We also compared the chances for a successful detection of a reflecting planet
with SPHERE/ZIMPOL in searching for a polarized signal or the intensity
signal. We find that a planet is easier to find with polarimetry. At small
separations, polarimetry is up to 30 times more efficient in suppressing the
strong speckle noise than the search of the corresponding intensity signal.
Because Earth (Stam 2008; Bazzon et al. 2013), Jupiter (Smith & Tomasko 1984),
or Uranus and Neptune (Schmid et al. 2006b; Buenzli & Schmid 2009) show a
scattering polarization of about 15 % or higher in the R band, a polarimetric
search of a planet with SPHERE/ZIMPOL is attractive.

The polarimetric contrast limits that were reached of about $C_{\rm P}\approx
1\cdot 10^{-8}$ at $\rho\approx 1\arcsec$ are still a factor of about ten
above the expected signal for the RV-planet $\epsilon$ Eri b. Steps to reduce
the gap between the planet signal and detection limit are discussed and we
conclude that a $3\,\sigma_{\mathcal{N}}$ contrast limit of $C_{\rm P}\approx
1\cdot 10^{-9}$ would require about 200 hours of VLT integration time under
good seeing conditions and a well-known planet orbit to optimize the observing
strategy in a follow-up search. This is very demanding but technically
feasible.

Contrast limits of $C_{\rm P}\approx 1\cdot 10^{-9}$ at $\rho\approx 1\arcsec$
are easier to achieve for planets around the nearest bright stars Sirius,
$\alpha$ Cen A and B, Altair, and a few others. For these objects it is
possible to measure a photon flux that is ten times higher with SPHERE/ZIMPOL
without harmful detector saturation effects and to reach contrasts of $C_{\rm
P}\approx 1\cdot 10^{-9}$ within 20 hours. Moreover, because $\alpha$ Cen is
so close, a contrast of $\approx 1\cdot 10^{-9}$ would allow the detection of
a planet with a radius of roughly 0.4 ${\rm R_{\rm J}}$, because $1\arcsec$
corresponds to a physical separation of only $d_{\rm p}=1.3$ au.

Such a programme with SPHERE/ZIMPOL (ESO programme ID 2107.C-5008) was
approved following the announcement of a planet candidate in $\alpha$ Cen A by
Wagner et al. (2021); however, because of scheduling issues during the corona
pandemic, only observations were executed for 4 hours. We analysed these
$\alpha$ Cen A observations very similar to the $\epsilon$ Eri data and
reached a $5\,\sigma_{\mathcal{N}}$ detection limit of $C_{\rm P}\approx
8\cdot 10^{-9}$ at $1\arcsec$ for these 4 h (Tschudi 2023). This is a similar
contrast to the 38.5 hours for $\epsilon$ Eri because of the higher photon
flux and the better seeing conditions for the $\alpha$ Cen A data. We could
not detect a point source in these data. Because the planet candidate around
$\alpha$ Cen A has not been confirmed yet, a future SPHERE/ZIMPOL programme
aiming for deeper observations would face the risks of a blind search, where
the location and brightness of the planet for a given epoch are unclear and
therefore could be very unfavourable if the observations are scheduled at the
’wrong’ time.

##### On the search of extended emission.

SPHERE/ZIMPOL is the only polarimeter regularly used for high-contrast imaging
polarimetry in the visual 500-900 nm range (e.g. Schmid 2021). It is therefore
very useful for the study of wavelength dependencies of the scattered
radiation of circumstellar disks and shells (e.g. Ma et al. 2023; Haubois et
al. 2023). For these applications the ZIMPOL performance is competitive with
respect to the state-of-the-art near-infrared polarimetric modes of
SPHERE/IRDIS (de Boer et al. 2020) or of GPI at Gemini (Perrin et al. 2015).
For dust around bright stars $R<5$ mag, ZIMPOL has the advantage that it was
designed for the search of planets around very bright stars and the instrument
can therefore fully exploit the photon collecting power of the VLT telescope
in broad-band filters without harmful detector saturation effects.

For $\epsilon$ Eri we added 38.5 hours of integration for the search of
extended polarized emission of dust. By averaging the signal over areas of
$0.11\arcsec\times 0.11\arcsec$, we could further reduce the statistical noise
to theoretical contrast limits of $\Delta{S\\!B}_{\rm p}\approx
17.7\leavevmode\nobreak\ {\rm mag/arcsec}^{2}$ at a separation of about
$1\arcsec$. This would be enough to see the expected dust scattering clearly
from the warm dust in $\epsilon$ Eri inferred from the $20\leavevmode\nobreak\
\mu$m infrared excess bump measured in SPITZER data (Backman et al. 2009).
However, we find that the contrast is limited by a weak systematic noise
pattern to about $\Delta{S\\!B}_{\rm p}\approx 15\leavevmode\nobreak\ {\rm
mag/arcsec}^{2}$. Even this contrast is much deeper than previous measurements
of the extended polarized emission in high-contrast imaging with SPHERE and
GPI. Typical contrast limits for one hour observations are about
$\Delta{S\\!B}_{\rm p}\approx 8$ to $10\leavevmode\nobreak\ {\rm
mag/arcsec}^{2}$ at $\rho\approx 0.4\arcsec$ for high inclination debris disks
(e.g. Engler et al. 2017, 2018; Esposito et al. 2020) or $\approx 10$ to
$12\leavevmode\nobreak\ {\rm mag/arcsec}^{2}$ around $1\arcsec$ for
protoplanetary disks (e.g. Avenhaus et al. 2018; Tschudi & Schmid 2021). We
are not aware of deep searches for polarized circumstellar emission that
combined long integrations of about 10 hours or more for a very bright star to
reach much fainter extended sources. The presented $\epsilon$ Eri data show
that much deeper limits $\Delta{S\\!B}_{\rm p}\approx 15\leavevmode\nobreak\
{\rm mag/arcsec}^{2}$ can be achieved with high-contrast imaging polarimetry
and the potential of such observations should be used more often for the
future investigation of faint circumstellar dust around bright stars.

###### Acknowledgements.

CT and HMS acknowledge the financial support by the Swiss National Science
Foundation through grant 200020_181983. SPHERE is an instrument designed and
built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg,
Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange
(Nice, France), INAF – Osservatorio di Padova (Italy), Observatoire de Gen‘eve
(Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France)
and ASTRON (Netherlands), in collaboration with ESO. SPHERE was funded by ESO,
with additional contributions from CNRS (France), MPIA (Germany), INAF
(Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received
funding from the European Commission Sixth and Seventh Framework Programmes as
part of the Optical Infrared Coordination Network for Astronomy (OPTICON)
under grant number RII3-Ct-2004-001566 for FP6 (2004–2008), grant number
226604 for FP7 (2009–2012) and grant number 312430 for FP7 (2013–2016). A.Z.
acknowledges support from ANID – Millennium Science Initiative Program –
Center Code NCN2021_080. This research is based on observations obtained with
ESO Telescopes at the La Silla Paranal Observatory under programme IDs:
0104.C-0178(A), 0104.C-0178(B), 0104.C-0178(C) and 106.2144.001.

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##  Appendix A Advanced improvements for the SPHERE/ZIMPOL data reduction

This appendix describes improvements in the ZIMPOL data reduction for the very
deep $\epsilon$ Eri observations presented in this work, which are new or go
beyond the procedures described previously (Hunziker et al. 2020; Schmid et
al. 2018). We had to improve our data analysis because this paper pushes the
limits of the ZIMPOL performance to deeper limits.

###  A.1 Camera 1 readout issue

The detector of ZIMPOL camera 1 had issues with the electronics during the
first eight nights of our observations. The analogue to digital converter of
the left read out register produced wrong results for pixels with a count
level of about 7000 ADUs as shown in Figure 16. The speckle pattern in the
coronagraphic image varies as result of the atmospheric turbulence and
therefore the affected pixels changed from frame to frame because different
regions had exposures levels near 7000 ADUs. Typically, there are around 1000
affected pixels located near the bright speckle ring or near the ’coronagraph
mask’. Fortunately, ZIMPOL takes data simultaneously with camera 1 and camera
2 and therefore the same image is taken twice with only a small scaling factor
difference because the ZIMPOL beamsplitter sends a few percent more light to
camera 2. This allowed a correction of the bad pixels in camera 1, identified
with an outlier detection procedure, by replacing them with corresponding,
scaled pixel values from camera 2. A corrected image is shown in Figure 16.

This reconstruction could save all the affected camera 1 data from the first
eight nights without producing spurious effects for the applied post-
processing procedures. Not correcting and not including the affected data in
our analysis would have reduced the effective exposure time of our programme
by about 28 % (56 $\%$ of the observing time is affected, however only one
camera, therefore 28 $\%$ of the ’photons’). Of course, the information of the
affected $\approx 1000$ pixels per camera 1 frame is lost and only duplicated
by camera 2 data, but this loss corresponds to only $\approx
0.1\leavevmode\nobreak\ \%$ of one camera 1 frame or only $\approx
0.05\leavevmode\nobreak\ \%$ of the total pixels for each integration
registered with the two detectors. ESO has solved the issue before the $\rm
9^{\rm th}$ night by replacing read-out electronics boards and two broken
cooling fans.

Figure 16: Illustration of the ZIMPOL camera 1 readout issue affecting the
left side of the image. Illustrated is one individual intensity image (two 5 s
sub-integrations = 10 s) with readout problem before and after correction of
the affected pixels.

###  A.2 Telescope polarization correction

Figure 17: Radial dependence of the telescope polarization parameters $p_{\rm
tel}$ and $\rm\delta_{\rm tel}$ for the Very Broad Band (VBB) filter. The
plotted curves are the mean curve for all $\epsilon$ Eridani observations
listed in Table 2.

For deep polarimetric observations with SPHERE/ZIMPOL it is important to apply
a correction for the residual telescope polarization as described in Schmid et
al. (2018). A polarization of about 4 % is introduced by the aluminium coated
M3 mirror in the VLT, which is for ZIMPOL polarimetry compensated with a
rotating half-wave plate and a ’crossed’ M4 mirror. The compensation is not
perfect, but the telescope effects are reduced to about $p_{\rm tel}\approx
0.5\leavevmode\nobreak\ \%$ or less. This residual polarization depends on the
parallactic angle $\theta_{\rm para}$ of the telescope and the measured values
lie in the Q/I-U/I plane on a circle with radius $p_{\rm tel}$ with position
angle $\theta_{\rm tel}=\theta_{\rm para}+\delta_{\rm tel}$ (see e.g. Hunziker
et al. 2020; Tschudi & Schmid 2021). The centre ($q_{\rm m},u_{\rm m}$) of the
circle can be offset from the origin (0,0) due to interstellar or intrinsic
polarization of the star. We measure for $\epsilon$ Eri with the VBB filter in
2019 and 2020 $p_{\rm tel}=0.26\pm 0.03$ and $\delta_{\rm tel}=32.1\pm
1.5^{\circ}$ and a centre of ($\rm q_{\rm m}=0.0032\pm
0.006\leavevmode\nobreak\ \%,u_{\rm m}=-0.0013\pm 0.004\leavevmode\nobreak\
\%$) with zero polarization as expected and in very good agreement with
previous high precision measurements for this target $\epsilon$ Eri $q_{\rm
m}:0.00284\pm 0.00056\leavevmode\nobreak\ \%$ and $u_{\rm m}:-0.00120\pm
0.00057\leavevmode\nobreak\ \%$ and $p\approx 30\cdot 10^{-6}$,
$\theta=168.5^{\circ}$ (Cotton et al. 2017). The expected polarization from
light scattering by the circumstellar dust around $\epsilon$ Eri is much
lower, less than $10^{-4}\leavevmode\nobreak\ \%$ (see Section 2.2). For such
objects without strong intrinsic polarization, a good first order correction
for the telescope polarization is obtained by the normalizations
$I_{0}=I_{90}$ and $I_{45}=I_{135}$ for each cycle. This is equivalent to
setting the integrated polarization to zero: $Q=0$ and $U=0$. Not correcting
for the telescope polarization would introduce an $I\rightarrow Q,U$ cross
talk and intensity speckles would be visible as polarized features with a
relative strength at the level of the telescope polarization $p_{\rm tel}$.

For $\epsilon$ Eri, we need to consider also second order effects of the
telescope polarization. The two parameters describing the telescope
polarization $p_{\rm tel}$ and $\delta_{\rm tel}$ depend on the wavelength
(Schmid et al. 2018) and this is an issue for very deep polarimetry in the VBB
filter with large bandwidth $590-890$ nm and therefore significantly different
instrument polarization for short and long wavelengths. This is shown in
Tschudi & Schmid (2021) for the $R^{\prime}$ and the $I^{\prime}$ filter which
cover roughly the short and long wavelength parts of the VBB filter,
respectively. In addition, the SPHERE AO PSF for the VBB filter is a
superposition of different radial profiles for the different wavelengths. This
produces a 2-dimensional polarization effect which cannot be corrected with a
single vector $(p_{\rm tel},\delta_{\rm tel})$ without leaving second order
calibration errors.

The dominant feature for the radial dependence of the instrument polarization
is the wavelength dependent location of the strong PSF speckle ring, which is
defined by the control radius ($20\,\lambda/D$) of the SPHERE AO system. This
ring is for the $R$-band at a separation of $\rho\approx 0.35^{\prime\prime}$
and for the $I$-band at $\rho\approx 0.45^{\prime\prime}$ (Figure 11 in Schmid
et al. 2018) and therefore the $R$-band instrument polarization dominates at
smaller separations while the $I$-band polarization contributes strongly for
$\rho\geq 0.45^{\prime\prime}$. We could measure clearly a corresponding
radial dependence of the telescope polarization $p_{\rm tel}(r)$ and
$\delta_{\rm tel}(r)$ with amplitudes of $\pm 0.01\leavevmode\nobreak\ \%$ and
$\pm 2^{\circ}$ for the mean radial curve as illustrated in Figure 17. The
curves $p_{\rm tel}(r)$ and $\delta_{\rm tel}(r)$ look similar in shape from
night to night but show small variations $\Delta p_{\rm
tel}<0.02\leavevmode\nobreak\ \%$ and $\Delta\delta_{\rm tel}<1^{\circ}$
because of PSF variations introduced by different observing conditions. We
don’t see a long term trend within the 13 months covered by our observations,
but on longer timescales one should expect systematic changes because of the
evolution of the coatings of the telescope mirror M3 and the first folding
mirror M4 in SPHERE.

For the general case with a ($q_{\rm m},u_{\rm m}$) offset from (0,0) we would
determine the mean $p_{\rm tel}(r)$ and $\delta_{\rm tel}(r)$ curves of each
night to correct the telescope polarization of that night. From these profiles
we construct two interpolated 2d maps to correct each polarization image
pixel-wise depending on the parallactic angle of that data. In the special
case of $\epsilon$ Eri with no offset polarization $(q_{\rm m},u_{\rm
m})=(0,0)$, we can measure Q/I, respective U/I in every image individually and
correct it to 0. We do this for each radial annuli separately to fully account
for the radial dependence as described above. The advantage of this method is
that PSF variations, which can happen within minutes, and higher order
telescope polarization effects are also corrected.

This second order correction for the telescope polarization is not crucial for
the search of a planet outside the speckle ring $\rho>0.6\arcsec$ where the
dependence of $p_{\rm tel}(r)$ and $\delta_{\rm tel}(r)$ is smooth. At small
separation ($<0.5\arcsec$), the residual pattern of strong speckles is
slightly reduced. The second order correction should however improve the
search of an extended weak signal from dust scattering. For example, the
effect is clearly seen in the $Q$ and $U$ images of the debris disk
observations of HIP 79 977 by Engler et al. (2017, Figure 3) as over-corrected
central area $r<0.5\arcsec$. At the time of that analysis the origin of this
calibration problem was unknown.

###  A.3 Beamshift correction

Figure 18: Beamshift correction parameters as a function of the pointing
(local sidereal time). Illustrated as example is the shift in X direction for
camera 2 and 0 phase images. (A): Beamshift as measured in the unsaturated
(ND2) PSF images. Different colour represent the different polarization images
$Q^{+}$, $Q^{-}$, $U^{+}$ and $U^{-}$. (B): Analogous figure as measured in
the coronagraphic images. The measurement accuracy is reduced especially for
bad observing conditions. (C): Example for the iterative process of Gaussian
process fitting and outlier detection to calculate the fit of this parameter
for $Q^{-}$.

It is important for polarimetric differential imaging (PDI) performance that
the differential aberrations between the opposite polarization directions
$I_{\parallel}$ and $I_{\perp}$ are very small so that unpolarized speckles
and other PSF features cancel out by the subtraction of the two images and the
polarized planet signal is easier to detect. The ZIMPOL design was optimized
to reduce such differential aberrations and the two polarization states are
for example recorded with the same detector pixels. However the inclined third
mirror M3 of the telescope, the 45∘ pupil tip-tilt mirror and the three image
derotator mirrors mainly introduce a, wavelength and telescope pointing
dependant, differential beamshift of up to 0.3 pixels (or $\approx$ 1 mas)
between $I_{\parallel}$ and $I_{\perp}$ Schmid et al. (2018); Hunziker et al.
(2020). For a given pointing of the telescope, meaning the same altitude and
parallactic angle, the beamshift effect is the same and therefore we can sort
the $\epsilon$ Eridani data and derive the beamshift parameter as a function
of their local sideral time (LST). There are many beamshift parameters as the
shifts are different in X and Y directions in the image, different for the
polarization images $Q^{+}$, $Q^{-}$, $U^{+}$, $U^{-}$, different for camera 1
and camera 2 and different for the $0$ and $\pi$ phase images. To measure the
beamshift accurately an unsaturated point source is required as available from
the regularly taken non-coronagraphic observation using the ND2 filter. The
used Lyot coronagraph (V$\\_$CLC$\\_$MT$\\_$WF) has a slightly transparent
mask and under good conditions it is possible to see the stellar PSF peak
through the mask and to measure the beamshift also in the coronagraphic
observations. In the top panel of Figure 18(A) the non-coronagraphic
measurements of the $\Delta X(\rm 0\ phase)$ beamshift parameters for $Q^{+}$,
$Q^{-}$, $U^{+}$, $U^{-}$ are displayed. The same parameters measured for the
coronagraphic data can be seen in the middle panel of Figure 18(B). It is
obvious that the dispersion is larger for the measurements with coronagraph,
however there are many more images and no time gaps between the images, but
sometimes the determination of the beamshift fails for the coronagraphic data
(many of these points are far outside the showed y-axis in the middle panel of
Figure 18(B)). Unfortunately, there exists no model for SPHERE/ZIMPOL which
could predict the beamshift effects. Therefore we use all the existing
$\epsilon$ Eri data and derive the shift corrections from the best fit to the
data. For this we calculate a Gaussian process fit, apply an outlier detection
and removal method and iterate the procedure a few times (see bottom panel of
Figure 18(C)). The fitting procedure is particularly important for data taken
under bad conditions (seeing ¿ 1$\arcsec$). All the images are visually
checked after applying the beam shift correction.

###  A.4 Chargetrap correction

Both ZIMPOL detectors consist of alternating open and masked rows (512 rows
each) with 1024 pixels. In the polarimetric ZIMPOL modes the charges created
in the illuminated rows are shifted up and down in synchronization with the
polarimetric modulation (frequency depends on fast or slow polarization mode)
(Schmid et al. 2018). The final detector frame consists of an “even rows”
subframe with one polarization state $\rm I_{\perp}$, and an “odd rows”
subframe for the opposite polarization state $\rm I_{\parallel}$. In column
direction the subframes are then interpolated in a flux conserving manner to
create 1024x1024 images. Although the charge transfer efficiency of the CCDs
are better than 99.9995 $\%$ there exist pixels which do not shift electron
charges perfectly. For example a pixel can block one electron which is not
down shifted but in the following up-shift it is transferred. Because of the
fast (de)modulation such a trap can cause a hole of many electrons in one
polarization state and a corresponding spike in the other. To get rid of this
effect in ZIMPOL polarimetric modes there are always an even number of
subintegrations and in every second subintegration the charge shifting is
reversed with respect to the polarization modulation. The images from zero to
$\pi$ phase subintegration are then combined to create a double difference in
which the charge trap effects are cancelled (Schmid et al. 2018; Gisler et al.
2004; Schmid et al. 2012). Unfortunately, for the intensity frame $\rm
I=I_{\perp}+I_{\parallel}$ derived from the polarimetric data the charge traps
do not vanish with the combination of the subintegrations. The charge traps
produce a negative-positive pattern with a very specific appearance in column
direction. First a pixel with too many counts, then an interpolated ’neutral’
pixel and then a pixel with too few counts or the other way around. Sometimes
secondary and tertiary pixels are also slightly affected. We search for this
pattern in each image and correct it by shifting counts from a ’spike’ to a
corresponding ’hole’ in a flux conserving manner as illustrated in Figure 19).
More than 99 $\%$ of the charge traps can be recognized and corrected with
this method. This correction is especially helpful for polarimetric p2 mode
when the field is fixed and one has only a few long exposures.

Figure 19: Illustration of the charge trap correction for ZIMPOL intensity
images [$\rm ct\ DIT^{-1}\ px^{-1}$] obtained in polarimetric mode.

##  Appendix B Detection maps

Figure 20: Polarized light $Q_{\phi}$ detection maps showing the Gaussian
significance of the individual pixels. North is up, east is to the left, and
the image field of view is the same as in Figure 7. Top: Detection maps of all
the individual nights. Bottom: Detection maps of the four epochs.


This verifies a weaker form of Conjecture 1.1(3) for the motive $M(\pi_f^\vee)(3)$ at the cost of supposing that  $\mathcal{I}_S(\Phi, \Psi^{[\lambda,\mu]} , 1)$ is non-zero for some $\mu $.
\end{remark}

\begin{remark}
By Proposition \ref{prop:Hdist}, an automorphic representation $\pi$ which satisfies the hypotheses of Theorem \ref{theoremcyclebetti2} is $\H$-distinguished, namely the map $\mathcal{P}_\H \in {\rm Hom}_{\H(\A)} ( \pi, \mathbf{1})$ defined by
\[ \Psi \mapsto \mathcal{P}_\H(\Psi) := \int_{Z(\A)\H(\Q) \backslash \H(\A)} \Psi(h) d h   \]
is not identically zero. Then, asking $\mathcal{I}_S(\Phi, \Psi^{[\lambda,\mu]} , 1) \ne 0$ for some $\mu$ is equivalent to asking that the map obtained as the composition of $\mathcal{P}_\H$ with an $\H(\R)$-equivariant embedding $\pi_\infty \to \pi$ restricts non-trivially to the minimal $K_\infty$-type of $\pi_\infty$.
\end{remark}

\begin{remark}
Let $\sigma$ be a cuspidal automorphic representation of $G_2(\A)$, unramified outside a finite set $S$ of primes, such that $\sigma_\infty$ is the quaternionic discrete series $\sigma_n$ for $n \geq 2$ of Definition \ref{def:quatdiscseries}, and  $\sigma_p$ is Steinberg at a finite prime $p \in S$. Then \cite[Theorem 3.1]{ganmultcubic} and Proposition \ref{Wallachprop} imply that there always exists a cubic totally real \'etale algebra $R$ such that $\sigma$ supports a Fourier coefficient associated to $R$. Assume that such algebra splits as $\Q \times F$.
By Theorem \ref{thm:thetaliftproperties}, $\Theta(\sigma)$ is a non-zero and cuspidal automorphic representation of $\PGSp_6(\A)$. If we let $\pi$ be an irreducible constituent of $\Theta(\sigma)$, then $\pi$ satisfies all the running assumptions of Theorem \ref{theoremcyclebetti2} up to asking that $\mathcal{I}_S(\Phi, \Psi^{[\lambda,\mu]} , 1) \ne 0$, where now $\lambda = (2n-4,n-2,n-2,0)$ and $\mu = n-2$.  By \cite[Proposition 5.1]{Pollack-Shah}, there exists a cusp form $\tilde{\Psi}^{[\lambda,\mu]}$ in $\pi$, with $\Psi_f$ and $\tilde{\Psi}_f$ equal away from $S$, and a Schwartz-Bruhat function $\Phi'$ such that
\[\mathcal{I}_S(\Phi', \tilde{\Psi}^{[\lambda,\mu]} , s) = \mathcal{I}_\infty(\Phi, \Psi^{[\lambda,\mu]} , s).\]
Hence, up to modifying the components of $\Psi$ at the finite places in $S$, our hypothesis reduces to the non-triviality of $\mathcal{I}_\infty(\Phi, \Psi^{[\lambda,\mu]}, 1)$. The explicit formulae for the Fourier coefficient $\Psi^{[\lambda,\mu]}_{U_P,\psi_D}$ given in Proposition \ref{prop:comparisonbetweenFC} allow us to write $\mathcal{I}_\infty(\Phi, \Psi^{[\lambda,\mu]}, s)$ as an archimedean integral involving the generalized $G_2$-Whittaker functions studied in [47] and a function $f_\infty$ in the minimal representation $\Pi_\infty$ of $E_7(\R)$, whose projection to $\pi_\infty$ equals $A^{[\lambda,\mu]} \cdot \Psi_\infty$.  The uniqueness of the former proved in \emph{loc.cit.} seems to suggest the possibility of calculating $\mathcal{I}_\infty(\Phi, \Psi^{[\lambda,\mu]}, s)$ up to determining $f_\infty \in \Pi_\infty$  explicitly. We hope to come back to this problem in the future.
\end{remark}

\subsection{Galois representations of $G_2$-type and a question of Gross and Savin}

By the work of [41], under some mild conditions, we can associate to the cuspidal automorphic representation $\sigma$ of $G_2(\A)$ a $G_2$-valued Galois representation $\rho_\sigma$ which satisfies the expected properties. Using the exceptional theta correspondence, this representation is realized in the middle degree cohomology $H^6_{\text{\'et}}(\Sh_{\G,\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))$ of the Siegel sixfold $\Sh_{\G}$, where we have denoted with $H^6_{\text{\'et}}(\Sh_{\G,\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))$ the direct limit of the cohomology at level $U$ as $U$ varies. This direct limit is a smooth admissible representation of $\G(\A_f)$, endowed with an action of $\mathrm{Gal}(\overline{\Q} / \Q)$ commuting with the one of $\G(\A_f)$. The theorem below for the compact form of $G_2$ is shown in \cite[Theorem 11.1 and Corollary 11.3]{KretShin}. The same proof works for the split form of $G_2$ as long as one has some information on its lift to $\mathrm{PGSp}_6$, and we only sketch it for the convenience of the reader.
\begin{theorem}\label{thmonsplitGalois}
Let $\sigma$ be an automorphic cuspidal representation of $G_2(\A)$ such that
\begin{itemize}
\item $\sigma_\infty$ is a quaternionic discrete series;
\item $\sigma_p$ is the Steinberg representation at some finite prime $p$;
\item $\sigma$ supports a Fourier coefficient to a cubic algebra of the form $\Q \times F$.
\end{itemize}
Then, for each prime $\ell$ and $\iota : \C \cong \overline{\Q}_\ell$, there exists a Galois representation $\rho_\sigma = \rho_{\sigma, \iota} : \mathrm{Gal}(\overline{\Q} / \Q) \to G_2(\overline{\Q}_\ell)$ such that
\begin{itemize}
\item For every finite place $v \neq \ell$ where $\sigma$ is unramified, $\rho_\sigma$ is unramified at $v$. Moreover, the semisimple part of $\rho_\sigma({\rm Frob}_v)$ is conjugate to the Satake parameter $\iota(s_{\sigma_v})$ in $G_2(\overline{\Q}_\ell)$.
\item $\rho_{\sigma_\ell}$ is de Rham, and it is crystalline if $\sigma$ if unramified at $\ell$.
\item $\zeta \circ \rho_\sigma = \rho_\pi$, where $\pi$ is a theta lift of $\sigma$, and $\zeta: G_2(\C) \to \mathrm{Spin}_7(\C)$ is the embedding appearing in \eqref{eq:diagramLF}.
\item the Zariski closure of the image of $\rho_\sigma$ maps onto either the image of a principal $\SL_2$ in $G_2$ or onto $G_2$.
\end{itemize}
\end{theorem}

\begin{proof}
By Theorem \ref{thm:thetaliftproperties}, the big theta lift $\Theta(\sigma)$ is non-zero and cuspidal. Moreover, if we let $\pi$ denote an irreducible subquotient of $\Theta(\sigma)$, its archimedean component $\pi_\infty$ is a discrete series of Hodge type $(3,3)$, and $\pi_p$ is the Steinberg representation. By \cite[Theorem A]{KretShin}, there exists a representation $\rho_\pi: \mathrm{Gal}(\overline{\Q} / \Q) \to {\rm Spin}_7(\overline{\Q}_\ell)$ attached to $\pi$. By the proof of \cite[Theorem 11.1]{KretShin}, one has that the image of $\rho_\pi$ is contained in $G_2(\overline{\Q}_\ell)$, and thus we have $\rho_\sigma$ such that $\zeta \circ \rho_\sigma = \rho_\pi$ for a suitable choice of embedding $\zeta: G_2(\C) \to \mathrm{Spin}_7(\C)$ fitting in the diagram \eqref{eq:diagramLF}.
Hence, by \cite[Theorem A]{KretShin} and Proposition \ref{propositiononlocalthetaprop}(1), the representation $\rho_\sigma: \mathrm{Gal}(\overline{\Q} / \Q) \to G_2(\overline{\Q}_\ell)$ satisfies the desired first three properties.
Finally, by \cite[Theorem A $(v)$]{KretShin}, the Zariski closure of $\rho_\pi$ must map onto either a principal $\SL_2$ in $\rm{ SO}_7 \cap G_2$, or $G_2$.
\end{proof}

\begin{remark}
The hypotheses of the Theorem above can be substantially weakened to asking that $\sigma_\infty$ is a discrete series and $\sigma_p$ is Steinberg. This follows from establishing a complete correspondence of Fourier coefficients between $\sigma$ and its theta lift and by the results of [41]. A proof of this fact is beyond the scope of this article and it will appear in forthcoming work.
\end{remark}

By construction, the composition of the Galois representation $\rho_\pi$ (and thus $\rho_\sigma$) with the Spin representation appears in the middle degree cohomology of $\Sh_\G$. If
$\sigma_\infty$ has Harish-Chandra parameter $x \epsilon_1 + y \epsilon_2$, with $x - 3 \geq y -1 \geq 0 $ and $x-y$ even, the archimedean component $\pi_\infty$ is a discrete series in the $L$-packet of $V^{\lambda}$, with $\lambda = ( x-3, \tfrac{x+y-4}{2},\tfrac{x-y-2}{2},0)$. By Lemma \ref{lemmaonBettiisotypicpart}, the $\pi_f^\vee$-isotypic component of $H^6_{\text{\'et},!}(\Sh_{\G,\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))$ is 8-dimensional, and we have
\[
H^6_{\text{\'et},!}(\Sh_{\G,\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))[\pi_f^\vee] \otimes \overline{\Q}_\ell = V_{{\rm Spin} \circ \rho_\pi} \otimes \pi_f^\vee
= V_{{\rm Spin} \circ \zeta \circ \rho_\sigma} \otimes \pi_f^\vee.
\]
If the image of $\rho_\sigma$ is Zariski dense in $G_2(\overline{\Q}_\ell)$, we have ${\rm Spin} \circ \zeta \circ \rho_\sigma = {\rm Std} \circ \rho_\sigma \oplus \mathbf{1}$, where ${\rm Std} \circ \rho_\sigma$ is the irreducible ``standard'' Galois representation attached to $\sigma$. If not, by Theorem \ref{thmonsplitGalois}, the image of $\rho_\sigma$ is Zariski dense onto a principal $\xi: \SL_2(\overline{\Q}_\ell) \to G_2(\overline{\Q}_\ell)$. Then, the  branching law of \cite[(7.1)]{GrossMinusculePrincipal} gives that ${\rm Spin} \circ \zeta \circ \rho_\sigma= {\rm Sym}^6 \circ \rho_\sigma \oplus \mathbf{1}$, where $ {\rm Sym}^6 \circ \rho_\sigma$ is the irreducible symmetric sixth power Galois representation attached to $\sigma$. Denote by $M_\ell(\pi_f)$ the Galois representation $V_{{\rm Spin} \circ \rho_\pi}$ and let $M_\ell(\sigma_f)$ be either the Galois representation $V_{{\rm Std} \circ \rho_\sigma}$ or $V_{{\rm Sym}^6 \circ \rho_\sigma}$. Then, we have that $  M_\ell(\sigma_f)^{G_\Q}  = 0$ and  $ M_\ell(\pi_f)$ decomposes as the direct sum of $M_\ell(\sigma_f)$ and the trivial representation $\mathbf{1}$ with multiplicity $1$.

\begin{remark}
In the case where $\rho_\sigma$ is not Zariski dense in $G_2(\overline{\Q}_\ell)$,  the Satake parameter $s_{\sigma_p} \in \xi(\SL_2(\C)) $ for any unramified prime $p$. By Langlands reciprocity principle, $\sigma$ should be the functorial lift of a cuspidal automorphic representation $\tau$ of $\mathrm{PGL}_2(\A)$, while $V_{{\rm Sym}^6 \circ \rho_\sigma}$ should be a geometric realization of the motive of the symmetric sixth power of $\tau$.
\end{remark}

Tate conjecture predicts the existence of a cycle which gives rise to the trivial representation appearing in the decomposition above. In [27], Gross and Savin, inspired by local computations, conjectured that this cycle should come from a Hilbert modular 3-fold inside $\Sh_\G$. The Theorem below confirms this expectation for cuspidal automorphic representations $\sigma$ of $G_2$  which satisfy the hypotheses of Theorem \ref{thmonsplitGalois}. In particular, we suppose that $\sigma$ supports a Fourier coefficient for a (necessarily totally real) cubic \'etale algebra $\Q \times F$. Let $\mathcal{Z}_{\H, \text{\'et}}^{[\lambda,\mu]} := \rm{cl}_{\text{\'et}}(\mathcal{Z}_{\H, \mathcal{M}}^{[\lambda,\mu]}) \in H^6_{\text{\'et}}(\Sh_\G(U)_{\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))^{G_\Q}$ be the \'etale realization of the motivic class $\mathcal{Z}_{\H, \mathcal{M}}^{[\lambda,\mu]}$ (see Definition \ref{etaleclass}), where $\H = \GL_2 \boxtimes \GL_{2,F}^*$. Since the construction of the class is compatible with pull-backs, the cycles at finite level are matched into a class $\mathcal{Z}_{\H, \text{\'et}}^{[\lambda,\mu]} \in H^6_{\text{\'et}}(\Sh_{\G,\overline{\Q}}, \mathcal{V}^\lambda_\mathfrak{l}(3))^{G_\Q}$.
By composing the projection to the $\pi_f^\vee$-isotypic component together with the projection given by the vector $\Psi_f \in \pi_f$, we get
\[ \mathcal{Z}_{\H, \text{\'et}}^{\sigma} : = \Psi_f( {\rm pr}_{\pi^\vee}(\mathcal{Z}_{\H, \text{\'et}}^{[\lambda,\mu]})) \in ( M_\ell(\sigma_f) \oplus \mathbf{1})^{G_\Q}=\mathbf{1}. \]
\begin{theorem} \label{TheoGS}
Let $\sigma$ be an automorphic cuspidal representation of $G_2(\A)$ such that
\begin{itemize}
\item $\sigma_\infty$ is a quaternionic discrete series;
\item $\sigma_p$ is the Steinberg representation at some finite prime $p$;
\item $\sigma$ supports a Fourier coefficient to a cubic algebra of the form $\Q \times F$.
\end{itemize}
Assume that the integral $\mathcal{I}_S(\Phi, \Psi^{[\lambda,\mu]} , 1)$ is non-zero for some Schwartz-Bruhat function $\Phi$ and $\Psi^{[\lambda,\mu]}$ as in \S \ref{subsec:cycleclassformula}. Then the class $ \mathcal{Z}_{\H, \text{\'et}}^{\sigma}$ generates the trivial sub-representation $\mathbf{1}$ of $M_\ell(\pi_f)$.
\end{theorem}

\begin{proof}
By the comparison theorem between \'etale and Betti cohomology \cite[Expos\'e XI, Theorem 4.4 (iii)]{sga} and Theorem \ref{theoremcyclebetti2}, we know that the projection   ${\rm pr}_{\pi^\vee} \mathcal{Z}_{\H, \text{\'et}}^{[\lambda, \mu]}$ to $M_\ell(\pi_f) \otimes \pi_f^\vee$ generates a one dimensional subspace, which is trivial for the action of the Galois group. As we have explained above, the image of $\rho_\sigma$ is either dense in $G_2(\overline{\Q}_\ell)$ or in $\SL_2(\overline{\Q}_\ell) \to  {\rm PGL}_2(\overline{\Q}_\ell) \hookrightarrow G_2(\overline{\Q}_\ell)$. In either case, the representation $M_\ell(\sigma_f)$ is irreducible and the trivial factor $\mathbf{1}$ in $M_\ell(\pi_f)$ is hence generated by the image of $ \mathcal{Z}_{\H, \text{\'et}}^{\sigma}$.
\end{proof}

%We conclude by rephrasing the above result in terms of Bloch-Kato conjecture.

%Let $\pi$ be as in Theorem \ref{theoremcyclebetti2}. And denote by $M_\ell(\pi_f)$ the Galois representation of $\pi$. Then
%\[ \dim H^1_f(\Q, M_\ell(\pi_f)) + \dim H^0(\Q, M_\ell(\pi_f)) \geq \mathrm{ord}_{s = 1} L(s, \pi, Spin). %\]

\begin{comment}
\subsection{The cycle class formula for theta lifts from $G_2$}
In this section, we look at the projection of the Tate classes in the isotypic components of the theta lifts studied in Theorem \ref{thm:thetaliftproperties}. We suppose that $\sigma$ is a cuspidal automorphic representation of $G_2(\A)$, unramified outside a finite set $S$ of primes, such that \begin{itemize}
\item $\sigma_\infty$ is the quaternionic discrete series $\sigma_n$ for $n \geq 2$ of Definition \ref{def:quatdiscseries};
\item $\sigma_p$ is Steinberg at a finite prime $p \in S$;
\item $\sigma$ supports a Fourier coefficient for the Heisenberg parabolic associated to $\Q \times F$, with $F$ a quadratic \'etale $\Q$-algebra.
\end{itemize}

\begin{remark}\label{remarkonFCqds}
If $\sigma_\infty$ is  the quaternionic discrete series $\sigma_n$ of Definition \ref{def:quatdiscseries}, \cite[Proposition 8.4]{Gan-Gross-Savin} and Proposition \ref{Wallachprop} imply that there always exists a cubic totally real \'etale $\Q$-algebra $R$ such that $\sigma$ supports a Fourier coefficient associated to $R$.  If $R$ does not split as $\Q \times F$, then the theta lift of $\sigma$ supports a Fourier coefficient for the Siegel parabolic associated to a symmetric matrix of rank three. Since we will not need this result, we leave the proof to the reader. Joint with the results of \cite[Theorem A.9]{harris-khare-thorne}, our results give a complete correspondence of Fourier coefficients of $\sigma$ and $\Theta(\sigma)$.
\end{remark}

By Theorem \ref{thm:thetaliftproperties}, $\Theta(\sigma)$ is a  non-zero and cuspidal automorphic representation of $\PGSp_6(\A)$; if we let $\pi$ be an irreducible constituent of $\Theta(\sigma)$, then  \begin{itemize}
\item $\pi_\infty$ is the discrete series of Hodge type $(3,3)$ in the L-packet of $V^\lambda$ with $\lambda = (n-2) (2,1,1,0)$;
\item $\pi_p$ is Steinberg;
\item $\pi$ supports a Fourier coefficient of type $(4 \, 2)$ associated to the totally real quadratic \'etale $\Q$-algebra $F$.
\end{itemize}

If we let $\Psi = \Psi_\infty \otimes \Psi_f$ be a cusp form in $\pi$ such that $\Psi_\infty$ is a highest weight vector of the minimal $K_\infty$-type $\tau_{(n, n, -2n)}$ of $\pi_\infty |_{\Sp_6(\R)}$, then the form \[\Psi^0:=\Psi^{[\lambda,n-2]} = \Psi^0_\infty \otimes \Psi_f,\]
with $\Psi^0_\infty$ a generator of the 1-dimensional weight $(0,0,0)$ space of $\tau_{(n,n,-2n)}$. By Proposition \ref{propositiononlocalthetaprop}, the Satake parameter $s_{\pi_\ell} \in \zeta(G_2(\C))$ for all $\ell \not \in S$, hence, by Proposition \ref{onsimplepoles}, the partial Spin $L$-function $L^S(s, \pi, {\rm Spin})$ has a simple pole at $s=1$ with residue proportional to $L^S(1, \sigma, {\rm Std})$.

Let $\H = \GL_2 \boxtimes \GL_{2,F}^*$; Theorem \ref{theoremcyclebetti2} gives the following.

\begin{corollary}
Suppose that $\mathcal{I}_S(\Phi, \Psi^0 , 1) \ne 0$. Then ${\rm pr}_{\pi^\vee} \mathcal{Z}_{\H, \mathcal{H}}^{[\lambda,n-2]}$ and its Hecke translates generate $H^1_\mathcal{H}(M(\pi_f^\vee)_{\R}(4))$.
\end{corollary}

This verifies Beilinson's conjecture for $M(\pi_f^\vee)_{\R}(4)$ at the cost of supposing that $\mathcal{I}_S(\Phi, \Psi^0 , 1)$ is non-zero.

\begin{remark}
By Proposition \ref{prop:Hdist}, we know that $\pi$ is $\H$-distinguished, namely the map $\mathcal{P}_\H \in {\rm Hom}_{\H(\A)} ( \pi, \mathbf{1})$ defined by
\[ \Psi \mapsto \mathcal{P}_\H(\Psi) := \int_{Z(\A)\H(\Q) \backslash \H(\A)} \Psi(h) d h   \]
is not identically zero. Then, asking $\mathcal{I}_S(\Phi, \Psi^0 , 1) \ne 0$ is equivalent to asking that the map obtained as the composition of $\mathcal{P}_\H$ with an $\H(\R)$-equivariant embedding $\pi_\infty \to \pi$ restricts non-trivially to the minimal $K_\infty$-type of $\pi_\infty$.
\end{remark}

In the remainder of the section, we study this integral. First of all, notice that \cite[Proposition 5.1]{Pollack-Shah} (see also the proof of \cite[Corollary 5.18]{CLR})
implies that there exists a cusp form $\tilde{\Psi}^0 = \Psi_\infty^0 \otimes \tilde{\Psi}_f$, with $\Psi_f$ and $\tilde{\Psi}_f$ equal away from $S$, and a Schwartz-Bruhat function $\Phi'$ such that
\[\mathcal{I}_S(\Phi', \tilde{\Psi}^0 , s) = \mathcal{I}_\infty(\Phi, \Psi^0 , s),\]
where \[\mathcal{I}_\infty(\Phi, \Psi^0, s) = \int_{ U_{B_\H}(\R)  Z_\G(\R) \backslash \H(\R)} f(h_1, \Phi_\infty, s)\Psi_{U_P,\psi_D}^0 (h) dh. \]
Hence, up to modifying the components of $\Psi$ at the finite places in $S$, we are left to study $\mathcal{I}_\infty(\Phi, \Psi^0, 1)$.

\begin{lemma}\label{lemmafromtechnical}
Suppose that $\Psi^0 = \Theta(f_{\rm min}, \varphi)$, then we have that \[ \Psi_{U_P,\psi_D}^0 (h) = C'' \cdot \int_{U_D(\Q_S)\backslash G_2(\Q_S)} f_{{\rm min},S}((g_S h)^{-1}A_0) \varphi_{U_H, \psi_{H,D}}(g_S) d g_S,  \]
where $C''$ is a non-zero constant depending on $A_0$.
\end{lemma}
\begin{proof}
This directly follows from the proof of Proposition \ref{prop:comparisonbetweenFC} and Lemma \ref{technical}.
\end{proof}

Let $T_\H(\R)$ denote the maximal diagonal torus of $\H(\R)/Z_\G(\R)$. Fix the isomorphism  $ \GL_1 \times \GL_{1,F} \simeq T_\H, \, (t,t') \mapsto \iota(t, t')$ as in \cite[Definition 2.1]{Pollack-Shah}. Then, thanks to Lemma \ref{lemmafromtechnical}, we can prove the following.

\begin{proposition}
Let $ \Psi^0 = \Theta(f_{\rm min}, \varphi)$. The integral $\mathcal{I}_\infty(\Phi, \Psi^0, s) $ equals (up to a non-zero constant) to
\[ \pi^{-s} \Gamma(s) \int_{U_D(\Q_S)\backslash G_2(\Q_S)}  \varphi_{U_H, \psi_{H,D}}(g_S) \int_{ \iota(t,t') \in T_\H(\R)}   |t|^{s-3}|t'|^2 f_{{\rm min},S}((g_S \iota(t,t'))^{-1}A_0) dt dt' dg_S.\]
\end{proposition}
\begin{proof}

The Lemma above implies that up to a non-zero constant $\mathcal{I}_\infty(\Phi, \Psi^0, s) $ equals
\begin{align*}
\int_{ U_{B_\H}(\R)  Z_\G(\R) \backslash \H(\R)} f(h_1, \Phi_\infty, s) \int_{U_D(\Q_S)\backslash G_2(\Q_S)} f_{{\rm min},S}((g_S h)^{-1}A_0) \varphi_{U_H, \psi_{H,D}}(g_S) d g_S dh.
\end{align*}
Changing the integrals, we can rewrite it as
\begin{align*}
\int_{U_D(\Q_S)\backslash G_2(\Q_S)}  \varphi_{U_H, \psi_{H,D}}(g_S)  \int_{ U_{B_\H}(\R)  Z_\G(\R) \backslash \H(\R)} f(h_1, \Phi_\infty, s)  f_{{\rm min},S}((g_S h)^{-1}A_0) dh d g_S.
\end{align*}
We now analyze the inner integral. By the compatibility between the local and global theta lift, the infinity component of $\Psi^0$ and thus of $f_{{\rm min},S}$ is fixed by the maximal compact of $\H(\R)$. Therefore, for a fixed $g_S \in G_2(\A_S)$, if we use the Iwasawa decomposition $B_\H(\R)K_\H = H(\R)$, the inner integral becomes
\begin{align}\label{innerintegralarch}\int_{ T_\H(\R)} \delta_{B_\H}^{-1}(t)f(t_1, \Phi_\infty, s)  f_{{\rm min},S}((g_S t)^{-1}A_0) dt.  \end{align}
Since $\Phi_\infty(x,y) = e^{-\pi(x^2+y^2)}$, we have \[ f(\left( \begin{smallmatrix} t & \\ & 1 \end{smallmatrix}\right), \Phi_\infty,s) = \pi^{-s} \Gamma(s) |t|^s, \]
so we can write \eqref{innerintegralarch} as \[ \pi^{-s} \Gamma(s)\int_{ (t,t') \in \R^\times \times \GL_{1,F}(\R)}   |t|^{s-3}|t'|^2 f_{{\rm min},S}((g_S \iota(t,t'))^{-1}A_0) dt dt',   \]
where we have used that $\delta_{B_\H}(\iota(t,t')) = |t|^3|t'|^{-2}$ (cf. \cite[Lemma 2.2]{Pollack-Shah}).
\end{proof}

\end{comment}

\begin{comment}
\subsection{Fourier coefficients for quaternionic discrete series}
In what follows, we describe a result of Pollack on the explicit description of Fourier coefficients for automorphic forms of $G_2$ whose infinity component lies in the the quaternionic discrete series $\sigma_n$ for some $n \geq 2$. We denote by $\tau_n$ their minimal $K_{G_2}$-type.

Let us consider \[ \varphi \in {\rm Hom}_{G_2(\R)}( \sigma_n , \mathcal{A}_{\rm cusp} ( G_2(\Q) \backslash G_2(\A))).\]
Restricting to $K_{G_2}$, $\varphi$ defines a $K_{G_2}$-equivariant map \[ \varphi : \tau_n \longrightarrow  \mathcal{A}_{\rm cusp} ( G_2(\Q) \backslash G_2(\A)),\]
which, by duality, gives \[ F_\varphi : G_2(\A) \longrightarrow \tau_n^\vee,
\; F_\varphi (g)(v):= \varphi(v)(g).\]

Recall that we have considered the unitary character \[\psi:= \psi_{H,D}: U_H(\Q)\backslash U_H(\A) \longrightarrow \C^\times \]
of the unipotent radical of the Heisenberg parabolic associated to the cubic algebra $\Q \times \Q(\sqrt{D})$. In view of \anto{Proposition 6.21?}, we are interested in calculating the values of the Fourier coeffcient

\[ F_{\varphi, \psi}(g):= \int_{[U_H]} F_\varphi(ug) \psi^{-1}(u) du,\]

which defines a function $ F_{\varphi, \psi}: G_2(\A) \to \tau_n^\vee$.  Recall the following.

\begin{theorem}[Wallach]
Let $\chi$ be a character of $U_H(\R) / [U_H,U_H](\R)$. There is (at most) a one dimensional space of $\chi$-equivariant linear functionals on $\sigma_n$ with moderate growth. In particular, \[ {\rm dim }\; {\rm Hom}_{U_H(\R)} (\sigma_n , \chi) = 1, \]
exactly when $\chi$ corresponds to an \'etale real cubic algebra of $\Q$.
\end{theorem}

Note that $\psi$ corresponds to an \'etale real cubic algebra of $\Q$, thus, by the Theorem above, there is hope to explicit the value of $F_{\varphi, \psi}$ at $g \in G_2(\R)$.
The function ${F_{\varphi, \psi}}_{|_{G_2(\R)}} $ belongs to the space $\mathcal{C}^\infty(U_H(\R) \backslash G_2(\R) / K_{G_2}, \psi, \tau_n^\vee)$ defined as \[\{f: G_2(\R) \to \tau_n^\vee \;:\; f(ugk)=\chi(u) (\tau_n^\vee(k) \cdot f(g)),\; u \in U_H(\R),\; k \in K_{G_2} \}.\]
Let $\mathcal{C}^\infty_{\rm mg}(U_H(\R) \backslash G_2(\R) / K_{G_2}, \psi, \tau_n^\vee)$
be the subspace consisting of functions with moderate growth.

By using the differential equations that arise from Schmid's
realization of discrete series, in [47] Pollack calculates explicitly a basis for this space. Before stating his result, we recall that using the Iwasawa decomposition \[ G_2(\R) = H(\R) K_{G_2}, \] elements of $\mathcal{C}^\infty_{\rm mg}(U_H(\R) \backslash G_2(\R) / K_{G_2}, \psi, \tau_n^\vee)$ are determined by their values at $L(\R) = \GL_2(\R)$.

\begin{theorem}[Pollack]
The function \[\mathcal{W}_\psi(\ell):= ... \]
generates $ \mathcal{C}^\infty_{\rm mg}(U_H(\R) \backslash G_2(\R) / K_{G_2}, \psi, \tau_n^\vee)$. In particular, ${F_{\varphi, \psi}}_{|_{L(\R)}} $ is a multiple of it.
\end{theorem}

\subsection{Fourier coefficients for discrete series of Hodge type (3,3)}

Let $\sigma=\sigma_n \otimes \sigma_f$ be a cuspidal automorphic representation of $G_2(\A)$ which is Steinberg at a finite place $p$, and let $\pi := \Theta(\sigma)$ be its Theta lift. Recall that $\pi= \pi_n \otimes \pi_f$ is Steinberg at $p$, while $\pi_n$ is the discrete series of Hodge type (3, 3) and minimal $K_{\PGSp_6}$-type $\gamma_n:=\tau_{(2n,-n,-n)}$. Let \[ \varphi' \in {\rm Hom}_{\PGSp_6(\R)}( \pi_n , \pi ). \]
As in the previous section, restricting to
$K_{\PGSp_6}$, $\varphi'$ determines a functional
\[ F_{\varphi'}: \PGSp_6(\A) \longrightarrow  \gamma_n^\vee, \]
which can be characterized as follows.
By compatibility of local and global theta lift, we have a commutative diagram
\[ \xymatrix{ \pi_n \ar@{^{(}->}[r] \ar[d]_{\varphi'} & \Pi_\infty \otimes \sigma_n^\vee \ar[d]^{ r \circ (\varphi, \varphi)} \\ \pi \ar@{=}[r] & \Theta(\sigma), } \]
where $\varphi_{\rm min} \in {\rm Hom}_{E_7(\R)}( \Pi_\infty, \Pi_{\rm min})$, $\varphi \in {\rm Hom}_{G_2(\R)}( \sigma_n^\vee, \sigma^\vee)$, and
\[ r \circ (\varphi_{\rm min}, \varphi)(v_1,v_2) (g) = \int_{[G_2]} \theta( \varphi_{\rm min}(v_1))(h g) \varphi(v_2)(h) dh,\;  v_1 \otimes v_2 \in \Pi_\infty \otimes \sigma_n^\vee. \]
As $\pi_n \otimes \sigma_n \hookrightarrow \Pi_\infty$, $\varphi_{\rm min}$ induces a map \[ \theta \circ \varphi_{\rm min}|_{\PGSp_6(\R) \times G_2(\R)} : \pi_n \otimes \sigma_n \to \Pi_{\rm min} \stackrel{\theta}{\longrightarrow} \mathcal{A}([E_7]),\]
which, after passing to the dual and then restricting to $K_{\PGSp_6} \times K_{G_2}$, yields a functional \[ F_{\varphi_{\rm min}} : \PGSp_6(\A) \times G_2(\A) \longrightarrow \gamma_n^\vee \otimes \tau_n^\vee,\] which on pure tensors is defined by \[F_{\varphi_{\rm min}}(gh)(v \otimes w) = \theta \circ \varphi_{\rm min} (v \otimes w)(gh).\]
Similarly, we have \[F_\varphi: G_2(\A) \longrightarrow \tau_n.\]
Furthermore, if we let $\langle \cdot , \cdot \rangle_{K_{G_2}}: \tau^\vee_n \otimes \tau_n \to \C$ denote  the $K_{G_2}$-equivariant pairing \anto{at a certain point need to fix it with right normalization} and denote in the same way the corresponding map \[ \langle \cdot , \cdot \rangle_{K_{G_2}}: \gamma_n^\vee \otimes \tau^\vee_n \otimes \tau_n \to \gamma_n^\vee,\]
the above diagram gives the equality
\[ F_{\varphi'}(g)  = \int_{[G_2]}\langle F_{\varphi_{\rm min}}(g h) , F_\varphi(h)  \rangle_{K_{G_2}} dh. \]

Let now $P=M N$ be the parabolic of $\GSp_6$ of Levi $M=GL_2 \times \GL_1^2$ and let $\psi:=\psi_D$ the character on $[N]$ of the \anto{to put reference}. We have the Fourier coefficient \[ F_{\varphi',\psi} : \PGSp_6(\A) \to \gamma_n^\vee.\]

\begin{proposition}
We have an equality \[ F_{\varphi',\psi}(g) = c_{A_0}  \int_{U_H(\A) \backslash G_2(\A)}\langle F_{\varphi_{\rm min},\psi}^{A_0}(g h) , F_{\varphi,\psi}(h)  \rangle_{K_{G_2}} dh.\]
\end{proposition}
\begin{proof}
TO DO + TO DEFINE everything. It should follows readily from the proposition 6.21
\end{proof}

The period of the Tate classes are related to the residue at $s=1$ of $\mathcal{I}(\Phi, \Psi, s),$ which turns out to be non-zero if
$\mathcal{I}_S(\Phi, \Psi, 1) \ne 0$ as the following shows.

\begin{corollary}
If $\mathcal{I}_S(\Phi, \Psi, 1) \ne 0$, then \[ {\rm Res}_{s=1} \mathcal{I}(\Phi, \Psi, s) = \mathcal{I}_S(\Phi, \Psi, 1)  L^S(\sigma_f, {\rm Std},1) \prod_{p \in S}(1-p^{-1})  \ne 0  \]
\end{corollary}
\begin{proof}
The residue has that shape because of the Corollary above. The fact that $L^S(\sigma_f, {\rm Std},1) \ne 0$ follows from the temperedness of $\pi$ and thus $\sigma$ thanks to a result of Kret-Shin. This is because $s=1$ is a critical value for $L(\sigma_f, {\rm Std},s)$, so one needs to know that the Euler factors at $S$ do not have poles at $s=1$. The latter follows as in our regulator paper - I'll add details.   \end{proof}

In the remaining of the section, we address the non-vanishing of this integral by using the formulae on the Fourier coefficients given in the previous sections.

\begin{proposition}
Let $g \in \PGSp_6(\A_S)$, then \[ \Psi_{N,\psi}(g) = c_{A_0} \int_{U_H(\A_S) \backslash G_2(\A_S)}  \langle F_{\varphi_{\rm min,S},\psi}^{A_0}(g h) , F_{\varphi,\psi}(h)  \rangle_{K_{G_2}}(v_0) dh. \]
\end{proposition}
\begin{proof}
This is Gan-Savin's final step.
\end{proof}

\end{comment}

\begin{comment}
Using Proposition \ref{PropFC}, one can write the integral
\[
\mathcal{I}(\Phi, \Theta(f, h), s) = \int_{ U_{B_\H(\A)} Z_\G(\A) \backslash \H(\A)} f(h_1, \Phi, s) \Theta(f, h)_{U_P, \psi_D}(h) dh \]
\[ c_{A_0} \int_{U_H(\A) \backslash G_2(\A)} \left ( \int_{ U_{B_\H(\A)} Z_\G(\A) \backslash \H(\A)} f(h_1, \phi, s) f_{\psi_{H,D}}(hg) dh \right ) h_{U_H, \psi_{H, D}}(g) dg. \\
\]

Recall the following.

\begin{proposition}[\cite{GinzburgJiang}, [27]]
Let $\sigma_p$ be an unramified generic representation of $G_2(\Q_p)$ of Satake parameter $s$; then $\Theta(\sigma_p)$ is the unramified representation of $\mathrm{PGSp}_6(\Q_p)$ of Satake parameter $\varphi(s)$, where $\varphi$ denotes the embedding $G_2 \to \mathrm{Spin}_7$ of $L$-groups. In particular,
\[ L(\Theta(\sigma_p), {\rm Spin} , s) = \zeta_p(s)  L(\sigma_p, {\rm Std} , s). \]
\end{proposition}

Fix a finite set of places $\Sigma$ containing $\infty$, the bad primes for $\Theta(\sigma)$ and for $f$;   at every $p \not \in \Sigma$, $\sigma_p$ is an unramified, generic representation of $G_2(\Q_p)$.

As a consequence of the main result of [48], we get the following.

\begin{proposition}
Let $h$ be a factorizable cusp form in $\sigma^\vee$ which is invariant under $G_2(\Z_p)$ for every $p \not \in \Sigma$. Then,
\[\mathcal{I}(\Phi, \Theta(f, h), s) = c_{A_0} I_\Sigma(\Phi,\Theta(f, h),s) \zeta^\Sigma(s)L^\Sigma(\sigma, {\rm Std} , s), \]
where \[I_\Sigma(\Phi,\Theta(f, h),s) :=\int_{U_{B_\H}(\A_\Sigma)\Z_\G(\A_\Sigma) \backslash \H(\A_\Sigma)}\!\! \!\! \! \! \! \! \!\!\! \! \! \! \! \!\! \! \! \! \! f(h_1,\Phi_\Sigma,s)   \int_{U_H(\A_\Sigma)
\backslash G_2(\A_\Sigma)} \!\! \! \! \! \! \! f_{\psi_{H,D}}^\Sigma(hg)   h_{U_H, \psi_{H, D}}(g) dg dh.\]
\end{proposition}

\begin{proof}
Given a finite set of primes $S$ containing $\Sigma$, define
\begin{equation} \label{semilcoaint} I_S(\Phi,\Theta(f, h),s):=\int_{U_{B_\H}(\A_S)\Z_\G(\A_S) \backslash \H(\A_S)}\!\! \!\! \! \! \! \! \!\!\! \! \! \! \! \!\! \! \! \! \! f(h_1,\Phi_S,s)   \int_{U_H(\A_S)
\backslash G_2(\A_S)} \!\! \! \! \! \! \! f_{\psi_{H,D}}^S(hg)   h_{U_H, \psi_{H, D}}(g) dg dh,
\end{equation}
where $\Phi_S=\prod_{v \in S} \Phi_v$ and \[f_{\psi_{H,D}}^S(g) = \prod_{v \in S} \int_{U_D(\Q_v) \backslash U_H(\Q_v)} f_v(ug^{-1} A_0) \psi_{H,D,v}^{-1}(u) du. \] Then, in the range of convergence \[ I(\Phi,\Theta(f,h),s)= \lim_{\Sigma \subseteq S}  I_S(\Phi,\Theta(f,h),s). \]
Now, let $p \not \in S$, then \[ I_{S \cup \{ p \}}(\Phi,\Theta(f,h),s)={L}(\Theta(\sigma_p), {\rm Spin} ,s)   I_S(\Phi,\Theta(f,h),s).\]
Indeed, $I_{S \cup \{ p \}}(\Phi,\Theta(f,h),s) $ equals to
\[ \int_{D_S}  \! \! f(h_1,\Phi_S,s)   \int_{D_S'}  \! f_{\psi_{H,D}}^S(hg)  \int_{D_p}\!\! f(h_{1,p},\Phi_p,s) \int_{D_p'}\! f_{\psi_{H,D}}^{\{p\}}(h_p h g_p g)  h_{U_H, \psi_{H, D}}(g_p g)dg_p dh_p dg dh, \]
where for any set of primes  $\Omega$ we have denoted $D_\Omega =U_{B_\H}(\A_\Omega)\Z_\G(\A_\Omega) \backslash \H(\A_\Omega)$ and $D_\Omega'=U_H(\A_\Omega)
\backslash G_2(\A_\Omega)$.
As $p \not \in \Sigma$, $\Theta(\sigma)_p=\Theta(\sigma_p)$ is unramified at $p$. Fix a spherical vector $v_0$ for $ \sigma_p^\vee $; as $h$ is invariant under the action of $G_2(\zp)$, there is a $G_2(\zp)$-equivariant map $\varphi_p: \sigma_p^\vee \to \sigma^\vee$ sending $v_0$ to $h$. Thus, for a fixed $m \in \G(\A_S)$ and $t \in G_2(\A_S)$, the functional $\Lambda_{m,t}: \Pi_{{\rm min},p} \otimes \sigma_p  \to \C$ defined by \[\Lambda_{m,t}(\tilde{f}_p,v):=\int_{D_p'}\! f_{\psi_{H,D}}^{\{p\}}(m g_p t)  \varphi_p(v)_{U_H, \psi_{H, D}}(g_p t)dg_p\]
\anto{We have to explain the compatibiity between local and global theta lifts in detail}
%and $\Lambda_h(h_p \cdot v_0)=\Psi'_{\chi, U_P}(h_p h)$. Proposition \ref{pollackshahgsp6}$(1)$ implies then
%\int_{U_{B_\H}(\qp)\Z_\G(\qp) \backslash \H(\qp)} &{} \!\! \!\!\! \! \!f(h_{1,p},\Phi_p,s) \Psi'_{\chi, U_P}(h_p h)dh_p \\
%&= \int_{U_{B_\H}(\qp)\Z_\G(\qp) \backslash \H(\qp)}\!\! \!\!\! \! \! f(h_{1,p},\Phi_p,s) \Lambda_h(h_p \cdot v_0) dh_p \\
%&={L}(\pi_p, {\rm Spin} ,s)  \Lambda_h(v_0) \\
%&= {L}(\pi_p, {\rm Spin} , s)  \Psi'_{\chi, U_P}(h), \end{align*}

\end{proof}

\end{comment}

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\end{thebibliography}

\end{document}