# Impact of atomic reconstruction on optical spectra of twisted TMD
homobilayers

Joakim Hagel<EMAIL_ADDRESS>Department of Physics, Chalmers
University of Technology, 412 96 Gothenburg, Sweden
Samuel Brem  Department of Physics, Philipps University of Marburg, 35037
Marburg, Germany
Johannes Abelardo Pineiro  Department of Physics, Philipps University of
Marburg, 35037 Marburg, Germany
Ermin Malic  Department of Physics, Philipps University of Marburg, 35037
Marburg, Germany
Department of Physics, Chalmers University of Technology, 412 96 Gothenburg,
Sweden

###### Abstract

Twisted bilayers of transition metal dichalcogenides (TMDs) have revealed a
rich exciton landscape including hybrid excitons and spatially trapped moiré
excitons that dominate the optical response of the material. Recent studies
have shown that in the low-twist-angle regime, the lattice undergoes a
significant relaxation in order to minimize local stacking energies. Here,
large domains of low energy stacking configurations emerge, deforming the
crystal lattices via strain and consequently impacting the electronic band
structure. However, so far the direct impact of atomic reconstruction on the
exciton energy landscape and the optical properties has not been well
understood. Here, we apply a microscopic and material-specific approach and
predict a significant change in the potential depth for moiré excitons in a
reconstructed lattice, with the most drastic change occurring in naturally
stacked TMD homobilayers. We show the appearance of multiple flat bands and a
significant change in the position of trapping sites compared to the rigid
lattice. Most importantly, we predict a multi-peak structure emerging in
optical absorption of WSe2 homobilayers - in contrast to the single peak that
dominates the rigid lattice. This finding can be exploited as an unambiguous
signature of atomic reconstruction in optical spectra of moiré excitons in
naturally stacked twisted homobilayers.

††preprint: APS/123-QED

##  I Introduction

The recent advances in the field of two-dimensional materials have opened up a
new platform to study many-particle physics in nanomaterials [1]. The field of
twistronics is of particular interest, where the relative twist angle between
two vertically stacked 2D materials can be used as an additional external knob
to tune the material properties [2, 3, 4]. Consequently, the introduction of a
twist angle in these layered systems can lead to intriguing many-particle
phenomena, such as unconventional superconductivity and correlated phases [5,
6]. Transition metal dichalcogenides (TMDs) have emerged as a significant and
noteworthy subclass of two-dimensional materials, garnering substantial
interest in recent years [7, 8]. Here, strongly bound electron-hole pairs,
known as excitons, form due to the two-dimensional confinement and the reduced
screening of the Coulomb interaction. They dominate the material’s optical
response even at room temperature [9, 10, 11, 12]. Furthermore, long-lived
interlayer excitons can form, which exhibit an out-of-plane dipole moment due
to the spatial separation of electrons and holes [13, 14, 15, 16, 17, 18, 19,
20]. Additionally, the large wave function overlap can lead to a significant
carrier tunneling between the layers, giving rise to hybrid states of intra-
and interlayer excitons [19, 21, 22, 23, 10, 24].

When introducing a twist angle between two vertically stacked TMDs, a periodic
superlattice emerges. This leads to a moiré pattern, i.e. a periodic
superlattice potential that can trap excitons at specific high-symmetry sites
[25, 26, 27, 28, 29, 30, 31, 21]. Furthermore, the emerging moiré potential
can be tuned by changing the twist angle which has been shown to have a large
impact on the optical response of these materials [25, 26, 32, 29, 33, 34].
Interestingly, recent studies have revealed that when the twist angle is
small, i.e. the period of the moiré pattern is large, the two monolayer
crystal lattices do no longer remain rigid, but instead undergo a deformation
to reduce the total adhesion energy [35, 36, 37, 38, 39, 40, 41, 42, 43].
Here, some energetically favorable high-symmetry sites grow to form large
domains, separated by typically narrow domain walls (cf. Fig. 1). It has been
predicted that this atomic reconstruction should have a major impact on the
electronic band structure and fundamentally change the potential landscape
[44, 45, 38, 46, 47]. However, it is still not thoroughly understood how
atomic reconstruction can be identified in the optical response of moiré
excitons.

In this work, we investigate photoluminescence (PL) and absorption spectra in
atomically reconstructed TMD bilayers. Our approach is based on the ab initio
derived continuum model for the deformation of the rigid lattice derived in
Refs. [45, 46] combined with an equation-of-motion approach for the optical
response of moiré excitons [19, 31, 30, 21]. This allows us to obtain
microscopic insights into the changes in the moiré exciton energy landscape
and the optical response in the presence of atomic reconstruction. The
deformation of the moiré supercell into large periodic domains induces strain
that can result in deeper potentials for moiré excitons. As a direct
consequence, we find more trapped states than in the rigid lattice and the
stacking region in which excitons are trapped can differ from the rigid
lattice system. Furthermore, we predict that the KK interlayer exciton becomes
lower in energy than the intralayer KK state due to the strain-induced
potentials at low twist angles. The lowest lying momentum-dark K$\Lambda$
exciton is also predicted to undergo a larger red-shift than in the rigid
lattice, visible in PL spectra. Most importantly, we find a qualitative change
in the absorption spectrum in naturally stacked bilayers reflecting the bright
KK excitons, where multiple peaks emerge as a consequence of the strain-
induced potentials - in stark contrast to the single peak that is present in
the rigid lattice. Consequently, we predict an unambiguous signature of atomic
reconstruction in the optical response of moiré excitons in naturally stacked
twisted TMD homobilayers.

##  II Moiré potential in atomically reconstructed TMD bilayers

To gain access to the moiré exciton energy landscape in atomically
reconstructed bilayers, we first describe the change in geometry in the
superlattice. Following the approach in Refs.[45, 46] we set up a functional
for the stacking energy, taking into account the strain energy from continuum
theory and a parameterized equation for the adhesion energy between the
layers, where the model parameters have been fitted to data from density
functional theory in Refs.[45, 47]. The functional can then be turned into an
optimization problem by expanding the displacement vectors
$\mathbf{u}^{l}(\mathbf{r})$ as a Fourier expansion
$\mathbf{u}^{l}(\mathbf{r})=\sum_{n}\mathbf{u}^{l}_{n}e^{i\mathbf{g}_{n}\mathbf{r}}$
and minimizing the integral with respect to the Fourier coefficients
$\mathbf{u}^{l}_{n}$. Here, $l$ is the layer index, $\mathbf{\mathbf{g}}_{n}$
are the reciprocal vectors of the moiré lattice and $\mathbf{r}$ is the real
space coordinate in the supercell. For further details, see the supplemental
part (SI) of the paper [48] (see also references [49, 50, 51, 52, 53, 54, 55,
56, 57] therein).

After obtaining the displacement vectors, we study the geometrical change in
the supercell. Here, we find that the energetically more favorable $H_{h}^{h}$
stacking grows into hexagonal domains (Kagome pattern) at small twist angles,
separated by a thin boundary wall - in good agreement with experimental
studies [35] and previous theoretical works [45]. This reconstruction is
illustrated in Fig. 1a showing the interlayer distance for H-type stacked WSe2
homobilayers at the twist angle $\theta=0.6^{\circ}$. Each high-symmetry
stacking is reflected by a specific interlayer distance, which can be computed
from DFT [58] and thus the interlayer distance can be used as an indicator for
a change in the local atomic registry. The blue domains denote the $H_{h}^{h}$
regions, which are significantly larger than in the rigid lattice, cf. Fig.
1.b. The other high-symmetry stackings ($H_{h}^{X}$ and $H^{M}_{h}$)
consequently shrink in size and form thin boundary networks (cf. Fig. 1.a). We
focus here on the examplary H-type stacked WSe2 homobilayer, but the same
approach holds for all vertically stacked TMDs. In R-type stacked
homobilayers, we instead obtain large triangular patterns as a result of the
atomic reconstruction, which yields a different potential landscape (see SI
for further details).

Figure 1:  Spatial map of the interlayer distance in (a) a reconstructed
lattice and (b) a rigid lattice at $\theta=0.6^{\circ}$ for H-type stacked
WSe2 homobilayers. Here, the interlayer distance is associated with a high-
symmetry stacking. The band edge variation for the conduction band in (c) the
bottom layer and (e) the top layer at the K-point, and the valance band edge
variation in (d) the bottom layer and (f) the top layer at the K-point. Note
the reversed sign between the layers and the difference in the potential depth
between valance and conduction band. Since this is a purely strain-induced
potential, it is not present in the rigid lattice. The shape of the potential
stems from the linear combination of the scalar strain potential and the piezo
potential. (g) Maximum exciton potential depth $\Delta E_{max}$ in the
supercell is shown as a function of twist angle for both intralayer excitons
(dashed) and interlayer excitons (solid).

The impact of atomic reconstruction on the exciton energy landscape does not
only stem from the geometrical change in the superlattice, but also from the
strain that accompanies the lattice deformation. Assuming small displacements,
the displacement vector $\mathbf{u}^{l}(\mathbf{r})$ can be associated to the
linear strain tensor $\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$, where
$u_{i,j}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})$ and $i(j)=(x,y)$.
Due to its D3h symmetry, we can decompose the dominating strain contributions
to the moiré potential into two parts (see SI) [59, 60, 61]: scalar potential
composed of inhomogeneous uniaxial strain in both x- and y-direction [62, 63]
and inhomogeneous vector gauge potential known as piezo potential [45, 47,
64]. Details about the strain potential model are provided in the SI.

In addition to the strain-induced potentials, there is a stacking-dependent
alignment shift of the two monolayer band structures, originating from charge
polarization between the two layers, that already exists in the non-
reconstructed system [30, 19]. In a homobilayer, these only affect the energy
of interlayer excitons [19]. Furthermore, it is only present in R-type stacked
bilayers due to the lack of inversion symmetry, which allows for local charge
transfers between the layers [27]. The remaining interlayer coupling mechanism
contributing to the total moiré potential is given by the interlayer
hybridization. For some electronic states, in particular in the $\Lambda$
valley in the conduction band, the large wave function overlap between the
layers leads to a strong mixing of states in both layers, which can be
described as a mixing of interlayer and intralayer excitons into hybrid
exciton states [19, 21]. In H-type stacking, the strongest hybridization
occurs around the $H^{h}_{h}$ stacking due its shorter interlayer distance
(cf. Fig. 1.a-b). To sum up, we have four different contributions to the moiré
potential: 1) scalar strain potential $S^{\mathbf{\xi}}_{L\mathbf{g}}$, 2)
piezo potential $P^{\mathbf{\xi}}_{L\mathbf{g}}$, 3) stacking-dependent
alignment potential $V^{\mathbf{\xi}}_{L\mathbf{g}}$ and 4) tunneling
$T^{\mathbf{\xi}}_{LL^{\prime}\mathbf{g}}$. The first three act as a periodic
renormalization of exciton energies, i.e. they are diagonal with respect to
the layer index, and can thus be modeled as an external potential acting on
free excitons
$M^{\mathbf{\xi}}_{L\mathbf{g}}=S^{\mathbf{\xi}}_{L\mathbf{g}}+P^{\mathbf{\xi}}_{L\mathbf{g}}+V^{\mathbf{\xi}}_{L\mathbf{g}}$,
where $L=(l_{e},l_{h})$ is a compound layer index and
$\mathbf{\xi}=(\xi_{e},\xi_{h})$ is the exciton valley index.

In Fig. 1.c-d, we show the resulting band edge variation for the conduction
band (Fig. 1.c) and the valance band (Fig. 1.d) in the bottom layer at
$\theta=0.6^{\circ}$. Similarly, the band edge variation for the top layer is
shown in Fig. 1.e-f, where the conduction band is given by Fig. 1.e and the
valance band is given by Fig. 1.e at $\theta=0.6^{\circ}$. Here, the only
potentials that are contributing are the scalar strain potential and the piezo
potential. In general, homobilayers mostly reconstruct due to atomic rotation
[37], which is associated with the piezo potential and will be concentrated at
the $H_{h}^{M}$/$H_{h}^{X}$ sites. However, there is also a significant scalar
strain potential close to the boundary regions of the $H_{h}^{h}$ domains
which also significantly contribute to the band edge variation. The
combination of scalar potential and piezo potential in turn gives the tilted
star-shape of the total band edge variation. Comparing the variation for
different layers (Fig. 1.e to Fig. 1.d ) we find that the minima of the
conduction band and the maxima of the valance band occur at the same sites,
thus yielding a very efficient band gap renormalization for interlayer
excitons. In contrast, comparing Fig. 1.c and Fig. 1.d we find that the minima
for the conduction band and the maxima for the valance band occur at different
sites in the supercell. This means that there is a certain driving force for
charge separation. However, the binding energies of the intralayer excitons
are very strong ($\sim$ -140 meV for KK) which in turn means that one would
need a very deep potential in order to have a complete charge separation. At
angles $\theta>0.5^{\circ}$, the formation of excitons is still favored in
comparison to the separated unbounded electron/holes. For much smaller
supercells, however, the formation of in-plane charge transfer excitons can
occur as reported in [38].

The effective exciton potential as shown in Fig. 1.g gives the calculated
potential depth for KK (blue) and K$\Lambda$ (yellow) excitons as a function
of twist angle, both for intra- (dashed) and interlayer (solid) excitons.
Since the piezo potential in H-type stacking is the same for both the valence
and the conduction band in both layers, the only contributing factor is the
scalar strain potential. The color map of this effective potential is given in
the supplemental part of the paper. Below $\theta=2^{\circ}$ we find a
significant decrease of the KK interlayer exciton potential (solid blue) due
to the efficient band gap renormalization stemming from the scalar strain
potential. The KK intralayer (dashed blue) potential has a much shallower
potential due to the conduction and valance band shifting in the same
direction, although with different rates. Comparing to the effective
K$\Lambda$ (solid yellow) potential, it only reaches half the potential depth
of the KK. This can be understood from different orbital compositions of the
valleys, which directly impact the change in the orbital overlap with scalar
strain [65]. Note that these effective exciton potentials are all in stark
contrast to the rigid lattice where they do not occur at all for H-type
stacking. For R-type stacking, we predict in general much weaker band edge
variations, having only a maximum variation of $\sim$40 meV. This difference
comes primarily from the lack of scalar strain in R-type stacking, were the
domain walls are formed purely from shear strain - in good qualitative
agreement with the experimentally observed values [37]. In turn, this means
that the effective intralayer exciton potential will be close to $0$ meV.
There will, however, be a significant addition to the interlayer potential in
the form of the piezo potential (see SI).

##  III Moiré exciton energy landscape in reconstructed TMD bilayers

Having a microscopic access to the potential landscape of moiré excitons, we
can now set up the complete moiré exciton Hamilton operator. Here, we start
from a decoupled monolayer basis in electron-hole picture and add the moiré
potential as periodic modifications to these energies [31, 30, 21].
Transforming to an exciton basis, the Hamiltonian then reads

$\begin{split}&H_{0}=\sum_{L\mathbf{Q}\mathbf{\xi}}E^{\mathbf{\xi}}_{L\mathbf{Q}}X^{\mathbf{\xi}\dagger}_{L\mathbf{Q}}X^{\mathbf{\xi}}_{L\mathbf{Q}}+\sum_{L\mathbf{Q}\mathbf{\xi}\mathbf{g}}M^{\mathbf{\xi}}_{L\mathbf{g}}X^{\mathbf{\xi}\dagger}_{L\mathbf{Q}+\mathbf{g}}X^{\mathbf{\xi}}_{L\mathbf{Q}}\\\
&+\sum_{LL^{\prime}\mathbf{Q}\mathbf{\xi}\mathbf{g}}T^{\mathbf{\xi}}_{LL^{\prime}\mathbf{g}}X^{\mathbf{\xi}\dagger}_{L\mathbf{Q}+\mathbf{g}}X^{\mathbf{\xi}}_{L^{\prime}\mathbf{Q}}+h.c,\end{split}$
(1)

where $E^{\mathbf{\xi}}_{L\mathbf{Q}}$ is the binding energy of the decoupled
monolayer as obtained from the generalized Wannier equation [66] and
$\mathbf{Q}$ is the center-of-mass momentum. The effective masses for
electrons and holes are obtained from Ref.[67]. Furthermore, $X^{(\dagger)}$
denotes the annihilation (creation) operator of excitons. Moreover,
$M^{\mathbf{\xi}}_{L\mathbf{g}}$ is the combined layer-diagonal moiré
potential component and $T^{\mathbf{\xi}}_{LL^{\prime}\mathbf{g}}$ is the
tunneling matrix element. The latter gives rise to the hybridization of intra-
and interlayer excitons, which corresponds to the off-diagonal component of
the moiré potential [58].

Figure 2:  Band structure and wave function of the bright KK exciton in H-type
stacked WSe2 bilayer at $\theta=0.6^{\circ}$ in (a,c) the reconstructed
lattice and (b,d) the rigid lattice. Energies are normalized to the lowest-
lying KK exciton in the rigid lattice. Note that the reconstruction
drastically changes the localization site of excitons and that the efficient
moiré potential for interlayer excitons shifts them below the intralayer KK
states. The band index $\eta=0$ indicates which wave function is shown.

The periodicity of the superlattice is taken into account by applying the
zone-folding scheme [31, 21, 30], where we restrict the summation over
$\mathbf{Q}$ to the first mini-Brillouin zone (mBZ), where states with larger
momentum are folded back via mBZ lattice vectors $\mathbf{g}$ giving rise to
the formation of mini-subbands. The final moiré exciton energies are then
obtained by first expanding the exciton operators into a moiré exciton basis
$Y^{\dagger}_{\mathbf{\xi}\eta\mathbf{Q}}=\sum_{\mathbf{g}L}\mathcal{C}^{\mathbf{\xi}\eta*}_{L\mathbf{g}}(\mathbf{Q})X^{\mathbf{\xi}\dagger}_{L,\mathbf{Q}+\mathbf{g}}$,
where $Y^{(\dagger)}$ are the moiré exciton annihilation (creation) operators,
$\eta$ is the new band index, and
$\mathcal{C}^{\mathbf{\xi}\eta*}_{L\mathbf{g}}(\mathbf{Q})$ are the mixing
coefficients that mix different sub-bands and layer configurations. Applying
this transformation to Eq. 1 we obtain the moiré exciton eigenvalue equation

$\begin{split}E^{\mathbf{\xi}}_{L\mathbf{Q+g}}\mathcal{C}^{\mathbf{\xi}\eta}_{L\mathbf{g}}(\mathbf{Q})+\sum_{\mathbf{g}^{\prime}}M^{\mathbf{\xi}}_{L}(\mathbf{g},\mathbf{g}^{\prime})\mathcal{C}^{\mathbf{\xi}\eta}_{L\mathbf{g}^{\prime}}(\mathbf{Q})\\\
+\sum_{L^{\prime}\mathbf{g}^{\prime}}T^{\mathbf{\xi}}_{LL^{\prime}}(\mathbf{g},\mathbf{g}^{\prime})\mathcal{C}^{\mathbf{\xi}\eta}_{L^{\prime}\mathbf{g}^{\prime}}(\mathbf{Q})=\mathcal{E}^{\mathbf{\xi}}_{\eta\mathbf{Q}}\mathcal{C}^{\mathbf{\xi}\eta}_{L\mathbf{g}}(\mathbf{Q}).\end{split}$
(2)

Here, $\mathcal{E}^{\mathbf{\xi}}_{\eta\mathbf{Q}}$ are the final moiré
exciton energies, which can be obtained by solving Eq. 2 numerically.

In Fig. 2, we show the calculated moiré exciton band structure for H-type
stacked hBN-encapsulated WSe2 bilayers (for R-type stacking see SI) with a
reconstructed lattice (Fig. 2.a) and with a rigid lattice (Fig. 2.b). We find
a large red-shift in the reconstructed lattice, where the lowest sub-band is
located $\sim$36 meV below the rigid lattice energies. Interestingly, due to
the efficient moiré potential for the interlayer exciton, the nature of the
lowest lying state has changed from an intralayer exciton to an interlayer
exciton. While in the rigid lattice (cf. Fig. 2.b) most bands have a
significant curvature, we find multiple ultra-flat bands in the reconstructed
lattice (cf. Fig. 2.a), both for the interlayer excitons (black) and the
intralayer excitons (blue). The spacing between the first two bands is about
15 meV, which is significantly larger than the small spacing (of about 3 meV)
in the rigid lattice indicating a much stronger spatial confinement. This is a
direct consequence of the additional strain potentials in the reconstructed
lattice. Note that also the hybridization of the higher-lying bands is
strongly suppressed as there is a lack of band crossing due to their flatness
in the reconstructed lattice, cf. the color gradient in Fig. 2.a-b denoting
the degree of hybridization. We need to go to higher energies to see some
hybridization in the reconstructed case (red color around 10 meV in Fig. 2.a).

The moiré exciton wave function in real space $|\Psi(\mathbf{r})|^{2}$ for the
lowest-lying states are shown in Fig. 2c-d. We find that in the reconstructed
lattice the wave function is localized close to the edges of the
$H_{h}^{h}$-domains of the of the moiré unit cell (cf. Fig. 2.c), reflecting
the potential minima from the strain-induced potential. In contrast, in the
rigid lattice with no additional strain potential (cf. Fig. 2.d) there is a
much larger probability distribution centered at the $H^{h}_{h}$ stacking
configuration (cf. Fig. 1.b). Here, the moiré potential only consists of the
tunneling and therefore the exciton localization is determined only by the
stacking with the strongest tunneling [19]. When increasing the twist angle,
the reconstructed lattice starts to revert back to the rigid lattice, making
the tunneling the dominating contribution of the moiré potential. This change
between different regimes implies the possibility of tuning the localization
of moiré excitons, which should have important consequences for exciton
transport in these materials [68, 69].

Figure 3:  Normalized photoluminescence spectrum at $4$K at different twist
angles, displaying phonon replicas of the energetically lowest K$\Lambda$
exciton in (a) H-type and (b) R-type WSe2 homobilyers. The spectra from the
reconstructed and the rigid lattice are shown in blue and gray, respectively.
A signature of lattice reconstruction is a more pronounced red-shift of the
exciton resonance for smaller twist angles.

##  IV Impact of reconstruction on optical spectra

Having access to the moiré exciton energy landscape, we can now determine the
optical response of TMD bilayers in presence of atomic reconstruction. In WSe2
homobilayers, the lowest-lying state in R-type stacking is the strongly
hybridized K$\Lambda$ exciton, while in H-type stacking K$\Lambda$ and
K${}^{\prime}\Lambda^{\prime}$ excitons are the lowest ones. Note that the
latter two are degenerate in H-type stacking, but they have a reversed layer
configuration, as a consequence of the reversed spin-orbit coupling the layers
[19, 21, 20]. We first focus on PL spectra, where these dark states are
expected to be visible via indirect phonon sidebands [70]. Here, the exciton
can not directly emit a photon due to the large momentum mismatch between the
valleys. Instead, it decays via a two-step process, first scattering to a
virtual bright state within the light cone via phonon emission followed by the
emission of a photon [70].

Figure 3.a-b shows the PL spectra for WSe2 with H-type (cf. Fig. 3.a) and
R-type stacking (cf. Fig. 3.b) at 4K at different twist angles. The reference
frame for the energies are the same as in Fig. 2.a-b. We observe sidebands
from momentum-dark states, where the smaller (larger) peak stems from optical
(acoustic) phonon modes [21, 19]. The energetic position of the
K$\Lambda$(K${}^{\prime}\Lambda^{\prime}$) phonon replicas will primarily
depend on the interlayer hybridization, stemming from the strong interlayer
tunneling of electrons around the $\Lambda$($\Lambda^{\prime}$) valley. This
gives rise to a twist-angle-dependent dehybridization, in turn yielding a blue
shift when increasing the twist angle (gray peaks in Fig. 3), i.e. the larger
the twist-angle the smaller is the wave function overlap [21, 29]. Since this
component of the moiré potential is already present in the rigid lattice we
find a very similar behavior in both the rigid and the reconstructed lattice
when changing the twist angle. There is, however, a clearly larger red-shift
for the reconstructed lattice at lower twist angles due to the increasing
strain-induced potentials. This is especially noticeable for R-type stacking
where the piezo potential contributes to the effective exciton potential.
Consequently, the change in the peak position closely resembles the change in
the effective exciton potential for K$\Lambda$ (SI Fig.3). In Fig. 3.b we can
also see that the peak position of the reconstructed lattice is converging
towards the peak position of the rigid lattice when increasing the twist
angle, indicating that the lattice becomes more rigid at larger angles.

Figure 4:  Twist-angle dependence of the normalized absorption in H-type WSe2
bilayers for (a) the rigid lattice and (b) the reconstructed lattice. The
emergence of multiple peaks in the latter is due to the strain-induced
intralayer potential and is an unambiguous signature of lattice
reconstruction.

So far we have been focusing on photoluminescence that is dominated by phonon
sidebands of the energetically lowest dark K$\Lambda$ excitons. These states
are strongly hybridized [21, 19], and are thus subject to a pronounced moiré
potential already in the rigid lattice. The situation is qualitatively
different for bright KK excitons that are accessible in absorption spectra.
Here, the reconstructed lattice gives rise to a deep strain-induced potential,
whereas the rigid lattice only experiences a small moiré potential due to the
weak periodic tunneling strength around the K-valley. In Fig. 2.a we have
already demonstrated the significant change occurring in the band structure of
KK excitons, where multiple flat-bands have emerged in the reconstructed
lattice for both the intralayer and the interlayer excitons.

In Fig. 4.a-b, we have calculated the absorption spectrum of H-type stacked
WSe2 bilayers as a continuous function of the twist angle for both the rigid
lattice (Fig. 4.a) and the reconstructed lattice (Fig. 4.b). Here, we
immediately observe a drastic change in the spectrum, where below
$\theta\approx 1.0^{\circ}$ multiple peaks appear in a reconstructed lattice -
in stark contrast to the single-peak spectra in the rigid lattice that are
mostly angle-independent. Furthermore, at smaller angles we find a noticeable
splitting around $\sim$15 meV between the two brightest peaks in the
reconstructed lattice - in good agreement with experimental observations [44].
We also find a small splitting in the rigid lattice due to the tunneling of
holes, which consequently traps the lowest state at $H_{h}^{h}$ (cf. Fig.
2.d). This state, however, is very close to the next non-trapped state (cf.
Fig. 2.b), and thus the trapping is not very efficient. In the reconstructed
lattice (cf. Fig. 4.b), we instead have a rich optical response, consisting of
multiple peaks. Here, the lowest branches stem from excitons trapped in the
deep strain potential pockets (cf. Fig. 2.a/c). Consequently, these peaks
undergo a non-linear blue-shift as we increase the twist angle, following the
non-linear twist angle dependence of the strain-induced moiré potential (cf.
Fig. 1.g). Furthermore, in Fig. 2.a, we find that the lowest lying KK states
are of pure interlayer character (black lines in Fig. 2.a). These are not
visible in absorption spectra due to their by orders of magnitude smaller
overlap between the electron and the hole wave function compared to intralayer
excitons [71]. When increasing the twist angle, the number of peaks is reduced
until it converges to the rigid lattice case around $\theta=1.2^{\circ}$ that
is characterized by only one resonance.

In R-type stacking (shown in the SI), we do not observe the same effect. Here,
the strain-induced potential for intralayer KK excitons is negligible and the
impact on the spectra only stems from the weak tunneling of carriers. We
predict this drastic change in optical absorption due to atomic reconstruction
to be similar for all naturally stacked TMD homobilayers. Note that the multi-
peak structure is also well-known from TMD heterostructures, such as
MoSe2-WSe2, already in the rigid lattice case due to deep moiré potentials
[30]. Here, the impact of lattice reconstruction is only of quantitative
nature with changes for the peak separation and amplitude. The situation is
qualitatively different in naturally stacked TMD homobilayers, where the
strain-induced potential in the reconstructed lattice is the dominant
mechanism and leads to the characteristic multi-peak structure for the bright
A exciton. The latter can thus be considered as an unambiguous optical
signature for the presence of lattice reconstruction.

##  V Conclusion

Overall, our work provides new microscopic insights into many-particle
mechanisms dominating the optical response in twisted TMD homobilayers. The
atomic reconstruction has a large impact on the moiré exciton energy landscape
where strain-induced potentials lead to significant red-shifts and multiple
new flat bands. Furthermore, we have calculated moiré exciton wave functions
and predict that excitons are trapped at different sites in the reconstructed
lattice reflecting the maxima of the strain potentials. Finally, we find a
significant change in optical spectra of naturally stacked TMD homobilayers
due to reconstruction. In particular, we predict the emergence of a multi-peak
structure in absorption spectra - in stark contrast to the single-peak
dominated spectra in a rigid lattice. Our work can help guide future
experiments in the growing field of 2D material twistronics.

##  VI Acknowledgments

We thank Maja Feierabend for insightful discussions. This project has received
funding from Deutsche Forschungsgemeinschaft via CRC 1083 (project B09) and
the regular project 512604469.

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# A parametrized set of explicit elements of $\Sha(E/{\mathbb{Q}})[3]$

Steven R. Groen  Department of Mathematics, University of Warwick, Coventry,
CV4 7AL, United Kingdom<EMAIL_ADDRESS>and  Jaap Top  Bernoulli
Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen, the
Netherlands<EMAIL_ADDRESS>
(Date: November 2020)

###### Abstract.

In this paper, we give explicit equations for homogeneous spaces corresponding
to a rational isogeny of degree $3$. An explicit set of elliptic curves with
elements of order $3$ in their Tate-Shafarevich group is constructed.
Combining this gives explicit examples of plane cubics over ${\mathbb{Q}}$
that have a point everywhere locally, but not globally.

00footnotetext:  2010 _Mathematics Subject Classification_ : 11G05, 11G07,
11R34, 11D2500footnotetext: Keywords and phrases: elliptic curves over global
fields, descent, Tate-Shafarevich groups

##  1\. Introduction

In this paper we consider elliptic curves defined over ${\mathbb{Q}}$ that
admit a rational isogeny of degree $3$. Associated to such an isogeny
$\psi\colon E\to\bar{E}$ is an exact sequence of ${\mathbb{F}}_{3}$-vector
spaces

$0\to\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))\to\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})\to\Sha(E/{\mathbb{Q}})[\psi]\to
0.$

The elements of $\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$ correspond to
genus-$1$ curves called _homogeneous spaces_. In section 2 we give explicit
equations for these curves. The fact that these curves are in the Selmer group
means they have a point everywhere locally. A global point on such a curve
gives rise to a point on $\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$.

In section 3, we construct an explicit set of elliptic curves defined over
${\mathbb{Q}}$ with elements of order $3$ in their Tate-Shafarevich group. We
define

$E_{h}:y^{2}=x^{3}-216(x-h(h-6))^{2}.$

We assume that $h\equiv 3\bmod 8$ and moreover that $|h|$, $|h-2|$, $|h-6|$
and $|h-8|$ are prime numbers. We show that $E_{h}$ has the following
properties:

  1. (1)

$E_{h}$ has a rational $2$-isogeny $\phi$ and a rational $3$-isogeny $\psi$.

  2. (2)

Using $2$-isogeny descent we show that $E_{h}({\mathbb{Q}})$ has rank zero.

  3. (3)

$\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})$ contains $9$ elements, all of
which must be in $\Sha(E_{h}/{\mathbb{Q}})[3]$.

This results in plane cubics that violate the Hasse principle. As an
application, the curves

$C^{h}_{\pm}\colon
18w^{2}z+432z^{3}+w^{3}+216wz^{2}+2(h-2)^{2}(h-8)=\pm(432z^{2}-6w^{2})$

have a point in every completion of ${\mathbb{Q}}$, but not in ${\mathbb{Q}}$
itself.

If Schinzel’s hypothesis H ([9]) holds,

$S=\left\\{h\in{\mathbb{Z}}\;:\;h\equiv 3\bmod
8\;\text{and}\;|h|,\;|h-2|,\;|h-6|,\;|h-8|\;\text{prime}\right\\}.$

is infinite and hence this set yields infinitely many examples.

##  2\. Homogeneous spaces corresponding to a rational $3$isogeny

This section describes the Galois cohomology associated to a rational isogeny
$\psi$ of degree $3$. This builds up to Theorem 2.5, giving explicit equations
for the homogeneous spaces in $\textup{WC}(E/{\mathbb{Q}})$ coming from
$\psi$.

###  2.1. Elliptic curves with a rational $3$-isogeny

Let $E$ be an elliptic curve over ${\mathbb{Q}}$ with a Galois invariant
subgroup $T$ of size $3$. The quotient map $\psi:E\to E/T=:\bar{E}$ is a
rational $3$-isogeny. By, e.g., [11, Ch. III Prop. 4.12] every rational
$3$-isogeny between elliptic curves is obtained in this way. Following [14],
one distinguishes the cases $j_{E}=0$ and $j_{E}\neq 0$ to obtain explicit
descriptions for these curves and maps.

In the case $j_{E}=0$, one gives $E$ by

$E\colon y^{2}=x^{3}+C$

(here $T$ is generated by a point with $x=0$), and the isogeny $\psi$ up to
possibly multiplication by $-1$ is

$\displaystyle\psi$
$\displaystyle:(x,y)\mapsto\left(\frac{y^{2}+3C}{x^{2}}\;,\;\frac{y(x^{3}-8C)}{x^{3}}\right),$
$\displaystyle\bar{E}$ $\displaystyle:\eta^{2}=\xi^{3}-27C.$

For brevity, we introduce the notation $\bar{C}:=-27C$.

In the case $j_{E}\neq 0$, one gives $E$ by

$E\colon y^{2}=x^{3}+A(x-B)^{2}.$

Again $T$ is generated by a point with $x=0$, and (up to $\pm 1$) here $\psi$
is given by

(2.1) $\displaystyle\psi$
$\displaystyle:(x,y)\mapsto\left(\frac{3(6y^{2}+6AB^{2}-3x^{3}-2Ax^{2})}{x^{2}}\;,\;\frac{27y(8AB^{2}-x^{3}-4ABx)}{x^{3}}\right),$
$\displaystyle\bar{E}$ $\displaystyle:\eta^{2}=\xi^{3}-27A(X-4A-27B)^{2}.$

For brevity, we introduce $\bar{A}:=-27A$ and $\bar{B}:=4A+27B$.

By $\hat{\psi}:\bar{E}\to E$ we will denote the dual isogeny of $\psi$, such
that $\hat{\psi}\circ\psi=[3]$.

###  2.2. Galois cohomology

For any field $K$, let $G_{K}=\textup{Gal}(K^{sep}/K)$ be its absolute Galois
group. The isogeny $\psi$ described above yields an exact sequence

(2.2) ${0}$${{E[\psi]}}$${E}$${\bar{E}}$${0}$$\scriptstyle{\psi}$

of $G_{\mathbb{Q}}$-modules. Taking $G_{\mathbb{Q}}$ invariants yields a long
sequence of Galois cohomology

${0}$${E({\mathbb{Q}})[\psi]}$${E({\mathbb{Q}})}$${\bar{E}({\mathbb{Q}})}$${H^{1}(G_{\mathbb{Q}},E[\psi])}$${H^{1}(G_{\mathbb{Q}},E)}$${H^{1}(G_{\mathbb{Q}},\bar{E}),}$$\scriptstyle{\psi}$$\scriptstyle{\delta}$$\scriptstyle{\psi_{*}}$

which we shorten to

(2.3)
${0}$${\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))}$${{H^{1}(G_{\mathbb{Q}},E[\psi])}}$${{H^{1}(G_{\mathbb{Q}},E)[\psi_{*}]}}$${0.}$$\scriptstyle{\delta}$

Recall that a _homogeneous space_ (sometimes called torsor) of
$E/{\mathbb{Q}}$ is a smooth curve $C/{\mathbb{Q}}$ with a simply transitive
action $E\times C\to C$. By [11, Theorem X.3.6], elements of
$H^{1}(G_{\mathbb{Q}},E)$ are in bijection with ${\mathbb{Q}}$-isomorphism
classes of homogeneous spaces. Hence the set of ${\mathbb{Q}}$-isomorphism
classes of homogenous spaces inherits the structure of a group, called the
Weil-Châtelet group $\textup{WC}(E/{\mathbb{Q}})\cong
H^{1}(G_{\mathbb{Q}},E)$. In this section we give an explicit equation for the
homogeneous space that comes from a point
$Q\in\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$ through the sequence (2.3).

Localizing (2.3) gives the commutative diagram with exact rows

(2.4)
${\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))}$${{H^{1}(G_{\mathbb{Q}},E[\psi])}}$${{\textup{WC}(E/{\mathbb{Q}})[\psi]}}$${\prod\limits_{v\in
M_{\mathbb{Q}}}\bar{E}({\mathbb{Q}}_{v})/\phi(E({\mathbb{Q}}_{v}))}$${\prod\limits_{v\in
M_{\mathbb{Q}}}H^{1}(G_{\mathbb{Q}},E[\psi])}$${\prod\limits_{v\in
M_{\mathbb{Q}}}\textup{WC}(E/{\mathbb{Q}}_{v})[\psi].}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

This allows one to define Selmer groups and Tate-Shafarevich groups, recalled
here:

######  Definition 2.1.

The _$\psi$ -Selmer group_ of $E$ over ${\mathbb{Q}}$, denoted
$\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$, is defined as the kernel of the
composite map $H^{1}(G_{\mathbb{Q}},E[\psi])\to{\prod_{v\in
M_{\mathbb{Q}}}\textup{WC}(E/{\mathbb{Q}}_{v})[\psi]}$ in the diagram (2.4).

######  Definition 2.2.

The _Tate-Shafarevich group_ of $E$ over ${\mathbb{Q}}$, denoted
$\Sha(E/{\mathbb{Q}})$, is the kernel of the localization map
$\textup{WC}(E/{\mathbb{Q}})\to\prod_{v\in
M_{\mathbb{Q}}}\textup{WC}(E/{\mathbb{Q}}_{v}).$

From (2.4) one infers an exact sequence

$0\to\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))\to\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})\to\Sha(E/{\mathbb{Q}})[\psi]\to
0.$

###  2.3. Explicit equations for homogeneous spaces

We define $L:={\mathbb{Q}}(\bar{E}[\hat{\psi}])$, the extension obtained by
adjoining $\sqrt{\bar{C}}$ or $\sqrt{\bar{A}}$. Note that
$E[\psi]\cong\mu_{3}$ as $G_{L}$-modules. Moreover, since
$\gcd(\\#\textup{Gal}(L/{\mathbb{Q}}),\\#E[\psi])=1$, all cohomology groups
$H^{k}(\textup{Gal}(L/{\mathbb{Q}}),E[\psi])$ are trivial, so the restriction
morphism $H^{1}(G_{\mathbb{Q}},E[\psi])\to H^{1}(G_{L},E[\psi])$ is an
isomorphism. Combining this with Hilbert’s Theorem 90 gives

$H^{1}(G_{\mathbb{Q}},E[\psi])\cong H^{1}(G_{L},E[\psi])\cong
H^{1}(G_{L},\mu_{3})\cong L^{\times}/{L^{\times}}^{3}.$

Under this sequence of isomorphisms, (2.3) becomes

(2.5)
${0}$${\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))}$${L^{\times}/{L^{\times}}^{3}}$${\textup{WC}(E/{\mathbb{Q}})[\psi]}$${0.}$$\scriptstyle{\delta}$

######  Lemma 2.3.

Let $E$ be given by $y^{2}=x^{3}+C$. The connecting morphism
$\delta:\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))\to L^{*}/L^{*3}$ is given
by

$\displaystyle\delta:\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$
$\displaystyle\to L^{\times}/{L^{\times}}^{3}$ $\displaystyle(\xi,\eta)$
$\displaystyle\mapsto\left(\eta+\sqrt{\bar{C}}\right)\cdot{L^{\times}}^{3}$
$\displaystyle O$ $\displaystyle\mapsto 1\cdot{L^{\times}}^{3}.$

If $\bar{C}$ is a square, then we have

$\delta:(0,\pm\sqrt{\bar{C}})\mapsto\left(4C\right)^{\mp 1}.$

If $E$ is given by $y^{2}=x^{3}+A(x-B)^{2}$, then the connecting morphism is
given by

$\displaystyle\delta:\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$
$\displaystyle\to L^{\times}/{L^{\times}}^{3}$ $\displaystyle(\xi,\eta)$
$\displaystyle\mapsto(\eta+(\xi-\bar{B})\sqrt{\bar{A}})\cdot{L^{\times}}^{3}$
$\displaystyle O$ $\displaystyle\mapsto 1\cdot{L^{\times}}^{3}$

If $\bar{A}$ is a square, then we have

$\delta:(0,\pm\sqrt{\bar{A}}\bar{B})\mapsto\left(2\bar{B}A^{2}\right)^{\mp
1}.$

###### Proof.

The connecting homomorphism
$\delta:\bar{E}({\mathbb{Q}})/\psi({\mathbb{Q}})\to
H^{1}(G_{\mathbb{Q}},E[\psi])$ sends a point
$P=(X,Y)\in\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$ to the cocycle
$\sigma\mapsto Q^{\sigma}-Q$, where $\psi(Q)=P$. The remainder of the proof is
a direct computation using the definition of $\psi$ and Hilbert’s Theorem 90.
∎

Let $S_{E,\psi}\subset M_{L}$ denote the set of bad primes of $E$, together
with the primes that divide the order of $\psi$ (in this case $3$) and the
infinite primes of $L$. We define

$\displaystyle L(S_{E,\psi},3)$ $\displaystyle:=\\{t\in L^{\times}/L^{\times
3}\;:\;v(t)\equiv 0\;\forall v\in M_{L}\setminus S_{E,\psi}\\}$ $\displaystyle
L(S_{E,\psi},3)^{*}$ $\displaystyle:=\\{t\in
L(S_{E,\psi},3)\;:\;N_{L/{\mathbb{Q}}}(t)\in{L^{\times}}^{3}\\}.$

It is well-known (see, for instance, [11, Corollary X.4.4]) that
$\textup{im}(\delta)$ is contained in $L(S_{E,\psi},3)$. This implies that
$\textup{im}(\delta)$ is finite, which is essentially the weak Mordell-Weil
theorem. It is also visible that $\textup{im}(\delta)$ is contained in
$L(S_{E,\psi},3)^{*}$.

######  Lemma 2.4.

Let $t\in{\mathcal{O}}_{L}$ be cubefree and suppose $t\in\textup{im}(\delta)$.
Then the ideal $t{\mathcal{O}}_{L}$ is factored into split primes and non-
principal ramified primes.

###### Proof.

Suppose an inert or principal ramified prime
${\mathfrak{p}}=\alpha{\mathcal{O}}_{L}$ were to divide $t{\mathcal{O}}_{L}$.
Then its norm, which is $\alpha^{2}$, would be a cube and hence $\alpha$ would
be a cube. But this contradicts the assumption that $t$ is cubefree. ∎

We now give explicit equations for the homogeneous spaces corresponding to
such an element $t$.

######  Theorem 2.5.

Let $t\in{\mathcal{O}}_{L}$ be a cubefree representative in
$L(S_{E,\psi},3)^{*}$ with $N_{L/{\mathbb{Q}}}(t)=s^{3}$. Through the map
$L(S_{E,\psi},3)^{*}\to\textup{WC}(E/{\mathbb{Q}})[\psi]$ in (2.5), $t$ is
mapped to a homogeneous space $C_{t}$. We give explicit equations for $C_{t}$
in $w$ and $z$ distinguishing four cases:

  * •

If $E$ is given by $y^{2}=x^{3}+C$ and $\bar{C}$ is a square, then we have

$C_{t}:t^{2}w^{3}-2t\sqrt{\bar{C}}=z^{3}.$

  * •

If $E$ if given by $y^{2}=x^{3}+C$ and $\bar{C}$ is not a square, write
$t=u+v\sqrt{\bar{C}}$. Then we have

$C_{t}:3uw^{2}z+\bar{C}uz^{3}+vw^{3}+3\bar{C}vwz^{2}=1.$

  * •

If $E$ is given by $y^{2}=x^{3}+A(x-B)^{2}$ and $\bar{A}$ is a square, we have

$C_{t}:t^{2}w^{3}-2t(wr-\bar{B})\sqrt{\bar{A}}=z^{3}.$

  * •

If $E$ is given by $y^{2}=x^{3}+A(x-B)^{2}$ and $\bar{A}$ is not a square,
write $t=u+v\sqrt{\bar{A}}$. We have

$C_{t}:3uw^{2}z+\bar{A}uz^{3}+vw^{3}+3\bar{A}vwz^{2}+\bar{B}=s(w^{2}-\bar{A}z^{2}).$

###### Proof.

We assume that $t$ comes from a point $P=(\xi,\eta)$ on
$\bar{E}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$, meaning that $t$ equals the
image of $\delta$ in Lemma 2.3 up to multiplication by a cube. For every case
this yields an explicit model of $C_{t}$.

In the first case, $t\in\textup{im}(\delta)$ implies
$w^{3}t=\eta+\sqrt{\bar{C}}$ for some $w\in{\mathbb{Q}}$. The restriction that
$\eta$ lies on $\bar{E}({\mathbb{Q}})$ yields that
$(w^{3}t-\sqrt{\bar{C}})^{2}=\eta^{2}=\xi^{3}+\bar{C}$. Setting
$z=\frac{x}{w}$ gives the equation of $C_{t}$.

The second case is similar, now we have

$(w+z\sqrt{\bar{C}})^{3}(u+v\sqrt{\bar{C}})=\eta+\sqrt{\bar{C}}.$

Equating the multiples of $\sqrt{\bar{C}}$ yields the equation of $C_{t}$.
$\eta$ automatically lies on the curve by letting $\xi^{3}$ be the norm of the
left-hand side (which is automatically a cube).

For the third case, $t\in\textup{im}(\delta)$ implies
$w^{3}t=\eta+\sqrt{\bar{A}}(\xi-\bar{B})$. Just like in the first case,
demanding that $\eta$ lies on $\bar{E}({\mathbb{Q}})$ gives the equation for
$C_{t}$.

In the fourth case, we have

$(w+z\sqrt{\bar{A}})^{3}(u+v\sqrt{\bar{A}})=\eta+\sqrt{\bar{A}}(\xi-\bar{B}).$

Equating multiples of $\sqrt{\bar{A}}$ yields

$3uw^{2}z+\bar{A}uz^{3}+vw^{3}+3\bar{A}vwz^{2}+B=\xi.$

Now, observe that

$\displaystyle\xi^{3}$
$\displaystyle=\eta^{2}-\bar{A}(\xi-\bar{B})^{2}=N_{L/{\mathbb{Q}}}(\eta+\sqrt{\bar{A}}(\xi-\bar{B}))$
$\displaystyle=N_{L/{\mathbb{Q}}}((w+z\sqrt{\bar{A}})^{3}(u+v\sqrt{\bar{A}}))=s^{3}(w^{2}-\bar{A}z^{2})^{3}.$

Substituting $\xi=s(w^{2}-\bar{A}z^{2})$ gives the equation for $C_{t}$. ∎

######  Remark 2.6.

Theorem 3.1 and Theorem 4.1 of [4] also describe the homogeneous spaces of a
rational $3$-isogeny, using the slightly different model

$\bar{E}:y^{2}=x^{3}+D(ax+b)^{2}.$

Theorem 3.1 treats the case $L={\mathbb{Q}}(\sqrt{D})={\mathbb{Q}}$, giving
the equation

$uX^{3}+\left(\frac{1}{u}\right)Y^{3}+2buZ^{3}-2abXYZ.$

This is equivalent to our first case (if $a=0$) and third case (if $a\neq 0$),
except the variables $(X,Y,Z):=(w^{2},-\xi,w)$ are used in [4]. Theorem 4.1
treats the case $L\neq{\mathbb{Q}}$, corresponding to the second case (if
$a=0)$ and fourth case (if $a\neq 0$) in Theorem 2.5. Introducing an extra
letter for the cube root of $N_{L/{\mathbb{Q}}}(t)$ is avoided by writing
$\textup{Gal}(L/{\mathbb{Q}})=\langle\tau\rangle$ and $t=v^{2}\tau(v)$ (which
is allowed because $t$ is chosen up to cubes), such that
$N_{L/{\mathbb{Q}}}(t)=(v\tau(v))^{3}$. This yields a different, but
equivalent model of the corresponding homogeneous space.

##  3\. A set of explicit elements of order $3$ in Shafarevic groups

In this section we exhibit a set $\\{E_{h}\\}$ of elliptic curves with a
rational $2$-isogeny $\phi$ and a rational $3$-isogeny $\psi$. By the well-
known method of $2$-isogeny descent we show that every $E_{h}$ has rank zero.
Then we show that the Selmer group $\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})$
must contain at least $9$ elements. These plane cubics, whose model is given
by Theorem 2.5, must then violate the Hasse principle; they have a point
everywhere locally (since they are in the Selmer group), but they don’t have a
point globally (since this would give rise to a global point on the elliptic
curve that we know doesn’t exist). We guarantee the existence of elements in
the $\psi$-Selmer group using the following result of Cassels.

######  Theorem 3.1.

Let $\psi:E/{\mathbb{Q}}\to\bar{E}/{\mathbb{Q}}$ be a rational isogeny and let
$\hat{\psi}:\bar{E}/{\mathbb{Q}}\to E/{\mathbb{Q}}$ be its dual. Then the
following relation between the corresponding Selmer groups holds ([3, Theorem
1.1]):

$\frac{\\#\textup{Sel}^{(\hat{\psi})}(\bar{E}/{\mathbb{Q}})}{\\#\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})}=\frac{\\#\bar{E}({\mathbb{Q}})[\hat{\psi}]\Omega_{E}\prod_{p}c_{E,p}}{\\#E({\mathbb{Q}})[\psi]\Omega_{\bar{E}}\prod_{p}c_{\bar{E},p}},$

where $\Omega_{E}:=\int_{E({\mathbb{R}})}\omega_{E}$ denotes the real period
of $E$ and $c_{p}:=\\#E({\mathbb{Q}}_{p})/E_{0}({\mathbb{Q}}_{p})$ denotes the
Tamagawa number of $E$ at $p$.

###  3.1. The elliptic curves $E_{h}$

Recall the general form of an elliptic curve with a rational $3$-isogeny and
non-zero $j$-invariant:

$E:y^{2}=x^{3}+A(x-B)^{2}.$

We wish to use descent by $2$-isogeny to show that $E$ has rank zero, so we
want to impose on the parameters $A$ and $B$ that $E$ admits a rational
$2$-isogeny. This is equivalent to the right-hand side polynomial having a
rational root $x=c$. This implies $A=-\frac{c^{3}}{(c-B)^{2}}$. Multiplying by
$(c-B)^{6}$, rescaling $x$ and $y$ and introducing $h:=B$ yields

$E:y^{2}=x^{3}-c^{3}(x-h(h-c)^{2})^{2}.$

We now make a strategic choice of $c$; we will later do arithmetic in
${\mathbb{Q}}(\bar{E}[\hat{\psi}]|)={\mathbb{Q}}(\sqrt{3c})$, so we pick $c=6$
to land in the convenient number field ${\mathbb{Q}}(\sqrt{2})$. This gives
the model of the curve we will use:

(3.1) $E_{h}:y^{2}=x^{3}-216(x-h(h-6)^{2})^{2}.$

Denote the rational $2$-isogeny by $\phi:E_{h}\to E_{h}^{\prime}$ and the
rational $3$-isogeny by $\psi:E_{h}\to\bar{E}_{h}$.

######  Remark 3.2.

Viewing $h$ as a variable, one can view $E_{h}/{\mathbb{Q}}(h)$ as the generic
fiber of an elliptic surface $\mathcal{E}\to\mathbb{P}^{1}$ defined over
${\mathbb{Q}}$. This turns out to be a quadratic twist of Beauville’s rational
modular elliptic surface associated to the congruence subgroup
$\Gamma_{0}(6)$. Indeed, Beauville in [1] describes this surface as the pencil
of plane cubics given by

$(x+y)(x+z)(y+z)+txyz=0.$

Using e.g. the basepoint/section $(0:0:1)$ on the curves in this pencil, one
readily obtains the Weierstrass equation

$y^{2}=x^{3}+((t+2)x+4t)^{2}$

for the latter surface. The Möbius transformation $h\mapsto t=-2h/(h-6)$
transforms this, after scaling, into

$y^{2}=x^{3}+(6x+h(h-6)^{2})^{2}$

which is the quadratic twist over ${\mathbb{Q}}(h)(\sqrt{-6})/{\mathbb{Q}}(h)$
of $E_{h}$.

As a consequence (see also [15, § 2.3.2] and No. 66 in the table of [8], as an
elliptic curve over the function field ${\mathbb{Q}}(h)$ one finds
$E_{h}({\mathbb{Q}}(h))\cong{\mathbb{Z}}/2{\mathbb{Z}}$ and
$E_{h}({\mathbb{Q}}(h)(\sqrt{-6}))\cong{\mathbb{Z}}/6{\mathbb{Z}}$. In [10, p.
167 Table 8.3] another equation defining Beauville’s example is presented.

We now consider values $h\in{\mathbb{Z}}$ and take $E=E_{h}/{\mathbb{Q}}$. In
order to maintain some control over the local computations, we minimize the
number of bad primes of $E$; we make sure that the discriminant

$\Delta_{E}=-2^{10}3^{9}h^{3}(h-2)^{2}(h-6)^{6}(h-8)$

has only $6$ bad primes, by demanding that $|h|$, $|h-2|$, $|h-6|$ and $|h-8|$
are prime numbers. Note that this implies $h\equiv 4\bmod 15$.

In view of Theorem 3.1, we wish to manipulate the Tamagawa numbers such that
$\prod_{p}c_{E,p}<\prod_{p}c_{\bar{E},p}$ and hence
$\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$ grows. Using the algorithm in [13], we
compiled Table 3.1.

| $h\equiv 1\bmod 8$ | $h\equiv 3\bmod 8$ | $h\equiv 5\bmod 8$ | $h\equiv 7\bmod 8$
---|---|---|---|---
$p=2$ | Additive | Additive | Additive | Additive
$p=3$ | Additive | Additive | Additive | Additive
$p=h-8$ | Split mult. | Non-split mult. | Non-split mult. | Split mult.
$p=h-6$ | Non-split mult. | Non-split mult. | Split mult. | Split mult.
$p=h-2$ | Split mult. | Split mult. | Non-split mult. | Non-split mult.
$p=h$ | Split mult. | Non-split mult. | Non-split mult. | Split mult.
$\prod_{p}c_{E_{h},p}$ | $48$ | $16$ | $48$ | $144$
$\prod_{p}c_{E_{h}^{\prime},p}$ | $48$ | $16$ | $48$ | $144$
$\prod_{p}c_{\bar{E}_{h},p}$ | $144$ | $48$ | $16$ | $48$
Table 3.1. A table representing the local properties of $E_{h}$.

The cases $h\equiv 1,3\bmod 8$ are favorable in view of Theorem 3.1. In order
to keep the local computations required for descent by $2$-isogeny manageable,
it helps to have small Tamagawa numbers, so we pick the case $h\equiv 3\bmod
8$. Combining this with the earlier congruence relation $h\equiv 4\bmod 15$
yields $h\equiv 19\bmod 120$, which we assume from here on out.

###  3.2. Descent by $2$ -isogeny

Let $E_{h}$ be given by (3.1) with $h\equiv 3\bmod 8$ and $|h|$, $|h-2|$,
$|h-6|$ and $|h-8|$ primes. In order to apply the classical theory of
$2$-isogeny descent, we move the $2$-torsion point to $(0,0)$. This yields the
models

(3.2) $\displaystyle E_{h}:y^{2}$
$\displaystyle=x^{3}+\left(18(h-6)^{2}\right)x^{2}+108(h-2)(h-6)^{3}x$ (3.3)
$\displaystyle E_{h}^{\prime}:Y^{2}$
$\displaystyle=X^{3}+\left(432-36(h-6)^{2}\right)X^{2}-108h^{3}(h-8)X$

######  Lemma 3.3.

$E_{h}^{\prime}({\mathbb{Q}})/\phi(E_{h}({\mathbb{Q}}))=\\{O,(0,0)\\}$.

###### Proof.

Following the classical descent by $2$-isogeny in [11, Proposition X.4.9],
there is an injective homomorphism

$\displaystyle\alpha:E_{h}^{\prime}({\mathbb{Q}})/\phi(E({\mathbb{Q}}))$
$\displaystyle\to{\mathbb{Q}}^{\times}/{{\mathbb{Q}}^{\times}}^{2}$
$\displaystyle(X,Y)$ $\displaystyle\mapsto X\cdot{{\mathbb{Q}}^{\times}}^{2}$
$\displaystyle(0,0)$
$\displaystyle\mapsto-108h^{3}(h-8)\cdot{{\mathbb{Q}}^{\times}}^{2}=-3h(h-8)\cdot{{\mathbb{Q}}^{\times}}^{2}.$

We have

$\langle-3h(h-8)\rangle\subseteq\textup{im}(\alpha)\subseteq\textup{Sel}^{(\phi)}(E/{\mathbb{Q}})\subseteq\langle-1,2,3,h,h-2,h-6,h-8\rangle.$

We prove that $\textup{Sel}^{(\phi)}(E/{\mathbb{Q}})=\\{1,-3h(h-8)\\}$ by
localizing to various primes.

First, note that the square-free part of $X$ has to divide $-216h^{3}(h-8)$ in
order to yield a point $(X,Y)\in E_{h}^{\prime}({\mathbb{Q}})$ (see [12, page
86-87] for a more elaborate argumentation). This gives
$\textup{im}(\alpha)\subseteq\langle-1,2,3,h,h-8\rangle$.

We restrict the image further by analyzing

$\alpha_{2}:E_{h}^{\prime}({\mathbb{Q}}_{2})/\phi(E_{h}({\mathbb{Q}}_{2}))\to{\mathbb{Q}}_{2}^{\times}/{{\mathbb{Q}}_{2}^{\times}}^{2}=\langle-1,5,2\rangle.$

The reduction of $(X,Y)$ is singular precisely when $X$ is even, in which case
it reduces to $(0,0)$. If $X$ is odd, then so is $Y$ and they satisfy the
homogeneous equation

$\displaystyle Y^{2}Z$
$\displaystyle=X^{3}+\left(432-36(h-6)^{2}\right)X^{2}Z-108h^{3}(h-8)XZ^{2}$
$\displaystyle\Rightarrow Z$ $\displaystyle\equiv X+4Z+4X\equiv X\bmod 8.$

Hence $X/Z=1\cdot{{\mathbb{Q}}_{2}^{\times}}^{2}$. Moreover, observe that

$\alpha_{2}((0,0))=-108h^{3}(h-8)\cdot{{\mathbb{Q}}_{2}^{\times}}^{2}=5{{\mathbb{Q}}_{2}^{\times}}^{2},$

and hence $\textup{im}(\alpha_{2})=\langle 5\rangle$. This gives the
restriction $\textup{im}(\alpha)\subseteq\langle-3,-h,-(h-8)\rangle$.

Now consider the localized map

$\alpha_{3}:E_{h}^{\prime}({\mathbb{Q}}_{3})/\phi(E_{h}({\mathbb{Q}}_{3}))\to{\mathbb{Q}}_{3}^{\times}/{{\mathbb{Q}}_{3}^{\times}}^{2}=\langle-1,3\rangle.$

The point $(X,Y)$ reduces to a singular point if and only if $3|X$ and
otherwise $X\equiv 1\bmod 3$. Moreover, $(0,0)$ is mapped to
$-3h(h-8)\cdot{{\mathbb{Q}}_{3}^{\times}}^{2}=3{{\mathbb{Q}}_{3}^{\times}}^{2}$.
We conclude that $\textup{im}(\alpha_{3})=\langle 3\rangle$, giving the
restriction $\textup{im}(\alpha)\subseteq\langle 3h,-(h-8)\rangle$.

The final restriction comes from the localized map

$\alpha_{h-2}:E_{h}^{\prime}({\mathbb{Q}}_{h-2})/\phi(E_{h}({\mathbb{Q}}_{h-2}))\to{\mathbb{Q}}_{h-2}^{\times}/{{\mathbb{Q}}_{h-2}^{\times}}^{2}=\langle-3,h-2\rangle.$

We see that
$-3h(h-8)\cdot{{\mathbb{Q}}_{h-2}^{\times}}^{2}=36\cdot{{\mathbb{Q}}_{h-2}^{\times}}^{2}=1\cdot{{\mathbb{Q}}_{h-2}^{\times}}^{2}.$
The algorithm in [13] shows that all points of
$E_{h}^{\prime}({\mathbb{Q}}_{h-2})$ have good reduction. Over
${\mathbb{F}}_{h-2}$, $E_{h}^{\prime}$ is given by

$E_{h}^{\prime}:Y^{2}=X(X-72)^{2}.$

Following [11, Proposition III.2.5], let
$\gamma=\sqrt{72}\in{\mathbb{F}}_{h-2}$ be the slope of a tangent line at the
node of $E_{h}^{\prime}({\mathbb{F}}_{h-2})$. There is an isomorphism

$\displaystyle E_{h}^{\prime}({\mathbb{F}}_{h-2})$
$\displaystyle\to{\mathbb{F}}_{h-2}^{\times}$ $\displaystyle(X,Y)$
$\displaystyle\mapsto\frac{Y+\gamma(X-72)}{Y-\gamma(X-72)}.$

This isomorphism maps the point $(18,27\gamma)$ to $-3$, which is not a
square, so this point cannot be a multiple of $2$. Finally, $18$ is a square
in ${\mathbb{Q}}_{h-2}$, so we infer $\textup{im}(\alpha_{h-2})=\\{1\\}$. This
gives the final restriction

$\textup{im}(\alpha)=\textup{Sel}^{(\phi)}(E/{\mathbb{Q}})=\\{1,-3h(h-8)\\}$

and therefore

$E_{h}^{\prime}(E/{\mathbb{Q}})/\phi(E_{h}({\mathbb{Q}}))=\\{O,(0,0)\\}.$

∎

######  Lemma 3.4.

$E_{h}({\mathbb{Q}})/\hat{\phi}(E_{h}^{\prime}({\mathbb{Q}}))=\\{O,(0,0)\\}$.

###### Proof.

The proof is similar to the proof of Lemma 3.3. There is an embedding

$\displaystyle\alpha^{\prime}:E_{h}({\mathbb{Q}})/\hat{\phi}(E_{h}^{\prime}({\mathbb{Q}}))$
$\displaystyle\to{\mathbb{Q}}^{\times}/{{\mathbb{Q}}^{\times}}^{2}$
$\displaystyle(x,y)$ $\displaystyle\mapsto x\cdot{{\mathbb{Q}}^{\times}}^{2}$
$\displaystyle(0,0)$ $\displaystyle\mapsto
108(h-2)(h-6)^{3}\cdot{{\mathbb{Q}}^{\times}}^{2}=3(h-2)(h-6)\cdot{{\mathbb{Q}}^{\times}}^{2}$

The image consists of divisors of $3(h-2)(h-6)$;
$\textup{im}(\alpha^{\prime})\subseteq\langle-1,2,3,h-2,h-6\rangle$.

An analysis of $\alpha_{2}^{\prime}$ shows that

$\alpha_{2}^{\prime}((0,0))=3(h-2)(h-6)\cdot{{\mathbb{Q}}_{2}^{\times}}^{2}=-1\cdot{{\mathbb{Q}}_{2}^{\times}}^{2},$

while the points of $E_{h}({\mathbb{Q}}_{2})$ with non-singular reduction have
an odd $x$-coordinate (up to squares). Combining this with the fact
$E_{h}({\mathbb{Q}}_{2})/E_{h,0}({\mathbb{Q}}_{2})=\\{O,(0,0)\\}$ implies that
the image of $\alpha^{\prime}$ must consist of odd numbers;
$\textup{im}(\alpha^{\prime})\subseteq\langle-1,3,h-2,h-6\rangle$.

A further restriction is obtained by computing the image of
$\alpha_{3}^{\prime}$. For points with non-singular reduction it follows that
$x\equiv 1\bmod{\mathbb{Q}}_{3}^{2}$. We obtain

$\textup{im}(\alpha_{3}^{\prime})=\langle\alpha_{3}^{\prime}((0,0))\rangle=\langle
3(h-2)(h-6)\rangle=\langle 6\rangle.$

This implies
$\textup{im}(\alpha^{\prime})\subseteq\langle-3,-(h-2),h-6\rangle$.

We then compute the image of $\alpha_{h-8}^{\prime}$. Note that $3(h-2)(h-6)$
is a square in ${\mathbb{Q}}_{h-8}$. $E_{h}$ has non-split multiplicative
reduction at $h$, so following [11, Ex.III.3.5] there is an isomorphism

$\displaystyle E_{h}({\mathbb{F}}_{h-8})$
$\displaystyle\cong\textup{ker}(N:{\mathbb{F}}_{(h-8)^{2}}^{\times}\to{\mathbb{F}}_{h-8}^{\times})$
$\displaystyle(x,y)$
$\displaystyle\mapsto\frac{y+\gamma(x-72)}{y-\gamma(x-72)},$

where $\gamma=\sqrt{72}\notin{\mathbb{F}}_{h-8}$ is the slope of a tangent
line at the node $(72,0)$. The point $(-72,864\sqrt{-2})$ is mapped to
$\sqrt{-1}$, which is a non-square in
$\textup{ker}(N:{\mathbb{F}}_{(h-8)^{2}}\to{\mathbb{F}}_{h-8})$. Note that
${\mathbb{F}}_{h-8}$ contains a square root of $-2$, since $h-8\equiv 3\bmod
8$. This also implies that $-72$ is a square in ${\mathbb{Q}}_{h-8}$ and hence
$\textup{im}(\alpha_{h-8}^{\prime})=\\{1\\}$. Therefore the image of
$\alpha^{\prime}$ must consist of squares in ${\mathbb{Q}}_{h-8}$;
$\textup{im}(\alpha^{\prime})\subseteq\langle-(h-2),-3(h-6)\rangle$.

Finally, we restrict the image one step further by analyzing the localization
$\alpha_{h}^{\prime}$. Note first that $\alpha^{\prime}((0,0))=3(h-2)(h-6)$ is
a square in ${\mathbb{Q}}_{h}$. The remaining points have non-singular
reduction. Again the reduction is non-split multiplicative, so we obtain

$\displaystyle E_{h}({\mathbb{F}}_{h})$
$\displaystyle\cong\textup{ker}(N:{\mathbb{F}}_{h^{2}}^{\times}\to{\mathbb{F}}_{h}^{\times})$
$\displaystyle(x,y)$
$\displaystyle\mapsto\frac{y+\gamma(x+216)}{y-\gamma(x+216)},$

where $\gamma^{2}=-216$, for $\gamma$ a slope of a tangent line at the node
$(-216,0)$. The point $(216,2592\sqrt{6})$ is mapped to $\sqrt{-1}$, a non-
square in
$\textup{ker}(N:{\mathbb{F}}_{h^{2}}^{\times}\to{\mathbb{F}}_{h}^{\times})$.
Note that $6$ is a square in ${\mathbb{F}}_{h}$, since $h\equiv 3\bmod 8$ and
$h\equiv 1\bmod 3$. The same observation yields that the $x$-coordinate is
$216$ is a square in ${\mathbb{F}}_{h}$ and hence
$\textup{im}(\alpha_{h}^{\prime})=\\{1\\}$. This gives the result

$\displaystyle\textup{im}(\alpha^{\prime})$
$\displaystyle=\textup{Sel}^{(\hat{\phi})}(E_{h}^{\prime}/{\mathbb{Q}})=\langle
3(h-2)(h-6)\rangle$ $\displaystyle\Rightarrow
E_{h}({\mathbb{Q}})/\hat{\phi}(E_{h}({\mathbb{Q}}))$
$\displaystyle=\\{O,(0,0)\\}$

∎

######  Corollary 3.5.

The Mordell-Weil groups $E_{h}({\mathbb{Q}})$, $E_{h}^{\prime}({\mathbb{Q}})$
and $\bar{E}_{h}({\mathbb{Q}})$ have rank zero.

###### Proof.

This is a consequence of the formula (see, e.g., [12, page 91])

$2^{\text{rank}_{\mathbb{Q}}(E_{h})}=\frac{\\#\textup{im}(\alpha)\\#\textup{im}(\alpha^{\prime})}{4}.$

Since $E_{h}^{\prime}$ and $\bar{E}_{h}$ are isogenous to $E_{h}$, these also
have rank zero over ${\mathbb{Q}}$. ∎

###  3.3. Descent by $3$-isogeny

We now compute $\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$. Corollary 3.5
guarantees that $E_{h}({\mathbb{Q}})$ has rank zero and our choice of equation
(3.1) shows that $E_{h}({\mathbb{Q}})$ does not have $3$-torsion, elements of
$\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})$ cannot come from points on
$\bar{E}_{h}({\mathbb{Q}})/\phi(E({\mathbb{Q}}))$.

######  Theorem 3.6.

$\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$ contains precisely $9$ elements and is
isomorphic to $\Sha(E_{h}/{\mathbb{Q}})[3]$.

###### Proof.

The lower bound for $\\#\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})$ comes from
Theorem 3.1. The only quantities yet unknown are the real periods of $E$ and
$\bar{E}$. We use [5, Lemma 7.4]:

(3.4)
$\frac{\Omega_{E_{h}}}{\Omega_{\bar{E}_{h}}}=\frac{\\#\textup{ker}\left(\psi:E_{h}({\mathbb{R}})\to\bar{E}_{h}({\mathbb{R}})\right)}{\\#\textup{coker}\left(\psi:E_{h}({\mathbb{R}})\to\bar{E}_{h}({\mathbb{R}})\right)}\cdot\left|\frac{\omega_{E_{h}}}{\psi^{*}\omega_{\bar{E}_{h}}}\right|,$

where $\omega_{E}$ denotes the invariant differential of a minimal model of an
elliptic curve $E$. Both the real kernel and the real cokernel of $\psi$ are
trivial, so we need only compute $\omega_{E_{h}}$ and
$\psi^{*}\omega_{\bar{E}_{h}}$. A minimal model of $\bar{E}_{h}$ is given by
$y^{2}=x^{3}+8(3x-(h-2)(h-6)^{2})^{2}$. We abbreviate $A:=8$ and
$B:=(h-2)(h-6)^{2}$, such that the explicit form (2.1) of $\psi$ gives

$\displaystyle\psi^{*}\omega_{\bar{E}_{h}}=\psi^{*}\left(\frac{d\xi}{2\eta}\right)$
$\displaystyle=\frac{d\left(3^{-3}(6y^{2}+6AB^{2}-3x^{3}-2Ax^{2})x^{-2}\right)}{2\cdot
3^{-3}(8AB^{2}-x^{3}-4ABx)x^{-3}}$
$\displaystyle=\frac{x^{3}d\left((6\left(x^{3}+A(x-B)^{2}\right)+6AB^{2}-3x^{3}-2Ax^{2})x^{-2}\right)}{2(8AB^{2}-x^{3}-4ABx)}$
$\displaystyle=\frac{x^{3}(3+12ABx^{-2}-24AB^{2}x^{-3})dx}{2(8AB^{2}-x^{3}-4ABx)}$
$\displaystyle=\frac{-3dx}{2y}=-3\omega_{E_{h}}.$

Equation (3.4) then reads

$\frac{\Omega_{E_{h}}}{\Omega_{\bar{E}_{h}}}=\frac{\\#\textup{ker}\left(\psi:E_{h}({\mathbb{R}})\to\bar{E}_{h}({\mathbb{R}})\right)}{\\#\textup{coker}\left(\psi:E_{h}({\mathbb{R}})\to\bar{E}_{h}({\mathbb{R}})\right)}\cdot\left|\frac{\omega_{E_{h}}}{\psi^{*}\omega_{\bar{E}_{h}}}\right|.$

Theorem 3.1 provides

$\frac{\\#\textup{Sel}^{(\hat{\psi})}(\bar{E}_{h}/{\mathbb{Q}})}{\\#\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})}=\frac{\\#\bar{E}_{h}({\mathbb{Q}})[\hat{\psi}]\Omega_{E_{h}}\prod_{p}c_{E_{h},p}}{\\#E_{h}({\mathbb{Q}})[\psi]\Omega_{\bar{E}_{h}}\prod_{p}c_{\bar{E}_{h},p}}=\frac{1\cdot
1\cdot 16}{1\cdot 3\cdot 48}=\frac{1}{9}.$

Thus $9|\\#\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})$.

An upper bound on $\\#\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})$ is obtained
by arithmetic in
$L={\mathbb{Q}}(\sqrt{2})={\mathbb{Q}}(E_{h}^{\prime}[\hat{\psi}])$. We
compute $L(S_{E,\psi},3)^{*}$, the elements of $L^{\times}/{L^{\times}}^{3}$
that only ramify at the bad primes and have cube norm. Recall that by Lemma
2.4 the inert primes and the principally ramified primes of
${\mathcal{O}}_{L}$ cannot contribute. The prime $2$ is ramified in
${\mathcal{O}}_{L}={\mathbb{Z}}[\sqrt{2}]$, a principal ideal domain. Since
$2$ is non-square in ${\mathbb{F}}_{3}$, ${\mathbb{F}}_{h}$,
${\mathbb{F}}_{h-6}$ and ${\mathbb{F}}_{h-8}$, these primes are inert. Only
the prime $|h-2|=(k+l\sqrt{2})(k-l\sqrt{2})$ is split. Thus
$L(S_{E,\psi},3)^{*}$ is generated by $(k+l\sqrt{2})^{2}(k-l\sqrt{2})$ and the
fundamental unit $1+\sqrt{2}$, but this gives $\\#L(S_{E,\psi},3)^{*}=9$. We
conclude

$9\leq\\#\textup{Sel}^{(\psi)}(E/{\mathbb{Q}})\leq\\#L(S_{E,\psi},3)^{*}\leq
9.$

Since $E_{h}({\mathbb{Q}})$ and $\bar{E}_{h}({\mathbb{Q}})$ have rank zero and
lack $3$-torsion, $\bar{E}_{h}({\mathbb{Q}})/\psi(E({\mathbb{Q}}))$ is
trivial. The exact sequence

$0\to\bar{E_{h}}({\mathbb{Q}})/\psi(E_{h}({\mathbb{Q}}))\to\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})\to\Sha(E_{h}/{\mathbb{Q}})[\psi]\to
0$

then implies
$\textup{Sel}^{(\psi)}(E_{h}/{\mathbb{Q}})\cong\Sha(E_{h}/{\mathbb{Q}})[\psi]$.
Finally, since $\textup{Sel}^{(\hat{\psi})}(E_{h}^{\prime}/{\mathbb{Q}})$ is
trivial by Theorem 3.1, we conclude
$\Sha(E_{h}/{\mathbb{Q}})[\psi]\cong\Sha(E_{h}/{\mathbb{Q}})[3]$, which
finishes the proof. ∎

######  Corollary 3.7.

Assume $h\in{\mathbb{Z}}$ satisfies the following conditions:

  * •

$h\equiv 3\bmod 8$.

  * •

$|h|$, $|h-2|$, $|h-6|$ and $|h-8|$ are prime numbers.

Then the plane cubics

$C_{\pm 1+\sqrt{2}}^{h}\colon
18w^{2}z+432z^{3}+w^{3}+216wz^{2}+2(h-2)^{2}(h-8)=\pm(432z^{2}-6w^{2})$

have a point in every completion of ${\mathbb{Q}}$, but not in ${\mathbb{Q}}$
itself.

###### Proof.

The equations for $C_{\pm 1+\sqrt{2}}^{h}$ are an application of Theorem 2.5
with $t=\pm 1+\sqrt{2}$. We use a minimal model of $\bar{E}_{h}$ with
$\bar{A}=72$ and $\bar{B}=(h-2)^{2}(h-8)/3$ to reduce the size of the
coefficients. Theorem 3.6 asserts that these curves represent non-trivial
elements of the Tate-Shafarevich group, meaning that they violate the Hasse
principle. ∎

An attentive reader may have already noticed that $h=19$ satisfies the
hypotheses, giving the homogeneous spaces

$C_{\pm 1+\sqrt{2}}^{19}\colon
18w^{2}z+432z^{3}+w^{3}+216wz^{2}+6358=\pm(432z^{2}-6w^{2})$

which violate the Hasse principle. Using Magma ([2]), the values of $h$
between the bounds $19-120000$ and $19+120000$ that satisfy these conditions
are:

$\begin{array}[]{c}-77261,-72221,-62981,-51341,-43781,-25301,-19421,-16061,-3461,-821,-101,\\\
19,3259,5659,15739,21019,55339,67219,69499,79699,88819,99139,116539,119299.\end{array}$

The set $\\{E_{h}\\}$ gives a sparser set of elements of order $3$ in a Tate-
Shafarevich group than the list in [6], but the advantage is that the examples
coming from $\\{E_{h}\\}$ are easier to generate; one only needs the
congruence $h\equiv 3\bmod 8$ and the primality of $h$, $h-2$, $h-6$ in $h-8$.
Under the assumption of the more general hypothesis H in [9], there are
infinitely many such integers $h$. It has been shown in [7] that the
$3$-primary part of Tate-Shafarevich groups can be arbitrarily large, but the
proof of this doesn’t yield explicit equations for the genus-$1$ curves.

## Acknowledgements

It is our pleasure to thank Steffen Müller and Damiano Testa for their
interest in this work, and for valuable suggestions.

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