# Nonlinear Color$-$Metallicity Relations of Globular Clusters. XI.
Nonlinearity Effect Revealed by NGC 5128 (Centaurus A) and NGC 4594 (Sombrero)
Galaxies

Hak-Sub Kim Both authors have contributed equally to this paper.  Department
of Physics and Astronomy, Sejong University, Seoul 05006, Republic of Korea
Suk-Jin Yoon Both authors have contributed equally to this paper.  Department
of Astronomy & Center for Galaxy Evolution Research, Yonsei University, Seoul
03722, Republic of Korea  Sang-Yoon Lee Department of Astronomy & Center for
Galaxy Evolution Research, Yonsei University, Seoul 03722, Republic of Korea
Sang-Il Han  Department of Science Education, Ewha Womans University, Seoul
03760, Republic of Korea

(Received 2022 January 10; Revised 2022 July 11; Accepted 2022 July 12)

###### Abstract

Metallicity distributions (MDs) of globular clusters (GCs) provide crucial
clues for the assembly and star formation history of their host galaxies. GC
colors, when GCs are old, have been used as a proxy of GC metallicities.
Bimodal GC color distributions (CDs) observed in most early-type galaxies have
been interpreted as bimodal MDs for decades, suggesting the presence of merely
two GC subpopulations within single galaxies. However, the conventional view
has been challenged by a new theory that nonlinear metallicity-to-color
conversion can cause bimodal CDs from unimodal MDs. The unimodal MDs seem
natural given that MDs involved many thousand protogalaxies. The new theory
has been tested and corroborated by various observational and theoretical
studies.

Here we examine the nonlinear nature of GC color$-$metallicity relations
(CMRs) using photometric and spectroscopic GC data of NGC 5128 (Centaurus A)
and NGC 4594 (Sombrero), in comparison with stellar population simulations. We
find that, with a slight offset in color, the overall shapes of observed and
modeled CMRs agree well for all available colors. Diverse color-depending
morphologies of GC CDs of the two galaxies are well reproduced based on their
observed spectroscopic MDs via our CMR models. The results corroborate the
nonlinear CMR interpretation of the GC color bimodality, shedding further
light on theories of galaxy formation.

Globular star clusters (656), Stellar populations(1622), Galaxy evolution
(594)

2022 August 25

##  1 Introduction

Globular clusters (GCs) are among the oldest stellar systems in the Universe.
They are thought to have formed during the major star formation events in
galactic history and survived relatively intact throughout the galactic
evolution. Since they retain the chemical properties of the surrounding
environment at the formation epoch, their metallicity distribution (MD)
provides a vital clue to the formation and evolution of galaxies (Bastian &
Lardo, 2018). Due to the limitations of observational instruments, GC MDs are
often inferred from photometric data rather than obtained directly through
spectroscopy. GC colors have been used as proxies for GC metallicity under the
assumption that, when GCs are old, colors correlate linearly with metallicity.

One of the most important features observed in GC color distribution (CD) is
bimodality, in that the CD of GCs belonging to one galaxy exhibits two peaks
(Brodie & Strader, 2006). Since this phenomenon does not appear only in
certain galaxies, but in most large galaxies, proper interpretation of this
phenomenon is crucial to the study of galaxy formation and evolution. With
empirical linear GC color–metallicity relations (CMRs), bimodal CDs have
generally been translated to bimodal MDs, implying the presence of two
populations of GCs within individual galaxies (e.g., Park & Lee, 2013; Lee &
Jang, 2016; Forbes & Remus, 2018; El-Badry et al., 2019). However, Yoon et al.
(2006, hereafter Paper I) give a plausible alternative explanation for the
origin of GC color bimodality, in which the nonlinear nature of GC CMRs can
lead to the bimodal CDs even from a single-peaked MD. In this theory, the
underlying GC MDs could be quite different from that of GC CDs characterized
by bimodality. Although many studies have attempted to examine the new
explanation by various means (e.g., Cantiello & Blakeslee, 2007; Kundu & Zepf,
2007; Strader et al., 2007; Peng et al., 2008; Spitler et al., 2008; Blakeslee
et al., 2010; Schuberth et al., 2010; Blakeslee et al., 2012; Brodie et al.,
2012; Chies-Santos et al., 2012; Park et al., 2012; Pota et al., 2013; Park &
Lee, 2013; Vanderbeke et al., 2014; Cantiello et al., 2014; Richtler et al.,
2015; Salinas et al., 2015; Cho et al., 2016; Villaume et al., 2019; Fahrion
et al., 2020), this issue is still under debate. Elucidating what the
nonlinearity is like in GC CMR and how it affects the color-to-metallicity
transformation is at the heart of this debate.

Since Paper I, a series of papers (Yoon et al., 2011a, b, 2013; Kim et al.,
2013; Chung et al., 2016; Kim & Yoon, 2017; Lee et al., 2019, 2020; Kim et
al., 2021, hereafter Papers II $-$ X) have been scrutinized the nonlinear CMR
theory, both observationally and theoretically. Papers II and IV reported
strong color-dependent variations in the CD morphology of M87 and of M84 GCs,
respectively, which were incomprehensible with linear, straight CMRs but were
well reproduced by the nonlinear CMRs. Paper III proposed a highly viable
solution based on the CMR nonlinearity for the perplexing conundrum of the MD
discrepancy between GCs and field halo stars in given parent galaxies. Papers
V and VII paid attention to bimodal distributions of absorption-line indices
such as Balmer and Mg$b$ lines for GCs in highly studied M31 and NGC 5128,
respectively. The bimodality is readily explained by nonlinear
index$-$metallicity relations, analogous to the nonlinear CMR projection.
Paper VI approached theoretically to the MDs derived using the Ca Triplet
index, a widely used metallicity indicator. They showed that the Ca
Triplet$-$color$-$metallicity relation is highly nonlinear, resulting in a
sharp difference between color and Ca Triplet distributions of old GCs. Paper
VIII proposed a simple, coherent explanation for the observed diversity in the
GC CDs of early-type galaxies in the Virgo and Fornax galaxy clusters, where
the diversity is due mainly to changes in the mean metallicity of GC systems.
Paper IX explained the difference in the radial number density profile between
blue and red GCs by combining the radial metallicity gradient of the GC system
and the nonlinear nature of metallicity-to-color transformation. Paper X
observed GCs in M87 with the Subaru/FOCAS multiobject spectroscopy mode. Their
high-quality, homogeneous dataset corroborated the theoretical prediction of
inflected CMRs.

As part of the series papers, we investigate the presence and degree of
nonlinearity along GC CMRs for various colors by combining multiband
photometric data with spectroscopic data of the two nearby massive galaxies,
NGC 5128 (Centaurus A) and NGC 4594 (Sombrero). The photometry of NGC 4594 GCs
is from our observations and the other data are taken from the literature. The
NGC 5128, the central giant elliptical galaxy in the Centaurus group of
galaxies, is one of the best targets to study the nature of GC CMRs thanks to
its proximity ($\sim$4 Mpc), the richness of GCs ($>3000$; Taylor et al.,
2017), and relatively low differential reddening of E($B-V$)=0.07 (Rejkuba et
al., 2002). The NGC 4594 is a lenticular galaxy with an unusually large bulge
and a prominent dust lane in the nearly edge-on disk. Since it is also nearby
($\sim$9.55 Mpc) and has a relatively large number of GCs ($\sim$1150; Larsen
et al., 2001), this galaxy’s GC system is well suited for GC CMR studies.

##  2 Observational Data and Theoretical Model

###  2.1 NGC 5128 (Centaurus A) Globular Clusters

The photometry for GCs in NGC 5128 was obtained from the compilation of
Woodley et al. (2007), which combined the $UBVRI$ photometry by Peng et al.
(2004) and the $CMT1$ photometry by Harris et al. (2004). We used the $UBVRI$
data because CMRs of $CMT1$ have too large scatter to examine the CMR
nonlinearity. To compare the photometry with theoretical predictions, we
applied a correction for Galactic extinction based on the COBE/DIRBE dust maps
(Schlegel et al., 1998) with the correction terms by Schlafly & Finkbeiner
(2011) and the extinction law by Cardelli et al. (1989).

The spectroscopic data were taken from Beasley et al. (2008). They obtained
integrated light spectra for 254 GCs in NGC 5128 using the 2-degree Field
instrument on the Anglo-Australian Telescope and provided 21 Lick indices for
146 GCs and empirical spectroscopic metallicities for 243 GCs. The left panel
of Figure 1 shows the spatial distribution of photometric data of Peng et al.
(2004, black dots) and Woodley et al. (2007, blue dots) and spectroscopic data
of Beasley et al. (2008, red dots).

We crossmatched these data using a matching radius of 1″, resulting in a total
number of 234 GCs with both the $UBVRI$ photometry and spectroscopic
metallicity. The empirical metallicity ([M/H]) provided by Beasley et al.
(2008) was transformed to [Fe/H] by subtracting 0.217 (Kim et al., 2002).
Table 1 gives astrometry, spectroscopic metallicity and $UBVRI$ photometry of
crossmatched GC sample in NGC 5128.

###  2.2 NGC 4594 (Sombrero) Globular Clusters

We carried out $UBVI$ photometry for the GC system in NGC 4594 on the nights
of 2008 May 4 and 5 (UT) with the Mosaic II CCD imager mounted on the prime
focus of the 4 m Blanco telescope at Cerro Tololo Inter-American Observatory
(CTIO). The observing conditions were clear and photometric with an average
seeing of around 1$\arcsec$. The Mosaic II consists of eight 2048$\times$4096
pixel CCDs with a pixel scale of 0$\farcs$27 providing a total field of view
of 36$\arcmin\times$36$\arcmin$. The total exposure times in $U$, $B$, $V$,
and $I$ were 18000, 4500, 3000, and 2000 seconds, respectively. We split our
observations in each band into five dithered exposures (two sets for $U$) to
fill in the gaps between the individual CCD chips in the mosaic, and to
minimize the impact on our photometry from CCD blemishes such as hot pixels
and bad columns.

All images were preprocessed using the MSCRED package (Valdes, 1998) in
IRAF111IRAF is distributed by the National Optical Astronomy Observatory,
which is operated by the Association of Universities for Research in
Astronomy, Inc., under a cooperative agreement with the National Science
Foundation.. We used the CCDPROC task to correct for cross-talk between the
CCD chips, trim the overscan, correct for the bias level, and apply the flat-
field correction. The MSCPIXAREA task is used to correct for the variations in
pixel scales across the frame produced by geometric distortion in the Mosaic
II imager. Each mosaic image was then split into eight individual images using
the MSCSPLIT task. We combined the images using the MONTAGE2 in the ALLFRAME
package (Stetson, 1994) to produce a clean high signal-to-noise coadded image
in each filter. Sources were detected from the combined images using the
DAOPHOT II/ALLSTAR (Stetson, 1987), producing a master catalog of the sources.
The master catalog was then input to the ALLFRAME along with all of the images
and their point-spread function (PSF) models for final PSF photometry.
Aperture corrections were computed through the DAOGROW (Stetson, 1990) using
bright and isolated point sources, and we applied them to our photometry
above. Finally, an error-weighted mean instrumental magnitude for each object
was obtained using the DAOMATCH/DAOMASTER (Stetson, 1993).

The photometric calibration was achieved using standard stars in the fields of
Landolt (1992) and Stetson (2000). We corrected for the foreground Galactic
extinction using the reddening maps by Schlegel et al. (1998) with the
correction terms by Schlafly & Finkbeiner (2011) and the extinction law by
Cardelli et al. (1989). Astrometric solutions were calculated using the USNO-B
1.0 catalog stars (Monet et al., 2003), resulting in the typical r.m.s. level
of $\sim$ 0$\farcs$2.

The spectroscopic data were taken from Alves-Brito et al. (2011), who measured
three metallicity-sensitive indices (CH, Mg$b$, & Fe5270) for 247 GC
candidates and transformed the measurements into metallicities using the
Brodie & Huchra (1990) method. They provided error-weighted mean metallicities
for 135 GC candidates for which all three indices are measured. From the
sample, we selected GCs using radial velocity and projected galactocentric
distance (see Figure 3 in Alves-Brito et al. 2011) and used their mean
metallicities for our analysis. The right panel of Figure 1 shows the spatial
distribution of photometric GC candidates (see Section 2.4, gray dots) and
spectroscopic data (yellow and red dots).

We crossmatched our $UBVI$ catalog with the spectroscopic data using a
matching radius of 1″, resulting in a total number of 122 GCs with both $UBVI$
photometry and spectroscopic metallicity. Table 2 gives GC ID, coordinate,
spectroscopic metallicity, and $UBVI$ photometry of crossmatched GC sample in
NGC 4594.

###  2.3 Theoretical Evolutionary Population Synthesis Model

The theoretical models used in this study are constructed using the Yonsei
Evolutionary Population Synthesis (YEPS) code (Chung et al., 2013b). The YEPS
model comprehensively deals with the effect of horizontal-branch stars of
simple stellar populations, which are the main source causing the nonlinearity
of GC CMRs. GC colors, when GCs are old, have been used as a proxy of GC
metallicities. We use the YEPS model of the best-matching age (12.5 Gyr) for
both NGC 5128 and NGC 4594. We adopt the enhanced $\alpha$-elements
([$\alpha$/Fe] = 0.3) and solar-scaled abundance ratios for other elements.
Readers are referred to Chung et al. (2013b, 2017, 2020) for details of the
ingredients and input parameters of the YEPS model. Table 3 provides the YEPS
predictions of metallicity and UBVRI colors for the 12.5 Gyr simple stellar
population, which is used in our present analysis.

###  2.4 Selection of Old Globular Clusters

In order to properly examine the nonlinearity of CMRs for old GCs, possible
young and intermediate-age GCs should be excluded from the sample. Both NGC
5128 and NGC 4594 have strong dust lanes across the central regions, hinting
at recent wet merger events and subsequent star formation. The cosmological
hydrodynamical simulations by Du et al. (2021) suggested that Sombrero-like
galaxies (halo-dominated galaxies with huge bulges) have experienced residual
star formation after major merger events during the formation history. It is
thus naturally expected that young/intermediate-age GCs are intermingled with
the dominant old GCs of the galaxies.

The GC age from integrated light is usually estimated from the comparison of
age-sensitive indices such as Balmer lines and metallicity-sensitive indices
in spectroscopy (Puzia et al., 2005, 2006; Worthey, 1994), or from two-color
diagrams of the combination of age- and metallicity-sensitive colors in
photometry (Puzia et al., 2002; Hempel et al., 2003; Yi et al., 2004; de Grijs
et al., 2005). We chose the photometric method to select old GCs from the
crossmatched data to secure as a large number of GCs as possible for the CMR
nonlinearity test.

Figures 2 and 3 show the two-color diagrams of GCs in NGC 5128 and in NGC
4594, respectively. The selection region is denoted by the polygon drawing by
thick black lines in each panel, and the YEPS models for ages from 1 Gyr
(blue) to 13 Gyr (red) at 2 Gyr intervals are overplotted for reference. The
model loci are slightly shifted to the direction of observations using the
measured offsets, which will be discussed in the following section. The use of
$U$\- and $B$-band data enables us to discern young/intermediate GCs from old
GCs, especially for metal-poor GCs, since young/intermediate GCs are bluer in
$U-B$ and $B-V$ colors. We determine the selection regions to exclude these
blue GCs in $U-B$ and $B-V$ colors and the outliers laid away from the GC
clustering in the two-color diagrams. However, we note that, as young GCs with
intermediate and high metallicities are indistinguishable from old GCs given
the observational uncertainties, they are eliminated only partially,
contaminating the sample to some extent.

##  3 Color–Metallicity Relations

It has been known that there are some offsets between observations and models
in the color–magnitude planes (VandenBerg & Clem, 2003; VandenBerg et al.,
2010; Worthey & Lee, 2011; Angelou et al., 2015) and in the integrated colors
(Barmby & Huchra, 2000; Lee et al., 2002; Conroy et al., 2009). The
discrepancy stems mainly from the incompleteness of models, such as imperfect
treatments of binaries, blue stragglers, horizontal-branch stars, and stars in
the late evolutionary stages (Chung et al., 2013a, 2017; Jang et al., 2021).
To compensate for the defect, we shift the model CMRs in the direction of
colors to match the observed CMRs, allowing a more direct comparison of the
model predictions with observations. The amount of shifts is determined using
the total least squares regression method with sigma clipping. After
transforming colors and metallicities into a normalized format, we calculate
the inversely error-weighted total least squares between observations and
models by moving the model loci by the interval of 0.01 mag in the color
direction. We then find the amount of shifts, which minimizes the sum of the
orthogonal residuals.

Figure 4 shows an example of the process of determining the offset value
between observations and models. The dotted line represents the original model
prediction of the GC CMR and the red solid line shows the shifted model locus
after applying the offset value denoted in the top-left corner of the figure.
The orange lines show the orthogonal distances from the data points to the
model CMR. The black circles are the final sample used in the total least
squares method and the open circles are outliers removed by 5$\sigma$
clipping.

Figure 5 shows the CMRs of NGC 5128 GCs in nine color combinations. The size
and opacity of each symbol are set to be inversely proportional to the
quadratic sum of errors calculated after normalization. The red dashed and
solid lines respectively represent the original 12.5 Gyr YEPS model
predictions and the models shifted by the offset value denoted on the top-left
corner of each panel. We verify that the overall shape of the CMRs agrees well
between observations and models for all the colors. The scatters may result
from both observational uncertainties and the spread of intrinsic properties
of GCs such as age and $\alpha$-element abundance. It is a common feature that
the slope of GC CMRs is steep in the blue region and decreases as the color
becomes redder, which agrees with the features of the YEPS model prediction.
The characteristics are also consistent with previously suggested nonlinear
empirical CMRs such as broken-line CMRs (e.g., Peng et al., 2006; Usher et
al., 2012) and high-order polynomial CMRs (e.g., Blakeslee et al., 2010;
Vanderbeke et al., 2014).

The presence of a quasi-inflection point along $BVRI$ CMRs is an essential
feature in view of the color bimodality origin debate. The observations,
although inconclusive, give hints of inflection along the $BVRI$ CMRs [panels
(d)-(i)]. A paucity of GCs occurs at the vicinity of the quasi-inflection
point where the color changes faster than the metallicity. The paucity near
the inflection point is evident along the color axis, which makes CDs bimodal
even from the unimodal MD (Paper I). We will investigate this effect further
in the next session.

Figure 6 shows the CMRs of NGC 4594 GCs for six color combinations. The
symbols and lines are the same as in Figure 5. Although the scatters are
rather larger compared to the NGC 5128 GC CMRs, there are similar features in
that the CMR slope is steeper in the blue and the slope-changing point appears
to be consistent with the model prediction.

Noticeably, there are outliers away from the model CMR predictions,
particularly among red GCs. To scrutinize the nature of these GCs, in Figure
7, we present their metallicities as a function of galactocentric distance
(left), Mg$b$ vs. [Fe/H] along with model predictions for different
$\alpha$-elements mixture (middle), and the $B-V$ CMR along with the 5 Gyr and
12.5 Gyr model predictions for [$\alpha$/Fe] = 0.0 and 0.3 (right). For sanity
check, the middle panel includes the MILES model predictions (Vazdekis et al.,
2015) for [$\alpha$/Fe] = 0.0 and 0.3. GC systems of massive galaxies usually
exhibit a metallicity gradient (e.g., Forbes & Remus, 2018; Lee et al., 2020),
in the sense that metal-rich GCs are more centrally concentrated than metal-
poor GCs. In the left panel, the inner-halo GCs (R $<$ 3′)222The effective
radius of NGC 4594 is 2′.6 (Jardel et al., 2011). show a large metallicity
spread. The outer-halo GCs (R $>$ 3′) are divided into two groups, one of
which appears to naturally follow the metallicity gradient while the other
group has unexpectedly high metallicity given the radial distance. We mark the
outer-halo, metal-rich GCs as red dots. The middle panel indicates that they
are less enhanced in $\alpha$ elements than the other GCs with similar
metallicities (at [Fe/H] = $-$0.5 $\sim$ 0.5). In the right panel, most of the
outer-halo, metal-rich GCs are located in the younger-age region of CMRs. As
we mentioned in Section 2.4, our sample could be contaminated by young GCs
with intermediate/high metallicities. We therefore suspect that these metal-
rich outliers are young GCs with a history different from that of most GCs in
the galaxy. For example, they might be created from metal-enriched gas in
other galaxies through a relatively prolonged star formation and later
accreted to the Sombrero. For the metal-poor red outliers, on the other hand,
no anomalous properties are found in terms of spatial distribution, radial
velocities, metallicity gradient, and $\alpha$-enhancement.

##  4 Nonlinear Metallicity-to-color Conversion and Color Bimodality

This section focuses on the effect of the nonlinear GC CMR on the metallicity-
to-color transformation. For the GC samples of NGC 5128 and NGC 4594 that are
crossmatched between photometry and spectroscopy, we present the MD and CDs
for each galaxy and analyze the morphologies of the distributions in terms of
the degree of bimodality. We then demonstrate that the nonlinear metallicity-
to-color conversion, combined with the observed GC MDs of NGC 5128 and NGC
4594, reproduces the observed bimodal CDs.

Figures 8 and 9 present the MD and CDs for NGC 5128 GCs and NGC 4594 GCs,
respectively. The MD and CDs in the lower panels of each figure are for the
identical crossmatched GC samples to those used for the CMR nonlinearity test
performed in Section 3. The upper panels of each figure show, for reference,
the CDs of photometrically selected old GC candidates, which have the same
properties as those of the crossmatched GC samples in terms of the magnitude
range and the radial coverage on the sky. Each histogram is smoothed using a
nonparametric kernel density estimate with the bandwidth determined by the
mean observational error. To mitigate spiky features in the histogram that
arises from the small number statistics, we use the same bandwidth for both
photometrically selected GCs (upper panel) and crossmatched GCs (lower panel).
We note that, for the MD of NGC 4594 GCs, Alves-Brito et al. (2011) mentioned
that their metallicity uncertainties might be underestimated, and the
bandwidth is taken as twice of the mean observational error. The smoothed
histograms are denoted as red lines for MDs and green lines for CDs. For NGC
5128 GCs, the MD can be described as a broad, unimodal distribution, but the
CDs appear clearly bimodal, with the detailed shape being different depending
on colors. The MD has a metal-rich peak but the CDs have prominent blue peaks.
Given that metal-rich GCs are red, it is interesting that the shape of the MD
is deformed dramatically after conversion into the CDs. For NGC 4954 GCs, the
MD is unimodal and slightly tilted toward the metal-poor side, and the CDs
exhibit more discernible bimodality and more prominent blue peaks and dips
compared to the MD. In agreement with NGC 5128 GCs, the degree of CD
bimodality of NGC 4594 GCs varies depending on color. This is consistent with
the result for the GC systems of M84, M87, and NGC 1399 (Papers II and IV; Kim
et al. 2013).

To evaluate the degree of bimodality of the MDs and CDs, we calculate the
bimodality coefficient (BC; SAS Institute Inc. 1990), defined as

${BC=\frac{m_{3}^{2}+1}{m_{4}+3\cdot\frac{(n-1)^{2}}{(n-2)(n-3)}}\,\,,}$ (1)

where $n$ refers to the number of samples, and $m_{3}$ and $m_{4}$ are
respectively the skewness and excess kurtosis of the distribution corrected
for sample bias. The BC exploits the skewness and the kurtosis of a
distribution, which measure asymmetry and central clustering of the
distribution, respectively. The basic idea is that a bimodal distribution
generally has higher skewness and lower kurtosis than a simple normal
distribution. The higher BC indicates stronger bimodality, and the lower BC
indicates weaker bimodality. Readers are referred to Knapp (2007), Pfister et
al. (2013), and references therein for further details. Table 4 presents the
BCs for MD and CDs of GCs in NGC 5128 and NGC 4594. The BCs alike indicate
that the CDs are more bimodal than the MD for both galaxies. This is
consistent with the prediction of the nonlinear CMR theory, in which the
inflection point along CMRs evokes or enhances bimodality by creating a dip in
the CDs. The BCs also suggest that the degree of CD bimodality varies with
color. Consistent with the visual impression above, both indices for $U-B$ CDs
of the two galaxies designate weaker bimodality compared to the other CDs,
except for the BC value for NGC 4594 GCs333Pfister et al. (2013) show that the
BC can give erroneous results in that highly skewed unimodal distributions
have larger BC values (i.e., more bimodal) than skewed bimodal distributions
(see also Kang & Noh, 2019)..

We also carry out two statistical tests; Hartigans’ dip test (Hartigan &
Hartigan, 1985) and the bimodality test using Gaussian Mixture Model (GMM)
procedure. The former test is widely used for assessing the unimodality of a
distribution by computing the maximum difference between the input
distribution and the unimodal (uniform) distribution. The latter test assumes
that the input distribution consists of multiple Gaussian components and
constructs the GMMs that best describe the input distributions by maximum
likelihood using the expectation–maximization algorithm. We use the $diptest$
(Maechler, 2021) and $mclust$ (Scrucca et al., 2016) packages implemented in R
(R Core Team, 2022) to perform the dip test and the GMM test (we refer to it
as GMMR) respectively. In the GMMR test, we select the ‘V’ model that assumes
the Gaussian components can have different variances. We also conduct a
revised GMM test provided by Muratov & Gnedin (2010, we refer to it as GMMMG),
that employs three different approaches (the ratio of the likelihood, the peak
separation, and the kurtosis) to get a more robust measure of bimodality.

All the tests provide the probability (i.e., $p$-value) for the null
hypothesis that a distribution is unimodal. We note that the two tests
occasionally give erroneous results. For example, the dip test hardly
distinguishes between multimodal and unimodal distribution when the sample
size is small and/or the significance level is set to be low (Kang & Noh,
2019). The GMMMG test often falsely indicates bimodal when the true
distribution is unimodal but skewed because of its sensitivity to the
assumption of Gaussian modes (Muratov & Gnedin, 2010). As these issues are
beyond the scope of this paper, we focus on the relative comparison of the
$p$-values for the MDs and CDs for each test.

Table 5 provides the $p$-values from the dip test and the GMM tests for the
MDs and CDs of NGC 5128 and NGC 4594 GCs. We remind that the smaller the
$p$-values, the stronger the bimodality. The dip test clearly shows that the
MDs are closest to a unimodal distribution, and the CDs are relatively close
to bimodal distribution for both galaxies. The GMMMG test gives three
probabilities of a unimodal distribution based on maximum likelihood, peak
separation, and kurtosis. They show that, with a few exceptions, the GC MDs
for both galaxies are closer to unimodal distributions than their GC CDs. The
GMMR test shows that, for NGC 5128, the $p$-value for MD is slightly larger,
but a bimodal distribution is preferred over a unimodal distribution for all
distributions. The $p$-values for NGC 4594 indicate that the MD is close to
unimodal and the CDs are close to bimodal.

Figure 10 presents Bayesian information criterion (BIC; Schwarz (1978)) and
integrated complete data likelihood criterion (ICL; Biernacki, Celeux, &
Govaert (2000)) values in the GMMR test as a function of the number of
Gaussian components $K$. BIC and ICL are usually used to select the optimal
model in a way that $K$-components GMM with the highest BIC/ICL value is
preferred. BIC tends to select the number of mixture components, whereas ICL
is more useful for selecting the number of discrete groups in the data
(Biernacki, Celeux, & Govaert, 2000). Given the properties of our data, it is
more appropriate to focus on BIC rather than on ICL. Consistent with the
$p$-values, BIC values are highest at $K=2$ for all distributions except for
the MD of NGC 4594 GCs. Note that, for the MD of NGC 5128, the difference in
BIC values between $K=1$ and $K=2$ is $\sim$1.5. The difference between BIC
values less than 2 is considered a “barely worth mentioning” difference
(Raftery, 1995). Thus, the GMMR test indicates that the MDs are relatively
unimodal in both galaxies, as opposed to the CDs being bimodal.

The statistical tests corroborate that the distributions are significantly
deformed when GC metallicities are converted to GC colors. The large variation
in $p$-values among different colors (e.g., the $p$-value of the dip test for
the NGC 4594 GC system) also indicates a significant change in their degree of
bimodality due to the color-dependent shape of the CMR.

In Figures 11 and 12, we perform Monte Carlo simulations aiming at the CDs of
NGC 5128 and NGC 4594 GCs to validate the nonlinear CMR effect on the
conversion of MDs into CDs. The top row of the figures shows the observed GC
CMRs as shown in Figures 5 and 6, overplotted with the YEPS model predictions.
The observed MDs are rotated and given on the $y$-axis. The second rows show
the observed color–magnitude diagrams of the crossmatched GCs (black dots) and
the photometrically selected GCs (gray dots). Observational uncertainties as a
function of $M_{V}$ are denoted by error bars on the left of each panel. The
third and the fourth rows show the CDs of the crossmatched GCs and
photometrically selected GCs, respectively (same as the Figures 8 and 9). The
bottom row of the figures shows the simulated GC CDs. We generate 100,000
model GCs with metallicities derived from the Kernel density estimation of the
observed MDs. We then transform the metallicity into colors using the CMRs
predicted from the YEPS model, taking into account the observational
uncertainties. For each color, we randomly select the same number of model GCs
with the observed GCs 10,000 times and calculate the 1$\sigma$ distribution
ranges, which are shown by the shaded bands.

The most conspicuous feature of the observed CDs is the prominent blue peak.
Our simulations with nonlinear CMRs can successfully reproduce the observed
prominent blue peak that cannot be explained by the conventional linear CMR
scheme given the unimodal MD with a metal-rich peak. Metal-poor GCs are
projected through the steeper part of the CMR into a narrow color region,
producing the sharp blue peak in the CD. By contrast, metal-rich GCs are
projected through the shallower part of the CMR into a wider color region,
resulting in an attenuated red peak in the CD. This even evokes a reversal of
peak intensity for the NGC 5128 GCs.

The simulations show that projections on nonlinear CMRs with inflection points
cause bimodal CDs from the observed MDs close to unimodal distributions. For
the $B-V$, $B-I$, and $V-I$ colors, the inflection point in the CMR causes a
dip between blue and red peaks, which makes a discernible bimodal feature in
the CDs. On the other hand, for the $U-B$ color, the shape of the CD shows a
skewed Gaussian without bimodality since there is no obvious inflection point
in the $U-B$ CMR. As a consequence, the simulation results for $U-B$, $B-V$,
$B-I$, and $V-I$ are consistent with the corresponding observations.

##  5 Summary and Discussion

In this study, we have examined the nonlinearity of GC CMRs and its effect on
the conversion of MD into CDs. We have used multiband photometric data
combined with spectroscopic data for NGC 5128 and NGC 4594 GCs. We have
demonstrated clear nonlinearity of the old GC CMRs for both galaxies, which is
in good agreement with the model predictions.

Our visual inspection and statistical tests confirm the significant changes in
the morphology of the CDs compared to the MD and even between the CDs. The
most remarkable findings for NGC 5128 are that the metal-rich peak of the GC
MD is significantly reduced into a weak red bump in the CDs, and that strong
blue peaks unexpectedly emerge in the ($U$-band-free) CDs. Also, a distinct
dip that is not obvious in the MD appears in between the blue and red peaks in
the CDs. Similarly, for NGC 4594 GCs, the blue peaks of the CDs are sharper
compared to the corresponding metal-poor peak in the MD, and the dips are
clearly visible in the CDs, making the CDs bimodal. We have demonstrated
through Monte Carlo simulations that all of these features arise from the
nonlinear conversion from the MDs to the CDs.

The CDs of photometrically selected old GCs are very similar to those of the
GC samples crossmatched between photometry and spectroscopy in both galaxies
(Figures 8 and 9). This implies that the MDs observed in this study are
representative MDs of old GCs that are detectable at the corresponding
galactocentric distance range. The shape of the GC MDs in both galaxies is
broad and unimodal. This is a natural expectation in the $\Lambda$ cold dark
matter paradigm, where large galaxies such as NGC 5128 and NGC 4954 form
through the hierarchical merging of numerous protogalaxies.

It is interesting to note that for NGC 5128, the shape of the GC MD is fairly
consistent with that of the halo field stars’ MD (Figure 1 of Rejkuba et al.
2011) and closely matches that of the MD inferred from the inverse
transformation of CDs using nonlinear CMRs (Figure 6 of Yoon et al. 2011a). In
the nonlinear CMR scenario, there is no discrepancy in MDs between GCs and
halo field stars, and no further explanation for the excess of blue GCs is
required. The readers are referred to Yoon et al. (2011a) for an in-depth
discussion about solving the apparent discrepancy between GC MDs and halo
stellar MDs.

Many studies on GC bimodality begin by dividing GCs into two groups in
CDs—blue and red GCs. This is because, in CDs, GCs are distinctively divided
into two groups by the dip. However, in our theory, the clear division in CDs
do not necessarily indicate the presence of the two groups in an MD in a
single galaxy. The crossmatched GC samples for NGC 5128 and NGC 4594 show that
the CDs are bimodal but the corresponding MDs are closer to unimodal than
bimodal. A crucial point of GC color bimodality is that it is a universal
phenomenon commonly observed in massive early-type galaxies (Brodie & Strader,
2006). The detailed formation process of these galaxies varies from case to
case, but there should be an essential common process to give rise to the
phenomenon. In this respect, the nonlinear CMR scenario has its own strength
in that the projection effect can simply solve the ubiquity of GC color
bimodality. This study provides further direct observational evidence (see
also Papers I–X) favoring the projection effect that significantly deforms MDs
into various CD morphologies. Therefore, we conclude that the nonlinear CMR
effect is the basic cause of the color bimodality phenomenon, and the detailed
characteristics of CDs vary depending on the formation history of each galaxy.

H.-S.K. acknowledges financial support from the Basic Science Research Program
(No. 2019R1A2C2086290) through the National Research Foundation of Korea
(NRF). S.-J.Y. acknowledges support from the Mid-career Researcher Program
(No. 2019R1A2C3006242) through the NRF. S.I.H. acknowledges the support
provided by the Basic Science Research Program through the NRF funded by the
Ministry of Education (No. 2020R1I1A1A01052358) and by the NRF grant funded by
the Korean government (MSIT) (No. 2021R1A2C1004117).

## References

  * Alves-Brito et al. (2011) Alves-Brito, A., Hau, G. K. T., Forbes, D. A., et al. 2011, MNRAS, 417, 1823. doi:10.1111/j.1365-2966.2011.19368.x
  * Angelou et al. (2015) Angelou, G. C., D’Orazi, V., Constantino, T. N., et al. 2015, MNRAS, 450, 2423. doi:10.1093/mnras/stv770
  * Barmby & Huchra (2000) Barmby, P. & Huchra, J. P. 2000, ApJ, 531, L29. doi:10.1086/312511
  * Bastian & Lardo (2018) Bastian, N. & Lardo, C. 2018, ARA&A, 56, 83. doi:10.1146/annurev-astro-081817-051839
  * Beasley et al. (2008) Beasley, M. A., Bridges, T., Peng, E., et al. 2008, MNRAS, 386, 1443. doi:10.1111/j.1365-2966.2008.13123.x
  * Biernacki, Celeux, & Govaert (2000) Biernacki, C., Celeux, G. and Govaert, G. 2000, ITPAM, 22, 719. doi:10.1109/34.865189
  * Blakeslee et al. (2010) Blakeslee, J. P., Cantiello, M., & Peng, E. W. 2010, ApJ, 710, 51. doi:10.1088/0004-637X/710/1/51
  * Blakeslee et al. (2012) Blakeslee, J. P., Cho, H., Peng, E. W., et al. 2012, ApJ, 746, 88. doi:10.1088/0004-637X/746/1/88
  * Brodie & Huchra (1990) Brodie, J. P. & Huchra, J. P. 1990, ApJ, 362, 503. doi:10.1086/169288
  * Brodie & Strader (2006) Brodie, J. P. & Strader, J. 2006, ARA&A, 44, 193. doi:10.1146/annurev.astro.44.051905.092441
  * Brodie et al. (2012) Brodie, J. P., Usher, C., Conroy, C., et al. 2012, ApJ, 759, L33. doi:10.1088/2041-8205/759/2/L33
  * Cantiello & Blakeslee (2007) Cantiello, M. & Blakeslee, J. P. 2007, ApJ, 669, 982. doi:10.1086/522110
  * Cantiello et al. (2014) Cantiello, M., Blakeslee, J. P., Raimondo, G., et al. 2014, A&A, 564, L3. doi:10.1051/0004-6361/201323272
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245. doi:10.1086/167900
  * Chies-Santos et al. (2012) Chies-Santos, A. L., Larsen, S. S., Cantiello, M., et al. 2012, A&A, 539, A54. doi:10.1051/0004-6361/201117169
  * Cho et al. (2016) Cho, H., Blakeslee, J. P., Chies-Santos, A. L., et al. 2016, ApJ, 822, 95. doi:10.3847/0004-637X/822/2/95
  * Chung et al. (2020) Chung, C., Yoon, S.-J., Cho, H., et al. 2020, ApJS, 250, 33. doi:10.3847/1538-4365/abb4e6
  * Chung et al. (2013a) Chung, C., Lee, S.-Y., Yoon, S.-J., et al. 2013, ApJ, 769, L3. doi:10.1088/2041-8205/769/1/L3
  * Chung et al. (2013b) Chung, C., Yoon, S.-J., Lee, S.-Y., et al. 2013, ApJS, 204, 3. doi:10.1088/0067-0049/204/1/3
  * Chung et al. (2016) Chung, C., Yoon, S.-J., Lee, S.-Y., et al. 2016, ApJ, 818, 201. doi:10.3847/0004-637X/818/2/201
  * Chung et al. (2017) Chung, C., Yoon, S.-J., & Lee, Y.-W. 2017, ApJ, 842, 91. doi:10.3847/1538-4357/aa6f19
  * Conroy et al. (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486. doi:10.1088/0004-637X/699/1/486
  * de Grijs et al. (2005) de Grijs, R., Anders, P., Lamers, H. J. G. L. M., et al. 2005, MNRAS, 359, 874. doi:10.1111/j.1365-2966.2005.08914.x
  * Du et al. (2021) Du, M., Ho, L. C., Debattista, V. P., et al. 2021, ApJ, 919, 135. doi:10.3847/1538-4357/ac0e98
  * El-Badry et al. (2019) El-Badry, K., Quataert, E., Weisz, D. R., et al. 2019, MNRAS, 482, 4528. doi:10.1093/mnras/sty3007
  * Fahrion et al. (2020) Fahrion, K., Lyubenova, M., Hilker, M., et al. 2020, A&A, 637, A27. doi:10.1051/0004-6361/202037686
  * Forbes & Remus (2018) Forbes, D. A. & Remus, R.-S. 2018, MNRAS, 479, 4760. doi:10.1093/mnras/sty1767
  * Harris et al. (2004) Harris, G. L. H., Geisler, D., Harris, W. E., et al. 2004, AJ, 128, 712. doi:10.1086/421847
  * Hartigan & Hartigan (1985) Hartigan, J. A. & Hartigan, P. M. 1985, AnSta, 13, 70. doi:10.1214/aos/1176346577
  * Hempel et al. (2003) Hempel, M., Hilker, M., Kissler-Patig, M., et al. 2003, A&A, 405, 487. doi:10.1051/0004-6361:20030598
  * Jang et al. (2021) Jang, S., Milone, A. P., Lagioia, E. P., et al. 2021, ApJ, 920, 129. doi:10.3847/1538-4357/ac1861
  * Jardel et al. (2011) Jardel, J. R., Gebhardt, K., Shen, J., et al. 2011, ApJ, 739, 21. doi:10.1088/0004-637X/739/1/21
  * Kang & Noh (2019) Kang, Y.-J. & Noh, Y. 2019, Math. Probl. Eng., 2019, 17. doi:10.1155/2019/4819475
  * Kim et al. (2002) Kim, Y.-C., Demarque, P., Yi, S. K., et al. 2002, ApJS, 143, 499. doi:10.1086/343041
  * Kim et al. (2013) Kim, H.-S., Yoon, S.-J., Sohn, S. T., et al. 2013, ApJ, 763, 40. doi:10.1088/0004-637X/763/1/40
  * Kim & Yoon (2017) Kim, S. & Yoon, S.-J. 2017, ApJ, 843, 43. doi:10.3847/1538-4357/aa7387
  * Kim et al. (2013) Kim, S., Yoon, S.-J., Chung, C., et al. 2013, ApJ, 768, 138. doi:10.1088/0004-637X/768/2/138
  * Kim et al. (2021) Kim, S., Yoon, S.-J., Lee, S.-Y., et al. 2021, ApJS, 256, 29. doi:10.3847/1538-4365/ac10c2
  * Knapp (2007) Knapp, T. R. 2007, JMASM, 6, 8. doi:10.22237/jmasm/1177992120
  * Kundu & Zepf (2007) Kundu, A. & Zepf, S. E. 2007, ApJ, 660, L109. doi:10.1086/518214
  * Landolt (1992) Landolt, A. U. 1992, AJ, 104, 340. doi:10.1086/116242
  * Larsen et al. (2001) Larsen, S. S., Forbes, D. A., & Brodie, J. P. 2001, MNRAS, 327, 1116. doi:10.1046/j.1365-8711.2001.04797.x
  * Lee et al. (2002) Lee, H.-. chul ., Lee, Y.-W., & Gibson, B. K. 2002, AJ, 124, 2664. doi:10.1086/344066
  * Lee et al. (2019) Lee, S.-Y., Chung, C., & Yoon, S.-J. 2019, ApJS, 240, 2. doi:10.3847/1538-4365/aaecd4
  * Lee et al. (2020) Lee, S.-Y., Chung, C., & Yoon, S.-J. 2020, ApJ, 905, 124. doi:10.3847/1538-4357/abc4e9
  * Lee & Jang (2016) Lee, M. G. & Jang, I. S. 2016, ApJ, 819, 77. doi:10.3847/0004-637X/819/1/77
  * Maechler (2021) Maechler, M. 2021, diptest: Hartigan’s Dip Test Statistic for Unimodality - Corrected. R package version 0.76-0 https://CRAN.R-project.org/package=diptest
  * Monet et al. (2003) Monet, D. G., Levine, S. E., Canzian, B., et al. 2003, AJ, 125, 984. doi:10.1086/345888
  * Muratov & Gnedin (2010) Muratov, A. L. & Gnedin, O. Y. 2010, ApJ, 718, 1266. doi:10.1088/0004-637X/718/2/1266
  * Park & Lee (2013) Park, H. S. & Lee, M. G. 2013, ApJ, 773, L27. doi:10.1088/2041-8205/773/2/L27
  * Park et al. (2012) Park, H. S., Lee, M. G., Hwang, H. S., et al. 2012, ApJ, 759, 116. doi:10.1088/0004-637X/759/2/116
  * Peng et al. (2004) Peng, E. W., Ford, H. C., & Freeman, K. C. 2004, ApJS, 150, 367. doi:10.1086/381144
  * Peng et al. (2006) Peng, E. W., Jordán, A., Côté, P., et al. 2006, ApJ, 639, 95. doi:10.1086/498210
  * Peng et al. (2008) Peng, E. W., Jordán, A., Côté, P., et al. 2008, ApJ, 681, 197. doi:10.1086/587951
  * Pfister et al. (2013) Pfister, R., Schwarz, K. A., Janczyk, M., et al. 2013, Front. Psychol., 4, 700. doi: 10.3389/fpsyg.2013.00700
  * Pota et al. (2013) Pota, V., Forbes, D. A., Romanowsky, A. J., et al. 2013, MNRAS, 428, 389. doi:10.1093/mnras/sts029
  * Puzia et al. (2006) Puzia, T. H., Kissler-Patig, M., & Goudfrooij, P. 2006, ApJ, 648, 383. doi:10.1086/505679
  * Puzia et al. (2005) Puzia, T. H., Kissler-Patig, M., Thomas, D., et al. 2005, A&A, 439, 997. doi:10.1051/0004-6361:20047012
  * Puzia et al. (2002) Puzia, T. H., Zepf, S. E., Kissler-Patig, M., et al. 2002, A&A, 391, 453. doi:10.1051/0004-6361:20020835
  * R Core Team (2022) R Core Team 2022, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria https://www.R-project.org/.
  * Raftery (1995) Raftery, A. E. 1995, Sociol. Methodol., 25, 111. doi:10.2307/271063.
  * Rejkuba et al. (2002) Rejkuba, M., Minniti, D., Courbin, F., et al. 2002, ApJ, 564, 688. doi:10.1086/324500
  * Rejkuba et al. (2011) Rejkuba, M., Harris, W. E., Greggio, L., et al. 2011, A&A, 526, A123. doi:10.1051/0004-6361/201015640
  * Richtler et al. (2015) Richtler, T., Salinas, R., Lane, R. R., et al. 2015, A&A, 574, A21. doi:10.1051/0004-6361/201424530
  * Salinas et al. (2015) Salinas, R., Alabi, A., Richtler, T., et al. 2015, A&A, 577, A59. doi:10.1051/0004-6361/201425574
  * SAS Institute Inc. (1990) SAS Institute Inc. 1990 SAS/STAT User’s Guide, Version 6, 4th Ed. (Cary, NC: SAS Institute), 561
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103. doi:10.1088/0004-637X/737/2/103
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525. doi:10.1086/305772
  * Schuberth et al. (2010) Schuberth, Y., Richtler, T., Hilker, M., et al. 2010, A&A, 513, A52. doi:10.1051/0004-6361/200912482
  * Schwarz (1978) Schwarz, G. 1978, Ann. Stat., 6, 461. doi:10.1214/aos/1176344136
  * Scrucca et al. (2016) Scrucca, L., Fop, M., Murphy, T. B., Raftery, A. E. 2016, 289, mclust 5: clustering, classification and density estimation using Gaussian finite mixture models., The R Journal https://CRAN.R-project.org/package=mclust doi:10.32614/RJ-2016-021
  * Spitler et al. (2008) Spitler, L. R., Forbes, D. A., & Beasley, M. A. 2008, MNRAS, 389, 1150. doi:10.1111/j.1365-2966.2008.13681.x
  * Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191. doi:10.1086/131977
  * Stetson (1990) Stetson, P. B. 1990, PASP, 102, 932. doi:10.1086/132719
  * Stetson (1993) Stetson, P. B. 1993, IAU Colloq. 136, Stellar Photometry–Current Techniques and Future Developments, ed. C. J. Butler & I. Elliot (Cambridge: Cambridge Univ. Press), 291
  * Stetson (1994) Stetson, P. B. 1994, PASP, 106, 250. doi:10.1086/133378
  * Stetson (2000) Stetson, P. B. 2000, PASP, 112, 925. doi:10.1086/316595
  * Strader et al. (2007) Strader, J., Beasley, M. A., & Brodie, J. P. 2007, AJ, 133, 2015. doi:10.1086/512770
  * Taylor et al. (2017) Taylor, M. A., Puzia, T. H., Muñoz, R. P., et al. 2017, MNRAS, 469, 3444. doi:10.1093/mnras/stx1021
  * Usher et al. (2012) Usher, C., Forbes, D. A., Brodie, J. P., et al. 2012, MNRAS, 426, 1475. doi:10.1111/j.1365-2966.2012.21801.x
  * Valdes (1998) Valdes, F. G. 1998, in ASP Conf. Ser. 145, Astronomical Data Analysis Software and Systems VII, ed. R. Albrecht, R. N. Hook, & H. A. Bushouse, 53
  * VandenBerg et al. (2010) VandenBerg, D. A., Casagrande, L., & Stetson, P. B. 2010, AJ, 140, 1020. doi:10.1088/0004-6256/140/4/1020
  * VandenBerg & Clem (2003) VandenBerg, D. A. & Clem, J. L. 2003, AJ, 126, 778. doi:10.1086/376840
  * Vanderbeke et al. (2014) Vanderbeke, J., West, M. J., De Propris, R., et al. 2014, MNRAS, 437, 1734. doi:10.1093/mnras/stt2012
  * Vazdekis et al. (2015) Vazdekis, A., Coelho, P., Cassisi, S., et al. 2015, MNRAS, 449, 1177. doi:10.1093/mnras/stv151
  * Villaume et al. (2019) Villaume, A., Romanowsky, A. J., Brodie, J., et al. 2019, ApJ, 879, 45. doi:10.3847/1538-4357/ab24d7
  * Woodley et al. (2007) Woodley, K. A., Harris, W. E., Beasley, M. A., et al. 2007, AJ, 134, 494. doi:10.1086/518788
  * Worthey (1994) Worthey, G. 1994, ApJS, 95, 107. doi:10.1086/192096
  * Worthey & Lee (2011) Worthey, G. & Lee, H.-. chul . 2011, ApJS, 193, 1. doi:10.1088/0067-0049/193/1/1
  * Yi et al. (2004) Yi, S. K., Peng, E., Ford, H., et al. 2004, MNRAS, 349, 1493. doi:10.1111/j.1365-2966.2004.07614.x
  * Yoon et al. (2006) Yoon, S.-J., Yi, S. K., & Lee, Y.-W. 2006, Sci, 311, 1129. doi:10.1126/science.1122294
  * Yoon et al. (2011a) Yoon, S.-J., Lee, S.-Y., Blakeslee, J. P., et al. 2011a, ApJ, 743, 150. doi:10.1088/0004-637X/743/2/150
  * Yoon et al. (2011b) Yoon, S.-J., Sohn, S. T., Lee, S.-Y., et al. 2011b, ApJ, 743, 149. doi:10.1088/0004-637X/743/2/149
  * Yoon et al. (2013) Yoon, S.-J., Sohn, S. T., Kim, H.-S., et al. 2013, ApJ, 768, 137. doi:10.1088/0004-637X/768/2/137

Table 1: $UBVRI$ Photometry and Spectroscopic Metallicity of Crossmatched GC
Sample in NGC 5128

ID | R.A. (J2000) | Decl. (J2000) | [Fe/H]aaThe metallicities are transformed from the empirical metallicity ([M/H]) provided by Beasley et al. (2008). |  $U_{0}$bbThe photometric data are from Woodley et al. (2007) and corrected for Galactic extinction (see text). |  $B_{0}$bbThe photometric data are from Woodley et al. (2007) and corrected for Galactic extinction (see text). |  $V_{0}$bbThe photometric data are from Woodley et al. (2007) and corrected for Galactic extinction (see text). |  $R_{0}$bbThe photometric data are from Woodley et al. (2007) and corrected for Galactic extinction (see text). |  $I_{0}$bbThe photometric data are from Woodley et al. (2007) and corrected for Galactic extinction (see text).
---|---|---|---|---|---|---|---|---
1 | 13:23:58.76 | –43:01:35.11 | –0.52 $\pm$ 0.08 | 19.91 $\pm$ 0.06 | 19.44 $\pm$ 0.02 | 18.54 $\pm$ 0.01 | 18.00 $\pm$ 0.01 | 17.37 $\pm$ 0.03
2 | 13:23:58.58 | –42:57:17.00 | –2.12 $\pm$ 0.17 | 20.03 $\pm$ 0.06 | 20.09 $\pm$ 0.02 | 19.52 $\pm$ 0.01 | 19.15 $\pm$ 0.01 | 18.75 $\pm$ 0.03
3 | 13:23:59.61 | –42:55:19.40 | –2.42 $\pm$ 0.48 | 20.01 $\pm$ 0.06 | 20.08 $\pm$ 0.02 | 19.53 $\pm$ 0.01 | 19.17 $\pm$ 0.01 | 18.80 $\pm$ 0.03
4 | 13:24:05.98 | –43:03:54.70 | –0.69 $\pm$ 0.18 | 21.04 $\pm$ 0.14 | 20.65 $\pm$ 0.03 | 19.79 $\pm$ 0.01 | 19.30 $\pm$ 0.01 | 18.78 $\pm$ 0.03
5 | 13:24:28.44 | –42:57:52.90 | –1.92 $\pm$ 0.24 | 20.59 $\pm$ 0.10 | 20.48 $\pm$ 0.03 | 19.79 $\pm$ 0.01 | 19.33 $\pm$ 0.01 | 18.91 $\pm$ 0.03
6 | 13:24:29.23 | –43:08:36.59 | –0.81 $\pm$ 0.13 | 21.27 $\pm$ 0.19 | 20.80 $\pm$ 0.04 | 19.90 $\pm$ 0.01 | 19.34 $\pm$ 0.01 | 18.80 $\pm$ 0.03
7 | 13:24:29.72 | –43:02:06.48 | –0.73 $\pm$ 0.27 | 20.68 $\pm$ 0.11 | 20.49 $\pm$ 0.03 | 19.74 $\pm$ 0.01 | 19.23 $\pm$ 0.01 | 18.77 $\pm$ 0.03
8 | 13:24:34.62 | –43:12:50.51 | –1.22 $\pm$ 0.34 | 21.17 $\pm$ 0.17 | 20.71 $\pm$ 0.04 | 19.82 $\pm$ 0.01 | 19.29 $\pm$ 0.01 | 18.69 $\pm$ 0.03
9 | 13:24:43.57 | –43:08:43.20 | –1.72 $\pm$ 0.11 | 19.11 $\pm$ 0.04 | 19.07 $\pm$ 0.01 | 18.45 $\pm$ 0.01 | 18.04 $\pm$ 0.01 | 17.66 $\pm$ 0.03
10 | 13:24:45.35 | –42:59:33.50 | –1.92 $\pm$ 0.15 | 19.38 $\pm$ 0.06 | 19.43 $\pm$ 0.02 | 18.98 $\pm$ 0.01 | 18.64 $\pm$ 0.01 | 18.13 $\pm$ 0.03
11 | 13:24:45.76 | –43:02:24.48 | –0.26 $\pm$ 0.09 | 21.32 $\pm$ 0.25 | 20.60 $\pm$ 0.04 | 19.56 $\pm$ 0.01 | 18.98 $\pm$ 0.01 | 18.37 $\pm$ 0.03
12 | 13:24:47.61 | –43:10:48.50 | –1.32 $\pm$ 0.25 | 20.34 $\pm$ 0.10 | 20.08 $\pm$ 0.03 | 19.32 $\pm$ 0.01 | 18.83 $\pm$ 0.01 | 18.37 $\pm$ 0.03
13 | 13:24:49.37 | –43:08:17.69 | –0.92 $\pm$ 0.31 | 20.93 $\pm$ 0.16 | 20.72 $\pm$ 0.04 | 19.99 $\pm$ 0.02 | 19.51 $\pm$ 0.01 | 19.02 $\pm$ 0.03
14 | 13:24:50.08 | –43:07:36.20 | –1.09 $\pm$ 0.42 | 20.92 $\pm$ 0.16 | 20.64 $\pm$ 0.04 | 20.11 $\pm$ 0.02 | 19.76 $\pm$ 0.02 | 19.28 $\pm$ 0.03
15 | 13:24:52.98 | –43:11:55.80 | –0.91 $\pm$ 0.31 | 20.63 $\pm$ 0.11 | 20.46 $\pm$ 0.03 | 19.77 $\pm$ 0.01 | 19.38 $\pm$ 0.01 | 18.92 $\pm$ 0.03
… | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | …

Note. — This table is available in its entirety in a machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.

Table 2: $UBVI$ Photometry and Spectroscopic Metallicity of Crossmatched GC
Sample in NGC 4594

ID | R.A. (J2000) | Decl. (J2000) | [Fe/H]aaThe metallicities are from Alves-Brito et al. (2011). | $U_{0}$ | $B_{0}$ | $V_{0}$ | $I_{0}$
---|---|---|---|---|---|---|---
1 | 12:40:10.48 | –11:45:20.56 | –0.58 $\pm$ 0.07 | 21.71 $\pm$ 0.02 | 21.27 $\pm$ 0.01 | 20.40 $\pm$ 0.01 | 19.28 $\pm$ 0.01
2 | 12:40:10.02 | –11:44:19.94 | –1.35 $\pm$ 0.08 | 22.64 $\pm$ 0.04 | 22.24 $\pm$ 0.01 | 21.37 $\pm$ 0.01 | 20.30 $\pm$ 0.01
3 | 12:40:08.50 | –11:43:24.49 | –1.41 $\pm$ 0.07 | 21.63 $\pm$ 0.02 | 21.32 $\pm$ 0.01 | 20.63 $\pm$ 0.01 | 19.68 $\pm$ 0.01
4 | 12:39:55.86 | –11:42:28.97 | –1.78 $\pm$ 0.06 | 22.28 $\pm$ 0.03 | 22.24 $\pm$ 0.01 | 21.58 $\pm$ 0.01 | 20.63 $\pm$ 0.01
5 | 12:40:05.67 | –11:42:18.26 | –1.31 $\pm$ 0.05 | 21.82 $\pm$ 0.02 | 21.72 $\pm$ 0.01 | 21.02 $\pm$ 0.01 | 20.08 $\pm$ 0.01
6 | 12:40:10.04 | –11:42:11.78 | –2.05 $\pm$ 0.05 | 22.41 $\pm$ 0.03 | 22.33 $\pm$ 0.01 | 21.68 $\pm$ 0.01 | 20.82 $\pm$ 0.01
7 | 12:40:11.71 | –11:41:56.65 | –1.30 $\pm$ 0.07 | 21.61 $\pm$ 0.02 | 21.39 $\pm$ 0.01 | 20.63 $\pm$ 0.01 | 19.66 $\pm$ 0.01
8 | 12:40:14.63 | –11:41:55.60 | –1.27 $\pm$ 0.06 | 22.52 $\pm$ 0.03 | 22.49 $\pm$ 0.01 | 21.80 $\pm$ 0.01 | 20.85 $\pm$ 0.02
9 | 12:40:12.63 | –11:41:29.59 | –1.45 $\pm$ 0.05 | 22.40 $\pm$ 0.03 | 22.34 $\pm$ 0.01 | 21.60 $\pm$ 0.01 | 20.64 $\pm$ 0.01
10 | 12:40:04.65 | –11:40:53.91 | –0.97 $\pm$ 0.05 | 22.91 $\pm$ 0.04 | 22.56 $\pm$ 0.01 | 21.71 $\pm$ 0.01 | 20.65 $\pm$ 0.02
11 | 12:40:01.36 | –11:40:52.32 | 0.32 $\pm$ 0.06 | 24.03 $\pm$ 0.10 | 23.32 $\pm$ 0.02 | 22.39 $\pm$ 0.02 | 21.12 $\pm$ 0.02
12 | 12:40:15.01 | –11:40:44.44 | –1.21 $\pm$ 0.05 | 22.40 $\pm$ 0.03 | 22.20 $\pm$ 0.01 | 21.47 $\pm$ 0.01 | 20.53 $\pm$ 0.01
13 | 12:40:07.23 | –11:40:25.56 | –1.03 $\pm$ 0.04 | 21.50 $\pm$ 0.02 | 21.20 $\pm$ 0.01 | 20.40 $\pm$ 0.01 | 19.40 $\pm$ 0.01
14 | 12:40:25.81 | –11:40:19.07 | –0.56 $\pm$ 0.05 | 23.32 $\pm$ 0.05 | 22.82 $\pm$ 0.01 | 21.92 $\pm$ 0.01 | 20.75 $\pm$ 0.02
15 | 12:40:01.23 | –11:40:07.90 | –2.13 $\pm$ 0.05 | 22.19 $\pm$ 0.03 | 22.13 $\pm$ 0.01 | 21.44 $\pm$ 0.01 | 20.53 $\pm$ 0.01
… | … | … | … | … | … | … | …
… | … | … | … | … | … | … | …
… | … | … | … | … | … | … | …

Note. — This table is available in its entirety in a machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.

Figure 1:  Spatial distribution of GC data set used in this study for NGC 5128
(left) and NGC 4594 (right). For NGC 5128, $UBVRI$ photometry is taken from
Peng et al. (2004, black dots) and Woodley et al. (2007, blue dots), and
spectroscopic metallicity is taken from Beasley et al. (2008, red dots). A
total of 234 GCs are crossmatched between the photometry and the spectroscopy.
For NGC 4594, $UBVI$ photometry is taken from our observations. Gray dots are
GC candidates selected using two-color diagrams (see Section 2.4).
Spectroscopic metallicity is taken from Alves-Brito et al. (2011), who provide
spectroscopic metallicity derived from three metallicity-sensitive indices
(CH, Mg$b$, & Fe5270). We only use the metallicities derived from all three
indices (red dots), except for metallicities from one or two indices (orange
dots). A total of 108 GCs are crossmatched between the photometry and the
spectroscopy.

Table 3: Theoretical $UBVRI$ Colors and Metallicity for a 12.5 Gyr GC

[Fe/H] | $U-B$ | $U-V$ | $U-R$ | $U-I$ | $B-V$ | $B-R$ | $B-I$ | $V-R$ | $V-I$ | $R-I$
---|---|---|---|---|---|---|---|---|---|---
–2.5 | –0.052 | 0.552 | 0.991 | 1.438 | 0.603 | 1.043 | 1.489 | 0.440 | 0.886 | 0.446
–2.4 | –0.030 | 0.582 | 1.023 | 1.464 | 0.612 | 1.053 | 1.494 | 0.440 | 0.881 | 0.441
–2.3 | –0.015 | 0.601 | 1.042 | 1.484 | 0.617 | 1.058 | 1.500 | 0.441 | 0.883 | 0.442
–2.2 | –0.004 | 0.617 | 1.059 | 1.504 | 0.621 | 1.062 | 1.508 | 0.442 | 0.887 | 0.445
–2.1 | 0.006 | 0.630 | 1.073 | 1.523 | 0.624 | 1.067 | 1.516 | 0.443 | 0.892 | 0.449
–2.0 | 0.017 | 0.646 | 1.091 | 1.543 | 0.629 | 1.074 | 1.526 | 0.445 | 0.897 | 0.452
–1.9 | 0.029 | 0.665 | 1.112 | 1.566 | 0.636 | 1.083 | 1.537 | 0.448 | 0.901 | 0.454
–1.8 | 0.048 | 0.690 | 1.141 | 1.596 | 0.643 | 1.093 | 1.548 | 0.450 | 0.905 | 0.455
–1.7 | 0.067 | 0.719 | 1.173 | 1.630 | 0.652 | 1.107 | 1.563 | 0.454 | 0.911 | 0.457
–1.6 | 0.086 | 0.750 | 1.209 | 1.669 | 0.664 | 1.124 | 1.584 | 0.460 | 0.920 | 0.460
–1.5 | 0.105 | 0.782 | 1.247 | 1.713 | 0.677 | 1.142 | 1.608 | 0.466 | 0.931 | 0.465
–1.4 | 0.138 | 0.821 | 1.290 | 1.760 | 0.683 | 1.152 | 1.622 | 0.469 | 0.939 | 0.470
–1.3 | 0.171 | 0.867 | 1.342 | 1.821 | 0.696 | 1.171 | 1.650 | 0.475 | 0.954 | 0.478
–1.2 | 0.198 | 0.915 | 1.400 | 1.890 | 0.717 | 1.201 | 1.692 | 0.484 | 0.975 | 0.491
–1.1 | 0.220 | 0.966 | 1.464 | 1.971 | 0.747 | 1.244 | 1.751 | 0.497 | 1.004 | 0.507
–1.0 | 0.244 | 1.025 | 1.537 | 2.064 | 0.780 | 1.293 | 1.820 | 0.513 | 1.040 | 0.527
–0.9 | 0.282 | 1.099 | 1.629 | 2.178 | 0.817 | 1.347 | 1.896 | 0.529 | 1.079 | 0.549
–0.8 | 0.321 | 1.169 | 1.712 | 2.282 | 0.848 | 1.391 | 1.960 | 0.543 | 1.113 | 0.570
–0.7 | 0.366 | 1.241 | 1.797 | 2.387 | 0.875 | 1.431 | 2.021 | 0.556 | 1.146 | 0.590
–0.6 | 0.420 | 1.323 | 1.893 | 2.502 | 0.903 | 1.472 | 2.082 | 0.570 | 1.179 | 0.610
–0.5 | 0.475 | 1.402 | 1.985 | 2.612 | 0.928 | 1.510 | 2.137 | 0.582 | 1.210 | 0.627
–0.4 | 0.529 | 1.477 | 2.070 | 2.711 | 0.949 | 1.541 | 2.182 | 0.593 | 1.233 | 0.641
–0.3 | 0.586 | 1.556 | 2.158 | 2.809 | 0.969 | 1.571 | 2.223 | 0.602 | 1.253 | 0.651
–0.2 | 0.649 | 1.639 | 2.251 | 2.909 | 0.990 | 1.601 | 2.259 | 0.611 | 1.270 | 0.658
–0.1 | 0.717 | 1.727 | 2.347 | 3.009 | 1.010 | 1.630 | 2.292 | 0.620 | 1.282 | 0.662
0.0 | 0.784 | 1.814 | 2.442 | 3.106 | 1.030 | 1.658 | 2.322 | 0.628 | 1.292 | 0.663
0.1 | 0.847 | 1.897 | 2.534 | 3.198 | 1.049 | 1.687 | 2.351 | 0.638 | 1.302 | 0.664
0.2 | 0.903 | 1.971 | 2.619 | 3.282 | 1.068 | 1.716 | 2.379 | 0.648 | 1.311 | 0.663
0.3 | 0.952 | 2.040 | 2.699 | 3.362 | 1.088 | 1.747 | 2.410 | 0.658 | 1.322 | 0.663
0.4 | 1.000 | 2.108 | 2.777 | 3.444 | 1.108 | 1.776 | 2.444 | 0.669 | 1.336 | 0.668
0.5 | 1.072 | 2.201 | 2.877 | 3.553 | 1.128 | 1.804 | 2.481 | 0.676 | 1.352 | 0.676

Note. — The spectrophotometric model data of the entire parameter space are
available at http://cosmic.yonsei.ac.kr/YEPS.htm.

Figure 2:  Two-color diagrams of crossmatched GC sample in NGC 5128. The YEPS
model predictions for GCs with ages from 1 Gyr (blue) to 13 Gyr (red) are
overlaid in 2 Gyr intervals. The model loci are slightly shifted to the
direction of observations (see section 3). We select old GCs using the
polygons marked by thick black lines. Note that young GCs with intermediate
and high metallicities are indistinguishable from old GCs given the
observational uncertainty and are partially removed, contaminating the sample
to some extent.  Figure 3:  Same as Figure 2, but for crossmatched GC sample
in NGC 4594.  Figure 4:  Determination of offsets between observations and
model predictions for B-I GC CMR (left) and V-I GC CMR (right). The offset
value denoted on the top-left corner of each panel is estimated by the
weighted total least squares method on the normalized plane of colors and
metallicities. The orange lines indicate orthogonal distances from the data
points to the model CMR. The black circles are the final sample used in the
total least squares method and the open circles are outliers removed by
5-sigma clipping. The dotted line represents the original model prediction of
the GC CMR and the solid line shows the shifted model locus after applying the
offset value, as shown by cyan horizontal arrows in the right panel.

Figure 5:  CMRs of NGC 5128 GCs. The size and opacity of each symbol are
inversely proportional to the quadratic sum of observational uncertainties
calculated after normalization. The red dashed and solid lines respectively
represent the original 12.5 Gyr YEPS model predictions and the models shifted
by the offset value denoted on the top-left corner of each panel. The overall
shape of the CMRs is in good agreement with the observations and models for
all colors, in that the slope of GC CMRs is steep in the blue region,
decreases with redder, and has a quasi-inflection point along $BVRI$ CMRs
[panels (d)–(i)].

Figure 6:  Same as Figure 5, but for NGC 4594 GCs. The scatters are rather
larger compared to the NGC 5128 GC CMRs, but similar features are shown: the
CMR slope is steeper in the blue and the slope-changing point appears to be
consistent with the model prediction.

Figure 7:  Elucidation of outliers in the CMR plane of NGC 4594 GCs. To
inspect the nature of the outliers, we present three plots showing GC
metallicities as a function of galactocentric distance (left panel), the Mg$b$
versus [Fe/H] correlation along with 12.5 Gyr model predictions of the YEPS
model (solid lines), and MILES model (dashed lines) for the different
$\alpha$-element mixture (middle panel), and the $B-V$ CMR along with the 5
Gyr and 12.5 Gyr YEPS model predictions for [$\alpha$/Fe] = 0.0 (blue lines)
and 0.3 (red lines) (right panel). In the left panel, the red circles indicate
outer-halo, metal-rich GCs that do not follow the metallicity gradient.

We suspect that they are young GCs with a different formation and evolutionary
history from most GCs in the galaxy, as they are less $\alpha$-element
enhanced than other GCs with similar metallicities in the middle panel and are
in the younger-age region of CMR in the right panel.

Figure 8:  MD and CDs for GCs in NGC 5128. Each histogram is smoothed using a
nonparametric kernel density estimate with the bandwidth corresponding to the
mean observational error. The smoothed histograms are denoted as red lines for
MDs and green lines for CDs. Upper panels: CDs for the GC candidates selected
only from the photometric data set. Old GC candidates are selected from the
photometric data of Peng et al. (2006) and Woodley et al. (2007) using the
same criteria as the two-color diagrams, magnitude range, and radial extent
used for the crossmatched GC sample. Lower panels: MD and CDs for the GCs
crossmatched between photometry and spectroscopy.

Figure 9:  MD and CDs for GCs in NGC 4594. Each histogram is smoothed using a
nonparametric kernel density estimate with the bandwidth corresponding to the
mean observational error. The kernel bandwidth of MD is set to be twice the
mean metallicity error since Alves-Brito et al. (2011) mentioned that the
metallicity error they provide may be underestimated. The smoothed histograms
are denoted as red lines for MDs and green lines for CDs. Upper panels: CDs
for the GC candidates selected only from the photometric data set. Old GC
candidates are selected from our photometric data using the same criteria as
the two-color diagrams, magnitude range, and radial extent used for the
crossmatched GC sample. Lower panels: MD and CDs for the GCs crossmatched
between photometry and spectroscopy.

Table 4: Bimodality Coefficient for Metallicity and Color Distributions of GCs in NGC 5128 and NGC 4594  GC system | [Fe/H] | $U-B$ | $B-V$ | $B-I$ | $V-I$
---|---|---|---|---|---
NGC 5128 | 0.487 | 0.513 | 0.536 | 0.568 | 0.559
NGC 4594 | 0.431 | 0.565 | 0.497 | 0.543 | 0.552
Table 5: Results from Dip Test and GMM Test (in the Heteroscedastic Case) for Metallicity and Color Distributions of GCs in NGC 5128 and NGC 4594  GC system | Statistic | [Fe/H] | $U-B$ | $B-V$ | $B-I$ | $V-I$
---|---|---|---|---|---|---
NGC 5128 | $p(diptest)$ | 0.215 | 0.144 | 0.006 | 0.124 | 0.129
$p(GMM_{\rm MG}:\chi^{2})$ | 0.001 | 0.001 | 0.001 | 0.001 | 0.001
$p(GMM_{\rm MG}:DD)$ | 0.389 | 0.352 | 0.128 | 0.104 | 0.153
$p(GMM_{\rm MG}:kurt)$ | 0.001 | 0.034 | 0.001 | 0.001 | 0.001
$p(GMM_{R})$ | 0.004 | 0.001 | 0.001 | 0.001 | 0.001
NGC 4594 | $p(diptest)$ | 0.912 | 0.850 | 0.792 | 0.492 | 0.364
$p(GMM_{\rm MG}:\chi^{2})$ | 0.157 | 0.001 | 0.001 | 0.001 | 0.001
$p(GMM_{\rm MG}:DD)$ | 0.421 | 0.476 | 0.249 | 0.147 | 0.136
$p(GMM_{\rm MG}:kurt)$ | 0.050 | 0.200 | 0.001 | 0.001 | 0.001
$p(GMM_{R})$ | 0.132 | 0.001 | 0.002 | 0.001 | 0.001

Figure 10: The BIC (solid red circle) and ICL (open blue circle) values as
functions of the number of Gaussian components $K$. The BIC and ICL values are
used as criteria for the selection of the best mixture model to fit the data
in the GMMR test. The models with the highest BIC/ICL values are preferred.
Note that BIC tends to select the number of Gaussian components, whereas ICL
is better suited for selecting discrete groups from data.

Figure 11: Monte Carlo simulations for the CDs of NGC 5128 GCs. Top row: same
as the CMRs in Figure 5. The error bars indicate the typical errors in [Fe/H]
and colors. The observed MD is rotated and given on the $y$-axes, and the
black solid line shows a kernel density estimate with a bandwidth set to the
mean observational error. Second row: the observed color-magnitude diagrams
for the GCs. Black and gray dots represent the crossmatched and
photometrically selected GCs, respectively. Observational uncertainties are
shown by error bars as a function of $M_{V}$. Third row: observed CDs of the
crossmatched GCs (see Figure 8). Fourth row: observed CDs of the
photometrically selected GCs. Bottom row: simulated CDs of model GCs. We
generate $10^{5}$ model GCs with metallicities derived from the kernel density
estimate of the observed MDs and convert the metallicities into colors via the
model CMRs presented in the top panel taking into account the observational
errors. Given that the number of observed GCs is much smaller, we estimate the
CD of 151 model GCs (solid line) and its 1$\sigma$ distribution range (gray
shaded bands) from the 10,000 bootstrap sampling.

Figure 12:  Monte Carlo simulations for the CDs of NGC 4594 GCs. Top row: same
as the CMRs in Figure 6. The error bars indicate the typical errors in [Fe/H]
and colors. We note that Alves-Brito et al. (2011) remark that their
metallicity errors might be underestimated. The observed MD is rotated and
given on the $y$-axes, and the black solid line shows a kernel density
estimate with a bandwidth set to twice the mean observational error. Second
row: the observed color-magnitude diagrams for the GCs. Black and gray dots
represent the crossmatched and photometrically selected GCs, respectively.
Observational uncertainties are shown by error bars as a function of $M_{V}$.
Third row: observed CDs of the crossmatched GCs (see Figure 9). Fourth row:
observed CDs of the photometrically selected GCs. Bottom row: simulated CDs of
model GCs. We generate $10^{5}$ model GCs with metallicities derived from the
kernel density estimate of the observed MDs and convert the metallicities into
colors via the model CMRs presented in the top panel taking into account the
observational errors. Given that the number of observed GCs is much smaller,
we estimate the CD of 108 model GCs (solid line) and its 1-$\sigma$
distribution range (gray shaded bands) from the 10,000 bootstrap sampling.


# Obstructions to applying the Baker–Bilu method for determining integral
points on curves

Aaron Landesman  Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, MA 02139-4307, USA  and  Bjorn Poonen  Department of
Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307,
USA<EMAIL_ADDRESS>http://math.mit.edu/~poonen/

(Date: June 14, 2023)

###### Abstract.

We prove that for every smooth projective integral curve $X$ of genus at least
$2$ over $\mathbb{C}$, there exists $x\in X(\mathbb{C})$ such that no
connected finite étale cover of $X-\\{x\\}$ admits a nonconstant morphism to
$\mathbb{G}_{m}$. This has implications for the applicability of Baker’s
method to determining integral points on curves.

###### Key words and phrases:

Baker’s method, integral point, finite étale cover, height

###### 2020 Mathematics Subject Classification:

Primary 14H30; Secondary 11G30, 11G50, 11J86, 14H40

B.P. was supported in part by National Science Foundation grant DMS-1601946
and Simons Foundation grants #402472 and #550033.

##  1\. Introduction

It is not known if there is an algorithm to find all the integer solutions to
an arbitrary polynomial equation in two variables. More generally, one can ask
about solutions in a ring of $S$-integers of a number field. Equivalently, one
can ask about effectively bounding the height of $S$-integral points on an
affine curve $U$. One can reduce to the case where $U$ is smooth and
geometrically integral; then $U=X-R$ for some nice curve $X$ and some finite
set $R$ of closed points of $X$. (Here, “nice” means smooth, projective, and
geometrically integral.) Let $g$ be the genus of $X$. We assume that the Euler
characteristic $\chi(U)=2-2g-\\#R$ is negative; in this case, the set of
$S$-integral points is finite by Siegel’s theorem [Sie29], but the question is
whether the result can be made effective.

Baker’s method together with Dirichlet’s $S$-unit theorem handles all cases
with $g\leq 1$ [Bak66, Bak68a, Bak68b, Bak68c, BC70]. It also handles some
$(X,R)$ with $g\geq 2$, such as those in which $X$ is a cyclic cover of
$\mathbb{P}^{1}$ and $R$ is an orbit of $\operatorname{Aut}(X/\mathbb{P}^{1})$
[Bak69]. Bilu [Bil95, Theorem E] generalized Baker’s approach to handle all
cases in which $U$ after base field extension has a connected finite étale
cover with a nondegenerate morphism to $\mathbb{G}_{m}\times\mathbb{G}_{m}$;
here nondegenerate means that the image is not contained in a coset of a
proper algebraic subgroup. Bilu’s theorem begs the question:

######  Question 1.0.

Does _every_ smooth integral affine curve of genus at least $2$ over
${\overline{\mathbb{Q}}}$ admit a connected finite étale cover with a
nondegenerate morphism to $\mathbb{G}_{m}\times\mathbb{G}_{m}$?

We conjecture that the answer is _no_. In fact, we conjecture the following
stronger statement:

######  Conjecture 1.0.

For _any_ nice curve $X$ of genus at least $2$, there exists $x\in
X({\overline{\mathbb{Q}}})$ such that the affine curve $X-\\{x\\}$ has no
connected finite étale cover with even a _single_ nonconstant morphism to
$\mathbb{G}_{m}$.

As evidence, we prove this statement with $\mathbb{C}$ in place of
${\overline{\mathbb{Q}}}$:

######  Theorem 1.1.

Let $X$ be a nice curve of genus at least $2$ over $\mathbb{C}$. Then there
exists $x\in X(\mathbb{C})$ such that $X-\\{x\\}$ has no connected finite
étale cover with a nonconstant morphism to $\mathbb{G}_{m}$.

Thus if, for some $g\geq 2$, Section 1 has a positive answer for all curves of
genus $g$ over ${\overline{\mathbb{Q}}}$, it cannot be because of a universal
algebraic construction of a cover of degree depending only on $g$, as was the
case for $g=1$ — instead the degree would have to depend on the height of the
curve as well, as happens in Belyi’s theorem [Bel79].

######  Remark 1.1.

For a smooth integral affine curve $U=X-R$ as above, having a nonconstant
morphism $U\to\mathbb{G}_{m}$ is equivalent to having a nontrivial integer
relation between the classes of the points of $R$ in $\operatorname{Pic}(X)$.

For any nice curve $X$ of genus at least $2$ over an algebraically closed
field $K$, let $X(K)_{\operatorname{special}}$ be the set of $x\in X(K)$ such
that $X-\\{x\\}$ has a connected finite étale cover with a nonconstant
morphism to $\mathbb{G}_{m}$. Theorem 1.1 is an immediate consequence of the
following, which we prove in Section 2.

######  Theorem 1.2.

Let $K$ be an algebraically closed field of characteristic $0$. Let $X$ be a
nice curve of genus at least $2$ over $K$. Then
$X(K)_{\operatorname{special}}$ is countable.

One approach to proving a negative answer to Section 1 (over
${\overline{\mathbb{Q}}}$) might be to prove that for a nice curve $X$ of
genus at least $2$ over ${\overline{\mathbb{Q}}}$, the set
$X({\overline{\mathbb{Q}}})_{\operatorname{special}}$ is of bounded height.
Our Theorem 3.2 implies the slightly weaker result that the set of $x\in
X({\overline{\mathbb{Q}}})$ such that $X-\\{x\\}$ admits such a cover of
degree $\leq d$ is a set of bounded height. Another approach to Section 1
might be to prove that $X({\overline{\mathbb{Q}}})_{\operatorname{special}}$
is not $p$-adically dense in $X({\overline{\mathbb{Q}}})$. Perhaps
$X({\overline{\mathbb{Q}}})_{\operatorname{special}}$ is even _finite_ for
most $X$.

######  Example 1.2 (Ihara, Tamagawa).

If $X$ is of genus $2$, then
$X({\overline{\mathbb{Q}}})_{\operatorname{special}}$ is infinite, because it
contains the set called $M$ just before Proposition 6.1 in [Bil95]. (Bilu
received this example in a letter from Yasutaka Ihara, who wrote that it
emerged in a discussion with Akio Tamagawa.)

######  Remark 1.2.

Theorem 1.1 is related to a number of other conjectures in algebraic geometry
and geometric topology.

Prill’s problem asked whether there is a nice curve $X$ of genus at least $2$
and a finite cover $f\colon Y\to X$ such that
$H^{0}(Y,\mathscr{O}_{Y}(f^{-1}(x)))\geq 2$ for every $x\in X$ [ACGH85, p.
268, Chapter VI, Exercise D]. Call such a cover Prill exceptional. Any Prill
exceptional cover has the property that for each $x\in X$, the curve
$Y-f^{-1}(x)$ has a nonconstant map to $\mathbb{A}^{1}$. It is known that for
every genus $2$ curve $X$ over $\mathbb{C}$, there is a finite étale Prill
exceptional cover of $X$ [LL22c], but it remains wide open whether there
exists any $X$ of genus at least $3$ with a Prill exceptional cover.

Moreover, by [LL22a, Lemma 5.5], any Prill exceptional cover $f:Y\to X$ of a
general curve $X$ of genus $g$ gives a counterexample to another conjecture in
geometric topology, the Putman–Wieland conjecture, as stated in [PW13,
Conjecture 1.2]. In fact, [LL22b, Lemma 6.10] implies that the Putman–Wieland
conjecture is equivalent to a statement about maps from covers of curves to
abelian varieties (in place of $\mathbb{G}_{m}$ as in Theorem 1.1): the
statement is that for each $g\geq 2$, $n\geq 0$, and abelian variety $A$ over
$\mathbb{C}$, a general $n$-pointed genus $g$ curve $X$ has no finite cover
$Y\to X$ branched only over the $n$ points with a nonconstant map $Y\to A$.
This statement is false for $g=2$ [Mar22, Theorem 1.3] but open for $g\geq 3$.
The Putman–Wieland conjecture, in turn, is closely related to another
longstanding conjecture in geometric topology, Ivanov’s conjecture [PW13,
Theorem 1.3].

##  2\. Proof of Theorem 1.2

######  Idea of proof 2.0.

Fix $K$ and $X$ as in Theorem 1.2. First, we reduce to proving countability of
the set $X(K)_{\operatorname{special},G}$ of $x\in X(K)$ such that $X-\\{x\\}$
has a $G$-_Galois_ finite étale cover with a nonconstant morphism to
$\mathbb{G}_{m}$, for each finite group $G$. Next, we construct a moduli space
$M$ parametrizing $G$-covers $f\colon Y\to X$ branched at a varying point
$x\in X$, together with a universal curve $S\to M$. We show that if there is a
relation between the points in $f^{-1}(x)$ for a very general $x$, then there
is a relation between certain divisors in $S$. Finally, we rule out such a
relation by showing that the intersection matrix of these divisors is
invertible.

######  Remark 2.0.

Since every finite étale cover is dominated by a Galois finite étale cover,
$\bigcup_{G}X(K)_{\operatorname{special},G}=X(K)_{\operatorname{special}}$.
From now on, we fix $G$. It remains to prove that
$X(K)_{\operatorname{special},G}$ is countable.

We next construct a moduli space of $G$-covers of $X$ branched over at most
one point.

######  Lemma 2.0.

There is a finite-type $K$-scheme $\mathscr{M}$ parametrizing
$(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})$, where $x$ is a point of $X$, $Y$ is
a nice curve, $f\colon Y\to X$ is a Galois cover with Galois group $G$ that is
étale above $X-\\{x\\}$ (at least), and $\mathcal{B}$ is a basis for $J[3]$,
where $J$ is the Jacobian of $Y$, and $y_{1},\ldots,y_{n}$ are the distinct
points of $f^{-1}(x)$ (here $n$ is constant on each irreducible component of
$\mathscr{M}$). Each irreducible component of $\mathscr{M}$ is a nice curve.

###### Proof.

This essentially follows from [Wew98, Theorem 4]. Let the $D\subset X\to S$ of
Wewers be the diagonal $\Delta\subset X\times X\to X$. By [Wew98, Theorem 4],
there is a finite-type algebraic stack $\mathcal{H}=\mathcal{H}_{X\times
X}^{G}(G)$ over $X$ such that for any $S^{\prime}\to X$, the groupoid
$\mathcal{H}(S^{\prime})$ parameterizes $G$-Galois finite locally free covers
$Y^{\prime}\to S^{\prime}\times_{X}(X\times X)$ that are tamely ramified above
$S^{\prime}\times_{X}\Delta$ and étale elsewhere. By [Wew98, Theorem 4],
$\mathcal{H}\to X$ is étale. It follows from [Wew98, Theorem 3.2.4] that
$\mathcal{H}\to X$ is proper.

By adding the data of $\mathcal{B}$ and the sections $y_{1},\ldots,y_{n}$ to
our moduli stack, we obtain a finite étale cover $\mathscr{M}$ of
$\mathcal{H}$. Each groupoid $\mathscr{M}(S^{\prime})$ is a setoid because
$\operatorname{Aut}Y\to\operatorname{Aut}J[3]$ is injective [KS99, 10.5.6],
and hence $\mathscr{M}$ is represented by an algebraic space. Since
$\mathscr{M}\to\mathcal{H}$ is finite étale and $\mathcal{H}\to X$ is proper
étale, $\mathscr{M}$ is proper étale over $X$. By the previous two sentences,
$\mathscr{M}$ is finite étale over $X$ and is therefore a scheme. Since
$\mathscr{M}$ is finite étale over the nice curve $X$, each irreducible
component of $\mathscr{M}$ is a nice curve. ∎

######  Notation 2.0.

Let $M$ be an irreducible component of $\mathscr{M}$. Then $M$ is a nice
curve. Let $\eta$ be the generic point of $M$. Let $\pi\colon S\to M$ be the
universal morphism whose fiber above
$m=(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})\in M$ is $Y$. Thus $S$ is a nice
surface, and $\pi\colon S\to M$ is a relative curve. Let $h\colon S\to X$ be
the morphism whose restriction to each fiber $Y$ of $\pi$ is the map $f\colon
Y\to X$. Let $\mu\colon M\to X$ be
$(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})\mapsto x$. Let $e$ be the positive
integer such that for any $m\in M$, the ramification index of the
corresponding map $f\colon Y\to X$ at any point above $x$ is $e$. Let
$s_{1},\ldots,s_{n}\colon M\to S$ be the sections such that $s_{i}(m)=y_{i}$
for each $i$. Let $D_{i}=s_{i}(M)\in\operatorname{Div}(S)$. Let
$F\in\operatorname{Div}(S)$ be a closed fiber of $S\to M$. Let
$\operatorname{NS}(S)$ be the Néron–Severi group of $S$.

######  Proposition 2.0.

The classes of $D_{1},\ldots,D_{n},F$ in $\operatorname{NS}(S)$ are
$\mathbb{Z}$-independent.

###### Proof.

It suffices to prove that the intersection matrix is nonsingular. We know
$D_{i}\cdot D_{j}=0$ for $i\neq j$, $D_{i}\cdot F=1$, $F\cdot F=0$, and
$D_{i}^{2}=d$ for some $d$ independent of $i$, because $G$ acts transitively
on $D_{1},\ldots,D_{n}$. Then the intersection matrix is

$\displaystyle\begin{pmatrix}d&0&0&\cdots&0&1\\\ 0&d&0&\cdots&0&1\\\
0&0&d&\cdots&0&1\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&d&1\\\ 1&1&1&\cdots&1&0\end{pmatrix},$

which has determinant $-nd^{n-1}$. In Section 2, we verify that $d\neq 0$. ∎

######  Lemma 2.0.

We have $D_{i}^{2}=(\deg\mu)(2-2g)/e<0$.

###### Proof.

Observe that $\mu=h\circ s_{i}$ because
$\mu(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})=x=h(y_{i})$ and
$y_{i}=s_{i}(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})$. Let $\delta\colon X\to
X\times X$ be the diagonal. The diagram

${S}$${M\times X}$${X\times X}$
${M}$${X}$$\scriptstyle{(\pi,h)}$$\scriptstyle{\mu\times\operatorname{id}}$$\scriptstyle{s_{i}}$$\scriptstyle{(\operatorname{id},\mu)}$$\scriptstyle{\mu}$$\scriptstyle{\delta}$
(2.1)

commutes, since $\pi\circ s_{i}=\operatorname{id}$ and $h\circ s_{i}=\mu$.

Let $\Delta=\delta(X)\subset X\times X$ be the diagonal. We will pull back the
class of $\Delta$ to a line bundle on $M$ along the two outer paths in (2.1).
First, we determine the pullback of $\Delta$ to $S$ by calculating its
restriction to each fiber of $\pi\colon S\to M$. If
$m=(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})\in M$, so $Y$ is the fiber of $S\to
M$ over $m$, then the composition

$Y\hookrightarrow S\xrightarrow{(\pi,h)}M\times
X\xrightarrow{\mu\times\operatorname{id}}X\times X$

is $Y\xrightarrow{(x,f)}X\times X$ (constant first coordinate), so
$(\pi,h)^{*}(\mu\times\operatorname{id})^{*}\Delta|_{Y}=f^{-1}(x)=e\sum_{j=1}^{n}y_{j}$
in $\operatorname{Div}Y$. Thus
$(\pi,h)^{*}(\mu\times\operatorname{id})^{*}\Delta=e\sum_{j=1}^{n}D_{j}$ in
$\operatorname{Div}S$, and pulling this back to $M$ along $s_{i}$ yields
$D_{i}\cdot(e\sum_{j=1}^{n}D_{j})\in\operatorname{Div}M$, which has degree
$eD_{i}^{2}$. On the other hand, the class of $\Delta\subset X\times X$ pulls
back to a divisor of degree $\Delta\cdot\Delta=2-2g$ on $X$, which pulls back
to a divisor of degree $(\deg\mu)(2-2g)$ on $M$. Thus

$eD_{i}^{2}=(\deg\mu)(2-2g),\qed$

which is negative, since $g\geq 2$.

######  Corollary 2.0.

The images of $D_{1},\ldots,D_{n}$ in $\operatorname{Pic}(S_{\eta})$ are
$\mathbb{Z}$-independent.

###### Proof.

If there were a relation, the exact sequence

${\displaystyle\bigoplus_{t\in
M(K)}\mathbb{Z}\cdot\pi^{-1}(t)}$${\operatorname{Pic}(S)}$${\operatorname{Pic}(S_{\eta})}$${0}$
(2.2)

would give a relation between $D_{1},\ldots,D_{n},F$ in
$\operatorname{NS}(S)$, since the class of each fiber $\pi^{-1}(t)$ in
$\operatorname{NS}(S)$ is $F$. ∎

###### End of proof of Theorem 1.2.

As mentioned in Section 2, it suffices to show
$X(K)_{\operatorname{special},G}$ is countable. It is the image under
$\mathscr{M}\to X$ of

$\mathscr{M}(K)^{\prime}\colonequals\\{(x,Y,\ldots)\in\mathscr{M}(K):\textup{$Y$
has a nonconstant morphism to $\mathbb{G}_{m}$}\\},$ (2.3)

which is the finite union of sets
$M(K)^{\prime}\colonequals\mathscr{M}(K)^{\prime}\cap M$. Each $M(K)^{\prime}$
is the union over nonzero $a=(a_{1},\ldots,a_{n})\in\mathbb{Z}^{n}$ of

$\displaystyle V_{a}:=\\{(x,Y,f,\mathcal{B},y_{1},\ldots,y_{n})\in
M(K):\sum_{i=1}^{n}a_{i}y_{i}=0\textup{ in $\operatorname{Pic}(Y)$}\\},$

so it suffices to prove that $V_{a}$ is finite. The set $V_{a}$ is the zero
locus of a section of the relative Picard scheme
$\operatorname{\textbf{Pic}}_{S/M}\to M$, so $V_{a}\subset M(K)$ is closed. If
$V_{a}=M(K)$, then $\sum_{i=1}^{n}a_{i}D_{i}=0$ in
$\operatorname{Pic}(S_{\eta})$, contradicting Section 2. Hence $V_{a}$ is
finite. ∎

##  3\. Bounded height

In this section, we set $K={\overline{\mathbb{Q}}}$ and prove in Theorem 3.2
that each set $X({\overline{\mathbb{Q}}})_{\operatorname{special},G}$ is of
_bounded height_.

First, we introduce notation for Theorem 3.1, which records known results
about heights and specialization. Let $\pi\colon S\to M$ be a morphism from a
nice surface to a nice curve over ${\overline{\mathbb{Q}}}$. For each field
extension $L\supset{\overline{\mathbb{Q}}}$ and $t\in M(L)$, let
$S_{t}=\pi^{-1}(t)$; if $S_{t}$ is a _nice_ curve, let $J_{t}$ be its
Jacobian, a principally polarized abelian variety over $L$. By a fibral
component we mean an irreducible component of $S_{t}$ for some $t\in
M({\overline{\mathbb{Q}}})$. Let $\eta$ be the generic point of $M$. We assume
that $S_{\eta}$ is a _nice_ curve (over the function field $k(M)$).

Let $k$ be a field with a product formula as in [Ser97, §2.1]. Given a
polarized abelian variety $A$ over $k$, one can enlarge $k$ so that the
polarization arises from a symmetric divisor on $A$ and then define a
canonical height pairing on $A({\overline{k}})$, or its subgroup $A(k)$, as in
[Sil83, pp. 200–201]. Applying this to $J_{\eta}$ over $k(M)$ yields a
(geometric) canonical height pairing $\langle\;,\;\rangle$ on
$J_{\eta}(k(M))=\operatorname{Pic}^{0}(S_{\eta})$ or on
$\operatorname{Div}^{0}(S_{\eta})$. For any $t\in M({\overline{\mathbb{Q}}})$
such that $S_{t}$ is a nice curve, applying this to $J_{t}$ (over a number
field to which it descends) yields a canonical height pairing
$\langle\;,\;\rangle_{t}$ on
$J_{t}({\overline{\mathbb{Q}}})=\operatorname{Pic}^{0}(S_{t})$ or on
$\operatorname{Div}^{0}(S_{t})$.

For each closed point $P\in S_{\eta}$, let $\overline{P}$ be its Zariski
closure in $S$. Extend $\mathbb{Z}$-linearly to define
$\overline{P}\in\operatorname{Div}S$ for any
$P\in\operatorname{Div}(S_{\eta})$; then for any $t\in
M({\overline{\mathbb{Q}}})$, define the specialization
$P_{t}\colonequals\overline{P}|_{S_{t}}\in\operatorname{Div}(S_{t})$. Let
$h\colon M({\overline{\mathbb{Q}}})\to\mathbb{R}$ be a Weil height associated
to a divisor of nonzero degree on $M$, as in [Sil83, p. 205].

######  Theorem 3.1.

Let $S$, $M$, $\pi$, $S_{\eta}$, $S_{t}$, $\langle\;,\;\rangle$,
$\langle\;,\;\rangle_{t}$, and $h$ be as above.

  1. n(a)

Let $P\in\operatorname{Div}^{0}(S_{\eta})$. Then there exists a
$\mathbb{Q}$-linear combination $\Phi$ of fibral components such that
$D_{P}\colonequals\overline{P}+\Phi\in\operatorname{Div}(S)\otimes\mathbb{Q}$
is orthogonal to each fibral component. Moreover, $\Phi$ is unique modulo
$\pi^{*}\operatorname{Div}(M)\otimes\mathbb{Q}$.

  2. n(b)

For $P,P^{\prime}\in\operatorname{Div}^{0}(S_{\eta})$, we have $\langle
P,P^{\prime}\rangle=-D_{P}\cdot D_{P^{\prime}}$.

  3. n(c)

Fix $P,Q\in\operatorname{Div}^{0}(S_{\eta})$. Then $\langle
P_{t},Q_{t}\rangle_{t}/h(t)\to\langle P,Q\rangle$ as $h(t)\to\infty$.

  4. n(d)

Let $P_{1},\ldots,P_{n}\in\operatorname{Div}^{0}(S_{\eta})$. If the matrix
$(\langle P_{i},P_{j}\rangle)$ is positive definite, then the set of $t\in
M({\overline{\mathbb{Q}}})$ such that $P_{1,t},\ldots,P_{n,t}$ are
$\mathbb{Z}$-dependent in $\operatorname{Pic}^{0}(S_{t})$ is of bounded
height.

###### Proof.

  1. n(a)

Let $t\in M({\overline{\mathbb{Q}}})$. Let $F_{1},\ldots,F_{n}$ be the
irreducible components of $S_{t}$. Let $F$ be the class of $S_{t}$ in
$\operatorname{Div}(S)$. The existence and uniqueness of the part of $\Phi$
supported above $t$ follows from the nondegeneracy of the intersection pairing
on $(\bigoplus\mathbb{Q}F_{i})/\mathbb{Q}F$, as in [Gro86, end of §3] (the
finiteness of the residue field assumed there is not needed for this).

  2. n(b)

This too follows from the same argument as in [Gro86, end of §3]. (The
relation between canonical heights and intersection numbers was discovered
earlier in the context of an abelian variety and its dual, by Manin [Man64]
and Néron [Nér65].)

  3. n(c)

This is a version of Silverman’s specialization theorem, which extended ideas
of Dem′janenko [Dem68] and Manin [Man69]. Specifically, the case $P=Q$ is
contained in [Sil83, Theorem B]. The general case follows from this case
applied to $P$, $Q$, and $P+Q$.

  4. n(d)

If $h(t)$ is sufficiently large, then (c) implies that $(\langle
P_{i,t},P_{j,t}\rangle)$ is positive definite too, so $P_{1,t},\ldots,P_{n,t}$
are $\mathbb{Z}$-dependent in $\operatorname{Pic}^{0}(S_{t})$.∎

######  Theorem 3.2.

Let $X$ be a nice curve of genus at least $2$ over ${\overline{\mathbb{Q}}}$.
Then for each finite group $G$, the set
$X({\overline{\mathbb{Q}}})_{\operatorname{special},G}$ is of bounded height.

###### Proof.

Let $S\to M$ and $D_{1},\ldots,D_{n}$ be as in Section 2. For $i=2,\ldots,n$,
let $P_{i}=(D_{i}-D_{1})|_{S_{\eta}}$. Since $S\to M$ has irreducible fibers,
we can always take $\Phi=0$ in Theorem 3.1(a). By Section 2, the matrix
$(D_{i}\cdot D_{j})_{1\leq i,j\leq n}$ is $dI$ for some $d<0$, so
$((D_{i}-D_{1})\cdot(D_{j}-D_{1}))_{2\leq i,j\leq n}$ is negative definite
too. By Theorem 3.1(b), $(\langle P_{i},P_{j}\rangle)$ is _positive_ definite.
For $t\in M({\overline{\mathbb{Q}}})$, the following are equivalent:

  * •

$t$ belongs to the subset $M({\overline{\mathbb{Q}}})^{\prime}$ defined in
(2.3);

  * •

$D_{1}|_{S_{t}},\ldots,D_{n}|_{S_{t}}$ are $\mathbb{Z}$-dependent in
$\operatorname{Pic}(S_{t})$;

  * •

$P_{2},\ldots,P_{n}$ are $\mathbb{Z}$-dependent in
$\operatorname{Pic}^{0}(S_{t})$.

By Theorem 3.1(d), the set of such $t$ is of bounded height. Thus
$M({\overline{\mathbb{Q}}})^{\prime}$ is of bounded height, and so is its
image in $X({\overline{\mathbb{Q}}})$. Taking the union over the finitely many
irreducible components $M$ of $\mathscr{M}$ shows that
$X({\overline{\mathbb{Q}}})_{\operatorname{special},G}$ is of bounded height.
∎

######  Remark 3.2.

To prove the weaker statement that
$X({\overline{\mathbb{Q}}})-X({\overline{\mathbb{Q}}})_{\operatorname{special},G}$
is infinite, one could use Néron’s specialization theorem [Nér52, Théorème 6]
in place of [Sil83, Theorem B].

######  Question 3.2.

Can we refine the proof of Theorem 3.2 to obtain a height bound that is
uniform in $G$? If so, then Section 1 has a negative answer.

## Acknowledgments

Part of the research for this article was carried out at the 2023 “Arithmetic,
Algebra, and Algorithms” workshop at the ICMS in Edinburgh. We heartily thank
Umberto Zannier for explaining to us the relevance of Silverman’s article
[Sil83] (building on the specialization theorems of Dem′janenko and Manin) for
proving Theorem 3.2, and for spending the time to write up for us a new, more
elementary proof of the specialization theorem in the case of constant abelian
schemes, inspired by the “alternative proof” of [BMZ08, Theorem 1]. We also
thank Yuri Bilu, Noam Elkies, Benedict Gross, Joseph Harris, Will Sawin,
Joseph Silverman, and Yuri Zarhin for helpful correspondence.

## References

  * [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris.  Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1985.
  * [Bak66] A. Baker.  Linear forms in the logarithms of algebraic numbers. I, II, III.  Mathematika, 13:204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228, 1966.
  * [Bak68a] A. Baker.  Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms.  Philos. Trans. Roy. Soc. London Ser. A, 263:173–191, 1968.
  * [Bak68b] A. Baker.  Contributions to the theory of Diophantine equations. II. The Diophantine equation $y^{2}=x^{3}+k$.  Philos. Trans. Roy. Soc. London Ser. A, 263:193–208, 1968.
  * [Bak68c] A. Baker.  The Diophantine equation $y^{2}=ax^{3}+bx^{2}+cx+d$.  J. London Math. Soc., 43:1–9, 1968.
  * [Bak69] A. Baker.  Bounds for the solutions of the hyperelliptic equation.  Proc. Cambridge Philos. Soc., 65:439–444, 1969.
  * [BC70] A. Baker and J. Coates.  Integer points on curves of genus 1.  Proc. Cambridge Philos. Soc., 67:595–602, 1970.
  * [Bel79] G. V. Belyĭ.  Galois extensions of a maximal cyclotomic field.  Izv. Akad. Nauk SSSR Ser. Mat., 43(2):267–276, 479, 1979.
  * [Bil95] Yuri Bilu.  Effective analysis of integral points on algebraic curves.  Israel J. Math., 90(1-3):235–252, 1995.
  * [BMZ08] Enrico Bombieri, David Masser, and Umberto Zannier.  Intersecting a plane with algebraic subgroups of multiplicative groups.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(1):51–80, 2008.
  * [Dem68] V. A. Dem′ janenko.  An estimate of the remainder term in Tate’s formula.  Mat. Zametki, 3:271–278, 1968.
  * [Gro86] Benedict H. Gross.  Local heights on curves.  In Arithmetic geometry (Storrs, Conn., 1984), pages 327–339. Springer, New York, 1986.
  * [KS99] Nicholas M. Katz and Peter Sarnak.  Random matrices, Frobenius eigenvalues, and monodromy, volume 45 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 1999.
  * [LL22a] Aaron Landesman and Daniel Litt.  Applications of the algebraic geometry of the Putman-Wieland conjecture.  arXiv preprint arXiv:2209.00718v2, 2022.
  * [LL22b] Aaron Landesman and Daniel Litt.  An introduction to the algebraic geometry of the Putman-Wieland conjecture.  arXiv preprint arXiv:2209.0071v1, 2022.
  * [LL22c] Aaron Landesman and Daniel Litt.  Prill’s problem.  arXiv preprint arXiv:2209.12958v1, 2022.
  * [Man64] Ju. I. Manin.  The Tate height of points on an Abelian variety, its variants and applications.  Izv. Akad. Nauk SSSR Ser. Mat., 28:1363–1390, 1964.
  * [Man69] Ju. I. Manin.  The $p$-torsion of elliptic curves is uniformly bounded.  Izv. Akad. Nauk SSSR Ser. Mat., 33:459–465, 1969.
  * [Mar22] Vladimir Marković.  Unramified correspondences and virtual properties of mapping class groups.  Bull. Lond. Math. Soc., 54(6):2324–2337, 2022.
  * [Nér52] André Néron.  Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps.  Bull. Soc. Math. France, 80:101–166, 1952.
  * [Nér65] A. Néron.  Quasi-fonctions et hauteurs sur les variétés abéliennes.  Ann. of Math. (2), 82:249–331, 1965.
  * [PW13] Andrew Putman and Ben Wieland.  Abelian quotients of subgroups of the mappings class group and higher Prym representations.  J. Lond. Math. Soc. (2), 88(1):79–96, 2013.
  * [Ser97] Jean-Pierre Serre.  Lectures on the Mordell-Weil theorem.  Aspects of Mathematics. Friedr. Vieweg & Sohn, Braunschweig, third edition, 1997.  Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre.
  * [Sie29] Carl Ludwig Siegel.  Über einige Anwendungen diophantischer Approximationen.  Abh. Preuß. Akad. Wissen. Phys.-math. Klasse, pages 41–69, 1929.  English translation: _On some applications of diophantine approximations_ , edited by Umberto Zannier, Scuola Normale Superiore Pisa, 2014\.
  * [Sil83] Joseph H. Silverman.  Heights and the specialization map for families of abelian varieties.  J. Reine Angew. Math., 342:197–211, 1983.
  * [Wew98] Stefan Wewers.  Construction of Hurwitz spaces.  Institut für Experimentelle Mathematik Essen, Ph.D. thesis, 1998.