# Synergistic Formulaic Alpha Generation for
Quantitative Trading based on Reinforcement Learning

Hong-Gi Shin, Sukhyun Jeong, Eui-Yeon Kim, Sungho Hong, Young-Jin Cho, and
Yong-Hoon Choi
Division of Robotics, Kwangwoon University, Seoul 01897, South Korea
Email: {ghdrl95, jayze3736, dmldussla93, gyogook608<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Mining of formulaic alpha factors refers to the process of discovering and
developing specific factors or indicators (referred to as alpha factors) for
quantitative trading in stock market. To efficiently discover alpha factors in
vast search space, reinforcement learning (RL) is commonly employed. This
paper proposes a method to enhance existing alpha factor mining approaches by
expanding a search space and utilizing pretrained formulaic alpha set as
initial seed values to generate synergistic formulaic alpha. We employ
information coefficient (IC) and rank information coefficient (Rank IC) as
performance evaluation metrics for the model. Using CSI300 market data, we
conducted real investment simulations and observed significant performance
improvement compared to existing techniques.

Keywords—Reinforcement learning, Formulaic alpha factor

##  I INTRODUCTION

Artificial intelligence plays a vital role in enhancing profitability in stock
investment and the financial sector. Specifically, quantitative trading
involves an automated approach to buying and selling stocks to maximize the
final net asset value. Machine learning models trained on technical stock
trading data (e.g., open, close, high, low, volume, etc.) actively participate
in the trading process. Research on quantitative trading utilizing
reinforcement learning considers the volatility and noise in stock data,
addressing gaps between signal-based trading decisions. Reinforcement learning
(RL) constructs end-to-end models, bypassing issues related to prediction.
This approach proves suitable for various quantitative trading tasks such as
algorithmic trading and portfolio management [1]. However, this field still
faces several limitations. For instance, given the importance of risk
management, there is a pressing need for explainable AI due to the black-box
nature of agents [2].

Formulaic alpha factor mining, a subset of artificial intelligence research in
the stock market, involves the process of generating formulas with high
correlations to future returns from raw features associated with stock
trading. Typically, it calculates correlations between alpha factor values and
future returns and identifies factors showing high correlations as ’alpha
factors,’ believed to bring returns exceeding the market performance. Symbolic
factors are equated with formulaic factors and are typically expressions
generated using various operators and operands. Traditionally, human
intervention was involved in expression creation, but recently, machine
learning models are used to automatically generate them. When conducting
symbolic factor mining based on machine learning, it requires a search space
that specifies which operators and operands to use in creating symbolic
factors, as well as a search algorithm to find the optimal symbolic factors.

There have been studies using genetic algorithms to find a single formulaic
alpha factor [3-7]. Genetic algorithms aim to start from various initial
factors and use an evolutionary mechanism to generate the optimal factor.
However, explaining the complex stock market with a single alpha factor is
challenging, so it is common to combine alpha factors with low correlations
among them. Existing methods for alpha factor generation have prioritized the
performance of individual alpha factors without considering the performance of
combined alpha factors. Therefore, there are limitations in finding a set of
alpha factors that synergistically contribute to each other.

To address these limitations, Yu et al. [8] proposed a new framework aimed at
optimizing the performance of alpha factor combination during formulaic alpha
factor mining. In their study, they explored the space for alpha factor
generation using proximal policy optimization (PPO)-based reinforcement
learning [9] and updated weights of generated alpha factor set through
gradient descent. The framework proposed in the paper demonstrated higher
correlations with future returns compared to a single alpha factor generated
by a genetic algorithm. However, unlike other studies on alpha factor
generation, the limited number of operators in the search space restricts the
ability to generate a wide variety of alpha combinations. These limitations
highlight the need to expand the search space. But excessively expanding the
search space raises complex issues, such as requiring a more sophisticated
policy architecture and a more efficient search algorithm.

In this paper, we build upon the research of Yu et al. [8] by proposing an
enhanced initialization method that defines a more extensive search space and
initializes it with pregenerated seed formulaic alpha set, thereby leveraging
the strengths of RL-based search algorithms. Our approach improves the
performance compared with previous synergistic formulaic alpha factor
methodologies. To assess performance against previous researches, we use
CSI300 market data for the same period. We sequentially apply the proposed
technique, observe improvements in performance, and evaluate investment
outcomes through investment simulations in comparison with previous methods.

The structure of this paper is as follows. In Section II, we present the
improvements compared to previous synergistic formulaic alpha factor mining
techniques. Section III provides a detailed explanation of data collection and
processing methods, experimental settings, and results. Finally, in Section
IV, we draw conclusions from this research and discuss directions for future
studies.

##  II PROPOSED METHOD

We adopt the alpha definition employed in [8]. We trade for a period of T days
and consider n stocks in the stock market. For each trading day
$\textit{t}\in\\{1,\ldots,\textit{T}\\}$ each stock i corresponds to a feature
vector $\textbf{x}_{i,t}\in\mathbb{R}^{mt}$ where, m is the number of raw
features such as opening and closing prices.

TABLE I: Information on the tokens used in the experiment.
Tokens that were added are indicated in bold. Category | Symbols
---|---
Features | open, close, high, low, volume, VWAP
Operator | Abs, Log, Add, Sub, Mul, Div, Greater, Less,
Ref, Mean, Std, Var, Sum, Max, Min, Med,
Mad, Delta, WMA, EMA, Sign, CSRank,
Product, Scale, Pow, Skew, Kurt, Rank
Delta, Argmax, Argmin, Cond
Times deltas |  5, 10, 20, 30, 40, 50, 60, 120, 252
Constant | -30.0, -10.0, -5.0, -2.0, -1.0, -0.5, -0.01, 0.5,
1.0, 2.0, 5.0, 10.0, 30.0
Sequence | BEG(begin), SEP(end of expression)
indicators |

With the feature vectors of all stocks on a trading day
$\textbf{X}\in\mathbb{R}^{n\times mt}$ consisting of $n$ feature vectors, the
alpha factor $f$ is defined as a mapping function that converts the feature
vectors $\textbf{X}\in\mathbb{R}^{n\times mt}$ to alpha values
$f(\textbf{X})\in\mathbb{R}^{n}$. Finally, alpha values z are obtained as
$\textbf{z}=\sum_{j=1}^{k}w_{j}f_{j}(\textbf{X})$ where $k$ is the number of
elements within the alpha set $\mathcal{F}=\\{f_{1},\ldots,f_{k}\\}$ and their
weights $\mathcal{W}=\\{w_{1},\ldots,w_{k}\\}$. To calculate the correlation
between the alpha values and the real stock trend
$\textbf{y}\in\mathbb{R}^{n}$, we used IC as an metric. Also, we based on [8]
as a method for generating alpha sets that can maximize IC. The alpha mining
system of [8] consists of an alpha generator that produces alpha factors and a
combination model that optimizes $\mathcal{W}$ to maximize the IC of the alpha
set over the training data.

We propose a search space expansion of the alpha generator for diversity of
alpha factors and initialization with seed alphas to effectively navigate a
large search space.

###  II-A Expanding Search Space

This paper utilizes reinforcement learning to generate a wide range of
formulaic alpha factors by exploring a much broader search space, thus
ensuring the creation of diverse factors. Table 1 depicts the search space
used in the paper. We have incorporated new operators presented in [10] to
those used in previous research [8], while excluding industry classification
data. Additionally, to gain a deeper understanding of trend changes, we have
included a constant range of {5, 60, 120, 252}, allowing for the reflection of
both short-term and long-term trends. Table 2 provides detailed descriptions
of the operators added in this paper.

TABLE II: Description of the operators added in the paper. Operator | Description
---|---
Sign($x$) | Returns 0 if the given x value is 0, 1 if it is
positive, and -1 if it is negative.
CSRank(x) | The cross-sectional rank (CSRank) is an
operator that returns the rank of the current
stock’s feature value x relative to the feature
values of all stocks on today’s date.
Product(x, t) | It returns the product of the feature values for
each date from the current date up to t days
ago.
Product(x, t) = $\prod_{i=0}^{t}\textit{x}_{t-i}$
Scale(x) | It returns the value obtained by dividing the
current feature value x by the total sum of the
absolute values of the feature.
Scale(x) = $\frac{\textit{x}}{\sum_{i}|\textit{x}_{i}|}$
Pow(x,y) | Pow(x, y) = $\textit{x}^{\textit{y}}$
Skew(x) | Skewness. It represents the asymmetry of a
data distribution and is expressed using the
third standard moment.
$\mu=E(\textit{x}),\quad\mu_{i}=E[(\textit{x}-\mu)^{i}],$
$\text{Skew}(\textit{x})=\left(\frac{\mu_{3}}{\mu_{2}^{1.5}}\right)=\left(\frac{E[(\textit{x}-\mu)^{3}]}{(E[(\textit{x}-\mu)^{2}])^{1.5}}\right)$
Kurt(x) | Kurtosis, a value indicating the peakedness of
a data distribution, represents how much the
observations are clustered around the mean.
Kurt(x) =
$\frac{\mu_{4}}{\mu_{2}^{2}}-3=\left(\frac{E[(\textit{x}-\mu)^{4}]}{(E[(\textit{x}-\mu)^{2}])^{2}}\right)-3$
Rank(x, t) | Time-series rank (Rank), an operator that
returns the rank of the current feature value x
among feature values from the current date up
to t days ago.
Delta(x, t) | An operator that returns the difference
between the current feature value x and the
feature value from t days ago.
$\text{Delta}(\textit{x, t})=\textit{x}-\text{Ref}(\textit{x, t})$
Argmax(x, t) | An operator that returns the date when the
feature value x was the highest within the
period from the current date up to t days ago.
Argmin(x, t) | An operator that returns the date when the
feature value x was the lowest within the
period from the current date up to t days ago.
Cond(x,y,t,f) | An operator that returns t if $\textit{x}>\textit{y}$ is true,
and f if it is false.

###  II-B Initialization with Seed Alphas

The policy must navigate the given search space to generate k alpha factors.
This implies that as we expand the search space, the complexity of the search
space that the policy must explore to generate optimal alpha factors
increases, necessitating more sophisticated architectural structures for the
policy and refinement of the search algorithm. This study sets pre-generated
formulaic alpha set as the initial seed alpha set and then performs alpha set
mining based on it. By storing a combination of pre-generated superior alpha
set in the replay buffer, it is possible to bias the search space that the
policy needs to explore, thereby reducing the complexity of the search and
enabling the creation of synergistic alphas in fewer steps. The process is
carried out in two stages. In the first stage, if there are no previously
known alpha factors, the alpha set is initialized as empty(w/o initial seed
alpha factor). Subsequently, mega alphas are generated through synergistic
formulaic alpha mining. In the second stage, if there are already known (or
generated) alpha factors, the alpha set is initialized with these factors
included. Then, by continuing to generate mega alphas in a complementary
manner, a more enhanced predictive model is ultimately constructed.

TABLE III: Main results on CSI 300. Values outside parentheses are the means, and values inside parentheses are the standard deviations across 10 runs. Method | CSI 300
---|---
IC($\uparrow$) | Rank IC($\uparrow$)
Baseline [1] | 0.045 | 0.058
(0.0036) | (0.0006)
Ours (Expanding search space) | 0.069 | 0.073
(0.0079) | (0.010)
Ours (Expanding search space + | 0.071 | 0.071
Initialization with 101 alpha) | (0.0086) | (0.008)
Ours (Expanding search space + | 0.085 | 0.087
Initialization with generated alpha set) | (0.003) | (0.003)

Figure 1:  Comparison of IC changes according to pool size variations in the
Combination model on CSI300. Display average IC after training five random
seeds for each pool size. Red: Original search space, Blue: Expanded search
space.

##  III EXPERIMENTS

###  III-A Experiment Environment

The data used in this study is from the Chinese A-shares market, which
includes six features: Open, Close, High, Low, Volume, and volume weighted
average price (VWAP). To prevent survivorship bias, the listing date of each
stock was used as its index inclusion date. Since this study only considers
long positions, the dates when stocks are excluded from the index were not
considered. The target variable is the stock price change percentage in 20
days later. The training set spans from January 1, 2009, to December 31, 2018,
the validation set from January 1, 2019, to December 31, 2019, and the test
set from January 1, 2020, to December 31, 2021. For the backtest framework,
Qlib [11] was utilized.

To evaluate the performance of the proposed technique, we sequentially applied
an expanded search space and initialization with the generated alpha set,
setting the model’s combination size to 20. Additionally, to verify accuracy,
five different random seeds were applied. As a performance metric, we use the
Pearson correlation coefficient to measure the correlation between the target
variable and alpha factor. Additionally, we use the Spearman rank correlation
coefficient to measure rank-based correlations.

TABLE IV: Top 5 formulas from Alpha 101 [10] with the highest IC. Alpha# | Expression | IC in
---|---|---
test set
Alpha | (-1 * Corr(open, volume, 10)) | 0.035
006
Alpha 099 | (Less(CSRank(Corr(Sum(((high + low) / | 0.032
2), 19.8975), Sum(Mean(volume, 60),
19.8975), 8.8136)), CSRank(Corr(low,
volume, 6.28259))) * -1)
Alpha 061 | Less(CSRank((vwap - Min(vwap, | 0.031
16.1219))), CSRank(Corr(vwap,
Mean(volume, 180), 17.9282)))
Alpha 014 | ((-1 * CSRank(Delta(Div(Sub(close, | 0.028
Ref(close, 1)), close), 3))) * Corr(open,
volume, 10))
Alpha 035 | ((Rank(volume, 32) * (1 - Rank(((close + | 0.024
high) - low), 16))) * (1 -
Rank(Div(Sub(close, Ref(close, 1)),
close), 32)))
|  |

Figure 2: IC change during the Test period when initializing the Alpha set
with Alpha 101’s formulaic alpha factor. Non-Init: Training without
initializing with a separate formulaic alpha. Blue: Original search space,
Red: Expanded search space. Green: IC for the test set of the existing Alpha
101 formula.

###  III-B Main Result

Table 3 presents the results of the performance comparison with [8]. Upon
examining the experimental results, we observed that the performance
differences among all models were minimal with changes in seed value. However,
performance improvements were observed when expanding the search space and
sequentially implementing the alpha set initialization strategy. While [8]
showed a significant value difference between IC and RankIC, the proposed
technique maintained value differences within the range of standard deviation.

Figure 3: IC change during the Test period when initializing the Alpha set with the created alpha set. Non-Init: Training without inserting any separate formulaic alpha. Blue: Original search space, Red: Expanded search space. Green: IC for the test set of alphas created when the pool size is 10. TABLE V: Parameters used in backtesting. Parameters | Value
---|---
Top K | 50
Swap N | 5
Minimum number of holding days $H$ | 20 days
Top K enter threshold $E_{th}$ | 0.0
Backtest dates | 2020-01-01 - 2021-12-31
Survivorship bias | Not included
Backtest platform | Qlib [11]
PPO agent seed | 0, 1, 2, 3, 4
Size of alpha pool | 20

###  III-C Case Study 1: Expanding Search Space

In this paper, we conducted experiments to investigate the impact of the
expansion of operators and operands on performance. The experiment observed
changes in IC by varying the pool sizes to 1, 10, 20, 50, and 100.

Figure 1 shows the results of IC changes for each pool size based on the
CSI300 data. These results demonstrate a general trend of performance
improvement with increasing pool sizes in [8]. However, the expanded search
space proposed in our study showed that even with smaller pool sizes, IC was
higher for the test period. In the expanded search space, we observed
performance improvement up to a pool size of 20, after which no further
changes in performance were noted.

###  III-D Case Study 2: Initialization with alpha 101

In this paper, we conducted experiments to determine whether initializing with
[10]’s alpha factors impacts the learning effectiveness. During the training
process, we selected 5 formulaic alpha factors from the top 101 alphas with
high IC for the CSI300 index, excluding alphas that use ’indneutralize’ and
’market cap’. The formulas used are presented in Table 4. The pool size for
the combination model was set to 20, ensuring that the formulas proposed in
[10] were included in the initial alpha set. At the same time, the setup
allowed for the possibility of these alpha factors being removed from the
alpha set by the combination model during training. For the comparison, we
named the original method without [10]’s seed alpha factors as ’Non-Init’ and
observed the performance differences depending on the presence or absence of
[10]’s seed alpha factors. We also observed the performance changes in both
the original search space and the expanded search space. In this experiment,
training was conducted with five different random seeds for the CSI300 index.

Figure 2 presents the results when the alpha set was initialized using the
formulas proposed in [10]. In the original search space, the average IC
increased by 0.0034, indicating a performance difference within the standard
deviation. Conversely, in the expanded search space, the average decreased by
0.0016, showing no significant changes in performance in both search spaces.
The formulas of alpha 101 used in the experiment displayed low ICs during the
test set period. In the original search space, only alpha 099 was included in
the alpha set for one of the five random seeds, while the other formulaic
alpha factors from alpha 101 were dropped during the generation process. These
results suggest that the formulaic alpha factors of alpha 101 were generally
removed from the alpha set creation process due to their overall low
performance.

Figure 4: Cumulative return changes during the test period with the created
mega alpha set. Set pool size to 20 and display individually after training
five random seeds. Black: CSI300 index for the test set period. Blue: alpha
set created with original search space, red: alpha set created with expanded
search space, green: alpha set created after initializing with alpha set
generated when pool size is 10.

###  III-E Case Study 3: Initialization with pre-generated alpha set

This experiment was designed to determine whether the initialization of alpha
set using synergistic formulaic alpha impacts the learning effectiveness.
During the expanded search space experiment, we measured the test ICs of the
alpha set generated when the combination model’s pool size was 10. To conduct
a comparative experiment with alpha 101, we selected a single formulaic alpha
factor from the alpha set that showed the highest IC in the test set for use
in alpha set initialization. The selected alpha set demonstrated an IC of
0.078 in the test set, and from this, a single formulaic alpha factor with an
IC of 0.07 was chosen.

The pool size of the combination model was set to 20, and 5 different random
seeds were used for training and evaluation of each technique. During the
training, selected formulaic alpha factors were added to the alpha set, and if
necessary, could be removed by the combination model. This experiment was
conducted on the CSI300 index, and we observed the impact of initialization
with seed alphas from the experiment by comparing performances based on
whether the empty alpha set, marked as ’Non-Init,’ were initialized or not.

Figure 3 shows the experimental results when alpha set with a pool size of 20
was initialized using alpha set with a pool size of 10. Initializing with a
single formulaic alpha factor resulted in an increase of 0.006 in IC compared
to initializing with empty alpha set with a pool size of 20, and an increase
of 0.005 compared to a pool size of 10. When initialized with 10 formulaic
alpha factors, there was an increase of 0.016 in IC compared to initializing
with empty alpha set with a pool size of 20, and an increase of 0.007 compared
to a pool size of 10. The significant IC differences observed in experiments
with the same pool size demonstrate that the method of initialization with the
initial seed alpha set can affect model performance.

###  III-F Backtesting

In this paper, the backtesting environment for the proposed investment
strategy utilized a Top-K/Swap-N based long only strategy. Top-K/Swap-N
involves selecting the top K stocks based on the highest alpha values to form
the portfolio. Daily, at the close of the stock market, the alpha values of
stocks in the portfolio are compared with those not included in the portfolio.
Up to N stocks are then sold, and up to N new stocks are purchased based on
this comparison. Additionally, stocks initially purchased must be held for at
least a minimum number of days (H) before they can be sold. A Top-K Enter
Threshold($E_{th}$) is also set, where the value must be higher than this
threshold at the time of purchase and lower at the time of sale. Table 5
presents the parameters used for backtesting.

Figure 4 displays the cumulative returns after backtesting for each technique
trained with 5 random seeds. It is observed that the proposed technique
consistently recorded higher cumulative returns across all seeds compared to
the existing techniques. While some seeds in the existing methods failed to
achieve excess returns over the CSI300 index during certain periods, the
application of the proposed technique resulted in excess returns over the
index for all seeds.

##  IV CONCLUSION

In this paper, we explored the efficiency of using reinforcement learning to
generate synergistic formulaic alpha collections, and confirmed the potential
of reinforcement learning in creating formulaic alpha factors from a vast
search space, demonstrating its capability to produce alpha set with high IC.
We found that initializing with pre-generated formulaic alphas led to the
creation of superior performing alpha set. Additionally, we expanded the
search space to integrate various operators and operands, and confirmed that
this expansion contributed to improved results. However, there are limitations
due to the complexity arising from the formula length restriction inherent in
the model structure. To address this, we proposed the use of predefined
auxiliary indicators. We also identified a problem where IC deteriorated at
the beginning of the training process due to not resetting the experience
buffer, suggesting the need to simultaneously initialize the alpha set and
experience buffer in future to resolve this issue.

Finally, we proposed the necessity of observing performance in other markets.
While this paper conducted experiments using the CSI300, further experiments
in various financial markets like the CSI500, NASDAQ, and KOSPI are planned to
additionally verify the universality of our approach.

## ACKNOWLEDGMENT

This work was supported in part by the National Research Foundation of Korea
(NRF) Grant funded by the Korea Government MSIT under Grant 2021R1F1A1064080.

## References

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# Restricting the Splitting Types of a Positive Density Set of Places in
Number Field Extensions

Brandon Alberts

###### Abstract.

We prove necessary and sufficient conditions for a finite group $G$ with an
ordering of $G$-extensions to satisfy the following property: for every
positive density set of places $A$ of a number field $K$ and every splitting
type given by a conjugacy class $c$ in $G$, $0\%$ of $G$-extensions avoid this
splitting type for each $p\in A$.

##  1\. Introduction

Let $K$ be a number field, $G\subseteq S_{n}$ a transitive group, and
$\mathcal{F}(K,G)$ the set of degree $n$ extensions $F/K$ with
$\textnormal{Gal}(\widetilde{F}/K)\cong G$. There has been significant study
into the proportion of $F\in\mathcal{F}(K,G)$ that satisfy certain local
conditions. That is, if $\Sigma_{p}$ is a set of isomorphism classes of
$G$-étale algebras over $K_{p}$ for each place $p$ of $K$, what proportion of
the elements $F\in\mathcal{F}(K,G)$ satisfy $F\otimes K_{p}\in\Sigma_{p}$ for
each $p$? More explicitly, determining the value of

(1) $\displaystyle\lim_{X\to\infty}\frac{\\#\\{F\in\mathcal{F}(K,G):\forall
p,F\otimes K_{p}\in\Sigma_{p},\ {\rm Ht}(F)\leq
X\\}}{\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq X\\}},$

where ${\rm Ht}:\mathcal{F}(K,G)\to\mathbb{R}^{+}$ is a height function
satisfying the Northcott property: finitely many fields have bounded height.
Most often ${\rm Ht}$ is taken to be the discriminant, the product of ramified
primes, or another multiplicative counting function as defined by Wood
[Woo09].

We give a partial answer to this question when $\Sigma=(\Sigma_{p})$ restricts
the splitting type at a positive proportion of places. Such a $\Sigma$ is
called nonadmissible by the author in [Alb23], due to the divergence of a
certain corresponding Dirichlet series. We measure the nonadmisibility of
$\Sigma$ with the following (lower) density:

$\delta_{\rm NA}(\Sigma)=\liminf_{x\to\infty}\frac{\\#\\{|p|\leq
x:\exists\text{ an unramified }G\text{-\'{e}tale algebra }F_{p}/K_{p}\text{
s.t. }F_{p}\not\in\Sigma_{p}\\}}{\pi_{K}(x)},$

where $|p|:={\rm Nm}_{K/\mathbb{Q}}(p)$ is the norm down to $\mathbb{Q}$ of
the prime, and $\pi_{K}(x)=\\#\\{p\text{ place of }K:|p|\leq x\\}$.

######  Theorem 1.1.

Let $K$ be a number field, $G\subset S_{n}$ a transitive group,
$\mathcal{F}(K,G)$ the set of degree $n$ extensions $F/K$ with
$\textnormal{Gal}(\widetilde{F}/K)\cong G$, and ${\rm
Ht}:\mathcal{F}(K,G)\to\mathbb{R}^{+}$ a height function. Suppose the inverse
Galois problem has a positive solution for $G$, that is
$\mathcal{F}(K,G)\neq\emptyset$. Then the following are equivalent:

  1. (i)

For every family of local conditions $\Sigma=(\Sigma_{p})$ with $\delta_{\rm
NA}(\Sigma)>0$,

$\lim_{X\to\infty}\frac{\\#\\{F\in\mathcal{F}(K,G):\forall p,F\otimes
K_{p}\in\Sigma_{p},\ {\rm Ht}(F)\leq X\\}}{\\#\\{F\in\mathcal{F}(K,G):{\rm
Ht}(F)\leq X\\}}=0.$

  2. (ii)

${\rm Ht}$ admits no accumulating subfields.

We say that ${\rm Ht}$ admits an accumulating subfield if there exists a
proper normal subgroup $H\trianglelefteq G$ and a $G/H$-extension $L/K$ such
that

$\limsup_{X\to\infty}\frac{\\#\\{F\in\mathcal{F}(K,G):\widetilde{F}^{H}=L,\
{\rm Ht}(F)\leq X\\}}{\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq X\\}}>0.$

In this case, the field $L$ occurs as a subfield of the Galois closure
$\widetilde{F}$ for a positive proportion of extensions
$F\in\mathcal{F}(K,G)$. Importantly, $L$ is a subfield of the Galois closure
but not necessarily of $F$.

### Acknowledgements

The author thanks the anonymous referees for helpful suggestions.

###  1.1. History

Previous work on the asymptotic proportion of $G$-extensions satisfying
certain local conditions has been primarily restricted to those $G$ for which
the asymptotic growth rate of $\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq
X\\}$ is known as $X$ tends to infinity. When this growth rate is understood
well enough, the techniques can be extended to evaluate the asymptotic rate of
growth of the numerator and denominator of (1) separately.

Such number field counting techniques have been used to compute the limit (1)
in the following cases:

  * •

$G$ abelian, ${\rm Ht}$ a multiplicative counting function, and $\Sigma$ is
such that, for all but finitely many places $p$, $\Sigma_{p}$ contains all
$G$-étale algebras over $K_{p}$ [Woo09].

  * •

$G=S_{n}$ for $n=3,4,5$, ${\rm Ht}=\textnormal{disc}$, and $\Sigma$ is such
that, for all but finitely many places $p$, $\Sigma_{p}$ contains all
$G$-étale algebras with squarefree discriminant over $K_{p}$ [BSW15].

  * •

$G=S_{3}$, ${\rm Ht}$ a multiplicative counting function, and $\Sigma$ is such
that, for all but finitely many places $p$, $\Sigma_{p}$ contains all
$G$-étale algebras over $K_{p}$ [ST22].

  * •

$G=D_{4}$, ${\rm Ht}$ the conductor, and $\Sigma$ is such that, for all places
$p$, $\Sigma_{p}$ contains all $G$-étale algebras of squarefree conductor
(with an extra condition at $p=2$) [ASVW21].

Moreover, under extra hypotheses in the above cases the limit (1) is proven to
be multiplicative of the form

$\left(\sum_{(F_{p})\in\bigoplus_{p\mid 2}\Sigma_{p}}\sigma_{2}((F_{p})_{p\mid
2})\right)\prod_{p\nmid
2}\left(\sum_{F_{p}\in\Sigma_{p}}\sigma_{p}(F_{p})\right)$

for explicit functions $\sigma_{p}:\Sigma_{p}\to\mathbb{R}_{\geq 0}$. These
extra assumptions require $\\{F\in\mathcal{F}(K,G):\forall p,\ F\otimes
K_{p}\in\Sigma_{p}\\}\neq\emptyset$, ${\rm Ht}$ is a so-called fair counting
function if $G$ is abelian, and $G$ is generated by minimal weight elements of
${\rm Ht}$ if $G=S_{3}$. The distinct behavior at primes above $2$ accounts
for certain local conditions that are not realizable by global extensions,
called inviable by Wood [Woo09]. When $G$ and ${\rm Ht}$ satisfy these
hypotheses, the corresponding cases of Theorem 1.1 follow by bounding $\Sigma$
above by larger families that, for all but finitely many places, contains all
$G$-étale algebras of $K_{p}$.

The primary strength of Theorem 1.1 is that it applies to all groups $G$ and
all heights ${\rm Ht}$, regardless of what is known about the asymptotic
growth rate of $\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq X\\}$. It is better
compared to theorems that determine number fields by their split primes, for
example the following result due to Bauer:

######  Proposition 1.2 (Bauer [Neu11, Proposition 13.9]).

If $L/K$ is Galois and $M/K$ is a finite extension, then

$\\{p\text{ split in }L/K\\}\supseteq\\{p\text{ lying under at least one
degree $1$ place $\beta$ of }M\\}$

if and only if $L\subseteq M$.

Suppose $\Sigma=(\Sigma_{p})$ has $\Sigma_{p}=\\{K_{p}^{\oplus n}\\}$ for all
$p$ split in some finite extension $M/K$. Then Bauer’s result implies

$\\{F\in\mathcal{F}(K,G):\forall p,\ F\otimes K_{p}\in\Sigma_{p},\ {\rm
Ht}(F)\leq X\\}\subseteq\\{F/K:\widetilde{F}\subseteq M\\},$

which is necessarily a finite set. With the numerator of (1) finite, the limit
necessarily exists and whether it is zero or not depends on whether there are
infinitely many $G$-extensions of $K$ or not. Theorem 1.1 is best understood
as a generalization of results like Bauer’s, following from the Chebotarev
Density Theorem.

###  1.2. Accumulating Subfields

The presence of accumulating subfields in Theorem 1.1 is worthy of
recognition, as this concept plays an important role in number field counting.

Malle predicted that

$\\#\\{F\in\mathcal{F}(K,G):|\textnormal{disc}(F/K)|\leq X\\}\sim
c(K,G)X^{1/a(G)}(\log X)^{b(K,G)-1}.$

as $X$ tends to infinity [Mal02, Mal04]. Malle gives $a(G)$ and $b(K,G)$
explicitly in terms of the index function
$\textnormal{ind}(g)=n=\\#\\{\text{orbits of }g\\}$. This is known to hold
when $G$ is abelian or $G=S_{n}$ with $n=3,4,5$, and is an important step for
studying (1) in these cases. However, it is notable that Malle’s conjecture is
wrong for some groups and base fields. Klüners famously demonstrated this by
proving that $G=C_{3}\wr C_{2}\subseteq S_{6}$ with $K=\mathbb{Q}$ is a
counter example [Klü05]. Malle predicted a growth rate $cX^{1/2}$, however
Klüners proved that

$\\#\\{F\in\mathcal{F}(\mathbb{Q},C_{3}\wr C_{2}):\widetilde{F}^{C_{3}\times
C_{3}}=\mathbb{Q}(\zeta_{3}),\ |\textnormal{disc}(F/\mathbb{Q})|\leq X\\}\gg
X^{1/2}\log X.$

The field $\mathbb{Q}(\zeta_{3})$ is an accumulating subfield.

There are more subtle examples where accumulating subfields are known to cause
issues with heuristic predictions. The leading constant $c(K,G)$ in Malle’s
prediction is more mysterious than $a(G)$ and $b(K,G)$. In the case that
$G=S_{n}$, Bhargava [Bha07] predicted that $c(K,S_{n})$ is a particular
convergent Euler product. Bhargava’s expression can be generalized to other
groups $G$, but it is known not to be equal to $c(K,G)$ in some cases with
accumulating subfields. Examples include

  * •

$G=D_{4}\subseteq S_{4}$ [CYDO02] and $G=H_{9}\subseteq S_{9}$ [FK21] for
which $c(\mathbb{Q},G)$ is explicitely expressed as a convergent sum of
special values of $L$-functions, where only the first summand agrees with
Bhargava’s construction.

  * •

The cases $G=C_{2}\wr H$ for certain groups $H$ [Klü12] and $G=S_{n}\times A$
for $n=3,4,5$ and $A$ abelian [Wan21, MTTW20]. The authors for these cases do
not give an explicit formula for the leading constant, but they do remark that
their methods imply that leading constant is given by a sum of Euler products,
only one of which agrees with Bhargava’s construction.

This is not an exhaustive list. At the time of this writing, all known
counting results for cases with accumulating subfields are known to either
disagree with Malle’s predicted growth rate or have a leading coefficient that
disagrees with Bhargava’s construction.

While accumulating subfields have been known to thwart existing heuristics,
previous examples have only been proven in cases which we know more about the
growth rate of $\\#\\{F\in\mathcal{F}(K,G):|\textnormal{disc}(F/K)|\leq X\\}$.
We will detail the existing heuristics for (1), which predict the limit will
be zero. Theorem 1.1 does not depend on knowledge of this growth rate, showing
that _any_ accumulating subfield can violate these heuristics.

##  2\. Heuristic Predictions

The Malle–Bhargava principle [Bha07, Woo19] generalizes Malle’s predictions
for counting $G$-extensions by proposing that the generating series

$\sum_{\begin{subarray}{c}F\in\mathcal{F}(K,G)\\\ \forall p,\ F\otimes
K_{p}\in\Sigma_{p}\end{subarray}}|\textnormal{disc}(F/K)|^{-s}$

should be arithmetically equivalent to the Euler product

$\prod_{p}\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{K_{p}},G)\\\
F_{p}\in\Sigma_{p}\end{subarray}}|\textnormal{disc}(F_{p}/K_{p})|^{-s}\right),$

where $F_{p}/K_{p}$ is the $G$-étale algebra corresponding to
$f_{p}\in\textnormal{Hom}(G_{K_{p}},G)$. Arithmetically equivalent here means
that both series have their rightmost pole at the same place of the same
order. Malle’s prediction is reproduced via a Tauberian theorem on the Euler
product.

The constant terms of each Euler factor is determined by the unramified étale
algebras in $\Sigma_{p}$. For all finite places, there are exactly $|G|$
possible unramified homomorphisms $f_{p}:G_{K_{p}}/I_{p}\to G$ corresponding
to unramified $G$-étale alebras, depending on the image of Frobenius. If
$\Sigma_{p}$ contains all unramified $G$-étale algebras then the constant term
is $1$, otherwise the constant term is $<1$. In the event that $\Sigma_{p}$ is
nonadmissible, infinitely many $p$ have constant term $<1$. An Euler product
of the form

$\prod_{p}\left(c_{p}+O(p^{-s})\right)$

with $c_{p}\leq 1$ for all $p$ and $c_{p}<1$ for infinitely many $p$
necessarily diverges to $0$ for all $s$ with positive real part. These cases
are not typically considered, in part because it is not clear how one should
interpret the divergence. Three possible interpretations come to mind
immediately:

  1. (1)

Divergence to $0$ predicts that there are zero such extensions.

  2. (2)

The function $0$ has no poles, which suggests the generating Dirichlet series
is holomorphic. A Tauberian theorem would then imply there are at most
finitely many such extensions.

  3. (3)

The Malle–Bhargava principle is only an expression of the “main term”. The
divergence to $0$ suggests that $0\%$ of $G$-extensions satisfy these local
conditions.

There is some credence to predicting that zero such extensions exist. There
are uncountably many ways to choose mutually disjoint nonadmissible $\Sigma$.
For each place $p$ for which $\Sigma_{p}$ does not contain all $G$-étale
algebras choose between $\Sigma_{p}$ and the complement $\Sigma_{p}^{c}$.
Given that there are only countable many number fields, comparing
cardinalities implies that uncountably many of these local restrictions are
satisfied by exactly zero number fields.

Theorem 1.1 suggests that the third interpretation applies best to _all_
families $\Sigma$, instead of just most families, as long as there are no
accumulating subfields. That the Malle–Bhargava principle is only a prediction
for the “main term” can be seen in other settings as well, such as for
counting cubic extensions. The number of $S_{3}$-cubic extensions of
$\mathbb{Q}$ with bounded discriminant admits a secondary term of order
$X^{5/6}$ [BST12, TT13], however the corresponding Euler product has no pole
at $s=5/6$. This example shows that, at best, the Malle–Bhargava principle can
make predictions only for the main term.

##  3\. The Proof

We call a sequence of field extensions $F_{1},F_{2},...,F_{m}$ containing $K$
independent over $K$ if

$\left(\prod_{i\neq j}\widetilde{F}_{i}\right)\cap\widetilde{F}_{j}=K$

for each $j=1,...,m$. These are precisely the sequences of fields for which

$\textnormal{Gal}(\widetilde{F}_{1}\widetilde{F}_{2}\cdots\widetilde{F}_{m}/K)\cong\prod_{i=1}^{m}\textnormal{Gal}(\widetilde{F}_{i}/K).$

######  Lemma 3.1.

Let $K$ be a number field, $G\subset S_{n}$ a transitive group, and
$\Sigma=(\Sigma_{p})$ a family of local conditions with $\delta_{\rm
NA}(\Sigma)>0$. Let $F_{1},...,F_{m}\in\mathcal{F}(K,G)$ be independent over
$K$ such that for all places $p$ and each $i=1,2,...,n$, $F_{i}\otimes
K_{p}\in\Sigma_{p}$. Then there exists a constant $C_{|G|,\delta_{\rm
NA}(\Sigma)}$ depending only on $|G|$ and $\delta_{\rm NA}(\Sigma)$ such that
$m\leq C_{|G|,\delta_{\rm NA}(\Sigma)}$.

###### Proof.

For each conjugacy class $c\subseteq G$, set

$A_{c}=\\{p:\text{ the unramified \'{e}tale algebra }F_{p}/K_{p}\text{ with
}\textnormal{Fr}_{p}(F_{p}/K_{p})\in c\text{ is in }\Sigma_{p}\\}.$

We will bound the density of one of these sets using the pigeonhole principle.
Certainly the upper density

$\delta^{+}(A_{c}):=\limsup_{x\to\infty}\frac{\\#\\{p\in A_{c}:p\leq
x\\}}{\pi_{K}(x)}\leq 1.$

Let $\kappa(G)$ be the number of conjugacy classes in $G$. We can bound

$\displaystyle\sum_{c}\delta^{+}(A_{c})$
$\displaystyle=\limsup_{x\to\infty}\frac{1}{\\#\pi_{K}(x)}\sum_{p\leq
x}|\Sigma_{p}|$
$\displaystyle\leq\kappa(G)-\liminf_{x\to\infty}\frac{1}{\\#\pi_{K}(x)}\sum_{p\leq
x}\left(\kappa(G)-|\Sigma_{p}|\right)$
$\displaystyle\leq\kappa(G)-\liminf_{x\to\infty}\frac{1}{\\#\pi_{K}(x)}\sum_{\begin{subarray}{c}p\leq
x\\\ \Sigma_{p}\text{ is not everything}\end{subarray}}1$
$\displaystyle=\kappa(G)-\delta_{\rm NA}(\Sigma).$

In particular,

$\displaystyle\delta_{\rm NA}(\Sigma)\leq\sum_{c}(1-\delta^{+}(A_{c}))$

is a sum of nonegative numbers. By the pigeonhole principle, at least one of
these is larger than $\frac{\delta_{\rm NA}(\Sigma)}{\kappa(G)}$, so that we
conclude that there exists a conjugacy class $c$ for which

$\delta^{+}(A_{c})\leq 1-\frac{\delta_{\rm NA}(\Sigma)}{\kappa(G)}.$

The compositum $M=F_{1}F_{2}\cdots F_{m}$ is necessarily an extension of
degree $n^{m}$ whose Galois closure has Galois group $G^{m}$ by independence.
Moreover, any prime for which $\rho_{i}(\textnormal{Fr}_{p}(M/K))\in c$ for at
least one projection map $\rho_{i}:G^{m}\to G$ necessarily belongs to $A_{c}$,
as each $F_{i}$ satisfies the local conditions of $\Sigma$. Thus, we compare

$\\{p:\rho_{i}(\textnormal{Fr}_{p}(M/K))\in c\text{ for at least one
}i\\}\subseteq A_{c}.$

By the Chebotarev density theorem, we can then compare the (upper) densities

$\frac{\\#\\{(g_{i})\in G^{m}:g_{i}\in c\text{ for at least one
}i\\}}{|G|^{m}}\leq 1-\frac{\delta_{\rm NA}(\Sigma)}{\kappa(G)}.$

The lefthand side is given by

$\displaystyle\frac{\\#\\{(g_{i})\in G^{m}:g_{i}\in c\text{ for at least one
}i\\}}{|G|^{m}}$ $\displaystyle=1-\frac{\\#\\{(g_{i})\in G^{m}:g_{i}\not\in
c\text{ for all }i\\}}{|G|^{m}}$
$\displaystyle=1-\frac{(|G|-|c|)^{m}}{|G|^{m}}$
$\displaystyle=1-\left(1-\frac{|c|}{|G|}\right)^{m}.$

In particular, this implies the lefthand side is an increasing function with

$\lim_{m\to\infty}\frac{\\#\\{(g_{i})\in G^{m}:g_{i}\in c\text{ for at least
one }i\\}}{|G|^{m}}=1>1-\frac{\delta_{\rm NA}(\Sigma)}{\kappa(G)}$

following from $\delta_{\rm NA}(\Sigma)>0$. Thus, $m$ is bounded in terms of
$|c|$, $\kappa(G)$, and $\delta_{\rm NA}(\Sigma)$. Given that there are at
most $|G|$ possibilities for $|c|$ and $\kappa(G)$, the dependence on $|c|$
and $\kappa(G)$ can be dropped in exchange for a dependence on $|G|$. ∎

We now prove Theorem 1.1. Suppose first that ${\rm Ht}$ admits no accumulating
subfields. Let $F_{1},F_{2},...,F_{m}$ be a maximal set of independent
$G$-extensions satisfying the local conditions in $\Sigma$, which exists by
Lemma 3.1. Then any other $G$-extension $F$ satisfying the local conditions in
$\Sigma$ has

$\left(\prod_{i=1}^{m}\widetilde{F}_{i}\right)\cap\widetilde{F}\neq K.$

In particular, the number of possible fields $F$ is bounded by

$\sum_{\begin{subarray}{c}H\trianglelefteq G\\\ H\neq
G\end{subarray}}\sum_{K\neq
L\leq\prod_{i}\widetilde{F}_{i}}\\#\\{F\in\mathcal{F}(K,G):\widetilde{F}^{H}=L,\
{\rm Ht}(F)\leq X\\}.$

The sums are finite. As ${\rm Ht}$ admits no accumulating subfields, the
result follows.

Conversely, suppose ${\rm Ht}$ has an accumulating subfield given by the
proper subgroup $H\trianglelefteq G$ and a $G/H$ extension $L/K$. Take
$\Sigma=(\Sigma_{p})$ such that

$\Sigma_{p}=\\{F_{p}/K_{p}:L\otimes K_{p}\subseteq\widetilde{F}_{p}\\}.$

Any prime which is not split in $L$ necessarily has $K_{p}^{\oplus
n}\not\in\Sigma_{p}$, so by $H\neq G$ and Chebotarev density we must have
$\delta_{\rm NA}(\Sigma)>0$. However, it is certainly the case that

$\\{F\in\mathcal{F}(K,G):\forall p,F\otimes K_{p}\in\Sigma_{p},\ {\rm
Ht}(F)\leq X\\}\supseteq\\{F\in\mathcal{F}(K,G):\widetilde{F}^{H}=L,\ {\rm
Ht}(F)\leq X\\}.$

Given that $L$ is an accumulating subfield, this implies

$\displaystyle\limsup_{x\to\infty}\frac{\\#\\{F\in\mathcal{F}(K,G):\forall
p,F\otimes K_{p}\in\Sigma_{p},\ {\rm Ht}(F)\leq
X\\}}{\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq X\\}}$
$\displaystyle\geq\limsup_{x\to\infty}\frac{\\#\\{F\in\mathcal{F}(K,G):\widetilde{F}^{H}=L,\
{\rm Ht}(F)\leq X\\}}{\\#\\{F\in\mathcal{F}(K,G):{\rm Ht}(F)\leq X\\}}$
$\displaystyle>0.$

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