2}\mid d$ or $p_{2}\nmid d$ and $p=p_{1}$. So, if $p_{2}\mid d$, we get
$v\left(\prod_{p\mid d^{\prime}}(1-\chi_{d}(p)p)\right)=n-s+1$, thus we must
have $v(L^{(D)}(\chi_{d},-1))\geq n+1$. For the remaining case when
$p_{2}\nmid d$, note that $v\left(\prod_{p\mid
d^{\prime}}(1-\chi_{d}(p)p)\right)=n-s$, we get $v(L^{(D)}(\chi_{d},-1))=n$.
However, the total number of $d$’s such that $d_{1}p_{2}$ does not divide $d$
is even (since $n\geq 4$), thus the sum of these $L^{(D)}(\chi_{d},-1)$ must
have $2$-adic valuation at least $n+1$.

Now, by noting the equality (3.15), we have $v(L(\chi_{D},-1))\geq n+1$, thus
(2c) follows and the proof for the Theorem is done. ∎

##  4\. Structure for $2$-primary part of $K_{2}{\mathcal{O}}_{F}$

Let $K={\mathbb{Q}}(\sqrt{d})$ be a quadratic number field and denote by
$\mathcal{C}_{K}$ the ideal class group of $K$. In this section, we first
collect some standard results on $2$-rank and $4$-rank for $\mathcal{C}_{K}$
and $K_{2}{\mathcal{O}}_{K}$. Then we give applications of results in section
§3 to determine the $2$-primary structure of $K_{2}{\mathcal{O}}_{F}$ for real
quadratic number fields $F$.

Let us denote by $D_{K}$ the discriminant of $K$ and $t$ the number of prime
divisors of $D_{K}$. Let $\left(\frac{\cdot}{\ell}\right)$ be the Legendre
symbol if $\ell$ is an odd prime and $\left(\frac{\cdot}{2}\right)$ the
Kronecker symbol. Let $p^{*}=\left(\frac{-1}{p}\right)p$ if $p$ is an odd
prime and $2^{*}=\pm 4$ or $\pm 8$ such that $D_{K}=\prod^{t}_{i=1}p^{*}_{i}$.
Put $r_{ij}=\frac{1}{2}\left(1-\left(\frac{p^{*}_{i}}{p_{j}}\right)\right)$,
where $1\leq i,j\leq t$ and $i\neq j$. Set $r_{kk}=\sum_{i\neq k}r_{ik}$. We
make the convention that $p_{1}=2$ if $2\mid D_{K}$. We then define the Redei
matrix $R_{K}$ associated to $K$ to be $R_{F}=(r_{ij})$. Here, the matrix
$R_{F}$ is viewed as a $t\times t$-matrix over the finite field
${\mathbb{F}}_{2}$ with $2$-elements. Denote by $r(R_{K})$ the rank of the
matrix $R_{F}$ over ${\mathbb{F}}_{2}$. Recall that the narrow class group
$\mathcal{C}^{+}_{K}$ of $K$ is the quotient of the group of fractional
${\mathcal{O}}_{K}$-ideals by the subgroup of principal ideals
$(x)=x{\mathcal{O}}_{K}$ with generator of positive norm $N(x)$. Here, we
write $N=N_{K}$ for the norm map from $K$ to ${\mathbb{Q}}$.

######  Proposition 4.1.

Keep the notations as above. Then

  1. (1)

$r_{2}(\mathcal{C}^{+}_{K})=t-1$.

  2. (2)

If $K$ is a real quadratic field and $r_{4}(\mathcal{C}_{K})=0$, then
$r_{2}(\mathcal{C}_{K})=t-1$ (resp. $t-2$) if $-1\in N(K^{\times})$ (resp.
$-1\notin N(K^{\times})$).

  3. (3)

$r_{4}(\mathcal{C}^{+}_{K})=t-1-r(R_{K})$. In particular, if the $4$-rank of
$\mathcal{C}^{+}_{K}$ is zero, then so is the ideal class group
$\mathcal{C}_{K}$.

###### Proof.

All the assertions are well-known. One can find a proof in [11, Theorems (2.1)
and (3.1)]. The in particular part in (3) can be shown in the following way.
Write $\mathfrak{P}$ (resp. $\mathfrak{P}^{+}$) for the group generated by
principal ideals (resp. principal ideals with positive norm generate). Then we
have the following exact sequence

$0\to\mathfrak{P}/\mathfrak{P^{+}}\to\mathcal{C}^{+}_{K}\to\mathcal{C}_{K}\to
0.$

Since $\mathfrak{P}/\mathfrak{P^{+}}$ is annihilated by $2$, we see that the
in particular part in (3) follows. ∎

We now write $F$ for a real quadratic number field with discriminant $D$ and
put $E={\mathbb{Q}}(\sqrt{-D})$. Let $S$ be the set of primes in $F$ lying
above $2$. We denote by ${\mathcal{O}}_{F,S}$ the ring of $S$-integers in $F$
and put $\mathcal{C}_{F,S}$ the ideal class group of ${\mathcal{O}}_{F,S}$. We
omit the proof for the below Proposition and one can find a proof in [2] and
[17] (see also [8]).

######  Proposition 4.2.

Keep the notions as above. Denote by $s=\\#(S)$, then we have

  1. (1)

$r_{2}(K_{2}{\mathcal{O}}_{F})=1+r_{2}(\mathcal{C}_{F,S})+s$.

  2. (2)

$\left|r_{4}(K_{2}{\mathcal{O}}_{F})-r_{4}(\mathcal{C}_{E})\right|\leq 1$.

Now for a finite abelian group $\mathfrak{A}$, we recall that
$\mathfrak{A}(2)$ is the $2$-primary subgroup of $\mathfrak{A}$. The following
two propositions prove the Theorem 1.2 in the introduction.

######  Proposition 4.3.

Assume that $D=p_{1}\cdots p_{n}$ satisfying $p_{i}$ are primes such that
$p_{i}\equiv 5\ \mathrm{mod}\ 8$ and $\left(\frac{p_{i}}{p_{j}}\right)=-1$ for
$1\leq i<j\leq n$. If $n$ is odd, then we have

$K_{2}{\mathcal{O}}_{F}(2)\simeq({\mathbb{Z}}/2{\mathbb{Z}})^{n+1}.$

###### Proof.

Since $n$ is odd, so $D\equiv 5\ \mathrm{mod}\ 8$. Then $2$ is inert in $F$.
Therefore, $s=1$ and $\mathcal{C}_{F,S}=\mathcal{C}_{F}$. We claim that $-1\in
N_{F}(F^{\times})$, in fact, by modulo every $p\mid D$ and noting that
$p\equiv 1\ \mathrm{mod}\ 4$, there exists integers $x,y,z$ not all zero
satisfying

$x^{2}-y^{2}D\equiv-z^{2}\ \mathrm{mod}\ p.$

Note also that $x^{2}-y^{2}D=-z^{2}$ has solutions in ${\mathbb{R}}$. Thus by
Legendre’s theorem (see [9, Lemma (2.1)]), $x^{2}-y^{2}D=-z^{2}$ has
nontrivial solutions in ${\mathbb{Z}}$. Therefore, the claim follows. By
Proposition 4.1, we get $r_{2}(\mathcal{C}_{F})=n-1$. Note Proposition 4.2,
then $r_{2}(K_{2}{\mathcal{O}}_{F})=n+1$. Now the Proposition follows from (2)
in Proposition 3.6 and Corollary 2.3. ∎

######  Proposition 4.4.

Let $F={\mathbb{Q}}(\sqrt{D})$ with discriminant $D=4\ell_{1}\cdots\ell_{n}$
satisfying the prime divisors $\ell_{1}\equiv 3\ \mathrm{mod}\ 8$ and
$\ell_{i}\equiv 5\ \mathrm{mod}\ 8\,(2\leq i\leq n)$. Assume that $n$ is odd
and $\left(\frac{\ell_{i}}{\ell_{j}}\right)=-1$ for $1\leq i<j\leq n$. Then
Then $K_{2}{\mathcal{O}}_{F}(2)$ is isomorphic to
$({\mathbb{Z}}/2{\mathbb{Z}})^{n-1}\times({\mathbb{Z}}/4{\mathbb{Z}})$ (resp.
$({\mathbb{Z}}/2{\mathbb{Z}})^{2}$) if $n\geq 3$ (resp. $n=1$).

###### Proof.

The case for $n=1$ is easy, we left to the reader. We now assume that $n\geq
3$. Since $\ell_{1}\equiv 3\ \mathrm{mod}\ 4$, by the same argument as in
Proposition 4.3, we have $-1\notin N(F^{\times})$. Thus from Proposition 4.1
we see that $r_{2}(\mathcal{C}_{F})=n-1$. Now, writing $D=p_{1}^{*}\cdots
p_{t}^{*}$ and recalling the convention that $p^{*}_{1}=-4$, we know the Redei
matrix $R_{F}$ is given as the following block matrix

$\left(\begin{array}[]{cc}A_{2}&\alpha_{1}\\\ \beta_{1}&B_{n-1}\\\
\end{array}\right),$

where $A_{2}$ (resp. $\beta_{1}$) is a $2\times 2$ (resp. $(n-1)\times 2$)
matrix with all entries equal to $1$, $B_{n-1}$ is a $(n-1)\times(n-1)$ matrix
with all non-diagonal entries equal to $1$ and all diagonal entries equal to
$0$, and $\alpha_{1}$ is a $2\times(n-1)$ matrix with all elements in the
first (resp. second) row equal to $0$ (resp. $1$). By applying an elementary
row operation to $R_{F}$, we see that $r(R_{F})=n$. Therefore, from
Proposition 4.1, we get $r_{4}(\mathcal{C}_{F})=0$. Note that $2$ is ramified
in $F$, so $s=1$. One can easily show that the unique prime $\mathfrak{p}$ in
$F$ lying above $2$ is not principal. So the ideal class of $\mathfrak{p}$
generates a subgroup in $\mathcal{C}_{F}$ isomorphic to
${\mathbb{Z}}/2{\mathbb{Z}}$. Therefore, we have
$r_{2}(\mathcal{C}_{F,S})=n-2$. From Proposition 4.2, we get
$r_{2}(K_{2}{\mathcal{O}}_{F})=n$. To get the $4$-rank for
$K_{2}{\mathcal{O}}_{F}$, we must study the $4$-rank for
$E={\mathbb{Q}}(\sqrt{-D})$. By Proposition 4.1, $r_{2}(\mathcal{C}_{E})=n-1$
and the Redei matrix $R_{E}$ is given by $B_{n}$. since $n$ is odd, we see
that $r(B_{n})=n-1$. Therefore, $r_{4}(\mathcal{C}_{E})=0$ and from
Proposition 4.2, we only have $r_{4}(K_{2}{\mathcal{O}}_{F})\leq 1$. On the
other hand, by (1) in Theorem 3.8 and Corollary 2.3, we have

$v(\\#(K_{2}{\mathcal{O}}_{F}))=n+1.$

Then by combining $r_{2}(K_{2}{\mathcal{O}}_{F})=n$ and
$r_{4}(K_{2}{\mathcal{O}}_{F})\leq 1$, we must have
$r_{4}(K_{2}{\mathcal{O}}_{F})=1$, $r_{8}(K_{2}{\mathcal{O}}_{F})=0$ and the
proposition follows. ∎

Now, we can prove the Theorem 1.1. It is the application of (2c) in
Proposition 3.8 mentioned in section §3.

###### Proof of Theorem 1.1.

The heart part of the proof is (2c) in Theorem 3.8, which we have already
proven. So to complete the proof, we only need to show that
$r_{4}(K_{2}{\mathcal{O}}_{F})\leq 1$ and $r_{2}(K_{2}{\mathcal{O}}_{F})=n$.
In fact, from (2c) in Theorem 3.8 and Corollary 2.3, then the isomorphism in
the Theorem follows.

Since $\ell_{1}\equiv 3\ \mathrm{mod}\ 8$ we know that $-1\notin
N(F^{\times})$. Thus from Proposition 4.1, we have
$r_{2}(\mathcal{C}_{F})=n-1$. The Redei matrix $R_{F}$ is given as

$R_{F}=\left(\begin{array}[]{cc}C&\gamma_{1}\\\ \gamma_{2}&B_{n-2}\\\
\end{array}\right)\qquad C\in M_{3}({\mathbb{F}}_{2}),$

where $B_{n-2}$ is the same matrix given in Proposition 4.4, and
$C,\gamma_{i}(i=1,2)$ can be explicitly determined. Here, we should note that
the sum of all elements in the each column is zero. We can show that $B_{n-2}$
is of full rank, and the rank of $R_{F}$ is $n$. Therefore, from Proposition
4.1, we have $r_{4}(\mathcal{C}_{F})=0$. Note that the ramified prime
$\mathfrak{p}$ of $F$ lying above $2$ is not principal. Therefore,
$r_{2}(\mathcal{C}_{F,S})=n-2$ and $s=1$. So we have
$r_{2}(K_{2}{\mathcal{O}}_{F})=n$ by Proposition 4.2. To show that
$r_{4}(K_{2}{\mathcal{O}}_{F})\leq 1$, we consider the field
$E={\mathbb{Q}}(\sqrt{-D})$, an easy calculation using Proposition 4.1 shows
that $r_{2}(\mathcal{C}_{E})=n-1$, $r(R_{E})=n-1$ and
$r_{4}(\mathcal{C}_{E})=0$. Thus, by Proposition 4.2 we must have
$r_{4}(K_{2}{\mathcal{O}}_{E})\leq 1$. The theorem now follows. ∎

In the final part of this section, we give some numerical data for the case
$n=4$ in the following table. The data shows that $\delta$ is always equal to
$3$, however, at present we can not prove it theoretically.

$\frac{D}{4}$ | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$ | $L(\chi_{D},-1)$ | $\delta$
---|---|---|---|---|---|---
7215 | 3 | 5 | 13 | 37 | 240480 | 3
26455 | 11 | 13 | 5 | 37 | 1997920 | 3
77415 | 3 | 5 | 13 | 397 | 8824224 | 3
119535 | 3 | 5 | 13 | 613 | 16692000 | 3
142935 | 3 | 5 | 13 | 733 | 22829088 | 3
153735 | 3 | 5 | 37 | 277 | 25086816 | 3
166335 | 3 | 5 | 13 | 853 | 28304352 | 3
171015 | 3 | 5 | 13 | 877 | 28115040 | 3
196359 | 3 | 29 | 37 | 61 | 37791648 | 3
226655 | 11 | 13 | 5 | 317 | 40287584 | 3
241215 | 3 | 5 | 13 | 1237 | 48586080 | 3
243295 | 19 | 13 | 5 | 197 | 53437792 | 3
257335 | 107 | 5 | 13 | 37 | 60717792 | 3
283855 | 11 | 13 | 5 | 397 | 67222496 | 3
311415 | 3 | 5 | 13 | 1597 | 70325856 | 3
315055 | 131 | 37 | 5 | 13 | 79864160 | 3
420135 | 3 | 5 | 37 | 757 | 113889888 | 3
430495 | 179 | 37 | 5 | 13 | 130245856 | 3
447135 | 3 | 5 | 13 | 2293 | 117673248 | 3
473415 | 3 | 5 | 37 | 853 | 128212896 | 3
475215 | 3 | 5 | 13 | 2437 | 131335968 | 3
490295 | 19 | 13 | 5 | 397 | 129126880 | 3
504295 | 11 | 173 | 5 | 53 | 167198304 | 3
545415 | 3 | 5 | 13 | 2797 | 160048800 | 3
550615 | 43 | 5 | 13 | 197 | 188399904 | 3
552695 | 11 | 13 | 5 | 773 | 147595872 | 3
553335 | 3 | 5 | 37 | 997 | 164792736 | 3
563695 | 11 | 277 | 5 | 37 | 187614944 | 3
568815 | 3 | 5 | 13 | 2917 | 177955488 | 3
592215 | 3 | 5 | 13 | 3037 | 186476448 | 3
603655 | 251 | 37 | 5 | 13 | 209029792 | 3
606615 | 3 | 5 | 37 | 1093 | 199213344 | 3
633399 | 3 | 149 | 13 | 109 | 216478368 | 3
657735 | 3 | 5 | 13 | 3373 | 220949088 | 3
665223 | 3 | 461 | 13 | 37 | 209098272 | 3
673215 | 3 | 5 | 37 | 1213 | 233606880 | 3
685815 | 3 | 5 | 13 | 3517 | 237620448 | 3
687895 | 19 | 13 | 5 | 557 | 251644512 | 3
727935 | 3 | 5 | 13 | 3733 | 245403744 | 3
751335 | 3 | 5 | 13 | 3853 | 264797280 | 3
755495 | 59 | 13 | 5 | 197 | 237803552 | 3
757055 | 19 | 13 | 5 | 613 | 232180384 | 3
790495 | 19 | 53 | 5 | 157 | 312845408 | 3
798135 | 3 | 5 | 13 | 4093 | 292536288 | 3
803751 | 3 | 557 | 13 | 37 | 315036576 | 3
807455 | 11 | 277 | 5 | 53 | 258772704 | 3
818935 | 43 | 5 | 13 | 293 | 328030688 | 3
833199 | 3 | 29 | 61 | 157 | 319335264 | 3
849615 | 3 | 5 | 13 | 4357 | 321461856 | 3
878415 | 3 | 5 | 157 | 373 | 324090528 | 3
884455 | 11 | 13 | 5 | 1237 | 376589472 | 3
886015 | 43 | 5 | 13 | 317 | 395428320 | 3
886335 | 3 | 5 | 37 | 1597 | 339119712 | 3
896415 | 3 | 5 | 13 | 4597 | 353689056 | 3
905255 | 19 | 13 | 5 | 733 | 316252704 | 3
911495 | 379 | 13 | 5 | 37 | 319484960 | 3
934935 | 3 | 5 | 157 | 397 | 377527776 | 3
961935 | 3 | 5 | 13 | 4933 | 386463456 | 3
973655 | 19 | 37 | 5 | 277 | 357620256 | 3
981695 | 11 | 13 | 5 | 1373 | 364695008 | 3
990015 | 3 | 5 | 13 | 5077 | 394553376 | 3

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Li-Tong Deng,
Department of Mathematical Sciences,
Tsinghua University, 100084
Beijing, China.
<EMAIL_ADDRESS>

Yong-Xiong Li,
Yau Mathematical Sciences Center,
Tsinghua University, 100084
Beijing, China.
<EMAIL_ADDRESS>