5(14,1) This is the preliminary version. Final version has been published in
The 15th Asia Information Retrieval Societies Conference proceeding (2019) and
can be obtained from
https://link.springer.com/chapter/10.1007/978-3-030-42835-8_6

11institutetext: Department of Systems Engineering and Engineering Management,
The Chinese University of Hong Kong,
Shatin, Hong Kong
11email: {olatunji,lixin<EMAIL_ADDRESS>

# Context-aware Helpfulness Prediction for Online Product Reviews††thanks: The
work described in this paper is substantially supported by a grant from the
Research Grant Council of the Hong Kong Special Administrative Region, China
(Project Code: 14204418).

Iyiola E. Olatunji  Xin Li  Wai Lam

###### Abstract

Modeling and prediction of review helpfulness has become more predominant due
to proliferation of e-commerce websites and online shops. Since the
functionality of a product cannot be tested before buying, people often rely
on different kinds of user reviews to decide whether or not to buy a product.
However, quality reviews might be buried deep in the heap of a large amount of
reviews. Therefore, recommending reviews to customers based on the review
quality is of the essence. Since there is no direct indication of review
quality, most reviews use the information that “X out of Y” users found the
review helpful for obtaining the review quality. However, this approach
undermines helpfulness prediction because not all reviews have statistically
abundant votes. In this paper, we propose a neural deep learning model that
predicts the helpfulness score of a review. This model is based on
convolutional neural network (CNN) and a context-aware encoding mechanism
which can directly capture relationships between words irrespective of their
distance in a long sequence. We validated our model on human annotated dataset
and the result shows that our model significantly outperforms existing models
for helpfulness prediction.

###### Keywords:

Helpfulness prediction context-aware Product review.

##  1 Introduction

Reviews have become an integral part of user’s experience when shopping
online. This trend makes product reviews an invaluable asset because they help
customers make purchasing decision, consequently, driving sales [7]. Due to
the enormous amount of reviews, it is important to analyze review quality and
to present useful reviews to potential customers. The quality of a review can
vary from a well-detailed opinion and argument, to excessive appraisal, to
spam. Therefore, predicting the helpfulness of a review involves automatic
detection of the influence the review will have on a customer for making
purchasing decision. Such reviews should be informative and self-contained
[6].

Votes: [1, 8]
---
HS: 0.13
HAS: 0.85
I received an updated charger as well as the updated replacement power supply
due to the recall. It looks as if this has solved all of the problems. I have
been using this charger for some time now with no problems. No more heat
issues and battery charging is quick and accurate. I can now recommend this
charger with no problem. I use this charger several times a week and much
prefer it over the standard wall type chargers. I primarily use Enelope
batteries with this charger.
Figure 1:  Example of a review text with helpfulness score. HS = helpfulness
score based on “X of Y approach” while HAS = human annotated helpfulness
score.

Review helpfulness prediction have been studied using arguments [12], aspects
(ASP) [25], structural (STR) and unigram (UGR) features [26]. Also, semantic
features such as linguistic inquiry and word count (LIWC), general inquirer
(INQ) [26], and Geneva affect label coder (GALC) [14] have been used to
determine review helpfulness. However, using handcrafted features is laborious
and expensive due to manual feature engineering and data annotation. Recently,
convolutional neural networks (CNNs) [10], more specifically, the character-
based CNN [5] has been applied to review helpfulness prediction and has shown
to outperform handcrafted features. However, it does not fully capture the
semantic representation of the review since different reviewers have different
perspective, writing style and the reviewer’s language may affect the meaning
of the review. That is, the choice of word of an experienced reviewer differs
from that of a new reviewer. Therefore, modelling dependency between words is
important.

Recent works on helpfulness prediction use the “X of Y approach” i.e. if X out
of Y users votes that a review is helpful, then the helpfulness score of the
review is X/Y. However, such simple method is not effective as shown in Figure
1. The helpfulness score (HS = 0.13) implies that the review is unhelpful.
However, this is not the case as the review text entails user experience and
provides necessary information to draw a reasonable conclusion (self-
contained). Hence, it is clear that the review is of high quality (helpful) as
pointed out by human annotators (HAS = 0.85). This observation demonstrates
that “X of Y” helpfulness score may not strongly correlate to review quality
[26] which undermines the effectiveness of the prediction output.

Similarly, prior methods assume that reviews are coherent [26][5]. However,
this is not the case in most reviews because of certain content divergence
features such as sentiment divergence (opinion polarity of products compared
to that of review) embedded in reviews. To address these issues, we propose a
model that predicts the helpfulness score of a review by considering the
context in which words are used in relation to the entire review. We aim to
understand the internal structure of the review text.

In order to learn the internal structure of review text, we learn dependencies
between words by taking into account the positional encoding of each word and
employing the self-attention mechanism to cater for the length of longer
sequences. We further encode our learned representation into CNN to produce an
output sequence representation.

In our experiment, our framework outperforms existing state-of-the-art methods
for helpfulness prediction on the Amazon review dataset. We conducted detailed
experiments to quantitatively evaluate the effectiveness of each designed
component of our model. We validated our model on the human annotated data.
The code is available.

##  2 Related Works

Previous works on helpfulness prediction can be categorized broadly into three
categories (a) score regression (prediction a helpfulness score between 0 and
1) (b) classification (classifying a review as either helpful or not helpful)
(c) ranking (ordering reviews based on their helpfulness score). In this paper
we define the problem of helpfulness prediction as a regression task. Most
studies tend to focus on extracting specific features (handcrafted features)
from review text.

Semantic features such as LIWC (linguistic inquiry and word count), general
inquirer (INQ) [26], and GALC (Geneva affect label coder) [14] were used to
extract meaning from the text to determine the helpfulness of the review.
Extracting argument features from review text has shown to outperform semantic
features [12] since it provides a more detailed information about the product.
Structural (STR) and unigram (UGR) features [26] have also been exploited.

Content-based features such as review text and star rating and context-based
features such as reviewer/user information are used for helpfulness prediction
task. Content-based features are features that can be obtained from the
reviews. They include length of review, review readability, number of words in
review text, word-category features and content divergence features. Context-
based features are features that can be derived outside the review. This
includes reviewer features (profile) and features to capture similarities
between users and reviews (user-reviewer Idiosyncrasy). Other metadata such as
the probability of a review and its sentences being subjective have been
successfully used as features [19][9][17][20][22][8]. Since review text are
mostly subjective, it is a more complicated task to model all contributing
features for helpfulness prediction. Thus, using handcrafted features has
limited capabilities and laborious due to manual data annotation.

Several methods have been applied to helpfulness prediction task including
support vector regression [9, 29, 26], probabilistic matrix factorization
[23], linear regression [13], extended tensor factorization models [16], HMM-
LDA based model [18] and multi-layer neural networks [11]. These methods
allows the integration of robust constraints into the learning process and
this in turn has improved prediction result. Recently, convolutional neural
network (CNN) has shown significant improvement over existing methods by
achieving the state-of-the-art performance [10][28]. CNNs automatically
extract deep feature from raw text. This capability can alleviate manual
selection of features. Furthermore, adding more levels of abstraction as in
character-level representations has further improved prediction results over
vanilla CNN. Embedding-gated CNN [3] and multi-domain gated CNN [4] are recent
methods for predicting helpfulness prediction.

Moreover, attention mechanism has also been employed to CNN such as the ABCNN
for modelling sentence pair [27]. Several tasks including textual entailment,
sentence representation, machine translation, and abstractive summarization
have applied self-attention mechanism and has shown significant result [1].
However, employing self-attention for developing context-aware encoding
mechanism has not been applied to helpfulness prediction. Using self-attention
mechanism on review text is quite intuitive because even the same word can
have different meaning based on the context in which it is being used.

Figure 2:  Proposed context-aware helpfulness prediction model

##  3 Model

We model the problem of predicting the helpfulness score of a review as a
regression problem. Precisely, given a sequence of review, we predict the
helpfulness score based on the review text. As shown in Figure 2, the sequence
of words in the review are embedded and concatenated with their respective
positional encoding to form the input features. These input features are
processed by a self-attention block for generating context-aware
representations for all tokens. Then such representation will be fed into a
convolutional neural network (CNN) which computes a vector representation of
the entire sequence (i.e the dependency between the token and the entire
sequence) and then we use a regression layer for predicting the helpfulness
score.

###  3.1 Context-aware Encoding

The context-aware component of our model consists of positional encoding and
self-attention mechanism. We augment the word embedding with positional
encoding vectors to learn text representation while taking the absolute
position of words into consideration.

Let $X=\left(x_{1},x_{2},...,x_{n}\right)$ be a review consisting of a
sequence of words. We map each word $x_{i}$ in a review $X$ to a
$l$-dimensional (word embedding) word vector $e_{x_{i}}$ stored in an
embedding matrix $\textbf{E}\in{\rm I\\!R}^{V\times l}$ where $V$ is the
vocabulary size. We initialize E with pre-trained vectors from GloVe [21] and
set the embedding dimension l to 100. A review is therefore represented as
$Y=\left(e_{x_{1}},e_{x_{2}},...,e_{x_{n}}\right)$. Since the above
representation learns only the meaning of each word, we need the position of
each word for understanding the context of each word. Let
$S=\left({s_{1}},{s_{2}},...,{s_{n}}\right)$ be the position of each word in a
sentence. Inspired by Vaswani et al [24], the positional encoding, denoted as
$PE\in{\rm I\\!R}^{n\times l}$, is a 2D constant matrix with position specific
values calculated by the sine and cosine functions below:

$\begin{split}PE\left({s_{k}},2i\right)=sin\left({s_{k}}/j^{2i/l}\right)\\\
PE\left({s_{k}},2i+1\right)=cos\left({s_{k}}/j^{2i/l}\right)\\\ \end{split}$
(1)

where $i$ is the position along the embedding vector $e$ and $j$ is a constant
representing the distance between successive peaks (or successive troughs) of
cosine and sine function. This constant is between $2\pi$ and 10000. Based on
a tuning process, we set $j$ to 1000. The sequence
$\overline{P}=\left(\overline{P_{1}},\overline{P_{2}},...,\overline{P_{n}}\right)$,
where $\overline{P_{s}}=PE(S)$ defined as the row vector corresponding to $S$
in the matrix $PE$ as in Equation 1.

The final representation $e^{\prime}$ is obtained by adding the word embedding
to the relative position values of each word in the sequence. Therefore,
$e^{\prime}=\left(e_{1}^{\prime},e_{2}^{\prime},...,e_{n}^{\prime}\right)$
where $e_{j}^{\prime}=\left(\overline{P_{j}}+e_{x_{j}}\right)$

Self-attention is employed in our model. Given an input $e^{\prime}$, self-
attention involves applying the attention mechanism on each
$\mathit{e_{i}^{\prime}}$ using $\mathit{e_{i}^{\prime}}$ query vector and
key-value vector for all other positions. The reason for using the self-
attention mechanism is to capture the internal structure of the sentence by
learning dependencies between words. The scaled dot-product attention is used
which allows faster computation instead of the standard additive attention
mechanism [2]. It computes the attention scores by:

$Attention(Q,K,V)=softmax(\frac{(Q)(K)^{T}}{\sqrt{l}})V$ (2)

where $\mathit{Q,K}$ and $\mathit{V}$ are the query, key and value matrices
respectively. The above equation implies that we divide the dot product of
queries with all keys by the key vector dimension to obtain the weight on the
values.

We used multi-head attention similar to [24]. The multi-head attention
mechanism maps the input vector $\mathit{e^{\prime}}$ to queries, keys and
values matrices by using different linear projections. This strategy allows
the self-attention mechanism to be applied $\mathit{h}$ times. Then all the
vectors produced by different heads are concatenated together to form a single
vector.

Concisely, our model captures context for a sequence as follows: We obtain the
relative position of the tokens in the sequence from Equation 1. The self
attention block, learns the context by relating or mapping different positions
$s_{1},s_{2},...,s_{n}$ of $\overline{P}$ via Equation 1 so as to compute a
single encoding representation of the sequence. By employing the multi-head
attention, our model can attend to words from different encoding
representation at different positions. We set heads h to 2. The query, key and
value used in the self-attention block are obtained from the output of the
previous layer of the context-aware encoding block. This design allows every
position in the context-aware encoding block to attend over all positions of
the input sequence. To consider positional information in longer sentences not
observed in training, we apply the sinusoidal position encoding to the input
embedding.

###  3.2 Convolutional Neural Network (CNN)

The output of the context-aware encoding representation $e^{\prime}$ is fed
into the CNN to obtain new feature representations for making predictions. We
employ multiple filters $f\in$ [1,2,3]. This method is similar to learning
uni-gram, bi-gram and tri-gram representations respectively. Specifically, for
each filter, we obtain an hidden representation
$r=Pool(Conv(e^{\prime},filterSize(f,l,c)))$ where $c$ is the channel size,
$Pool$ is the pooling operation and $Conv(.)$ is the convolution operation. In
our experiment, we use max pooling and average pooling. The final
representation $h$ is obtained by concatenating all hidden representation.
i.e., $h=[r_{1},r_{2},r_{3}]$. These features are then passed to the
regression layer to produce the helpfulness scores.

##  4 Dataset

We used two datasets for our experiments. The first dataset called D1 is
constructed from the Amazon product review dataset [15]. This dataset consists
of over 142 million reviews from Amazon between 1996 to 2014. We used a subset
of 21,776,678 reviews from 5 categories, namely; health, electronics, home,
outdoor and phone. We selected reviews with over 5 votes as done in [5, 26].
The statistics of the dataset used are shown in table 1. We removed reviews
having less than 7 words for experimenting with different filter sizes. Note
that this is the largest dataset used for helpfulness prediction task.

The second dataset called D2 is the human annotated dataset from Yang et al.
[26]. This dataset consists of 400 reviews with 100 reviews selected randomly
from four product categories (outdoor, electronics, home and books). The
reason for using the human annotated dataset is to verify that our model truly
learns deep semantics features of review text. Therefore, our model was not
trained on the human annotated dataset but only used for evaluating the
effectiveness of our model. We used only three categories for our experiment
and performed cross-domain experiment on categories not in D2.

Table 1:  Data statistic of Amazon reviews from 5 different categories. We used Health instead of Watches as done by Chen et al. [5] because it is excluded from newly published Amazon dataset.  Product category |  # of reviews with 5+ votes |  Total # of reviews
---|---|---
Phone |  261,370 |  3,447,249
Outdoor |  491,008 |  3,268,695
Health |  550,297 |  2,982,326
Home |  749,564 |  4,253,926
Electronics |  1,310,513 |  7,824,482

##  5 Experimental Setup

Following the previous works [5, 26, 25], all experiments were evaluated using
correlation coefficients between the predicted helpfulness scores and the
ground truth scores. We split the dataset D1 into Train / Test / Validation
(70, 20, 10). We used the same baselines as state-of-the-art convolutional
model for helpfulness prediction [5] i.e. STR, UGR, LIWC, INQ [26], ASP [25].
CNN is the CNN model by [10] and C_CNN is the state-of-the-art character-based
CNN from [5]. We added two additional variants (S_Attn and S_Avg) to test
different components of our model. S_Attn involves using only self-attention
without CNN while S_Avg is self-attention with CNN using average pooling and
finally our model uses max pooling with context-aware encoding. We re-
implemented all baselines as well as C_CNN as described by [5] but excluded
the transfer learning part of their model since it is for tackling
insufficient data problem. We used RELU for non-linearity and set dropout rate
to 0.5 (for regularization). We used Adaptive moment estimation (Adam) as our
optimizer. The learning rate was set to 0.001 and $l$ to 100. We experimented
with different filter sizes and found that $f\in$ [1,2,3] produces the best
result. Also we tried using Recurrent Neural Network (RNN) such as LSTM and
BiLSTM but they performed worse than CNN.

Table 2:  Experimental result for the dataset D1  |  Phone |  Outdoor |  Health |  Home |  Electronics
---|---|---|---|---|---
STR |  0.136 |  0.210 |  0.295 |  0.210 |  0.288
UGR |  0.210 |  0.299 |  0.301 |  0.278 |  0.310
LIWC |  0.163 |  0.287 |  0.268 |  0.285 |  0.350
INQ |  0.182 |  0.324 |  0.310 |  0.291 |  0.358
ASP |  0.185 |  0.281 |  0.342 |  0.233 |  0.366
CNN |  0.221 |  0.392 |  0.331 |  0.347 |  0.411
C_CNN |  0.270 |  0.407 |  0.371 |  0.366 |  0.442
S_Attn |  0.219 |  0.371 |  0.349 |  0.358 |  0.436
S_Avg |  0.194 |  0.236 |  0.336 |  0.318 |  0.382
Ours |  0.373 |  0.461 |  0.428 |  0.402 |  0.475
Table 3:  Experimental result for the dataset D2. (Fusion_all = STR + UGR + LIWC + INQ)  |  Outdoor |  Home |  Electronics
---|---|---|---
Fusion_all |  0.417 |  0.596 |  0.461
CNN |  0.433 |  0.521 |  0.410
C_CNN |  0.605 |  0.592 |  0.479
Ours |  0.748 |  0.758 |  0.699
Table 4:  Cross-domain investigation  |  D1-Phone
D2-Home |  D1-Health
D2-Electronics
---|---|---
C_CNN |  0.389 |  0.436
Ours |  0.586 |  0.654

##  6 Results

As shown in Table 2, our context-aware encoding based model using max pooling
outperforms all handcrafted features and CNN-based models including C_CNN with
a large margin on D1. This is because by applying attention at different
positions of the word embedding, different information about word dependencies
are extracted which in turn handles context variation around the same word.
However, using self-attention alone (S_Attn) (table 2) performs poorly than
CNN as learning word dependencies alone is not sufficient for our task. We
further need to understand the internal structure of the review text. Since
self-attention can handle longer sequence length than CNN when modelling
dependencies, we resolve to capturing the dependencies using self-attention
and then encode the dependencies into a vector representation using CNN to
further extract the positional invariant features. Two variants are presented
using average pooling (S_Avg) and max pooling. S_Avg performs comparable to
handcrafted features probably due to its tendency of selecting tokens having
low attention scores. Our proposed model with max-pooling produces the best
result on D1 (Table 2) and significantly on D2 (Table 3) since it selects the
best representation with most attention. It implies that our model can capture
the dependency between tokens and the entire sequence. Likewise, our model
understands the internal structure of review and has a high correlation to
human score.

Since D2 does not include the Phone and Health category, we tested our
proposed model trained on the Phone and Health category from D1 on the Home
and Electronics category respectively on D2. Specifically, we used the
training data of the Phone category from D1 to train our proposed model and
used the data of the Home category from D2 for testing. Similarly, we used the
training data of the Health category from D1 to train our proposed model and
tested the model using the data from the Electronics category of D2.

As shown in Table 4, the result is quite surprising. This shows that our
proposed model can effectively learn cross domain features and is robust to
“out-of-vocabulary” (OOV) problem by predicting reasonable helpfulness score
having a high correlation to human score.

##  7 Conclusions

Predicting review helpfulness can substantially save a potential customer’s
time by presenting the most useful review. In this paper, we propose a
context-aware encoding based method that learns dependencies between words for
understanding the internal structure of the review.Experimental results on the
human annotated data shows that our model is a good estimator for predicting
the helpfulness of reviews and robust to the “out-of-vocabulary” (OOV)
problem. In the future, we aim to explore some learning to rank models to
effectively rank helpfulness score while incorporating some other factors that
may affect helpfulness prediction including the types of products.

## References

  * [1] Ambartsoumian, A., Popowich, F.: Self-Attention : A Better Building Block for Sentiment Analysis Neural Network Classifiers. In: Proceedings of the 9th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis. pp. 130–139 (2018)
  * [2] Bahdanau, D., Cho, K., Bengio, Y.: Neural machine translation by jointly learning to align and translate. In: Proc. of ICLR. pp. 1–15 (2015)
  * [3] Chen, C., Qiu, M., Yang, Y., Zhou, J., Huang, J., Li, X., Bao, F.S.: Review Helpfulness Prediction with Embedding-Gated CNN. arXiv (2018)
  * [4] Chen, C., Qiu, M., Yang, Y., Zhou, J., Huang, J., Li, X., Bao, F.S.: Multi-domain gated cnn for review helpfulness prediction. In: Proc. of WWW. pp. 2630–2636 (2019)
  * [5] Chen, C., Yang, Y., Zhou, J., Li, X., Bao, F.S.: Cross-Domain Review Helpfulness Prediction based on Convolutional Neural Networks with Auxiliary Domain Discriminators. In: Proc. of NAACL-HLT. pp. 602–607 (2018)
  * [6] Diaz, G.O., Ng, V.: Modeling and Prediction of Online Product Review Helpfulness : A Survey. In: Proc. of ACL. pp. 698–708 (2018)
  * [7] Duan, W., Gu, B., Whinston, A.B.: The dynamics of online word-of-mouth and product sales — An empirical investigation of the movie industry. Journal of Retailing 84, 233–242 (2008)
  * [8] Ghose, A., Ipeirotis, P.G.: Estimating the helpfulness and economic impact of product reviews: Mining text and reviewer characteristics. IEEE Transactions on Knowledge and Data Engineering 23(10), 1498–1512 (2011)
  * [9] Kim, S.M., Pantel, P., Chklovski, T., Pennacchiotti, M.: Automatically assessing review helpfulness. In: Proc. of EMNLP. pp. 423–430 (2006)
  * [10] Kim, Y.: Convolutional Neural Networks for Sentence Classification. In: Proc. of EMNLP. pp. 1746–1751 (2014)
  * [11] Lee, S., Choeh, J.Y.: Predicting the helpfulness of online reviews using multilayer perceptron neural networks. Expert Systems with Applications 41(6), 3041 – 3046 (2014)
  * [12] Liu, H., Gao, Y., Lv, P., Li, M., Geng, S., Li, M., Wang, H.: Using Argument-based Features to Predict and Analyse Review Helpfulness. In: Proc. of EMNLP. pp. 1358–1363 (2017)
  * [13] Lu, Y., Tsaparas, P., Ntoulas, A., Polanyi, L.: Exploiting social context for review quality prediction. In: In Proc. of WWW. pp. 691–700 (2010)
  * [14] Martin, L., Pu, P.: Prediction of Helpful Reviews Using Emotions Extraction. In: Proc. of AAAI. pp. 1551–1557 (2014)
  * [15] Mcauley, J., Targett, C., Hengel, A.V.D.: Image-based Recommendations on Styles and Substitutes. In: Proc. of SIGIR (2015)
  * [16] Moghaddam, S., Jamali, M., Ester, M.: Etf: Extended tensor factorization model for personalizing prediction of review helpfulness. In: Proceedings of the Fifth ACM International Conference on Web Search and Data Mining. pp. 163–172 (2012)
  * [17] Mudambi, S.M., Schuff, D.: Research note: What makes a helpful online review? a study of customer reviews on amazon.com. MIS Quarterly 34(1), 185–200 (2010)
  * [18] Mukherjee, S., Popat, K., Weikum, G.: Exploring latent semantic factors to find useful product reviews. In: Proceedings of the 2017 SIAM International Conference on Data Mining. pp. 480–488 (2017)
  * [19] Otterbacher, J.: ’helpfulness’ in online communities: A measure of message quality. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. pp. 955–964 (2009)
  * [20] Pan, Y., Zhang, J.Q.: Born unequal: A study of the helpfulness of user-generated product reviews. Journal of Retailing 87(4), 598 – 612 (2011)
  * [21] Pennington, J., Socher, R., Manning, C.D.: GloVe : Global Vectors for Word Representation. In: Proc. of EMNLP. pp. 1532–1543 (2014)
  * [22] Salehan, M., Kim, D.J.: Predicting the performance of online consumer reviews: A sentiment mining approach to big data analytics. Decision Support Systems 81, 30 – 40 (2016)
  * [23] Tang, J., Gao, H., Hu, X., Liu, H.: Context-aware review helpfulness rating prediction. In: Proceedings of the 7th ACM Conference on Recommender Systems (2013)
  * [24] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention Is All You Need. In: Proc. of NIPS (2017)
  * [25] Yang, Y., Chen, C., Bao, F.S.: Aspect-Based Helpfulness Prediction for Online Product Reviews. In: Proc. of International Conference on Tools with Artificial Intelligence (ICTAI). pp. 836–843 (2016). https://doi.org/10.1109/ICTAI.2016.0130
  * [26] Yang, Y., Yan, Y., Qiu, M., Bao, F.S.: Semantic Analysis and Helpfulness Prediction of Text for Online Product Reviews. In: Proc. of ACL-IJCNLP. pp. 38–44 (2015)
  * [27] Yin, W., Schutze, H., Xiang, B., Zhou, B.: ABCNN : Attention-Based Convolutional Neural Network for Modeling Sentence Pairs. Transactions of the Association for Computational Linguistics 4, 259–272 (2016)
  * [28] Zhang, X., Zhao, J., LeCun, Y.: Character-level convolutional networks for text classification. In: Proc. of NIPS. pp. 649–657 (2015)
  * [29] Zhang, Z., Varadarajan, B.: Utility scoring of product reviews. In: Proceedings of the 15th ACM International Conference on Information and Knowledge Management. pp. 51–57 (2006)


# Homotopy groups of spectra and $p$-adic $L$-functions over totally real
number fields

Guillem Sala Fernandez

###### Abstract.

The goal of this paper is to illustrate different approaches to understand
Euler characteristics in the setting of totally real commutative and non-
commutative Iwasawa theory. In addition to this, and in the spirit of [Hes18]
and [Mit05], we show how these objects interact with homotopy theoretic
invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic homology.

## Introduction

### Summary of the article and structure

The main motivation for this paper was Hesselholt’s expository note [Hes18].
In his article, he argues that if $p$ is an odd number and $F_{p}$ denotes the
$p$-completion of the homotopy fiber of the cyclotomic trace map
$\mathrm{tr}:K(\mathbf{Z})\to\mathrm{TC}(\mathbf{Z};p)$, then, among others,
but more interestingly to us:

  1. (1)

The value of the Kubota-Leopoldt $p$-adic $L$-function
$L_{p}(\mathbf{Q},\omega^{-2k},1+2k)$ is non-zero if and only if
$\pi_{4k+1}(F_{p})$ is zero, and

  2. (2)

If $L_{p}(\mathbf{Q},\omega^{-2k},1+2k)\neq 0$, then $\pi_{4k}(F_{p})$ and
$\pi_{4k-1}(F_{p})$ are finite and

$\left|L_{p}(\mathbf{Q},\omega^{-2k},1+2k)\right|_{p}=\frac{\pi_{4k-1}(F_{p})}{\pi_{4k}(F_{p})},$

where $|-|_{p}$ denotes the $p$-adic absolute value,

These results follow from combining the Iwasawa main conjecture for
$F=\mathbf{Q}$, together with some well-known results by, for example, Bayer-
Neukirch in [BN78] and Schneider in [Sch83], and more (homotopy) $K$-theoretic
results.

Our first goal – and the goal of this paper – was to find an alternative proof
of the result, as well as an extension of the result to the case of the
Deligne-Ribet $p$-adic $L$-function, that is, to the case where $F$ is a
totally real field. As it will be seen in §3, this was performed successfully,
thanks to results by Blumberg and Mandell in [BM15].

The second goal was to see if similar results could be reached in the totally
real and non-commutative setting. Unfortunately, it was not possible to find a
formula that related the (conjectural) Artin $p$-adic $L$-function with
homotopy-theoretic invariants like in the commutative case. It was possible,
however, to complete the results involving Euler characteristics in [BV05],
that is, a cohomological description of the $p$-adic norm for certain values
of the Artin $p$-adic $L$-function was found.

The third goal was to see if similar results held in other relevant number
theoretic settings, such as the case of imaginary quadratic number fields, or
even algebraic function fields, where a $K$-theoretic description of the
special values of Katz’s $p$-adic $L$-function has already been found. These
will be addressed in the second part of this series of two articles.

Finally, one of the tangential motivations of this paper has been to introduce
the number theorist to the field of higher algebra, and the higher algebraist
to some of the main topics in Iwasawa theory. For this same reason, a brief
but concise introduction to both fields is provided in sections §1 and §2.
Afterwards, §3 addresses the commutative and totally real case, and §4
addresses the non-commutative counterpart.

### Acknowledgments

The author would like to thank Lars Hesselholt, Jonas McCandless, Andrew
Blumberg, Dustin Clausen, Francesc Bars, Francesca Gatti, Armando Gutierrez,
Oscar Rivero, Jeanine Van Order, and specially his thesis supervisor Victor
Rotger for their help and comments on the article. In addition to this, the
project was funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No.
682152).

##  1\. Number theoretic background

###  1.1. $\mathbf{Z}_{p}^{r}$-extensions and the Iwasawa algebra

Let $F$ be a number field, and let $p$ be an odd prime. Recall that, given
$r\geq 1$, a $\mathbf{Z}_{p}^{r}$-extension of $F$ is a pro-finite abelian
field extension $F_{\infty}$ of $F$ so that
$\Gamma=\mathrm{Gal}(F_{\infty}|F)\simeq\mathbf{Z}_{p}^{r}$.

######  Example I.1.1.

Let $F=\mathbf{Q}$, and note that
$\mathrm{Gal}(\mathbf{Q}(\mu_{p^{n+1}})|\mathbf{Q})\simeq\Delta\times\Gamma_{n}$,
where $\Delta$ is a cyclic group of order $p-1$, and $\Gamma_{n}$ is cyclic of
order $p^{n}$. We can then let $\mathbf{Q}_{n}$ denote the fixed field of
$\mathbf{Q}(\mu_{p^{n+1}})$ by $\Delta$ so that the direct limit
$\mathbf{Q}^{\mathrm{cyc}}=\mathbf{Q}_{\infty}$ is a
$\mathbf{Z}_{p}$-extension of $\mathbf{Q}$. By the Kronecker-Weber theorem,
this is the unique $\mathbf{Z}_{p}$-extension of $\mathbf{Q}$. Furthermore,
given any number field $F$, we can construct the cyclotomic
$\mathbf{Z}_{p}$-extension of $F$ as
$F^{\mathrm{cyc}}=F\cdot\mathbf{Q}^{\mathrm{cyc}}$.

######  Example I.1.2.

Let $F=\mathcal{K}$ be an imaginary quadratic number field. By class field
theory, there exists a $\mathbf{Z}_{p}^{2}$-extension $\mathcal{K}_{\infty}$
of $\mathcal{K}$ sitting inside the maximal extension of $\mathcal{K}$ that is
unramified outside $p$, which is maximal among the
$\mathbf{Z}_{p}^{d}$-extensions of $\mathcal{K}$. The Galois group
$\mathrm{Gal}(\mathcal{K}|\mathbf{Q})$ acts on
$\Gamma=\mathrm{Gal}(\mathcal{K}_{\infty}|\mathcal{K})$ via conjugation, and
if we denote the only non-trivial automorphism of
$\mathrm{Gal}(\mathcal{K}|\mathbf{Q})$ by $\sigma$, its action gives us a
decomposition

$\Gamma=\Gamma_{+}\oplus\Gamma_{-}$

where $\sigma\cdot g=\sigma g\sigma^{-1}=g^{\pm 1}$, for each
$g\in\Gamma_{\pm}$. We then recover the cyclotomic $\mathbf{Z}_{p}$-extension
taking the fixed field $\mathcal{K}^{\mathrm{cyc}}$ of $\mathcal{K}_{\infty}$
by $\Gamma_{-}$ and obtain the anticyclotomic $\mathbf{Z}_{p}$-extension by
taking the fixed field of $\mathcal{K}_{\infty}$ by $\Gamma_{+}$, which we
denote by $\mathcal{K}^{\mathrm{acyc}}$. Alternatively, let us assume that $p$
splits in $\mathcal{O}_{\mathcal{K}}$ as $p=\wp_{+}\wp_{-}$, then we can
construct the $\mathbf{Z}_{p}$-extensions
$\mathcal{K}_{\pm}\subset\mathcal{K}_{\infty}$ of $\mathcal{K}$ which are
unramified outside $\wp_{\pm}$.

Now, let $\mathcal{O}$ be the valuation ring of some finite algebraic
extension of $\mathbf{Q}_{p}$. The main goal of Iwasawa theory is to study the
structure of modules over the Iwasawa $\mathcal{O}$-algebra
$\Lambda_{\mathcal{O}}(\Gamma)=\mathcal{O}\llbracket\Gamma\rrbracket$, which
is defined as

$\mathcal{O}\llbracket\Gamma\rrbracket=\lim_{U\unlhd\Gamma}\mathcal{O}[G/U],$

where the limit runs through all the open normal subgroups of $G$ with respect
to the restriction maps $\mathcal{O}[G/V]\to\mathcal{O}[G/U]$, for $V\leq U$.
Note that whenever the base ring $\mathcal{O}$ is understood, we will simply
denote $\Lambda(\Gamma)$, or even $\Lambda$.

Modules over the Iwasawa algebra are also known as Iwasawa modules, and the
next fundamental theorem aims at simplifying the description of such modules.

######  Theorem I.1.3.

(Iwasawa-Serre, [Sha16, Thm 2.3.9]) Let $G$ be a profinite group with
$G\simeq\mathbf{Z}_{p}^{r}$, and fix a generating set $\\{\gamma_{i}\\}_{1\leq
i\leq r}$ of $G$. Then, there exists a unique topological isomorphism
$\mathcal{O}\llbracket G\rrbracket\to\mathcal{O}\llbracket
T_{1},\dots,T_{r}\rrbracket$ sending $\gamma_{i}-1$ to $T_{i}$.

######  Remark I.1.4.

The previous theorem allows us to think of modules over the Iwasawa
$\mathcal{O}$-algebra in terms of modules over a formal power series ring. For
that reason, we will refer indistinctly to the Iwasawa algebra and its
associated formal power series ring, taking into account that this
identification carries the extra data of an isomorphism between both algebras.

###  1.2. The characteristic polynomial of an Iwasawa module

Now that we know how to think of modules over the Iwasawa algebra, we would
like to find invariants that simplify their study. To do so, let $\varpi$ be a
uniformizer for the valuation ring $\mathcal{O}$, and let $M$ be a finitely-
generated, torsion $\Lambda$-module. It is a classical result that $\Lambda$
is a noetherian Krull domain whose only height 1 primes are either those
generated by powers of $\varpi$ or by polynomials
$f\in\mathcal{O}[T_{1},\dots,T_{d}]$ that are coprime with $\varpi$, so
[Bou98, Theorems 4 & 5,§VII.4.4] can be applied to obtain the existence of a
map

(1.1) $\displaystyle M\to\bigoplus_{n\geq
0}\Lambda/\varpi^{\mu_{n}}\oplus\bigoplus_{m\geq
0}\Lambda/(f_{m}^{\lambda_{m}}),$

which is a quasi-isomorphism, meaning that it has finite kernel and cokernel.

######  Remark I.2.1.

Classically, the sum of the powers $\mu_{n}$ of $\varpi$ give an invariant of
$M$, the so-called $\mu$-invariant, which we denote by
$\mu_{\mathcal{O},\Gamma}(M)$, or simply by $\mu(M)$, whenever the base
Iwasawa algebra is understood. This invariant will vanish in the cases that
will appear throughout this paper, although this might not always be the case.

######  Definition I.2.2.

The characteristic ideal of $M$ is the product of ideals

(1.2)
$\displaystyle\mathrm{Ch}_{\Lambda}(M)=\varpi^{\mu(M)}\prod_{m=1}^{N}(f_{m}^{\lambda_{m}}).$

######  Remark I.2.3.

The choice of a generator for $\mathrm{Ch}_{\Lambda}(M)$ provides a
characteristic element for $M$, and we denote any such choice by
$\mathrm{ch}_{\Lambda}(M)$.

###  1.3. Twists of Galois modules

From now on, we denote the $p$-adic cyclotomic character of a number field $F$
by $\chi_{\mathrm{cyc},F}$, or simply by $\chi_{\mathrm{cyc}}$ whenever the
base field $F$ is understood. Recall this is the map
$\chi_{\mathrm{cyc}}:G_{F}\to\mathbf{Z}_{p}^{\times}$ defined by
$\sigma(\zeta)=\zeta^{\chi_{\mathrm{cyc}}(\sigma)}$ for all
$\zeta\in\mu_{p^{\infty}}$. One can then define the Tate module
$\mathbf{Z}_{p}(1)$ to be the $\mathbf{Z}_{p}[G_{F}]$-module whose action is
given by

$\sigma\cdot a=\chi_{\mathrm{cyc}}(\sigma)\cdot a,\qquad\forall\sigma\in
G_{F},\forall a\in\mathbf{Z}_{p}.$

More generally, note that the $n$-th twist $\mathbf{Z}_{p}(n)$ of the Tate
module is obtained by replacing $\chi_{\mathrm{cyc}}(\sigma)$ in the above
definition by $\chi_{\mathrm{cyc}}(\sigma)^{n}$, and with this, given any
$\mathbf{Z}_{p}[G_{F}]$-module $M$, one may define

$M(n)=M\otimes_{\mathbf{Z}_{p}}\mathbf{Z}_{p}(n),\qquad\sigma\cdot
m=\chi_{\mathrm{cyc}}(\sigma)^{n}(\sigma\cdot m),\qquad\forall\sigma\in
G_{F},\forall m\in M.$

######  Example I.3.1.

Given a number field $F$ and a choice of a compatible system $\zeta_{p^{n}}$
of primitive $p^{n}$-th roots of unity, one obtains isomorphisms of
$\mathbf{Z}_{p}[G_{F}]$-modules
$\mathbf{Z}_{p}(1)\simeq\varprojlim\mu_{p^{n}}$ and
$\mathbf{Q}_{p}/\mathbf{Z}_{p}(1)\simeq\mu_{p^{\infty}}$.

######  Remark I.3.2.

It is a well-known result that the cyclotomic character factors through
$\mathrm{Gal}(F(\mu_{p^{\infty}})|F)$. Thus, given that there is an
isomorphism $\mathrm{Gal}(F(\mu_{p^{\infty}})|F)\simeq\Delta\times\Gamma$,
where $\Gamma$ is the Galois group of the cyclotomic
$\mathbf{Z}_{p}$-extension from Example I.1.1, whenever we are given a module
$M$ over the Iwasawa algebra $\Lambda$, we can form $M(n)$ by twisting the
action of $\Gamma$ by the $\Gamma$-part of $\chi_{\mathrm{cyc}}$.

The “twist” provided by the cyclotomic character is the most commonly known,
although from the definition one can tell that this can be extended to more
general characters. For that reason, we provide the following definition.

######  Definition I.3.3.

Let $\chi:G_{F}\to\mathcal{O}^{\times}$ a continuous character, possibly of
infinite order, and let $\mathcal{O}(\chi)$ be the free $\mathcal{O}$-module
of rank 1 whose action of $G_{F}$ is given by $\chi$. Then, for any
$\mathcal{O}[G_{F}]$-module $M$, we define
$M(\chi)=M\otimes_{\mathcal{O}}\mathcal{O}(\chi)$.

######  Lemma I.3.4.

([Rub14, §IV, Lemma 1.2]) Let $M$ be a finitely generated torsion
$\Lambda$-module, and let $\chi:\Gamma\to\mathcal{O}^{\times}$ be a character.
Then, $M(\chi)$ is also a finitely generated torsion $\Lambda$-module.

###  1.4. Pontryagin Duality

Recall that one can define an endofunctor on the category of locally compact
abelian groups, which is usually denoted by $\mathrm{LCA}$, via the rule
$A\mapsto A^{\vee}:=\mathrm{Hom}_{\mathrm{cts}}(A,\mathbf{Q}/\mathbf{Z})$,
where the image is endowed with the compact-open toplogy. We refer to
$A^{\vee}$ as the Pontryagin dual of $A$. It is then a well-known result that
the contra-variant functor $(-)^{\vee}:\mathrm{LCA}\to\mathrm{LCA}$ produces
an equivalence of categories (cf. [Sha16, 2.5.2]).

######  Remark I.4.1.

Note that if $A$ is pro-$p$ or discrete $p$-torsion, then

$A^{\vee}=\mathrm{Hom}_{\mathrm{cts}}(A,\mathbf{Q}_{p}/\mathbf{Z}_{p}),$

and if $A$ is $\mathbf{Z}_{p}$-finitely generated, then since every
$\mathbf{Z}_{p}$-linear homomorphism is continuous, we have that

$A^{\vee}=\mathrm{Hom}_{\mathbf{Z}_{p}}(A,\mathbf{Q}_{p}/\mathbf{Z}_{p}).$

######  Remark I.4.2.

If $A$ has the additional structure of a topological $G$-module, then it can
be endowed with the continuous action given by

$(g\cdot f)(a)=f(g^{-1}\cdot a),\qquad\forall g\in G,\forall a\in A,\forall
f\in A^{\vee}.$

######  Example I.4.3.

Clearly, one has that $\mathbf{Z}_{p}^{\vee}=\mathbf{Q}_{p}/\mathbf{Z}_{p}$.
Furthermore, note that if we endow $\mathbf{Z}_{p}(1)^{\vee}$ with the action
from the previous remark, then we obtain
$\mathbf{Z}_{p}(1)^{\vee}=\mathbf{Q}_{p}/\mathbf{Z}_{p}(-1)$.

######  Lemma I.4.4.

([How02, (5)]) Let $M$ be a finitely generated $\Lambda(G)$-module. Then, the
Pontryagin dual induces a canonical isomorphism between group homology and
group-cohomology, namely,

$\mathrm{H}^{n}(G,M^{\vee})\cong\mathrm{H}_{n}(G,M)^{\vee}.$

###  1.5. Eigenspaces

Let us consider $\hat{\Delta}$, the group of $p$-adic characters of $\Delta$,
where $\Delta$ is a finite abelian group, and let $\mathcal{O}$ be the
$\mathbf{Z}_{p}$-algebra generated by the roots of unity of order dividing the
exponent of $\Delta$. Also, given $\psi\in\hat{\Delta}$, we denote the
$\mathbf{Z}_{p}$-algebra generated by the values of $\psi$ by
$\mathcal{O}_{\psi}$.

Note that $\psi$ extends to a map
$\mathbf{Z}_{p}[\Delta]\to\mathcal{O}_{\psi}$, which in turn extends to a map
$\mathcal{O}[\Delta]\to\mathcal{O}$. In particular, given an
$\mathcal{O}[\Delta]$-module, we define
$A^{\psi}:=A\otimes_{\mathcal{O}[\Delta]}\mathcal{O}$ using the previous map,
and define

$e_{\psi}=\frac{1}{\sharp\Delta}\sum_{\delta\in\Delta}\psi(\delta)\delta^{-1}\in\mathrm{Frac}(\mathcal{O}_{\psi})[\Delta].$

It is then easy to see that $e_{\psi}\mathcal{O}[\Delta]=e_{\psi}\mathcal{O}$,
and thus we obtain the following result.

######  Proposition I.5.1.

([Sha16, Prop 2.8.6]) Assume $p$ does not divide $\sharp\Delta$. Then, for any
$\mathcal{O}[\Delta]$-module $A$, we have that $A^{\psi}=e_{\psi}A$, and there
is a decomposition

$A\cong\bigoplus_{\psi\in\hat{\Delta}}A^{\psi}.$

In the case $\mathcal{O}=\mathbf{Z}_{p}$, we define
$A^{(\psi)}:=A\otimes_{\mathbf{Z}_{p}[\Delta]}\mathcal{O}_{\psi}$, and note
that in order to be able to define an analogue of $e_{\psi}$, we need to work
over a field containing the images $\psi$. For that reason, we define

$\tilde{e}_{\psi}=\frac{1}{\sharp\Delta}\sum_{\delta\in\Delta}\mathrm{Tr}_{\mathrm{Frac}(\mathcal{O}_{\psi})|\mathbf{Q}_{p}}(\psi(\delta))\delta^{-1}\in\mathbf{Q}_{p}[\Delta],$

We write $\psi\sim\varphi$ whenever there exists $\sigma\in
G_{\mathbf{Q}_{p}}$ so that $\psi=\sigma\circ\varphi$. Note that under this
notation, we have that if $\psi\sim\varphi$, then
$\mathcal{O}_{\psi}=\mathcal{O}_{\varphi}$. Thus, we can now state the
relationship between working with $\mathbf{Z}_{p}[\Delta]$-modules and
$\mathcal{O}[\Delta]$-modules.

######  Proposition I.5.2.

([Sha16, Lem 2.8.14 & Prop 2.8.15]) Let $A$ be a
$\mathbf{Z}_{p}[\Delta]$-module.

  1. (1)

One has that
$A^{(\psi)}\otimes_{\mathcal{O}_{\psi}}\mathcal{O}\cong(A\otimes_{\mathbf{Z}_{p}}\mathcal{O})^{\psi}$,
and if $p$ does not divide $\sharp\Delta$, then

$A^{(\psi)}\otimes_{\mathbf{Z}_{p}}\mathcal{O}\cong\bigoplus_{\varphi\in[\psi]}(A\otimes_{\mathbf{Z}_{p}}\mathcal{O})^{\varphi}.$

  2. (2)

If $p$ does not divide $\sharp\Delta$, then

$A\cong\bigoplus_{[\psi]\in\hat{\Delta}/\sim}A^{(\psi)}$

###  1.6. Twists and characteristic ideals

Our goal now is to see what is the effect of twisting a module on its
characteristic ideal. In order to do so, let
$\chi:G_{F}\to\mathcal{O}^{\times}$ be a character (possibly of infinite
order), and define the $\mathcal{O}$-linear isomorphism
$\mathrm{Tw}_{\chi}:\Lambda\to\Lambda$ given by the rule
$\gamma\mapsto\chi(\gamma)\gamma$. We have the following result, which is
possible due to I.3.4.

######  Lemma I.6.1.

([Rub14, §VI, Lemma 1.2] Let $M$ be a finitely generated and torsion
$\Lambda$-module. Then,

$\mathrm{Tw}_{\chi}\left(\mathrm{Ch}_{\Lambda}(M(\chi))\right)=\mathrm{Ch}_{\Lambda}(M).$

######  Example I.6.2.

Let us restrict our attention to the case where $F=\mathbf{Q}$. Then, if we
let $M$ be a finitely generated torsion $\Lambda$-module, we can conclude from
the previous result that, after a choice of a generator $\gamma$ of $\Gamma$,
the Iwasawa-Serre isomorphism produces an equality

$\mathrm{ch_{\Lambda}}\left(M(-n)\right)(T)\equiv\mathrm{ch}_{\Lambda}(M)(u^{n}(T+1)-1)\mod\mathbf{Z}_{p}^{\times},$

which holds for any pair of characteristic elements for $M$ and $M(n)$
respectively, where $u=\chi_{\mathrm{cyc}}(\gamma)$.

###  1.7. Euler characteristics of Iwasawa modules

Finally, we introduce Euler characteristics of Iwasawa modules.

######  Definition I.7.1.

Let $M$ be a finitely-generated $\Lambda(G)$-module, for $G$ the Galois group
of a $\mathbf{Z}_{p}^{d}$-extension. Then, $M$ is said to have finite
$G$-Euler characteristic if all the (group) homology groups
$\mathrm{H}_{i}(G,M)$ are finite for all $0\leq i\leq d$. Furthermore, if $M$
has finite $G$-Euler charactersistic, then the $G$-Euler characteristic of $M$
is defined as the product

$\mathrm{EC}(G,M):=\prod_{0\leq i\leq d}\sharp\mathrm{H}_{i}(G,M)^{(-1)^{i}}.$

Note that given a finitely generated $\Lambda=\Lambda(\Gamma)$-module, one can
define analogously its $\Gamma$-Euler characteristic, and the following result
illustrates the relation between both modules. The result can be found in,
e.g, [Coa+12, (31), p.14]

######  Theorem I.7.2.

Given a finitely generated $\Lambda(G)$-module $M$, $M$ has finite $G$-Euler
characteristic if and only if the $\Gamma$-modules $\mathrm{H}_{i}(\Gamma,M)$
have finite $\Gamma$-Euler characteristic for all $0\leq i<d$. Whenever any of
the two equivalent conditions holds, one also has the equality

$\mathrm{EC}(G,M)=\prod_{0\leq
i<d}\mathrm{EC}(\Gamma,\mathrm{H}_{i}(\Gamma,M))^{(-1)^{i}}.$

##  2\. Background in homotopy theory

The following section will serve the purpose of illustrating the subject of
stable homotopy theory using the language of $\infty$-categories. By no means
this intends to be a precise introduction, our main goal here is to allow the
reader that is not familiarized with the language to follow the rest of this
paper up to certain technicalities. The main references are [Lur09] and
[Lur16].

###  2.1. The language of $\infty$-categories

For our purposes in this paper, we can think of an $\infty$-category as a
category that is weakly enriched over spaces, in the sense that there is a
space of maps between every pair of objects, and the composition law is
defined only up to coherent homotopy. Homotopy coherence is the problem one
faces when trying to lift diagrams in the homotopy category of spaces to the
ordinary category of spaces. It is easy to see that the existence of such a
lift is equivalent to the existence of certain higher-degree homotopies, and
this is precisely the problem that $\infty$-categories tackle, as we are
asking their sets of maps to be spaces, and therefore contain homotopical
information in all degrees. Although we are asking for the composition law to
be only unique up to coherent homotopy, which is a much stronger condition
than being unique up to homotopy, this is still sufficient to provide a robust
theory of $\infty$-categories, in which all constructions for $1$-categories
(functors, limits, colimits…) can also be performed.

######  Remark II.1.1.

One can also look at the weaker concept of an $\infty$-groupoid, whose main
difference with $\infty$-categories is that all maps admit an inverse. As
explained in [Lur09, Example 1.1.1.4], one can think of $\infty$-groupoids as
categories behaving as spaces, and from it one defines the $\infty$-category
of spaces $\mathcal{S}$, which is the $\infty$-categorical analogue of the
classical category of spaces.

###  2.2. Stable $\infty$-categories

We will be working with stable $\infty$-categories most of the time. These are
pointed111An $\infty$-category is said to be pointed if it admits a zero
object, that is, an object that is both initial and final. $\infty$-categories
with finite limits, and whose loop functor gives an equivalence222The
condition of being pointed and having finite limits is enough to construct a
loop space functor $\Omega$ and a suspension functor $\Sigma$. Furthermore,
the stability condition implies that the adjunction $(\Sigma,\Omega)$ gives an
equivalence.. In addition, there is a precise notion of fiber and cofiber
sequences just as in classical homotopy theory. By [Lur16, Theorem 1.1.2.14],
the homotopy category of every stable $\infty$-category admits a triangulated
structure, which illustrates another reason why one would be interested in
working the $\infty$-categorical setting: the complexity of triangulated
categories could be justified by its being a “decategorification” of a
structure that is natural in higher category theory.

Moreover, the homotopy category of a stable $\infty$-category $\mathcal{C}$
can be endowed with a $t$-structure, although it might not be unique. Roughly,
a $t$-structure is the additional data in the homotopy category of
$\mathcal{C}$ that axiomatizes the phenomenon of truncation of complexes and
spaces and, therefore, taking cohomology or homotopy groups. From a
$t$-structure, one can obtain the heart $\mathcal{C}^{\heartsuit}$ of
$\mathcal{C}$. This is an abelian subcategory of the homotopy category of
$\mathcal{C}$ which, in the case of the derived $\infty$-category333The
derived $\infty$-category admits a canonical $t$-structure., can be regarded
as those complexes concentrated in degree 0. A consequence of this
construction is that fiber sequences in $\mathcal{C}$ induce long exact
sequences in $\mathcal{C}^{\heartsuit}$ after taking homotopy groups.

###  2.3. The $\infty$-category of Spectra

For our purposes, we will think of the $\infty$-category of spectra, which is
denoted by $\mathrm{Sp}$, as the higher analogue of the 1-category of abelian
groups. More precisely, there exist adjoint functors $\Omega^{\infty}$ and
$\Sigma^{\infty}_{+}$ giving rise to the following diagram

$\textstyle{\mathcal{A}b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\mathrm{Sp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\Omega^{\infty}}$$\textstyle{{\mathcal{S}et}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{Z}(-)}$$\textstyle{\mathcal{S}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{+}}$$\scriptstyle{\pi_{0}}$

######  Remark II.3.1.

The functor $\pi_{0}:\mathrm{Sp}\to\mathcal{A}b$ comes from the fact that
$\mathrm{Sp}$ admits a canonical $t$-structure, and its heart can be naturally
identified with (the nerve of) the 1-category of abelian groups. At the same
time, this implies that any fibration in $\mathrm{Sp}$ gives rise to a long
exact sequence of abelian groups just as in classical homotopy theory.
However, note that in this case one can take negative homotopy groups.

An alternative way to visualize $\mathrm{Sp}$ is given by Brown’s
representability theorem [Lur16, Theorem 1.4.1.2], which roughly states that
any generalized cohomology theory is representable by a spectrum. Thus,
$\mathrm{Sp}$ can be seen as the $\infty$-category containing all cohomology
theories, although we warn the reader that the embedding is not always
faithful. Furthermore, given a space $X$, one can define
$E^{n}(X)=[\Omega^{n}E,\Sigma^{\infty}_{+}X]$, and check that the assignment
$X\mapsto E^{n}(X)$ gives a generalized cohomology theory. In fact, if $X$ is
also a spectrum, one can define $E^{n}(X)=[\Omega^{n}E,X]$.

######  Example II.3.2.

We give a list of some spectra that we will use during this article.

  1. nosep

A central object in the study of spectra is the sphere spectrum, which is the
$\infty$-suspension of the point, $\mathbb{S}=\Sigma_{+}^{\infty}(\ast)$. This
object is sometimes referred to as the stable sphere, and the reason is that
its homotopy groups agree with the stable homotopy groups of spheres.

  2. nosep

As in the classical case, given an abelian group $G$, one can always construct
its Eilenberg-MacLane spectrum, denoted by $HG$, although when the context of
spectra is understood, we will simply denote it by $G$. It is characterized by
the fact that its homotopy groups are concentrated in degree 0.

  3. nosep

We will also use the Moore spectrum $SG$, which is characterized by having
$\pi_{i}SG=0$ for $i<0$, $\pi_{0}SG=G$, and $\pi_{i}(SG\otimes
H\mathbf{Z})=H_{i}(SG;\mathbf{Z})=0$ for $i>0$. The importance of this
spectrum will become visible in the Bousfield localization section below,
where we will work with non-connective spectra, that is, spectra with
homotopical information in negative degrees.

  4. nosep

Given a spectrum $X$, one can associate to it its $K$-theory spectrum $K(X)$.
In particular, when $X=HR$ for some ring $R$, $\pi_{n}K(R)\simeq K_{n}(R)$,
where the right hand side denotes the classical $n$-th K-theory group.

Finally, just as the category of abelian groups admits a symmetric monoidal
category structure, one can endow $\mathrm{Sp}$ with a symmetric monoidal
$\infty$-category structure given by the smash product $\otimes$ and whose
unit is the sphere spectrum. This provides the possibility of forming
algebraic structures over $\mathrm{Sp}$, whence the name Higher Algebra.

Ordinary Algebra |  $\infty$-Categorical Algebra
---|---
Set | Space
Abelian group | Spectrum
Tensor product of abelian groups | Smash product of spectra
Associative ring |  $\mathbb{E}_{1}$-ring
Commutative ring |  $\mathbb{E}_{\infty}$-ring
444This table was obtained from [Lur16, Chapter 7].

###  2.4. Bousfield localization of spectra and Iwasawa theory

Let us fix a spectrum $E\in\mathrm{Sp}$. Another spectrum $X$ is said to be
$E$-acyclic if $X\otimes E\simeq 0$. On the other hand, $X$ is said to be
$E$-local if $Y\to X$ is null-homotopic (i.e, factors through a zero object)
whenever $Y$ is acyclic. Then, there exists a functor $L_{E}$ from
$\mathrm{Sp}$ called Bousfield localization, which is characterized up to
equivalence by the following two properties, namely that $L_{E}X$ is $E$-local
and that there is a natural map $X\to L_{E}X$ and its fiber is $E$-acyclic. We
then define the $p$-adic completion $X_{p}^{\wedge}$ of a spectrum $X$ to be
the Bousfield localization of $X$ with respect to the Moore spectrum
$S\mathbf{Z}/p$.

Now, let $X$ be a space, and let $\mathrm{KU}^{\ast}(X)$ be the ring of
(periodic) complex $K$-theory. Since $\mathrm{KU}^{\ast}$ defines a
generalized cohomology theory, it corresponds to a spectrum $\mathrm{KU}$ by
Brown’s theorem. We refer to $\mathrm{KU}$ as the complex $K$-theory spectrum.
Finally, we define the $K(1)$-local $K$-theory spectrum as
$L_{K(1)}K=\mathrm{KU}_{p}^{\wedge}$.

######  Remark II.4.1.

In fact, the $K(1)$-local adjective comes from the first Morava $K$-theory
spectrum, which satisfies that
$L_{K(1)}K(X)\simeq\mathrm{KU}(X)_{p}^{\wedge}$. In general, one can construct
the $n$-th Morava $K$-theory spectrum $K(n)$, which is a central tool in
stable homotopy theory.

As explained in [Mit05], $K(1)$-local $K$-theory already has a direct relation
with Iwasawa theory: One can identify the group of Adams operations on
$K(1)$-local $K$-theory with $\mathrm{Gal}(\mathbf{Q}_{\infty}|\mathbf{Q})$
via the cyclotomic character and obtain
$(L_{K(1)}K)^{\ast}(L_{K(1)}K)\simeq\Lambda(\Gamma)[\Delta]$.

##  3\. The commutative and totally real case

From now on, let us fix an odd prime $p$ and let $F$ be a totally real number
field. Let $\Sigma$ denote a finite set of primes of $F$ containing all primes
dividing $p$ and let $F_{\Sigma}$ denote the maximal extension of $F$ which is
unramified outside $\Sigma$ and the infinite places of $F$. Also, given an
extension $L$ of $F$ inside $F_{\Sigma}$, we denote the maximal abelian
$p$-extension of $L$ contained inside $F_{\Sigma}$ by $M(L)$, and its Galois
group over $L$ by $X(L)=\mathrm{Gal}(M(L)|L)$.

######  Definition III.0.1.

A Galois extension $F_{\infty}|F$ is said to be a totally real admissible
$p$-adic Lie extension if

  1. nosep

$F_{\infty}$ is totally real.

  2. nosep

The Galois group of $F_{\infty}|F$ is a $p$-adic Lie group, that is, a
profinite topological group of finite rank containing an open pro-$p$
subgroup.

  3. nosep

$F_{\infty}|F$ is unramified outside $\Sigma$.

  4. nosep

$F_{\infty}$ contains $F^{\mathrm{cyc}}$, the maximal
$\mathbf{Z}_{p}$-extension of $F$ contained in $F(\mu_{p^{\infty}})$.

Finally, and following [Coa+12, p. 6], we will work under the assumption that
$G=\mathrm{Gal}(F_{\infty}|F)$ is an abelian $p$-adic Lie group of dimension
equal to one – which, according to Leopoldt’s conjecture should always be the
case –, and denote $H=\mathrm{Gal}(F_{\infty}|F^{\mathrm{cyc}})$, which is a
finite group under our running hypotheses.

###  3.1. The main conjecture

Let us start by noting that if we denote
$\Gamma=\mathrm{Gal}(F^{\mathrm{cyc}}|F)$, $G$ can be written as
$G=H\times\Gamma$. If we denote by $K$ the fixed field of $\Gamma$, which
satisfies that $K\cap F^{\mathrm{cyc}}=F$, then $F_{\infty}$ can be written as
the composite field $KF^{\mathrm{cyc}}$. Moreover, we will refer to the
$p$-adic cyclotomic character
$\chi_{\mathrm{cyc}}:\mathrm{Gal}(F(\mu_{p^{\infty}})|F)\to\mathbf{Z}_{p}^{\times}$
in a slightly unconventional way, namely as a character of $G$ that is trivial
on $H$, given that $\Gamma$ is a subgroup of
$\mathrm{Gal}(F(\mu_{p^{\infty}})|F)$ and thus we can look at the restriction
of $\chi_{\mathrm{cyc}}$ to $\Gamma$, as it was commented in I.3.2.

Following the notations from 1.5, let
$\psi\in\hat{H}\setminus\\{\mathbf{1}\\}$ be of type S, that is, so that
$F_{\psi}$, the field cut out by $\ker(\psi)$, or perhaps more precisely, the
minimal extension of $F$ through which $\psi$ factors, is totally real and
$F_{\psi}\cap F^{\mathrm{cyc}}=F$. For simplicity, we will also assume that
the order of $\psi$ is coprime to $p$. Furthermore, fix a topological
generator $\gamma\in\Gamma$, whose image through the cyclotomic character
$\chi_{\mathrm{cyc}}(\gamma)$ we denote by $u$,

Now, let $L^{\Sigma}(\psi,s)$ denote the $L$-function with Euler factors at
$\Sigma$ in the Euler product expression removed, whose rationality is
justified in [Wil90, (1.2)] by using a result by Klingen-Siegel. We then have
the following theorem.

######  Theorem III.1.1.

(Deligne-Ribet [DR80], Cassou-Noguès [CN79], Barsky [Bar77]) Let $n\geq 1$ be
an integer with $n\equiv 0\mod[F(\mu_{p}):F]$. There exists a unique formal
power series $L_{p}(\psi)\in\mathcal{O}_{\psi}\llbracket T\rrbracket$ such
that

$L_{p}(\psi,u^{n}-1)=L^{\Sigma}(\psi,1-n).$

######  Remark III.1.2.

As explained in [Wil90], the theorem can also be stated for $\mathbf{1}$ and
characters of type $W$, that is, characters in which $F_{\psi}\subseteq
F^{\mathrm{cyc}}$, multiplying the left hand side by
$(\psi(\gamma)u^{n}-1)^{-1}$ but for simplicity we only focus on the type $S$
case.

######  Remark III.1.3.

As explained in [Coa+12, p. 7] the theorem can also be stated regarding
$L_{p}(\psi)$ as a pseudo-measure on $G$, so that if $n>0$ is an integer with
$n\equiv 0\mod[F(\mu_{p}):F]$,

$L_{p}(\psi,\chi_{\mathrm{cyc}}^{n}):=\int_{G}\left(\psi\otimes\chi_{\mathrm{cyc}}^{n}\right)dL_{p}(\psi)=L^{\Sigma}(\psi,1-n).$

This is why we will denote $L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})$ and
$L_{p}(\psi,u^{n}-1)$ indistinctly.

For the rest of this section, let us fix $\psi$ of type $S$; denote
$X=X(F_{\infty})$, and let
$X_{\psi}=e_{\psi}(X\otimes_{\mathbf{Z}_{p}}\mathcal{O}_{\psi})$. By a
classical theorem of Iwasawa, $X_{\psi}$ is finitely generated and torsion as
an $\mathcal{O}_{\psi}\llbracket\Gamma\rrbracket$-module, and therefore we can
look at a characteristic power series
$C_{\psi}:=\mathrm{ch}(X_{\psi})\in\mathcal{O}_{\psi}\llbracket T\rrbracket$
for $X_{\psi}$. It is easy to check that

(3.1) $\displaystyle C_{\psi}(T)\mathcal{O}_{\psi}\llbracket
T\rrbracket=\varpi_{\psi}^{\mu_{\psi}}\det\left(T-(\gamma-1)|X_{\psi}\otimes
F_{\psi}\right)\mathcal{O}_{\psi}\llbracket T\rrbracket,$

where $\varpi_{\psi}$ is any fixed local parameter for $\mathcal{O}_{\psi}$,
and $\mu_{\psi}$ is the algebraic $\mu$-invariant, which by the work of Wiles
agrees with the analytic invariant obtained after applying the Weierstrass
preparation theorem to $L_{p}(\psi,T)$ above. With this, one can state the
algebraic part of the IMC as follows.

######  Theorem III.1.4.

(IMC for totally real fields) For all $\psi\in\hat{H}$ of order prime to $p$,

$L_{p}(\psi,T)\mathcal{O}_{\psi}\llbracket
T\rrbracket=C_{\psi}(T)\mathcal{O}_{\psi}\llbracket T\rrbracket.$

###  3.2. The group-cohomological Euler characteristic formula

Now, let us note that $G$ acts on $\mathbf{Z}_{p}(n)$ as long as $n\equiv
0\mod[F(\mu_{p}):F]$, so let $n$ be an integer – positive or negative – of
that form. The determinant part in equation (3.1) admits a cohomological
interpretation by the work of Bayer-Neukirch [BN78, Theorem 6.1] and Schneider
[Sch83, Theorem §4.4]. Namely, the determinant does not vanish at $T=0$ if and
only if the $\Gamma$-invariants (or equivalently, the $\Gamma$-coinvariants)
of $X_{\psi}$ are finite dimensional. When one of these three equivalent
conditions holds, one has

(3.2)
$\displaystyle|C_{\psi}(u^{n}-1)|_{p}=\frac{\sharp\mathrm{H}^{0}\left(\Gamma,X_{\psi}(-n)\right)}{\sharp\mathrm{H}^{1}\left(\Gamma,X_{\psi}(-n)\right)},$

where the $(-n)$-th Tate twist of $X_{\psi}$ appearing in the right hand side
follows from I.6.2.

Now, combining III.1.1, III.1.4 and (3.2), we obtain the following well-known
result.

######  Proposition III.2.1.

Let $n$ be an integer with $n\equiv 0\mod[F(\mu_{p}):F]$. Then,

$|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})|_{p}=\frac{\sharp\mathrm{H}^{0}\left(\Gamma,X_{\psi}(-n)\right)}{\sharp\mathrm{H}^{1}\left(\Gamma,X_{\psi}(-n)\right)}.$

######  Remark III.2.2.

In particular, note that when $n>0$ is an integer such that $n\equiv
0\mod[F(\mu_{p}):F]$, we have a group-cohomological interpretation of the
$p$-adic valuation of the $L$-function, that is,

$\left|L^{\Sigma}(\psi,1-n)\right|_{p}=\frac{\sharp\mathrm{H}^{0}\left(\Gamma,X_{\psi}(-n)\right)}{\sharp\mathrm{H}^{1}\left(\Gamma,X_{\psi}(-n)\right)}.$

###  3.3. The étale-cohomological Euler characteristic formula

Let $n$ be an integer as in III.2.1. We can now use Shapiro’s lemma to extend
the formula above as follows

(3.3)
$\displaystyle|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})|_{p}=\frac{\\#\mathrm{H}^{0}\left(\Gamma,X_{\psi}(-n)\right)}{\\#\mathrm{H}^{1}\left(\Gamma,X_{\psi}(-n)\right)}=\frac{\\#\mathrm{H}^{0}\left(G,X(-n)\right)}{\\#\mathrm{H}^{1}\left(G,X(-n)\right)}.$

Let us now recall that we can identify $X$ with
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F_{\infty}}[1/p];\mathbf{Q}_{p}/\mathbf{Z}_{p})^{\vee}$.
This is due to the fact that since $X$ is the Galois group of the maximal
abelian $p$-extension of $F^{\mathrm{cyc}}$ that is unramified outside $p$, it
can be identified with the maximal abelian pro-$p$ factor group of the
algebraic fundamental group of $\mathrm{Spec}(\mathcal{O}_{F_{\infty}}[1/p])$.
Taking this into account,

$\displaystyle\mathrm{H}^{\nu}(G,X(-n))$
$\displaystyle=\mathrm{H}^{\nu}\left(G,\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F_{\infty}}[1/p];\mathbf{Q}_{p}/\mathbf{Z}_{p}(n))^{\vee}\right)$
$\displaystyle=\mathrm{H}^{1-\nu}\left(G,\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F_{\infty}}[1/p];\mathbf{Q}_{p}/\mathbf{Z}_{p}(n))\right)^{\vee},\qquad\nu=0,1.$

where the first equality follows from the fact that the Pontryagin dual
inverts the action of the cyclotomic character, and the second equality
follows from the proof of [Sch83, Lemma §I.2.4]. Combining these equalities
with 3.3, we obtain

$|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})|_{p}=\frac{\\#\mathrm{H}^{1}\left(G,\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F_{\infty}}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)\right)}{\\#\mathrm{H}^{0}\left(G,\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F_{\infty}}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)\right)}.$

Following [Sch83, Lemma §II.4.1] we can use the Hochschild-Serre spectral
sequence

$E^{i,j}_{2}=\mathrm{H}^{i}\left(G,\mathrm{H}^{j}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F_{\infty}}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)\right)\Rightarrow\mathrm{H}^{i+j}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right).$

Schneider studies the groups appearing in this spectral sequence and observes
that it degenerates in the second page, and from there he deduces that, since
the weak Leopoldt conjecture is proven in the case of totally real number
fields, and that is equivalent to
$\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F_{\psi}}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p})=0$,
one has the following classical result.

######  Theorem III.3.1.

([BN78, Theorem 6.1],[Sch83, Theorem §4.4]) Let $n$ be an integer with
$n\equiv 0\mod[F(\mu_{p}):F]$. Then, the following equality holds

$\left|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})\right|_{p}=\frac{\\#\mathrm{H}^{0}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)}{\\#\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)}.$

###  3.4. ($K(1)$-local) $K$-theory and étale descent

Following the same notation as in the previous sections, as well as the one
introduced in §2, we can state the following theorem.

######  Theorem III.4.1.

(Blumberg-Mandell, [BM15, Theorem 1 & Corollary 1]) The homotopy fiber
$\mathrm{fib}(\kappa)$ of the completion map

$\kappa:L_{K(1)}K(\mathcal{O}_{F}[1/p])\to\prod_{v\in\Sigma}L_{K(1)}K(F_{v})$

in $K(1)$-local algebraic $K$-theory is weakly equivalent to $\Omega
I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathcal{O}_{F}[1/p])$, where $I_{\mathbf{Z}_{p}}$
denotes the Anderson duality functor for $p$-local spectra. Furthermore, there
is a canonical map of spectra from the homotopy fiber of the $p$-adic
cyclotomic trace map

$\mathrm{tr}_{p}^{\wedge}:K(\mathcal{O}_{F};\mathbf{Z}_{p})\to\mathrm{TC}(\mathcal{O}_{F};\mathbf{Z}_{p})$

to $\Omega I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathcal{O}_{F})$ that induces an
isomorphism of homotopy groups on degrees $\geq 2$.

######  Remark III.4.2.

Note that, thanks to [BCM20, Theorem 1.1], we know that
$L_{K(1)}K(\mathcal{O}_{F}[1/p])\simeq L_{K(1)}K(\mathcal{O}_{F})$, which
allows us to identify both fibers in the specified degrees.

We would like to relate the étale cohomology groups showing up in Theorem
III.3.1 and ($K(1)$-local) $K$-theory. To do so, we use the Thomason descent
spectral sequence which, given a ring $R$ that we assume to be either
$\mathcal{O}_{F}[1/p]$ or $F_{v}$ for any $v\in\Sigma$, has the following form

$E^{p,q}_{2}=\mathrm{H}^{p}_{\mathrm{\acute{e}t}}(R,\mathbf{Z}_{p}(-q/2))\Rightarrow\pi_{-p-q}L_{K(1)}K(R),$

where the groups on the left are assumed to be zero whenever $q$ is not
divisible by $2$.

Using Thomason’s descent spectral sequence as in [BM15] or [DM98], if we
denote the Moore spectrum associated to $\mathbf{Q}_{p}/\mathbf{Z}_{p}$ by
$M(\mathbf{Q}_{p}/\mathbf{Z}_{p})$, we have the following table.

$X$ | $\pi_{2k+1}X$ | $\pi_{2k}X$
---|---|---
$L_{K(1)}K(R)$ |  | $\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(R,\mathbf{Z}_{p}(k))$
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(R,\mathbf{Z}_{p}(k+1))$ | $\oplus$
| $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(R,\mathbf{Z}_{p}(k+1))$
$L_{K(1)}\left(K(R)\otimes M_{\mathbf{Q}_{p}/\mathbf{Z}_{p}}\right)$ |  | $\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(R,\mathbf{Q}_{p}/\mathbf{Z}_{p}(k))$
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(R,\mathbf{Q}_{p}/\mathbf{Z}_{p}(k+1))$ | $\oplus$
| $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(R,\mathbf{Q}_{p}/\mathbf{Z}_{p}(k+1))$

######  Remark III.4.3.

As explained in Blumberg-Mandell’s article, one can use III.4.1 to identify
(non-canonically) the Poitou-Tate exact sequence

${0\to\mathrm{H}_{\mathrm{\acute{e}t}}^{0}(\mathcal{O}_{F}[1/p],\mathbf{Z}_{p}(k))}$${\prod_{v\in\Sigma}\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(F_{v},\mathbf{Z}_{p}(k))}$${\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(1-k))^{\vee}}$${\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(\mathcal{O}_{F}[1/p],\mathbf{Z}_{p}(k))}$${\prod_{v\in\Sigma}\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(F_{v},\mathbf{Z}_{p}(k))}$${\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(1-k))^{\vee}}$${\mathrm{H}_{\mathrm{\acute{e}t}}^{2}(\mathcal{O}_{F}[1/p],\mathbf{Z}_{p}(k))}$${\prod_{v\in\Sigma}\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(F_{v},\mathbf{Z}_{p}(k))}$${\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(1-k))^{\vee}\to
0,}$$\scriptstyle{\Pi_{0}}$$\scriptstyle{\Pi_{1}}$$\scriptstyle{\Pi_{2}}$

with the long exact sequence induced by the taking homotopy groups of the
fibration in $K(1)$-local $K$-theory

${\cdots\to\pi_{l}\mathrm{fib}(\kappa)}$${\pi_{l}L_{K(1)}K(\mathcal{O}_{F}[1/p])}$${\prod_{v\in\Sigma}\pi_{l}L_{K(1)}K(F_{v})}$${\pi_{l-1}\mathrm{fib}(\kappa)}$${\pi_{l-1}L_{K(1)}K(\mathcal{O}_{F}[1/p])}$${\prod_{v\in\Sigma}L_{K(1)}K(F_{v})\to\cdots.}$

This is due to Anderson duality, which provides an isomorphism

(3.4)
$\displaystyle\left(\pi_{-1-l}\left(L_{K(1)}K(\mathcal{O}_{F}[1/p])\otimes
M_{\mathbf{Q}_{p}/\mathbf{Z}_{p}}\right)\right)^{\vee}\simeq\pi_{l+1}(I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathcal{O}_{F}[1/p])).$

The second term is isomorphic to $\pi_{l}\mathrm{fib}(\kappa)$ by Theorem
III.4.1, and the first term can be compared with the Pontryagin dual of the
corresponding étale cohomology group in the table above, thus giving the
identification between both exact sequences.

###  3.5. The $K(1)$-local $K_{\ast}$-theoretic description of the special
values

We can move on to applying all the machinery we have introduced to give
another formula for the special values of the $p$-adic $L$-function, and the
results we provide here are an extension, as well as an alternative proof of a
result by Hesselholt, who studied the case $F=\mathbf{Q}$ in [Hes18].

######  Corollary III.5.1.

Let $n$ be such that $n\equiv 0\mod[F(\mu_{p}):F]$. Then,

$\left|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})\right|_{p}=\frac{\\#\pi_{-1-2n}\mathrm{fib}(\kappa)}{\\#\pi_{-2n}\mathrm{fib}(\kappa)}.$

Furthermore, if $n\leq-1$,

$\left|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})\right|_{p}=\frac{\\#\pi_{-1-2n}\mathrm{fib}\left(\mathrm{tr}_{p}^{\wedge}\right)}{\\#\pi_{-2n}\mathrm{fib}\left(\mathrm{tr}_{p}^{\wedge}\right)}$

###### Proof.

Let us start by observing that

$\displaystyle\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n))^{\vee}$
$\displaystyle\simeq\pi_{2n}L_{K(1)}\left(K(\mathcal{O}_{F}[1/p])\otimes
M_{\mathbf{Q}_{p}/\mathbf{Z}_{p}}\right)^{\vee}$
$\displaystyle\simeq\pi_{-1-2n}\Omega
I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathcal{O}_{F}[1/p])$
$\displaystyle\simeq\pi_{-1-2n}\mathrm{fib}(\kappa),$

where the middle equality follows from (3.4) above.

On the other hand, and as mentioned in the previous section, the weak Leopoldt
conjecture is true in the case of totally real and imaginary quadratic fields,
and that is equivalent to the vanishing of
$\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p})$.
Thanks to this, if we use the table above again, we obtain that

$\displaystyle\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n))^{\vee}$
$\displaystyle\simeq\pi_{2n-1}L_{K(1)}\left(K(\mathcal{O}_{F}[1/p])\otimes
M_{\mathbf{Q}_{p}/\mathbf{Z}_{p}}\right)^{\vee}$
$\displaystyle\simeq\pi_{-2n}\mathrm{fib}(\kappa),$

where the last equivalence is obtained by mimicking the argument above. The
degrees on which the homotopy groups of $\mathrm{fib}(\kappa)$ and
$\mathrm{fib}(\mathrm{tr}_{p}^{\wedge})$ come from Theorem III.4.1 above. ∎

We can also use the $K(1)$-local $K$-theoretic Poitou-Tate duality of
Blumberg-Mandell to give a description of the $p$-adic valuation of the
special values without using the fiber.

######  Corollary III.5.2.

Let $n$ be an integer with $n\equiv 0\mod[F(\mu_{p}):F]$. Then,

$|L_{p}(\psi,\chi_{\mathrm{cyc}}^{n})|_{p}=\frac{\\#\pi_{-2n+1}L_{K(1)}K(\mathcal{O}_{F}[1/p])}{\\#\pi_{-2n}L_{K(1)}K(\mathcal{O}_{F}[1/p])}\cdot\prod_{v\in\Sigma}\frac{\\#\pi_{-2n}L_{K(1)}K(F_{v})}{\\#\pi_{-2n+1}L_{K(1)}K(F_{v})}$

###### Proof.

We can use the table above together with the fact that the étale cohomology
group
$\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p})^{\vee}$
must vanish by the weak Leopoldt conjecture, to conclude that the classical
long exact sequence on homotopy groups can be split as follows

${0\to\pi_{1-2k}L_{K(1)}K(\mathcal{O}_{F}[1/p])}$${\prod_{v\in\Sigma}\pi_{1-2k}L_{K(1)}K(F_{v})}$${\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(1-k))^{\vee}}$${\pi_{2-2k}L_{K(1)}K(\mathcal{O}_{F}[1/p])}$${\prod_{v\in\Sigma}\pi_{2-2k}L_{K(1)}K(F_{v})}$${\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(1-k))^{\vee}\to
0,}$$\scriptstyle{\Pi_{1}}$$\scriptstyle{\Pi_{2}}$

Since all terms in the exact sequence are finite, we obtain the result. ∎

###  3.6. A note on Hesselholt’s expository note

According to Hesselholt, for $k>1$,

$\left|L_{p}\left(\mathbf{Q},\psi,\chi_{\mathrm{cyc}}^{-2k}\right)\right|_{p}=\frac{\\#\pi_{4k-1}\mathrm{fib}\left(\mathrm{tr}_{p}^{\wedge}:K(\mathbf{Z};\mathbf{Z}_{p})\to\mathrm{TC}(\mathbf{Z};\mathbf{Z}_{p})\right)}{\\#\pi_{4k}\mathrm{fib}\left(\mathrm{tr}_{p}^{\wedge}:K(\mathbf{Z};\mathbf{Z}_{p})\to\mathrm{TC}(\mathbf{Z};\mathbf{Z}_{p})\right)}$

This formula is recovered for those values of $k$ in our result III.5.1. The
problem is that we would like to give the formula above in terms of the
cyclotomic fiber for all $k$, instead of focusing on $k>1$. To do so, observe
that

$\left|L_{p}\left(F,\psi,\chi_{\mathrm{cyc}}^{n}\right)\right|_{p}=\frac{\\#\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)}{\\#\mathrm{H}^{0}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)}=\frac{\\#\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)^{\vee}}{\\#\mathrm{H}^{0}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)^{\vee}},$

and let us also remember that, for all $n$,

$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p],\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)^{\vee}\simeq\pi_{2n}\mathrm{fib}\left(L_{K(1)}\kappa:L_{K(1)}K(\mathcal{O}_{F}[1/p])\to\prod_{v\in\Sigma}L_{K(1)}K(F_{v})\right)$
$\mathrm{H}^{0}_{\mathrm{\acute{e}t}}\left(\mathcal{O}_{F}[1/p];\mathbf{Q}_{p}/\mathbf{Z}_{p}(n)\right)^{\vee}\simeq\pi_{1-2n}\mathrm{fib}\left(L_{K(1)}\kappa:L_{K(1)}K(\mathcal{O}_{F}[1/p])\to\prod_{v\in\Sigma}L_{K(1)}K(F_{v})\right)$

On the other hand, the following diagram can be used to identify the fiber of
$\kappa$ with the connective cover of the fiber of $\mathrm{tr}_{p}^{\wedge}$,
where the equivalences are argued in [Hes18a]:

However, using Quillen’s localization sequence will only give us information
in degrees greater or equal than $2$, and that is why we will only have the
formula for positive values of $k$.

Instead of following that path, we realize we also have the following diagram,
obtained after $K(1)$-localization, and substituting $\mathbf{Z}$ and
$\mathbf{Z}_{p}$ by their respective generalizations to $F$, namely

where $(\ast)$ is an equivalence by [BCM20, Cor. 1.3]. With this, we obtain
the desired equivalence

$\mathrm{fib}\left(L_{K(1)}\mathrm{tr}\right)\simeq\mathrm{fib}(L_{K(1)}\kappa),$

which, when combined with III.5.1, gives the following result.

######  Corollary III.6.1.

Let $n$ be an intger with $n\equiv 0\mod[F(\mu_{p}):F]$. Then,

$\left|L_{p}\left(F,\psi,\chi_{\mathrm{cyc}}^{-n}\right)\right|_{p}=\frac{\\#\pi_{2n-1}\mathrm{fib}\left(L_{K(1)}\mathrm{tr}:L_{K(1)}K(\mathcal{O}_{F})\to
L_{K(1)}\mathrm{TC}(\mathcal{O}_{F})\right)}{\\#\pi_{2n}\mathrm{fib}\left(L_{K(1)}\mathrm{tr}:L_{K(1)}K(\mathcal{O}_{F})\to
L_{K(1)}\mathrm{TC}(\mathcal{O}_{F})\right)}$

##  4\. The non-commutative and totally real case

###  4.1. Initial considerations

As in the commutative case, let $F_{\infty}|F$ be an arbitrary admissible
$p$-adic Lie extension – cf. III.0.1 – with Galois group $G$. Following
[Coa03], there exists a closed subgroup $H$ such that
$\Gamma:=G/H\simeq\mathbf{Z}_{p}$.

######  Remark IV.1.1.

For simplicity and without loss of generality, we assume that $G$ contains no
element of order $p$. This makes most discussions much simpler, given that
under this hypothesis $\Lambda(G)$ has finite global dimension, and $G$ also
has finite $p$-homological dimension, which agrees with the dimension of $G$
as a $p$-adic Lie group (cf. [Bur, Remark 1.4]).

Let us define the following sets, following [Coa+04, Section 2]

$\mathfrak{S}=\left\\{f\in\Lambda(G):\Lambda(G)/(f)\in\mathcal{M}\mathrm{od}_{\Lambda(H)}^{\mathrm{fg}}\right\\},\qquad\mathfrak{S}^{\ast}=\varinjlim_{n}p^{n}\mathfrak{S},$

where $\mathcal{M}\mathrm{od}^{\mathrm{fg}}_{\Lambda(H)}$ is the category of
finitely generated $\Lambda(H)$-modules.

From now on, we let
$\mathfrak{S}^{\prime}\in\\{\mathfrak{S},\mathfrak{S}^{\ast}\\}$. Then, the
following properties are known:

  1. (1)

[Coa03, Proposition 2.3] If $M$ is a finitely generated $\Lambda(G)$-module,
then $M$ is finitely generated over $\Lambda(H)$ if and only if $M$ is
$\mathfrak{S}$-torsion.

  2. (2)

[Coa03, Theorem 2.4] $\mathfrak{S}^{\prime}$ is multiplicatively closed, and
is a (left or right) Ore set in $\Lambda(G)$.

  3. (3)

[Coa03, Theorem 2.4] The elements of $\mathfrak{S}^{\prime}$ are non-zero
divisors in $\Lambda(G)$.

Thus, one can define $\mathcal{M}_{\mathfrak{S}^{\prime}}(G)$, the set of
finitely generated $\Lambda(G)$-modules with the property that
$\Lambda(G)_{\mathfrak{S}^{\prime}}\otimes_{\Lambda(G)}M$ vanishes or,
equivalently, that are $\mathfrak{S}^{\prime}$-torsion.

######  Remark IV.1.2.

Generally, it is not known whether $X=X(F_{\infty})$ lies in
$\mathcal{M}_{\mathfrak{S}^{\prime}}(G)$. For that reason, one works with
extensions $F_{\infty}|F$ satisfying the generalized Iwasawa conjecture,
meaning that there exists a finite extension $F^{\prime}|F$ in $F_{\infty}$
such that $\mathrm{Gal}(F_{\infty}|F^{\prime})$ is pro-$p$, and
$X(F^{\prime\mathrm{cyc}})$ is a finitely generated $\mathbf{Z}_{p}$-module,
or with the same hypotheses as Burns in [Bur, Theorem 9.1].

###  4.2. $K$-theoretic characteristic elements

From now on, we denote we use $\Lambda$ to refer to $\Lambda(G)$, unless it is
otherwise stated. Before being able to state an Iwasawa main conjecture for
the non-commutative case, the concept of a characteristic element must be
introduced. To do so, recall that there is a localization sequence in
$K$-theory of the form

$\cdots\to K_{1}(\Lambda)\to
K_{1}(\Lambda_{\mathfrak{S}^{\prime}})\xrightarrow{\partial_{G}}K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\prime}})\to
K_{0}(\Lambda)\to K_{0}(\Lambda_{\mathfrak{S}^{\prime}})\to 0,$

where $K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\prime}})$ is the 0-th relative
$K$-theory group of $\Lambda\to\Lambda_{\mathfrak{S}^{\prime}}$.

######  Remark IV.2.1.

As it will be observed later, it is important to note that as explained in
[Bur, §1.1.2], under the hypothesis that $G$ has no element of order $p$,
there is a natural isomorphism between
$K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\prime}})$ and the Grothendieck group
$G_{0}(\mathcal{M}_{\mathfrak{S}^{\prime}}(G))$.

As mentioned in [Bur, Remark 1.1], under our hypotheses, the map
$\partial_{G}$ is always surjective. For this reason, we can provide the
following definition.

######  Definition IV.2.2.

Given $X\in K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\prime}})$, a characteristic
element for $X$ is $\mathfrak{c}(X)\in K_{1}(\Lambda_{\mathfrak{S}^{\prime}})$
such that $\partial_{G}\mathfrak{c}(X)=X$.

###  4.3. Evaluation at $K$-theoretic characteristic elements

This section is explained in more generality in [Bur, §1.1.3]. Of course, in
order to link this definition to the point-wise property in the Iwasawa main
conjectures, we need to define a way to evaluate the elements $\xi\in
K_{1}(\Lambda_{\mathfrak{S}^{\prime}})$. To fulfill this analogy, let
$\rho:G\to\mathrm{Aut}_{\mathbf{Z}_{p}}(\Gamma)$ be a (continuous) Artin
representation of $G$. It is clear that $\rho$ extends canonically to a map
$\Lambda_{\mathfrak{S}^{\prime}}\to\mathrm{End}_{\mathbf{Z}_{p}}(Q)$, where
$Q=\mathrm{Frac}(\Lambda)$. If we then apply to it the functor $K_{1}$, we
obtain a map

$\rho_{\ast}:K_{1}(\Lambda_{\mathfrak{S}^{\prime}})\to Q^{\times}.$

Localizing at the kernel $\kappa=\ker(\pi)$ of the projection map
$\pi:\Lambda\to\mathbf{Z}_{p}$ produces a map
$\pi_{\kappa}:\Lambda_{\kappa}\to\mathbf{Q}_{p}$.

######  Definition IV.3.1.

Following the notation from the previous paragraph, and if $\xi\in
K_{1}(\Lambda_{\mathfrak{S}^{\prime}})$ and $\rho\in\mathrm{Art}(G)$, the
image of $\xi$ at $\rho$ is defined by

$\xi(\rho)=\begin{cases}\pi_{\kappa}\left(\rho_{\ast}(\mathfrak{c})\right),&\rho_{\ast}(\xi)\in\Lambda_{\kappa},\\\
\infty&\text{otherwise.}\end{cases}$

###  4.4. $p$-adic Artin $L$-functions

During this subsection, we follow [Bur, §9.1] almost exactly. Let $\rho$ be a
finite-dimensional complex representation of any finite quotient of a compact
$p$-adic Lie extension $F_{\infty}|F$ that is unramified outside a finite set
of places $\Sigma$ of $F$. Denote by $\mathrm{Art}^{+}(G)$ the subset of
$\mathrm{Art}(G)$ of representations that are trivial on the Galois group of
some intermediate field $F^{\prime}$ of $F_{\infty}|F$ that is totally real.
Also, let $\Sigma^{\prime}$ be a finite set of places of $F$ and let
$L^{\Sigma^{\prime}}(s,\rho)$ be the Artin $L$-function of $\rho$ truncated by
removing the Euler factors of all non-archimedean places in $\Sigma^{\prime}$.

######  Definition IV.4.1.

For each $\rho\in\mathrm{Art}^{+}(G)$, the $\Sigma$-truncated $p$-adic Artin
$L$-function $L_{p}^{\Sigma}(s,\rho)$ is defined as the unique $p$-adic
meromorphic function on $\mathbf{Z}_{p}$ such that for all integers $n<0$ and
all field isomorphisms $j:\mathbf{C}_{p}\to\mathbf{C}$,

$L^{\Sigma_{p}}\left(n,(\rho\otimes\omega_{F}^{n-1})^{j}\right)=j\left(L_{p}^{\Sigma}(n,\rho)\right),$

where $\Sigma_{p}$ is the union of $\Sigma$ and all places of $F$ that lie
above $p$, and $\omega_{F}$ denotes the Teichmüller character.

###  4.5. Twisting in the non-commutative setting

Let $X$ be a perfect complex of $\Lambda$-modules, and let
$\rho:G\to\mathrm{GL}_{n}(\mathcal{O})$ be a continuous representation of $G$.
Then, the complex $X(\rho):=\mathcal{O}^{n}\otimes_{\mathbf{Z}_{p}}X$ can be
endowed with a $G$-action by setting

$g(a\otimes x):=\rho(g)(a)\otimes g(x),\qquad g\in G,a\in\mathcal{O}^{n},x\in
X^{j}.$

###  4.6. Bockstein cohomology

As it is mentioned in [BV05, §3.1], given a bounded complex of
$\Lambda$-modules $X$, the choice of a generator $\gamma$ of $\Gamma$ induces
a morphism $\theta_{\gamma}$ in $D^{\flat}(\Lambda)$ of the form $X\to X[1]$.
Given a continuous representation $\rho:G\to\mathrm{GL}_{n}(\mathcal{O})$
where $\mathcal{O}$ denotes the valuation ring of some finite extension $L$ of
$\mathbf{Q}_{p}$, $\theta_{\gamma}$ induces
$T_{\rho}\otimes_{\Lambda}^{\mathbf{L}}X\to
T_{\rho}\otimes_{\Lambda}^{\mathbf{L}}X[1]$.

######  Definition IV.6.1.

The homomorphisms induced by $\theta_{\gamma}$

$\beta_{\nu}:\mathrm{Tor}_{\nu}^{\Lambda}(T_{\rho},X)\to\mathrm{Tor}_{\nu-1}^{\Lambda}(T_{\rho},X)$

are referred to as the Bockstein homomorphisms of $(X,T_{\rho},\gamma)$.
Moreover, these homomorphisms induce a complex, the Bockstein complex
associated to $X\in D^{\mathrm{perf}}(\Lambda(G))$, and $X$ is said to be
semi-simple at $\rho$ the cohomology of the complex is
$\mathbf{Z}_{p}$-torsion in each degree. Additionally, the Bockstein
cohomology groups $\mathrm{H}_{\beta}^{i}(G,X(\rho))$ are defined as the
cohomology groups of the complex given by the homomorphisms

$\beta_{-\nu}:\mathrm{Tor}_{-n}^{\Lambda}(T_{\rho},X)\to\mathrm{Tor}_{-n-1}^{\Lambda}(T_{\rho},X).$

###  4.7. The totally real non-commutative IMC

Following [Bur, §9.2], given a $G$-module $M$, $M^{\sharp}$ is the
$G_{F}$-module whose action by $\sigma\in G_{F}$ is given by
$\chi(\sigma)\overline{\sigma}^{1}$, where $\overline{\sigma}$ is the image of
$\sigma$ in $G$. We then define the derived complex

$C_{\infty,\Sigma}:=\mathbf{R}\Gamma_{\mathrm{\acute{e}t},c}\left(\mathrm{Spec}(\mathcal{O}_{F_{\infty}}[1/p],\Lambda^{\sharp}(1)\right)\in
D^{\flat}(\Lambda)$

of étale cohomology with compact support, which is in fact perfect as stated
in the citation. Moreover, according to [op. cit, §9.3], the $n$-th cohomology
groups of $C_{\infty,\Sigma}$ can be canonically identified as

$\mathrm{H}^{n}(C_{\infty,\Sigma})\cong\begin{cases}X(F_{\infty}),&n=2\\\
\mathbf{Z}_{p},&n=3\\\ 0&n\neq 2,3.\end{cases}$

######  Theorem IV.7.1.

([Bur, Thm 9.1]) There exists an element $\mathcal{L}_{p}\in
K_{1}(\Lambda_{\mathfrak{S}})$ such that

  1. (1)

For all $\rho\in\mathrm{Art}(G)$,

$\mathcal{L}_{p}(\rho\otimes\chi)=L_{p}^{\Sigma}(1,\rho\otimes\chi)\mod\mathbf{Q}_{p}^{\times},$

for almost all $\mathbf{1}\neq\chi\in\hat{\Gamma}$.

  2. (2)

The composition of the boundary map
$\partial_{G}:K_{1}(\Lambda_{\mathfrak{S}})\to
K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\ast}})$ with the isomorphism
$K_{0}(\Lambda,\Lambda_{\mathfrak{S}^{\ast}})\cong
G_{0}(\mathcal{M}_{\mathfrak{S}^{\ast}}(G))$ sends
$\mathcal{L}_{p}\mapsto[X(F_{\infty})]-[\mathbf{Z}_{p}]$.

###  4.8. The Bockstein-cohomological Euler characteristic formula

Let $X\in\mathcal{M}_{\mathfrak{S}^{\ast}}(G)$ be a semi-simple complex at
$\rho\in\mathrm{Art}(G)$, and such that
$\mathbf{Q}_{p}\otimes_{\mathbf{Z}_{p}}A(\rho)$ is acyclic. Then, it follows
from [BV05, Proposition 3.16] that there is a commutative diagram

$\textstyle{K_{1}(\Lambda,\Sigma_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ch}_{G,X}}$$\scriptstyle{(-)(\rho)}$$\textstyle{K_{1}(\Lambda_{\mathfrak{S}^{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)(\rho)}$$\textstyle{L^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\times},}$

where the map $\mathrm{ch}_{G,X}$ and the subcategory $\Sigma_{X}$ of
$C^{\mathrm{perf}}(\Lambda)$ is constructed in [BV05, §3.2.3]. For a detailed
description of the map, we refer the reader to [BV11, Appendix A.]

The following statement is the non-commutative counterpart of Corollary
III.6.1, that is, the main result of the previous section.

######  Corollary IV.8.1.

([BV05, Proposition 3.19] corrected using [BV11, Appendix C.]) If $X$ is semi-
simple at $\rho$, then for any $\xi\in\mathrm{Im}(\mathrm{ch}_{G,X})\leq
K_{1}(\Lambda_{\mathfrak{S}^{\ast}})$,

$\left|\xi(\rho)\right|_{p}^{[L:\mathbf{Q}_{p}]}=\prod_{\nu\in\mathbf{Z}}\sharp\mathrm{H}^{i}_{\beta}\left(G,X(\mathrm{Ad}(\rho))\right)^{(-1)^{\nu}}.$

######  Remark IV.8.2.

In the classical, commutative case, one has that $G$ is of rank one, and
$\rho=\chi_{\mathrm{cyc}}$. Given a finitely generated and torsion
$\Lambda$-module $M$, the differentials induced by the Hochschild-Serre
spectral sequence

$\mathrm{H}^{i}(G,M^{\vee})\to\mathrm{H}^{i+1}(G,M^{\vee})$

are manipulated in order to obtain the equality in III.3.1. According to the
result above, however, one only needs to use a projective resolution $P$ of
$M^{\vee}$ to obtain the same formula, or simply observe that

$(d^{i})^{\vee}=\beta_{i+1}:\mathrm{Tor}_{i+1}^{\Lambda}(\mathbf{Z}_{p},P)\to\mathrm{Tor}_{i}^{\Lambda}(\mathbf{Z}_{p},P).$

######  Corollary IV.8.3.

For all $\rho\in\mathrm{Art}(G)$ and almost all
$\mathbf{1}\neq\chi\in\hat{\Gamma}$,

$\left|\mathcal{L}_{p}(\rho\otimes\chi)\right|_{p}=\left|L_{p}^{\Sigma}(1,\rho\otimes\chi)\right|_{p}=\prod_{\nu\in\mathbf{Z}}\sharp\mathrm{H}^{i}_{\beta}\left(G,C_{\infty,\Sigma}\left(\mathrm{Ad}(\rho\otimes\chi)\right)\right)^{(-1)^{\nu}}.$

###### Proof.

The proof of this result follows directly from combining the IMC in the non-
commutative setting IV.7.1 together with IV.8.1. ∎

## References

  * [Bar77] Daniel Barsky  “Fonctions zêta $p$-adiques d’une classe de rayon des corps de nombres totalement réels” talk:16  In _Groupe de travail d’analyse ultramétrique_ 5 Secrétariat mathématique, 1977-1978  URL: http://www.numdam.org/item/GAU_1977-1978__5__A9_0
  * [BCM20] Bhargav Bhatt, Dustin Clausen and Akhil Mathew  “Remarks on $K(1)$-local $K$-theory”, 2020  arXiv:2001.11622 [math.KT]
  * [BM15] Andrew J. Blumberg and Michael A. Mandell  “The fiber of the cyclotomic trace for the sphere spectrum”, 2015  arXiv:1508.00014 [math.KT]
  * [BN78] Pilar Bayer and Jürgen Neukirch  “On values of zeta functions and $p$-adic Euler characteristics”  In _Inventiones mathematicae_ 50.1, 1978, pp. 35–64  DOI: 10.1007/BF01406467
  * [Bou98] N. Bourbaki  “Commutative Algebra”  Springer Science & Business Media, 1998
  * [Bur] David Burns  “On main conjectures in non-commutative Iwasawa theory and related conjectures”  In _Journal für die reine und angewandte Mathematik_
  * [BV05] David Burns and Otmar Venjakob  “On the leading terms of Zeta isomorphisms and p-adic L-functions in non-commutative Iwasawa theory”, 2005  arXiv:math/0511672 [math.NT]
  * [BV11] David Burns and Otmar Venjakob  “On descent theory and main conjectures in non-commutative Iwasawa theory”  In _Journal of the Institute of Mathematics of Jussieu_ 10.1 Cambridge University Press, 2011, pp. 59–118
  * [CN79] Pierrette Cassou-Noguès  “Valeurs aux entiers négatifs des fonctions zêta et fonctions zêtap-adiques”  In _Inventiones mathematicae_ 51.1, 1979, pp. 29–59  DOI: 10.1007/BF01389911
  * [Coa03] Schneider Peter Sujatha Ramdorai Coates  “Links between cyclotomic and Iwasawa theory.”  In _Documenta Mathematica_ Extra Vol. Universi t Bielefeld, Fakult t f r Mathematik, 2003, pp. 187–215  URL: http://eudml.org/doc/54395
  * [Coa+04] J. Coates et al.  “The $GL_{2}$ main conjecture for elliptic curves without complex multiplication”, 2004  arXiv:math/0404297 [math.NT]
  * [Coa+12] John Coates, Peter Schneider, Ramdorai Sujatha and Otmar Venjakob  “Noncommutative Iwasawa main conjectures over totally real fields: Münster, April 2011”  Springer Science & Business Media, 2012
  * [DM98] WG Dwyer and SA Mitchell  “On the K-theory spectrum of a ring of algebraic integers”  In _K-theory_ 14.3 Dordrecht; Boston: D. Reidel Pub. Co., 1987-, 1998, pp. 201–264
  * [DR80] Pierre Deligne and Kenneth A. Ribet  “Values of abelianL-functions at negative integers over totally real fields”  In _Inventiones mathematicae_ 59.3, 1980, pp. 227–286  DOI: 10.1007/BF01453237
  * [Hes18] Lars Hesselholt  “Algebraic K-theory and the $p$-adic $L$-function”, 2018  URL: http://web.math.ku.dk/~larsh/papers/s06/L-function.pdf
  * [Hes18a] Hesselholt, Lars  “Algebraic K-theory and the p-adic L-function”, http://web.math.ku.dk/~larsh/papers/s06/, 2018
  * [How02] Susan Howson  “Euler characteristics as invariants of Iwasawa modules”  In _Proceedings of the London Mathematical Society_ 85.3 Cambridge University Press, 2002, pp. 634–658
  * [Lur09] Jacob Lurie  “Higher Topos Theory (AM-170)”  Princeton University Press, 2009
  * [Lur16] Jacob Lurie  “Higher Algebra”, http://www.math.harvard.edu/~lurie/papers/HA.pdf, 2016
  * [Mit05] Stephen A. Mitchell  “K(1)-Local Homotopy Theory, Iwasawa Theory and Algebraic K-Theory”, 2005
  * [Rub14] Karl Rubin  “Euler Systems.(AM-147), Volume 147”  Princeton University Press, 2014
  * [Sch83] Peter Schneider  “Iwasawa $L$-functions of varieties over algebraic number fields”  In _Inventiones mathematicae_ 71.2, 1983, pp. 251–293  DOI: 10.1007/BF01389099
  * [Sha16] Romyar Sharifi, 2016  URL: https://www.math.ucla.edu/~sharifi/iwasawa.pdf
  * [Wil90] A. Wiles  “The Iwasawa Conjecture for Totally Real Fields”  In _Annals of Mathematics_ 131.3 Annals of Mathematics, 1990, pp. 493–540  URL: http://www.jstor.org/stable/1971468