Vincent Huon
email: huon@lmgc.univ-montp2.fr

Bertrand Wattrisse
email: wattriss@lmgc.univ-montp2.fr

Moulay Saïd

El Youssoufi

; Moulay

Saïd El

André Chrysochoos

Mechanical Behavior of Terra Cotta Ceramics Characterized by Kinematic Full-Field Measurements

Keywords: Mechanical properties, Clays, Measurement, Construction materials

teaching and research institutions in France or abroad, or from public or private research centers.

Mechanical behavior of terra cotta ceramics characterized by kinematic full-field measurements

Introduction

Terra cotta ceramics are often used in residential house building. They are nearly always associated with other civil engineering materials, e.g., terra cotta beams with a core of prestressed concrete are widely used for building floors, terraces, or flat roofs. In this kind of composite element, terra cotta serves as a sacrificial formwork, i.e., concrete is considered to be the only material that supports prestressing forces. However, from a mechanical standpoint, terra cotta has remarkable strengths ͑i.e., often better than those of standard concrete͒ that are often not effectively utilized. The work presented in this paper is a part of an overall study carried out to characterize the anisotropic thermohygromechanical behavior of terra cottas used as building material in civil engineering projects. From an industrial standpoint, the goal is to develop numerical computer-aided design tools that can be used to optimize structural elements. This optimization must simultaneously take into account parameters related to thermal comfort and those associated with the mechanical strength of the designed structure.

In the recent past, the classical use of terra cotta bricks did not require in-depth knowledge on their mechanical properties, since they were mainly loaded in compression. Now, the use of the terra cotta as a composite structural component calls for a more rigorous analysis of its contribution to the overall behavior of the structure. Thus, we studied the heterogeneous and anisotropic properties of this material using strain field measurements. These measurements were obtained by digital image correlation ͑DIC͒ techniques with the aim of checking the degree of heterogeneity of the ceramic structures and characterizing the anisotropy of the terra cotta. This paper is structured as follows: First, we review the material properties that led us to propose a transverse isotropic model to describe the elasticity of ceramics. Second, we show how the elastic parameters were locally derived from tests conducted on elementary structures considered as representative volumes of the material. Finally, to check the influence of the discrepancy noted in the elastic constants, we compare strain patterns predicted by a three-dimensional ͑3D͒ finite element ͑FE͒ computation with those obtained by the DIC.

Terra Cotta

Argillaceous soils are the primary products for manufacturing terra cotta. In most cases, they are used with additives ͑sand, limestone, etc.͒ to enhance the characteristics of structural elements, or to modify the functional characteristics or the aspect of the finished products. Clays are hydrated aluminosilicates whose lamellar structure can fix a certain quantity of water between folia. Four stages are required to obtain the end product: 1. Preparation: To obtain an argillaceous mixture after proportioning and crushing the components; 2. Forming: Generally by extrusion; 3. Drying: To eliminate almost all of the water used during the forming stage; and 4. Firing: The duration depends on the size of the terra cotta components, generally within the 800-1,200°C temperature range. These different stages induce an anisotropic and heterogeneous thermomechanical behavior of the terra cotta structural elements. The extrusion process involves a privileged direction and leads to transverse isotropy of the material. This also induces gradients of elastic properties throughout the structure, especially between the boundary and the core of the extruded parts. These gradients are often amplified and disturbed by the drying and firing processes, because of mass and heat transfers, by the material heterogeneity due to the irregular distribution of the pores, and the varied size grains of the mixture despite the preparation stage. We used sets of specimens randomly extracted from an extruded structure to estimate discrepancies in the elastic parameters.

To define transverse isotropy axes, we introduced L as the direction of extrusion, T as the direction across the layers of terra cotta, and R as the direction in the plane of the layers, i.e., orthogonal with L and T ͑Fig. 1͒. The five elastic constants to be determined are then: Young's moduli E L = E R and E T ; Poisson's ratios LR and LT = RT ; Shear moduli G LT = G RT . Due to the isotropy of the plane of the layers, the shear modulus G LR can be written as G LR = E L / ͓2͑1+ LR ͔͒. Using engineering notations, the strain-stress relationship may be written as

LL RR TT LR RT LT = ΄ 1 E L - LR E L - LT E L 00 0 - LR E L 1 E L - LT E L 00 0 - LT E L - LT E L 1 E T 00 0 000 1+ LR E L 00 000 0 1 2G LT 0 000 0 0 1 2G LT ΅ LL RR TT LR RT LT ͑1͒
where IJ and IJ ϭmatrix components of the strain and stress tensors with respect to the LRT frame.

Experimental Protocol

Material and Specimens

An electron probe microanalysis ͑EPM͒ was used to determine the grain size and nature. The maximum grain size was 1 mm, including:

• Quartz grains ͑the most common͒;

• Calcite grains of up to 200 m in size ͑the least common͒; and

• A low percentage of small feldspath grains.

The chemical composition of the terra cotta studied in this paper is given in Table 1. Building engineering standards do not give any dimensional specifications for terra cotta specimens. Moreover, in the scientific literature, we found no examples of terra cotta specimens adapted for classical mechanical tests such as tension compression tests, shearing, and bending tests. Thus, we based our study on guidelines for the characterization of cements or mortars with granular sizes similar to that of terra cotta. Although compression tests are generally performed on specimens with a ratio of 2, we decided to use cubic samples of reduced sizes ͑sides: 15 mm͒ in order to ensure an adequate homogeneity of these elementary structures at the risk of getting a triaxiality effect due to the compression plates. The specimens were machined using a water jet cutting process to avoid geometrical defects. The compression tests, performed on cubic samples with transverse isotropy axes, gave access to the three elastic moduli and the three Poisson ratios.

The shear modulus G LR was determined in the shearing zone of the Y-shaped specimen whose cross section is given in Fig. 2. For this type of geometry, it is easy to obtain a layer plane parallel

Experimental Setup

The experimental setup involved a 100 kN tension-compression servomechanical testing machine. Digital images were recorded during the test by a charge coupled device ͑CCD͒ camera set in front of the sample, Fig. 3. The lens axis of the camera was fixed according to the frame of the testing machine and remained perpendicular to the surface of the sample. The CCD sensor had eight square megapixels distributed in a 3,500 line ϫ 2,300 column grid. The camera provided maximum space resolution from 0.01-0.02 mm/ pixel with the "macro" lens used during the tests.

For images of maximum size, the camera can capture up to five images a second digitized in eight bits.

Displacement and Strain Measurements

The inplane displacement components were computed by a direct DIC algorithm on each point M 0 ͑i 0 , j 0 ͒ of a virtual grid defined in the reference configuration. The position of the discrete maximum of the discrete intercorrelation function gives the displacement of point M 0 with a one pixel resolution. Between two images, I 1 and I 2 , separated by a small strain increment, is written as

͑k,l͒ ͑k,l͒ ͓ -RZ 2 , RZ 2 ͔ = ͚ i-i 0 =-CZ/2 CZ/2 ͚ j-j 0 =-CZ/2 CZ/2 I 1 ͑i, j͒ • I 2 ͑i + k, j + l͒ ͱ ͚ i-i 0 =-CZ/2 CZ/2 ͚ j-j 0 =-CZ/2 CZ/2 I 1 2 ͑i, j͒ • ͱ ͚ i-i 0 =-CZ/2 CZ/2 ͚ j-j 0 =-CZ/2 CZ/2 I 2 2 ͑i + k, j + l͒

͑2͒

In Eq. ͑2͒, CZ stands for the correlation zone ͓i.e., the M 0 neighborhood defining the optical signature of "point" M 0 ͑i 0 , j 0 ͔͒, and RZϭresearch zone ͑i.e., the M 0 neighborhood where the optical signature is tracked͒. These zones correspond to the domain of variation of ͑i , j͒ and ͑k , l͒, respectively. To obtain subpixel measurements, we used a polynomial interpolation of in the vicinity of its discrete maximum ͑Oulamara et al. 1988͒. Heterogeneous strain field analysis considers small "gauge lengths" that induce a poor signal-to-noise ratio. Consequently, a local least-squares fitting of the displacement data is performed before any differentiation: The displacement field is locally ap-proximated in the neighborhood of each point M 0 by a given function. Both the shape of the approximation function and the size of the approximation zone ͑AZ͒ may affect the accuracy of the strain measurement. Here, bilinear functions were used as they are associated with a locally constant deformation. For more information, the reader may refer to ͑Wattrisse et al. 2001͒. The image processing performance was tested, respectively, in experimental and analytic cases corresponding to rigid body motion and to homogeneous or heterogeneous strain ͑Wattrisse et al. 2002͒. Indeed, analytical checks were necessary, because it is impossible to impose a given strain field to a real material specimen.

Experimental Results

Global Response of the Samples

The compression tests, performed at constant velocity ͑0.1 mm s -1 ͒, were initially conducted on cubic specimens in the three directions L, R, and T. Fig. 4 illustrates the results obtained. This feature is consistent with the transverse isotropy hypothesis. At the beginning of the loading process, the cubic structure has an elastic response until the maximum load was reached. Then, the softening part of the curve corresponds to the propagation of microcracks throughout the sample. Compression in the direction T was performed perpendicular to the plane of the layers. This configuration gave the structure greater compliance and seemed to be less favorable for the propagation and coalescence of microscopic cracks.

Material Response

Based on previous results, the tests were carried out, while limiting maximum compressive loading in order to ensure an elastic response of the structure. These tests were performed at constant velocity ͑0.1 mm s -1 during loading and 0.5 kN s -1 during un-loading͒. The strain field measurements were first used to check the boundary conditions during loading. Some examples of strain patterns obtained by DIC are shown in Fig. 5 and correspond to components LL and TT of the strain tensor. Both fields were extracted from a compression test in direction L at maximum loading ͑Ϸ11 kN͒.

We observed slightly heterogeneous deformation fields, which were not in full agreement with the kinematics of a simple compression test. The strain level curve distributions highlighted that this nonuniformity was partly due to the boundary conditions at the specimen surfaces in contact with the compression plates. The irregular patterns noted for level curves could also have been associated with the material heterogeneities and, of course, with the noise on strain measurements. The fields could be interrelated with the FE calculation using displacement boundary conditions corresponding to those observed experimentally. The stress-strain curve corresponding to measurements obtained in the central area of the specimen is plotted in Fig. 6. The compression strain LL is directly derived from speckle image processing, while the compression stress LL is determined by assuming a uniform stress distribution over the cross section. Then, a simple regression is used to estimate the elasticity modulus ͑E L in Fig. 6͒. All results for the estimation of Young's modulus are given in Table 2. Note that the standard deviations obtained on moduli using a random set of about ten samples were about 4 GPa for mean values of about 20 GPa. This discrepancy is very substantial and illustrates the heterogeneity between specimens. Fig. 7 presents estimates of Poisson ratio LT =-TT / LL for different LL . As previously underlined, this computation assumes a uniform compression state in the central part of the sample. The increasing discrepancy of measurements observed when strain components were around zero was associated with degradation of the signal to noise ratio. Therefore, only estimates of the Poisson's ratios for ͉ LL ͉ Ͼ 5 ϫ 10 -4 were considered. Tests were performed on Y-shaped specimens to determine the shear modulus G LR ͑see Fig. 2͒. These tests were performed at constant velocity ͑0.01 mm s -1 during loading and 0.5 kN s -1 during unloading͒. We noted two phases in the structural response: The first corresponded to a quasielastic behavior, while the second started with the inception of the first macroscopic crack. Fig. 8 illustrates the potentials of correlation methods regarding microcrack detection. The displacement field features ͑here, component U L ͒ enabled us to visualize crack onset and propagation at the sample surface as soon as its opening was not parallel to the direction of the chosen displacement component. The amplitude of the displacement discontinuity may be related to the crack opening.

Fig. 10 presents the RL distribution and shows the significant sliding ͑"shearing"͒ zones. In these zones, we estimated a mean

Validity Checks

Cubic Specimens

The elastic modulus estimates were checked by comparing the local stress values obtained in the central area by 3D FE computations with the compression stress deduced from the loading data. The numerical model takes transverse isotropy elasticity into account and supposes that the material is homogeneous. The elasticity tensor takes the predetermined mean values into account. It also supposes that the axis of compression is one of the transverse isotropy axes. Loading was modeled by displacement fields imposed at the surfaces in contact with the compression plates. These boundary conditions were directly extracted from the dis-placement data obtained by DIC. Fig. 9 presents the results obtained for a simulated compressive test in the L direction. Considering a cross section placed in the medium part of the specimen, computations enabled us to estimate, at maximum elastic loading, a mean value for the compression stress pattern ͑ ¯LL ͒ comp of about 44.5 MPa. The standard deviation associated with this distribution was about 0.6 MPa. Hence, the stress state, imposed by the experimental boundary conditions, was uniform in a first good approximation. Compression stress ͑ LL ͒ exp can also be estimated assuming, as usual, a uniform stress state over the cross section and using the load measurements. We found that ͑ LL ͒ exp was about 43.5 MPa. The difference between ͑ LL ͒ exp and ͑ ¯LL ͒ comp remained small, thus, partly confirming the previous approximations.

Y-Shaped Specimens

We checked the overall consistency of the characterization procedure by comparing strain fields obtained in Y-shaped samples with those resulting from FE computations. Loading was modeled by displacements obtained by DIC imposed at surfaces in contact with the compression plates. The computed and experimental strain patterns are compared in Fig. 10. We noted a satisfactory global correlation between both distributions, regardless of the strain tensor component. Moreover, we noted a close correlation between the measurement and the calculation when comparing the overall loading applied to the structure ͑Table 3͒.

Conclusion

The results presented in this paper highlight the benefits of digital correlation techniques for characterizing and identifying the mechanical behavior of a traditional civil engineering material. Full field measurements were initially useful for checking the quality of the tests by analyzing the characteristics of the displacement fields near surfaces where loading boundary conditions were imposed. The kinematic data were also useful for the evaluation of elasticity constants of a transverse isotropic behavior model. A comparison between experimental and numerical results obtained on a Y-shaped terra cotta brick element showed that the set of identified elastic constants gave satisfactory results. To estimate the different elastic parameters and to perform 3D FE computations, we assumed that the material was and remained homogeneous during mechanical transformation. Other very promising approaches will be available in the near future ͑Geymonat et al. 2002;[START_REF] Claire | Identification de conductivités thermiques et de propriétés élastiques locales par analyse de champs[END_REF][START_REF] Bonnet | Inverse problems in elasticity[END_REF] to consider the local elastic properties of the material and to account for the structural heterogeneities leading to property gradients in civil engineering structures. The capabilities of these inverse approaches are presently being tested on academic ͑numerical͒ examples ͑Latourte et al. 2005͒. In the near future, they will be applied to noisy and discrete data fields obtained by DIC.

Fig. 1 .

 1 Fig. 1. Directions of transverse isotropy

Fig. 2 .

 2 Fig. 2. Y-shaped specimen

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Compression tests according to the directions R, L, and T-Sample responses

Fig. 7 .

 7 Fig. 7. Evolution of the Poisson's ratio LT associated with the loading in Fig. 6

Fig. 8 .Fig. 9 .

 89 Fig. 8. Displacement pattern of u L ͑a͒ fissuring; ͑b͒ propagation

Fig. 10 .

 10 Fig. 10. Fields of strain: ͑1͒ derived from digital image correlation; ͑2͒ 3D FE calculation

Table 1 .

 1 Chemical Composition of Terra Cotta

	Elements	Atomic %
	Silicon dioxide ͑SiO 2 ͒	57.9
	Alumina ͑Al 2 O 3 ͒	15.4
	Lime ͑CaO͒	14.4
	Iron oxide ͑Fe 2 O 3 ͒	4.9
	Titania ͑TiO 2 ͒	0.4
	Magnesia ͑MgO͒	1.4
	Potash ͑K 2 O͒	4.3
	Soda ͑Na 2 O͒	1.3

Table 2 .

 2 Average Value and Standard Deviation of Experimental ResultsObtained for E L , E R , and E T

			Average value	Standard deviation
		Number of tests	͑GPa͒	͑GPa͒
	E L	10	20.2	2.8
	E R	6	19.8	3.1
	E T	8	5	0.3

Table 3 .

 3 Compression Load Supported by the Y-shaped Sample: Measurement and Computation

		F exp ͑kN͒	E comp ͑kN͒
	t =6 s	0.32	0.37
	t =12 s	2.17	2.36

Acknowledgments

The writers would like to thank Saverdun Terre Cuite ͑STC͒,a French terra cotta building component manufacturer, for partly supporting this research work.


domain: scco sde shs stat qfin

text:
Angelo Leogrande

Alberto Costantiello

The Labor Force Participation Rate in the Context of ESG Models at World Level

Keywords: Analysis of Collective Decision-Making, General, Political Processes: Rent-Seeking, Lobbying, Elections, Legislatures, and Voting Behaviour, Bureaucracy, Administrative Processes in Public Organizations, Corruption, Positive Analysis of Policy Formulation, Implementation. JEL Classification: D7, D70, D72, D73, D78

In this article we analyze the impact of Labor Force Partecipation Rate-LFPR in the context of the Environmental, Social and Governance-ESG model at world level. We use data from the ESG dataset of the World Bank for the period 2011-2020. We use Panel Data with Fixed Effects, Panel Data with Random Effects, Pooled OLS, Dynamic Panel. We find that the level of LFPR is positively associated among others to "Ratio of Female to Male Labor Force Participation Rate" and "Life Expectancy at Birth", and negatively associated among others, to "Unemployment" and "Agricultural Land". Furthermore, we have applied a clusterization with the k-Means algorithm optimized with the Silhouette coefficient, and we found the presence of three clusters. Finally, we confront eight different machine learning algorithms to predict the value of LFPR. We find that the best predictor is the Linear Regression. Linear Regression predicts an increase in LFPR equal to 0.42% on average for the analyzed countries.

1) Introduction-Research Question

In the following article, we analyze the role of LFPR in the context of ESG models worldwide using data of the World Bank. The role of work in the current economic systems is subjected to many pressures and criticisms. On the one hand, in fact, technology and capital tend to reduce the role of labour by favoring IT systems and the work of machines. On the other hand, labour seems to be increasingly a privilege rather than a right in developed and underdeveloped economies due to gender and racial discrimination. However, work remains an essential component for value added both in countries with low per capita incomes and in countries with high per capita incomes. The value of labour, and in particular the value of LFPR tends to reflect all the various discriminations present at the social level. Added to these discriminations are those produced by educational qualifications with the contrast between high-skilled workers and low-skilled workers. Labour is therefore the most relevant context for analyzing the social and economic contradictions in terms of income inequality and opportunities. In our case, we analyzed the LFPR variable in the context of ESG models. This analysis adds an element of originality to the study. In fact, if the various determinants of the labor market have been extensively analyzed in the scientific literature, there are few studies that have taken into consideration the relationship between the LFPR variable and ESG models. The article continues as follows: the second section refers to the analysis of the literature, the third section presents the econometric model, the fourth section shows the results of the clustering, the fifth section presents the results of the analysis with the machine algorithms learning, the sixth section concludes.

2) Literature Review

A brief review of the literature related to LFPR is presented below. The articles discussed are not exhaustive of the scientific debate in terms of LFPR. The citations reported have the sole purpose of introducing the topic by highlighting the salient points of recent research.

LFPR, Covid 19 and health issues. Low levels of LFPR have been positively associated to high levels of child abuse and neglect during the Covid 19 pandemic at Los Angeles [START_REF] Barboza | A spatiotemporal analysis of the impact of COVID-19 on child abuse and neglect in the city of Los Angeles, California[END_REF]. The increase in drug abuse during the Covid 19 pandemics has reduced the level of LFPR in the post-pandemic [START_REF] Greenwood | Did Substance Abuse During the Pandemic Reduce Labor Force Participation?[END_REF]. There is a negative relationship between depression and anxiety disorders on one side and LFPR on the other side [START_REF] Banerjee | Effects of psychiatric disorders on labor market outcomes: a latent variable approach using multiple clinical indicators[END_REF]. The presence of health and psychological problems reduces the population's ability to actively participate in the labor market. The distinction between people who enjoy good health and people who have both physical and mental health problems creates a further discrimination in the workplace. Furthermore, the fact that people actively participate in the labor market also reduces the likelihood that they are involved in criminal activity or violent behavior. High levels of LFPR are therefore necessary either to ensure a better health status of the population either to reduce the incidence of domestic and social violence.

LFPR and developing countries. Remittances reduces the female LFPR but do not affect male LFPR in a set of 122 developing countries in the period 1990-2015 [START_REF] Azizi | The impacts of workers' remittances on human capital and labor supply in developing countries[END_REF]. There is a negative relationship between the level of female education and the female LFPR in rural India [START_REF] Afridi | Why are fewer married women joining the work force in rural India? A decomposition analysis over two decades[END_REF]. A study on individuals in the age 55-64 in Turkey shows that the main determinants for LFPR are education and marital status [START_REF] Şentürk | Aging and Employment: A Study on the Labor Force Participation Decision of the Elderly in Turkey[END_REF]. The increase of LFPR has a positive effect on income distribution in India [START_REF] Angeles | The Effect of Gender Inequality in Education, Labor Force Participation and Economic Opportunity on the Income Distribution of India[END_REF]. There is a positive relationship between female LFPR and female salaries in Pakistan [START_REF] Afridi | Female Labor Force Participation in Pakistan and Central Asian States: A Comparative Analysis[END_REF]. There is a negative relationship between female LFPR and natality in Brazil [START_REF] De Souza | The Changing Effects of Fertility on Women's Education and Labor Supply: Natural Experiments at Different Parities in Brazil[END_REF]. Female LFPR and occupational segregation grown together in 66 developing countries in the period 1980-2011 [START_REF] Borrowman | Drivers of gendered sectoral and occupational segregation in developing countries[END_REF]. Female LFPR in Middle East is low despite the increase in the level of female education; the presence of a male's veto power on the decision of woman to work reduces female LFPR [START_REF] Majbouri | Preferences and the Puzzle of Female Labor Force Participation in the Middle East[END_REF]. The analysis shows some counterfactual elements. The fact that there is an inverse relationship between female education and female LFPR levels in India may be due to the low incomes in rural markets especially for highskilled female workers. But, in general, except for this case, the dynamics of LFPR and especially female LFPR in developing and low-income countries is similar to that of high-income countries.

LFPR and democracy and discriminations.

There is a positive relationship between female LFPR and the participation in USA democratic elections i.e. the number of voters increases with female LFPR [START_REF] Cebula | Female Labor Force Participation and Voter Turnout: Evidence from the American Presidential Elections[END_REF]. The black-white racial gap in LFPR is greater than the hispanic-white gap LFPR in USA [START_REF] Cajner | Racial gaps in labor market outcomes in the last four decades and over the business cycle[END_REF]. The presence of low skilled immigrants has a positive effect on LFPR for low skilled coloured native born workers in South Africa [START_REF] Broussard | Immigration and the labor market outcomes of natives in developing countries: A case study of South Africa[END_REF]. Labor participation rates have a positive impact in terms of democratic activism. A person who works might also have a greater interest in voting at political elections. There is therefore a positive relationship between the value of participation in the labor market and the value of participation in democratic life, at national level. However, for the same reasons, racial discriminations could have a negative retroactive effect on the solidity of democratic institutions. In fact, the case of the USA shows that both the Afro-American community and the Hispanic community, despite showing a proclivity for activism in the labor market, still have reduced levels of employability, due to racial discrimination.

LFPR policies. The offering of afterschool care increases labor force participation rate by 7% in Chile [START_REF] Martínez | Childcare effects on maternal employment: Evidence from Chile[END_REF]. The promotion of childcare services has increased mother's LFPR by 0.2% in Germany, especially for mothers with medium-high education [START_REF] Müller | Does subsidized care for toddlers increase maternal labor supply. Evidence from a large-scale expansion of early childcare[END_REF]. The investment in labor-intensive industries can improve female LFPR in Botswana and Namibia [START_REF] Matandare | Botswana unemployment rate trends by gender: Relative analysis with upper middle income Southern African countries (2000-2016)[END_REF]. There are policies that can have a positive effect on the growth of the value of LFPR, especially in the case of female LFPR. Specifically, it is possible to develop care services for children, to ensure that women can actively participate in the labor market. However, in some cases, such as in the case of people with low levels of education, the offer of childcare services should also be accompanied by a set of educational and financial interventions to support mothers in the pathways of entry into the labor market. Economic policies that invest in labor-intensive sectors can help LFPR growth.

LFPR miscellaneous. There is a positive relationship between the increase in female entrepreneurship and the reduction in male vs. female LFPR [START_REF] Ribes-Giner | Domestic economic and social conditions empowering female entrepreneurship[END_REF]. Internet has increased female LFPR by 4.1% for married woman thanks to tele working [START_REF] Dettling | Broadband in the labor market: The impact of residential high-speed internet on married women's labor force participation[END_REF]. A positive relationship was found between labour supply and LFPR in Germany during the period 2003-2010 [START_REF] Burda | Reevaluating the German labor market miracle[END_REF]. The promotion of female entrepreneurship can reduce the gap between men and women in the sense of LFPR. However, it is necessary to invest in promoting women's economic rights, freedom, and capabilities to increase the value of LFPR through female entrepreneurship. One of the variables that increases women's participation in work is information technology. In fact, if IT makes it possible to improve the work-life balance, then it is possible that the number of women participating in the labor market will increase. In addition, there are demographic conditions that can lead to an increase in the value of LFPR. However, it is said that an increase in the value of LFPR is not always accompanied by an improvement in the condition of the workers in the sense of income and the quality of workplace.

3) The Econometric Model for the Estimation of the Value of LFPR

Below we present a regression analysis aimed at identifying the LFPR determinants within the context of the ESG dataset of the World Bank. The data refer to 193 countries in the period between 2011 and 2020. The data were analyzed using the following econometric models or: Panel Dat with Fixed Effects, Panel Data with Random Effects, Pooled OLS, Dynamic Panel. In summary, we estimated the following equation or:

= + + + + . + + +
Where = and = [ ; ]. We found that LFPR is positively associated to:

• #$ % : it is a variable that considers the value of carbon dioxide emissions deriving from the combustion of fossil fuels and from the production of cement. They include carbon dioxide produced during the burning of solid, liquid, and gaseous fuels and gas flaring. There is a positive relationship between the value of LFPR and #$ % . This positive relationship is due to the fact that the countries where there is the greatest production of #$ % are also the countries that have the highest levels of LFPR, i.e. the western countries. Countries that have high levels of LFPR are also countries that have an active industrial and manufacturing system, and which therefore tend to have significant #$ % production. However, this positive relationship is very likely to change in the future due to the emphasis placed by European governments especially on climate change. The incentives that are offered to the European industrial system try to transform production through the application of sustainable methodologies that could allow an increase in the LFPR in the reduction of the #$ % value in the future [START_REF] Costantiello | The Determinants of CO2 Emissions in the Context of ESG Models at World Level[END_REF]. • LEAB: is a variable that considers life expectancy at birth as the number of years a newborn would live if the prevailing patterns of mortality at the time of birth remained the same throughout life. There is a positive relationship between the LEAB value and the LFPR value. Such a relationship tends to be paradoxical. Countries that have high levels of LFPR are either low-income countries from Central and Southern Africa, either high-income countries i.e. North America, North Europe and Oceania. Otherwise, there are a set of countries that show a low level of LFPR i.e. Mediterranean and South Asia countries. However, the fact that there is a positive relationship between LFPR and LEAB suggests that the positive effect on life expectancy in high-income countries that have high levels of LFPR tends to largely offset the negative effect on life expectancy in countries lower-middle-income with a high level of LFPR. • PM2.5: is a variable that considers the weighted exposure for the population to environmental pollution from PM2.5 defined as the average level of exposure of the population of a nation to concentrations of suspended particles measuring less than 2.5 microns in aerodynamic diameter, which are able to penetrate deep into the respiratory tract and cause great harm to health. harm. Exposure is calculated by weighting the average annual concentrations of PM2.5 by population in urban and rural areas. There is a positive relationship between the PM2.5 value and the LFPR value. In fact, it must be considered that many of the countries that have high levels of LFPR are low-middle income countries that also have high levels of pollution from PM2.5 emissions. In fact, countries in Central and Southern Africa that have high levels of LFPR also have high levels of PM2.5. • RFTM: is a variable that considers the percentage of the population aged 15 or over who is economically active: all the people who provide work to produce goods and services during a given period. The ratio of female to male labor force participation rates is calculated by dividing the female labor force participation rate by the male labor force participation rate and multiplying by 100. Looking at the map of countries by RFTM value many countries that have high LFPR levels also have high RFTM values. This is the case, for example, of the countries of Central-Southern Africa, of North America, of Northern Europe, of South America and of some parts of Oceania. Hence the positive relationship between the LFPR value and the RFTM value derives. However, it must also be considered that there are some countries which have low LFPR value, and which nonetheless have high RFTM values such as Italy, Spain, China for example. • RQ: captures perceptions of government's ability to formulate and implement robust policies and regulations that enable and promote private sector development. The estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5. There is a positive relationship between the RQ value and the LFPR value. This relationship is since many of the countries that have high levels of LFPR such as the countries of North America and Northern Europe also have high levels of RQ. However, there are also countries that have high LFPR lifelines and low RQ levels such as, for example, the countries of Central-Southern Africa. It therefore follows that the positive impact of high-middle-income per capita countries with high LFPR levels on the RQ value tends to more than offset the negative impact of low-middle-income countries with high LFPR levels on the RQ value [START_REF] Costantiello | The Regulatory Quality and ESG Model at World Level[END_REF]. LFPR is negatively associated to:

• AL: refers to the share of arable land, with permanent crops and permanent pastures. Arable land includes land defined by FAO as land under temporary crops (double harvested areas are counted once), temporary grassland for mowing or grazing, land under market or market gardens and land temporarily fallow. There is a positive relationship between the AL value and the LFPR value. In fact, many countries that have high levels of AL have low levels of LFPR such as for example India and many countries in the Mediterranean area. On the contrary, many Anglo-Saxon countries that have a high level of LFPR, such as the USA, Canada, Australia have low levels of AL. • UT: refers to the share of the labor force that is out of work but available and looking for work. There is a negative relationship between the UT value and the LFPR value. That is, unemployment tends to rise as the value of LFPR decreases. This is because many countries that have high LFPR values are also low unemployment countries such as North America, Northern Europe, and Australia.

4) Clusterization with k-Means Algorithm Optimized with the Silhouette Coefficient

In the following analysis we apply the k-Means clustering algorithm to check for clustering within the data. Since the k-Means algorithm is an unsupervised algorithm then we will apply the Silhouette coefficient to choose the optimal number of clusters. The result shows the presence of three clusters.

Considering the value of the median of the clusters, the following ordering of the clusters results: #3 = 69,978 > #1 = 59,79 > #2 = 43,677.

• We can therefore note that although these countries have the same capacity in terms of LFPR value, they nevertheless have important differences in terms of GDP value both in absolute value and in per capita value. This analysis highlights the low value of labor compared to capital in creating the conditions for economic growth. In fact, even if the population actively participates in the labor market, this does not necessarily lead to growth in GDP. Otherwise, it is the capital endowment that allows, LFPR ceteris paribus, to create the conditions for economic growth either in the sense of growth in the value of per capita gross domestic product either as growth in GDP in absolute terms. Furthermore, we must also underline the enormous heterogeneity of the conditions of workers in the various countries even if they are in the same cluster for the median level of LFPR. In fact, working class conditions in Mongolia are certainly not comparable to those in Sweden, despite Mongolia and Sweden participate in the same cluster in the sense of LFPR. And in fact, perhaps it would be necessary to create a new indicator capable of measuring LFPR adjusted for the level of workers conditions. • Cluster 2: Afghanistan, Bosnia and Herzegovina, Comoros, Djibouti, Algeria, Egypt Arab Rep., Gabon, Greece, Croatia, India, Iran, Iraq, Italy, Jordan, Lebanon, Libya, Morocco, Moldova, Mauritania, Nepal, Pakistan, Papua New Guinea, Puerto Rico, West Bank and Gaza, Sudan, Senegal, Somalia, Serbia, Eswatini, Syrian Arab Republic, Tajikistan, Turkmenistan, Tunisia, Turkey, Yemen. It is the last cluster by value of LFPR. C2 is a very heterogeneous cluster as it is made up of countries from various continents, which have different levels of per capita income. All these countries have low participation in the labor market. However, it must be considered that the LFPR variable could underestimate the presence of undeclared, irregular, and informal workers in the economy. In Italy, for example, there is a significant percentage of the population working in the irregular and in the informal economy. These are workers who however appear as inactive in the official labour statistics. It is probable that similar conditions also occur in other countries of C2. However, net of this effect, it is highly probable that there is a problem of inefficient incentives in these countries. The workers of the C2 countries do not find in the labor market a feasible opportunity to improve their life and their social condition through the active participation in the job market. One of the reasons for the low participation of workers in the labor market could be the low level of labor income. Other reasons could be connected to the lack of development of industrial and labor systems capable of offering jobs considered financially and technically adequate. These countries suffer for a double problem in the labor market: on the one hand, low wages and on the other, insufficient working conditions. C2 is the only cluster that does not include high-income countries. This condition indicates that low levels of LFPR could in the long run compromise per capita income and productivity levels at country level. • Cluster 3: Angola, United Arab Emirates, Burundi, Benin, Bahrain, The Bahamas, Bolivia, Central African Republic, Switzerland, China, Cameroon, Congo Rep., Colombia, Eritrea, Ethiopia, Ghana, Indonesia, Iceland, Kazakhstan, Kenya, Cambodia, Kuwait, Liberia, St. Lucia, Macao SAR, Madagascar, Mali, Mozambique, Malawi, Niger, New Zealand, Oman, Peru, North Korea, Paraguay, Qatar, Singapore, Solomon Islands, South Sudan, Thailand, Timor-Leste, Tanzania, Uganda, Vietnam, Vanuatu. This is the first cluster in the sense of LFPR. The cluster is made up of countries with low-medium per capita income, except for the following countries: Switzerland, Iceland, New Zealand, Singapore, Qatar, Macao, United Arab Emirates. The other countries that are in the cluster, all have low per capita incomes. It must be considered that many of these countries are very significant nations from the point of view of absolute GDP value as in the case of China and Indonesia. Other countries have a reduced GDP value both from a per capita and absolute perspective. Many of the C3 countries are African and Latin American ones. It must therefore be considered that there is a real dichotomy considering the relationship between the LFPR and the value of GDP either in absolute value either per capita. Indeed, the LFPR tends to be high both in countries with high levels of per capita income and in countries with low levels of per capita income. The C3 therefore presents a very significant level of polarization between rich and poor countries.

Both are characterized by the presence of high labor participation of the population. This condition makes us understand how much work is in some ways irrelevant. In fact, the real distinction between rich and poor countries, as indicated in the case of C3, does not consist in the difference in terms of LFPR, but in the capital endowment. This fact is so evident that even with the same values of LFPR there are still significant differences in terms of per capita income, which are entirely attributable to differences in capital endowments.

5) Machine Learning and Prediction for the Estimation of the Future Value of LFPR

Below we present an analysis for predicting the future value of LFPR. Specifically, we compare eight different machine learning algorithms. The algorithms are compared according to their ability to maximize the R-squared value and to minimize the MAE-Mean Average Error, MSE-Mean Standard Error, RMSE-Root Mean Standard Error value. The algorithms were trained with 70% of the available data while the remaining 30% was used for the actual prediction. To identify the best performing algorithm, four rankings were created for each of the four statistical indicators presented. The score of each algorithm in the individual rankings was added up. The algorithm with the lowest aggregate score was chosen, i.e. the algorithm that totaled the highest places in the single rankings presented. In this way, the following ordering of the algorithms was obtained, i.e.:

• Linear Regression with a payoff equal to 5;

• Random Forest Regression with a payoff value of 7;

• Tree Ensemble Regression with a payoff value of 13;

• Gradient Boosted Tree Regression with a payoff value of 15;

• PNN-Probabilistic Neural Network with a payoff value of 20;

• ANN-Artificial Neural Network with a payoff value of 24;

• Simple Regression Tree with a payoff value of 29;

• Polynomial Regression with a payoff value of 31. Therefore, by applying the best predictor algorithm or Linear Regression it is possible to verify that there are some countries for which an increase in the LFPR value is expected, while there are other countries for which a reduction is predicted. There are some countries for which a growth in the value of LFPR is predicted, namely: Philippines with a value of +9.1%; Venezuela with +7.88%; Ecuador with 6.61%; Lebanon with 5.9%; Cuba with 5.67%; Honduras with a value of 5.58%; El Salvador with a value of 5.24%; Uruguay with a value of 3.95%; The Bahamas with a value of 3.48%; Bolivia with 3.26%; Mali with 3.21%; Trinidad and Tobago with 2.53%; Gabon with 2.33%; Iceland with 2.24%; Kuwait with 2.05%, Bhutan with 1.96%,Vietnam with 1.91%; Canada with 0.86%; Belize with 0.82%; Equatorial Guinea with 0.81%; Algeria with 0.5%; Germany 0.34%; United States '0.11%; North Korea with 0.08%; Ukraine with 0.06%. By applying the best predictor algorithm or Linear Regression it is also possible to obtain the following values for the countries that are losers or: Nicaragua with -0.07%; Guyana with -0.14%; Sri Lanka with -0.22%; Madagascar with -0.24%; Kazakhstan with -0.28%; Mozambique with -0.32%; Greece with -0.36%; Congo Dem Rep with -0.4%; Ghana with -0.47%; Lesotho with -0.59%; Myanmar with -0.61%; Vanuatu with -0.78%; Guam with -0.95%; Guinea-Bissau with -1.11%; French Polynesia with -1.18%; United Kingdom with -1.36%; Denmark with -1.37%; Norway with -1.39%; Mauritania with -1.45%; Virgin Islands with -1.5%; Georgia with -1.53%; Belgium with -1.71%; Poland with -1.78%; Malaysia with -1.96%; Croatia with -1.97%; Netherlands with -2.13%; Slovenia with -2.14%; Hungary with -2.77%; Zambia with -4.16%; Malta with -5.09%; Azerbaijan with -5.26%; Jordan with -6.13%.

6) Conclusions

In this article, we analyzed the relationship between LFPR and ESG using a dataset from the World Bank over the period 2011-2020. We found that the value of LFPR is negatively connected to the E-Environment component, and positively connected to the S-Social and G-Governance components within the ESG model. Thus, econometric analysis shows that LFPR growth tends to be costly for the Environment even if it is compatible with Social and Governance issues within the ESG model. The cluster analysis performed with the k-Means algorithm optimized with the Silhouette Coefficient shows the existence of three clusters in the sense of LFPR. The k-Means algorithm shows that within the same cluster it is possible to find either countries with low per capita income either countries with high per capita income. LFPR growth by itself does not increase GDP. The positive relationship between LFPR and GDP per capita is mediated through the investment in capital structure. Therefore, to ensure that the increase in the LFPR in low-income countries has a significant positive impact in terms of GDP, it is necessary to invest in capital endowments in developing countries. Finally, the predictive analysis based on machine learning algorithms shows a positive trend in the value of the LFPR in the future with an average growth of 0.42% for the countries analyzed.

7) Declarations

Data Availability Statement. The data presented in this study are available on request from the corresponding author.

Funding. The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Competing Interest. The authors declare that there is no conflict of interests regarding the publication of this manuscript. In addition, the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication.

Software. The authors have used the following software: Gretl for the econometric models, Orange for clusterization and network analysis, and KNIME for machine learning and predictions. They are all free version without licenses.

9) Appendix

Figure 1 .

 1 Figure 1. Average value of regressions and ESG decomposition with average value.

Figure 2 .

 2 Figure 2. Structure of the clusters with the k-Means algorithm optimized with the Silhouette Coefficient.

Figure 3 .

 3 Figure 3. Ranking of algorithms based on the maximization of R-squared and Minimization of MAE, MSE and RMSE.

Figure 4 .

 4 Figure 4. Winners: countries for which is predicted an increase in LFPR.

Figure 15 .

 15 Figure 15. Losers. Countries for which is predicted a reduction in LFPR.

  = 44 Test for AR(1) errors: z = -1,63743 [0,1015] Test for AR(2) errors: z = 0,0789429 [0,9371] Sargan over-identification test: Chi-square(35) = 1363,95 [0,0000] Wald (joint) test: Chi-square(8) = 17245,1 [0,0000] Random-effects (GLS), using 1930 observations Using Nerlove's transformation Included 193 cross-sectional units Time-series length = 10 Dependent variable: A33

  Cluster 1: Albania, Argentina, Armenia, Australia, Austria, Azerbaijan, Belgium, Burkina Faso, Bangladesh, Bulgaria, Belarus, Belize, Brazil, Barbados, Brunei Darussalam, Bhutan, Botswana, Canada, Channel Islands, Chile, Cote d'Ivoire, Congo Dem. Rep., Cabo Verde, Costa Rica, Cuba, Cyprus, Czechia, Germany, Denmark, Dominican Republic, Ecuador, Spain, Estonia, Finland, Fiji, France, United Kingdom, Georgia, Guinea, The Gambia, Guinea Bissau, Equatorial Guinea, Guatemala, Guam, Guyana, Hong Kong SAR, Honduras, Haiti, Hungary, Ireland, Israel, Jamaica, Japan, Kyrgyz Republic, South Korea, Lao PDR, Sri Lanka, Lesotho, Lithuania, Luxembourg, Latvia, Maldives, Mexico, North Macedonia, Malta, Myanmar, Montenegro, Mongolia, Mauritius, Malaysia, Namibia, New Caledonia, Nigeria, Nicaragua, Netherlands, Norway, Panama, Philippines, Poland, Portugal, French Polynesia, Romania, Russian Federation,

Rwanda, Saudi Arabia, Sierra Leone, El Salvador, Sao Tome and Principe, Suriname, Slovak Republic, Slovenia, Sweden, Chad, Togo, Tonga, Trinidad and Tobago, Ukraine, Uruguay, United States, Uzbekistan, St. Vincent and the Grenadines, Venezuela, Virgin Islands, Samoa, South Africa, Zambia, Zimbabwe. It is the second cluster for median value of LFPR. It is a very large cluster made up of various countries that are either uppermiddle income, such as the USA, Australia, Austria, Hong Kong, either lower-middle income countries such as Mongolia, Namibia, Zimbabwe.

Professor of Economics at LUM University Giuseppe Degennaro and Researcher at LUM Enterprise s.r.l. Email: leogrande.cultore@lum.it, Strada Statale 100 km 18, Casamassima, Bari, Puglia, Italia.

Professor of Economics at LUM University Giuseppe Degennaro. Email: costantiello@lum.it. Strada Statale 100 km 18, Casamassima, Bari, Puglia, Italia.

Acknowledgements. We are grateful to the teaching staff of the LUM University "Giuseppe Degennaro" and to the management of the LUM Enterprise s.r.l. for the constant inspiration to continue our scientific research work undeterred.


ON THE MAHLER MEASURE OF (1 + x)(1 + y) + z FRANC ¸OIS BRUNAULT

Keywords: 2020 Mathematics Subject Classification. Primary 19F27, Secondary 11F67, 11G16, 11R06, 19E15 Mahler measure, motivic cohomology, modular curve, regulator, L-function

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The (logarithmic) Mahler measure of a complex Laurent polynomial P (x 1 , . . . , x n ) is defined as the average of log |P | over the torus

T n ∶ |x 1 | = . . . = |x n | = 1, m(P ) = 1 (2πi) n ∫ T n log |P | dx 1 x 1 . . . dx n x n .
For a monic one-variable polynomial P (x) = d ∏ i=1 (x -α i ), Jensen's formula gives

m(P ) = d ∑ i=1 |α i |≥1 log |α i |.
Originally, the multivariate Mahler measure was introduced as a height function for polynomials, in relation with transcendental number theory. It was later realised that the Mahler measure appears naturally in other contexts. For example, Smyth discovered in 1981 the following formulas [START_REF] Beilinson | Higher regulators of modular curves[END_REF] m(1

+ x + y) = 3 √ 3 4π
L(χ -3 , 2), m(1

+ x + y + z) = 7 2π 2 ζ(3), where χ -3 (n) = ( -3
n ) is the Dirichlet character modulo 3. The Mahler measure of integer polynomials turns out to have deep links with special values of L-functions. We mention here some aspects of this connection, referring to [START_REF] Boyd | The many aspects of Mahler's measure[END_REF][START_REF] Boyd | Explicit formulas for Mahler measure[END_REF][START_REF] Bertin | Women in numbers 2: research directions in number theory[END_REF][START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF] for more complete surveys.

A combination of experiments and theoretical insights led Boyd and Deninger to conjecture the identity [START_REF] Bertin | Women in numbers 2: research directions in number theory[END_REF] m(x +

1 x + y + 1 y + 1) = L ′ (E, 0),
where E ∶ x + 1 x + y + 1 y + 1 = 0 is an elliptic curve of conductor 15. This was proved some 15 years later by Rogers and Zudilin [START_REF] Rogers | On the Mahler measure of 1 + X + 1/X + Y + 1/Y , Intern[END_REF].

The identity (2) can be conceptually explained using Beilinson's theory of regulators, and Deninger gave in [START_REF] Deninger | Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions[END_REF] a general framework to relate Mahler measures and cohomology. More precisely, let P (x 1 , . . . , x n ) be a complex Laurent polynomial, which we assume to be monic in x n . Applying Jensen's formula with respect to x n , we may write m(P ) as an integral [START_REF] Bosma | The Magma algebra system. I. The user language. Computational algebra and number theory[END_REF] ∫

Γ P η(x 1 , . . . , x n ),
where η is a differential (n -1)-form on the zero locus V P of P in (C × ) n , and Γ P is the (n -1)dimensional Deninger chain,

Γ P = {(x 1 , . . . , x n ) ∈ V P ∶ |x 1 | = ⋯ = |x n-1 | = 1, |x n | ≥ 1}.
We make here all necessary assumptions for this integral to make sense [START_REF] Deninger | Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions[END_REF]Assumptions 3.2], in particular Γ P must avoid the singular points of V P . Assume now that Γ P is closed. Then (3) can be given a cohomological interpretation, since the class of η in de Rham cohomology is the image under the Beilinson regulator map of the cup-product {x 1 , . . . , x n } in the motivic cohomology group H n M (V P , Q(n)). This situation is favourable and under certain conditions, the Beilinson conjectures predict a link between m(P ) and some L-value associated to V P . The identity [START_REF] Bertin | Women in numbers 2: research directions in number theory[END_REF] is an example of this phenomenon (in reality, in this case Γ P is not a closed path, but symmetries can be used to "close Γ P ").

A more mysterious situation is when the form η is exact, in which case we say P is exact. Stokes's formula reduces the Mahler measure m(P ) to an (n -2)-dimensional integral over the boundary ∂Γ P , but Deninger's theory does not provide an intrinsic cohomological interpretation of this integral. Maillot suggested in 2003 that, in the exact case, m(P ) should be related to the cohomology of the variety

W P ∶ P (x 1 , . . . , x n ) = P ( 1 x 1 , . . . , 1 x n ) = 0.
What makes it plausible is that ∂Γ P is contained in W P , because V P ∩T n = W P ∩T n . The relevant motivic cohomology group is now

H n-1 M (W P , Q(n))
, which is harder to deal with, as we cannot use cup-products. The identities (1) are of this type. For example, the motivic cohomology group for the polynomial 1 + x + y is isomorphic to the algebraic K-group K 3 (Q( √ -3)) ⊗ Q, which is not a Milnor K-group. For general exact polynomials, this non-Milnor character makes it difficult to handle the Mahler measure. Following Maillot's insight, Boyd and Rodriguez Villegas discovered in 2003 several identities involving 3-variable exact polynomials [START_REF] Boyd | The many aspects of Mahler's measure[END_REF][START_REF] Boyd | Explicit formulas for Mahler measure[END_REF][START_REF] Boyd | Mahler's measure and L-functions of elliptic curves evaluated at s = 3[END_REF]. One example is: Conjecture 1 (Boyd and Rodriguez Villegas [START_REF] Boyd | Explicit formulas for Mahler measure[END_REF]). We have the equality

(4) m((1 + x)(1 + y) + z) ? = -2L ′ (E, -1)
,

where E ∶ (1 + x)(1 + y)(1 + 1 x )(1 + 1 y ) =
1 is an elliptic curve of conductor 15. Here E arises as the Maillot variety W P for P = (1 + x)(1 + y) + z, and L ′ (E, -1) is related in a simple way to L(E, 3) by the functional equation of the L-function [13, 7.3.6]. The first result towards Conjecture 1 was obtained by Lalín [START_REF] Lalín | Mahler measure and elliptic curve L-functions at s = 3[END_REF], who expressed m(P ) as the regulator of a cocycle in the Goncharov complex Γ(E, 3) (see Section 3 for the definition of this complex). We write γ E = ∂Γ P for the boundary of Deninger's chain Γ P ; this is a closed path in E.

Theorem 2 (Lalín [START_REF] Lalín | Mahler measure and elliptic curve L-functions at s = 3[END_REF]). We have m [START_REF] Bosma | The Magma algebra system. I. The user language. Computational algebra and number theory[END_REF], and r 3 (2) is the Goncharov regulator map.

(P ) = 1 4π 2 ∫ γ E r 3 (2)(ξ E ), where ξ E is the class of the cocycle {-x} 2 ⊗ y -{-y} 2 ⊗ x in Γ(E,
In essence, Lalín's theorem reduces Conjecture 1 to the Beilinson conjecture for L ′ (E, -1). Beilinson proved a weak form of his conjecture for L-values of modular forms by considering regulators associated to Eisenstein symbols [START_REF] Beilinson | Higher regulators of modular curves[END_REF], but the Goncharov regulator here is of different nature. In this article, we compute the latter regulator, leading to the following theorem.

Theorem 3. The Boyd and Rodriguez Villegas conjecture (4) is true.

Another fascinating conjecture by Rodriguez Villegas concerns the Mahler measure of the polynomials 1 + x 1 + . . . + x n for n = 4 and n = 5. These polynomials are also exact and their Mahler measures are expected to involve L-values of cusp forms of weight 3 and 4, respectively [START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF]Section 6.2]. Partial results have been obtained by Shinder and Vlasenko [START_REF] Shinder | Linear Mahler measures and double L-values of modular forms[END_REF]. We found recently a (conjectural) identity which naturally generalises (4):

m((1 + x)(1 + y)(1 + z) + t) ? = -6L ′ (f 7 , -1) - 48 7 ζ ′ (-2),
where f 7 (τ ) = η(τ ) 3 η(7τ ) 3 is the unique CM newform of weight 3 and level 7.

The main ingredient in the proof of Theorem 3 is the computation by Zudilin and the author [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF] of the Goncharov regulator of explicit classes ξ 1 (a, b) in the motivic cohomology of the modular curve Y 1 (N ), which were introduced in [START_REF] Brunault | On the K 4 group of modular curves[END_REF]. A key fact here is that E is isomorphic to the modular curve X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], something we make precise in Section 2. In Section 3, we recall Goncharov's theory of polylogarithmic complexes in weight 2 and 3 and, for modular curves, we define subcomplexes built out of modular units. These complexes are amenable to computation, and we partly implemented the weight 3 complex in PARI/GP [START_REF]PARI/GP version 2.15[END_REF]; the scripts are available at [START_REF] Brunault | K 4 of modular curves[END_REF]. In Sections 4 and 5, we express Lalín's class ξ E and the path γ E in purely modular terms. The final computation is performed in Section 6, using the results of [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF]. In the appendix, we give tables of (conjectural) identities relating 3-variable Mahler measures and L(E, 3) for a number of elliptic curves E over Q.

Acknowledgements. I am grateful to Matilde Lalín, Riccardo Pengo, Wadim Zudilin and the International Groupe de travail on differential equations in Paris for exchanges which have been helpful in several parts of this paper. I would also like to thank Berend Ringeling for checking numerically several Mahler measure identities from the appendix.

The modular parametrisation

Consider the polynomial P (x, y, z) = (1 + x)(1 + y) + z. We keep the same notations as in the introduction, so that the Maillot variety W P in (C × ) 3 is defined as

W P ∶ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (1 + x)(1 + y) + z = 0, (1 + 1 x )(1 + 1 y ) + 1 z = 0. Eliminating z, we see that W P is isomorphic to the smooth curve in (C × ) 2 given by (5) C ∶ (1 + x) 2 (1 + y) 2 = xy.
Let E denote the closure of C in P 1 (C) × P 1 (C). We view E as a smooth projective curve defined over Q. It turns out that E is isomorphic to an elliptic curve of conductor 15 [22, (4.2)].

The PARI/GP commands E = ellfromeqn((1+x)^2*(1+y)^2-x*y) ellidentify(ellinit(E)) confirm that E is isomorphic to the elliptic curve with Cremona label 15a8. On the other hand, we know that the modular curve X 1 (15) is isomorphic to 15a8, since they are both elliptic curves of conductor 15, and the period lattice of X 1 (15) can be computed using modular symbols, agreeing with that of 15a8. Note that Stevens's conjecture [29, Conjecture II] is known in this case by [START_REF] Stevens | Stickelberger elements and modular parametrizations of elliptic curves[END_REF]Section 7].

In Proposition 4 below we give an explicit isomorphism X 1 (15) ≅ E (note that the proof does not rely on floating point computations). An important feature of this parametrisation is that the functions -x and -y correspond to modular units on X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]. This is crucially used in Section 4 to relate Lalín's class ξ E and the modular classes ξ 1 (a, b) from [9, Section 6]. Even more, we need the functions -x and -y to be of the form u 1 (a, b, c, d), a class of modular units introduced in [START_REF] Brunault | On the K 4 group of modular curves[END_REF] and whose definition we now recall.

Let N ≥ 1 be an integer. For any a = (a 1 , a 2 ) ∈ (Z/N Z) 2 / ± 1, a ≠ (0, 0), we define

℘ a (τ ) = ℘(τ ; a 1 τ + a 2 N ) (τ ∈ C, Im(τ ) > 0),
where ℘(τ ; z) is the Weierstraß function. The function ℘ a is a modular form of weight 2 on the principal congruence group Γ(N ). For any distinct elements a, b, c, d of (Z/N Z) 2 / ± 1, we then define u(a, b, c, d) as the cross-ratio

[℘ a , ℘ b , ℘ c , ℘ d ],
with the convention ℘ 0 = ∞. This is a modular unit on Γ(N ). For distinct elements a, b, c, d of (Z/N Z)/ ± 1, we use the shortcut

u 1 (a, b, c, d) = u((0, a), (0, b), (0, c), (0, d))
, which is a modular unit on Y 1 (N ) defined over Q.

The properties of u(a, b, c, d) needed in this article can be found in [9, Section 3].

In the following proposition, we take N = 15.

Proposition 4. The curve E is parametrised by the following modular units on Γ 1 (15):

(6) x(τ ) = -u 1 (1, 2, 3, 7)(τ ), y(τ ) = -u 1 (2, 4, 6, 1)(τ ).
Moreover, the map τ ↦ (x(τ ), y(τ )) induces an isomorphism φ ∶ X 1 (15)

≅ → E defined over Q.

Proof. Let us show that u = -u 1 (1, 2, 3, 7) and v = -u 1 (2, 4, 6, 1) satisfy (1+u) 2 (1+v) 2 = uv. For this we may replace u and v by their transforms under the Atkin-Lehner involution W 15 ∶ τ ↦ -1/15τ on X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], as this does not affect the equation. The units ũ = u ○ W 15 and ṽ = v ○ W 15 can be expressed in terms of Siegel units of level 15 using [9, eq. ( 6)]:

ũ = - g2 g4 g1 g7 = -1 -q + q 4 + q 5 -q 7 + O(q 8 ), ṽ = - g4 g7 g1 g2 = -q -2 -q -1 -2 -2q -2q 2 -2q 3 -2q 4 + O(q 5 ) (7)
where, for a ∈ Z/N Z, a ≠ 0,

ga (τ ) = q N B 2 (â/N )/2 ∏ n≥1 n≡a mod N (1 -q n ) ∏ n≥1 n≡-a mod N (1 -q n ) (q = e 2πiτ ).
Here B 2 (t) = t 2 -t + 1 6 is the Bernoulli polynomial and â is the lift of a in {1, . . . , N -1}. We are now going to compute the divisors of ũ and ṽ. To this end, we recall the description of the cusps of the modular curve X 1 (N ). There is a bijection [START_REF] Diamond | Modular forms and modular curves[END_REF]Example 9.1.3]

{cusps of X 1 (N )(C)} ≅ → {(c, d) ∶ c ∈ Z/N Z, d ∈ (Z/(c, N )Z) × }/ ± 1
which associates to a cusp γ∞ with γ ∈ SL 2 (Z), the class of the bottom row (c, d) of γ. Moreover, by [15, Section 9.3, p. 79], the Galois action on the cusps is described as follows: for σ ∈ Aut(C), we have σ ⋅ (c, d) = (c, χ(σ)d), where χ(σ) ∈ (Z/N Z) × is characterised by σ(e 2πi/N ) = e 2πiχ(σ)/N . As a consequence, a complete set of representatives of the Galois orbits is provided by the cusps

1 k = ( 1 0 k 1 )∞ with 0 ≤ k ≤ ⌊ N 2 ⌋. Now we can compute the divisor of u 1 (a, b, c, d) for distinct a, b, c, d ∈ (Z/N Z)/ ± 1 as follows.
Since this unit is defined over Q, it suffices to determine its order of vanishing at the cusps 1/k just described. By [9, Proposition 3.6], we have

u 1 (a, b, c, d)| ( 1 0 k 1 ) = u((ka, a), (kb, b), (kc, c), (kd, d)).
The order of vanishing of this unit at ∞ is deduced from the expression of u(a, b, c, d) in terms of Siegel units [9, Proposition 3.7], taking into account that it should be computed with respect to the uniformising parameter q (k,N )/N . Applying this in our situation, we obtain

div(u) = -2[1/2] + 2[1/7], div(v) = -2[0] + 2[1/4].
These cusps are rational and we see that 2 -uv has poles of order at most 4 at 0 and 1/2, and is regular elsewhere. Moreover, we compute from (7) that F (-1/15τ ) = O(q 5 ) when Im(τ ) → +∞. Therefore F vanishes at order ≥ 5 at 0, and consequently F = 0. It remains to show that φ ∶ X 1 (15) → E is an isomorphism. Since X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF] and E are smooth, it suffices to check that φ is a birational map. We know that u has degree 2 as a function on X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], while x has degree 2 as a function on E. It follows that φ is birational. □ 3. The weight 3 complex of the modular curve Y 1 (N )

F = (1 + u) 2 (1 + v)
Goncharov has defined in [START_REF] Goncharov | Geometry of configurations, polylogarithms, and motivic cohomology[END_REF] polylogarithmic complexes which are expected to compute the motivic cohomology of arbitrary fields. We define in this section a modular complex C N (3), which is a subcomplex of the weight 3 polylogarithmic complex attached to the modular curve Y 1 (N ). It is generated (in a suitable sense) by the Siegel units and the modular units u 1 (a, b, c, d) from Section 2. Our construction can be seen as a weight 3 analogue of the weight 2 Euler complex E • N introduced by Goncharov in [START_REF] Goncharov | Euler complexes and geometry of modular varieties[END_REF]. We also explain how to manipulate C N (3) using PARI/GP. The constructions below work with no more effort for the modular curve Y (N ) with full level N structure, using parameters in (Z/N Z) 2 instead of Z/N Z. However we have not implemented it, as the case of Y 1 (N ) suffices for our application.

We briefly recall Goncharov's polylogarithmic complexes in weight 2 and 3. Let F be any field. Define B 2 (F ) to be the quotient of Q[F × /{1}] by the subspace generated by the 5-term relations [START_REF] Goncharov | Geometry of configurations, polylogarithms, and motivic cohomology[END_REF]Section 1.8]. The group B 3 (F ) is defined similarly as an explicit quotient of Q[F × /{1}] [17, Section 1.8], whose definition will not be needed here. For x ∈ F × /{1} and n ∈ {2, 3}, we denote by {x} n the image of the generator [x] in B n (F ). Then the complex Γ(F, 2), in degrees 1 and 2, is defined as

Γ(F, 2) ∶ B 2 (F ) Λ 2 F × ⊗ Q {x} 2 (1 -x) ∧ x,
and the complex Γ(F, 3), in degrees 1 to 3, is defined as

Γ(F, 3) ∶ B 3 (F ) B 2 (F ) ⊗ F × ⊗ Q Λ 3 F × ⊗ Q {x} 3 {x} 2 ⊗ x {x} 2 ⊗ y (1 -x) ∧ x ∧ y. Goncharov conjectures that H i (Γ(F, n)) is isomorphic to H i M (F, Q(n)).
In the case F is the function field of a smooth curve Y over a field k, these complexes are endowed with residue maps Γ(F, n) → Γ(k(x), n -1)[-1] for every closed point x ∈ Y . Goncharov then defines the complex Γ(Y, n) as the simple of the morphism of complexes Γ(F, n) → ⊕ x∈Y Γ(k(x), n-1)[-1], and he conjectures that

H i (Γ(Y, n)) is isomorphic to H i M (Y, Q(n)) [17, Section 1.15(b)
]. We will consider these complexes in the case Y is the modular curve Y 1 (N ), and F is its function field. We will see, in particular, that they have natural subcomplexes built out of modular units. Definition 5. Fix an integer N ≥ 1. We introduce the following sets of modular units on Y 1 (N ):

• U 1 consists of the Siegel units g 0,a , a

∈ (Z/N Z)/{0}, in O(Y 1 (N )) × ⊗ Q; • U 2 consists of the modular units u 1 (a, b, c, d) in O(Y 1 (N )) ×
, where a, b, c, d are distinct elements of (Z/N Z)/ ± 1. Moreover, we associate to them the following spaces:

• ⟨U 1 ⟩ is the Q-span of U 1 in F × ⊗ Q; • ⟨U 2 ⟩ is the Q-span of {u} 2 , u ∈ U 2 , in B 2 (F ); • ⟨U 2 ⟩ 3 is the Q-span of {u} 3 , u ∈ U 2 , in B 3 (F ).
With these definitions, the weight 2 modular complex can be defined as

C N (2) ∶ ⟨U 2 ⟩ Λ 2 ⟨U 1 ⟩, {u} 2 ↦ (1 -u) ∧ u.
This complex is well-defined because U 2 is contained in ⟨U 1 ⟩ by [9, Proposition 3.8], and U 2 is stable under u ↦ 1-u from the definition of u 1 (a, b, c, d) as a cross-ratio. It would be interesting to compare C N (2) with the Euler complex E • N defined by Goncharov in [19, Section 2.5]. We are now ready to introduce a version of the weight 3 modular complex. Definition 6. The complex C N (3) is the following subcomplex of Γ(F, 3) in degrees 1 to 3:

C N (3) ∶ ⟨U 2 ⟩ 3 ⟨U 2 ⟩ ⊗ ⟨U 1 ⟩ Λ 3 ⟨U 1 ⟩.
We warn the reader that the group ⟨U 2 ⟩ 3 in degree 1 may not be the right one. Indeed, the unit u 1 (a, b, c, d) is by definition a cross-ratio, hence is a natural argument for the dilogarithm, but a priori not for the trilogarithm. However, the complex C N (3) will suffice for our needs.

Since the construction of C N (3) involves only modular units, the elements of ⟨U 2 ⟩ 3 , ⟨U 2 ⟩ ⊗ ⟨U 1 ⟩ and Λ 3 ⟨U 1 ⟩ have trivial residues at every point of Y 1 (N ). In particular, C N (3) embeds as a subcomplex of Γ(Y 1 (N ), 3), and we have natural maps in cohomology

H i (C N (3)) → H i (Γ(Y 1 (N ), 3)) in degree i ∈ {1, 2, 3}. The case of interest to us is i = 2.
We have implemented part of the complex C N (3) in PARI/GP, with the specific aim of comparing cocycles in degree 2. Firstly, the following lemma gives a natural way to represent modular units in ⟨U 1 ⟩. Lemma 7. A basis of ⟨U 1 ⟩ is given by the Siegel units g 0,a with 1 ≤ a ≤ ⌊N /2⌋.

Proof. We have g 0,-a = g 0,a in F × ⊗ Q, and by [START_REF] Streng | Generators of the group of modular units for Γ 1 (N ) over the rationals[END_REF], the units g 0,a with 1

≤ a ≤ ⌊N /2⌋ form a basis of (O(Y 1 (N )) × /Q × ) ⊗ Q. □
Each unit in U 2 can be written in the basis of Lemma 7 using [9, Proposition 3.8]. Note that no computation of divisor is needed here, thanks to this choice of basis. We actually need to determine U 2 as a set, and so to check whether two given units u 1 (a, b, c, d) and u 1 (a ′ , b ′ , c ′ , d ′ ) are equal. We remark that the leading coefficient of u 1 (a, b, c, d) at the cusp 0 is equal to 1 by the discussion after [9, Proposition 3.8]. Combining this with Lemma 7, we see that two such units are equal if and only if their coordinates in the basis of ⟨U 1 ⟩ are equal.

We now consider the free vector space Q[U 2 ], and we quotient it by the following subspaces, encoding the relations between the symbols {u 1

∑ j∈Z/5Z {u 1 (a j , a j+1 , a j+2 , a j+3 )} 2 = 0 in B 2 (F ), (9)
for any family (a j ) j∈Z/5Z of distinct elements of (Z/N Z)/ ± 1. We denote by R 2 the subspace of Q[U 2 ] generated by the antisymmetry relations (8) and the 5-term relations [START_REF] Brunault | On the K 4 group of modular curves[END_REF]. Finally, we denote by Q the subspace of Q[U 2 ] ⊗ ⟨U 1 ⟩ generated by the symbols [u] ⊗ u with u ∈ U 2 , which correspond to the degree 2 coboundaries in C N (3).

In practice, in order to reduce the size of the objects, we only compute:

• a set U ′ 2 of representatives of the quotient U 2 /S 3 ; • the subspace R ′ 2 of Q[U ′ 2 ]
generated by the 5-term relations;

• the subspace Q ′ of Q[U ′ 2 ]
⊗ ⟨U 1 ⟩ of degree 2 coboundaries. The corresponding scripts are contained in the file K4-modular-complex.gp from [START_REF] Brunault | K 4 of modular curves[END_REF]. They can be applied in the following way. Say we have two degree 2 cocycles ξ and ξ ′ in Γ(Y 1 (N ), 3). Assume that they are both linear combinations of symbols {u 1 (a, b, c, d)} 2 ⊗ g 0,x . We may then represent ξ -ξ ′ by an element of Q[U 2 ] ⊗ ⟨U 1 ⟩, and we check whether this element belongs to the subspace R 2 ⊗ ⟨U 1 ⟩ + Q. If so, then we can deduce that ξ and ξ ′ are cohomologous, and thus have the same image in K does not belong to the subspace, we cannot conclude anything, as R 2 and Q may not contain all the relations in the respective groups.

The linear system involved in the above computation has size O(N 5 ) × O(N 6 ). Experimentally, we have found that the cardinality of U 2 for N = p prime is (p 2 -1)(p 2 -25)/192, which is smaller by a factor of about 3 than what we could expect, namely 6( (p+1)/2

4

). Furthermore, it seems that the dimension of Q[U 2 ]/R 2 is equal to (p -1)(p -5)/12, which is also the number of triples (a, b, c) with 0 < a < b < c < p and a + b + c ≡ 0 mod p, where 2b < p; see [23, Sequence A242090]. If true, there should be a way to bypass the step of quotienting by R 2 . This would result in a much smaller linear system for the comparison of cocycles.

It would be extremely interesting to extend the construction of the modular complex in this section to higher weight complexes C N (n) with n > 3, as well as in higher dimension, replacing the modular curve Y 1 (N ) by Kuga-Sato modular varieties in order to deal with higher weight modular forms.

The Lalín class

Recall that Lalín's theorem (Theorem 2) expresses the Mahler measure of (1 + x)(1 + y) + z as the regulator of the class ξ E of the cocycle ξE in Γ(E, 3) defined by ξE ∶= {-x} 2 ⊗ y -{-y} 2 ⊗ x.

Our aim in this section is to relate ξ E to the classes ξ 1 (a, b) on X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], which were introduced in [9, Section 6]. This is a purely algebraic computation making use of our implementation of the weight 3 complex of X 1 (15) explained in Section 3.

We first pull back ξ E to the modular curve X 1 (15) using the modular parametrisation φ. Using Proposition 4 and its proof, we have in the degree 2 cohomology of Γ(Y 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], 3)

(10) φ * ξE = {u 1 (1, 2, 3, 7)} 2 ⊗ ( g 4 g 7 g 1 g 2 ) -{u 1 (2, 4, 6, 1)} 2 ⊗ ( g 2 g 4 g 1 g 7 ),
with the shortcut g k = g 0,k for k ∈ Z/15Z. Let us denote by ξ15 the right-hand side of [START_REF] Brunault | K 4 of modular curves[END_REF]. Lalín has shown that the cocycle ξE has trivial residues [22, Section 4.1, p. 213], so that ξ15 has trivial residues at the cusps of X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]. The next task is to express ξ15 in terms of the cocycles ξ1 (a, b) with a, b ∈ Z/15Z. We do this using the modular complex C 15 (3) from Section 3. Using the function find_xi1ab from K4-modular-complex.gp [START_REF] Brunault | K 4 of modular curves[END_REF], we detect the following simple expression for ξ15 .

Proposition 8. We have the equality of cocycles ξ15 = -20 ξ1 (1, 4) modulo coboundaries {u} 2 ⊗u with u ∈ U 2 . In particular, we have φ * (ξ E ) = -20ξ 1 (1, 4).

The linear system involved in the proof of Proposition 8 has size 273 × 288.

The integration path

In Theorem 2, the integration path γ E = ∂Γ P is a closed path in E, and we would like to express it in terms of modular symbols on X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], via the modular parametrisation of Section 2. This is a crucial ingredient in the computation of the regulator integral on E. We will do this carefully in order to certify the relation (Proposition 9).

Lalín [START_REF] Lalín | Mahler measure and elliptic curve L-functions at s = 3[END_REF]Section 4.1] has shown that γ E is a generator of H 1 (E, Z) + , where (⋅) + denotes the subgroup of invariants under complex conjugation. So we first search for a generator γ 15 of H 1 (X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], Z) + . We do this with the help of SageMath [START_REF] Sagemath | the Sage Mathematics Software System (Version 9.4), The Sage Developers[END_REF]; see the notebook ModularSymbolGamma15.ipynb in [START_REF] Brunault | K 4 of modular curves[END_REF]. For any g ∈ SL 2 (Z), denote by [g] = {g0, g∞} the associated Manin symbol, viewed in the relative homology group H 1 (X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], {cusps}, Z). We obtain [START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF]

γ 15 = 2 [( 1 9 2 19 )] -[( 0 -1 1 11 )] -[( 0 -1 1 4 )] + 2 [( 0 -1 1 2 )] .
We therefore have γ E = ±φ * (γ 15 ). The precise sign is not strictly needed in what follows, as the Mahler measure is a positive real number and the final identity fixes the sign for us.

However, we want to sketch a method to determine the sign rigorously, as it could be useful in more general situations, where the integration path γ need not be a generator of the homology group. In such a scenario, one wishes to ascertain an identity of the form γ = c ⋅ φ * (γ 0 ), where φ is the modular parametrisation, γ 0 is a modular symbol, and c ∈ Z is to be determined. The idea is to integrate an invariant differential form over the cycles to be compared. By [START_REF] Lalín | Mahler measure and elliptic curve L-functions at s = 3[END_REF]Section 4.1], an invariant differential form on E is given by

ω E ∶= -dx 2(x + 1) 2 (y + 1) -x .
Using [START_REF] Boyd | The many aspects of Mahler's measure[END_REF], we can compute the Fourier expansion of the pull-back of ω E to X 1 (15):

W * 15 (φ * ω E ) = -(q -q 2 -q 3 + O(q 4 )) dq q .
A basis of Ω 1 (X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]) is given by ω 15 ∶= 2πif 15 (τ )dτ , where f 15 = q -q 2 -q 3 + O(q 4 ) is the newform of weight 2 on Γ 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]. Therefore W * 15 (φ * ω E ) = -ω 15 . Moreover, the involution W 15 has a fixed point τ = i/ √ 15 in the upper half-plane, so it must act on the complex torus underlying X 1 (15) as z ↦ z 0 -z for some z 0 (it cannot be a translation). It follows that W 15 acts as -1 on Ω 1 (X 1 (15)), and we conclude that φ * ω E = ω 15 . Now let us integrate the forms ω E and ω 15 , and compare the signs of the integrals. Following [22, Section 4.1], the path γ E is described using polar coordinates x = e iθ , y = e iψ with θ, ψ ∈ [-π, π], and is given by the equation cos(θ/2) cos(ψ/2) = 1/4. Since the orientation of the Deninger chain Γ P is induced by the product orientation of [-π, π] 2 , its boundary γ E is oriented counterclockwise in this square (see Figure 1). We can use the symmetries of γ E to reduce the integration path. For any automorphism σ of E defined over R, we have σ * ω E = ε(σ)ω E , where ε(σ) = 1 if σ preserves the orientation of E(R), and ε(σ) = -1 otherwise. Equivalently, ε(σ) = 1 if and only if σ = id or σ has no fixed point. Applying this with the symmetries (x, y) ↦ (1/x, y) and (x, y) ↦ (x, 1/y), which reverse the orientation of E(R) as well as that of γ E , we obtain that ∫ γ E ω E is 4 times the integral over the path γ ′ E pictured in Figure 1.

Figure 1. The Deninger chain Γ P , its boundary γ E and the path γ ′ E . After some computation, we get [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF] ∫

γ E ω E = 4 ∫ γ ′ E ω E = 4 ∫ 2 arccos(1/4) 0 dθ √ 16 cos 2 (θ/2) -1 > 0.
Now with the modular curve X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], we wish to determine the sign of ∫ γ 15 ω 15 . For this, consider the linear map H 1 (X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], {cusps}, Z) → H 1 (X 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], Q) provided by the Manin-Drinfeld theorem [START_REF] Drinfeld | Two theorems on modular curves[END_REF]. Again with SageMath, we compute that the image of {0, ∞} is equal to -1 16 γ 15 (see ModularSymbolGamma15.ipynb [START_REF] Brunault | K 4 of modular curves[END_REF]). It follows that [START_REF] Cohen | A course in computational algebraic number theory[END_REF] ∫ To be fully accurate (and in order to handle more general situations), ascertaining this equality requires to compute numerically the integrals ( 12) and ( 13) with rigorous error bounds. This suffices since the ratio of these integrals is known to be an integer. The integral ( 12) is a complete elliptic integral which can be dealt with the Arb library [START_REF] Johansson | Arb: efficient arbitrary-precision midpoint-radius interval arithmetic[END_REF][START_REF] Johansson | Numerical integration in arbitrary-precision ball arithmetic[END_REF]. On the other hand, [START_REF] Cohen | A course in computational algebraic number theory[END_REF] involves integrating a modular form over a modular symbol. We can do it in the present situation thanks to the rapidly convergent series. In general, although PARI/GP [START_REF]PARI/GP version 2.15[END_REF] and Magma [START_REF] Bosma | The Magma algebra system. I. The user language. Computational algebra and number theory[END_REF] can evaluate such integrals efficiently, we are not aware of implementations that prove error bounds for them.

Final computation

We denote by r 3 (2) the Goncharov regulator map in degree 2 for the weight 3 complex of a smooth complex curve [START_REF] Goncharov | Explicit regulator maps on polylogarithmic motivic complexes[END_REF]. It sends a degree 2 cocycle to an explicit closed 1-form on this curve. By [START_REF] Brunault | K 4 of modular curves[END_REF] and Proposition 9, we have [START_REF] Deninger | Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions[END_REF] ∫

γ E r 3 (2)(ξ E ) = ∫ φ * (γ 15 ) r 3 (2)( ξE ) = ∫ γ 15 r 3 (2)(φ * ξE ) = ∫ γ 15 r 3 (2)( ξ15 ).
Note that the differential form r 3 (2)( ξ15 ) is defined only on the open modular curve Y 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]. However, it has trivial residues at the cusps since the same is true for ξ15 , see Section 4. We may therefore compute the integral by choosing the representative of γ 15 given by [START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF]. Note that this integral involves cusps but it is absolutely convergent by [START_REF] Brunault | On the K 4 group of modular curves[END_REF]Corollary 7.3]. The technical details of this procedure are explained at the end of [9, Section 8].

Lemma 10. Let u be a modular unit on X 1 (N ) such that 1 -u is also a modular unit. For any two cusps α ≠ β in P 1 (Q), we have ∫

β α r 3 (2)({u} 2 ⊗ u) = L3 (u(β)) -L3 (u(α))
, where L3 ∶ P 1 (C) → R is the single-valued trilogarithm defined in [18, Section 2.1].

Proof. By [18, Theorem 2.2], we have

r 3 (2)({u} 2 ⊗ u) = r 3 (2)(δ({u} 3 )) = dr 3 (1)({u} 3 ) = d L3 (u). □
Since the path γ 15 is closed, Lemma 10 implies that ∫ γ 15 r 3 (2)({u} 2 ⊗ u) = 0 for any u ∈ U 2 . Using Proposition 8, the computation (14) continues as [START_REF] Diamond | Modular forms and modular curves[END_REF] ∫

γ E r 3 (2)(ξ E ) = -20 ∫ γ 15 r 3 (2)( ξ1 (1, 4
)).

We are now in position to apply the main result of [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF], which computes

G(a, b) ∶= ∫ ∞ 0 r 3 (2)( ξ(a, b)) (a, b ∈ (Z/N Z) 2 ),
under the assumption that the coordinates of a, b and a + b are non-zero. We may integrate along Manin symbols [g] = {g0, g∞} as well, noting that

∫ g∞ g0 r 3 (2)( ξ(a, b)) = ∫ ∞ 0 r 3 (2)( ξ(ag, bg)) = G(ag, bg) (g ∈ SL 2 (Z)).
Recall also that ξ1 (a, b) = ξ((0, a), (0, b)). Expanding [START_REF] Diamond | Modular forms and modular curves[END_REF], we get The assumption on the coordinates of the parameters is satisfied, and [12, Theorem 1] gives ( 16)

∫ γ E r 3 (2)(ξ E ) = -
∫ γ E r 3 (2)(ξ E ) = π 2 L ′ (F, -1) with F = -8(G 2,1 G 8,-4 + G 2,-1 G 8,4 ) + 4(G 1,14 G 4,-11 + G 1,-14 G 4,11 ) + 4(G 1,1 G 4,-4 + G 1,-1 G 4,4 ) -8(G 1,8 G 4,-2 + G 1,-8 G 4,2 ). (17)
Here G a,b is a shortcut for the Eisenstein series G defined in [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF]Introduction] for arbitrary level N by

G (1);N a,b (τ ) = a 0 (G (1);N a,b ) + ∑ m,n≥1 (m,n)≡(a,b) mod N q mn/N - ∑ m,n≥1 (m,n)≡-(a,b) mod N q mn/N (a, b ∈ Z/N Z).
In our situation the indices a, b are non-zero modulo 15, so that the constant terms a 0 (G a,b ) all vanish. The functions G a,b are Eisenstein series of weight 1 on Γ [START_REF] Diamond | Modular forms and modular curves[END_REF]. Note that the products G x G y appearing in ( 17) are actually power series in q, because x 1 x 2 + y 1 y 2 is divisible by 15 for each such product. It follows that F belongs to M 2 (Γ 1 (15)).

We have written a script K4-reg-Lvalue.gp [START_REF] Brunault | K 4 of modular curves[END_REF] to automate the application of [12, Theorem 1] and compute the q-expansion of the resulting modular form to arbitrary precision. We find that F = -8f 15 + O(q 21 ), where f 15 is the newform associated to E. Moreover, the Sturm bound for the space M 2 (Γ 1 [START_REF] Diamond | Modular forms and modular curves[END_REF]) is equal to 16 (apply [28, Sturm's theorem, 9.4.1.2] with the group ±Γ 1 [START_REF] Diamond | Modular forms and modular curves[END_REF], which has index 96 in SL 2 (Z)). This means that if two modular forms F 1 and F 2 in this space satisfy F 1 = F 2 + O(q 17 ), then F 1 = F 2 . In our situation, this allows us to certify that F = -8f 15 . Using Theorem 2 and ( 16), the Mahler measure finally equals

m(P ) = 1 4π 2 ∫ γ E r 3 (2)(ξ E ) = 1 4π 2 ⋅ π 2 L ′ (-8f 15 , -1) = -2L ′ (E, - 1
). This concludes the proof of Theorem 3.

Appendix. Tables of 3-variable Mahler measures

We would like to give here a list of conjectural identities for 3-variable Mahler measures involving L(E, 3) for several elliptic curves E over Q. It is possible that our methods can be applied to prove at least some of these identities. The success of the approach will depend very much on the modular parametrisation of the elliptic curve; in our case, Proposition 4 was crucial. This is similar to what happens for the 2-variable Mahler measures, where the proofs using the Rogers-Zudilin method require the curve to be parametrised by modular units [START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF]Section 8.4 and Chapter 9].

Boyd and Rodriguez Villegas [START_REF] Boyd | Explicit formulas for Mahler measure[END_REF] discovered several identities of type m(P (x, y, z)) = r ⋅ L ′ (E, -1) with r ∈ Q × by looking at polynomials of the form P = A(x) + B(x)y + C(x)z where A, B, C are products of cyclotomic polynomials. Boyd found further examples in [START_REF] Boyd | Mahler's measure and L-functions of elliptic curves evaluated at s = 3[END_REF][START_REF] Boyd | Conjectural explicit formulas for the Mahler measure of some three variable polynomials[END_REF]. We extended Boyd's search with A, B, C of degree up to 5 and found a few other examples, see Table 1 below (we do not claim to have spotted all identities for this range of A, B, C). Table 2 displays two Mahler measures which involve a combination of L(E, 3) and ζ(3). Note that a ζ(3) term also appears in the main result of [START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF] (see Theorem 1 there).

In the tables below, the curve E is given by its Cremona label, and the integer g is the genus of the Maillot variety W P (or a component of it) whose Jacobian has E as an isogeny factor.

We also looked at polynomials P (x, y, z) which have degree 1 in each variable x, y, z, and all of whose coefficients are ±1 (or zero). It seems to be the case that every such polynomial

= r ⋅ L ′ (E, -1) + s ⋅ ζ ′ (-2)
is exact. The identities found are collected in Table 3. The first entry in this table is not of this shape but we include it for completeness; it already appears in [START_REF] Brunault | Many Variations of Mahler Measures: A Lasting Symphony[END_REF]. Ringeling computed numerically the Mahler measures in Table 3, and the identities seem to hold to at least 100 digits.

A particular feature of Table 3 is the appearance of the elliptic curve 36a1, which has complex multiplication. The elliptic curve 450c1 is also the first example with a curve of rank 1.

P

  (a, b, c, d)} 2 . From the definition of u 1 (a, b, c, d) as a cross-ratio, the symmetric group S 4 acts on U 2 by permuting the indices, and this action factors through S 3 . Moreover, because of the relations {1/u} 2 = {1 -u} 2 = -{u} 2 in B 2 (F ) [31, VI, Lemma 5.4], we have the antisymmetry property: (8) {u 1 (a σ(1) , a σ(2) , a σ(3) , a σ(4) )} 2 = ε(σ){u(a 1 , a 2 , a 3 , a 4 )} 2 (σ ∈ S 4 ), for all distinct parameters a i in (Z/N Z)/ ± 1, where ε(σ) = ±1 is the signature. It thus suffices to consider those parameters satisfying 0 ≤ a < b < c < d ≤ ⌊N /2⌋. The elements {u 1 (a, b, c, d)} 2 are also subject to the 5-term relations [9, Lemma 4.7]:

( 3 ) 4 (

 34 Y 1 (N )) under De Jeu's map [9, Theorems 5.3 and 5.4]. If ξ -ξ ′

γ 15 ω 15 = -16 ∫ ∞ 0 ω 15 = 15 φ 15 ωProposition 9 .

 151501515159 16L(f 15 , 1) > 0.That L(f 15 , 1) is positive can be ascertained witout much effort using the rapidly convergent series L(f 15 , 1) = 2 ∑ ∞ n=1 a n e -2πn/ √ 15 /n [13, Proposition 7.5.8], where the a n are the Fourier coefficients of f 15 . Namely, one may use the bound |a n | ≤ n for n ≥ 1, which follows from the Hasse bound on E and the inspection of a n for small n. Combining[START_REF] Brunault | Modular regulators and multiple Eisenstein values[END_REF] and∫ φ * (γ 15 ) ω E = ∫ γ * ω E = ∫ γ 15 > 0,we come to the following conclusion. We have γ E = φ * (γ 15 ).

  20(2G((2, 4), (8, 1)) -G((1, 11), (4, 14)) -G((1, 4), (4, 1)) + 2G((1, 2), (4, 8))).

Table 1 .

 1 Conjectural identities m(P ) -x + 1 + (x + 1)(y + z) 30a1 -2/9 -476/27 1[START_REF] Brunault | Unpublished list of conjectural identities for 3-variable Mahler measures[END_REF] (x -1) 3 + (x + 1) 3 (y + z) 108a1 -2/15 112/15 2 [5, p. 21],[START_REF] Brunault | Unpublished list of conjectural identities for 3-variable Mahler measures[END_REF]

	P	E	r	g		Source
	(x -1) 3 + (x + 1)(y + z)	14a4	-6	2	[5, p. 21], [8]
	(1 + x) 2 + (1 -x)(y + z)	20a1	-2	1	[6], [11, p. 81]
	x + 1 + (x -1)y + (x 2 -1)z	20a1 -3/2 3		[8]
	(1 + x) 2 (1 + y) + z	21a4 -3/2 1	[6], [11, p. 81]
	1 + x + y -xy + z	21a1 -5/4 1 [6], [11, Section 6.3]
	(1 + x) 2 + y + z	24a4	-1	1 [7, Section 8], [6]
	(x + 1) 2 + (x 2 -1)y + (x -1) 2 z 48a1 -2/5 3		[8]
	(x + 1) 2 + (x -1) 2 y + z	225c2 -1/48 1		[6, 8]
				? = r ⋅ L ′ (E, -1)
	P	E	r	s	g	Source
	x 2

Table 2 .

 2 Conjectural identities m(P )

	?

Table 3 .

 3 Conjectural identities m(P )ÉNS Lyon, Unité de mathématiques pures et appliquées, 46 allée d'Italie, 69007 Lyon, France Email address: francois.brunault@ens-lyon.fr URL: https://perso.ens-lyon.fr/francois.brunault

	E	r	g	Source

1 + x + y + z + xy + xz + yz 14a4 -5/2 1 [8] 1 + x + y + z + xy + xz + yz -xyz 36a1 -1? = r ⋅ L ′ (E, -1)

The author was supported by the research project "Motivic homotopy, quadratic invariants and diagonal classes" (ANR-21-CE40-0015) operated by the French National Research Agency (ANR).

1