# A Readily Implemented Atmosphere Sustainability Constraint for Terrestrial
Exoplanets Orbiting Magnetically Active Stars

Evangelia Samara  Royal Observatory of Belgium, 1180 Brussels, Belgium  Centre
for Mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium  Spiros
Patsourakos  Department of Physics, University of Ioannina, 45110 Ioannina,
Greece  Manolis K. Georgoulis  Research Center of Astronomy and Applied
Mathematics (RCAAM) of the Academy of Athens, 11527 Athens, Greece

###### Abstract

With more than 4,300 confirmed exoplanets and counting, the next milestone in
exoplanet research is to determine which of these newly found worlds could
harbor life. Coronal Mass Ejections (CMEs), spawn by magnetically active,
superflare-triggering dwarf stars, pose a direct threat to the habitability of
terrestrial exoplanets as they can deprive them from their atmospheres. Here
we develop a readily implementable atmosphere sustainability constraint for
terrestrial exoplanets orbiting active dwarfs, relying on the magnetospheric
compression caused by CME impacts. Our constraint focuses on a systems
understanding of CMEs in our own heliosphere that, applying to a given
exoplanet, requires as key input the observed bolometric energy of flares
emitted by its host star. Application of our constraint to six famous
exoplanets, (Kepler-438b, Proxima-Centauri b, and Trappist-1d, -1e, -1f and
-1g), within or in the immediate proximity of their stellar host’s habitable
zones, showed that only for Kepler-438b might atmospheric sustainability
against stellar CMEs be likely. This seems to align with some recent studies
that, however, may require far more demanding computational resources and
observational inputs. Our physically intuitive constraint can be readily and
en masse applied, as is or generalized, to large-scale exoplanet surveys to
detect planets that could be sieved for atmospheres and, perhaps, possible
biosignatures at higher priority by current and future instrumentation.

Exoplanets — Habitable zone — Solar coronal mass ejections — Solar flares —
Stellar coronal mass ejections — Stellar flares

††facilities: The properties of the exoplanets and host-stars used in this
study are available at the NASA Exoplanet Archive
(https://exoplanetarchive.ipac.caltech.edu/).††software: The code to implement
the mASC for any terrestrial-like exoplanet lying within the tidally locked
regime of its host star is available in the following repository:
https://github.com/SamaraEvangelia/mASC-method

##  1 Introduction

Planets beyond our solar system have become an object of fascination in recent
decades, with nearly regular references in headlines and the popular media.
Only recently have observational capabilities evolved to the point where
potential terrestrial planets are detected around M-type dwarf stars. However,
young M-type dwarfs are known to be magnetically active, often more than our
middle-aged Sun. Superflares in them is a common occurrence (e.g., Maehara et
al., 2012; Armstrong et al., 2016) that should be resulting in fast and
massive coronal mass ejections (CMEs; see, e.g., Khodachenko et al., 2007;
Lammer et al., 2007; Vidotto et al., 2011, and several others). CMEs are
gigantic, eruptive expulsions of magnetized plasma and helical magnetic fields
from the solar and stellar coronae at speeds that may well surpass local
Alfvénic and sound speeds, severely but temporarily disrupting stellar winds
and generating shocks around their bodily ejecta.

Contrary to solar flares that are known since the 19th century (Carrington,
1859), CMEs were only observed well into the space age (Howard, 2006) due to
their much fainter magnitude compared to the bright solar photospheric disk.
Stellar CME detections are notoriously ambiguous, although recent efforts
offer reasonable hope (Argiroffi et al., 2019). However, in strongly
magnetized stellar coronae CMEs are inevitable. In case of intense stellar
magnetic activity and the existence of an atmosphere that shields a planet,
extreme pressure effects by CMEs owning to stellar mega-eruptions can, under
certain circumstances, cause intense atmospheric depletion via ionization-
triggered erosion (e.g., Zendejas et al., 2010). In the solar system, results
from NASA’s Mars Atmosphere and Volatile Evolution (MAVEN) mission seem to
establish that the sustained eroding effect of solar interplanetary CMEs
(ICMEs) may be responsible for the thin Martian atmosphere (Jakosky et al.,
2015) after the planet’s magnetic field weakened.

Our objective here is the introduction of a practical and reproducible
(magnetic) atmosphere sustainability constraint (mASC), reflected on a
positive, rational number $\mathcal{R}$ and relying on CME and planetary
magnetic pressure effects. $\mathcal{R}$ is a dimensionless ratio that, in
tandem with the considered habitability zone (HZ; e.g., Kopparapu et al.,
2013), can provide an understanding of which terrestrial exoplanets warrant
further screening for the existence of a possible atmosphere
($\mathcal{R}<1$), or otherwise ($\mathcal{R}>1$). Our mASC becomes fully
constrained in case of tidally locked exoplanets (e.g., Grießmeier et al.,
2004) by means of an estimated planetary magnetic field, while if no tidal
locking is assumed the planetary magnetic field can be replaced by a known
benchmark field, such as Earth’s or other. Only magnetic pressure effects are
taken into account in this initial study, but our mASC $\mathcal{R}$ can be
generalized at will. Given that pressure effects are extensive and additive,
however, adding more terms (i.e., kinetic, thermal) to the CME pressure will
only increase the adversity of possible atmospheric erosion effects for a
studied exoplanet (e.g., Ngwira et al., 2014).

##  2 (Magnetic) Atmosphere Sustainability Constraint (mASC)

The magnetic activity of the exoplanets’ host stars reflects on several
observational facts, including the bolometric energy of their flares. From it,
and assuming a Sun-as-a-Star analogue further reflected on the solar magnetic
energy – helicity (EH) diagram (Tziotziou et al., 2012), we estimate the
magnetic helicity of stellar CMEs and a corresponding stellar CME magnetic
field based on the fundamental principle of magnetic helicity conservation in
high magnetic Reynolds number plasmas (Patsourakos et al., 2016; Patsourakos &
Georgoulis, 2017). As explained in Appendix A, CMEs are inevitable in strongly
magnetized stellar coronae as they relieve stars from excess helicity that
cannot be shed otherwise.

The near-star CME magnetic field is propagated self-similarly in the
astrosphere until it reaches exoplanet orbits (Patsourakos & Georgoulis,
2017). The mASC introduced in this study achieves a precise, quantitative
assessment of whether the magnetic pressure of stellar ICMEs alone can be
balanced by the estimated (tidally locked) or guessed (in the general case)
magnetic pressure of _observed_ terrestrial exoplanets at a magnetopause
distance large enough to avert erosion effects of a possible atmosphere.

There are two conceptual pillars of the mASC introduced here: first, it relies
on observed stellar flare energies but does not perform a case-by-case stellar
eruption analysis. In other words, it does not look at the particular eruption
but points to the collective effect of numerous similar eruptions over the
$\sim$1 Gyear, or significant fractions thereof, of the young star’s activity.
In this sense, the suitable orientation required for an enhanced ICME planeto-
effectiveness, namely, the ICMEs’ ability to reconnect with the planetary
magnetic field causing magnetic storms, is ignored: it is implicitly assumed
that numerous such ICMEs will have the correct orientation. Second, our mASC
ratio relies explicitly on a worst-case scenario (i.e., largest possible ICME
magnetic field strength) magnetic pressure for stellar CMEs ($\sim
B_{worst}^{2}$) _and_ a best-case scenario planetary magnetic field (i.e.,
largest possible planetary magnetic field) generated in the planet’s core due
to internal dynamo action ($\sim B_{best}^{2}$). Then, our mASC becomes the
ratio of planetary magnetic intensities relating to these pressure terms: the
minimum planetary magnetic field (equal to $B_{worst}$) able to balance the
worst-case CME magnetic field and the best-case planetary magnetic field
$B_{best}$ as per the modeled planetary characteristics, i.e.,

$\mathcal{R}={{B_{worst}}\over{B_{best}}}$ (1)

The two planetary magnetic field strengths are taken at a critical
magnetopause distance from the studied planet in terms of atmospheric erosion
effects (Sections 2.1, 2.2). It is then understood that if $\mathcal{R}>1$,
the planet’s magnetosphere will be compressed beyond the critical threshold,
presumably leading to atmospheric erosion (after processes such as thermal,
nonthermal or hydrodynamic escape, catastrophic erosion and others, take
place; see, for example, Melosh & Vickery (1989); Lundin et al. (2004);
Barabash et al. (2007) for more details). Assuming that, statistically, the
presumed ICME is not a unique occurrence, the planet may undergo this
atmospheric stripping for hundreds of millions of years due to its star’s
magnetic activity. Magnetic helicity conservation, on the other hand, dictates
that at least some (or even most, or all) magnetic eruptions in the star will
unleash powerful CMEs to shed away the excess helicity generated in the star’s
magnetized atmosphere that is otherwise conserved and remains accumulated in
the star’s corona. In such situations, one casts doubt on the viability of an
atmosphere in the otherwise terrestrial planet, even if the planet is seated
well into the HZ of its astrosphere. The opposite is the case for
$\mathcal{R}<1$.

###  2.1 The worst-case CME-equipartition magnetic field

A critical magnetopause distance of $r_{mp}=2\;R_{p}$ (i.e., two planetary
radii, or one radius away from the surface of the planet, see Khodachenko et
al. (2007); Lammer et al. (2007)) was adopted as the minimum planetocentric
distance in which atmospheric ionization and erosion can still be averted
during extreme magnetospheric compression. Then, the _equipartition_ planetary
magnetic field $B_{eq}$ (to be viewed as $B_{worst}$) that balances the ICME
magnetic pressure at $r_{mp}=2\;R_{p}$ can be estimated as (see Appendix B for
a complete description)

$B_{eq}=8\;B_{icme}\;\;,$ (2)

where $B_{icme}$ is the worst-case ICME axial magnetic field at
$r_{mp}=2R_{p}$.

To infer $B_{icme}$ at any given astrocentric distance $r_{icme}$, we first
need to constrain the axial magnetic field $B_{0}$ of the CME at a near-star
distance $r_{0}$ (see Appendix A for a derivation). $B_{0}$ is constrained by
observational facts and more specifically by assuming a given flux rope model
and a corresponding magnetic helicity formula depending on the radius $R$ and
length $L$ of the flux rope, that can then be solved for $B_{0}$ (Patsourakos
et al., 2016). Patsourakos & Georgoulis (2017) tested several linear (LFF),
nonlinear (NLFF) and non-force-free flux rope models and determined that the
worst-case scenario $B_{0}$ for near-Sun CME flux ropes was obtained by the
LFF Lundquist flux-rope model (values estimated by other models range between
2-80 $\%$ of the Lundquist values) that gives a magnetic helicity of the form

$H_{m}={{4\pi B_{0}^{2}L}\over{\alpha}}\int_{0}^{R}J_{1}^{2}(\alpha
r)rdr\;\;,$ (3)

where $\alpha$ is the constant force-free parameter and $J_{1}()$ is the
Bessel function of the first kind. Parameter $\alpha$ is inferred by the
additional constraint $\alpha R\simeq 2.405$, imposed by the first zero of the
Bessel function of the zero-th kind, $J_{0}()$, in the Lundquist model. The
flux rope radius $R$ corresponds to the CME front that is assumed to have a
circular cross-section with maximum area.

We use the Lundquist flux-rope model throughout this analysis, as this is a
standard ICME model for the inner heliosphere. It gives the strongest near-Sun
CME axial magnetic field $B_{0}$, provided that the twist is not excessive
(see Patsourakos & Georgoulis (2017)). Adopting the fundamental helicity
conservation principle for high magnetic Reynolds number plasmas (e.g.,
Berger, 1984) implies a fixed $H_{m}$ and dictates that as the CME expands,
$B_{0}$ decreases self-similarly as a function of $1/r^{2}$, with $r$ being
the heliocentric distance. We maintain this quadratic scaling for distances
relatively close to the Sun (e.g., up to the Alfvénic surface where the solar
wind speed matches the local Alfvén speed at $\sim 10\;R_{\odot}$). Beyond
that surface this analysis continues to adhere to helicity conservation but
adopts a power-law radial fall-off index $a_{B}=1.6$ for the propagation of
Lundquist-flux-rope solar CMEs within the astrospheres (see Appendix A for a
derivation of the exponent). As a result, $B_{icme}$ at a given astrocentric
distance $r_{icme}$ is given by

$B_{icme}=B_{0}({{r_{0}}\over{r_{icme}}})^{1.6}$ (4)

where $r_{0}$ is the near-star distance up to which the CME axial magnetic
field scales as $(1/r^{2})$ and $B_{0}$ is this magnetic field at that
distance.

###  2.2 The best-case planetary magnetic field

The ’defense’ line in the ICME pressure effects for any given planet is being
held primarily by the planet’s magnetic pressure. The planet’s dipole magnetic
moment $\mathcal{M}$ gives rise to a planetary magnetic field

$B_{p}={{\mathcal{M}}\over{r_{mp}^{3}}}$ (5)

for the dayside magnetopause occurring at a planetocentric distance $r_{mp}$.
To identify the best-case scenario, we examined several prominent models for
the magnetic moment $\mathcal{M}$ (Busse, 1976; Stevenson et al., 1983;
Mizutani et al., 1992; Sano, 1993) –see also Christensen (2010) for a review–
to determine which would provide the strongest $\mathcal{M}$, focusing
particularly on the tidally locked regime. We concluded that an upper-case
$\mathcal{M}$ is provided by Stevenson et al. (1983) and a model variant of
Mizutani et al. (1992), namely

${\mathcal{M}}=\mathcal{M}_{Stev}\simeq
A\;\rho_{c}^{1/2}\omega^{1/2}R_{c}^{3}\;\sigma^{-1/2}.$ (6)

The other models provided values ranging between 18-62 $\%$ of the Stevenson
value. In Equation (6), $A\simeq 3.45\times 10^{5}$ A$\cdot$s$\cdot$kg-0.5 is
the proportionality constant, $\omega$ corresponds to the planet’s angular
rotation and $\rho_{c}$, $R_{c}$ and $\sigma$ correspond to the planetary
core’s mean density, radius and electrical conductivity, respectively (see
Appendix C for more details on the calculations and assumptions made). $B_{p}$
for $\mathcal{M}$ = $\mathcal{M}_{Stev}$ ($B_{Stev}$, hereafter) is to be
viewed as $B_{best}$.

Summarizing, our mASC ratio of Equation (1) translates to
$\mathcal{R}=(B_{eq}/B_{Stev})$. $B_{eq}$ can be estimated in case of known
bolometric stellar flare energies, while $B_{Stev}$ is fully constrained for
tidally locked exoplanets.

##  3 Application of the mASC method

Figure 1: The mASC ratio $\mathcal{R}=(B_{eq}/B_{Stev})$ for Earth twins, but
with different equatorial magnetic fields, lying on the inner (red) and outer
(blue) HZ boundary inferred by Kopparapu et al. (2013) for different stellar
flare energies, each represented by a different curve thickness. The
astrocentric distance in the abscissas implicitly includes the stellar mass
shown in Figure 2. The limit $\mathcal{R}=1$ (dotted lines) separates an
apparent non-viability of an atmosphere ($\mathcal{R}>1$) from an apparently
likely atmosphere ($\mathcal{R}<1$) for both cases.

Assuming tidal locking to fully determine $B_{Stev}$, Figure 1 provides the
nominal values of $\mathcal{R}$ for different stellar flare energies and for
an Earth-like exoplanet lying precisely at the inner (red) and outer (blue) HZ
boundary of Kopparapu et al. (2013). This plot represents a different
conception of Figure 2 that shows a number of exoplanets provided by the NASA
Exoplanet Archive lying within and without the tidally locked regime and
within and without the inner and outer HZ limits. Nevertheless, Figure 1 now
includes $\mathcal{R}$ as a function of stellar flare energies while stellar
masses are implicitly included in the astrocentric distances $d_{inner}$ and
$d_{outer}$. It comes as no surprise that higher flare energies, statistically
resulting in more helical CMEs and stronger axial magnetic fields, require the
planet to be orbiting its host star at a larger orbital distance to achieve
$\mathcal{R}<1$. For flare energies higher than $10^{33}$ erg it appears that
planets located precisely on the inner HZ boundary may be incapable to sustain
an atmosphere while this is the case for energies higher than $10^{34}$ erg
for planets located on the outer HZ boundary.

Figure 2: An ensemble of 1,771 confirmed exoplanets by the NASA Exoplanet
Archive plotted on a diagram of stellar mass vs. astrocentric distance. We
restrict the analysis to stars less or equally massive to the Sun while
exoplanets shown have a confirmed orbital semi-major axis. Inner and outer HZ
limits are indicated by solid lines enclosing the pink-shaded HZ area, while
the black and gray dashed lines indicate internal and potential external
tidal-locking limits, respectively. The locations of nine exoplanets, six of
which are studied here due to their proximity to, or presence within, the HZ,
are also highlighted.

Importantly, our mASC method can also be applied to case studies of
terrestrial exoplanets, provided that flares from their host stars are
observed. If these planets are –or are assumed to be– tidally locked, then
$\mathcal{R}$ becomes fully constrained. In Figure 3, we examine six popular
cases of terrestrial exoplanets, all within the tidal-locking zone and either
within, or close to, the respective HZ of their host stars. These cases are
Kepler-438b (K438b), Proxima-Centauri b (PCb), and four Trappist-1 (Tp1)
exoplanets, namely Tp1d, Tp1e, Tp1f and Tp1g. Actual mASC $\mathcal{R}$-values
and applicable uncertainties (see Appendix D) are provided in Table 1. Figure
3 offers a graphical representation of these results where observed flare
energies from host stars have been taken from Armstrong et al. (2016), Howard
et al. (2018) and Vida et al. (2017) for Keppler-438, Proxima Centauri and
Trappist-1, respectively.

Figure 3: Testing the viability of a potential atmosphere from the value of
the mASC $\mathcal{R}$ for six confirmed exoplanets, namely (clockwise from
top left) K438b, PCb, Tp1e, Tp1g, Tp1f and Tp1d. Ordinates correspond to
stellar flare energies and abscissas to astrocentric distances. The color
scales correspond to $\log({\mathcal{R}})$. Each planet is represented by a
point corresponding to coordinates set by the maximum observed flare energies
from its host star and confirmed orbital distances, highlighted by vertical
lines.

The uncertainty analysis mentioned above and described in Appendix D aims to
determine under which circumstances could a conclusion of $\mathcal{R}<1$ or
$\mathcal{R}>1$ be reversed due to applicable uncertainties. In particular, we
define an _equipartition_ scaling index $a_{B_{(\mathcal{R}=1})}$ for which
$\mathcal{R}=1$, along with its uncertainty $\delta a_{B}$. If (i)
$|a_{B}-a_{B_{(\mathcal{R}=1})}|>\delta a_{B}$ for the nominal $a_{B}=1.6$ of
Patsourakos & Georgoulis (2016, 2017) or (ii) $a_{B_{(\mathcal{R}=1})}$ is
steeper than 2 beyond applicable $\delta a_{B}$, with 2 being the ’vacuum’
radial fall-off index near the star and $a_{B}<2$ reflecting an astrosphere
with its own MHD environment, then our conclusion of $\mathcal{R}<1$ or
$\mathcal{R}>1$ is unlikely to change. If uncertainties are large enough to
preclude a safe conclusion, then our main result on an exoplanet is likely to
change because of these uncertainties. Analogous uncertainties may be sought
in the case of additional pressure effects included in $\mathcal{R}$.

Table 1 shows that for all six presumed tidally locked exoplanets the 1:1
spin-orbit resonance results in planetary rotations that are slow enough to
allow $\mathcal{R}>1$, rendering a sustainable atmosphere unlikely. In all
cases, $a_{B_{(\mathcal{R}=1)}}$ is well above the ’vacuum’ value of 2, with
PCb and the four Tp-1 exoplanets showing $a_{B_{(\mathcal{R}=1)}}>2$ beyond
applicable uncertainties. Hence, for five out of six potentially tidally
locked exoplanets, our result for $\mathcal{R}>1$, meaning a non-sustainable
atmosphere, seems robust. The result could be reversed for K438b as the
difference between $a_{B_{(\mathcal{R}=1)}}$ and 1.6 is within applicable
uncertainties, meaning that if $a_{B}$ systematically lies in the range
$(1.6,2.0)$ then the exoplanet might conceivably sustain an atmosphere.

Exoplanet | Abridged | $\mathcal{R}$ | Atmosphere | $a_{B_{(\mathcal{R}=1)}}$ | $\delta a_{B}$ | Result
---|---|---|---|---|---|---
|  |  | likely? |  |  | robust?
Keppler-438b | K438b | 5.46 | No | 2.48 | 0.99 | No
Proxima Centauri b | PCb | 42.84 | No | 3.48 | 0.96 | Yes
Trappist-1d | Tp1d | 95.35 | No | 4.95 | 1.40 | Yes
Trappist-1e | Tp1e | 77.87 | No | 4.24 | 1.15 | Yes
Trappist-1f | Tp1f | 70.61 | No | 3.80 | 0.99 | Yes
Trappist-1g | Tp1g | 48.53 | No | 3.43 | 0.90 | Yes
Table 1: Six case studies of the mASC $\mathcal{R}$-value as depicted in
Figure 3. The values of $\mathcal{R}$ correspond to our nominal radial power-
law fall-off index $a_{B}=1.6$. The index $a_{B_{(\mathcal{R}=1)}}$
corresponds to the equipartition $a_{B}$-value for which $\mathcal{R}=1$,
while $\delta a_{B}$ corresponds to the uncertainty of
$a_{B_{(\mathcal{R}=1)}}$. The last column assesses whether our result is
robust as per the difference $a_{B_{(\mathcal{R}=1)}}\pm\delta a_{B}$ from the
nominal $a_{B}$-value.

The above underline the potential value of the mASC $\mathcal{R}$, even in its
simplest form involving only magnetic pressure effects: lying in the HZ of
their host stars does not necessarily make exoplanets capable of sustaining an
atmosphere, a favorable Earth Similarity Index (ESI; Schulze-Makuch et al.
(2011)) notwithstanding. This would be the case for PCb and Tp1d,e,f, and g
although virtually all lie in the HZ. Conversely, K438b is slightly beyond the
inner HZ that might inhibit an atmosphere and possible liquid water on its
surface but, at the same time, due to its larger orbital distance it might be
relatively immune to at least plausible, as per flare observations, space
weather from its host star.

##  4 Conclusions

This versatile and highly reproducible analysis shows that space weather
cannot be left out of considerations for planetary habitability in stellar
systems. It carries both value and promise: consider the European Space Agency
CHEOPS (Characterizing Exoplanet Satellite) mission, for example ( e.g., Sulis
et al., 2020). The mission is designed to characterize only _selected_ ,
confirmed exoplanets, ranging from super-Earth to Neptune sizes, aiming toward
studies that extend into their potential atmospheres. Although the mission is
already in orbit, analyses such as this could help assess future observing
priorities. The same applies to optimizing exoplanet selection for
biosignature analysis in the framework of the upcoming James Webb mission.

In anticipation of these exciting observations that will give the ultimate
test to our method, we hereby supply a first round of tentative tests aiming
towards its validation. For example, it was recently shown that LHS 3844b
(Vanderspek et al., 2019; Kane et al., 2020), a rocky exoplanet orbiting a
M-dwarf star within its tidally locked zone, lacks an atmosphere. Kane et al.
(2020) suggest that the mother star of LHS 3844b exhibited an active past,
comparable to that of Proxima Centauri. By adopting, therefore, a maximum
super-flare energy identical to the one of Proxima Centauri, we obtained R =
251.08 ($>$ 1, atmosphere unlikely) which agrees with the observations. Also,
our result seems robust because $|a_{B}-a_{B_{(\mathcal{R}=1})}|>\delta
a_{B}$.111For this specific exoplanet which orbits very close to its mother
star, we maintain the quadratic scaling of $B_{0}$ with astrocentric distance
($~{}1/r^{2}$) for distances up to 7$R{{}_{*}}$ and not 10$R{{}_{*}}$. Beyond
7$R{{}_{*}}$, we adopt the same power-law radial fall-off index $a_{B}$ = 1.6
for the propagation of Lundquist-flux-rope solar CMEs within the astrospheres.
Our mASC could be indirectly validated as well, by checking if other studies –
focusing also on tidally locked, terrestrial worlds and having similar
objectives – converge on similar results. This said, results presented here
are in qualitative agreement with MHD simulations of stellar winds, for
example by Garraffo et al. (2017), that find extreme magnetospheric
compressions below $\approx$ 2.5 planetary radii for Tp1d-g and a magnetopause
distance of $[1.5,4.5]$ planetary radii for PCb. Recent extensive (and
computationally expensive) MHD models of stellar CMEs (e.g., Lynch et al.,
2019) along with semi-empirical approaches (e.g., Kay et al., 2016) have
emerged, and both approaches take as input maps of the stellar surface
magnetic field inferred by Zeeman-Doppler imaging reconstructions (e.g.,
Donati & Brown, 1997). While detailed, these studies may be rather impractical
for bulk application to a large number of exoplanets. On the other hand, our
introduced mASC could provide guidance to large-scale MHD simulations of
stellar CMEs, by efficiently scanning and bracketing the corresponding
parameter space so that the simulations could be performed only to pertinent
cases.

Concluding, we reiterate that mASC $\mathcal{R}$ can be generalized at will
with additional pressure terms, albeit mainly from the ICME side. In its most
general form, $\mathcal{R}=P_{eq}/P_{planet}$, with worst- and best-case
scenario pressure terms $P_{eq}$ and $P_{planet}$, respectively. We note, in
particular, the study of Moschou et al. (2019) where a (flare) energy vs.
(CME) kinetic energy diagram is inferred from stellar observations and
modeling. Such statistics could be integrated into the energy-helicity diagram
of this study to revise the $P_{eq}$-term. The energy-helicity diagram used
here is also an entirely solar one, so any possibilities to extend it to
better reflect the magnetic activity of dwarf, planet-prolific stars are well
warranted. Equally meaningful $P_{planet}$-terms could be introduced to
provide a far more sophisticated, but still readily achieved, mASC ratio
$\mathcal{R}$ for the screening of alien terrestrial worlds for potential
atmospheres and, ultimately, life.

The authors would like to thank the anonymous referee for his comments and
suggestions that improved the manuscript. This work was inspired by and
originated during the M.Sc. Thesis of ES, implemented at the University of
Athens and the Research Center for Astronomy and Applied Mathematics of the
Academy of Athens, Greece. We thank both institutions for their support and
encouragement. The authors would also like to extend their acknowledgements to
Prof. Dr. Stefaan Poedts from the Centre of Mathematical Plasma-Astrophysics
(CMPA) KU Leuven, for his support and great incitement on this work.

##  Appendix A Near- Sun/star CMEs and helio-/astro-spheric propagation

A physically meaningful expression of magnetic helicity in the Sun and
magnetically active stars is the relative helicity, related to the absence of
vacuum and hence the flow of electric currents in the solar and stellar
coronae, along with the topological settings of solar (and stellar) magnetic
fields that are only partially observed and detected, on and above the stars’
surface. By construction, the relative helicity must be quantitatively
connected to the excess or free magnetic energy that is also explicitly due to
the presence of electric currents (e.g., Sakurai, 1981). An attempt to
correlate the free magnetic energy with the relative magnetic helicity in
solar active regions was taken by Georgoulis & LaBonte (2007) for linear
force-free (LFF) magnetic fields, and then by Georgoulis et al. (2012) for
nonlinear force-free (NLFF) ones.

The NLFF (magnetic) energy - (relative) helicity correlation resulted in the
energy - helicity (EH) diagram of Tziotziou et al. (2012). There, $\sim$160
vector magnetograms of observed solar active regions were treated in the
homogeneous way of Georgoulis et al. (2012), aiming toward a scaling between
the NLFF free magnetic energy $E_{c}$ and relative helicity $H_{m}$. They
found a robust scaling of the form (CGS units)

$\log\mid H_{m}\mid\propto 53.4-0.0524(\log
E_{c})^{0.653}\exp{{{97.45}\over{\log E_{c}}}}\;\;.$ (A1)

Here $\mid H_{m}\mid$ refers to the magnitude of the relative helicity
$H_{m}$, that can be right- (+) or left- (–) handed. A similar expression to
Equation (A1) provides a simpler, power-law dependence between $\mid
H_{m}\mid$ and $E_{c}$, of the form (CGS units)

$\mid H_{m}\mid\propto 1.37\times 10^{14}E_{c}^{0.897}.$ (A2)

The above EH scaling was shown to hold for typical active-region free energies
in the range $E_{c}\sim(10^{30},10^{33})$ erg and respective relative helicity
budgets $\mid H_{m}\mid\sim(10^{40},10^{44})$ Mx2. The robustness of the EH
diagram scaling was validated in multiple cases that involved not only active
regions but also quiet-Sun regions and magnetohydrodynamic models (Tziotziou
et al., 2013).

Combining the EH scaling with the conservation principle of the relative
magnetic helicity, $E_{c}$ in Equations (A1) and (A2) is dissipated in every
instability (solar flare, CME, etc.), while $H_{m}$ is only shed away from the
Sun via CMEs. If an active region was to completely relax (i.e., return to the
vacuum energy state) by a single magnetic eruption, then for a given free
magnetic energy $E_{c}$ it would expel helicity $H_{m}$. This is the core
reasoning behind a worst-case-scenario solar eruption originating from a given
solar source. Typically, up to $\sim$10% of the total free energy and up to 30
– 40% of the total magnetic helicity of the source are dissipated and ejected,
respectively, in a solar eruption (see, e.g., Nindos et al., 2003; Moraitis et
al., 2014).

The majority of CME ejecta are observed to be in the form of a magnetic flux
rope (e.g., Vourlidas et al., 2013, 2017), with this geometry surviving the
CMEs’ inner heliospheric propagation all the way to the Sun-Earth libration
point L1 and probably beyond (e.g., Zurbuchen & Richardson, 2006; Nieves-
Chinchilla et al., 2018). Based on the flux rope CME geometry, tools such as
the Graduated Cylindrical Shell (GCS) model of Thernisien et al. (2009);
Thernisien (2011), processed observations by the STEREO/SECCHI coronagraph
(Howard et al., 2008) to obtain the aspect ratio $k=R/L$ between the radius
$R$ and along with the half-angular width $w$ of the CME flux rope at some
distance (typically, 10 solar radii, $R_{\odot}$) away from the Sun. The
length $L$ corresponds to the perimeter of the flux rope while the radius $R$
corresponds to the CME front, that is assumed to have a circular cross-section
with maximum area.

The worst-case scenario $B_{0}$ for near-Sun CME flux ropes was provided by
the LFF Lundquist flux-rope model which was used throughout this analysis and
gives a magnetic helicity of the form

$H_{m}={{4\pi B_{0}^{2}L}\over{\alpha}}\int_{0}^{R}J_{1}^{2}(\alpha
r)rdr\;\;,$ (A3)

where $\alpha$ is the constant force-free parameter and $J_{1}()$ is the
Bessel function of the first kind. Parameter $\alpha$ is inferred by the
additional constraint $\alpha R\simeq 2.405$, imposed by the first zero of the
Bessel function of the zero-th kind, $J_{0}()$, in the Lundquist model.

As already mentioned, for helicity conservation imposing a fixed $H_{m}$, the
Lundquist model requires that as the CME expands, $B_{0}$ decreases self-
similarly as a function of $1/r^{2}$ for distances close to the Sun. In this
analysis however and for distances away from the Sun (see also Patsourakos &
Georgoulis (2016)), the self-similar expansion was not a priori assumed to be
quadratic (i.e., $1/r^{2}$), but more generally, $(1/r^{a_{B}})$, with $a_{B}$
being the absolute value of the power-law radial fall-off index. This
different index, still under helicity conservation that dictates respective
power laws for the increase (expansion) of the CME flux rope $R$ and $L$,
aimed to include all effects present during heliospheric propagation and the
interaction of ICMEs with the ambient solar wind (see, e.g., Manchester et
al., 2017, for an account of these effects). Prominent among them is the CME
flattening (see, e.g., Raghav & Shaikh, 2020, and references therein) that
tends to distort the CME geometry due to plasma draping or flux-pileup as the
CME pushes through the heliospheric spiral. As a result, inner-heliospheric
propagation implies that $B_{icme}$ at a given heliocentric distance
$r_{icme}$ is given by

$B_{icme}=B_{0}({{r_{0}}\over{r_{icme}}})^{a_{B}}$ (A4)

where $r_{0}$ is the (near-Sun) distance up to which the CME axial magnetic
field scales as $(1/r^{2})$ and $B_{0}$ is this magnetic field at that
distance.

This analysis adopts $a_{B}=1.6$ in Equation (A4) for the propagation of
Lundquist-flux-rope stellar CMEs within their astrospheres (where $a_{B}$ can
vary in the range [1.34, 2.16], see Salman et al. (2020)). A Monte Carlo
simulation for various stellar $(k,w)$-pairs is adopted, while the stellar CME
$\mid H_{m}\mid$ is inferred by the bolometric, observed stellar flare
energies via Equation (A2). A valid question is where in the near-star space
are we to apply the model $(k,w)$, that were taken at $10\;R_{\odot}$:
assuming a $10\;R_{*}$ astrocentric distance, where $R_{*}$ is the radius of
the host star, we should apply a ’fudge-factor’ correction to $B_{0}$ as
follows:

$B_{0_{(10\;R_{*})}}=B_{0_{(10\;R_{\odot})}}({R_{\odot}\over{R_{*}}})^{2}\;\;.$
(A5)

This factor is adopted for the cases of this study’s six exoplanets and their
host stars. In the general case, or where no assessment is taken for the
stellar radius, one may start the astrospheric propagation at a physical
astrocentric distance of $10\;R_{\odot}$, independently from the stellar
radius (provided, of course, that this radius does not exceed 10 $R_{\odot}$).

Figure 4: Mean (main plot) and median (inset) values of the near-star CME
axial magnetic field $B_{0}$ at a physical distance of 10 $R_{\odot}$ from the
star. The modeled data are shown by blue points, while least-squares power-law
best fits are shown by red curves. Standard deviations around each
$B_{0}$-value are represented by cyan segments.

Figure 4 provides the mean $B_{0_{(10\;R_{\odot})}}$ of the above-mentioned
Monte Carlo simulation for different flare energies $E_{c}$, along with a
standard deviation around these values (cyan ranges). The inset in Figure 4
provides the respective median values. We notice that both mean and median
$B_{0_{(10\;R_{\odot})}}$ can be adequately modeled by power laws of the flare
energy $E_{c}$ of the form

$B_{0_{(10\;R_{\odot})}}\;(G)\;=\;f\;E_{c}^{0.23}\;(erg)\;\;,$ (A6)

where the proportionality constant $f$ is $3.30\times 10^{-9}$ (median) or
$5.94\times 10^{-9}$ (mean), with an uncertainty amplitude 1.29 $\times
10^{-8}$.

##  Appendix B ICME equipartition magnetic field

Figure 5 provides the worst-case scenario axial magnetic field of ICME
magnetic flux ropes (as virtually all CMEs/ICMEs are expected to be) as a
function of solar/stellar flare energies and helio-/astro-centric distances.
The ICME magnetic pressure effects on the planet will be determined from this
magnetic field strength.

Figure 5: Worst-case scenario axial magnetic field for ICME magnetic flux
ropes as a function of source flare energy and astrocentric distance, up to
one astronomical unit (AU). A wide range of flare energies are provided, from
flares observed in the Sun (i.e., up to $10^{33}$ erg) to orders of magnitude
stronger superflares with energies up to $10^{36}$ erg.

Textbook physics dictates that either nominal solar/stellar winds or ’stormy’
ICMEs exert pressure that comprises magnetic (i.e., $B^{2}/(8\pi)$), kinetic
(i.e., ram; $\rho\upsilon^{2}$) and thermal (i.e., $n\;k\;T$) terms as per the
local magnetohydrodynamical (MHD) environment (plasma number density, mass
density and speed, $n$, $\rho$ and $\upsilon$, respectively, and magnetic
field $B$). At the planet’s dayside at few to several planetary radii, the
only non-negligible pressure term is magnetic pressure stemming from the
planetary magnetosphere. This is because a possible atmosphere typically wanes
at the planetary thermosphere or exosphere, already at small fractions of a
planetary radius. In a seminal early work, Chapman & Ferraro (1930) considered
a spherical magnetosphere interfacing with interplanetary ejecta. Adopting
this working hypothesis and taking only the magnetic pressure term of the ICME
determined by its characteristic field $B_{icme}$, the existence of a
magnetopause at planetocentric distance $r_{mp}$ implies an equilibrium of the
form (Patsourakos & Georgoulis, 2017)

${{B_{icme}^{2}}\over{8\pi}}={{B_{mp}^{2}}\over{8\pi}}({{1}\over{r_{mp}}})^{6}\;\;,$
(B1)

where $r_{mp}$ is expressed in planetary radii and $B_{mp}$ is the planetary
magnetic field at the magnetopause. We now adopt a critical magnetopause
distance, that is, the minimum planetocentric distance at extreme compression
in which atmospheric ionization and erosion can still be averted at
$r_{mp}=2\;R_{p}$ (i.e., two planetary radii, or one radius away from the
surface of the planet). This is a limit already adopted by several previous
studies (Khodachenko et al., 2007; Lammer et al., 2007). Therefore, the
_equipartition_ planetary magnetic field $B_{eq}$ that balances the ICME
magnetic pressure at $r_{mp}=2\;R_{p}$ is

$B_{eq}=8\;B_{icme}\;\;,$ (B2)

from Equation (B1). This equipartition field for the same range of flare
energies and astrocentric distances (viewed in this case as the exoplanets’
mean orbital distances in circular orbits) as in Figure 5 is provided in
Figure 6. Equation (B2) and Figure 6, therefore, provide the requirement for
the planetary magnetic field such that planets avoid atmospheric erosion due
to a given ICME magnetic pressure.

It would worth mentioning at this point, that the toroidal component of the
interplanetary magnetic field could become important in the case of fast
rotators such as M-stars (e.g., Petit et al. (2008), Kay et al. (2016),
Villarreal D’Angelo et al. (2019)). In such case, an extra term is added in
the pressure-balance equation at the sub-stellar point. However, we note that
in the case of active M-dwarf stars the notion of a Parker spiral for the
associated interplanetary magnetic field may not be fully applicable, given
the much frequent CME activity than in the solar case, that could keep the
spiral significantly perturbed virtually at all times.

Figure 6: Same as in Figure 5 but showing the equipartition magnetic field of
the planet at the adopted atmospheric-erosion critical threshold of two
planetary radii. An indication labeled ’Earth’ in the color bar shows the
uncompressed terrestrial equatorial magnetic field, for reference.

##  Appendix C Best-case planetary magnetic field in tidally locked regimes

The planet’s dipole magnetic moment $\mathcal{M}$ gives rise to a planetary
magnetic field

$B_{p}={{\mathcal{M}}\over{r_{mp}^{3}}}$ (C1)

for the dayside magnetopause occurring at a planetocentric distance $r_{mp}$.
The magnetic moment and resulting magnetic field in exoplanets is generally
unknown and one may need to use a known planetary field benchmark (i.e.,
Earth’s or another) to estimate the equilibrium magnetopause (or stand-off)
distance between a planet and the stellar wind it encounters. For Earth, in
particular, this distance is $\sim 10\;R_{\oplus}$ for the unperturbed solar
wind, where $R_{\oplus}$ is Earth’s radius. Patsourakos & Georgoulis (2017)
concluded that the terrestrial magnetopause cannot be compressed to a value
smaller than $\sim 5\;R_{\oplus}$ by the magnetic pressure of ICMEs, an
estimate which aligns with other findings of extreme magnetospheric
compression (e.g., Russell et al., 2000).

While our knowledge of exoplanet magnetic fields is virtually non-existent,
for the subset of exoplanets that lie in the tidally-locked zone of their host
stars (Grießmeier et al., 2004; Khodachenko et al., 2007) we have an
additional constraint: the 1:1 spin-orbit resonance (see Figure 2 to visually
locate all such exoplanets). It implies that tidally locked exoplanets have a
self-rotating speed equal to the rotational speed around their host stars, in
a synchronous rotation. While exceptions are possible, this study adopts the
1:1 spin-orbit resonance because it affords us an estimate of the angular
self-rotation of the planet in case the orbital period around its host star is
known from observations. Synchronous rotation statistically weakens the
planetary magnetic dipole moment, but at the same time it enables its first-
order estimation, further enabling an estimation of the planetary magnetic
field that is crucial for this analysis. In case of planets that are not
tidally locked to their mother star, or we do not employ tidal locking, the
planetary magnetic field (Bbest) will have to be hypothesized by assigning an
ad hoc magnetic dipole moment in Equation C1. No scaling law is employed then,
but we can use a benchmark planetary field, such as Earth’s, for example.

Emphasizing on terrestrial planets, we calculate the upper-case $\mathcal{M}$
provided by Stevenson et al. (1983) model by assuming a core conductivity
$\sigma=5\times 10^{5}\;S/m$ as per Stevenson (2003). The mean core density
$\rho_{c}$ that enters the relationship is further assumed equal to the mean
density of the planet, i.e., $\rho_{c}=\rho=3M_{p}/(4\pi R_{p}^{3})$ where
$M_{p}$ and $R_{p}$ are the planetary mass and radius, respectively. The
assumption allows a density estimate based on direct or implicit observational
facts relevant to the terrestrial planet under study. Moreover, an angular
rotation ($\omega$) equal to the planet’s angular rotation around its host
star is adopted, for presumed tidally locked planets. In case of planets that
are not tidally locked, as explained initially, ad hoc planetary field
benchmarks must be used.

Another obvious unknown is the planetary core radius, $R_{c}$. For Earth, we
have $R_{c}\simeq 0.55\;R_{\oplus}$ while for Mercury the fraction is
significantly larger ($R_{c}\simeq 0.85\;R_{\mercury}$). Regardless, an upper
limit of $R_{c}$ is the radius of the studied planet. In the frame of adopting
the best-case scenario for the planetary magnetic pressure, we will use this
upper limit, adopting $R_{c}=R_{p}$. As an example, the above settings and
Equation (6) for Earth imply an equatorial magnetic field $\sim 0.309\;G$,
almost identical to the nominal terrestrial equatorial field of $\sim
0.305\;G$.

##  Appendix D Sensitivity analysis

As already explained, we are essentially interested in the ratio
$\mathcal{R}=B_{eq}/B_{Stev}$ of the equipartition magnetic field $B_{eq}$ of
a planet at magnetopause distance $r_{mp}=2\;R_{p}$ to the expected upper-
limit, Stevenson magnetic field ($B_{Stev}$) for the planet. This paragraph
details our approach to determine whether applicable uncertainties are capable
of changing the outcome of $\mathcal{R}>1$ or $\mathcal{R}<1$ for a given
exoplanet.

We start by asking what value of the radial fall-off power-law index $a_{B}$
is required for $\mathcal{R}=1$. Under this condition and Equation (4), we
find

$a_{B_{(\mathcal{R}=1)}}={{\log(B_{stev}/(8\;B_{0}))}\over{\log(r_{0}/r_{icme})}}\;\;.$
(D1)

Evidently, if we find $\mathcal{R}>1$ for the nominal $a_{B}=1.6$, then
$a_{B_{(\mathcal{R}=1)}}>a_{B}$, meaning that the ICME magnetic field must
decay more abruptly than assumed to achieve $\mathcal{R}=1$ at $2\;R_{p}$. The
opposite is the case if we find $\mathcal{R}<1$ for $a_{B}=1.6$ (i.e.,
$a_{B_{(\mathcal{R}=1)}}<a_{B}$).

Let us now assume an uncertainty $\delta B_{0}$ of the near-star magnetic
field of the CME, $B_{0}$. This relates to the uncertainty $\delta a_{B}$ on
$a_{B_{(\mathcal{R}=1)}}$ as follows:

${{\delta
a_{B}}\over{a_{B_{(\mathcal{R}=1)}}}}={{1}\over{ln(8B_{0}/B_{stev})}}{{\delta
B_{0}}\over{B_{0}}}\;\;.$ (D2)

From Equation (A3) we can relate $\delta B_{0}$ to the uncertainty in the CME
helicity, $\delta H_{m}$, as

${{\delta H_{m}}\over{\mid H_{m}\mid}}=2{{\delta B_{0}}\over{B_{0}}}\;\;,$
(D3)

and by using the EH diagram of Equation (A2) we can find another expression
for $\delta H_{m}$, namely

${{\delta H_{m}}\over{\mid H_{m}\mid}}=[\beta^{2}({{\delta
E_{c}}\over{E_{c}}})^{2}+(\ln{E_{c}})^{2}\delta\beta^{2}]^{1/2}\simeq
ln{E_{c}}\;\delta\beta\;\;,$ (D4)

for $(\delta E_{c}/E_{c})\lesssim 4$ and typical superflare energies
$E_{c}\geq 10^{30}$ erg. In Equation (D4) we have propagated the uncertainties
for the free energy of the eruptive flare, $\delta E_{c}$, and the power-law
index in the EH diagram of Equation (A2), $\delta\beta\simeq 0.05$. In
Equation (D3) we have further assumed that the forward-modeled GCS geometrical
properties $R$ and $L$ of the CME carry no uncertainties, as even assuming
$(\delta k/k)=(\delta w/w)=1$ the dominant error term is still $(\delta
B_{0}/B_{0})$. As further shown in Equation (D4), given the term depending on
the logarithm of $E_{c}$ we may ignore the contribution of $\delta E_{c}$ for
typical superflare energies $E_{c}\in(10^{30},10^{38})$ erg, thus simplifying
the uncertainty equation.

Combining Equations (D3) and (D4), we eliminate $(\delta H_{m}/\mid
H_{m}\mid)$ and solve for $\delta B_{0}$ to find

${{\delta B_{0}}\over{B_{0}}}\simeq{1\over 2}\;ln{E_{c}}\;\delta\beta\;\;.$
(D5)

Then, substituting Equation (D5) to Equation (D2) we reach the desired
expression for $\delta a_{B}$ on $a_{B_{(R=1)}}$ as follows:

${{\delta
a_{B}}\over{a_{B_{(\mathcal{R}=1)}}}}\simeq{{lnE_{c}}\over{2ln(8B_{0}/B_{stev})}}\delta\beta\;\;.$
(D6)

Equation (D6) allows us to assign an uncertainty to $a_{B_{(\mathcal{R}=1)}}$
and the relevant question is whether $a_{B_{(R=1)}}\neq 1.6$ beyond this
uncertainty. In this case, our conclusion on $\mathcal{R}$ is unlikely to
change. In addition, if $\mathcal{R}>1$ and $a_{B_{(\mathcal{R}=1)}}>2$, then
$a_{B_{(\mathcal{R}=1)}}$ may be deemed unrealistic because the ICME magnetic
field is required to fall more abruptly in the astrosphere than the
’unobstructed’ $a_{B}=2$ case for the near-star CME expansion in order to
achieve $\mathcal{R}=1$. If $a_{B_{(\mathcal{R}=1)}}>2$ beyond applicable
uncertainties, then our conclusion for $\mathcal{R}>1$ is considered solid,
whereas in case $\mid a_{B_{(\mathcal{R}=1)}}-2\mid\leq\delta a_{B}$ our
conclusion is again unlikely to change but it is conceivable that
$\mathcal{R}\leq 1$ for a significantly steeper than 1.6 decrease of the ICME
magnetic field in the astrosphere.

Figure 7: Normalized uncertainties $(\delta a_{B}/a_{B_{(\mathcal{R}=1)}})$ of
the radial power-law fall-off index $a_{B}$ required for an equipartition
($\mathcal{R}=1$) between the ICME and planetary magnetic pressures at
planetocentric distance equal to 2 planetary radii. $B_{0}$-values shown
correspond to the mean near-Sun CME axial magnetic fields (Figure 4) for
$E_{c}=10^{33}$ erg (a) and $E_{c}=10^{34}$ erg (b). Diamonds in images and
indications in the color bar show $(\delta a_{B}/a_{B_{(\mathcal{R}=1)}})$ for
Earth for the two cases.

Figure 7 provides two examples of the ratio $(\delta
a_{B}/a_{B_{(\mathcal{R}=1)}})$ of Equation (D6) for a wide range of
superflare energies and Stevenson planetary fields ranging from 0 to the
$B_{Stev}$-value for Earth ($B_{Stev_{\oplus}}\simeq 0.309$ G). Examples are
based on two nominal $B_{0}$-values; one for $E_{c}=10^{33}$ erg (Figure 7a)
and another for $E_{c}=10^{34}$ erg (Figure 7b). These are mean values
stemming from the Monte Carlo simulations of Figure 4 for these two energies.
Both are toward the upper end of the distribution for near-Sun CME magnetic
fields as found in Patsourakos & Georgoulis (2016). Stronger $B_{0}$ increase
the value of $a_{B_{(\mathcal{R}=1)}}$ (Equation (D1)) but correspondingly
decrease its uncertainty fraction (Equation (D6)).

As an example, for the strong terrestrial magnetic field we find
$a_{B_{(\mathcal{R}=1)}}=0.40\pm 0.63$ (Equations (D1) and (D6)) assuming
$E_{c}=10^{33}$ erg and a nominal $r_{0}=10\;R_{\odot}$. The respective value
for $E_{c}=10^{34}$ erg is $a_{B_{(\mathcal{R}=1)}}=0.63\pm 0.64$. These are
both very flat $a_{B}$-values, flatter beyond uncertainties than the nominal
$a_{B}=1.6$ that gives $\mathcal{R}\simeq 0.026$ and $\simeq 0.052$,
respectively, under the same settings.

## References

  * Argiroffi et al. (2019) Argiroffi, C., Reale, F., Drake, J. J., et al. 2019, Nature Astr., 3, 742, doi: 10.1038/s41550-019-0781-4
  * Armstrong et al. (2016) Armstrong, D. J., Pugh, C. E., Broomhall, A. M., et al. 2016, MNRAS, 455, 3110, doi: 10.1093/mnras/stv2419
  * Barabash et al. (2007) Barabash, S., Fedorov, A., Lundin, R., & Sauvaud, J.-A. 2007, Science, 315, 501, doi: 10.1126/science.1134358
  * Berger (1984) Berger, M. A. 1984, Geophysical and Astrophysical Fluid Dynamics, 30, 79, doi: 10.1080/03091928408210078
  * Busse (1976) Busse, F. H. 1976, Phys. Earth Planet. Interiors, 12, 350, doi: 10.1016/0031-9201(76)90030-3
  * Carrington (1859) Carrington, R. C. 1859, MNRAS, 20, 13, doi: 10.1093/mnras/20.1.13
  * Chapman & Ferraro (1930) Chapman, S., & Ferraro, V. C. A. 1930, Nature, 126, 129, doi: 10.1038/126129a0
  * Christensen (2010) Christensen, U. R. 2010, Space Sci. Rev., 152, 565, doi: 10.1007/s11214-009-9553-2
  * Donati & Brown (1997) Donati, J. F., & Brown, S. F. 1997, A&A, 326, 1135
  * Garraffo et al. (2017) Garraffo, C., Drake, J. J., Cohen, O., Alvarado-Gómez, J. D., & Moschou, S. P. 2017, ApJ, 843, L33, doi: 10.3847/2041-8213/aa79ed
  * Georgoulis & LaBonte (2007) Georgoulis, M. K., & LaBonte, B. J. 2007, Astrophys. J., 671, 1034, doi: 10.1086/521417
  * Georgoulis et al. (2012) Georgoulis, M. K., Tziotziou, K., & Raouafi, N.-E. 2012, Astrophys. J., 759, 1, doi: 10.1088/0004-637X/759/1/1
  * Grießmeier et al. (2004) Grießmeier, J. M., Stadelmann, A., Penz, T., et al. 2004, Astron. Astrophys., 425, 753, doi: 10.1051/0004-6361:20035684
  * Howard (2006) Howard, R. A. 2006, Washington DC American Geophysical Union Geophysical Monograph Series, 165, 7, doi: 10.1029/165GM03
  * Howard et al. (2008) Howard, R. A., Moses, J. D., Vourlidas, A., et al. 2008, Space Sci. Rev., 136, 67, doi: 10.1007/s11214-008-9341-4
  * Howard et al. (2018) Howard, W. S., Tilley, M. A., Corbett, H., et al. 2018, ApJ, 860, L30, doi: 10.3847/2041-8213/aacaf3
  * Jakosky et al. (2015) Jakosky, B. M., Grebowsky, J. M., Luhmann, J. G., et al. 2015, Science, 350, 0210, doi: 10.1126/science.aad0210
  * Kane et al. (2020) Kane, S. R., Roettenbacher, R. M., Unterborn, C. T., Foley, B. J., & Hill, M. L. 2020, The Planetary Science Journal, 1, 36, doi: 10.3847/PSJ/abaab5
  * Kay et al. (2016) Kay, C., Opher, M., & Kornbleuth, M. 2016, ApJ, 826, 195, doi: 10.3847/0004-637X/826/2/195
  * Khodachenko et al. (2007) Khodachenko, M. L., Ribas, I., Lammer, H., et al. 2007, Astrobiology, 7, 167, doi: 10.1089/ast.2006.0127
  * Kopparapu et al. (2013) Kopparapu, R. K., Ramirez, R., Kasting, J. F., et al. 2013, ApJ, 765, 131, doi: 10.1088/0004-637X/765/2/131
  * Lammer et al. (2007) Lammer, H., Lichtenegger, H. I. M., Kulikov, Y. N., et al. 2007, Astrobiology, 7, 185, doi: 10.1089/ast.2006.0128
  * Lundin et al. (2004) Lundin, R., Barabash, S., Andersson, H., et al. 2004, Science, 305, 1933.  http://www.jstor.org/stable/3837868
  * Lynch et al. (2019) Lynch, B. J., Airapetian, V. S., DeVore, C. R., et al. 2019, ApJ, 880, 97, doi: 10.3847/1538-4357/ab287e
  * Maehara et al. (2012) Maehara, H., Shibayama, T., Notsu, S., et al. 2012, Nature, 485, 478, doi: 10.1038/nature11063
  * Manchester et al. (2017) Manchester, W., Kilpua, E. K. J., Liu, Y. D., et al. 2017, Space Sci. Rev., 212, 1159, doi: 10.1007/s11214-017-0394-0
  * Melosh & Vickery (1989) Melosh, H. J., & Vickery, A. M. 1989, Nature, 338, 487, doi: 10.1038/338487a0
  * Mizutani et al. (1992) Mizutani, H., Yamamoto, T., & Fujimura, A. 1992, Adv. Space Res., 12, 265, doi: 10.1016/0273-1177(92)90397-G
  * Moraitis et al. (2014) Moraitis, K., Tziotziou, K., Georgoulis, M. K., & Archontis, V. 2014, Solar Phys., 289, 4453, doi: 10.1007/s11207-014-0590-y
  * Moschou et al. (2019) Moschou, S.-P., Drake, J. J., Cohen, O., et al. 2019, ApJ, 877, 105, doi: 10.3847/1538-4357/ab1b37
  * Ngwira et al. (2014) Ngwira, C. M., Pulkkinen, A., Kuznetsova, M. M., & Glocer, A. 2014, Journal of Geophysical Research (Space Physics), 119, 4456, doi: 10.1002/2013JA019661
  * Nieves-Chinchilla et al. (2018) Nieves-Chinchilla, T., Vourlidas, A., Raymond, J. C., et al. 2018, Solar Phys., 293, 25, doi: 10.1007/s11207-018-1247-z
  * Nindos et al. (2003) Nindos, A., Zhang, J., & Zhang, H. 2003, Astrophys. J., 594, 1033, doi: 10.1086/377126
  * Patsourakos & Georgoulis (2016) Patsourakos, S., & Georgoulis, M. K. 2016, Astron. Astrophys., 595, A121, doi: 10.1051/0004-6361/201628277
  * Patsourakos & Georgoulis (2017) —. 2017, Solar Phys., 292, 89, doi: 10.1007/s11207-017-1124-1
  * Patsourakos et al. (2016) Patsourakos, S., Georgoulis, M. K., Vourlidas, A., et al. 2016, Astrophys. J., 817, 14, doi: 10.3847/0004-637X/817/1/14
  * Petit et al. (2008) Petit, P., Dintrans, B., Solanki, S. K., et al. 2008, MNRAS, 388, 80, doi: 10.1111/j.1365-2966.2008.13411.x
  * Raghav & Shaikh (2020) Raghav, A. N., & Shaikh, Z. I. 2020, Monthly Notices Royal Astron. Soc, 493, L16, doi: 10.1093/mnrasl/slz187
  * Russell et al. (2000) Russell, C. T., Le, G., Chi, P., et al. 2000, Advances in Space Research, 25, 1369, doi: 10.1016/S0273-1177(99)00646-8
  * Sakurai (1981) Sakurai, T. 1981, Solar Phys., 69, 343, doi: 10.1007/BF00149999
  * Salman et al. (2020) Salman, T. M., Winslow, R. M., & Lugaz, N. 2020, Journal of Geophysical Research (Space Physics), 125, e27084, doi: 10.1029/2019JA027084
  * Sano (1993) Sano, Y. 1993, J. Geomagn. Geoelectr., 45, 65, doi: 10.5636/jgg.45.65
  * Schulze-Makuch et al. (2011) Schulze-Makuch, D., Méndez, A., Fairén, A. G., et al. 2011, Astrobiology, 11, 1041, doi: 10.1089/ast.2010.0592
  * Stevenson (2003) Stevenson, D. J. 2003, Earth and Planet. Sci. Lett., 208, 1, doi: 10.1016/S0012-821X(02)01126-3
  * Stevenson et al. (1983) Stevenson, D. J., Spohn, T., & Schubert, G. 1983, Icarus, 54, 466, doi: 10.1016/0019-1035(83)90241-5
  * Sulis et al. (2020) Sulis, S., Lendl, M., Hofmeister, S., et al. 2020, A&A, 636, A70, doi: 10.1051/0004-6361/201937412
  * Thernisien (2011) Thernisien, A. 2011, Astrophys. J. Suppl. Series, 194, 33, doi: 10.1088/0067-0049/194/2/33
  * Thernisien et al. (2009) Thernisien, A., Vourlidas, A., & Howard, R. A. 2009, Solar Phys., 256, 111, doi: 10.1007/s11207-009-9346-5
  * Tziotziou et al. (2013) Tziotziou, K., Georgoulis, M. K., & Liu, Y. 2013, Astrophys. J., 772, 115, doi: 10.1088/0004-637X/772/2/115
  * Tziotziou et al. (2012) Tziotziou, K., Georgoulis, M. K., & Raouafi, N.-E. 2012, Astrophys. J. Lett., 759, L4, doi: 10.1088/2041-8205/759/1/L4
  * Vanderspek et al. (2019) Vanderspek, R., Huang, C. X., Vanderburg, A., et al. 2019, ApJ, 871, L24, doi: 10.3847/2041-8213/aafb7a
  * Vida et al. (2017) Vida, K., Kővári, Z., Pál, A., Oláh, K., & Kriskovics, L. 2017, ApJ, 841, 124, doi: 10.3847/1538-4357/aa6f05
  * Vidotto et al. (2011) Vidotto, A. A., Jardine, M., Opher, M., Donati, J. F., & Gombosi, T. I. 2011, MNRAS, 412, 351, doi: 10.1111/j.1365-2966.2010.17908.x
  * Villarreal D’Angelo et al. (2019) Villarreal D’Angelo, C., Jardine, M., Johnstone, C. P., & See, V. 2019, MNRAS, 485, 1448, doi: 10.1093/mnras/stz477
  * Vourlidas et al. (2017) Vourlidas, A., Balmaceda, L. A., Stenborg, G., & Dal Lago, A. 2017, Astrophys. J., 838, 141, doi: 10.3847/1538-4357/aa67f0
  * Vourlidas et al. (2013) Vourlidas, A., Lynch, B. J., Howard, R. A., & Li, Y. 2013, Solar Phys., 284, 179, doi: 10.1007/s11207-012-0084-8
  * Zendejas et al. (2010) Zendejas, J., Segura, A., & Raga, A. C. 2010, Icarus, 210, 539, doi: 10.1016/j.icarus.2010.07.013
  * Zurbuchen & Richardson (2006) Zurbuchen, T. H., & Richardson, I. G. 2006, Space Sci. Rev., 123, 31, doi: 10.1007/s11214-006-9010-4


# Finite group schemes of essential dimension one

Najmuddin Fakhruddin  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>
###### Abstract.

We prove that if a finite group scheme $G$ over a field $k$ has essential
dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit
classification of all infinitesimal group schemes of essential dimension one
over any field and a characterisation of all finite group schemes of essential
dimension one over algebraically closed fields.

###### Key words and phrases:

Essential dimension, finite group schemes

###### 1991 Mathematics Subject Classification:

14L20, 14L30, 12F05

##  1\. Introduction

Let $G$ be a group scheme over a field $k$. The essential dimension of $G$—see
e.g. [8] or [7] for the definition—is a non-negative integer defined using
$G$-torsors over all extensions of $k$. For example, if $G=GL_{n/k}$ then all
$G$ torsors are trivial and the essential dimension is zero. The essential
dimension of a non-trivial finite group scheme $G$ is always positive, so
those having essential dimension one are of particular interest. The main aim
of this note is to prove the following:

######  Theorem 1.1.

Let $G$ be a finite group scheme over a field $k$.

  1. (1)

If the essential dimension of $G$ over $k$ is one then $G$ can be embeded in
$PGL_{2/k}$ and $\dim_{k}(\operatorname{Lie}(G))\leq 1$.

  2. (2)

If $G$ is infinitesimal then it has essential dimension one over $k$ iff it
can be embedded in $PGL_{2/k}$, $\dim_{k}(\operatorname{Lie}(G))=1$, and $G$
lifts to $GL_{2/k}$. Such group schemes exist over $k$ iff
$\operatorname{char}(k)=p>0$ and a list of such group schemes is as follows:

    1. (a)

$\alpha_{p^{n}}$ for all $n>0$;

    2. (b)

$\mu_{p^{n}}$ for all $n>0$;

    3. (c)

any form of $\mu_{p^{n}}$, $n>0$, which becomes isomorphic to $\mu_{p^{n}}$
over a quadratic extension of $k$ if $p\neq 2$.

Ledet proved in [5] that a constant finite group $G$ has essential dimension
one iff it embeds in $PGL_{2/k}$ and lifts to $GL_{2/k}$. A complete list of
such $G$ was given by Chu, Hu, Kang and Zhang [3]. It is likely that by
combining these results with Theorem 1.1 one can classify all finite group
schemes of essential dimension one. Theorem 4.1 gives such a classification
for group schemes over perfect fields with constant etale quotients.

Tossici has conjectured [9, Conjecture 1.4] that for a finite commutative
unipotent group $G$, $\operatorname{ed}_{k}(G)\geq n_{V}(G)$, where $n_{V}(G)$
is the order of nilpotence of the Verschiebung morphism of $G$. Theorem 1.1,
and the fact that $\operatorname{ed}_{k}(\mathbb{Z}/p^{2}\mathbb{Z})=2$ (over
a field of characteristic $p$), implies that $\operatorname{ed}_{k}(G)\geq 2$
if $n_{V}(G)\geq 2$. In particular, we see that the $p$-torsion of a
supersingular elliptic curve has essential dimension two (Example 4.2).

For a constant group scheme $G$ with essential dimension one over $k$ it is
immediate that $G$ embeds in $PGL_{2/k}$; this is because any rationally
defined action of a finite group on a smooth projective curve extends to a
regular action. However, this extension property is far from being true for
infinitesimal group actions and the key to our classification is a simple
criterion (Proposition 2.2) for the existence of such extensions. Aside from
this, we also use some basic structure theory of finite group schemes,
especially in the case $\operatorname{char}(k)=2$.

### 1.1.

For the basic definitions in the theory of essential dimension we refer the
reader to [8] or [7]; we only need the definition of essential dimension of a
group scheme over a field $k$, denoted by $\operatorname{ed}_{k}(G)$ below,
the $p$-essential dimension for a prime $p$, denoted by
$\operatorname{ed}_{k}(G;p)$ and the notion of versal torsors [7, §3d]. For
the particular case of infinitesimal group schemes the reader may consult
[10]. Our basic reference for the theory of (finite) group schemes is [4].

###  1.2. Acknowledgements

I thank Patrick Brosnan and Zinovy Reichstein for helpful comments on an
earlier version of this note, and Dajano Tossici for pointing out an error in
the original proof of the main theorem when $p=2$ and for other useful
comments.

##  2\. Preliminary results

###  2.1. An extension criterion

######  Lemma 2.1.

Let $\iota:A\to B$ be an inclusion of noetherian integral domains with $A$
normal and with the map $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$
being surjective (or with image containing all height one primes). If $b\in B$
is such that $b\in A_{(0)}$ (the quotient field of $A$), then $b\in A$.

###### Proof.

Since A is normal, it suffices to show that the valuation of $b$ at any height
one prime of $A$ is non-negative, so by localisation we may assume that $A$ is
a dvr. If $b\notin A$ then $b^{-1}\in m_{A}$. By surjectivity of the map
$\operatorname{Spec}(B)\to\operatorname{Spec}(A)$, it follows that $b^{-1}\in
P$ for some prime ideal $P$ of $B$. But this implies that $1=b\cdot b^{-1}\in
P$, a contradiction. ∎

######  Proposition 2.2.

Let $k$ be a field, let $G$ be an infinitesimal group scheme over $k$ and let
$Y$ be a normal projective curve with a generically defined action of $G$. If
there exists $X$, a normal projective variety over $k$ with a regular action
of $G$, and a dominant rational map $f:X\dashrightarrow Y$ compatible
(generically) with the $G$-actions, then the rational action of $G$ extends
uniquely to a regular action on $Y$. If $k$ is perfect or $Y$ is smooth, then
$G$ can be taken to be any finite group scheme.

It is easy to see that rational actions of infinitesimal group schemes on
smooth projective curves do not always extend to regular actions, so our
hypothesis on the existence of the equivariant rational map
$f:X\dashrightarrow Y$ is not superfluous; see Example 2.4 below.

###### Proof.

Assume first that $k$ is arbitrary and $G$ is infinitesimal. Let $U$ be the
maximal open subset of $X$ on which the rational map $f$ restricts to a
morphism $f|_{U}:U\to Y$. The map $f|_{U}$ is surjective, because by normality
$X\backslash U$ is of codimension at least two in $X$ so a general complete
intersection curve $C\subset X$ will lie in $U$ and $f|_{C}:C\to Y$ is
surjective since $C$ is proper and also general.

Let $y\in Y$ be any closed point and let $u\in U$ be a (closed) point such
that $f(u)=y$. Let $V=\operatorname{Spec}(A)$ be any open affine subset of $Y$
with $y\in V$ and such that the $G$-action on $Y$ is defined at all points of
$V\backslash\\{y\\}$. Let $W\subset U$ be any affine open subset such that
$u\in W$ and $f(W)\subset V$. Since $G$ is infinitesimal, $W$ is $G$-invariant
and $G$ acts generically on $V$. Let $W=\operatorname{Spec}(B)$, so $f$
induces an inclusion $\iota:A\to B$.

Let $G=\operatorname{Spec}(R)$. Then $R$ is a Hopf algebra and the $G$-actions
correspond to maps $c_{1}:B\to R\otimes_{k}B$ and $c_{2}:A_{(0)}\to R\otimes
A_{(0)}$ (satisfying the usual properties). For any $a\in A$, we have that
$c_{1}(\iota(a))=c_{2}(a)$, where the equality holds in $R\otimes B_{(0)}$.
Choosing a basis of $R$ over $k$ and then applying Lemma 2.1 coordinatewise,
we see that $c_{2}(a)\in R\otimes A$. This proves that the action of $G$ on
$V\backslash\\{y\\}$ extends to all of $V$—the necessary identities hold
because the map $A\to A_{(0)}$ is an injection—and this extension is clearly
unique. Since $y\in Y$ was arbitrary, it follows that the generically defined
action of $G$ extends uniquely to all of $Y$.

Now suppose that $k$ is perfect and $G$ is any finite group scheme over $k$.
By [4, II, §5, 2.4], $G$ is a semidirect product of its identity component
$G^{0}$ and its etale quotient $G^{et}$. The action of $G^{0}$ extends to all
of $Y$ and the generic action of an etale group scheme (over any field) on a
normal projective curve extends uniquely to a regular action. This is clear
for a constant group scheme and the case of an etale group scheme follows from
this by Galois descent. The uniqueness of the two extensions then implies that
the action of $G$ also extends.

Now suppose that $k$ is arbitrary, $G$ is any finite group scheme, and $Y$ is
smooth. We know that an extension of the $G^{0}$ action exists over $k$ and an
extension of the $G$ action exists after we base change to any perfect field
$K/k$, since $Y_{K}$ is also smooth. For $V=\operatorname{Spec}(A)$ any
nonempty affine open subset of $Y$ as above, consider the action map
$c_{2}:A_{(0)}\to R\otimes A_{(0)}$ as above. Using a basis of $R$ over $k$
and writing $c_{2}(a)$ in terms of coordinates, we see that an extension of
the $G$ action actually exists over $k$ since any element of $A_{(0)}$ which
lies in $A\otimes_{k}K$ must also lie in $A$. The extension is unique since
this holds over $K$. ∎

######  Remark 2.3.

The assumption that $Y$ is a curve was only used in the proof to deduce the
surjectivity of $f|_{U}$. If instead of $f$ being a rational map, we assume
that it is regular and surjective (in codimension one), then the proof in the
case of infinitesimal $G$ works for any normal $Y$ (without any
projectivity/properness hypothesis on $Y$ or $X$).

######  Example 2.4.

Let $Z$ be a smooth projective ordinary curve of genus $g\geq 2$ over an
algebraically closed field $k$ of charactersitic $p>0$. Let $T$ be a non-
trivial $\mu_{p}$-torsor over $Z$ corresponding to non-zero element of
$\operatorname{Pic}(Z)[p]$. Such a torsor is non-trivial over the generic
point of $Z$, so $T$ must be integral at its generic point. However, there is
no $\mu_{p}$-equivariant completion of this torsor with total space a smooth
projective curve since such a curve would map to $Z$, so would have genus at
least $2$, and a smooth projective curve of genus $g\geq 2$ has no non-zero
vector fields.

We do not know the answer to the following:

######  Question 2.5.

Does the action of a finite group scheme on a function field of transcendence
degree one always extend to a proper model of the function field?

######  Lemma 2.6.

Let $k$ be any field and $G$ an affine group scheme of finite type over $k$.
Then there exists a smooth projective (geometrically irreducible) rational
variety $X$ over $k$ with a generically free action of $G$.

###### Proof.

Since $G$ is of finite type it can be embedded in $GL_{n}$ for some $n>0$, so
$G$ acts linearly and generically freely on $M_{n}$, the space of $n\times n$
matrices. We may then take $X$ to be the projective completion of $M_{n}$,
i.e., $\mathbb{P}(M_{n}\oplus k)$. ∎

###  2.2. Infinitesimal group schemes with one dimensional Lie algebra

In this section we assume that $\operatorname{char}(k)=p>0$. We denote the Lie
algebra of group schemes $G$, $H$, …, by $\mathfrak{g}$, $\mathfrak{h}$, ….

######  Lemma 2.7.

Let $G$ be an infinitesimal group scheme over a field $k$ with
$\dim_{k}(\mathfrak{g})=1$. If $H$ is any subquotient of $G$, then
$\dim_{k}(\mathfrak{h})\leq 1$.

###### Proof.

The statement is clear for subgroup schemes so it suffices to prove it for
quotients. Let $K\subset G$ be a normal subgroup scheme and let $H=G/K$.
Clearly $\mathcal{O}(G)\cong B=k[x]/(x^{p^{n}})$ for some $n\geq 0$, and
$\mathcal{O}(H)\cong A$, where $A$ is a $k$-subalgebra of $B$ such that $B$ is
flat over $A$. If $A=k$ the lemma is clear, so we may assume that the maximal
ideal $m_{A}$ of $A$ is nonzero.

Let $r$ be the smallest integer such that $m_{A}$ contains an element $a\in
m^{r}_{B}\backslash m^{r+1}_{B}$. Then the elements $1,x,\dots,x^{r-1}$ of $B$
are linearly independent in $B/m_{A}B$, so they must be part of a basis of $B$
over $A$. In particular, we get that $p^{n}=\dim_{k}(B)\leq r\dim_{k}(A)$. On
the other hand, the structure of the ring $B$ implies that
$1,a,\dots,a^{\lceil\frac{p^{n}}{r}\rceil}$ are $k$-independent elements of
$A$. This implies that $r|p^{n}$ and $a$ generates $m_{A}$, so $\mathfrak{h}$
is one dimensional.

∎

######  Remark 2.8.

The lemma holds with $1$ replaced by $n$ for arbitrary finite type group
schemes $G$ over a field $k$, but we do not give the details here since we do
not need this.

######  Proposition 2.9.

Let $G$ be an infinitesimal group scheme over a field $k$ with
$\dim_{k}(\mathfrak{g})=1$. Then either $G$ is multiplicative or it is
unipotent.

###### Proof.

Since Lie algebras are compatible with field extensions and so are the notions
of multiplicative and unipotent groups, we may assume that $k$ is
algebraically closed.

Let $G_{1}$ be the height one subgroup scheme of $G$ corresponding to
$\mathfrak{g}$ (viewed as $p$-Lie algebra); see for example [4, II, §7, 4.3].
This is a characteristic subgroup scheme of $G$ of order $p$, so we may form
the quotient $G/G_{1}$. By Lemma 2.7 and induction on the order of $G$, we get
a filtration

$\\{1\\}=G_{0}\subset G_{1}\subset G_{2}\subset\dots\subset G_{n}=G$

where each $G_{i}$ is a characteristic subgroup of $G$ and $G_{i}/G_{i-1}$ is
of order $p$ for $1\leq i\leq n$, so isomorphic to $\alpha_{p}$ or $\mu_{p}$.
To prove the lemma it suffices by [4, IV, §1, 4.5] to show that all these
quotients are of the same type.

If $G$ is neither unipotent nor multiplicative, it follows from the above that
$G$ has a subquotient of order $p^{2}$ which is an extension of $\mu_{p}$ by
$\alpha_{p}$ or an extension of $\alpha_{p}$ by $\mu_{p}$.

In the first case, [4, IV, §2, 3.3] implies that the extension is a
trigonalisable group scheme. Then by [4, IV, §2, 3.5] the extension splits,
i.e., it is a semidirect product. In the second case, since $k$ is
algebraically closed, so perfect, it follows from [1, Théorème 6.1.1 (B)] that
the extension is trivial.

Lemma 2.7 leads to a contradiction in both cases, so the proposition is
proved. ∎

######  Remark 2.10.

Multiplicative group schemes $G$ with $\dim_{k}(\mathfrak{g})=1$ are just
forms of $\mu_{p^{n}}$ for some $n>0$. Can one classify all unipotent group
schemes with one dimensional Lie algebras? (It seems likely that they are all
commutative.)

##  3\. Proof of the main theorem

###### Proof of Theorem 1.1.

For any non-trivial finite group scheme $G$, $\operatorname{ed}_{k}(G)\geq 1$
since there exist $G$-torsors over any extension of $k$ whose underlying
scheme is integral.

Now suppose $G$ is an arbitrary finite group scheme with
$\operatorname{ed}_{k}(G)=1$. By Lemma 2.6, there exists a smooth projective
rational variety $X$ with a generically free action of $G$. Let $\emptyset\neq
U\subset X$ be an open subset on which $G$ acts freely and let $V=X/G$, so $U$
is the total space of a $G$-torsor over $V$. The induced torsor over
$\operatorname{Spec}(k(V))$ does not have essential dimension $0$, since any
$G$-torsor over $\operatorname{Spec}(k)$ becomes trivial over an algebraic
closure $\bar{k}$ of $k$ but the torsor over
$\operatorname{Spec}(\bar{k}(V_{\bar{k}}))$ is non-trivial. Thus, since
$\operatorname{ed}_{k}(G)=1$, there is a normal integral curve $Z$ over $k$, a
$G$-torsor $T$ over $Z$, and a dominant rational map $V\dashrightarrow Z$ such
that $X$ is generically equal to $V\times_{Z}T$ (which makes sense over the
generic point of $V$) as a $G$-torsor.

Since $X$ is integral, the fibre of $T$ over the generic point of $Z$ must
also be integral. Let $Y$ be the unique normal projective curve with function
field equal to $k(T)$ (the residue field at the generic point of $T$), so $Y$
has a generically defined, generically free, action of $G$ and we have a
$G$-equivariant dominant rational map from $X$ to $Y$. Since $X$ is rational,
Luroth’s theorem implies that $Y\cong\mathbb{P}^{1}_{k}$; in particular, $Y$
is smooth. By Proposition 2.2, the generic $G$-action extends to a regular
action of $G$ on $Y$. The generic freeness of the $G$-action on $Y$ then gives
an embedding $G\hookrightarrow\operatorname{Aut}(Y)\cong PGL_{2/k}$. Finally,
since the $G$-action on $Y$ is generically free and $Y$ is a smooth curve, it
follows that $\dim_{k}(\mathfrak{g})\leq 1$.

Now let $G\subset PGL_{2/k}$ be an infinitesimal subgroup scheme with
$\dim_{k}(\mathfrak{g})=1$. If $G$ lifts to a subgroup scheme of $GL_{2/k}$,
then $G$ acts generically freely on $\mathbb{A}^{2}_{k}$ and the map
$\mathbb{A}^{2}_{k}\to\mathbb{A}^{2}_{k}/G$ gives rise to a versal $G$-torsor
over a non-empty open subset of $\mathbb{A}^{2}_{k}/G$. We also have a
$G$-torsor corresponding to the quotient map
$\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}/G$, and this torsor gives a one
dimensional compression of the versal $G$-torsor. Thus
$\operatorname{ed}_{k}(G)\leq 1$ and if $G$ is non-trivial then we must have
$\operatorname{ed}_{k}(G)=1$.

To prove that $G$ lifts to $GL_{2/k}$ if $\operatorname{ed}_{k}(G)=1$, we
first consider the case that $\operatorname{char}(k)\neq 2$. Then we have an
etale isogeny $SL_{2/k}\to PGL_{2/k}$, so if $G$ is infinitesimal it lifts
(uniquely) to $SL_{2/k}$, therefore to $GL_{2/k}$.

Suppose that $G$ is multiplicative (with $\operatorname{char}(k)$ arbitrary)
and let $C(G)$ be the centralizer of $G$ in $PGL_{2/k}$. By [1, XI, Corollaire
2.4], the centralizer of a group scheme of multiplicative type in any smooth
affine group scheme is smooth, so $C(G)$ is smooth; let $C^{0}(G)$ be its
identity component. Now $C(G)$ is a proper subgroup scheme of $PGL_{2/k}$
since the centre of $PGL_{2/k}$ is trivial. Also, $G$ is contained in the
centre of $C^{0}(G)$, which must therefore be positive dimensional. If
$C^{0}(G)$ is not a torus then it must contain a smooth one dimensional
unipotent subgroup $U$. However, since $G\subset G^{0}(G)$ and $G$ is
multiplicative, it follows that in this case $C^{0}(G)$ must be two
dimensional. Therefore, $C^{0}(G)$ must be a Borel subgroup of $PGL_{2/k}$ and
$U$ must be its unipotent radical. However, the centre of a Borel subgroup of
$PGL_{2/k}$ is trivial, so this contradicts the fact that $G$ is contained in
the centre of $C^{0}(G)$. Thus, any infinitesimal multiplicative subgroup $G$
of $PGL_{2/k}$ in any characteristic must be contained in a torus $T$.

Let $T^{\prime}$ be the inverse image of $T$ in $GL_{2/k}$, so $T^{\prime}$ is
a two dimensional torus. It is a maximal torus, of which the non-split ones
are classified by separable quadratic extensions of $k$: the action of the
Galois group on the character group $\cong\mathbb{Z}^{2}$ is given by
switching coordinates. It follows that $T^{\prime}$ is a quadratic twist of
$\mathbb{G}_{m}$. If $\operatorname{char}(k)>2$ this gives the claimed
classification of all multiplicative $G$ with $\operatorname{ed}_{k}(G)=1$.

If $T$ is non-split then the above description shows that it contains a unique
subgroup scheme of order $2$. If $\operatorname{char}(k)=2$, it follows from
this that $G\subset T^{\prime}$ lifts to a subgroup of $GL_{2/k}$ iff $T$
(equivalently $T^{\prime}$) is split; in particular, $G$ must be isomorphic to
$\mu_{2^{n}}$ for some $n$.

By [6, Proposition 6.1], if $G$ is a non-trivial quadratic twist of
$\mu_{2^{n}}$ then $\operatorname{ed}_{k}(G;2)=2$. Since
$\operatorname{ed}_{k}(G)\geq\operatorname{ed}_{k}(G;p)$ for any prime $p$, we
conclude that if $\operatorname{char}(k)=2$ the only infinitesimal
multiplicative group schemes $G$ with $\operatorname{ed}_{k}(G)=1$ are the
$\mu_{2^{n}}$ for $n>0$.

Now suppose $G\neq\\{1\\}$ is unipotent and assume that the $G$-action on
$\mathbb{P}^{1}_{k}$ gives a versal $G$-torsor.

######  Claim.

$G$ preserves a point in $\mathbb{P}^{1}_{k}(k)$.

###### Proof of claim.

Consider a faithful representation of $G$ on a finite dimensional $k$-vector
space $V$. Since $G$ is unipotent, by [4, IV, §2, 2.5] $V$ has a complete
filtration by $k$-subspaces

$0=V_{0}\subset V_{1}\subset\dots\subset V_{n-1}\subset V_{n}=V$

which is preserved by $G$. By induction on $n$, we may assume that the action
on $V_{n-1}$ is not faithful.

Since $\operatorname{Lie}(G)$ is one dimensional, the action of $G$ on $V$
(which we now think of as an affine variety) is generically free. By the
assumed versality of the $G$-action on $\mathbb{P}^{1}_{k}$, there exists a
rational $G$-equivariant morphism $f:V\dashrightarrow\mathbb{P}^{1}_{k}$.
Since $\mathbb{P}^{1}_{k}$ is proper, $f$ is defined at the generic point of
$V_{n-1}$. The restriction of $f$ to $V_{n-1}$ cannot be dominant since the
action of $G$ on $V_{n-1}$ is not faithful, so $f(V_{n-1})$ must be a rational
point $x\in\mathbb{P}^{1}_{k}(k)$. Since $V_{n-1}$ is $G$-invariant, it
follows that so is $x$, and the claim is proved. ∎

The stabilizer in $PGL_{2/k}$ of any point in $\mathbb{P}^{1}_{k}(k)$ is Borel
subgroup so $G$ is contained in a Borel $B$. Since $G$ is unipotent, it is in
fact contained in the unipotent radical of $B$ which lifts to $GL_{2/k}$.
Thus, $G$ also lifts to $GL_{2/k}$. 111When $p=2$ there do exist infinitesimal
unipotent subgroups of $PGL_{2/k}$ which do not lift to $GL_{2/k}$; this was
pointed out to us by D. Tossici.

Any unipotent subgroup of $GL_{2/k}$ in any characteristic preserves a line,
so we see that any infinitesimal unipotent group $G$ with
$\operatorname{ed}_{k}(G)=1$ must be isomorphic to a subgroup of
$\mathbb{G}_{a}$, so must be $\alpha_{p^{n}}$ for some $n>0$. ∎

######  Remark 3.1.

If $\operatorname{char}(k)=2$ and $\mu_{2}$ is embedded in $PGL_{2/k}$ as a
subgroup of a torus (which is uniquely determined as the connected
centralizer), then the $\mu_{2}$ action on $\mathbb{P}^{1}_{k}$ has a fixed
point iff the torus is split. It follows that the embedding gives a versal
$\mu_{2}$-torsor iff the torus is split.

##  4\. Applications of the main theorem

###  4.1. Finite group schemes of essential dimension one

Let $G$ be any finite group scheme over $k$ with $\operatorname{ed}_{k}(G)=1$.
By Theorem 1.1, $G$ embeds in $PGL_{2/k}$ and we also have
$\dim_{k}(\mathfrak{g})=1$. If $G$ is any finite subgroup scheme of
$PGL_{2/k}$ with $\dim_{k}(\mathfrak{g})=1$, then a sufficient condition for
$\operatorname{ed}_{k}(G)=1$ is that $G$ lifts to $GL_{2/k}$. If this
condition were also necessary—we know this is the case for constant as well as
infinitesimal group schemes—then we would have a complete classification of
all finite group schemes $G$ with $\operatorname{ed}_{k}(G)=1$.

As a first step, one should verify this for etale group schemes. This can
presumably be done using the classification of constant groups $G$ with
$\operatorname{ed}_{k}(G)=1$ in [3], but we do not do this here and proceed
after making some simplifying assumptions.

Recall that for any finite group scheme $G$ we have an exact sequence

$1\to G^{0}\to G\to G^{et}\to 1,$

where $G^{0}$ is infinitesimal and $G^{et}$ is etale. This sequence splits
when $k$ is perfect, i.e., $G\cong G^{0}\rtimes G^{et}$ [4, II, §5, 2.4].

######  Theorem 4.1.

A finite group scheme over a perfect field $k$ with $G^{et}$ constant has
$\operatorname{ed}_{k}(G)\leq 1$ iff $G$ can be embedded in $PGL_{2/k}$,
$\dim_{k}(\mathfrak{g})\leq 1$ and $G$ lifts to $GL_{2/k}$.

Note that if $k$ is algebraically closed this gives a classification of all
finite group schemes over $k$ with $\operatorname{ed}_{k}(G)=1$. For general
perfect $k$ the method of proof, together with the classification results of
[3], can be used to make a more explicit list of all such group schemes.

###### Proof.

The conditions are clearly sufficient and we have already seen that the first
two are necessary, so we need to show that if $\operatorname{ed}_{k}(G)=1$
then $G$ has an embedding in $PGL_{2/k}$ which lifts to $GL_{2/k}$. We may
assume that $G^{0}$ is non-trivial since otherwise the theorem is a
consequence of [5].

If $\operatorname{ed}_{k}(G)=1$ then there is an embedding $G\hookrightarrow
PGL_{2/k}$ which makes the quotient map
$\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}/G$ generically into a versal
$G$-torsor. This implies (as is well known) that
$\operatorname{ed}_{k}(G^{0})=1$ and $\operatorname{ed}_{k}(G^{et})$ and the
corresponding quotient maps are also versal torsors. By Theorem 1.1 and [5,
Theorem 8], both $G^{0}$ and $G^{et}$ lift to $GL_{2/k}$; call the lifts
$\widetilde{G^{0}}$ and $\widetilde{G^{et}}$.

If $\operatorname{char}(k)\neq 2$, we saw in the proof of Theorem 1.1 that we
can assume that $\widetilde{G^{0}}\subset SL_{2/k}$ and such a lift is unique.
It follows that $\widetilde{G^{et}}$ normalizes $\widetilde{G^{0}}$ and the
subgroup $\widetilde{G^{0}}\rtimes\widetilde{G^{et}}\subset GL_{2/k}$ is a
lift of $G$.

If $\operatorname{char}(k)=2$ and $G^{0}$ is unipotent then
$\widetilde{G^{0}}$ is still unique, so again
$\widetilde{G^{0}}\rtimes\widetilde{G^{et}}\subset GL_{2/k}$ is a lift of $G$.
Finally, suppose that $G^{0}$ is multiplicative, so it is isomorphic to
$\mu_{2^{n}}$ for some $n>0$. We may assume that $\widetilde{G^{0}}$ is a
subgroup of the group $T^{\prime}$ of diagonal matrices in $GL_{2/k}$ and
$\widetilde{G^{0}}\cap Z(GL_{2/k})=\\{1\\}$. If
$\widetilde{G^{0}}\neq\\{1\\}$, one easily sees that the normalizer of
$\widetilde{G^{0}}$ in $GL_{2/k}$ is $T^{\prime}$. Thus, $G$ has a lift to
$GL_{2/k}$ iff $G\subset T$, the image of $T^{\prime}$ in $PGL_{2/k}$. If this
condition is not satisfied then $G$ contains a subgroup
$G_{1}\cong\mu_{2}\times\mathbb{Z}/2\mathbb{Z}$. By [2, Theorem 3.1]
$\operatorname{ed}_{k}(\mu_{2}\times\mathbb{Z}/2\mathbb{Z})=2$—one may give an
elementary proof of this particular case by using that the action of $G_{1}$
on $\mathbb{P}^{1}_{k}$ does not have any fixed points222This, together with
the reference in [2], was pointed out by Z. Reichstein.— and this implies that
$\operatorname{ed}_{k}(G)>1$.

We conclude that if $\operatorname{ed}_{k}(G)=1$ then $G$ lifts to $GL_{2/k}$.
∎

###  4.2. Some finite group schemes of essential dimension two

Using Theorem 1.1 we may compute $\operatorname{ed}_{k}(G)$ for various other
group schemes. The point is that if one knows that
$\operatorname{ed}_{k}(G)\leq 2$ for some group scheme $G$ not occurring in
the list of group schemes with $\operatorname{ed}_{k}=1$ then we must have
$\operatorname{ed}_{k}(G)=2$.

######  Example 4.2.

Let $G=E[p]$, the $p$-torsion of a supersingular elliptic curve $E$ over a
field $k$ with $\operatorname{char}(k)=p>0$. It sits in an exact sequence

$0\to\alpha_{p}\to E[p]\to\alpha_{p}\to 0,$

so by [10, Theorem 1.4] we have $\operatorname{ed}_{k}(E[p])\leq 2$. However,
$E[p]$ is not isomorphic to $\alpha_{p^{2}}$, so we conclude that
$\operatorname{ed}_{k}(E[p])=2$. The group scheme $E[p]$ is trigonalizable,
but $2=\operatorname{ed}_{k}(E[p])>\dim_{k}(\operatorname{Lie}(E[p]))=1$, so
it is not _almost special_ in the sense of [10, Definition 4.2]. This answers
a question discussed in [10, Example 4.8].

## References

  * [1] Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin, 1962/1964.
  * [2] A. Babic and V. Chernousov, Lower bounds for essential dimensions in characteristic 2 via orthogonal representations, Pacific J. Math., 279 (2015), pp. 37–63.
  * [3] H. Chu, S.-J. Hu, M.-C. Kang, and J. Zhang, Groups with essential dimension one, Asian J. Math., 12 (2008), pp. 177–191.
  * [4] M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970.  Avec un appendice Corps de classes local par Michiel Hazewinkel.
  * [5] A. Ledet, Finite groups of essential dimension one, J. Algebra, 311 (2007), pp. 31–37.
  * [6] R. Lötscher, M. MacDonald, A. Meyer, and Z. Reichstein, Essential dimension of algebraic tori, J. Reine Angew. Math., 677 (2013), pp. 1–13.
  * [7] A. S. Merkurjev, Essential dimension: a survey, Transform. Groups, 18 (2013), pp. 415–481.
  * [8] Z. Reichstein, Essential dimension, in Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188.
  * [9] D. Tossici, Essential dimension of inifinitesimal commutative unipotent group schemes, Bollettino dell’Unione Matematica Italiana, (2019).
  * [10] D. Tossici and A. Vistoli, On the essential dimension of infinitesimal group schemes, Amer. J. Math., 135 (2013), pp. 103–114.