# Radio detections of two unusual cataclysmic variables in the VLA Sky Survey

M. E. Ridder,1 C. O. Heinke,1 G. R. Sivakoff,1 & A. K. Hughes1
1 University of Alberta, Physics Dept., CCIS 4-183, Edmonton, AB T6G 2E1,
Canada
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report two new radio detections of cataclysmic variables (CVs), and place
them in context with radio and X-ray detections of other CVs. We detected QS
Vir, a low accretion-rate CV; V2400 Oph, a diskless intermediate polar (IP);
and recovered the polar AM Her in the Very Large Array Sky Survey (VLASS) 2–4
GHz radio images. The radio luminosities of these systems are higher than
typically expected from coronal emission from stars of similar spectral types,
and neither system is expected to produce jets, leaving the origin of the
radio emission a puzzle. The radio emission mechanism for these two CVs may be
electron-cyclotron maser emission, synchrotron radiation, or a more exotic
process. We compile published radio detections of CVs, and X-ray measurements
of these CVs, to illustrate their locations in the radio - X-ray luminosity
plane, a diagnostic tool often used for X-ray binaries, active galactic
nuclei, and radio stars. Several radio-emitting CVs, including these two newly
detected CVs, seem to lie near the principal radio/X-ray track followed by
black hole X-ray binaries (BHXBs) at low luminosity, suggesting additional
complexity in classifying unknown systems using their radio and X-ray
luminosities alone.

###### keywords:

cataclysmic variables — stars: dwarf novae — stars: individual: QS Vir —
stars: individual: V2400 Oph — radio continuum: stars

††pubyear: 2022††pagerange: Radio detections of two unusual cataclysmic
variables in the VLA Sky Survey–Radio detections of two unusual cataclysmic
variables in the VLA Sky Survey

##  1 Introduction

Cataclysmic variables (CVs) are binary systems in which a white dwarf (WD)
accretes from an M or K type donor star (e.g., see review in Warner, 2003).
Accretion begins when the donor reaches the edge of its Roche lobe, but the
accretion geometry is determined by the WD’s magnetic field. If the WD is
weakly magnetic, then matter from the donor fills an accretion disk around the
WD. These kinds of CVs periodically go into outburst (dwarf novae) caused by
an increased accretion rate, or remain at a high accretion rate for long
periods of time (novalikes). The moderately magnetic case (intermediate
polars; IPs) results in a truncated accretion disk, where the inner region of
the disk follows the field lines onto the poles. In the most extreme case
(polars), there is no accretion disk and matter only flows along the WD’s
magnetic field lines.

There are currently 33 known CVs detected in the radio at $\geq$3$\sigma$
confidence and 16 marginal detections. Of these 33 CVs, 23 are magnetic
(Barrett et al., 2020) and 10 are non-magnetic (Körding et al., 2008;
Coppejans et al., 2015, 2016; Russell et al., 2016). The main hypotheses for
the origin of radio emission from CVs are synchrotron emission from a jet
(Russell et al., 2016; Mooley et al., 2017; Fender et al., 2019), or electron-
cyclotron maser emission (ECME) near the WD or donor (Barrett et al., 2020).
Even within a given class (e.g. novalike), there may be multiple processes
responsible for the radio emission (Coppejans et al., 2015, 2016).

SS Cyg is a well-studied non-magnetic CV with a radio counterpart (Russell et
al., 2016; Mooley et al., 2017; Fender et al., 2019). Multiple radio flares
coincide with its dwarf nova (DN) outbursts, and the evolution of the spectral
index during one flare suggests a jet launched by the WD in SS Cyg (Fender et
al., 2019). Unfortunately, there are not enough data for other DNe and
novalikes to determine if their radio emission is also jet-powered (Coppejans
et al., 2015, 2016). The prototypical polar, AM Her, is another well-known CV
that has been observed at radio frequencies since the early 1980s (Chanmugam &
Dulk, 1982). The radio source in magnetic systems such as this has been
suggested to be ECME, which is the result of a plasma instability (Barrett et
al., 2020). In the weakly relativistic regime, ECME produces narrow band,
highly circularly polarized emission. In order to produce the instability, the
plasma environment needs to be low-density ($n<10^{-12}$ cm-3; Barrett et al.,
2020) and highly magnetized, which is the case in the corona of the donor star
or near the WD, possibly adjacent to the accretion column.

Among the other radio bright binaries containing white dwarfs are AR Sco
(Marsh et al., 2016) and AE Aqr (Bookbinder & Lamb, 1987). These are CV-like
with a M-K type secondary and WD primary, but unlike CVs, there is little or
no accretion, and the observed electromagnetic radiation is thought to be
produced by more complex processes. AR Sco is commonly referred to as a white
dwarf pulsar, as its WD’s magnetic poles sweep over the companion with a
period of 1.95 minutes, which produces emission at the WD spin period and the
beat period between the spin and orbital periods (Marsh et al., 2016; Buckley
et al., 2017; Marcote et al., 2017; Garnavich et al., 2019). The leading
explanation for AR Sco’s emission mechanism, based on the high luminosity of
the system (larger than either star should be), the beat-period signal
(suggesting reprocessing of radiation from the WD striking the companion), and
the rapid spindown of the WD (indicating a high B field of 100–500 MG, Buckley
et al., 2017), is that the emission is largely synchrotron radiation from
electrons accelerated by the WD magnetic field and/or interaction between the
WD and companion magnetic fields (e.g. Marsh et al., 2016; Takata et al.,
2018).

AE Aqr is a propeller system, meaning that as material from the donor falls
towards the WD, it is flung out of the system by the magnetic field of the
rapidly spinning WD (Eracleous & Horne, 1996; Wynn et al., 1997), with radio
emission thought to be produced by acceleration of electrons in the outflowing
material in shocks (Bastian et al., 1988; Meintjes & Venter, 2003). The only
known AE Aqr twin is LAMOST J024048.51+195226.9 (hereafter J0240, Thorstensen,
2020; Garnavich et al., 2021; Pretorius et al., 2021; Pelisoli et al., 2022).
Since we do not yet know the exact process that produces radio emission in
CVs, we include AE Aqr, LAMOST J0240, and AR Sco in our plots as references.

In this paper we discuss two new detections of unusual CVs in Epochs 1 and 2
of the Very Large Array Sky Survey (VLASS), QS Vir and V2400 Oph. QS Vir is a
low accretion-rate ($\dot{M}=$ 1.7$\times 10^{-13}$ M⊙ yr-1), eclipsing CV,
sometimes referred to as a pre-CV (O’Donoghue et al., 2003; Matranga et al.,
2012). V2400 Oph is a diskless IP that may undergo diamagnetic blob accretion,
in which blobs of matter from the donor star orbit the WD and eventually
accrete or transfer back to the donor (Langford et al., 2022). Based on the
VLASS data, V2400 Oph has shown the brightest radio emission of any CV to
date.

In Section 2 we discuss the archival radio observations of QS Vir and V2400
Oph. In Section 3 we explain the analysis of these radio images as well as
archival X-ray data. In Section 4 we discuss the implications of our results.
Throughout this paper, we quote errors at 1-$\sigma$ unless otherwise noted.

##  2 VLASS observations

Table 1: Coordinates of our detected sources, from Gaia EDR3 and in VLASS, and Gaia distances (Bailer-Jones et al., 2021). The errors quoted on VLASS distances are 1-sigma. Offsets are rounded to the nearest 0.05″. Name |  Gaia RA (deg) |  Gaia Dec (deg) | VLASS RA (deg) | VLASS Dec (deg) | Offset (arcsec) | Distance (pc)
---|---|---|---|---|---|---
QS Vir | 207.466870 | -13.226865 | 207.46694(9) | -13.22678(11) | 0.40 | 50.1${}^{+0.07}_{-0.06}$
V2400 Oph | 258.151755 | -24.245759 | 258.15156(8) | -24.24585(8) | 0.75 | 700${}^{+10}_{-11}$
AM Her (1.1) | 274.054911 | 49.868113 | 274.05489(11) | 49.86816(9) | 0.20 | 87.9$\pm$0.2
AM Her (2.1) | 274.054911 | 49.868113 | 274.05496(10) | 49.86810(9) | 0.10 | 87.9$\pm$0.2

The Very Large Array Sky Survey (VLASS; Lacy et al., 2020) is conducted using
the VLA S band (2–4 GHz) in a raster scan, with about 4.5 seconds exposure
time on any source in each epoch, assuming a scan speed of 3.31”/s and a prime
beam of 45/$\nu_{GHz}$. Upon completion, the full sky (above -40 declination)
will be visited three times by VLASS; at the time of writing, both the first
and second epochs of observations of the sky above $-40$ degrees declination
are now available. Rapid CLEAN processing by NRAO produces Quick-Look images,
which have a typical noise of 120 $\mu$Jy (Gordon et al., 2021). Using the
Canadian Initiative for Radio Astronomy Data Analysis111https://cirada.ca/
(CIRADA) image cutout service, an online tool that generates cutout images of
radio and IR surveys, we retrieved images of each object from the list of 1681
CVs and related systems in Ritter & Kolb (2003), which were crossmatched with
Gaia EDR3 (Gaia Collaboration et al., 2021). We queried all VLASS epochs in
late 2021 when observations up to Epoch 2.1 were available. We identified
three radio counterparts; the polar AM Her (previously known to be radio-
emitting, Chanmugam & Dulk 1982), QS Vir, and V2400 Oph. When Epoch 2.2 was
released, we checked for new observations of these three CVs, and when
astrometric corrections were provided for Epoch 1, we updated our data. After
astrometric corrections are applied, the absolute astrometric systematic
uncertainty is $\sim$0.25″(Lacy, 2021), which is better than that discussed in
Gordon et al. (2020).

The radio images of QS Vir, V2400 Oph, and AM Her are shown in Figures 1, 2,
and 3. Both QS Vir and V2400 Oph were detected only in the first epoch, while
AM Her was detected in both epochs. As our search was performed only on the
Epochs 1.1 – 2.1, and both QS Vir and V2400 Oph appear to be transient in the
radio, it seems plausible that additional CVs, including those in Ritter &
Kolb (2003), may be detected in the second and third epochs of VLASS.

Based on visual inspections of the images, we do not believe the sources are
likely to arise from image artifacts. We obtained source positions and fluxes
with Common Astronomy Software Applications (CASA) 6.5 (McMullin et al.,
2007), software written by the National Radio Astronomy Observatory (NRAO) for
analyzing radio data. The imfit algorithm fits a two-dimensional Gaussian
distribution to a source inside a given position and radius, then returns a
source position, flux, and flux error. To obtain the position error, we
calculated the width of the beam in RA and declination. Since the synthesized
beam is reported in units of the full-width-half-maximum, we divided the
projected beam sizes by 2 $\times$ SNR to determine the statistical error in
centroiding. Because there is a systematic error of 1/10 of the beam in VLASS,
we added this term to the last in quadrature to find the total error. We found
the position of the source in Figure 1(a) is 0.4″away from the coordinates for
QS Vir in Gaia EDR3. For Figure 2(a), the source is 0.7″away from the Gaia
EDR3 position of V2400 Oph. The radio source near AM Her is offset by
0.2″(Figure 3(a)) and 0.1″(Figure 3(b)), in the two epochs respectively. The
positions of the radio counterparts in these images are consistent within 2
$\sigma$ of the Gaia EDR3 coordinates of each object.

With the imfit algorithm described above, we centered a region on each
source’s Gaia position and fixed all Gaussian components to the shape of the
synthesized beam for the fit. We characterized the background with the
function imstat and estimated the noise from the root-mean-square statistic in
an annulus around each source. The measured fluxes of QS Vir and V2400 Oph can
be found in Table 2. Each CV is detected at least once at $>$5$\sigma$.

Table 2: VLASS flux measurements (2 – 4 GHz) of all sources when they were observed. The value for QS Vir in Epoch 2.2 and V2400 Oph in Epoch 2.1 are three sigma upper limits. Name | Flux ($\mu$Jy) | Epoch | Observation date
---|---|---|---
QS Vir | 970$\pm$160 | 1.2 | 2019 April 25
QS Vir |  $\leq$ 510 | 2.2 | 2022 January 31
V2400 Oph | 640$\pm$120 | 1.1 | 2018 February 5
V2400 Oph |  $\leq$ 620 | 2.1 | 2020 October 23
AM Her | 640$\pm$120 | 1.1 | 2017 September 9
AM Her | 880$\pm$150 | 2.1 | 2020 August 11

(a) 25 April 2019

(b) 31 January 2022

Figure 1: VLASS-QL cutouts of QS Vir on 2019 April 25 and 2022 January 31. The
images are 30”$\times$30” and the red circles have a 5 arcsecond radius,
centered on QS Vir’s Gaia coordinates.

(a) 5 February 2018

(b) 23 October 2020

Figure 2: VLASS-QL cutouts of V2400 Oph on 2018 February 5 and 2020 October
23. The images are 30″$\times$30″and the red circles have a 5″radius, centered
on V2400 Oph’s Gaia coordinates.

(a) 9 September 2017

(b) 11 August 2020

Figure 3: VLASS-QL cutouts of AM Her on 2017 September 9 and 2020 August 11.
The images are 30”$\times$30” and the red circles have a 5 arcsecond radius,
centered on AM Her’s Gaia coordinates.

##  3 Analysis

###  3.1 Chance coincidence?

To calculate the likelihood that our radio detections are simply spurious
detections of background AGN, we first need to estimate the density of
background AGN. We take two approaches to this. First, we find the number of
radio sources in the vicinity of QS Vir and V2400 Oph, using VLASS. Using the
VLASS catalog of source from Gordon et al. (2020), there are 179 and 133
reliable components ($Duplicate\\_flag<2$ and $Quality\\_flag==0$) from Gordon
et al. (2020) within a 1 degree radius around QS Vir and V2400 Oph,
respectively. We thus derive an average background source density of
3.8$\times$10-6 sources per square arcsecond.

We searched for VLASS emission near the 1681 objects from the Ritter & Kolb
(2003) catalog that also had Gaia EDR3 positions. Given typical astrometric
errors in 5-$\sigma$ detected VLASS sources of 0.35″, we consider matches out
to $3\sigma$, or 1.05″. Given the 1681 potential objects, we estimate the
probability of a chance coincidence at 0.022.

Since we were able to search for radio sources fainter than included in the
(conservative) catalog of Gordon et al. (2020), this chance coincidence
calculation may be an underestimate. A more conservative estimate uses the
expected total VLASS source number of 5.3 million anticipated detections (Lacy
et al., 2020), divided by the 33885 square degrees to be surveyed. Since this
number is for the stacked combination of all three epochs, we use Owen &
Morrison (2008) to estimate that there will be about 2.3 million anticipated
detections in a single VLASS Epoch, and estimate that the background source
density should be about 5.2$\times$10-6 sources per square arcsecond. Given
the 1681 error circles searched, we estimate the probability of a chance
coincidence at 0.029.

We conclude that it is unlikely that either of our newly detected radio
sources are chance coincidences, and that we have detected radio counterparts
to QS Vir and V2400 Oph.

###  3.2 X-ray flux measurements

Comparison of X-ray and radio luminosities can be helpful in understanding
accretion behaviour. Although we did not have (near)-simultaneous X-ray
observations, non-simultaneous X-ray observations can still be useful,
especially if the X-ray flux is not known to vary by orders of magnitude.
Polars, for instance, appear to be only moderately variable, except for their
drops into "low states" with little accretion, which appear to occur about
half the time (Ramsay et al., 2004b). AM Her, for example, shows variability
of about an order of magnitude between the Second ROSAT All-Sky Survey Source
Catalog (2RXS; Boller et al., 2016) and the XMM-Newton Serendipitous Source
Catalog (4XMM-DR11; Webb et al., 2020), $\sim 10^{-13}$ and $\sim 10^{-12}$
erg s-1 cm-2 respectively, from 0.5 - 1 keV. However, there is more
uncertainty in DNe because of the X-ray flux’s dependence on whether the
object is in outburst or quiescence. While the radio observations of the DNe
in our dataset were taken during outburst (Coppejans et al., 2016), we cannot
be certain whether the same is true for X-ray catalog data. Similarly, we
cannot determine what accretion state the polars in our dataset were in during
the X-ray or radio observations. We estimate the X-ray variability of our
sources by comparing their detections in the improved Swift XRT Point Source
Catalog (2SXPS; Evans et al., 2020) and 2RXS (Figure 4), which allows us to
estimate a dispersion of $\sim$3 ct s-1 between multiple X-ray observations of
the same source.

Figure 4: ROSAT (2RXS) vs. Swift (2SXPS) count rates of the objects in Tables
3 and 4. We find a typical scatter in X-ray count rates of $\sigma_{X}\sim 3$
ct s-1.

To obtain catalog X-ray data, we queried the HEASARC database using the Gaia
EDR3 coordinates of QS Vir and V2400 Oph, which returned detections in 4XMM-
DR11. To compare to other radio-detected CVs (Coppejans et al., 2015, 2016;
Barrett et al., 2020), we retrieved (generally non-simultaneous) X-ray data
from their positions. These objects were observed in 4XMM-DR11, 2SXPS, the
ROSAT All-Sky Survey Bright Source Catalog (RASS-BSC; Voges et al., 1999), and
2RXS. Only SS Cyg has published simultaneous X-ray and radio light curves
during outbursts, for which we chose to plot points at the highest radio flux
and the highest X-ray count rate from Russell et al. (2016). To obtain the
exact Swift flux, we fit SS Cyg’s spectra using the fitting tool provided by
the UK Swift Science Data Centre222https://www.swift.ac.uk/user_objects/
(Evans et al., 2009).

To convert count rates from these catalogs into luminosities, we searched the
literature for spectral fits of magnetic CVs and non-magnetic CVs. For the few
that had a spectral fit, we used the reported model and absorption column
density $N_{H}$ to obtain the unabsorbed flux with CXC
PIMMS333https://cxc.harvard.edu/toolkit/pimms.jsp. When only catalog data were
available, we used CXC Colden444https://cxc.harvard.edu/toolkit/colden.jsp to
estimate $N_{H}$ provided by Dickey & Lockman (1990). For magnetic CVs, we
followed the convention in Ramsay et al. (1994), choosing a thermal
bremsstrahlung model with a 30 keV temperature for energies greater than 0.5
keV and a blackbody model with a 30 eV temperature for energies between 0.1
and 0.4 keV. In order to do this when RASS-BSC or 2RXS data were available, we
used the hardness ratio between the two bands 0.1–0.4 keV and 0.5–2 keV to
estimate the count rates in both. When 2SXPS data was available we used the
hardness ratio between the bands 0.3–1 keV and 1–2 keV to determine whether
there was a significant soft component and if so, applied a blackbody model to
the 0.3–1 keV count rate. If not, a thermal bremsstrahlung model was used for
that energy range. Above 1 keV, we assumed the bremsstrahlung model would
dominate in all cases.

Of the novalikes in Coppejans et al. (2015) and Hewitt et al. (2020), only TT
Ari had an X-ray spectral fit (Mauche & Mukai, 2002). Thus we assumed the
other novalikes should be fit with a similar model and characteristic
temperature; we assumed a single MEKAL model with an average temperature of 7
keV. Of the dwarf novae in Coppejans et al. (2016), only Z Cam (Saitou et al.,
2012) and SU UMa (Collins & Wheatley, 2010) had X-ray spectral fits that could
be converted with PIMMS. For these, we assumed a MEKAL model (as used in both
of the fits) and an average characteristic temperature of 11 keV.

When fluxes were available in 4XMM-DR11, we made an approximate conversion to
1–10 keV by taking the full band (Band 8, 0.2–12 keV) and subtracting two soft
bands (Band 1, 0.2–0.5 keV; Band 2, 0.5–1 keV). Assuming very few photons were
detected between 10 and 12 keV, the full band from 4XMM-DR11 approximates the
0.1–10 keV flux.

For data from a single band in RASS-BSC and 2RXS, we use the error given in
each catalog. For data from 4XMM-DR11, we assumed the largest error quoted for
the full band would be roughly the same as that for 1–10 keV. Some of the
errors on Swift’s 2–10 keV and 0.3–1 keV bands were of the same magnitude so
these were added in quadrature.

###  3.3 Radio - X-ray luminosity plane

We compare the radio and X-ray luminosities of the CVs, extracted from the
literature as above, in Figure 5. The data points here are also shown in
Tables 3 and 4. We assumed a flat spectrum up to 5 GHz, thus multiplying the
flux density by ($5\times 10^{9}$ Hz $\times$ $4\pi d^{2}$) to infer a radio
luminosity. While some CVs have been detected at higher frequencies than 5 GHz
(e.g. EQ Cet at 18-26.5 GHz), we do not have clear measurements of the shapes
of CV spectra. Using a flat spectrum up to 5 GHz (while logically inconsistent
for CVs detected at higher frequencies) has the benefit of allowing comparison
between CVs (assuming they all have the same spectrum), and with the
literature, both of CVs and X-ray binaries (e.g. Bahramian et al., 2018). We
have also plotted the well-known relationship between the X-ray and radio
luminosity of BHXBs (Gallo et al., 2003, black dotted line in Figure 5 of this
paper). Included in this figure are our detections, previously known radio
CVs, and the unique systems, AR Sco and AE Aqr. V2400 Oph stands out in the
upper right corner, far brighter in radio and X-ray than most magnetic and
non-magnetic CVs. The second brightest CV in the radio is V1323 Her, another
IP (Barrett et al., 2020). QS Vir is near the bottom left, near several
magnetic CVs. For comparison of our data to X-ray binaries, we have plotted a
transitional millisecond pulsar (tMSP; PSR J1023+0038; Bahramian et al., 2018;
Deller et al., 2015; Bogdanov, 2018) and three BHXBs (A0620-00, XTE J1118+480,
Cen X-4; Bahramian et al., 2018; Fender et al., 2010; Gallo et al., 2014;
Tudor et al., 2017).

Sources in Barrett et al. (2020) with radio detections below 3-$\sigma$ (Cas
1, FL Cet, Hya 1, HS0922+1333, and Her 1) were excluded in this plot. We
plotted the highest and lowest radio fluxes, for objects with multiple radio
detections, as a range denoted with a dashed line.

###  3.4 Radio luminosity and orbital period

Figure 7 shows our detections compared to previously known radio CVs, AR Sco,
AE Aqr, and the candidate propeller system LAMOST J0240 (note that the latter
is not yet X-ray-detected, and thus not shown in Figure 5). V2400 Oph’s
orbital period is much shorter than the propellers, but it is similarly
bright, as is the other IP, V1323 Her. QS Vir is within the cluster of
magnetic and non-magnetic CVs, including AM Her.

Figure 5: X-ray vs. radio luminosities of CVs (and related systems) with
detections in both bands. Two locations of simultaneous data (filled symbols)
are plotted for SS Cyg, one at maximum X-ray luminosity and one at maximum
radio luminosity (Russell et al., 2016). All other data are non-simultaneous
(open symbols). Multiple radio detections of the same object at different
luminosities are connected with dashed lines. The black dotted line marks the
LX-LR relationship for BHXBs (Gallo et al., 2003). Figure 6: Comparing the
effects of using the 1–10 keV X-ray luminosity vs. the 0.1–10 keV X-ray
luminosity for the positions of radio-detected CVs in the radio/X-ray plane.
The black dotted line marks the LX-LR relationship for BHXBs (Gallo et al.,
2003). Figure 7: The orbital period in hours for our sample vs. specific radio
luminosity. Here we plot each individual observation of magnetic and non-
magnetic CVs in Barrett et al. (2020); Coppejans et al. (2015, 2016); Hewitt
et al. (2020), unlike Figure 5. LAMOST J0240 is an AE Aqr twin that we do not
show in Figure 5 since it currently is undetected in X-rays.

##  4 Discussion

The proximity of our detections and the rest of the CV population to the hard-
state BH X-ray binary relationship in Figure 5 suggests a possible problem for
radio/X-ray identification of quiescent BHXBs. The radio-X-ray relation has
often been used to suggest that a given radio source near the BHXB line is
likely to be a BHXB (e.g. Bahramian et al., 2020; Zhao et al., 2020;
Shishkovsky et al., 2018). However, our results show that there could be
significant CV contamination among sources near this line. We note that CVs
often show substantial X-ray variability of an order of magnitude or more,
and, based on the limited observations so far, radio variability seems common
at well. Simultaneous X-ray and radio observations would be very useful. We
provide a series of radio observations of AM Her in Figure 8 to illustrate the
observed variability on a timescale of years. Except for a flare of 9.7 $\pm$
2.3 mJy (Dulk et al., 1983) and a 100 $\mu$Jy upper limit detected by Bastian
et al. (1985), the flux varies between $\sim$200 to $\sim$700 $\mu$Jy.
However, as most of our data is non-simultaneous, interpreting Figure 5 should
be done with caution.

Figure 8: 4.9 GHz radio observations of AM Her from 1981 to 2013 (MJD 44900 –
56600) (Dulk et al., 1983; Bastian et al., 1985; Gawroński et al., 2018) and
the 2 – 4 GHz VLASS observations from 2017 and 2020 (MJD 58000 – 59100). Upper
limits are at the 1-$\sigma$ level. For the majority of its history, the radio
flux varies by a factor of $\sim$ 4; the outlying points are the 9.7 $\pm$ 2.3
mJy flare and lower limit of 100 $\mu$Jy.

Given the limited VLASS data at present, we cannot be certain of the emission
mechanism behind the radio counterparts of V2400 Oph and QS Vir. In the case
of QS Vir, the extremely low accretion rate paired with the known lack of an
accretion disc suggests radio emission originates in ECME rather than jets.
Radio flares on isolated M dwarfs are fairly common, and Hallinan et al.
(2008) propose that the dominant radio emission mechanism in late type dwarfs
is ECME. QS Vir has displayed coronal flares from its M-type donor star in the
past (O’Donoghue et al., 2003), suggesting the radio flares could be from the
donor. However, Güdel (2002) shows that M and K dwarf radio detections in the
cm band generally reach $10^{15}$ erg/s/Hz at most, which for 5 GHz gives a
maximum $L_{R}=5\times 10^{23}$ erg/s; a factor of 20 below QS Vir’s
detection. This does not prove that the radio emission cannot come from the
donor star, as there is a candidate detection of brighter radio emission from
a young M-type brown dwarf (2MASS J16044075–1936525) in Ling et al. (2022), at
$L_{R}$(5 GHz)=$1.1\times 10^{27}$ erg/s. However, such radio emission is
quite rare; Ling et al. (2022) surveyed 2600 (mostly young) stellar objects
and detected only 6 of them in radio. Ling et al. (2022) produced a stacked 5
GHz radio luminosity upper limit of $<1.3\times 10^{23}$ erg/s for the RECONS
sample (821 M stars within $\sim$70 pc from Earth, Henry et al. 2018).
Similarly, Launhardt et al. (2022) observed 170 young stars at 6 GHz, finding
a half dozen detections near $10^{8}$ W/Hz, or $L_{R}$(5 GHz)=$5\times
10^{24}$ erg/s, but only one subgiant (V875 Per) brighter than QS Vir.

The source of the radio emission in V2400 Oph is even more difficult to
explain. Its larger (by a factor of 100) radio luminosity (larger than any
known M or K dwarf) would make ECME from the companion very surprising, while
the lack of an accretion disc in the system makes the jet hypothesis unlikely.
Langford et al. (2022) suggest that the accretion flow in V2400 Oph consists
of a series of disconnected diamagnetic blobs (King, 1993; Littlefield et al.,
2021), which may either ride down the magnetic field lines to the magnetic
poles of the white dwarf (generating the clearly observed spin periodicity),
or gain energy from the magnetic field and be expelled, likely back to the
donor star. This suggests the possibility of a similarity in the radio
emission mechanism between V2400 Oph and the propeller systems AE Aquarii and
J0240, which also expel material and have similar radio luminosity to V2400
Oph. On the other hand, V2400 Oph has not shown evidence for $\sim$3000 km/s
motion of propeller-launched blobs, as seen in AE Aquarii and J0240.
Alternatively, Kennedy et al. (2020) identify evidence of high-velocity
(600–1000 km/s) features during a low state of the IP FO Aquarii, which they
suggest may be attributed to a jet or fast outflow. IPs lack a disk near the
WD surface, which makes the formation of a jet in such systems challenging to
justify theoretically. If this is indeed ECME, the source could be akin to the
model for polars in Barrett et al. (2020), near the short-lived accretion
stream caused by the blobs.

Future work on these CVs will include longer radio observations of these
systems to measure the spectral index, polarization, and (with simultaneous
X-ray observations) place the system firmly on the Lx/Lr plot. We do not know
if there is a pattern to the radio flares, nor how frequently they recur, so a
series of several 1-hour observations on different days may be the best option
to catch their radio emission and advance our understanding of these unusual
systems.

Table 3: The fluxes for magnetic CVs used to generate Figure 5 and 6, including AR Sco, AE Aqr, AM Her, and V2400 Oph. Refer to Table 4 for non-magnetic CVs. Name | RA | Dec | Distance | 1–10 keV flux | 0.1–10 keV flux | Radio band | Radio flux | X-ray ref. | Radio ref.
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (pc) | (erg s-1 cm-2) | (erg s-1 cm-2) | (GHz) | ($\mu$Jy) |  |
EQ Cet |  022.21913 | $-23.66237$ | 283 ${}^{+5}_{-6}$ | 9$\pm 4\times 10^{-13}$ | 1.2$\pm 0.4\times 10^{-12}$ | 18 – 26.5 | 96$\pm 30$ | 9 | 2
BS Tri |  032.37423 |  $-$28.54133 | 277 ${}^{+8}_{-7}$ | 1.7$\pm 0.6\times 10^{-12}$ | 1.9$\pm 0.6\times 10^{-12}$ | 8 – 12 | 57$\pm 9$ | 22 | 2
EF Eri |  048.55430 |  $-22.5948$0 | 163 ${}^{+66}_{-50}$ | 3.7$\times 10^{-14}$ | 6.0$\times 10^{-14}$ | 8 – 12 | 87$\pm 15$ | 19 | 2
UZ For |  053.86947 | $-25.73939$ | 238 $\pm 3$ | 2.5$\pm 0.7\times 10^{-13}$ | 3.1$\pm 0.7\times 10^{-13}$ | 4 – 8 | 78$\pm 9$ | 20 | 2
RXJ0502.8+1624 |  075.71247 |  $-$16.40595 | 217 ${}^{+13}_{-10}$ | 1.0$\pm 0.2\times 10^{-10}$ | 1.1$\pm 0.2\times 10^{-10}$ | 8 – 12 | 105$\pm 32$ | 3 | 2
LW Cam | 106.04185 |  $-$62.05777 | 549 $\pm 19$ | 2.3$\pm 0.5\times 10^{-12}$ | 2.8$\pm 0.5\times 10^{-12}$ | 8 – 12 | 50$\pm 5$* | 22 | 2
VV Pup | 123.77841 | $-19.05524$ | 135.3 $\pm 0.7$ | 2.7$\times 10^{-12}$ | 3.5$\times 10^{-12}$ | 8 – 12 | 79$\pm 14$ | 4 | 2
FR Lyn | 133.55839 |  $-$39.09354 | 480 ${}^{+50}_{-40}$ | 2.4$\pm 1.1\times 10^{-13}$ | 6.5$\pm 1.1\times 10^{-13}$ | 8 – 12 | 28$\pm 4$ | 22 | 2
WX LMi | 156.61466 |  $-$38.75055 | 96.8 ${}^{+0.5}_{-0.6}$ | 8.7$\times 10^{-15}$ | 4.2$\times 10^{-14}$ | 4 – 8 | 73$\pm 12$ | 23 | 2
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 8 - 12 | 52$\pm 14$ |  $\cdots$ |  $\cdots$
ST LMi | 166.41570 |  $-$25.10783 | 114.2 $\pm 0.9$ | 4.0$\pm 0.5\times 10^{-14}$ | 6.0$\pm 0.6\times 10^{-14}$* | 8 – 12 | 153$\pm 15$* | 24 | 2
AR UMa | 168.93529 |  $-$42.97289 | 98.6 $\pm 0.5$ | $2.4\times 10^{-13}$ | $3.2\times 10^{-13}$ | 4 – 8 | 489$\pm 49$* | 21 | 2
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.3\times 10^{-12}$ | $1.7\times 10^{-12}$ | 18 – 26.5 | 317$\pm 32$* |  $\cdots$ |  $\cdots$
EU UMa | 177.48200 |  $-$28.75199 | 287 ${}^{+16}_{-15}$ | 6${}^{+2}_{-5}\times 10^{-13}$ |  $\cdots$ | 8 – 12 | 39$\pm 5$ | 14 | 2
V1043 Cen | 198.32121 | $-32.98694$ | 172.9 $\pm 0.9$ | 7.9$\pm 0.8\times 10^{-13}$* | 1.0$\pm 0.1\times 10^{-12}$* | 8 – 12 | 20$\pm 5$ | 24 | 2
J1503-2207 | 225.97494 | $-22.11956$ | 387$\pm$ 18 | 1.5$\pm 0.2\times 10^{-12}$ | 2.0$\pm 0.2\times 10^{-12}$ | 8 – 12 | 29$\pm$5 | 9 | 2
BM CrB | 235.26945 |  $-$36.04798 | 423 ${}^{+14}_{-16}$ | 1.6$\pm 1.5\times 10^{-14}$ | 2.1$\pm 1.5\times 10^{-14}$ | 8 – 12 | 43$\pm 15$ | 24 | 2
MR Ser | 238.19645 |  $-$18.94162 | 131.2 ${}^{+0.5}_{-0.7}$ | $1.5\times 10^{-12}$ | $1.8\times 10^{-12}$ | 4 – 8 | 239$\pm 24$* | 13 | 2
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 8 – 12 | 116 $\pm 15$ |  $\cdots$ |  $\cdots$
MQ Dra | 238.37941 |  $-$55.27067 | 180.7 ${}^{+2.1}_{-1.7}$ | 2.9$\pm 0.8\times 10^{-14}$ | 7.5$\pm 0.8\times 10^{-14}$ | 8 – 12 | 17 $\pm 4$ | 24 | 2
AP CrB | 238.55136 |  $-$27.36463 | 199.6 ${}^{+1.8}_{-2.1}$ | 4.9$\pm 1.6\times 10^{-13}$ | 8.7$\pm 1.6\times 10^{-13}$ | 8 – 12 | 24 $\pm 4$ | 22 | 2
AR Sco | 245.44706 | $-22.88645$ | 116.5 ${}^{+0.4}_{-0.5}$ | $1.80\times 10^{-12}$ |  $\cdots$ | 5.5 | 7000${}^{+5000}_{-2000}$ | 11 | 11
V2400 Oph | 258.15176 | $-24.24576$ | 700${}^{+10}_{-11}$ | 4.4$\pm 0.4\times 10^{-11}$* |  $\cdots$ | 2 – 4 | 640$\pm$120 | 24 | 25
V1007 Her | 261.02625 |  $-$41.23535 | 470 ${}^{+40}_{-30}$ | 1.6$\pm 0.7\times 10^{-13}$ | 9.4$\pm 0.9\times 10^{-13}$* | 8 – 12 | 38 $\pm 9$ | 22 | 2
V1323 Her | 270.91524 |  $-$40.20558 | 1830 ${}^{+180}_{-140}$ | 1.6$\pm 0.2\times 10^{-12}$* | 2.0$\pm 0.2\times 10^{-12}$* | 4 – 8 | 43 $\pm 9$ | 24 | 2
AM Her | 274.05491 |  $-$49.86811 | 87.9 $\pm 0.2$ | 1.1$\pm 0.1\times 10^{-10}$* |  $\cdots$ | 2 – 4 | 640 $\pm 120$ | 24 | 25
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 880$\pm 150$ |  $\cdots$ |  $\cdots$
V1432 Aql | 295.04755 | $-10.42382$ | 450 $\pm 7.0$ | $2.1\times 10^{-11}$ | $4.1\times 10^{-11}$ | 8 – 12 | 15 $\pm 5$ | 15 | 2
J1955+0045 | 298.80199 |  $-$0.76011 | 165.5 ${}^{1.9}_{-1.5}$ | 1.1$\pm 0.1\times 10^{-11}$ | 1.3$\pm 0.1\times 10^{-11}$ | 8 – 12 | 79$\pm$ 8* | 24 | 2
AE Aqr | 310.03848 | $-0.87079$ | 91.34 ${}^{+0.12}_{-0.13}$ | 5.1$\pm 0.5\times 10^{-12}$* |  $\cdots$ | 4.9 | 7500$\pm 750$* | 8 | 1
HU Aqr | 316.99219 | $-5.29488$ | 189.6 $\pm$ 1.5 | $3.3\times 10^{-14}$ | $5.4\times 10^{-14}$ | 8 – 12 | 44 $\pm 13$ | 18 | 2
V388 Peg | 329.38469 |  $-$8.92079 | 720 ${}^{+80}_{-50}$ | 4.1$\pm 0.8\times 10^{-12}$ | 6$\pm 2\times 10^{-11}$ | 8 – 12 | 34 $\pm 5$ | 9 | 2
Note. — The coordinates and distances in this table were found in Gaia EDR3
and Bailer-Jones et al. (2021) respectively, except for EF Eri, which did not
appear in that catalog. That information was taken from Thorstensen (2003).
RXJ0502.8+1624 is equivalent to Tau 4 in Barrett et al. (2020). Where there
are multiple entries for the same object, the row will begin with a “ $\cdots$
” in the “Name” column and anywhere the values are the same as the previous
row. Otherwise, a “ $\cdots$ ” indicates the information is unavailable, such
as for our detections and the highlighted objects in Figure 5, since those
were not included in Figure 6. In the case of EU UMa, the model from Ramsay et
al. (2004a) did not provide enough information to find the flux between 0.1
$-$ 10 keV, so that entry was left blank. Errors on the X-ray and radio flux
columns are assumed to be at least 10% and were raised (and marked with an
asterisk) if quoted as less in their references. The references in the final
two columns are numbered as follows: [1] Abada-Simon et al. (1993), [2]
Barrett et al. (2020), [3] Boller et al. (2016), [4] Bonnet-Bidaud et al.
(2020), [5] Collins & Wheatley (2010), [6] Coppejans et al. (2015), [7]
Coppejans et al. (2016), [8] Eracleous et al. (1991), [9] Evans et al. (2020),
[10] Hewitt et al. (2020), [11] Marsh et al. (2016), [12] Mauche & Mukai
(2002), [13] Ramsay et al. (1994), [14] Ramsay et al. (2004a), [15] Rana et
al. (2005), [16] Russell et al. (2016) [17] Saitou et al. (2012), [18] Schwarz
et al. (2009), [19] Schwope et al. (2007), [20] Still & Mukai (2001), [21]
Szkody et al. (1999), [22] Voges et al. (1999), [23] Vogel et al. (2007), [24]
Webb et al. (2020), [25] this paper.
Table 4: The fluxes for non-magnetic CVs used to generate Figure 5 and 6, including QS Vir and SS Cyg. Refer Table 3 for magnetic CVs and the list of citations in the table note. Name | RA | Dec | Distance | 1–10 keV flux | 0.1–10 keV flux | Radio band | Radio flux | X-ray ref. | Radio ref.
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (pc) | (erg s-1 cm-2) | (erg s-1 cm-2) | (GHz) | ($\mu$Jy) |  |
CM Phe |  005.38842 | $-51.70967$ | 308$\pm 3$ | 3.3$\pm$0.3$\times 10^{-12}$* | 4.2$\pm$0.4$\times 10^{-12}$ | 0.43 – 2.1 | < 30 | 9 | 10
RX And |  016.14809 |  $-$41.29928 | 196.6 $\pm 0.9$ | 7$\pm 2\times 10^{-13}$ | 1.0$\pm 0.1\times 10^{-11}$* | 8 – 12 | 19.6 $\pm 4.4$ | 9 | 7
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 13.6 $\pm 3.2$ |  $\cdots$ |  $\cdots$
TT Ari |  031.72112 |  $-$15.29490 | 247 $\pm 2$ | $2.40\times 10^{-11}$ | $3.10\times 10^{-11}$ | 4 – 8 | 39.6 $\pm 4.2$ | 12 | 6
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 240 $\pm 24$* |  $\cdots$ |  $\cdots$
IM Eri |  066.17152 | $-20.12003$ | 185.5 ${}^{+1.0}_{-0.9}$ | 4.3$\pm 1.1\times 10^{-12}$ | 5.5$\pm 1.0\times 10^{-12}$ | 0.43 – 2.1 | 99 $\pm 26$ | 3 | 10
V347 Pup |  092.64028 | $-48.74036$ | 290.3$\pm$1.2 | 1.7$\pm$0.4$\times 10^{-13}$ | 2.1$\pm$0.6$\times 10^{-13}$ | 0.43 – 2.1 | < 30 | 9 | 10
U Gem | 118.77167 |  $-$22.00122 | 93.1 $\pm 0.3$ | 1.0$\pm 0.1\times 10^{-11}$* | 1.2$\pm 0.1\times 10^{-11}$* | 8 – 12 | 12.7 $\pm 2.8$ | 24 | 7
YZ Cnc | 122.73616 |  $-$28.14233 | 233.5 ${}^{+1.5}_{-1.8}$ | 8.5$\pm 0.8\times 10^{-12}$* | 1.5$\pm 0.2\times 10^{-11}$* | 8 – 12 | 17.4 $\pm 3.7$ | 24 | 7
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 26.8 $\pm 5.2$ |  $\cdots$ |  $\cdots$
IX Vel | 123.82884 | $-49.22283$ | 90.13$\pm$0.14 | 2.2$\pm$0.2$\times 10^{-12}$* | 4.2$\pm$0.4$\times 10^{-12}$* | 0.43 – 2.1 | < 42 | 24 | 10
SU Uma | 123.11785 |  $-$62.60612 | 220.1 $\pm 1.1$ | $5.40\times 10^{-12}$ | $7.60\times 10^{-12}$ | 8 – 12 | 19.1 $\pm 4.9$ | 5 | 7
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 58.1 $\pm 5.8$* |  $\cdots$ |  $\cdots$
Z Cam | 126.30486 |  $-$73.11083 | 213.5 ${}^{+1.2}_{-1.3}$ | 4.7$\pm 0.5\times 10^{-12}$* | 5.7$\pm 0.6\times 10^{-12}$* | 8 – 12 | 25.0 $\pm 3.1$ | 17 | 7
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 40.3 $\pm 5.2$ |  $\cdots$ |  $\cdots$
RW Sex | 154.98584 | $-8.69899$ | 221.6 ${}^{+1.1}_{-1.0}$ | 4.4$\pm 0.5\times 10^{-12}$ | 5.6$\pm 0.6\times 10^{-12}$* | 0.86 – 1.7 | 233 $\pm 36$ | 9 | 10
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4 – 8 | 26.8 $\pm 3.3$ |  $\cdots$ | 2
QS Vir | 207.46687 | $-13.22687$ | 50 ${}^{+0.07}_{-0.06}$ | 8.5$\pm 0.9\times 10^{-13}$* |  $\cdots$ | 2 – 4 | 970$\pm$160 | 24 | 25
V1084 Her | 250.94041 |  $-$34.04427 | 465 $\pm 4$ | 1.6$\pm 0.2\times 10^{-12}$* | 1.8$\pm 0.2\times 10^{-12}$* | 4 – 8 | < 10.2 $\pm 3.4$ | 24 | 6
LS IV -08 3 | 254.12335 |  0$-8.57738$ | 211$\pm 2$ | 4.0$\pm$0.7$\times 10^{-12}$ | 5.1$\pm$0.9$\times 10^{-12}$ | 0.43 – 2.1 | < 33 | 3 | 10
V341 Ara | 254.42249 | $-63.21105$ | 155.3$\pm$0.8 | 1.2$\pm$0.2$\times 10^{-12}$ | 1.5$\pm$0.3$\times 10^{-12}$ | 0.43 – 2.1 | < 27 | 9 | 10
V603 Aql | 282.22770 |  $-$0.58408 | 315 ${}^{+3}_{-4}$ | 6.7$\pm 1.1\times 10^{-13}$ | 4.3$\pm 0.4\times 10^{-12}$* | 4 – 8 | 22 $\pm$ 3 | 9 | 6
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.86 – 1.7 | 233 $\pm 36$ |  $\cdots$ | 10
V3885 Sgr | 296.91905 | $-42.00751$ | 128.6 ${}^{+0.6}_{-0.5}$ | 4.8$\pm 0.5\times 10^{-12}$* | 6.1$\pm 0.6\times 10^{-12}$* | 0.43 – 2.1 | 256 $\pm 26$* | 9 | 10
V5662 Sgr | 301.46284 | $-29.58358$ | 169$\pm$4 | 3.3$\pm$0.9$\times 10^{-12}$ | 4.2$\pm$1.1$\times 10^{-12}$ | 0.43 – 2.1 | < 24 | 3 | 10
SS Cyg | 325.67904 |  $-$43.58622 | 112.3 ${}^{+0.4}_{-0.3}$ | 9.0$\pm 0.9\times 10^{-10}$* |  $\cdots$ | 4.6 | 180 $\pm 20$* | 16 | 16
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4.1${}^{+0.5}_{-0.4}\times 10^{-11}$* |  $\cdots$ |  $\cdots$ | 810 $\pm 81$* |  $\cdots$ |  $\cdots$

## Acknowledgements

COH is supported by NSERC Discovery Grant RGPIN-2016-04602. GRS and AJH are
supported by NSERC Discovery Grant RGPIN-2021-04001. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. This
research has made use of the CIRADA cutout service at URL cutouts.cirada.ca,
operated by the Canadian Initiative for Radio Astronomy Data Analysis
(CIRADA). CIRADA is funded by a grant from the Canada Foundation for
Innovation 2017 Innovation Fund (Project 35999), as well as by the Provinces
of Ontario, British Columbia, Alberta, Manitoba and Quebec, in collaboration
with the National Research Council of Canada, the US National Radio Astronomy
Observatory and Australia’s Commonwealth Scientific and Industrial Research
Organisation. This research also made use of TOPCAT (Taylor, 2005), software
designed to handle large astronomical datasets with virtual observatory tools.

## Data Availability

VLASS Quicklook data, including those used for this paper, are available via
https://cirada.ca/vlasscatalogueql0.

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An intrinsic characterization of Bruhat-Tits buildings inside analytic groups

Bertrand Rémy, Amaury Thuillier and Annette Werner

Abstract: Given a semisimple group over a complete non-Archimedean field, it
is well known that techniques from non-Archimedean analytic geometry provide
an embedding of the corresponding Bruhat-Tits builidng into the analytic space
associated to the group; by composing the embedding with maps to suitable
analytic proper spaces, this eventually leads to various compactifications of
the building. In the present paper, we give an intrinsic characterization of
this embedding.

Keywords: algebraic group, valued field, Berkovich analytic geometry, Bruhat-
Tits building.

AMS classification (2000): 20E42, 51E24, 14L15, 14G22.

Dedicated to Gopal Prasad with admiration

###### Contents

  1. 1 Affinoid subgroups potentially of Chevalley type
  2. 2 Galois-fixed points in buildings
  3. 3 Apartments
  4. 4 A reformulation in terms of norms

## Introduction

Bruhat-Tits theory ([BrT72], [BrT84]) deals with reductive groups over valued
fields. Given such a group ${\rm G}$ over a complete non-Archimedean field
$k$, the theory provides a certain cell complex $\mathcal{B}({\rm G},k)$,
called a _Euclidean_ or _affine building_ , on which the group of rational
points ${\rm G}(k)$ acts in a very balanced way (see [BrT84, Introduction] for
more precise conditions on the ground field). The space $\mathcal{B}({\rm
G},k)$ is large enough with respect to the group ${\rm G}(k)$ in the sense
that the group action is proper (i.e. cell stabilizers are compact subgroups),
and the group ${\rm G}(k)$ is large enough with respect to the space
$\mathcal{B}({\rm G},k)$ in the sense that the group action is _strongly
transitive_ (i.e. the action is transitive on the inclusions of a maximal cell
– called an _alcove_ – in a maximal flat subspace – called an _apartment_).

The latter property implicitly uses the fact that Euclidean buildings are
complete metric spaces with non-positive curvature properties, which makes
them relevant to geometric group theory [BH99]: this is the geometric side of
Bruhat-Tits theory. The second aspect of the theory is of arithmetic nature.
Namely, in order to perform the construction of the buildings, F. Bruhat and
J. Tits define and study integral models of the group ${\rm G}$: a posteriori,
each cell in the building – called a _facet_ – gives rise to such a model
whose group of integral points is, possibly up to finite index, the stabilizer
of the facet. These integral models turned out to be very useful in various
mathematical fields, such as representation theory ([MP94], [MP96]), so that
the viewpoint of integral models can be chosen as main viewpoint for Bruhat-
Tits theory, as shown by the upcoming book [KP21].

Bruhat-Tits buildings are singular spaces (from the viewpoint of differential
geometry for instance) since they are products of simplicial complexes.
Therefore it was a big surprise when V. Berkovich noticed that they could be
related to non-Archimedean analytic geometry from his perspective [Ber90,
Chapter 5]. We developed this approach and this led us to the construction of
families of compactifications by analogy with those associated to Riemannian
symmetric spaces ([RTW10], [RTW12], [RTW17], [Cha20]). For a semisimple group
${\rm G}$ over a complete non-Archimedean local field $k$, these
compactifications are defined to be the closures of embedding maps from
$\mathcal{B}({\rm G},k)$ to analytifications of proper ${\rm G}$-schemes (flag
varieties ${\rm G}/{\rm P}$ or the wonderful compactification $\overline{\rm
G}$ associated to ${\rm G}$). More precisely, the required maps are obtained
by constructing first a canonical ${\rm G}(k)$-equivariant embedding
$\vartheta:\mathcal{B}({\rm G},k)\rightarrow{\rm G}^{\rm an}$, where ${\rm
G}(k)$ acts on the analytification ${\rm G}^{\rm an}$ by conjugation, and then
by composing it with the analytication of the maps ${\rm G}\^{A}~\to{\rm
G}/{\rm P}$ (where ${\rm P}$ is a suitable parabolic subgroup) or of the de
Concini-Procesi map ${\rm G}\^{A}~\hookrightarrow\overline{\rm G}$.

In this paper, we go back to the canonical ${\rm G}(k)$-equivariant embedding
$\vartheta:\mathcal{B}({\rm G},k)\rightarrow{\rm G}^{\rm an}$ itself and
address the following basic problem about the relationship between Bruhat-Tits
and Berkovich theories: give an intrinsic description of the building
$\mathcal{B}({\rm G},k)$ as a subspace of the analytic group ${\rm G}^{\rm
an}$. Our answer is the content of Theorem 3.4; it is also rephrased in
Theorem 4.2. It is intrinsic in the sense that the information we use from a
point $z\in{\rm G}^{\rm an}$ to decide whether it belongs to the image of
$\vartheta$ or not, is its _holomorphic envelope_ in ${\rm G}^{\rm an}$
defined as ${\rm G}(z)=\\{y\in{\rm G}^{\rm an}\ ;\ \forall
f\in\mathcal{O}({\rm G}),\ |f(y)|\leqslant|f(z)|\\}$.

######  Theorem A

The image of the canonical embedding $\vartheta:\mathcal{B}({\rm
G},k)\hookrightarrow{\rm G}^{\rm an}$ is the subset of points $z\in{\rm
G}^{\rm an}$ whose holomorphic envelope ${\rm G}(z)$ is a $k$-affinoid
subgroup potentially of Chevalley type for which there exists a maximal torus
${\rm T}$ of ${\rm G}$ containing a maximal split torus and such that ${\rm
G}(z)\cap{\rm T}^{\rm an}$ is the maximal affinoid subgroup ${\rm T}^{1}$ of
${\rm T}^{\rm an}$.

Being potentially of Chevalley type is our first (natural) necessary
condition; it is explained in Section 1. In Section 2, we show why this
condition cannot be sufficient: this is related to the fact, well known when
performing descent of the ground field in Bruhat-Tits theory, that Galois
fixed points sets are in general bigger than the expected rational buildings.
The second condition deals with tori, or equivalently with apartments in the
building: the existence of a torus intersecting nicely with ${\rm G}(z)$
corresponds to the fact that the point $z$ does not belong to the undesirable
part of Galois fixed point sets. In Section 3 we explain that the combination
of these two conditions is a sufficient one in order to belong to the image of
$\vartheta$. In Section 4, we provide a reformulation of the above theorem
according to which the building of ${\rm G}(k)$ has a natural realization as a
space of multiplicative $k$-norms on the coordinate ring $\mathcal{O}({\rm
G})$ of ${\rm G}$.

In what follows, we denote by $k$ a field which is complete with respect to a
given non-Archimedean absolute value and we denote by ${\rm G}$ a semisimple
group defined over $k$. We use the functoriality of the building
$\mathcal{B}({\rm G},k)$ with respect to non-Archimedean extensions of $k$; it
is satisfied at least when ${\rm G}$ is split or when $k$ is discretely valued
with perfect residue field (see [RTW10, 1.3.3]). Each of the latter two
conditions also allow us to use the following fact: for any maximal split
torus ${\rm S}$ in ${\rm G}$ there are a maximal torus ${\rm T}$ containing
${\rm S}$ and a finite Galois extension $k^{\prime}/k$ splitting ${\rm T}$
such that, for the map given by the previous functoriality, the apartment of
${\rm T}\otimes_{k}k^{\prime}$ contains the apartment of ${\rm S}$ (this is
clear when ${\rm G}$ is split and it follows from [BrT84, Corollaire 5.1.12]
in the second case). At last, we denote by ${\rm G}^{\rm an}$ the analytic
space attached to ${\rm G}$: it is a $k$-analytic space in the sense of
Berkovich whose underlying topological space consists of the multiplicative
seminorms on the coordinate ring $\mathcal{O}({\rm G})$ inducing the initial
absolute value on $k$.

Acknowledgement: The third author was partially supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and
Arithmetic of Uniformized Structures, project number 444845124.

##  1 Affinoid subgroups potentially of Chevalley type

As already mentioned in the introduction, recall [RTW10, Theorem 2.1] that to
any point $x$ of $\mathcal{B}({\rm G},k)$ can be attached a $k$-affinoid
subgroup ${\rm G}_{x}$ of ${\rm G}^{\rm an}$ satisfying the following
condition: for any non-Archimedean extension $K/k$, the subgroup ${\rm
G}_{x}(K)$ of ${\rm G}(K)$ is the stabilizer of $x$ seen in the building
$\mathcal{B}({\rm G},K)$. By definition, the point $\vartheta(x)$ is the
unique element of the Shilov boundary of ${\rm G}_{x}$, i.e. the only point of
${\rm G}_{x}$ such that $|f(y)|\leqslant|f(\vartheta(x))|$ for any $y\in{\rm
G}_{x}$ and any $f\in\mathcal{O}({\rm G})$. Conversely, one can recover ${\rm
G}_{x}$ from $\vartheta(x)$ as its _holomorphic envelope_ [RTW10, Proposition
2.4 (ii)], which is to say:

${\rm G}_{x}=\\{y\in{\rm G}^{\rm an}\ ;\ \forall f\in\mathcal{O}({\rm G}),\
|f(y)|\leqslant|f(\vartheta(x))|\\}.$

This can be phrased equivalently in terms of multiplicative norms on
$\mathcal{O}({\rm G})$ by saying that one recovers the affinoid algebra of
${\rm G}_{x}$ as the completion of the normed $k$-algebra
$\left(\mathcal{O}({\rm G}),|.|(\vartheta(x)\right)$.

Let us say that a $k$-affinoid subgroup ${\rm H}$ of ${\rm G}^{\rm an}$ is of
_Chevalley type_ (or a _Chevalley $k$-affinoid subgroup_) if there exists a
$k^{\circ}$-Chevalley semi-simple group $\mathcal{H}$ [SGA 3, Exposé XXIII §5]
with $\mathcal{H}\otimes_{k^{\circ}}k\simeq G$ and such that $H$ is the
generic fibre of the formal completion of $\mathcal{H}$ along its special
fibre. More generally, we will say that an affinoid subgroup ${\rm H}$ of
${\rm G}^{\rm an}$ is _potentially_ of Chevalley type if there exists an
affinoid extension $K/k$ such that ${\rm H}\widehat{\otimes}_{k}K$ is a
Chevalley affinoid subgroup of ${\rm G}^{\rm an}\widehat{\otimes}_{k}K$ ; by
an _affinoid_ extension, we simply mean that $K$ is a non-Archimedean field
which is a $k$-affinoid algebra (see [RTW10, Appendix A]; this restriction
allows to recover $k$-affinoid algebras from $K$-affinoid algebras equipped
with a descent datum). By construction, the $k$-affinoid subgroup ${\rm
G}_{x}$ attached to a point $x$ of $\mathcal{B}({\rm G},k)$ is always
potentially of Chevalley type.

For a point $z$ of ${\rm G}^{\rm an}$, let us define its _holomorphic
envelope_ by

${\rm G}(z)=\\{y\in{\rm G}^{\rm an}\ ;\ \forall f\in\mathcal{O}({\rm G}),\
|f(y)|\leqslant|f(z)|\\}.$

The above discussion brings out a first condition fulfilled by any point of
${\rm G}^{\rm an}$ belonging to the image of $\vartheta$.

First condition.— _The holomorphic envelope of $z$ is a $k$-affinoid subgroup
potentially of Chevalley type._

It is easily checked that a point satisfying this condition does appear in the
image of $\vartheta$ over some non-Archimedean extension of $k$, as the next
Lemma shows.

######  Lemma 1.1

Let $z$ be a point of ${\rm G}^{\rm an}$ whose holomorphic envelope is a
$k$-affinoid subgroup potentially of Chevalley type.

  * (i)

For every non-Archimedean extension $K/k$, the Shilov boundary of the
$K$-affinoid domain $G(z)\widehat{\otimes}_{k}K$ is a singleton $\\{z_{K}\\}$.

  * (ii)

There exists a non-Archimedean extension $K/k$ such that the point $z_{K}$
belongs to the image of $\vartheta_{K}$.

_Proof.—_ (i) Let $K/k$ be a non-Archimedean extension. By assumption, there
exists an affinoid extension $K_{0}/k$ such that ${\rm
G}(z)\widehat{\otimes}_{k}K_{0}$ is a $K_{0}$-affinoid subgroup of Chevalley
type in ${\rm G}^{\rm an}\widehat{\otimes}_{k}K_{0}$. We consider a non-
Archimedean extension $L/k$ containing both $K_{0}$ and $K$. The set

${\rm G}(z)\widehat{\otimes}_{k}L=({\rm
G}(z)\widehat{\otimes}_{k}K_{0})\widehat{\otimes}_{K_{0}}L$

is a $L$-affinoid subgroup of Chevalley type in ${\rm G}^{\rm
an}\widehat{\otimes}_{k}L$, hence is the generic fibre of a formal scheme with
geometrically integral special fibre; it implies that the Shilov boundary of
${\rm G}(z)\widehat{\otimes}_{k}L$ is a singleton $\\{z_{L}\\}$. Since the
canonical projection map sends the Shilov boundary of ${\rm
G}(z)\^{A}~\widehat{\otimes}_{k}L$ onto the Shilov boundary of ${\rm
G}(z)\widehat{\otimes}_{k}K$, the latter is also a singleton. (ii) We consider
again an affinoid extension $K/k$ such that ${\rm G}(z)\widehat{\otimes}_{k}K$
is a $K$-affinoid subgroup of Chevalley type in ${\rm G}^{\rm
an}\widehat{\otimes}_{k}K$.

The $K$-affinoid Chevalley subgroup ${\rm G}(z)\widehat{\otimes}_{k}K$ is the
stabilizer of a unique point $x$ of $\mathcal{B}({\rm G},K)$, hence $({\rm
G}_{K})_{x}={\rm G}(z)\widehat{\otimes}_{k}K$ and therefore
$\vartheta_{K}(x)=z_{K}$. We used the fact that any $K$-affinoid Chevalley
subgroup of ${\rm G}^{\rm an}\widehat{\otimes}_{k}K$ occurs as the stabilizer
of some point in the building. To see this, just pick a special vertex; its
stabilizer is a $K$-affinoid Chevalley subgroup of ${\rm G}^{\rm
an}\widehat{\otimes}_{k}K$, and any two of them are ${\rm G}(K)$-conjugate.
$\square$

##  2 Galois-fixed points in buildings

It is clear that the above condition does not suffice to characterize the
image of $\vartheta$. Indeed, consider a finite Galois extension
$k^{\prime}/k$ and pick a point $x^{\prime}$ in $\mathcal{B}({\rm
G},k^{\prime})$ which is fixed under the natural action of ${\rm
Gal}(k^{\prime}|k)$ on the building. Let $z$ denote the image of
$\vartheta_{k^{\prime}}(x^{\prime})$ under the canonical projection of ${\rm
G}_{k^{\prime}}^{\rm an}$ onto ${\rm G}^{\rm an}$. The $k^{\prime}$-affinoid
subgroup $({\rm G}_{k^{\prime}})_{x^{\prime}}$ is equipped with a Galois
descent datum, from which one deduces: $({\rm
G}_{k^{\prime}})_{x^{\prime}}={\rm G}(z)\otimes_{k}k^{\prime}$. It follows
that ${\rm G}(z)$ is a $k$-affinoid subgroup potentially of Chevalley type.
Now, if the field extension $k^{\prime}/k$ is wildly ramified, then the
inclusion of $\mathcal{B}({\rm G},k)$ into the set of Galois-fixed points in
$\mathcal{B}({\rm G},k^{\prime})$ is strict in general (see example below);
therefore, if we pick a Galois-fixed point $x^{\prime}$ outside
$\mathcal{B}({\rm G},k)$, then $z$ does not belong to the image of
$\vartheta$. Therefore we see, up to illustrating the strict inclusions
mentioned above, that this argument shows that we need additional conditions
to describe the image of $\vartheta$ in ${\rm G}^{\rm an}$.

We want to illustrate this discussion by looking at an elementary example. Let
us consider the group ${\rm G}={\rm SL}_{2}$ over some discretely valued field
$k$ and pick a finite Galois extension $k^{\prime}$ of $k$. Via its canonical
embedding in $\mathbf{P}^{1,{\rm an}}_{k^{\prime}}$, the building
$\mathcal{B}({\rm G},k^{\prime})$ can be identified with the convex hull of
$\mathbf{P}^{1}(k^{\prime})$ inside $\mathbf{P}^{1,{\rm an}}_{k^{\prime}}$
with $\mathbf{P}^{1}(k^{\prime})$ omitted, i.e. with the subset

$\bigcup_{a\in k}\eta_{a}\left(\mathbf{R}_{>0}\right),$

where $\eta_{a}$ denotes the map from $\mathbf{R}_{>0}$ to $\mathbf{A}^{1,{\rm
an}}_{k}$ sending $r$ to the maximal point of the ball of radius $r$ centered
in $a$. The Galois action on $\mathcal{B}({\rm G},k^{\prime})$ is induced by
the Galois action on $\mathbf{P}^{1,{\rm an}}_{k^{\prime}}$, and the sub-
building $\mathcal{B}({\rm G},k)$ is the image of paths $\eta_{a}$ with $a\in
k$. Since the field $k$ is discretely valued – hence spherically complete –
there exists a well-defined Galois-equivariant retraction

$\tau:\mathbf{P}^{1,{\rm an}}_{k^{\prime}}-\mathbf{P}^{1,{\rm
an}}(k)\longrightarrow\mathcal{B}({\rm G},k)$

defined by sending a point $x$ to the maximal point of the smallest ball with
center in $k$ containing $x$.

Using this picture, one easily sees how a Galois-fixed point can appear in
$\mathcal{B}({\rm G},k^{\prime})-\mathcal{B}({\rm G},k)$. It suffices to find
an element $\alpha$ of $k^{\prime}$ such that all the paths
$\eta_{\alpha^{g}}(\mathbf{R}_{>0})$ issued from conjugates $\alpha^{g}$ of
$\alpha$ intersect at some point distinct form $\tau(\alpha)$; since the
Galois action permutes these paths, their meeting point $x^{\prime}$ will be
fixed. Note that we have

$x^{\prime}=\eta_{\alpha}(r)\ \ \ {\rm and}\ \ \
\tau(\alpha)=\eta_{\alpha}(r^{\prime}),$

where $r=\max\\{|\alpha^{g}-\alpha|\ ;\ g\in{\rm Gal}(k^{\prime}|k)\\}$ is the
diameter of the Galois orbit of $\alpha$ and $r^{\prime}=\min\\{|\alpha-a|\ ;\
a\in k\\}$ is the distance from $\alpha$ to $k$.

Let $k^{\prime}$ be any totally ramified finite Galois extension of $k$. It is
well-known that $k^{\prime}$ can be realized as the splitting field of some
Eisenstein polynomial ${\rm P}({\rm T})={\rm T}^{e}+a_{e-1}{\rm
T}^{e-1}+\ldots+a_{1}{\rm T}+a_{0}$, where $|a_{i}|\leqslant|a_{0}|<1$ for all
$i$ and $|k^{\times}|=|a_{0}|^{\mathbf{Z}}$ (see for example Chapter 7,
Theorem 7.1, in [Cas86]). The group $|{k^{\prime}}^{\times}|$ is generated by
$|\alpha|=|a_{0}|^{1/e}$ for any root $\alpha$ of ${\rm P}$. We have ${\rm
d}(\alpha,k)=|\alpha|$ and all conjugates of $\alpha$ are contained in the
closed ball ${\rm E}(\alpha,|\alpha|)={\rm E}(0,|\alpha|)$. The endomorphism
of $\mathbf{A}^{1,{\rm an}}_{k^{\prime}}$ defined by ${\rm P}({\rm T})$ maps
this ball onto the closed ball ${\rm E}(0,|a_{0}|)$. In order to study the
induced map ${\rm E}(0,|\alpha|)\rightarrow{\rm E}(0,|a_{0}|)$, set ${\rm
U}={\rm T}/\alpha$ and write

${\rm Q}({\rm U})=\frac{1}{a_{0}}{\rm P}(\alpha{\rm
U})=\frac{\alpha^{e}}{a_{0}}{\rm U}^{e}+\frac{a_{e-1}\alpha^{e-1}}{a_{0}}{\rm
U}^{e-1}+\ldots+\frac{a_{1}}{a_{0}}\alpha{\rm U}+1.$

Since $|a_{i}|.|\alpha|^{i}<|a_{i}|\leqslant|a_{0}|$ for any
$i\in\\{1,\ldots,e-1\\}$, the polynomial ${\rm Q}$ reduces to
$\widetilde{\frac{\alpha^{e}}{a_{0}}}{\rm U}^{e}+1=1-{\rm U}^{e}$ in
$\widetilde{k^{\prime}}[{\rm U}]$. It follows that the following four
conditions are equivalent:

  * -

all paths $\eta_{\alpha^{g}}(\mathbf{R}_{>0})$, for $g\in{\rm
Gal}(k^{\prime}|k)$, intersect outside $\mathcal{B}({\rm G},k)$;

  * -

all roots of ${\rm P}$ are contained in the _open_ ball ${\rm
D}(\alpha,|\alpha|)$;

  * -

all roots of ${\rm Q}$ are contained in the _open_ ball ${\rm D}(1,1)$;

  * -

$e$ vanishes in $\widetilde{k}$.

$\alpha^{g^{\prime}}$$\alpha^{g}$$\eta_{0}(|\alpha|)$Case $e\neq 0$ in
$\widetilde{k}$$\alpha$$\alpha^{g}$$\alpha^{g^{\prime}}$$\alpha$$\eta_{0}(|\alpha|)$Case
$e=0$ in $\widetilde{k}$$0$$0$$\infty$$\infty$

In particular, for any totally ramified (finite) Galois extension
$k^{\prime}/k$, the building $\mathcal{B}({\rm G},k)$ is strictly smaller than
the set of Galois-fixed points in $\mathcal{B}({\rm G},k^{\prime})$ if and
only if $[k^{\prime}:k]$ is divisible by the residue characteristic. We refer
to [Pra20] for a recent contribution to the question of describing fixed-point
sets of finite group actions in Bruhat-Tits theory.

######  Example 2.1

Let $k=\mathbf{Q}_{2}$ and $k^{\prime}=\mathbf{Q}_{2}(\alpha)$, where
$\alpha^{2}=2$. The two paths $\eta_{\alpha}(\mathbf{R}_{>0})$ and
$\eta_{-\alpha}(\mathbf{R}_{>0})$ intersect $\mathcal{B}({\rm G},k)$ along the
image of $[2^{-1/2},\infty)$, whereas they meet along the image of
$[2^{-3/2},\infty)$. The whole interval
$\eta_{\alpha}\left([2^{-3/2},2^{-1/2})\right)$ consists of Galois-fixed
points lying outside $\mathcal{B}({\rm G},k)$. In general, Rousseau gave an
upper bound for the distance of a Galois-fixed point in $\mathcal{B}({\rm
G},k^{\prime})$ to $\mathcal{B}({\rm G},k)$ in terms of the ramification of
$k^{\prime}/k$ [Rou77, Proposition 5.2.7].

##  3 Apartments

As we have just seen, the characterization of the image of the Bruhat-Tits
building of $\rm G$ inside ${\rm G}^{\rm an}$ requires an additional
condition. The goal of this section is to formulate such a condition which
involves maximal tori of ${\rm G}$. We will have to make use of the following
fact.

######  Lemma 3.1

Let ${\rm T}$ be a torus over $k$.

  * (i)

Its analytification ${\rm T}^{\rm an}$ contains a largest bounded subgroup
${\rm T}^{1}$. This is an affinoid subgroup, which coincides with the affinoid
domain cut out by the equations $|\chi|=1$ for all $\chi\in{\rm X}^{*}({\rm
T})$ if ${\rm T}$ is split.

  * (ii)

The Shilov boundary of ${\rm T}^{1}$ is reduced to a single point.

We will write ${\rm o}_{\rm T}$ for the unique Shilov boundary point of ${\rm
T}^{1}$.

_Proof.—_ We consider first the case of a split torus. If
$\chi_{1},\ldots,\chi_{n}$ is a basis of characters of ${\rm T}$, then the
equations $|\chi_{1}|=1,\ldots,|\chi_{n}|=1$ cut out a $k$-affinoid subgroup
${\rm T}^{1}$ of ${\rm T}^{\rm an}$ over which $|\chi|=1$ for any character
$\chi\in{\rm X}^{*}({\rm T})$. Let $K/k$ be a non-Archimedean extension and
$\Gamma$ a bounded subgroup of ${\rm T}(K)$. For any character $\chi$ of ${\rm
T}$, both $\chi(\Gamma)$ and $(-\chi)(\Gamma)=\chi(\Gamma)^{-1}$ are bounded
subgroups of $K^{\times}$, hence $|\chi(\Gamma)|=1$ and $\Gamma\subset{\rm
T}^{1}({\rm K})$. The Shilov boundary of ${\rm T}^{1}$ is reduced to a point
since the reduction of ${\rm T}^{1}$, a torus over $\widetilde{k}$, is
irreducible.

In general, pick a finite Galois extension $k^{\prime}/k$ splitting ${\rm T}$
and set ${\rm T}_{k^{\prime}}={\rm T}\otimes_{k}k^{\prime}$. The affinoid
subgroup ${\rm T}_{k^{\prime}}^{1}$ of ${\rm T}_{k^{\prime}}^{\rm an}$ is
stable under the natural Galois action on ${\rm T}_{k^{\prime}}^{\rm an}$,
hence its descends to a $k$-affinoid subgroup ${\rm T}^{1}$ of ${\rm T}$ such
that ${\rm T}_{k^{\prime}}^{1}=({\rm T}^{1})\otimes_{k}k^{\prime}$. Finally,
since the Shilov boundary of ${\rm T}^{1}\otimes_{k}k^{\prime}$ is the
preimage of the Shilov boundary of ${\rm T}^{1}$ under the canonical
projection, we see that ${\rm T}^{1}$ contains a unique Shilov boundary point.
$\Box$

Let us now go back to our discussion toward a characterization of the building
inside ${\rm G}^{\rm an}$. We assume temporarily that the group ${\rm G}$ is
split. Given a point $z$ in ${\rm G}^{\rm an}$ whose holomorphic envelope,
again denoted by ${\rm G}(z)$, is a $k$-affinoid subgroup potentially of
Chevalley type, let us consider a non-Archimedean extension $K/k$ such that
the canonical lifting $z_{K}$ of $z$ belongs to the image of $\vartheta_{K}$
and denote by $x$ its preimage: $z_{K}=\vartheta_{K}(x)$. Since the group
${\rm G}$ is split, the embedding $\mathcal{B}({\rm
G},k)\hookrightarrow\mathcal{B}({\rm G},K)$ identifies the left-hand side with
the union of apartments of all maximal split tori in ${\rm G}_{K}$ which are
defined over $k$. Therefore, in order to guarantee that the point $z$ itself
belongs to the image of $\vartheta$, we should require that $x$ belongs to the
apartment of a maximal split torus defined over $k$. The next proposition
translates this additional condition into appropriate terms.

######  Proposition 3.2

Let ${\rm S}$ be a maximal split torus and let $x$ be a point of
$\mathcal{B}({\rm G},k)$. The following conditions are equivalent:

  * (i)

the point $x$ belongs to the apartment ${\rm A}({\rm S},k)$;

  * (ii)

for every non-Archimedean extension $K/k$, the point $x$ is fixed by the ${\rm
S}^{1}(K)$-action on $\mathcal{B}({\rm G},K)$;

  * (iii)

the affinoid subgroup ${\rm G}_{x}$ of ${\rm G}^{\rm an}$ contains ${\rm
S}^{1}$.

Note that the proof will provide another equivalent statement, namely:

  * (ii)′

_for every finite unramified extension $k^{\prime}/k$, the point $x$ is fixed
by the ${\rm S}^{1}(k^{\prime})$-action on $\mathcal{B}({\rm G},k^{\prime})$_.

Proof.— Equivalence of assertions (ii) and (iii) follows immediately from the
definition of the affinoid group ${\rm G}_{x}$.

For any non-Archimedean extension $K/k$, the action of ${\rm S}(K)$ on
$\mathcal{B}({\rm G},K)$ preserves the apartment ${\rm A}({\rm S},K)$ and the
induced action of the maximal bounded subgroup ${\rm S}^{1}(K)$ is trivial
[RTW15, Proposition 4.18 (iii)]. Hence (i) implies (ii).

The only non trivial point to prove is that (ii) implies (i). We argue by
contraposition: given a point $x$ of the building which does not belong to the
apartment ${\rm A}={\rm A}({\rm S},k)$, we will exhibit a finite unramified
extension $k^{\prime}/k$ and an element $s$ of ${\rm S}^{1}(k^{\prime})$ such
that $s\cdot x\neq x$ in $\mathcal{B}({\rm G},k^{\prime})$.

Recall that the building $\mathcal{B}({\rm G},k)$ is also a complete metric
space satisfying a non-positive curvature property [BrT72, 2.5 and 3.2],
called by now the _${\rm CAT}(0)$ -property_ (see [BH99, Part II]). Since the
apartment ${\rm A}$ is a closed and convex subset, it contains a unique point
$x^{\prime}$ with

${\rm d}(x,{\rm A})={\rm d}(x,x^{\prime}).$

Moreover, there exists a unique geodesic segment $[x,x^{\prime}]$ between
those two points and

$[x,x^{\prime}]\cap{\rm A}=\\{x^{\prime}\\}.$

Consider any apartment ${\rm A}^{\prime}$ containing both $x$ and
$x^{\prime}$; it contains $[x,x^{\prime}]$ and intersects ${\rm A}$ along a
closed convex subset ${\rm C}$. The point $x^{\prime}$ coincides with the
projection in ${\rm A}^{\prime}$ of $x$ to ${\rm C}$, hence it lies in the
boundary of ${\rm C}$ since $x\not\in{\rm A}$. Finally, since ${\rm C}$ is the
intersection of a finite number of half-apartments [Rou09, Proposition 9.1],
we conclude that $x^{\prime}$ belongs to the wall ${\rm H}$ in ${\rm A}$
defined by the vanishing of some affine root $a$. Let ${\rm H}^{+}$ denote the
half-apartment on which $a$ is non-negative and write $a=\alpha+\lambda$,
where the linear part $\alpha$ belongs to the root system $R({\rm T},{\rm G})$
and $\lambda$ is an element of ${\rm log}|k^{\times}|$.

Recall now that the unipotent root group ${\rm U}_{\alpha}$ is endowed with a
separated, exhaustive and decreasing filtration $\\{{\rm
U}_{\alpha,s}\\}_{s\in\mathbf{R}}$ by affinoid subgroups [RTW15, 4.2.1]. The
subgroup ${\rm U}_{a}={\rm U}_{\alpha,\lambda}$ corresponding to the affine
root $a=\alpha+\lambda$ has the following geometric interpretation: for every
non-Archimedean extension $K/k$, the action of ${\rm U}_{a}(K)$ on
$\mathcal{B}({\rm G},K)$ fixes pointwise the half-apartment ${\rm H}^{+}$ and
is transitive on the set of apartments which intersect ${\rm A}$ along ${\rm
H}^{+}$. Moreover, if we set

${\rm U}_{\alpha,\lambda}^{+}=\bigcup_{r>\lambda}{\rm U}_{\alpha,r},$

then ${\rm U}_{\alpha,\lambda}(K)/{\rm U}_{\alpha,\lambda}^{+}(K)$ is in
bijection with equivalence classes of apartments in $\mathcal{B}({\rm G},K)$
containing ${\rm H}^{+}$, where two such apartment are said to be equivalent
they intersect along a neighborhood of ${\rm H}^{+}$.

The torus ${\rm S}$ acts on ${\rm U}_{\alpha}$ by conjugation. The bounded
subgroup ${\rm S}^{1}$ preserves each step of the filtration and there is a
non-canonical bijection

${\rm U}_{\alpha,\lambda}(K)/{\rm
U}_{\alpha,\lambda}^{+}(K)\simeq\widetilde{K}$

such that the action of an element $s\in{\rm S}^{1}(K)$ on the left-hand side
corresponds to multiplication by
$\widetilde{\alpha(s)}\in\widetilde{K}^{\times}$ on the right-hand side. Note
that this condition implies that the unit element on the left-hand side
corresponds to $0$ on the right-hand side.

To conclude the proof, observe that there exists a unique element
$\xi\in\widetilde{k}$ satisfying the following property: for every $u\in{\rm
U}_{\alpha,\lambda}(k)$ whose class modulo ${\rm U}^{+}_{\alpha,\lambda}(k)$
corresponds to $\xi$, the apartment $u\cdot{\rm A}$ contains a germ of
$[x,x^{\prime}]$ at $x^{\prime}$, i.e. we may write $u\cdot{\rm
A}\cap[x,x^{\prime}]=[x,y]$ for some $y\in[x,x^{\prime})$. We have $\xi\neq 0$
since $x$ does not belong to ${\rm A}$. Now, if $x$ is fixed by ${\rm
S}^{1}(k)$, then $\widetilde{\alpha(s)}\cdot\xi=\xi$ and therefore
$\widetilde{\alpha(s)}=1$ for any $s\in{\rm S}^{1}(k)$. This means that the
character $\widetilde{\alpha}$ of the $\widetilde{k}$-torus $\widetilde{\rm
S^{1}}$ is trivial at the level of $\widetilde{k}$-points. Since the character
$\widetilde{\alpha}$ is non-trivial, there exists a finite separable extension
$\ell$ of $\widetilde{k}$ such that $\widetilde{\alpha}\neq 1$ on
$\widetilde{\rm S}(\ell)$. Denoting by $k^{\prime}$ the corresponding
unramified extension of $k$, we conclude that $x$ is not fixed by the whole
group ${\rm S}^{1}(k^{\prime})$. $\square$

######  Remark 3.3

The arguments above are close to the ones proving that the apartment ${\rm
A}({\rm S},k)$ coincides with the fixed-point set of ${\rm S}^{1}(k)$ if
$\widetilde{k}$ contains at least $4$ elements [BrT84, 5.1.37].

In the split case, the discussion above shows precisely which additional
condition is required in order to characterize the image of $\vartheta$ in
${\rm G}^{\rm an}$: _there exists a maximal split torus ${\rm S}$ in ${\rm G}$
such that ${\rm G}(z)\cap{\rm S}={\rm S}^{1}$_. In general, we impose the
following:

Second condition.— _There exists a maximal torus ${\rm T}$ in ${\rm G}$ which
contains a maximal split torus and such that ${\rm G}(z)\cap{\rm T}={\rm
T}^{1}$._

Once this second condition has been introduced naturally, we are in good
position to characterize the image of $\vartheta$ in the analytic space ${\rm
G}^{\rm an}$ of ${\rm G}$.

######  Theorem 3.4

The image of the canonical embedding $\vartheta:\mathcal{B}({\rm
G},k)\hookrightarrow{\rm G}^{\rm an}$ is the subset of points $z$ satisfying
the following two conditions:

  * 1.

the holomorphic envelope ${\rm G}(z)$ of $z$ is a $k$-affinoid subgroup
potentially of Chevalley type;

  * 2.

there exists a maximal torus ${\rm T}$ of ${\rm G}$ containing a maximal split
torus and such that ${\rm G}(z)\cap{\rm T}^{\rm an}$ is the maximal affinoid
subgroup ${\rm T}^{1}$ of ${\rm T}^{\rm an}$.

_Proof.—_ We have already seen that the first condition is necessary. The same
holds for the second one. Given a point $x\in\mathcal{B}({\rm G},k)$, there
exists a maximal split torus ${\rm S}$ and a maximal torus ${\rm T}$
containing ${\rm S}$ such that $x\in{\rm A}({\rm S},k)\subset{\rm A}({\rm
T},k^{\prime})$, where $k^{\prime}$ is a finite extension of $k$ which splits
${\rm T}$. It follows that ${\rm G}_{x}\otimes_{k}k^{\prime}$ contains the
bounded torus ${\rm T}^{1}\otimes_{k}k^{\prime}$, hence ${\rm
T}^{1}\subset{\rm G}_{x}$.

Now, let us show that the two conditions are sufficient. We consider a point
$z\in{\rm G}^{\rm an}$ satisfying the two conditions and pick a finite Galois
extension $k^{\prime}/k$ spliting the maximal torus $T$ given by the second
condition, as well as a non-Archimedean extension $K/k^{\prime}$ and a point
$w\in\mathcal{B}({\rm G},K)$ such that $z_{K}=\vartheta_{K}(w)$, where $z_{K}$
denotes the (unique) Shilov boundary point of $G(w)\widehat{\otimes}_{k}K$.

Since

${\rm T}^{\rm an}_{K}\cap\left({\rm G}_{K}\right)_{x}={\rm T}^{\rm
an}_{K}\cap{\rm G}(z)_{K}=\left({\rm T}^{\rm an}\cap{\rm
G}(z)\right)_{K}=\left({\rm T}^{1}\right)_{K}=\left({\rm T}_{K}\right)^{1},$

it follows from Proposition 3.2 that $w$ belongs to the apartment of ${\rm
T}_{K}$. The canonical embedding of $\mathcal{B}({\rm G},k^{\prime})$ into
$\mathcal{B}({\rm G},K)$ induces a bijection between ${\rm A}({\rm
T},k^{\prime})$ and ${\rm A}({\rm T},K)$, hence $w$ comes from a point $y$ of
$\mathcal{B}({\rm G},k^{\prime})$ contained in ${\rm A}({\rm T},k^{\prime})$.
Moreover $({\rm G}_{k^{\prime}})_{y}=H_{k^{\prime}}$, for both sides coincide
with $({\rm G}_{K})_{w}$ after base change to $K$. In particular, this shows
that $y$ is fixed by ${\rm Gal}(k^{\prime}|k)$. Since ${\rm A}({\rm
T},k^{\prime})^{{\rm Gal}(k^{\prime}|k)}$ is the image of ${\rm A}(S,k)$ in
$\mathcal{B}({\rm G},k^{\prime})$, we get a point $x$ in ${\rm A}({\rm S},k)$
such that

$({\rm G}_{z_{k^{\prime}}})\otimes_{k}k^{\prime}=({\rm
G}_{k^{\prime}})_{y}=H_{k^{\prime}},$

hence $H={\rm G}_{x}$ and $z=\vartheta(x)$. $\Box$

##  4 A reformulation in terms of norms

The above characterization of points of ${\rm G}^{\rm an}$ lying inside the
building $\mathcal{B}({\rm G},k)$ (identified with its image by the canonical
map $\vartheta$) can be conveniently rephrased in terms of (multiplicative)
$k$-norms on the coordinate algebra $\mathcal{O}({\rm G})$. As we are going to
explain, we need here to make the additional assumption that the group ${\rm
G}$ splits over a _tamely ramified_ extension of $k$.

Let us start by recalling the notion of a _universal point_ 111This notion was
introduced by Berkovich, who used the adjective _peaked_ [Ber90, 5.2]; its
study was carried on by Poineau, who preferred the adjective _universal_
[Poi13].. Let $z$ be a point in ${\rm G}^{\rm an}$, seen as a multiplicative
$k$-seminorm on $\mathcal{O}({\rm G})$. For a given non-Archimedean field
extension $K/k$, there is a natural $K$-seminorm $||.||=x\otimes 1$ on
$\mathcal{O}({\rm G})\otimes_{k}K$, defined by

$||a||=\inf\max_{i}|a_{i}(z)|\cdot|\lambda_{i}|$

where the infimum is taken over the set of all expressions
$\sum_{i}a_{i}\otimes\lambda_{i}$ representing $a$, with
$a_{i}\in\mathcal{O}({\rm G})$ and $\lambda_{i}\in K$. The point $z$ is said
to be _universal_ if, for any non-Archimedean field extension $K/k$, the above
$K$-seminorm on $\mathcal{O}({\rm G})\otimes_{k}K$ is multiplicative. One then
writes $z_{K}$ for the corresponding point in ${\rm G}^{\rm
an}\widehat{\otimes}_{k}K$.

######  Remarks 4.1

  1. 1.

Obviously, points of ${\rm G}^{\rm an}$ coming from $k$-rational points of
${\rm G}$ are universal.

  2. 2.

Let $x\in{\rm G}^{\rm an}$ be universal. For any finite Galois extension
$k^{\prime}/k$, the canonical extension $x_{k^{\prime}}$ of $x$ to ${\rm
G}^{\rm an}\otimes_{k}k^{\prime}$ is invariant under the action of ${\rm
Gal}(k^{\prime}/k)$: indeed, the $k^{\prime}$-norm $x\otimes 1$ on
$\mathcal{O}({\rm G})\otimes_{k}k^{\prime}$ is Galois invariant.

  3. 3.

If ${\rm G}$ is split, then $\vartheta(x)$ is universal for every point $x$ of
$\mathcal{B}({\rm G},k)$, as follows readily from the explicit description of
$\vartheta(x)$ as a norm on the coordinate algebra of a big cell [RTW10,
Proposition 2.6]. More generaly, the same is true if ${\rm G}$ splits over a
tamely ramified extension $k^{\prime}/k$ by [RTW10, Lemma A.10] and [RTW15,
Erratum].

  4. 4.

Given a torus ${\rm T}$ over $k$, we denote by $\sigma_{{\rm T}}$ the (unique)
Shilov boundary point of the maximal affinoid subtorus ${\rm T}^{1}$ of ${\rm
T}^{\rm an}$. This point is universal if $T$ splits over a tamely ramified
extension of $k$, but not in general as checked by Mayeux [May20, Proposition
10.2].

Setting — _As already mentioned, the semisimple group ${\rm G}$ considered in
this last section is assumed to split over a tamely ramified extension of $k$,
so that all points in the image of $\vartheta$ are universal._ Let ${\rm
G}^{\rm an}_{u}$ denote the subset of universal points in ${\rm G}^{\rm an}$.
Following Berkovich [Ber90, 5.2], there is a natural _monoid_ structure on
${\rm G}^{\rm an}_{u}$ extending the group structure on ${\rm G}(k)$. Given
any two points $g,h\in{\rm G}_{u}^{\rm an}$, the seminorm $g\otimes h$ on
$\mathcal{O}({\rm G})\otimes_{k}\mathcal{O}({\rm G})$ is multiplicative and
one defines $g\ast h$ as the induced multiplicative seminorm on
$\mathcal{O}({\rm G})$ via the comultiplication map $\Delta:\mathcal{O}({\rm
G})\rightarrow\mathcal{O}({\rm G})\otimes_{k}\mathcal{O}({\rm G})$. This
binary operation is associative, with unit the element $1\in{\rm G}(k)$;
moreover, it is obvious that we recover the group law if $g$ and $h$ belong to
${\rm G}(k)$.

More generally, given an (analytic) action of ${\rm G}^{\rm an}$ on some
$k$-analytic space ${\rm X}$, one can define in a similar way an action of the
monoid ${\rm G}^{\rm an}_{u}$ on the topological space underlying ${\rm X}$,
which extends the action of ${\rm G}(k)$. In particular, $x\ast y$ is well-
defined for any two points $x,y$ in ${\rm G}^{\rm an}$ such that one of them
is universal.

Finally, we also recall that one can define a partial order on the set
underlying ${\rm G}^{\rm an}$ as follows:

$x\preccurlyeq y\ \ \ {\rm if\ and\ only\ if}\ \ \forall a\in\mathcal{O}({\rm
G}),\ |a(x)|\leqslant|a(y)|.$

We can now give the following description of the building inside ${\rm G}^{\rm
an}$.

######  Theorem 4.2

Let $x\in{\rm G}^{\rm an}$. Then the point $x$ belongs to the image of the
canonical embedding $\vartheta:\mathcal{B}({\rm G},k)\hookrightarrow{\rm
G}^{\rm an}$ if, and only if, the following conditions are satisfied:

  * (i)

$x$ is universal;

  * (ii)

$x\ast x\preccurlyeq x$ and ${\rm inv}(x)\preccurlyeq x$;

  * (iii)

there exists a maximal torus ${\rm T}$ containing a maximal split torus such
that ${\rm o}_{\rm T}\preccurlyeq x$;

  * (iv)

$x$ is maximal with respect to the three conditions above.

######  Remarks 4.3

  1. 1.

Conditions (i)-(iv) imply that $x$ is a multiplicative $k$-_norm_ on
$\mathcal{O}({\rm G})$.

  2. 2.

One way to understand condition (ii) is to say that $x$ defines a
$k$-multiplicative norm on the commutative Hopf algebra $\mathcal{O}({\rm G})$
with respect to which commultiplication and the antipode are bounded. One
should observe that (ii) obviously implies $1_{\rm G}\preccurlyeq x$, so that
the counit is also bounded.

  3. 3.

By Lemma 4.4 below, the condition $1_{\rm G}\preccurlyeq x$ (resp. ${\rm
inv}(x)\preccurlyeq x$) implies $x=1_{\rm G}\ast x\preccurlyeq x\ast x$ (resp.
$x\preccurlyeq{\rm inv}(x)$), hence condition (ii) could be replaced by $x\ast
x=x$ and ${\rm inv}(x)=x$.

######  Lemma 4.4

Let $x,y,x^{\prime},y^{\prime}$ be four points in ${\rm G}^{\rm an}$ such that
both sets $\\{x,y\\}$ and $\\{x^{\prime},y^{\prime}\\}$ contain at least one
universal point (so that $x\ast y$ and $x^{\prime}\ast y^{\prime}$ are well-
defined). If $x\preccurlyeq x^{\prime}$ and $y\preccurlyeq y^{\prime}$, then
$x\ast y\preccurlyeq x^{\prime}\ast y^{\prime}$ and ${\rm
inv}(x)\preccurlyeq{\rm inv}(x^{\prime})$.

_Proof.—_ This follows directly from the formulas

$\forall a\in\mathcal{O}({\rm G}),\ \ \ |a(x\ast
y)|=\inf\max_{i}|b_{i}(x)|\cdot|c_{i}(y)|\ \ {\rm and}\ \ |a(x^{\prime}\ast
y^{\prime})|=\inf\max_{i}|b_{i}(x^{\prime})|\cdot|c_{i}(y^{\prime})|,$

where the infimum is taken over the set of all expressions
$\sum_{i}b_{i}\otimes c_{i}$ representing $\Delta(a)$, and

$\forall a\in\mathcal{O}({\rm G}),\ \ \ |a({\rm inv}(x))|=|{\rm
inv}(a)(x)|\leqslant|{\rm inv}(a)(x^{\prime})|=|a({\rm inv}(x^{\prime}))|$.
$\Box$

######  Lemma 4.5

Let $x$ be a universal point of ${\rm G}^{\rm an}$. We again denote by ${\rm
G}(x)=\\{z\in{\rm G}^{\rm an}\ ;\ z\preccurlyeq x\\}$ its holomorphic
envelope. Then the following conditions are equivalent:

  * (i)

The envelope ${\rm G}(x)$ is a subgroup object of ${\rm G}^{\rm an}$, i.e.
${\rm G}(x)({\rm K})$ is a subgroup of ${\rm G}({\rm K})$ for any non-
Archimedean extension $K/k$;

  * (ii)

The point $x$ satisfies: $1_{\rm G}\preccurlyeq x,\ \ \ {\rm
inv}(x)\preccurlyeq x\ \ {\rm and}\ \ \ x\ast x\preccurlyeq x$.

Moreover, ${\rm G}(x)$ is bounded in ${\rm G}^{\rm an}$.

_Proof.—_ Assume that ${\rm G}(x)$ is a subgroup object of ${\rm G}^{\rm an}$.
Since ${\rm G}(x)(k)$ is a subgroup of ${\rm G}(k)$, it contains the unit
element $1_{\rm G}$ and therefore $1_{\rm G}\preccurlyeq x$. Now we consider
the canonical point $\underline{x}\in{\rm G}(\mathcal{H}(x))$ lying over $x$;
we have $|a(x)|=|a(\underline{x})|$ for any $a\in\mathcal{O}({\rm G})$, as
well as

$|a(x\ast x)|=|a(\underline{x}\cdot\underline{x})|\ \ \ {\rm and}\ \ \ |a({\rm
inv}(x))|=|a(\underline{x}^{-1})|.$

Since $\underline{x}\cdot\underline{x}$ and $\underline{x}^{-1}$ belong to
${\rm G}(x)(\mathcal{H}(x))$, it follows that

$|a(x\ast x)|\leqslant|a(x)|\ \ \ {\rm and}\ \ \ |a({\rm
inv}(x))|\leqslant|a(x)|$

for all $a\in\mathcal{O}({\rm G})$, which is to say $x\ast x\preccurlyeq x$
and ${\rm inv}(x)\preccurlyeq x$.

We assume now that $x$ is a universal point of ${\rm G}^{\rm an}$ satisfying
the conditions of (ii). Obviously, ${\rm G}(x)$ contains the $k$-rational
point $1_{\rm G}$. Given a non-Archimedean extension $K/k$ and elements
$g,h\in{\rm G}(x)(K)$, we have for all $a\in\mathcal{O}({\rm G})$:

$|a(g^{-1})|=|{\rm inv}(a)(g)|\leqslant|{\rm inv}(a)(x)|=|a({\rm inv}(x))|$

and

$|a(gh)|=|\Delta(a)(g,h)|\leqslant\inf\max_{i}|b_{i}(g)|\cdot|c_{i}(h)|\leqslant\inf\max_{i}|b_{i}(x)|\cdot|c_{i}(x)|=|a(x\ast
x)|,$

where the infimum is taken over the set of all expressions
$\sum_{i}b_{i}\otimes c_{i}$ representing $\Delta(a)$. Since ${\rm
inv}(x)\preccurlyeq x$ and $x\ast x\preccurlyeq x$, we deduce

$|a(g^{-1})|\leqslant|a(x)|\ \ \ {\rm and}\ \ \ |a(gh)|\leqslant|a(x)|,$

hence $g^{-1},gh\in{\rm G}(x)({\rm K})$. This proves that ${\rm G}(x)$ is a
subgroup object of ${\rm G}^{\rm an}$.

Finally, boundedness is obvious: if $f_{1},\ldots,f_{n}$ is a finite set
generating $\mathcal{O}({\rm G})$ as a $k$-algebra, then we have
$|f_{i}(y)|\leqslant\max_{i}|f_{i}(x)|$ for any point $y\in{\rm G}(x)$. $\Box$

_Proof of Theorem 4.2.—_ In what follows, we identify the building
$\mathcal{B}({\rm G},k)$ with its image in ${\rm G}^{\rm an}$ by the embedding
$\vartheta$.

If a point $x$ of ${\rm G}^{\rm an}$ belongs to $\mathcal{B}({\rm G},k)$, then
$x$ is universal by Remark 4.1 and ${\rm G}(x)$ is a $k$-affinoid subgroup of
${\rm G}^{\rm an}$, hence $x\ast x\preccurlyeq$ and ${\rm inv}(x)\preccurlyeq
x$ by Lemma 4.5. Moreover, there exists a maximal torus ${\rm T}$ containing a
maximal split torus and such that ${\rm G}(x)\cap{\rm T}^{\rm an}={\rm
T}^{1}$, which amounts to saying that $x$ dominates the distinguished point
${\rm o}_{\rm T}$ of ${\rm T}$. Finally, consider a universal point $z\in{\rm
G}^{\rm an}$ satisfying condition (ii) and dominating $x$ (which implies that
$z$ dominates $1_{\rm G}$). For any non-Archimedean extension $K/k$, Lemma 4.5
implies that ${\rm G}(z)(K)$ is a bounded subgroup of ${\rm G}(K)$ containing
${\rm G}(x)(K)$; by maximality of the latter, we deduce ${\rm G}(z)(K)={\rm
G}(x)(K)$, hence ${\rm G}(x)={\rm G}(z)$ and $z=x$. We have thus checked that
$x$ satisfies conditions (i)-(iv).

Conversely, let $x$ be a point in ${\rm G}^{\rm an}$ satisfying conditions
(i)-(iv). We observe that condition (iv) implies that $x$ dominates $1_{\rm
G}$, hence it follows from Lemma 4.5 that ${\rm G}(x)(K)$ is a bounded
subgroup of ${\rm G}(K)$ for any non-Archimedean extension $K/k$. We are going
to see that all these subgroups fix a common point in $\mathcal{B}({\rm
G},k)$. By condition (iii), there exists a maximal torus ${\rm T}$ of ${\rm
G}$ containing a maximal split torus ${\rm S}$ and such that
$\sigma_{T}\preccurlyeq x$, hence ${\rm T}^{1}(K)\subset{\rm G}(x)(K)$ for
every non-Archimeden extension $K/k$.

We first assume that ${\rm T}$ is split and that $\widetilde{k}$ contains at
least 4 elements. For any non-Archimedean field extension $K/k$, we identify
${\rm A}({\rm T},k)$ with ${\rm A}({\rm T},K)$ in $\mathcal{B}({\rm G},K)$,
and we let $\mathcal{F}(K)$ denote the fixed-point set of ${\rm G}(x)(K)$ in
$\mathcal{B}({\rm G},K)$. This is a non-empty closed subset, which is
contained in the apartment ${\rm A}({\rm T},k)$ since ${\rm G}(x)(K)$ contains
the group of units ${\rm T}^{1}(K)$ and there are at least 4 elements in
$\widetilde{K}$. Considering in particular the extension $\mathcal{H}(x)/k$,
we get a point $z\in{\rm A}({\rm T},k)$ fixed under the canonical element
$\underline{x}\in{\rm G}(\mathcal{H}(x))$ lifting $x$. This means that
$\underline{x}$ belongs to ${\rm G}(z)(\mathcal{H}(x))$, or equivalently that
$x$ is contained in the $k$-affinoid subgroup ${\rm G}(z)$. This amounts to
saying that $z$ dominates $x$, hence $x=z$ by maximality.

In general, we consider a finite Galois extension $k^{\prime}/k$ which splits
${\rm T}$ and such that $\widetilde{k^{\prime}}$ contains at least four
elements. It follows from the argument above that the canonical extension
$x_{k^{\prime}}$ of $x$ to ${\rm G}^{\rm an}\otimes_{k}k^{\prime}$ belongs to
the apartment of ${\rm T}\otimes_{k}k^{\prime}$ in $\mathcal{B}({\rm
G},k^{\prime})$. Since this point is invariant under the action of ${\rm
Gal}(k^{\prime}/k)$ by Remark 4.1 2, $x_{k^{\prime}}$ belongs to the image of
${\rm A}({\rm S},k)$ in ${\rm A}({\rm T},k^{\prime})$, and therefore $x$
belongs to ${\rm A}({\rm S},k)$. $\Box$

## References

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Bertrand Rémy

Unité de Mathématiques Pures et Appliquées (UMR 5669)

École normale supérieure de Lyon / CNRS / Inria

46 allée d’Italie

F-69364 Lyon cedex 07

<EMAIL_ADDRESS>
Amaury Thuillier

Institut Camille Jordan (UMR 5208)

Université de Lyon / Université Lyon 1 / CNRS

43 boulevard du 11 novembre 1918

F-69622 Villeurbanne cedex

<EMAIL_ADDRESS>
Annette Werner

Institut für Mathematik

Goethe-Universität Frankfurt

Robert-Mayer-Str. 6-8

D-60325 Frankfurt-am-Main

<EMAIL_ADDRESS>