# VIDA: A simulation model of domestic VIolence in times of social DistAncing

Lígia Mori Madeira  Department of Political Science at Federal University of
Rio Grande do Sul (UFRGS)  National Council for Technological Development
(CNPq)  Bernardo Alves Furtado  National Council for Technological Development
(CNPq)  Institute for Applied Economic Research (Ipea)  Alan Rafael Dill
Department of Political Science at Federal University of Rio Grande do Sul
(UFRGS)  National Council for Technological Development (CNPq)

###### Abstract

Violence against women occurs predominantly in the family and domestic
context. The COVID-19 pandemic led Brazil to recommend and, at times, impose
social distancing, with the partial closure of economic activities, schools,
and restrictions on events and public services. Preliminary evidence shows
that intense coexistence increases domestic violence, while social distancing
measures may have prevented access to public services and networks,
information, and help. We propose an agent-based model (ABM), called VIDA, to
illustrate and examine multi-causal factors that influence events that
generate violence. A central part of the model is the multi-causal stress
indicator, created as a probability trigger of domestic violence occurring
within the family environment. Two experimental design tests were performed:
(a) absence or presence of the deterrence system of domestic violence against
women and (b) measures to increase social distancing. VIDA presents
comparative results for metropolitan regions and neighbourhoods considered in
the experiments. Results suggest that social distancing measures, particularly
those encouraging staying at home, may have increased domestic violence
against women by about 10%. VIDA suggests further that more populated areas
have comparatively fewer cases per hundred thousand women than less populous
capitals or rural areas of urban concentrations. This paper contributes to the
literature by formalising, to the best of our knowledge, the first model of
domestic violence through agent-based modelling, using empirical detailed
socioeconomic, demographic, educational, gender, and race data at the
intraurban level (census sectors).

Keywords:  Domestic Violence, Violence against Women, Agent-based Models
(ABMs), Pandemics, Simulation, Metropolitan Regions

##  1 Introduction

Domestic violence is a global public health and human rights violation issue
(OPAS and WHO, 2017). It is estimated that around 30% of women in the world
suffer, or have suffered violence, usually committed by intimate male
partners. Many cases result in homicides, with almost 40% of female murders
committed by their partners. UN Women estimates that in countries like France,
Cyprus, Singapore, and Argentina, social distancing has increased cases of
domestic violence by 25% to 30% (Women, 2020). A report by WHO and PAHO also
denounces that domestic violence against women tends to increase in times of
emergency, including epidemics, victimising more vulnerable female groups such
as the elderly, women with disabilities, refugees, and those living in
conflict-affected areas (WHO and OPAS, 2020).

Under normal circumstances, this panorama reveals the seriousness with which
the problem must be addressed. During the pandemic with the recommendations
for isolation and social distancing, authorities, media, and researchers have
pointed to an increase in domestic violence rates in many countries. This is
also the case in Brazil, if not worse (Warken, 2020). Data released by the
National Ombudsman for Human Rights, office linked to the Ministry of Women,
Family, and Human Rights, indicate that calls to the hotline grew 14%, with an
increase of 37.6% in April compared to the same month in 2019 (Antunes, 2020).

Social distancing alters the internal dynamics of domestic violence against
women by intensifying the factors at the root of this violence, such as gender
inequality, the patriarchal system, the macho culture, and misogyny. Also,
domestic violence may increase in the context of the pandemic due to the
consequent economic impact, the overload of reproductive labour on women,
stress and other emotional effects, alcohol or drug abuse, and the reduction
of helplines and services (Alencar et al., 2020).

This study seeks to illustrate situations of domestic violence against women
before and after the COVID-19 pandemic using agent-based simulation modelling,
reproducing the main findings in the literature regarding the causes of
domestic violence. The model proposed (called VIDA) contributes to
understanding the mechanisms behind the hypothesis that staying at home as a
social distancing measure affects the magnitude of numbers related to domestic
violence. Our experimental design introduces deterrence and social distancing
measures in the model, to compare with the baseline model that reflected the
context before the pandemic.

The simulation demonstrated that the difference in the effects of deterrence
in each cohort due to the availability of the criminal justice system and
informal control networks (community) was the mechanism guiding the
explanation for the increase in cases of domestic violence against women while
following social distancing measures. Specifically, the study estimated
metrics for the economic decline and for the behaviour of staying at home as a
social distancing measure that allows comparing Brazilian metropolitan
regions. Therefore, the simulation could grasp regional differences while
contributing to understanding the main factors around social distancing and
economic decline, as well as the existence and features of deterrence
measures.

Section two provides an overview of the literature on the causes of domestic
violence committed by intimate partners. Section three describes the
methodology and presents the model VIDA. We explain the model’s idea,
assumptions, and implementation based on the construction of the indicator of
stress and the characterisation of the artificial population. Also, we discuss
the model’s validation, calibration, and tests of policies that illustrate
domestic violence, before and during the application of social distancing
measures. Section four shows the general results and analysis, based on
weighted areas, for each of the Brazilian metropolitan regions. Section five
closes this article, presenting our final considerations.

##  2 Domestic violence and femicides: a brief literature review

The literature review on partner abuse perpetration (Rothman, 2018) lists
several causes and determinants of this type of violence. The discussion
starts from the definition of the very concept of dating abuse, now understood
as physical, sexual, verbal, and emotional abuse directed at a partner
(present or ex), occurring in relationships of people of the opposite or same
sex (Center for Disease Control, 2019), not distinguishing between the type of
abuse, its severity, the consequences of these acts, the frequency with which
they occur, or differentiating the gender of perpetrator and victim. This
concept does not include violence characterised as being chronic, severe,
resulting in physical or sexual injury, caused by a person in a position of
advantage over the victim. These types generally characterise the domestic
violence against women and are taken by social movements as a core element in
their definition, given the characteristic of power and control, involving
what the literature calls patriarchal terrorism, of the abuser towards the
victim.

Based on a sociological perspective, Johnson (1995) demonstrates two different
ways to approach the issue of domestic violence. One is the family violence
perspective, in which the object refers to family conflicts due to their
frequency, the role of stress, and the adherence to normative standards that
would accept some form of violence in a family context. The other is the
feminist perspective, which restricts the focus to specific factors of the
perpetration of violence against women by intimate partners, prevailing the
historical, cultural, and social patterns of patriarchy. Other concepts
observed in the literature are intimate terrorism, a form of violence aimed at
controlling the relationship; and situational violence between partners, which
does not result from patterns of abuse and control but escalates to violence
in the emergence of specific situations of conflict (Johnson and Leone, 2005).

According to Stark (2012), the conceptualisation and, consequently, how to
address domestic violence, are divided into two models, the “violent incident
model” and the “coercive control model”. The first defines violence between
partners as a criminal offence since the late 1990s. It considers measures
such as restricting access to victims, children, and firearms, and is based on
shelter programs, legal assistance, and other support for victims. In the
violent incident model, incarceration is the preferred response for offenders,
and the level of danger observed is the main factor determining how the police
respond to domestic abuse. Abusers who continue their actions – with
sufficient time between each event – are considered, by analogy to criminals
in general, as repeat offenders. Despite the contribution to reducing severe
forms of violence, scholars question whether incarceration and judicial
protection can have a long-term impact.

The coercive control model is defended as a more comprehensive approach. It
considers that most female victims of violence are subjected to a pattern of
dominance that includes techniques of isolation, degradation, exploitation,
control, and physical violence. It is a pattern of psychological and emotional
abuse that the literature calls patriarchal or intimate terrorism (Johnson,
2008), or coercive control (Stark and Hester, 2019; Stark, 2020). Stark shares
that “this gap between what the law defines as the crime of domestic violence
and the actual tactics abusers use to subjugate their partners severely limits
the efficacy of even the most dedicated and well-trained police” (2012, p.
201). For advocates of this model, it is necessary to change from a pattern
based on specific incident response to a proactive response that redefines
partner abuse as ongoing conduct. It is crucial to apply appropriate sanctions
to stop the course of such conduct and the risk of escalation. Studies have
advanced in understanding the risk to children living in environments of
domestic violence (Katz, 2016). Despite criticism, scholars argue that police
intervention in events of domestic violence has prevented thousands of
injuries and deaths, in addition to changing the normative pattern regarding
violence against women (Stark, 2012).

The concern with causal explanations for violence between intimate partners
began in the last century, involving different mechanisms, ranging from
psychoanalytic theories, explanations of intergenerational transmission, and
social learning, to theories of situational background. Bell and Naugle (2008)
review the main theories of intimate partner violence (IPV) with variables of
interest and theoretical limitations. The main explanations encompass (a) the
idea that frustration in one area of life can lead to aggression – for
example, low autonomy in the workplace could lead to abusive compensation at
home –, (b) the assumption that childhood and the environment where a person
was raised play a role in predicting future abusive partners, and (c) the
theories mixing explanations of distal background, such as childhood abuse,
exposure to interparental aggression, and situations that trigger violence
(alcohol consumption, jealousy, and other conflicts with the partner). The
authors propose a framework that, based on different theories, offers a
contextual analysis of IPV including distal and proximal aspects, proposing
that the perpetration of abuse can be explained by antecedent factors,
motivators, unwritten rules, and legal and social consequences of violent
behaviour (Bell and Naugle, 2008). According to the authors, the advantage of
this integrating theoretical framework is the flexibility in presenting common
points between apparently contradictory findings.

Sociocultural theories range from economic explanations to approaches of
structural inequality (present in explanations of feminist theories). Models
based on inequality consider that domestic abuse reflects the power structure
in society, taking into account income, unemployment, culture, race, and
gender. Feminist theories share the understanding that an unequal society
influences the emergence of incidents of violence. They emphasise, however,
that male privilege overcomes any other social inequality in a society with a
patriarchal structure. Feminist theories originally proposed that the primary
reasons for domestic violence were related to a reinforcement of gender
standards, but later adjustments suggested the need to accommodate inter-
gender relations. Because of the prevalence of feminists in the political
coalitions that formulated programs against this type of violence, these
initiatives tend to assume that gender norms and traditional roles lead to
perpetration and victimisation.

Criminology theories have taken advantage of recent developments in the study
of urban crime and criminal “hot spots” and have relied on the routine
activity theory of criminal events and on the explanations related to target
or victim suitability to understand domestic and intimate violence (Mannon,
1997). These approaches consider crime as predatory, involving a motivated
offender, an available target, and the absence of capable guardians. Family
isolation also contributes to increasing victimisation, making the home one of
the most dangerous places for women and children.

Finally, socio-ecological models organise multiple determinants at different
levels (individual, family, peer, community, institutional, and societal).
Violent behaviour cannot be attributed to a single factor but to a wide range
of possible grouped factors, including biological and psychological risk
factors, childhood experiences, economic status, substance use, social
characteristics (friends, for instance), aspects of the intimate relationship
with the partner, workplace culture, and level of involvement with education,
public policies, disadvantaged neighbourhood, violence in the community,
access to weapons, cultural norms related to aggression, the woman’s social
status, intersectional nature, and multiple forms of oppression (Rothman,
2018).

More recent studies have distinguished the lethal risk of IPV from domestic
and family violence (DFV). The prediction of each of these types of violence
has different numbers and causes. For Ferguson and McLachlan (2020), many
tools to access risk are designed to predict violence and its escalation,
including re-victimisation and repeated offences, not necessarily to predict
homicides. While IPV is consistently identified as one of the high-risk
factors for femicides, the risk factors may not be the same, for example, for
the use and abuse of drugs and alcohol. The prevalence of drugs and alcohol
abuse is much higher in cases of physical violence than in the prediction of
femicides (Ferguson and McLachlan, 2020). On the other hand, having a gun
increases the risk of femicide by 1000%, but such a risk would not be
confirmed in cases of domestic violence. Also, the characteristics of the
relationship, the presence of children, marital status, and the duration of
the relationship do not appear to be significant risk factors for femicides,
although they may contribute to the risk of intimate violence.

##  3 Methodology

This section explains the concept of agent-based models (ABMs) and describes
the proposed model, VIDA. After this introductory part, subsection 3.1 will
address the model’s intuition, followed by subsection 3.2 that describes the
generation of the indicator of stress. Subsection 3.3 explains the
experimental design and the proposed changes in parameters related to measures
of social distancing and deterrence of domestic violence. Finally, subsection
3.4 presents the model’s sensitivity analysis and validation.

Agent-based models (ABMs) are built by describing agents (individual and
active subjects) and the interrelationship between them, as well as the
environment where these interactions occur (Epstein and Axtell, 1996). ABMs
are artificial simulations made in a computational environment. They mimic the
core mechanisms of a phenomenon under examination (Abdou et al., 2012).
Therefore, ABMs are simulacra, artificial, in silico, reduced to core aspects
of the phenomenon, and their purpose is well-defined.

Epstein and Axtell (1996) synthesise ABMs as the analysis of the functions of
transformation of agents and environment, as a result of the present
interaction between the agents and the environment. These transformation
algorithms, the sequence, and how interactions occur, are formally described
and can be verified and reproduced through the computational code provided,
which is considered a best practice (Grimm et al., 2010, 2020). ABMs are
subject to verification and scrutiny of the background theory, and the
processes and empirical data adopted.

Understanding criminal processes is an area of criminology in which ABMs can
be useful and contribute to the construction of public policies and the action
of public managers. Among the aspects of criminology, opportunity theories
investigate the motivations of aggressors to understand the environmental
contexts in which crimes occur. When developing models to test these
opportunities, it is necessary to predict the dynamic interactions of
individuals involved in each criminal event, their interactions with other
agents and the environment.

A criminal system is guided by a wide range of interrelated factors. These
factors include and are not limited to the individual perception of the
offender, the configuration and knowledge of the physical environment, the
convenience and attractiveness of the target or victim, the cognitive
representation of the environment, and other factors related to the community.

The criticisms and importance of ABMs result from the difficulty of
criminology theories hitherto addressing the importance of individual
incidents located in specific time and space. The theories have always been
concerned with general and aggregate standards, making it difficult to draw
conclusions regarding the behaviour of victims and offenders that could affect
crime rates and occurrence. It is necessary to examine the individual actors
who play essential roles in criminal events in order to understand the
patterns and characteristics of crimes better.

###  3.1 The Intuition of the Model VIDA

In normal circumstances, the reality of families encompasses individual
members participating in professional or social activities, most of the time
outside the home. During a pandemic where society is upstanding social
distancing measures, family reality changes, and individuals are at home most
of the time, even when carrying out work activities. Environmental and
socioeconomic factors continue to contribute to this experience, intensifying
social and pathological factors such as the use and abuse of drugs and
alcohol.

Regarding the issue of domestic violence against women, it is noted that the
deterrent mechanisms are compromised in a context of social distancing by at
least two aspects. First, it is hard or even impossible to be distant from the
abuser. Second, there are fewer opportunities to denounce the abusive
situation to the community, as well as less access to the police and reduced
response by the justice system.

The purpose of the model VIDA is to illustrate causal elements intrinsic to
the domestic and family environment that favours the likelihood of violence
against women. It contributes to the increase in the theory’s explanatory
capacity and offers quantitative elements to support the validity of the
causal elements embedded in the model. When the general behaviour of
conditions for violence is calibrated according to the figures available for
Brazil, it is possible to construct indicators of the magnitude of adverse
effects resulting from a specific contextual situation. In this study, VIDA
explicitly tested the increase in families’ permanence at home due to social
distancing measures taken to prevent the spread of the virus Sars-CoV-2 in the
context of the COVID-19 pandemic and the restriction to deterrence systems
available to women victims of domestic abuse.

The model is based on the following family situation: male abuser and female
victim, with or without children. The characteristics of men, women, and the
family environment are derived as samples of the population as observed in the
census of 2010 conducted by the Brazilian Institute of Geography and
Statistics (IBGE), for Brazilian metropolitan regions. Based on the family
nucleus, we build a risk indicator (stress) that accumulates additively (Berge
et al., 2014) theoretical hypotheses that contribute as a trigger to domestic
violence. This risk indicator serves as a probabilistic factor that leads to
violence.

###  3.2 Construction of the indicator of stress from familial and model
context

The indicator of stress is the central part of the model and aims at
systematically weighing in the findings of the literature. The indicator is
created successively adding: (a) exogenous characteristics of each family
member (model’s agents), (b) endogenous agents’ variables that accumulate
throughout the simulation, and (c) variables chosen by the modeller.

Aspects related to income, gender, years of education, age, race, and the
typical size of families are exogenously included. The variables are collected
from census sectors and weighted statistical areas. Endogenous variables
result from other variables or the interaction promoted by the model’s
algorithm. The modeller may change some variables to assess the model’s
response. These variables include the proportion of those staying at home
(before and during the pandemic), the relevance of gender to the indicator of
stress, level of employment, the possession or not of weapons, and presence of
substance use disorder.

The starting point to establish the indicator of stress is gender, considering
that being male leads to a higher indicator (see table 1). After that, we add
individual, household and per capita income influence, specifying that lower
income leads to comparatively higher indicators. Household income works as a
proxy for neighbourhood quality and income per capita as a proxy for domestic
(dis)comfort.

The literature review in section two further suggests four hypotheses related
to the composition of the indicator of stress. The variable years of schooling
is added to the indicator based on the hypothesis that fewer than six years of
study leads to a higher indicator of stress of 60%, and the greater the number
of years of schooling, the lower the indicator. Another hypothesis is that
being older than 18 and younger than 29 years old is a characteristic relevant
to the indicator of stress. The third hypothesis concerns the victim’s race.
In this case, black agents receive a proportional increase of 30% in the
indicator of stress. Finally, the fourth hypothesis leads to an increase in
the indicator of stress for women who are employed.

Some factors undergo multiplicative proportionality of the value to be added,
according to their relevance (see table 1). Those considered highly relevant
are multiplied by ten, and medium relevant, by five. For years of schooling
(which ranges from 0 to 17) and history of domestic abuse (number of events of
violence), values are divided by a constant (10). These values were chosen to
maintain a sensible balance among factors influencing the indicator of stress
and were validated via the sensitivity analysis performed.

Access to firearms and substance use disorder are included as aggravating
factors. Firearms are considered twice as highly relevant. Access to firearms,
once confirmed, is deterministic (always added to the indicator). Substance
use disorder, in turn, is implemented as a random additional factor because it
may or may not occur.

As such, the indicator of stress is the result of the addition and
multiplication of a number of composition individual and familial census
characteristics that follow existing literature reasoning and degree of
relevance. The composite measure is then used probabilistically as a violence
trigger that accumulates endogenously. This tentative construction captures
the subjective findings of the literature in a systemic way and enables our
implementation of a deterrence system mechanism.

Indicator of stress | Values added | Observations | Source | Relevance
---|---|---|---|---
Gender | Male 0.8, female 0.2 | Initial value | Modeller |
Income | 1 – income |  | IBGE 2010 | High
Household income | – household income |  | IBGE 2010 | Medium
Income per capita | 1 – income pc |  | IBGE 2010 | Medium
Yrs. of schooling | 1 – (schooling / 10) | \+ 60% if <6 | IBGE 2010 | High
Age | 1, If >18 age <29 |  | IBGE 2010 | High
Spouse’s race | 1, if black female | \+ 30% | IBGE 2010 |
Employment | 1, if employed | Def.: 80% pop. | Modeller | Medium
Staying at home | 0.67, if no work; 0.34 |  | Endogenous | Medium
Firearms | 1, if access; 0 | Def.: 10% pop. | Modeller | High * High
Substance use | 1, if addicted; 0 | Def.: 10% pop. | Modeller | High, random
Hist. domestic abuse | Events violence / 10 |  | Endogenous | High
Table 1: The table lists the components of the indicator of stress, how they
affect it, the source of the data and the relevance of the multiplicative
proportionality.

VIDA is a model with simple dynamics. In each simulation, VIDA starts by
generating the artificial population sample according to the choices made by
the modeller. The model then runs for a brief period of 10 steps. During this
period, there is some stochastic neutral volatility associated with employment
and income. Moreover, there is enough time for an endogenous history of
violence to build up. After 200 steps, the violence levels have stabilized and
the model is interrupted and output is generated. The model also includes a
fixed model scale that divides the indicator by a 1,000 so to calibrate the
cases of attacks to the available empirical data. The execution of each step
of the model consists of:

  1. 1.

Update of the indicator of stress for all agents.

  2. 2.

Verification of the violence trigger.

  3. 3.

Verification of the need to seek help from public policies against violence,
the deterrence process.

The full model is available at the COmSES platform and GitHub. It is built
upon the Mesa framework (Masad and Kazil, 2015). The reader interested in
using VIDA should focus on the ’violence’ folder. Basically, the construction
of the stress indicator is included as an agent step at agents.py and the
creation of the sampled families from data is made within model.py, using data
and processes from the folder ’input’. Generalization.py iterates the model
200 times and tests the experimental design. Outputs are also included in the
repository.

###  3.3 Experimental design: social distancing and deterrence of domestic
violence

When the model appropriately represents the phenomena researched, it is
possible to ask what-if questions. Therefore, it is possible to test the
likely effects of interventions that have not yet taken place. Our
experimental design includes two consequences of the pandemic: (a) the
imposition of social distancing measure of staying at home, and (b) the access
(or lack thereof) of the victim to protective measures in case of domestic
violence (deterrence system).

Victims searching for help when suffering from domestic violence typically
have three similarly-sized behaviour patters, according to data compiled by
the Brazilian Senate. The first group is of victims that never denounce.
Victims from the second group in contrast denounce immediately after the first
case of violence. Victims from the third group denounce after the third event
of violence. Following this segmentation, the victims in the model VIDA are
distributed equally among these three groups and act accordingly.

The model then considers that women who denounce have a 50% chance of
obtaining protection. Among those who denounce and obtain protection, 50% have
the chance to see their abuser convicted. The victim’s situation (denounce,
protection, or abuser conviction) is included in the model as an element that
reduces the abuser’s indicator of stress (see table 2. Therefore, it works to
reduce the chances of further violence cases, and each request for support in
the deterrence system reduces the likelihood of future violent events.

Indicator of stress | Values added | Source | Relevance
---|---|---|---
Denounce | – 1 | Endogenous | Medium
Protective measure | – 1 | Endogenous | High
Conviction | – 1 | Endogenous | High
Table 2: Influence of deterrence system on aggressor’s indicator of stress.

###  3.4 Comments on calibration, validation and sensitivity analysis

VIDA is simulated with different sampled data. The results presented refer to
the median of results obtained after the 200-fold simulation. The model was
calibrated in order to approximate the number of notifications of domestic
violence against women made by health agencies in 2011, using data from the
2010 census. In 2011, data from the Senate indicated 73.7 notifications per
100,000 women.

The literature offers different concepts of model validation (Galán et al.,
2009; Guerini and Moneta, 2017; Moss, 2008; Ngo and See, 2012; Wilensky and
Rand, 2015). However, there is some consensus around the idea that models are
validated according to their purpose, context and ontology (Edmonds and Meyer,
2017). A model proposed for predicting is expected to offer accurate and
confirmed results that were not part of the model beforehand. A model designed
to help to understand a given mechanism does not have to prove a predicted
empirical data. In any case, a model must be anchored in its purpose and
reflect the available literature.

Accordingly, VIDA’s purpose is to illustrate causal elements that are
subjectively present in the literature, thus helping understand the phenomena
and enable the implementation of what-if analysis. As a first tentative to
systematically represent a multi-causal intricate theoretical mechanism – that
of domestic violence trigger – and the empirical foundation of the input of
the model, we believe VIDA results are sufficiently validated to enable
comparisons among metropolitan regions and insights into intraurban
differences.

The sensitivity analysis results presented in the next section contributes to
demonstrate the robustness of the model as it assesses how the variation of
parameters and processes interfere in the results. Moreover, the analysis of
parameter changes contributes to comprehending the effects these changes
generate.

The modeller running VIDA can control the variation of the parameters that are
not endogenous or derived directly from the IBGE population sampling. Thus,
the modeller may alter data on: relevance of gender; employment percentage
(and consequently the population staying at home and whether receiving
salaries); access to firearms, and substance use disorder. It is possible to
test volatility of employment and income value, increasing the chances of
moving from employed to unemployed and vice versa, as well small increases or
decreases in income.

##  4 Results

First, we present the results of the sensitivity analysis, where the
parameters chosen by the modeller are submitted to variation. Second, we
present the baseline results and those for the experimental design. Then, we
show a comparison of results across 46 metropolitan regions in Brazil.
Finally, we detail intraurban results for Brasília and Porto Alegre. Once the
model was calibrated with nationwide data from 2011, showing 73.7 reports of
violence per 100,000 women, it was possible to observe the differences
resulting from the implicit variation of race, gender, age, family size,
presence of weapons, and economic capacity of families in different
metropolitan regions and intrametropolitan neighbourhoods.

###  4.1 Sensitivity analysis

The results presented adopt the standard parameters of the model for the
metropolitan region of Brasília (see table 3), with the presence of deterrence
and absence of social distancing measures, that is, the situation before the
pandemic. The model runs 200 simulations for each tested parameter.

The indicator of stress according to gender, fixed at 0.2 for women and
varying for men between 0.1 and 0.9, has little influence on the results
related to increasing cases and denounces. The percentage increase between the
lowest and highest value was 2.24% for cases of violence and 2.48% for
denounces.

The percentage of employment between 0.1 and 0.9, led to an increase of 4.85%
in domestic violence against women, and an increase of 5.36% in the number of
denounces. This slight increase is in accordance with the suggestion of the
literature that working women are more subject to domestic violence.

As expected, the introduction of a high * high relevance multiplicative
proportionality for access to firearms into the model resulted in an increase
in cases and denouncing. Varying from 0.1 to 0.9, access to firearms increased
violence cases significantly by 72.43% and denouncing by 66.52%.

The occurrence of cases of domestic violence related to substance use disorder
included an additional random factor, considering that such cases may or may
not occur. Thus, the increase (from 0.01 to 0.5) in the chances of being
addicted resulted in an increase in violence cases of 7.44% and the number of
denounces of 7.86%.

Indicator component level | Cases | Denounces
---|---|---
Gender stress |  |
0.1 | 169.57 | 60.05
0.44 | 172.28 | 61.72
0.9 | 173.45 | 61.58
Percentage of employed |  |
0.1 | 165.66 | 58.65
0.44 | 169.14 | 60.81
0.9 | 174.11 | 61.98
Firearms |  |
0.1 | 173.33 | 62.4
0.44 | 368.05 | 115.29
0.9 | 628.79 | 186.41
Substance use disorder |  |
0.01 | 170.99 | 60.94
0.22 | 178.09 | 63.68
0.5 | 184.73 | 66.14
Table 3: Sensitivity analysis: results of cases and denounces per 100,000
women when varying modeller choices of indicator os stress components.

###  4.2 Simulation results

The model VIDA with typical parameters (see table 1) was simulated 200 times
to report the average numbers obtained. The average number of denounces was
61.73, in an environment with 174.49 domestic violence cases per 100,000
women.

Table 4 shows the results for the experimental design tests. Social distancing
measures, which forces the agents to stay at home, regardless of whether they
are working, leads to an increase in violence cases of 8.86%. At the same
time, denounces reduced by almost a third (29.25%), compared with the baseline
simulated model with the default parameters.

Additionally, the inclusion of the deterrence system in the model demonstrated
a reduction in domestic violence cases by only 3.53%, when compared with its
absence (178.89 versus 172.79 cases per 100,000 women). These findings
corroborate the literature (Campbell et al., 2003), which states that judicial
protection and incarceration have a greater long-term impact and work to
reduce severe forms of violence, especially femicides. According to the
literature, other aspects contribute to reducing the risk of severe forms of
domestic violence, such as the expansion of employment, preventing substance
use disorder, and restricting access to weapons.

Deterrence | Social distancing | Cases | Denounces
---|---|---|---
FALSE | FALSE | 180.38 | 0
FALSE | TRUE | 197.35 | 0
TRUE | FALSE | 173.62 | 61.97
TRUE | TRUE | 190.74 | 47.38
Table 4: Results per 100,000 women for different alternatives of social
distancing and deterrence system.

A relevant contribution of the simulation is the ability to compare the
results across different metropolitan regions with varying composing
attributes, such as age and gender, family size, average wages, years of
schooling, and race distribution, according to the 2010 demographic census
sample data. Figure 1 shows the number of cases of domestic violence per
100,000 women in the high-populated areas in Brazilian metropolitan regions.
The data simulated in the model suggest that larger and wealthier metropolitan
regions have proportionately fewer cases of domestic violence against women
than regions with smaller and poorer populations.

Figure 1: Number of domestic violence cases per 100,000 women in Brazilian
high-populated areas metropolitan regions as classified by the IBGE.

Empirical analysis for Brazil have shown increasing rates in the regions North
and Northeast, and decreasing rates in the South and Southeast of Brazil in
the past decade. When shifting the focus of analysis to the size of the
municipalities, there is a general decrease in homicides in large cities (with
more than 500,000 inhabitants) and an increase in medium-sized (between
100,000 and 500,000 inhabitants) and small municipalities (with less than
100,000 inhabitants) (Cerqueira et al., 2019). Despite their sizes, another
important explanatory element that may affect the results is the lack of
infrastructure and deterrence systems in regions with a larger number of
domestic violence cases (Madeira et al., 2018). Our simulation results reveal
patterns that are similar to the empirical data, showing notably fewer cases
in metropolitan regions of the South and Southeast. When cases are more
frequent in these regions, they are identified in metropolitan regions that
encompass smaller cities, located inland and with less access to deterrence
apparatus.

Further, we compared intraurban simulation results with the current reality
(see figure 2). The comparison data was retrieved from the database of
Indicators of Violence Against Women — Maria da Penha Law, for the year 2019.
111Retrieved from https://www.ssp.rs.gov.br/indicadores-da-violencia-contra-a-
mulher The analysis from the municipalities located in the metropolitan region
of Porto Alegre 222Retrieved from
https://atlassocioeconomico.rs.gov.br/regiao-metropolitana-de-porto-alegre-
rmpa suggest that results show similar patterns of domestic violence when
comparing the simulation and reality. Neighbourhoods corresponding to
municipalities with high rates of violence, such as Viamão, Taquara,
Cachoeirinha and Guaíba figure in the simulation among the groups with the
highest incidence. Alvorada, with a proven record of violence, appears in the
simulation with an western portion that is in the second tier of highest
violence, along with an eastern, more sparsely inhabited area that is less
violent. The small municipality of Nova Santa Rita which is comprised by a
single neighbourhood (statistically weighted area) shows a dispersed average
and stands out from a simulated low number of attacks, as compared to
intermediate levels in the obtained data.

Figure 2: Cases per 100,000 women. Mean obtained from 500 simulations for
domestic violence cases per 100,000 women in weighted areas (IBGE), for the
metropolitan region of Porto Alegre

Figure 3 shows the violence cases in the metropolitan region of Brasília
(Federal District). In this case, it appears that the eastern, very large and
sparsely populated rural portion of the region and some peripheral urban
centres are more susceptible to violence against women than in the most
populated areas. Our results show neighboring areas of high violence, side by
side with less violent areas out of the Federal District bounds, in peripheral
areas of the south, namely Valparaíso de Goías and Luziânia, and Águas Lindas
de Goiás to the west. Those results suggest the possibility of high intraurban
distinctions of violence that are not captured by municipalities averages, but
that may be corroborated by police hot-spot analysis.

Figure 3: Cases per 100,000 women. Mean obtained from 500 simulations for
domestic violence cases per 100,000 women in weighted areas (IBGE), for the
metropolitan region of Brasilia.

This exercise can be replicated for all metropolitan regions. The adoption of
real data allows studying specific weighted areas and simulating the
geospatial distribution of domestic violence cases, which contributes to the
formulation of policies to address this issue.

##  5 Final considerations

This article presents a first attempt to formalise a multi-causal model of
domestic violence, based on a series of family characteristics and comparable
artificial data, generated from intra-urban (large neighbourhoods) census
data. The model illustrates theoretical mechanisms and quantifies explanatory
aspects of violence against women. Additionally, it incorporates aspects that
emerged with the COVID-19 pandemic – forced coexistence in the home
environment due to social distancing measures – and elements of the deterrence
system to address domestic abuse. As such, the paper contributes to the
understanding of the complex mechanisms encompassing violence against women in
Brazil.

The results suggest an increase in violence against women of about 10% with
the implementation of social distancing measures emphasising staying at home,
which is confirmed by recent data that point to a 30% decrease in denounces in
several countries and 25% in some Brazilian states (Bueno et al., 2020). More
densely populated metropolitan regions appear to have fewer violence cases per
100,000 women compared to smaller metropolitan regions. The same comparative
pattern appears in the intra-metropolitan analysis between high-populated
areas and rural or peripheral areas.

In addition, the model, its codification, and the process of generating
artificial population samples based on census data are publicly available.
This practice aims to facilitate the dissemination of the model and its use by
the scientific community, analysts, and public policy managers.

Future work involves detailing empirical data in the model, specifically
identifying possession of firearms per household, and feedback effects within
the neighbourhood. Data on the access of firearms can greatly contribute to
the reliability of the proposed spatial analysis because the presence of
firearms seems to be the strongest among the causes of femicides.

The elaboration of public policies depends on the right diagnoses capable of
supporting appropriate interventions. The area of violence and crime routinely
deals with the problem of lack of data, even more so in cases of high
underreporting such as domestic violence. The ethical limitations of producing
experiments in the area can also be overcome by using agent-based simulating
models, which make it possible to replicate reality and illustrate different
effects in an auspicious way. However, researchers who have been trying to
understand the risks related to domestic abuse, especially in the context of
the pandemic, have suggested the need to use data parsimoniously since it
takes time to understand the changes in the dynamics of collecting and
recording information. The model demonstrated to be useful to simulate and
anticipate data, allowing to illustrate this type of violence and its
georeferenced disparities. This tool proves to be of great importance in this
particular moment where all kinds of inequalities stand out in the country.

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# On the Torsion Congruence for Zeta Functions of Totally Real Fields

Yubo Jin  Department of Mathematical Sciences
Durham University
South. Rd.
Durham, DH1 3LE, U.K<EMAIL_ADDRESS>
###### Abstract.

In this short note, we study the special values for zeta functions of totally
real fields using the Shintani’s cone decomposition. We prove certain
congruence between the special values for zeta functions under the prime
degree field extension. This congruence implies the ‘torsion congruence’
proved by Ritter-Weiss ([14]) which is crucial in the proof of the
noncommutative Iwasawa main conjecture for totally real fields.

###### Key words and phrases:

Special zeta values; Shintani’s cone decomposition; Noncommutative Iwasawa
main conjectures

###### 2020 Mathematics Subject Classification:

11R42; 11R23

##  1\. Introduction and the Main Theorem

Generalizing the classical commutative Iwasawa main conjectures, the
noncommutative Iwasawa main conjectures are proposed in [5, 9] (see also [10,
Section 2] for totally real fields). By the algebraic result of Burns [2] (see
also [10, 12, 13] for totally real fields) on the $K$-theory of noncommutative
Iwasawa algebras, the proof of the noncommutative Iwasawa main conjectures are
reduced to the proof of certain congruences between special $L$-values and the
commutative Iwasawa main conjectures.

For totally real fields, certain congruences and the noncommutative Iwasawa
main conjectures are proved in [10, 14]. We recall the congruence in [14]. Let
$F$ be a totally real number field of degree $n$ over $\mathbb{Q}$ with ring
of integers $\mathcal{O}_{F}$. Fix an odd prime $p$ and let $K$ be a totally
real Galois extension over $F$ of degree $p$ with ring of integers
$\mathcal{O}_{K}$. Fix a finite set $S$ of nonarchimedean places of $F$
containing all places above $p$ and all places ramify in $K$. Let $F_{S}$
(resp. $K_{S}$) be the maximal abelian extension of $F$ (resp. $K$) unramified
at all nonarchimedean places outside $S$. Denote
$\Sigma=\mathrm{Gal}(K/F),G_{S}=\mathrm{Gal}(F_{S}/F),H_{S}=\mathrm{Gal}(K_{S}/K)$.
Let $\lambda_{F}\in\mathbb{Z}_{p}[[G_{S}]]$ (resp.
$\lambda_{K}\in\mathbb{Z}_{p}[[H_{S}]]$) be the Serre’s pesudomeasures [16]
interpolating the special values of zeta functions for $F$ (resp. $K$). Two
pesudomeasures can be compared by the transfer map $\mathrm{ver}:G_{S}\to
H_{S}$. The ‘torsion congruence’ in [14] is stated in the following theorem.

######  Theorem 1.1.

Let $\mathcal{I}$ be the ideal in the ring $\mathbb{Z}_{p}[[H_{S}]]^{\Sigma}$
of $\Sigma$-fixed points of $\mathbb{Z}_{p}[[H_{S}]]$ consisting elements of
the form
$\sum_{\sigma\in\Sigma}\alpha^{\sigma},\alpha\in\mathbb{Z}_{p}[[H_{S}]]$. Then
for $g\in G_{S},h=\mathrm{ver}(g)\in H_{S}$,

(1.1)
$\mathrm{ver}\left((1-g)\lambda_{F}\right)\equiv(1-h)\lambda_{K}\qquad\text{
mod }\mathcal{I}.$

The proof of this theorem based on two steps. First is reducing the theorem to
a rather simpler congruence of twisted partial zeta functions ([14,
Proposition 4]). Second is proving this congruence (in [14, Proposition 4]) by
studying the zeta functions through constant terms of Eisenstein series as in
[8]. The ‘torsion congruence’ (1.1) can be also stated by replacing
$\lambda_{F},\lambda_{K}$ for other $p$-adic measures interpolating special
$L$-values of motives over $F,K$. In [18, 19], the $p$-adic $L$-functions for
Hilbert modular forms are also studied via the constant terms of Eisenstein
series. However, proving the ‘torsion congruence’ for Hilbert modular forms
using the similar method seems much more complicated as the Eisenstein series
used [18, 19] are of ‘Klingen type’ whose Fourier coefficients are more
difficult to study.

The aim of this paper is to give another proof of the ‘torsion congruence’
(1.1). That is, we study the zeta functions using the Shintani’s cone
decomposition which is also used in [3] constructing $p$-adic measures. This
method also has a cohomological explanation in [4]. We hope this new proof can
provide a new approach on studying the ‘torsion congruence’. In particular, we
expect the ‘torsion congruence’ for Hilbert modular forms can be studied via
the cohomological approach as in [1].

We now introduce the main theorem proved in this paper. Fix an integral ideal
$\mathfrak{f}\subset\mathcal{O}_{F}$ such that $p|\mathfrak{f}$. Denote
$\mathrm{Cl}_{F}^{+}(\mathfrak{f})$ be the narrow ray class group modulo
$\mathfrak{f}$. Denote $F^{+}(\mathfrak{f})=\\{\alpha\in F:\alpha\equiv
1\text{ mod }\mathfrak{f},\alpha\gg 0\\}$ and
$U_{F}^{+}(\mathfrak{f})=\mathcal{O}_{F,+}^{\times}\cap F^{+}(\mathfrak{f})$.
Let $\mathfrak{f}^{\prime}=\mathfrak{f}\mathcal{O}_{K}$ and similarly denote
$\mathrm{Cl}_{K}^{+}(\mathfrak{f}^{\prime}),K^{+}(\mathfrak{f}),U_{K}^{+}(\mathfrak{f})$.
Denote $N_{K/F},N_{K}:=N_{K/\mathbb{Q}},N_{F}:=N_{F/\mathbb{Q}}$ and
$\mathrm{Tr}_{K/F},\mathrm{Tr}_{K}:=\mathrm{Tr}_{K/\mathbb{Q}},\mathrm{Tr}_{F}:=\mathrm{Tr}_{F/\mathbb{Q}}$
for norms and traces between number fields. We abuse the same notation for
both norm of elements and norm of ideals. For integral ideals
$\mathfrak{a}\subset\mathcal{O}_{F}$ (resp.
$\mathcal{A}\subset\mathcal{O}_{K}$), we define the partial zeta functions

(1.2) $\displaystyle\zeta_{\mathfrak{f}}^{F}(\mathfrak{a}^{-1},s)$
$\displaystyle=\sum_{\mathfrak{b}\sim_{\mathfrak{f}}\mathfrak{a}^{-1}}N_{F}(\mathfrak{b})^{-s},$
$\displaystyle\zeta_{\mathfrak{f^{\prime}}}^{K}(\mathcal{A}^{-1},s)$
$\displaystyle=\sum_{\mathcal{B}\sim_{\mathfrak{f}^{\prime}}\mathcal{A}^{-1}}N_{K}(\mathcal{B})^{-s},$

where the sum is over all integral ideals $\mathfrak{b}$ (resp. $\mathcal{B}$)
that are in the same class of $\mathrm{Cl}_{F}^{+}(\mathfrak{f})$ (resp.
$\mathrm{Cl}_{K}^{+}(\mathfrak{f}^{\prime})$) as $\mathfrak{a}^{-1}$ (resp.
$\mathcal{A}^{-1}$). Fix prime ideals $\mathfrak{c}\subset\mathcal{O}_{F}$ and
$\mathfrak{c}^{\prime}\subset\mathcal{O}_{K}$ relatively prime to
$\mathfrak{f}$ with residue degree $1$ above the prime number $c$. The twisted
partial zeta functions are defined by

(1.3) $\displaystyle\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},s)$
$\displaystyle=N_{F}(\mathfrak{c})^{1-s}\zeta_{\mathfrak{f}}^{F}((\mathfrak{ac})^{-1},s)-\zeta_{\mathfrak{f}}^{F}(\mathfrak{a}^{-1},s),$
$\displaystyle\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1};s)$
$\displaystyle=N_{K}(\mathfrak{c}^{\prime})^{1-s}\zeta_{\mathfrak{f^{\prime}}}^{K}((\mathcal{A}\mathfrak{c}^{\prime})^{-1},s)-\zeta_{\mathfrak{f^{\prime}}}^{K}(\mathcal{A}^{-1},s).$

Our main theorem, rephrased from [14, Proposition 4], is stated as follow.

######  Theorem 1.2.

Assume $\mathcal{A}^{\sigma}=\mathcal{A}$ for all $\sigma\in\mathrm{Gal}(K/F)$
and take $\mathfrak{a}=\mathcal{A}\cap\mathcal{O}_{F}$. Then, for all integers
$k\geq 1$,

(1.4)
$\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)\equiv\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},1-pk)\qquad\text{
mod }p\mathbb{Z}_{p}.$

We review the Shintani’s cone decomposition as well as the Colmez’s explicit
construction in Section 2. The key ingredient is a relative version of the
cone decomposition in Proposition 2.1, 2.2. In Section 3, we summarize the
computations for the special values of zeta functions using the cone
decomposition following [15, Section 3] and the proof of Theorem 1.2 is then
completed in Section 4.

##  2\. Shintani’s Cone Decomposition

Let $F$ be a totally real number field of degree $n$ over $\mathbb{Q}$ with
$\mathrm{Gal}(F/\mathbb{Q})=\\{\tau_{1},...,\tau_{n}\\}$. Then each $\tau_{i}$
defines an embedding $F\to\mathbb{R}$. We can thus embed $F$ into
$\mathbb{R}^{n}$ by sending $x\mapsto(\tau_{1}(x),...,\tau_{n}(x))$. Denote
$\mathcal{O}_{F,+}^{\times}$ be the group of totally positive units whose
action on $\mathbb{R}_{>0}^{n}$ is given by the componentwise multiplication
$\epsilon.(x_{1},...,x_{n})=(\epsilon x_{1},...,\epsilon x_{n})$. In [17],
Shintani provide a cone decomposition for the fundamental domain
$\mathbb{R}_{>0}^{n}/\mathcal{O}_{F,+}^{\times}$ (see also [11, (9.3)]). A
more explicit construction of the fundamental domain is given by Colmez in [6]
(see also [7, Section 2]).

We first review the explicit construction of Colmez. Let
$\\{\epsilon_{1},...,\epsilon_{n-1}\\}$ be a set of linearly independent
totally positive units generates $\mathcal{O}_{F,+}^{\times}$. For a
permutation $\tau\in S_{n-1}$ of indices $\\{1,...,n-1\\}$, set

(2.1)
$f_{i,\tau}=\epsilon_{\tau(1)}\epsilon_{\tau(2)}...\epsilon_{\tau(i-1)}=\prod_{j=1}^{i-1}\epsilon_{\tau(j)},\qquad
1\leq i\leq n,\tau\in S_{n-1},$

where $f_{1,\tau}$ is understood as $1=(1,...,1)\in\mathbb{R}_{+}^{n}$. We
make the following assumption (called the Unit Assumption $F$):

(2.2)
$\mathrm{sgn}(\tau)=\mathrm{sgn}\left(\det(f_{1,\tau},f_{2,\tau},...,f_{n,\tau})\right)\qquad\text{
for all }\tau\in S_{n-1}.$

Here $\mathrm{sgn}(\tau)$ is the signature of the permutation $\tau$ and
$\mathrm{sgn}\left(\det(f_{1,\tau},f_{2,\tau},...,f_{n,\tau})\right)$ is the
signature of the determinant of the $n\times n$ matrix whose rows are
$f_{i,\tau},1\leq i\leq n$.

For $\tau\in S_{n-1}$ and $J$ a non-empty subset of $\\{1,...,n\\}$, we
consider open cones

(2.3) $C_{\tau,J}=\sum_{i\in J}\mathbb{R}_{>0}\cdot f_{i,\tau}.$

Define an equivalence relation $C_{\tau,J}\sim C_{\tau^{\prime},J^{\prime}}$
if there exists some $\epsilon\in\mathcal{O}_{F,+}^{\times}$ such that
$C_{\tau,J}=\epsilon C_{\tau^{\prime},J^{\prime}}$. Let
$\mathcal{C}_{F}=\left\\{C_{\tau,J}\right\\}/\sim$ be a class of open cones.
Then the Shintani decomposition is given by

(2.4)
$\mathbb{R}_{>0}^{n}=\coprod_{\epsilon\in\mathcal{O}_{F,+}^{\times}}\coprod_{C\in\mathcal{C}_{F}}\epsilon\cdot
C.$

Fix an odd prime $p$. Let $K$ be a totally real Galois extension of degree $p$
over $F$ with ring of integers $\mathcal{O}_{K}$. Take $\sigma$ to be the
nontrivial generator of $\mathrm{Gal}(K/F)$, then
$\mathrm{Gal}(K/\mathbb{Q})=\\{\tau_{i}^{j}:1\leq i\leq n,0\leq j\leq p-1\\}$
with $\tau_{i}^{j}=\tau_{i}\sigma^{j}$. We can thus embed $K$ into
$\mathbb{R}^{pn}$ by sending
$x\mapsto(\tau_{1}^{0}(x),...,\tau_{n}^{0}(x),...,\tau_{1}^{p-1}(x),...,\tau_{n}^{p-1}(x))$.
We also embed $\mathbb{R}^{n}$ into $\mathbb{R}^{pn}$ by sending

(2.5)
$(x_{1},...,x_{n})\mapsto\iota(x_{1},...,x_{n}):=(x_{1},...,x_{n},x_{1},..,x_{n},...,x_{1},...,x_{n})$

with $x_{1},...,x_{n}$ repeating $p$ times. The action of the group of totally
positive units $\mathcal{O}_{K,+}^{\times}$ on $\mathbb{R}_{>0}^{pn}$ is given
by the componentwise multiplication as for the field $F$. Replace $F$ by $K$
above, we have a cone decomposition for the fundamental domain
$\mathbb{R}_{>0}^{pn}/\mathcal{O}_{K,+}^{\times}$. For our purpose, we need to
compare two decomposition $\mathbb{R}_{>0}^{n}/\mathcal{O}_{F,+}^{\times}$ and
$\mathbb{R}_{>0}^{pn}/\mathcal{O}_{K,+}^{\times}$.

######  Proposition 2.1.

There is a finite class of open cones $\mathcal{C}_{F}$ in
$\mathbb{R}_{+}^{n}$ and a finite class of open cones $\mathcal{C}_{K}$ in
$\mathbb{R}_{+}^{pn}$ such that

(2.6) $\displaystyle\mathbb{R}_{>0}^{n}$
$\displaystyle=\coprod_{\epsilon\in\mathcal{O}_{F,+}^{\times}}\coprod_{C\in\mathcal{C}_{F}}\epsilon\cdot
C,$ $\displaystyle\mathbb{R}_{>0}^{pn}$
$\displaystyle=\coprod_{\epsilon\in\mathcal{O}_{K,+}^{\times}}\coprod_{C\in\mathcal{C}_{K}}\epsilon\cdot
C.$

Moreover, we have
$\mathcal{C}_{K}=\mathcal{C}_{K}^{0}\sqcup\mathcal{C}_{K}^{\prime}$ such that:
(1) The cones $C\in\mathcal{C}_{K}^{0}$ are obtained from cones
$C_{0}\in\mathcal{C}_{F}$ by the map
$\iota:\mathbb{R}_{>0}^{n}\to\mathbb{R}_{>0}^{pn}$, i.e. $C=\iota(C_{0})$.
Hence $x^{\sigma}=x$ for any $x\in C$ and $\sigma\in\mathrm{Gal}(K/F)$,
(2) The cones $C\in\mathcal{C}_{K}^{\prime}$ satisfy $x^{\sigma}\neq x$ for
any $x\in C$ and $\sigma\in\mathrm{Gal}(K/F)$.

###### Proof.

We first take $\\{\epsilon_{1},...,\epsilon_{n-1}\\}$ generates
$\mathcal{O}_{F,+}^{\times}$ satisfying the Unit Assumption $F$. The existence
of these units is proved by [6, Lemme 2.1]. We recall the idea of the proof.
For any $M>0$, denote

$l^{F}_{i}(M)=\left(\frac{-M}{n-1},...,\frac{-M}{n-1},M,\frac{-M}{n-1},...,\frac{-M}{n-1}\right)\in\mathbb{R}^{n},\qquad
1\leq i\leq n$

with $M$ in the $i$-th entry. There is a constant $r_{F}$ depending only on
$\mathcal{O}_{F,+}^{\times}$ such that for all $M>0$ there exists
$\epsilon_{1},...,\epsilon_{n-1}\in\mathcal{O}_{F,+}^{\times}$ such that
$\mathrm{Log}\epsilon_{i}\in B(l^{F}_{i}(M),r_{F})$. To satisfy the Unit
Assumption $F$, it suffices to take such
$\\{\epsilon_{1},...,\epsilon_{n-1}\\}$ for some large enough $M$. Here
$\mathrm{Log}:\mathbb{R}_{>0}^{n}\to\mathbb{R}^{n}$ is the map
$(x_{1},...,x_{n})\mapsto(\mathrm{log}x_{1},...,\mathrm{log}x_{n})$ and
$B(x,r)$ is the open ball in $\mathbb{R}^{n}$ with origin $x$ and radius $r$.

We use the same notation $\mathrm{Log}:\mathbb{R}_{>0}^{pn}\to\mathbb{R}^{pn}$
and $B(x,r)$ the open ball in $\mathbb{R}^{pn}$ with origin $x$ and radius $r$
for the field $K$. For $1\leq i\leq n$ and $M^{\prime}>0$, let
$l_{i}^{K}(M^{\prime})=\iota(l_{i}^{F}(M^{\prime}))$ be the embedding of
$l_{i}^{F}(M^{\prime})$ under the map (2.5). For $n+1\leq i\leq pn$, we set

$l^{K}_{i}(M^{\prime})=\left(\frac{-M^{\prime}}{pn-1},...,\frac{-M^{\prime}}{pn-1},M^{\prime},\frac{-M^{\prime}}{pn-1},...,\frac{-M^{\prime}}{pn-1}\right)\in\mathbb{R}^{pn},$

with $M^{\prime}$ in the $i$-th entry. Then $\\{l_{i}^{K}(M^{\prime}):1\leq
i\leq qn\\}$ forms a basis of the trace zero hyperplane
$\\{(x_{1},...,x_{pn})\in\mathbb{R}^{pn}:x_{1}+...+x_{pn}=0\\}$ in
$\mathbb{R}^{pn}$. There is a constant $r_{K}$ depending only on
$\mathcal{O}_{K,+}^{\times}$, such that for all $M^{\prime}>0$ there exists
$\epsilon_{1}^{\prime},...\epsilon^{\prime}_{pn-1}\in\mathcal{O}_{K,+}^{\times}$
such that $\mathrm{Log}\epsilon_{i}^{\prime}\in
B(l_{i}^{K}(M^{\prime}),r_{K})$. We enlarge $r_{K}$ if necessary to assume
$r_{K}>r_{F}$. The same computations in [6, Lemme 2.1] show that to satisfy
the Unit Assumption $K$, it suffices to take such
$\\{\epsilon^{\prime}_{1},...,\epsilon^{\prime}_{pn-1}\\}$ for some large
enough $M^{\prime}$.

Take $\epsilon_{1},...,\epsilon_{n-1}\in\mathcal{O}_{F,+}^{\times}$ satisfying
$\mathrm{Log}\epsilon_{i}\in B(l_{i}^{F}(M),r_{F})$ for some $M$. Then,
viewing as elements in $\mathcal{O}_{K,+}^{\times}$, they satisfy
$\mathrm{Log}\epsilon_{i}\in B(l_{i}^{K}(M^{\prime}),r_{K})$ if

$r_{F}-r_{K}-M<M^{\prime}<r_{K}-r_{F}+M.$

Therefore, taking $M,M^{\prime}$ large enough satisfying above inequality, we
can find $\\{\epsilon_{1},...,\epsilon_{pn-1}\\}$ generates
$\mathcal{O}_{K,+}^{\times}$ satisfying the Unit Assumption $K$ and
$\\{\epsilon_{1},...,\epsilon_{n-1}\\}$ generates $\mathcal{O}_{F,+}^{\times}$
satisfying the Unit Assumption $F$. The proposition then follows from Colmez’s
explicit constructions of the cones. ∎

Let

(2.7)
$C(\lambda_{1},...\lambda_{g}):=\mathbb{R}_{>0}\lambda_{1}+...+\mathbb{R}_{>0}\lambda_{g},$

be a cone of $\mathbb{R}_{>0}^{pn}$ generated by
$\lambda_{1},...,\lambda_{g}\in K$. We define the action of
$\mathrm{Gal}(K/F)$ on this cone by

(2.8)
$C(\lambda_{1},...,\lambda_{g})^{\sigma}:=C(\lambda_{1}^{\sigma},...,\lambda_{g}^{\sigma}).$

######  Proposition 2.2.

The decomposition of $\mathbb{R}_{>0}^{pn}$ in (2.6) can be further refined
such that

(2.9)
$\mathbb{R}_{>0}^{pn}=\coprod_{\epsilon\in\mathcal{O}_{K,+}^{\times}}\coprod_{C\in\mathcal{C}_{K}}\epsilon\cdot
C,\qquad\mathcal{C}_{K}=\mathcal{C}_{K}^{0}\sqcup\mathcal{C}_{K}^{1}\sqcup\mathcal{C}_{K}^{2}$

with
$\mathcal{C}_{K}^{0},\mathcal{C}^{\prime}_{K}=\mathcal{C}_{K}^{1}\sqcup\mathcal{C}_{K}^{2}$
as in Proposition 2.1 and
(1) The cones $C\in\mathcal{C}_{K}^{1}$ satisfy $C^{\sigma}=C$ for any
$\sigma\in\mathrm{Gal}(K/F)$,
(2) If $C\in\mathcal{C}_{K}^{2}$, then $C\neq
C^{\sigma}\in\mathcal{C}_{K}^{2}$ for any $\sigma\in\mathrm{Gal}(K/F)$.

###### Proof.

Write $\mathcal{C}_{K}^{\prime}=\\{C_{1},...,C_{m}\\}$ and let $\sigma$ be a
nontrivial generator of $\mathrm{Gal}(K/F)$. For $1\leq i_{1},...,i_{p}\leq
m$, denote

$C_{(i_{1},...,i_{p})}=C_{i_{1}}\cap C_{i_{2}}^{\sigma}\cap
C_{i_{3}}^{\sigma^{2}}\cap...\cap C_{i_{p}}^{\sigma^{p-1}}.$

Note that

$\mathcal{O}_{K,+}^{\times}\left(C_{1}\cup...\cup
C_{m}\right)=\mathcal{O}_{K,+}^{\times}\left(C_{1}^{\sigma}\cup...\cup
C_{m}^{\sigma}\right)=...=\mathcal{O}_{K,+}^{\times}\left(C_{1}^{\sigma^{p-1}}\cup...\cup
C_{m}^{\sigma^{p-1}}\right),$

we have

$\mathcal{O}_{K,+}^{\times}\left(C_{1}\cup...\cup
C_{m}\right)=\mathcal{O}_{K,+}^{\times}\coprod_{1\leq i_{1},...,i_{p}\leq
m}C_{(i_{1},...,i_{p})}.$

Therefore, the proposition follows by taking

$\displaystyle\mathcal{C}_{K}^{1}$
$\displaystyle=\\{C_{(i_{1},...,i_{p})}:i_{1}=...=i_{p}\\},$
$\displaystyle\mathcal{C}_{K}^{2}$
$\displaystyle=\\{C_{(i_{1},...,i_{p})}:i_{1},...,i_{p}\text{ not identically
same}\\}.$

∎

##  3\. Special Values of Zeta Functions

In this section, we summarize the computations in [15, Section 3] for the
special values of zeta functions using the cone decomposition. Let $F$ be a
totally real number field and take ideals
$\mathfrak{f},\mathfrak{c},\mathfrak{a}$ as in Section 1. We consider the
twisted partial zeta function
$\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},s)$ defined in (1.3). For $0\leq
j\leq c$, denote $\xi^{F}_{j}:\mathcal{O}_{F}\to\mathbb{C}^{\times}$ be the
additive character given by $\xi^{F}_{j}(x)=e^{2\pi
i\frac{j\mathrm{Tr}_{F}(x)}{c}}$. Then by [15, Proposition 3.6],

(3.1)
$\zeta^{F}_{\mathfrak{f,c}}(\mathfrak{a}^{-1};s)=N_{F}(\mathfrak{a})^{s}\sum_{j=1}^{c}\sum_{\alpha\in\left(\mathfrak{a}\cap
F^{+}(\mathfrak{f})\right)/U^{+}_{F}(\mathfrak{f})}\xi^{F}_{j}(\alpha)N_{F}(\alpha)^{-s}.$

Recall that we have a decomposition

(3.2)
$\mathbb{R}_{>0}^{n}=\coprod_{\epsilon\in\mathcal{O}_{F,+}^{\times}}\coprod_{C\in\mathcal{C}_{F}}\epsilon\cdot
C.$

By multiplying a certain scalar on generators of $C$, we assume each
$C\in\mathcal{C}_{F}$ can be written as

(3.3)
$C=C(\lambda_{1},...,\lambda_{g}):=\mathbb{R}_{>0}\cdot\lambda_{1}+...+\mathbb{R}_{>0}\cdot\lambda_{g}\qquad\text{
with }\lambda_{1},...,\lambda_{g}\in\mathfrak{af}$

and for such cone $C$ we denote

(3.4) $\displaystyle C^{1}$
$\displaystyle:=C^{1}(\lambda_{1},...,\lambda_{g})=\\{t_{1}\lambda_{1}+...+t_{g}\lambda_{g}:0<t_{1},...,t_{g}\leq
1\\},$ $\displaystyle C^{0}(\beta)$
$\displaystyle:=C^{0}(\beta;\lambda_{1},...,\lambda_{g})=\beta+\mathbb{Z}_{>0}\lambda_{1}+...+\mathbb{Z}_{>0}\lambda_{g}\qquad\text{
for }\beta\in C^{1}.$

Then by [3, Lemma 1] (see also [15, Section 3.3]),

(3.5) $\mathfrak{a}\cap F^{+}(\mathfrak{f})=\coprod_{\epsilon\in
U_{F}^{+}(\mathfrak{f})}\coprod_{\begin{subarray}{c}C\in\mathcal{C}_{F}\\\
\beta\in\mathfrak{a}\cap F^{+}(\mathfrak{f})\cap
C^{1}\end{subarray}}\epsilon\cdot C^{0}(\beta).$

Hence

(3.6)
$\zeta^{F}_{\mathfrak{f,c}}(\mathfrak{a}^{-1};s)=N_{F}(\mathfrak{a})^{s}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{F}\\\
\beta\in\mathfrak{a}\cap F^{+}(\mathfrak{f})\cap
C^{1}\end{subarray}}\sum_{\alpha\in
C^{0}(\beta)}\xi_{j}^{F}(\alpha)N_{F}(\alpha)^{-s}.$

For a cone $C^{0}(\beta):=C^{0}(\beta;\lambda_{1},...\lambda_{g})$ and
$\xi^{F}_{j}$ an additive character of $\mathcal{O}_{F}$ as above, we define a
power series

(3.7)
$P_{F}(C^{0}(\beta),\xi^{F}_{j};T)=\frac{\xi^{F}_{j}(\beta)(1+T)^{N_{F}(\beta)}}{\prod_{i=1}^{g}\left(1-\xi^{F}_{j}(\lambda_{i})(1+T)^{N_{F}(\lambda_{i})}\right)}.$

As proved in [15, Corollary 4.2], this is indeed a power series in
$\mathbb{Z}_{p}[[T]]$. Let $\Delta=(1+T)\frac{\partial}{\partial T}$ be the
differential operator acting on $\mathbb{Z}_{p}[[T]]$. The computations in
[15, Section 3.4, 3.5] show that

(3.8)
$\zeta^{F}_{\mathfrak{f,c}}(\mathfrak{a}^{-1},1-k)=N_{F}(\mathfrak{a})^{1-k}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{F}\\\
\beta\in\mathfrak{a}\cap F^{+}(\mathfrak{f})\cap
C^{1}\end{subarray}}\Delta^{k-1}P_{F}(C^{0}(\beta),\xi^{F}_{j};T)|_{T=0},$

for any integer $k\geq 1$. We remark that in [15], the special zeta value
$\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},1-k)$ is first interpolated by a
power series in $\mathbb{C}[[T_{1},...,T_{n}]]$. After applying the projection
operator $\Omega:\mathbb{C}[[T_{1},...,T_{n}]]\to\mathbb{C}[[T]]$ defined in
[15, Section 3.5], we then obtained above one variable power series. This is
important for us as now we can compare the power series obtained from
different fields $F$ and $K$ and thus the special zeta values
$\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},1-pk)$ and
$\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)$.

##  4\. Proof of the Main Theorem

We now start the proof of our main Theorem 1.2.

Let $K/F$ be a totally real Galois extension of degree $p$ and take ideals
$\mathfrak{f^{\prime}},\mathfrak{c^{\prime}},\mathcal{A}$ as in Section 1. In
particular, we assume $\mathcal{A}^{\sigma}=\mathcal{A}$ for any
$\sigma\in\mathrm{Gal}(K/F)$ and $\mathfrak{a}=\mathcal{A}\cap\mathcal{O}_{F}$
as in Theorem 1.2. Take $\mathcal{C}_{F}$ be a class of open cones in
$\mathbb{R}_{>0}^{n}$ and
$\mathcal{C}_{K}=\mathcal{C}_{K}^{0}\sqcup\mathcal{C}_{K}^{1}\sqcup\mathcal{C}_{K}^{2}$
a class of open cones in $\mathbb{R}_{>0}^{pn}$ as in Proposition 2.2.
Replacing $F$ by $K$ in the previous section we have

(4.1)
$\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)=N_{K}(\mathcal{A})^{1-k}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{K}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}\Delta^{k-1}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)|_{T=0}$

where for cone $C=C(\lambda_{1},...,\lambda_{g})\in\mathcal{C}_{K}$, $\beta\in
C^{1}$ and $\xi_{j}^{K}$ an additive character of $\mathcal{O}_{K}$,

(4.2)
$P_{K}(C_{0}(\beta),\xi_{j}^{K};T)=\frac{\xi^{K}_{j}(\beta)(1+T)^{N_{K}(\beta)}}{\prod_{i=1}^{g}\left(1-\xi^{K}_{j}(\lambda_{i})(1+T)^{N_{K}(\lambda_{i})}\right)}.$

We further write

(4.3)
$\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)=\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K,0}(\mathcal{A}^{-1},1-k)+\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K,1}(\mathcal{A}^{-1},1-k)+\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K,2}(\mathcal{A}^{-1},1-k),$

where for $i=0,1,2$,

(4.4)
$\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K,i}(\mathcal{A}^{-1},1-k)=N_{K}(\mathcal{A})^{1-k}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}^{i}_{K}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}\Delta^{k-1}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)|_{T=0}.$

######  Lemma 4.1.

For $i=1,2$ and any $1\leq j\leq c$,

(4.5) $\sum_{\begin{subarray}{c}C\in\mathcal{C}^{i}_{K}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)\in
p\mathbb{Z}_{p}[[T]].$

###### Proof.

Note that by the definition of $P_{K}(\beta,C,\xi_{j}^{K};T)$, we have

$P_{K}(C_{0}(\beta),\xi_{j}^{K};T)=P_{K}(C_{0}(\beta)^{\sigma},\xi_{j}^{K};T)\qquad\text{
for }\sigma\in\mathrm{Gal}(K/F).$

For $i=1$, since for any $\sigma\in\mathrm{Gal}(K/F)$ we have
$\mathcal{A}^{\sigma}=\mathcal{A}$ by assumption and $C^{\sigma}=C$ for any
$C\in\mathcal{C}_{K}^{1}$ by Proposition 2.2, the Galois group
$\mathrm{Gal}(K/F)$ acts transitively on $\mathcal{A}\cap
K^{+}(\mathfrak{f}^{\prime})\cap C^{1}$ and each orbit has $p$ elements. Thus,

$\displaystyle\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}\\\
\beta\in\left(\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\right)/\mathrm{Gal}(K/F)\end{subarray}}\sum_{\sigma\in\mathrm{Gal}(K/F)}P_{K}(C^{0}(\beta^{\sigma}),\xi^{K}_{j};T)$
$\displaystyle=$ $\displaystyle
p\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}\\\
\beta\in\left(\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\right)/\mathrm{Gal}(K/F)\end{subarray}}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)\in
p\mathbb{Z}_{p}[[T]].$

For $i=2$, since $C\neq C^{\sigma}\in\mathcal{C}_{K}^{2}$ for any
$C\in\mathcal{C}_{K}^{2}$ and $\sigma\in\mathrm{Gal}(K/F)$ by Proposition 2.2,
the Galois group $\mathrm{Gal}(K/F)$ acts transitively on
$\mathcal{C}_{K}^{2}$ and each orbit has $p$ elements. Thus,

$\displaystyle\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta),\xi^{K}_{j};T)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}/\mathrm{Gal}(K/F)\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}\sum_{\sigma\in\mathrm{Gal}(K/F)}P_{K}(C^{0}(\beta)^{\sigma},\xi^{K}_{j};T)$
$\displaystyle=$ $\displaystyle
p\sum_{\begin{subarray}{c}C\in\mathcal{C}^{1}_{K}/\mathrm{Gal}(K/F)\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta)^{\sigma},\xi^{K}_{j};T)\in
p\mathbb{Z}_{p}[[T]].$

∎

Hence,

(4.6)
$\displaystyle\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)$
$\displaystyle\equiv\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K,0}(\mathcal{A}^{-1},1-k)$
$\displaystyle\equiv
N_{F}(\mathfrak{a})^{1-pk}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{K}^{0}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}\Delta^{k-1}P_{K}(C^{0}(\beta),\xi_{j}^{K};T)|_{T=0}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
mod }p.$

By Proposition 2.1 on $\mathcal{C}_{K}^{0}$,

(4.7) $\sum_{\begin{subarray}{c}C\in\mathcal{C}_{K}^{0}\\\
\beta\in\mathcal{A}\cap K^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta),\xi_{j}^{K};T)=\sum_{\begin{subarray}{c}C\in\mathcal{C}_{F}\\\
\beta\in\mathfrak{a}\cap F^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}P_{K}(C^{0}(\beta),\xi_{j}^{K};T),$

where on the write hand side we are writing $C_{0}(\beta)$ for its image in
$\mathbb{R}_{>0}^{pn}$ under the embedding (2.5).

######  Lemma 4.2.

Let $C\in\mathcal{C}_{F}$ and $\beta\in\mathfrak{a}\cap
F^{+}(\mathfrak{f})\cap C^{1}$. Then

(4.8)
$\sum_{j=1}^{c}\Delta^{k-1}P_{K}(C_{0}(\beta),\xi_{j}^{K};T)|_{T=0}\equiv\sum_{j=1}^{c}\Delta^{pk-1}P_{F}(C_{0}(\beta),\xi_{j}^{F};T)|_{T=0}\text{
mod }p.$

###### Proof.

For integers $m$ and $M$ with $0\leq m\leq M$ define the rational function

$B_{m,M}(x)=(-1)^{m}\sum_{i=m}^{M}\left(\begin{array}[]{c}i\\\
m\end{array}\right)\left(\frac{x}{x-1}\right)^{n}.$

Then by [15, Theorem 4.1], we have

$\displaystyle P_{K}(C_{0}(\beta),\xi_{j}^{K};T)$ $\displaystyle\equiv$
$\displaystyle\frac{\xi_{j}^{K}(\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{K}(\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}(1+T)^{N_{K}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{K}(\lambda_{i}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
mod }T^{k+1}.$

Hence the left hand side of (4.8) equals

$\displaystyle\sum_{j=1}^{c}\frac{\xi_{j}^{F}(p\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{F}(p\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}N_{K}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})^{k-1}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{F}(p\lambda_{i}))$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{c}\frac{\xi_{j}^{F}(\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{F}(\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}N_{F}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})^{p(k-1)}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{F}(\lambda_{i}))$
$\displaystyle\equiv$
$\displaystyle\sum_{j=1}^{c}\frac{\xi_{j}^{F}(\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{F}(\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}N_{F}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})^{pk-1}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{F}(\lambda_{i}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
mod }p.$

Applying [15, Theorem 4.1] for the right hand side of (4.8), we have

$\displaystyle P_{F}(C_{0}(\beta),\xi_{j}^{F};T)$ $\displaystyle\equiv$
$\displaystyle\frac{\xi_{j}^{F}(\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{F}(\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}(1+T)^{N_{F}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{F}(\lambda_{i}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
mod }T^{pk+1}.$

Hence the right hand side of (4.8) equals

$\sum_{j=1}^{c}\frac{\xi_{j}^{F}(\beta)}{\prod_{i=1}^{g}\left(1-\xi_{j}^{F}(\lambda_{i})\right)}\sum_{m_{1},...,m_{g}=0}^{kpn}N_{F}(\beta+m_{1}\lambda_{1}+...+m_{g}\lambda_{g})^{pk-1}\prod_{i=1}^{g}B_{m_{i},kpn}(\xi_{j}^{F}(\lambda_{i})),$

which completes the proof of the lemma. ∎

Therefore, Theorem 1.2 follows from

(4.9)
$\displaystyle\zeta_{\mathfrak{f^{\prime},c^{\prime}}}^{K}(\mathcal{A}^{-1},1-k)$
$\displaystyle\equiv
N_{F}(\mathfrak{a})^{1-pk}\sum_{j=1}^{c}\sum_{\begin{subarray}{c}C\in\mathcal{C}_{F}\\\
\beta\in\mathfrak{a}\cap F^{+}(\mathfrak{f}^{\prime})\cap
C^{1}\end{subarray}}\Delta^{pk-1}P_{F}(C^{0}(\beta),\xi_{j}^{F};T)|_{T=0}$
$\displaystyle\equiv\zeta_{\mathfrak{f,c}}^{F}(\mathfrak{a}^{-1},1-pk)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
mod }p.$

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