††thanks: Current address: Heinz Maier-Leibnitz Zentrum (MLZ), Technische
Universität München, 85748 Garching, Germany

# Field-Tuned Quantum Effects in a Triangular-Lattice Ising Magnet

Yayuan Qin  State Key Laboratory of Surface Physics and Department of Physics,
Fudan University, Shanghai 200433, China  Yao Shen  State Key Laboratory of
Surface Physics and Department of Physics, Fudan University, Shanghai 200433,
China  Condensed Matter Physics and Materials Science Department, Brookhaven
National Laboratory, Upton, New York 11973, USA  Changle Liu  Hongliang Wo
Yonghao Gao  State Key Laboratory of Surface Physics and Department of
Physics, Fudan University, Shanghai 200433, China  Yu Feng  Institute of High
Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China
Spallation Neutron Source Science Center, Dongguan 523803, China  Materials
and Life Science Division, J-PARC Center, Tokai, Ibaraki 319-1195, Japan
Xiaowen Zhang  Gaofeng Ding  Yiqing Gu  State Key Laboratory of Surface
Physics and Department of Physics, Fudan University, Shanghai 200433, China
Qisi Wang  State Key Laboratory of Surface Physics and Department of Physics,
Fudan University, Shanghai 200433, China  Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Shoudong Shen  State Key
Laboratory of Surface Physics and Department of Physics, Fudan University,
Shanghai 200433, China  Helen C. Walker  Robert Bewley  ISIS Facility,
Rutherford Appleton Laboratory, STFC, Chilton, Didcot, Oxon OX11 0QX, United
Kingdom  Jianhui Xu  Helmholtz-Zentrum Berlin für Materialien und Energie
GmbH, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany  Martin Boehm  Paul
Steffens  Institut Laue-Langevin, 71 Avenue des Martyrs, 38042 Grenoble Cedex
9, France  Seiko Ohira-Kawamura  Naoki Murai  Materials and Life Science
Division, J-PARC Center, Tokai, Ibaraki 319-1195, Japan  Astrid Schneidewind
Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum
(MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, 85748 Garching,
Germany  Xin Tong  Institute of High Energy Physics, Chinese Academy of
Sciences (CAS), Beijing 100049, China  Spallation Neutron Source Science
Center, Dongguan 523803, China  Gang Chen<EMAIL_ADDRESS>Department of Physics and HKU-UCAS Joint Institute for Theoretical and
Computational Physics at Hong Kong, The University of Hong Kong, Pokfulam
Road, Hong Kong, China  State Key Laboratory of Surface Physics and Department
of Physics, Fudan University, Shanghai 200433, China  Institute of
Nanoelectronics and Quantum Computing, Fudan University, Shanghai 200433,
China  Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing, 210093, China  Jun Zhao<EMAIL_ADDRESS>State Key
Laboratory of Surface Physics and Department of Physics, Fudan University,
Shanghai 200433, China  Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing, 210093, China

###### Abstract

We report thermodynamic and neutron scattering measurements of the triangular-
lattice quantum Ising magnet TmMgGaO4 in longitudinal magnetic fields. Our
experiments reveal a quasi-plateau state dressed with quantum fluctuations.
This state exhibits an unconventional non-monotonic field and temperature
dependence of the magnetic order and excitation gap. In the high field regime
where the quantum fluctuations are largely suppressed, we observed a
disordered state with coherent magnon-like excitations despite the suppression
of the spin excitation intensity. Through the combination of quantum and semi-
classical analysis, we are able to understand these behaviors rather
quantitatively from the subtle competition among quantum fluctuations,
frustrated Ising interactions, and thermal fluctuations.

Ising model is a stereotype model in modern statistical physics and has
revolutionarily advanced our understanding of phase transitions Onsager . It
can be realized in quantum magnets with a strong easy-axis anisotropy that
pins the spin moments towards a fixed axis. Due to the pinning effect, the
quantum effect is suppressed, and the physics becomes rather classical
Ising_Ramirez . To introduce quantum effects, one usually applies an external
magnetic field along the transverse direction and creates a quantum Ising
model Quantum_Sachdev ; TranField_Moessner ; TranField_Savary . Representative
examples of this type include the quasi-one-dimensional magnets CoNb2O6 (Ref.
CoNb2O6_Coldea, ), BaCo2V2O8 (Refs. BCVO_Faure, ; BCVO_Matsuda, ; BCVO_Zou, ),
and SrCo2V2O8 (Refs. SCVO_Wang1, ; SCVO_Wang2, ), in which quantum criticality
and novel transitions have been reported. Nevertheless, the experimental
realization of quantum Ising model in two dimensional systems is rare despite
decades of theoretical efforts TranField_Moessner ; TFIM_Moessner ;
TFIM_Isakov ; TFIM_Damle ; TFIM_Biswas ; TFIM_Wang . In contrast to this
conventional route, nature provides a distinct example of quantum Ising
magnets that build quantum mechanics intrinsically in the system. The
observation is that, although the exchange interaction is primarily Ising
like, the intrinsic crystal electric field (CEF) splitting of the Ising moment
naturally creates quantum effects out of these classically interacting degrees
of freedom TranField_Wang ; TranField_Gang ; TranField_Dun . These systems are
quoted as “intrinsic quantum Ising magnets”.

The triangular-lattice antiferromagnet TmMgGaO4 (ref. TMGO_Cava, ) is a
promising candidate for such requirements. Due to the strong spin-orbit
coupling (SOC) and CEF splitting, the CEF ground state wave function of Tm3+
ions is dominated by $J^{z}=\pm 6$, leading to a large magnetic moment of 6.59
$\mu_{\mathrm{B}}$/Tm3+ and Ising spin nature TMGO_Yao . Thus, quantum effects
are expected to be significantly suppressed. Interestingly, however, the
TmMgGaO4 CEF ground state is a quasi-doublet composed of two singlets
separated by a small energy gap that can be mapped into an effective
transverse field $h_{y}$ (Fig. 1a). In this case, the transverse field, which
is intrinsic in origin and homogeneous in general, leads to quantum tunneling
effects among various Ising spin configurations and strongly compete with the
Ising-type interactions TranField_Wang ; TranField_Gang ; TranField_Dun .
Different from the coherent quantum fluctuations in quantum spin liquids that
lead to long-range entanglement QSL_Balents ; QSL_Zhou ; QSL_Yao1 ; QSL_Yao2 ,
the quantum effects here are of single-ion level. The magnetic properties can
be effectively described by the transverse field Ising model (TFIM) TMGO_CLiu
; TMGO_Yao

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{ij}J^{zz}_{ij}S^{z}_{i}S^{z}_{j}-\sum_{i}(h_{y}S^{y}_{i}+B_{z}S^{z}_{i}),$
(1)

where the Ising exchange interactions can be kept within the first few
neighbors and are geometrically frustrated. Intriguingly, due to the large
Landé $g$-factor in TmMgGaO4 ($g_{\parallel}\sim$ 13.18) TMGO_YLi1 , the
contribution from external longitudinal field,
$B_{z}=g_{\parallel}\mu_{\mathrm{B}}B$, is comparable to the Ising
interactions and quantum fluctuations so that the interplay between quantum
and classical contributions can be easily tuned by external field. Therefore,
TmMgGaO4 provides a unique platform to manipulate the quantum effects in a
controlled manner.

Moreover, in rare-earth materials, due to the complex SOC and CEF splitting,
the pseudo-spins can host multipolar behaviors Multipole_Santini ;
Multipole_Alistair ; Multipole_CLiu . In the case of TmMgGaO4, we find that
the transverse components of the pseudo-spins, $S^{x}$ and $S^{y}$, behave as
multipoles that cannot be directly detected by neutron diffraction while the
longitudinal one, $S^{z}$, remains dipolar. Its zero-field magnetic ground
state is an intertwined dipolar and multipolar order in which the dipolar
$S^{z}$ forms a three-sublattice clock phase and the multipolar $S^{x}/S^{y}$
components are ferro-aligned TMGO_CLiu ; TMGO_Yao . The observed spin
excitations are in a reasonable agreement with the linear spin wave (LSW)
theory, in which, however, only $S_{zz}$ channel is detectable, rising from
the coherent spin wave excitations and fluctuations of the multipolar
components Multipole_CLiu ; Multipole_Kadowaki . In this paper, we continue
this research and study the evolution of intrinsic quantum properties of
TmMgGaO4 in longitudinal fields.

Figure 1: Heat capacity and magnetic phase diagram of TmMgGaO4. (a) Schematic
of the low energy CEF levels of Tm3+ ions in TmMgGaO4. The ground-state quasi-
doublet is well separated from the high-energy excited state ($\Delta\sim$ 38
meV), indicating an effective spin-1/2 state at low temperature TMGO_YLi1 ;
TMGO_Yao . (b) Sketch of the reciprocal space in $xy$ plane. The dashed lines
denote Brillouin zone boundaries and the wavevector is defined as
$\textbf{Q}=H\textbf{a}^{*}+K\textbf{b}^{*}+L\textbf{c}^{*}$; r.l.u.,
reciprocal lattice unit. (c) Magnetic heat capacity in various longitudinal
fields ($H\parallel z$). The phonon contributions were subtracted by measuring
the non-magnetic iso-structural LuMgGaO4 single crystal. The transition
temperatures are marked by black triangles. (d) Magnetic phase diagram
consisting of field dependent $T_{\mathrm{N}}$ determined from heat capacity,
and spin gaps at K points determined from INS data below 0.15 K. The phase
diagram can be divided into three parts. Phase I: zero-field phase; Phase II:
intermediate phase; Phase III: high-field phase. The dashed line is a guide to
the eye. (e)-(g) Schematics of the semi-classical spin structures regarding
the three phases in (d). The dashed lines denote the virtual $y$ direction and
different colors indicate three different sublattices. Figure 2: Neutron
diffraction data as a function of field and temperature. (a),(b) Temperature
dependence of the peak intensities and full-width at half-maximum (FWHM) of
the magnetic Bragg peak at K1 point. (c) Qscans near the magnetic peak, Q =
(1/3, 1/3, 0), in different longitudinal fields. (d) Field dependence of the
fitted amplitude of the magnetic peaks at different temperatures. (e)
Representative Qscans near the magnetic peak at different temperatures in
external field of 1.5 T. (f) Temperature dependence of the fitted peak
amplitude at Q = (1/3, 1/3, 0) in different fields. The solid lines in (c) and
(e) are the fitting results with Gaussian profiles. Data in (c), (e) and (f)
are collected at FLEXX spectrometer. a.u., arbitrary unit; cts/s, counts per
second; cts/min, counts per minute; error bars, 1 s.d.

We start by reviewing the phase transitions of TmMgGaO4 in absence of external
field. Although no $\lambda$-shaped transition is present in the heat capacity
data (Fig. 1c), a well-defined magnetic Bragg peak associated with the three-
sublattice spin order is observed at the K point, Q = (1/3, 1/3, 0), at low
temperature (Fig. 2c). With temperature decreasing from 20 K, the intensity
gradually increases with reduced peak width (Fig. 2a, b). The most abrupt
change takes place around 1 K, which corresponds to a shoulder-like anomaly in
the heat capacity (Fig. 1c). Both the peak intensity and peak width saturate
below $\sim$0.4 K. When a small magnetic field is applied along the
longitudinal $z$ direction, the heat capacity shows a sharp anomaly,
indicating a new magnetic phase transition (Fig. 1c). The transition
temperature, $T_{\mathrm{N}}$, increases with increasing field, reaches a
maximum at $\sim$1.5 T, and then decreases with further increasing field (Fig.
1d). The transition completely vanishes above $\sim$2.7 T.

Figure 3: Spin excitations in different longitudinal fields at 0.12 K. (a)-(e)
INS spectra of the spin excitations in TmMgGaO4 along the high symmetry
directions marked by the black arrows in Fig. 1b in the indicated fields.
(f)-(j) Calculated spin wave dispersions using the model specified in the main
text. Intensities in (e) and (j) are multiplied by 6.

To elucidate the nature of the phase transitions in longitudinal fields, we
performed neutron diffraction measurements at various temperatures. As shown
in Fig. 2c, d, the magnetic Bragg peak intensity at K point with temperatures
of 0.04, 0.12 and 0.8 K also shows a maximum around 1.5 T, and disappears at
$\sim$2.7 T. This indicates that the phase transition seen in heat capacity
measurements is also associated with the K point order. Moreover, the peak
intensities in magnetic fields of 0.5, 1, and 1.5 T are enhanced in the
intermediate temperature range (0.2 $\lesssim T\lesssim$ 0.8 K) below
$T_{\mathrm{N}}$, distinct from a conventional order parameter behavior (Fig.
2e, f). The unusual enhancement of the magnetic peak on warming is due to the
competition between thermal and quantum fluctuations, as will be discussed
subsequently.

Figure 4: Polarized neutron scattering data and the field dependence of the
spin excitations. (a) Energy scans at K1 point and 1.7 K in all the three
spin-flip (SF) channels which correspond to different excitation components
supp . The tiny peak in SFz channel may come from the leakage of non-spin-flip
channel. (b) Energy dependence of the scattering function integrated through
the whole Brillouin zone (-0.5 $<L<$ 0.5) in different external fields at 0.12
K. The inset shows the field dependence of the total local moment $\langle
m^{2}\rangle$ (All) and the separated elastic and inelastic contributions supp
. (c) The external field term, $g_{\parallel}\mu_{\mathrm{B}}B$, used during
the LSW calculation versus the experimentally applied field values,
$\mu_{0}H$. The dashed red line denotes the linear fitting of the data points
in Phase III, which gives the $g$-factor along the $z$ direction.

The enhancement of magnetic order in longitudinal fields has been interpreted
either as the consequence of quantum fluctuations in TFIM TMGO_CLiu , or large
CEF randomness induced by structural disorder TMGO_YLi1 . To further
distinguish these two scenarios, we used inelastic neutron scattering (INS) to
measure the detailed field dependence of the spin excitations in TmMgGaO4.
Figure 3a-e exhibit a series of neutron spectra along the high symmetry
directions in various fields. Compared with the zero-field data in Fig. 3a,
the overall spin excitations in external field of 1.5 T tend to shift towards
higher energies, leading to an enlarged spin gap (Fig. 3b). This is clearly
inconsistent with the CEF randomness picture in which it was suggested that
the system would be essentially gapless in external field TMGO_YLi1 . Instead,
it can be readily understood without introducing much structural disorder.
Below 1.5 T, the spin gap increases with increasing field (Fig. 1d). Thus, the
system needs more energy to excite the quasi-particles from the magnetic
ground state, implying a more stable order. Consequently, the magnetic Bragg
peaks at K points are getting stronger and $T_{\mathrm{N}}$ is higher. Above
1.5 T, the spin gap gradually decreases with field and both the magnetic peak
intensity and $T_{\mathrm{N}}$ are reduced (Fig. 1d, 2d). When the external
field goes across the transition of $\sim$2.7 T into the high-field regime,
the spin gap reappears, indicating that the external field overcomes the spin-
spin correlations and the Zeeman term dominates the spin dynamics, making the
excitations less dispersive (Fig. 3e). The sharp excitations in 5 T further
confirm that the structural disorder is not significant in TmMgGaO4 TMGO_YLi1
.

The Ising interactions arise from the exchange and dipole-dipole interactions
among Tm3+ local moments. For a classical Ising model, the excitations
correspond to Ising spin flipping and are non-dispersive. In TmMgGaO4, the
excitations are dressed with quantum fluctuations and become dispersive. To
further determine the nature of the spin excitations, we carried out polarized
neutron scattering measurements on TmMgGaO4. It is shown that the spin
excitations are essentially longitudinally polarized in the $S_{zz}$ channel
while the transverse $S_{+-}$ component is absent (Fig. 4a) supp . This is
completely different from the classical spin wave with only transverse
excitations at low energies. Therefore, the longitudinal channel of INS
spectra, $S_{zz}$, manifests the quantumness of the excitations that are
strictly forbidden for classical spins.

After confirming the quantum fluctuations in TmMgGaO4, we can understand the
phase diagram comprehensively. Dressed with single-ion quantum fluctuations,
the Ising moments are no longer homogeneous but modulated from site to site,
the order of which resembles the spin density wave in quantum spin chains
BCVO_Grenier . In zero field, the competition between classical Ising
interactions and quantum fluctuations drives the system into a three-
sublattice phase with up-down-zero configuration in each triangle (Phase I in
Fig. 1d). The quantum fluctuation in this phase is so strong that the magnetic
moments are significantly reduced. In weak longitudinal fields, the classical
triangular-lattice Ising magnet would evolve into a two-up-one-down plateau
phase XXZ_Sellmann ; XXZ_Yamamoto . In TmMgGaO4, the plateau phase is dressed
with quantum fluctuations, leading to a quasi-plateau phase with modulated
spin moments (Phase II). Here, the external field acts as an extra classical
contribution and suppresses the quantum effects. As a result, the magnetic
order is enhanced with stronger magnetic peaks at K points, higher
$T_{\mathrm{N}}$ and a larger spin gap. Since the spin dynamics here are of
quantum origin, the INS intensity is weaker (Fig. 4b). Moreover, at higher
temperatures, the quantum effects are partially quenched due to the thermal
fluctuations, leading to enhanced magnetic Bragg peaks at higher temperatures
(Fig. 2e, f). When the field is further increased towards the transition field
$\mu_{0}H_{\mathrm{c}}\sim$ 2.7 T, the quantum fluctuation is enhanced again
and the rebuilt quantum effects suppress the static magnetic order, reduce the
spin gap and lower the transition temperature while the quantum excitations
grow in intensity (Fig. 1d, Fig. 4b). Above $\mu_{0}H_{\mathrm{c}}$, the
three-sublattice magnetic order is completely eliminated and the system enters
the nearly polarized state (Phase III), making the magnetic state more
classical. Meanwhile, the spin gap increases with field and the quantum
excitations are progressively suppressed (Fig. 3c-e).

To quantitatively describe the field tuned quantum effects, we utilize mean-
field approach to solve TFIM by introducing the virtual $y$ axis. In this
semi-classical picture, the pseudo-spin has a uniform size but becomes tilted
from the Ising axis to $y$ direction owing to the transverse field. The
corresponding spin structures of the three phases are illustrated in Fig.
1e-g. Since the transverse components of the effective spins, $S^{x}$ and
$S^{y}$, transform as multipoles, only the moment projection in $z$ direction
contributes to the magnetic Bragg peak, making the peak intensity at K point
varies with longitudinal field (Fig. 2d). Regarding the INS process, measuring
$S^{z}$ moment will flip the multipolar components and trigger coherence spin
wave excitations in $S_{zz}$ channel Multipole_CLiu ; pyrochlore_YLi ,
consistent with our polarized neutron measurements. Based on this, we use LSW
theory to simulate the quantum excitations of TFIM in longitudinal fields. The
Ising interactions and transverse field term are determined by fitting the
zero-field spin excitation spectra TMGO_Yao . We show that the LSW theory can
describe the observed quantum excitations throughout the phase diagram (Fig.
3f-j). The field-dependent evolution of the spin gap, overall spectra
intensity and dispersions are reproduced reasonably well by the LSW theory
supp . We note that the simulated magnetic field is slightly smaller than the
applied field (Fig. 4c); this is probably due to the quantum effect that is
not considered in the LSW calculations. More sophisticated calculation such as
the renormalized spin wave theory that takes into account the quantum
correction to ordered moments may further refine this analysis. A recent
quantum Monte-Carlo calculations actually suggested a slightly larger exchange
coupling constants than the LSW theory calculations, but the ratio of
$J_{2}/J_{1}$ remains nearly the same, which is not surprising and does not
qualitatively change the magnetic structure and phase diagram based on semi-
classical calculations TMGO_WLi ; TMGO_CLiu . When the external field is
relatively large, quantum effects are largely suppressed and the LSW
calculation becomes more accurate. We can extract the effective $g$-factor of
13.79 from the high field fitting (Fig. 4c), which is close to the reported
value determined by magnetization measurement TMGO_YLi1 .

It has also been suggested theoretically that Phase I will melt in a two-step
manner through two Berezinskii-Kosterlitz-Thouless (BKT) transitions and the
intermediate phase hosts a emergent U(1) symmetry TFIM_Isakov ; TFIM_Damle ;
TFIM_Biswas ; TMGO_CLiu ; TMGO_WLi . Across the BKT transitions, the heat
capacity and the order parameter indicated by the Bragg peaks at K points are
expected to evolve without discontinuity TFIM_Isakov ; KT_Chaikin , consistent
with our data (Fig. 1c, Fig. 2a). Nevertheless, the predicted divergence of
susceptibility is not observed in our magnetization measurements supp . A
recent work reports susceptibility data and numerical simulation showing BKT
behavior NMR_BKT  which, however, are actually collected deep into Phase II of
our phase diagram. A Monte-Carlo calculation also suggested that a new
magnetic peak at M point should appear at high temperatures owing to
proliferated vortex-antivortex pairs TMGO_WLi , which is also not observed in
our neutron scattering measurements supp . We note that the BKT phase is a
compressible quasi-long-range ordered phase with divergent correlation lengths
in 2D that is fragile to perturbations such as inter-layer couplings or weak
disorder.

To summarize, we have performed neutron scattering measurements on the
intrinsic quantum Ising magnet TmMgGaO4 in longitudinal fields. In weak field,
the zero-field three-sublattice order is replaced by a intermediate quasi-
plateau phase, in which both the static moments and spin gap behave in a non-
monotonic manner as a function of external field. In high fields, the system
is driven into the field-induced polarized state. By combining quantum and
semi-classical analysis, we show that the observed magnetic order and the
associated quantum excitations can be well described by TFIM throughout the
phase diagram with the subtle competition among quantum fluctuations,
frustrated Ising interactions, and thermal fluctuations. Our results
demonstrate that TmMgGaO4, as a rare experimental realization of 2D TFIM, is a
remarkable platform to illustrate the interplay between quantum and classical
interactions, and that the associated quantum states can be manipulated
through external fields in a highly controlled manner.

##  I Acknowledgments

This work was supported by the Innovation Program of Shanghai Municipal
Education Commission (grant no. 2017-01-07-00-07-E00018), the National Key R&D
Program of the MOST of China (grant nos. 2016YFA0300203, 2016YFA0300500,
2016YFA0301001, 2018YFE0103200), the National Natural Science Foundation of
China (grant No. 11874119), Shanghai Municipal Science and Technology Major
Project with Grant No.2019SHZDZX04, and the Hong Kong Research Grants Council
(GRF no. 17303819, 17306520). Y.F. and X.T. was supported by the National
Nature Science Foundation of China (No. 11875265), the Scientific Instrument
Developing Project of the Chinese Academy of Sciences (3He based neutron
polarization devices), the Institute of High Energy Physics, and the Chinese
Academy of Science. The datasets for the inelastic neutron scattering
experiment on the time-of-flight LET spectrometer are available from the ISIS
facility, Rutherford Appleton Laboratory data portal
(10.5286/ISIS.E.RB1910127). The datasets for the polarized neutron scattering
experiment on the cold triple-axis ThALES spectrometer are available from the
Institute Laue-Langevin data portal (https://doi.ill.fr/10.5291/ILL-
DATA.4-05-755). The neutron experiment at the Materials and Life Science
Experimental Facility of the J-PARC was performed under a user program
(Proposal No. 2019A0116). All other data that support the plots within this
paper and other findings of this study are available from the corresponding
authors upon reasonable request. Y.Q. and Y.S. contributed equally to this
work.

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Supplementary Material: Field-Tuned Quantum Effects in a Triangular-Lattice
Ising Magnet

This document provides details of the sample synthesis and neutron
experiments, polarized neutron scattering data, magnetic susceptibility
measurements and more inelastic neutron scattering results.

##  II Sample synthesis and Neutron scattering experiments.

Large size TmMgGaO4 single crystals were synthesized using the floating-zone
technique. The X-ray Laue and X-ray diffraction measurements indicate the high
crystallization quality of the sample sTMGO_Yao . The neutron diffraction
measurements were carried out on the cold three axes spectrometer PANDA at the
Heinz Maier-Leibnitz Zentrum, Garching, Germany sPANDA , and the FLEXX cold
triple-axis spectrometer in the BER-II reactor at Helmholtz-Zentrum Berlin,
Germany sFLEXX . For the PANDA experiment, a vertically focused PG(002) was
used as a monochromator and analyzer. We fixed the final neutron energy at
$E_{\mathrm{f}}$ = 4.06 meV, leading to an energy resolution of around 0.1
meV. In order to avoid the contamination from neutrons with higher orders, a
Be filter was placed after the sample. One piece of single crystal with mass
of 3 grams was aligned in the ($H$, $K$, 0) plane. A closed-cycle refrigerator
equipped with a 3He insert was used to reach the base temperature of 0.485 K.
For the FLEXX experiment, we used a PG(002) as both the monochromator and
analyzer, and the final neutron energy was fixed at $E_{\mathrm{f}}$ = 3.5
meV. A velocity selector installed in front of the monochromator was used to
remove the higher-order neutrons. One piece of single crystal (1.7 g) was
aligned in the ($H$, $K$, 0) plane for the measurement. The sample was
attached to a dilution insert and put into a vertical magnet to reach the base
temperature of 40 mK.

The inelastic neutron scattering measurements were performed on the cold
neutron multi-chopper spectrometer LET at the Rutherford Appleton Laboratory,
Didcot, UK, and the AMATERAS cold neutron disk chopper spectrometer at the
Japan Proton Accelerator Research Complex sAMATERAS . For the LET experiment,
the incident energies are chosen to be 6, 3.15 and 1.94 meV with energy
resolutions of 0.185, 0.069 and 0.031 meV, respectively. A dilution insert was
used to reach the base temperature of 0.12 K, which was equipped with a
vertical magnet to provide external fields. The same pieces of single crystals
with a total mass of 17.2 g were used as in Ref. sTMGO_Yao, , which were co-
aligned in the ($H$, $K$, 0) plane. For the AMATERAS experiment, we choose the
incident energy to be 2.6 and 5.9 meV and the corresponding energy resolutions
are 0.045 and 0.155 meV,respectively. A dilution insert with a vertical magnet
is used in this experiment. One piece of single crystal was aligned in the
($H$, $K$, 0) plane which could rotate around the vertical $z$ axis. The data
were analysed using the Horace-Matlab suite sHORACE .

The polarized neutron scattering experiments were carried out on the ThALES
cold triple-axis spectrometer at the Institut Laue-Langevin, Grenoble, France.
In this experiment, we used a focusing Heusler monochromator and analyzer to
produce and analyse the polarized neutrons with the fixed final energy of
$E_{\mathrm{f}}$ = 4.662 meV. Longitudinal polarization analysis was done
using the Helmholtz coils. One piece of single crystal with mass of $\sim$0.3
g was used for the measurement.

##  III Polarized neutron scattering measurements.

For the polarized neutron scattering measurements, we used the conventional
notations to define the polarization directions, $x$, $y$ and $z$, in
reciprocal space. To be more specific, $x$ is defined in such a way that it
keeps parallel to the momentum transfer Q, and $y$ is perpendicular to $x$ but
remains in the scattering plane while $z$ is perpendicular to both $x$ and
$y$, thus perpendicular to the scattering plane (Fig. S1a). During the
measurements, we polarized the incident neutrons along one of the basic
directions, $x$, $y$ or $z$, and selectively measured the scattered neutrons
parallel or anti-parallel to the incident neutrons, which are marked as non-
spin-flip (NSF) and spin-flip (SF) channels, respectively. Correspondingly,
there are six different channels in total, SF x, SF y, SF z, SF x, SF y and SF
z.

FIG. S 1: Polarized neutron scattering measurements of a TmMgGaO4 single
crystal at 1.7 K. (a) Sketch of the reciprocal space in the ($H$, $K$, 0)
plane. The red arrows indicate the polarization directions at a representative
Q. (b) Qscans near the magnetic Bragg peak K1 point at $E$ = 0 meV. (c),(d)
Energy scans at the high symmetry points, M2 and $\Gamma_{1}$. FIG. S 2:
Magnetic susceptibility in magnetic fields along the $z$ axis. (a) Field
dependence of the DC differential susceptibility d$M$/d$H$ above 2 K. (b) Real
part of the ac susceptibility $\chi$’ as a function of external field at
different temperatures. The measurement was performed with ac frequency of
469.2 Hz. No divergence or enhancement related to the BKT phase is observed.
FIG. S 3: Polarized neutron scattering measurements around the M2 point at the
indicated temperatures and $E$ = 0 meV. FIG. S 4: Spin wave dispersion in
different external longitudinal fields measured at LET and 0.12 K. (a)-(c)
Spin excitations along the high symmetry reciprocal directions marked by the
black arrows in Fig. 1b in the indicated fields. (c)-(f) Calculated spin wave
dispersions. The intensities in (d) and (f) are multiplied by 60. FIG. S 5:
Spin wave dispersion in different external fields at 0.06 K. (a)-(d) Spin
excitations along the high symmetry reciprocal directions marked by the black
arrows in Fig. S1a in the indicated fields. (e)-(h) Calculated spin wave
dispersions. The intensities in (d) and (h) are multiplied by 2.

In the neutron scattering measurements, the neutrons are only sensitive to the
magnetic components perpendicular to Q. Moreover, the spin of a neutron can
only be flipped by magnetic components perpendicular to it with a flipping
ratio determined by the instrument, leading to leakage of the signal from SF
channel to NSF channel and vice versa. For elastic measurements, the
scattering process probes the static spins along the $z$ direction ($S^{z}$)
or perpendicular to $z$ ($S^{\perp}$) while the inelastic measurements detect
the spin-spin correlations out of the basic plane ($S_{zz}$) and within the
plane ($S_{+-}$). In the configuration of our experiments, the SF x channel
corresponds to the components that are perpendicular to Q, which are
$S^{z}$/$S_{zz}$ and part of $S^{\perp}$/$S_{+-}$. Meanwhile, the SF y channel
contains the components that are perpendicular to both Q and $y$, which are
$S^{z}$/$S_{zz}$ only, and the SF z channel detects the components in the $xy$
plane but perpendicular to Q, which are part of $S^{\perp}$/$S_{+-}$ but no
$S^{z}$/$S_{zz}$.

We start by the polarization analysis of the elastic signals. In Fig. S1b, we
present the momentum scans across the magnetic Bragg peak K1 in all three SF
channels. With signals absent in SF z channel, the peaks in SF x and SF y
channels are comparable which means that only $S^{z}$ component is observed in
elastic neutron measurements, consistent with our model that the transverse
$S^{\perp}$ component behaves as multipoles and cannot be detected directly.
Moreover, we performed the constant Q scans at the high symmetry points, K1,
M2 and $\Gamma_{1}$, and only magnetic excitations of $S_{zz}$ components are
present in these scans (Fig. S1c, d, Fig. 4a). The peak-like features close to
the zero energy in Fig. S1c, d may come from the background or leakage since
they do not obey the fundamental relationship: SF y + SF z = SF x. For
conventional localized systems, the magnetic excitations run perpendicular to
the static spins. Here, the spin dynamics occur in the longitudinal channels,
indicating quantum effect and multipolar nature of the transverse spin
components.

##  IV Susceptibility measurements.

It has been theoretically proposed that in the TFIM on a triangular lattice,
the low-temperature three-sublattice ground state will melt in a two-step
manner with an intermediate BKT phase sTFIM_Isakov ; sTFIM_Damle ;
sTFIM_Biswas ; sTMGO_WLi ; sChangle . For the TFIM on the triangular lattice,
it has been predicted that the uniform susceptibility will diverge in an
external field along the $c$ direction in the BKT phase sTFIM_Damle ;
sTFIM_Biswas . According to the theoretical calculations, the susceptibility
tends to diverge as $\chi(B)\sim|B|^{-[(4-18\eta)/(4-9\eta)]}$ where
$\eta(T)\in(1/9,2/9)$ (Ref. sTFIM_Damle, ; sTFIM_Biswas, ). To test this
result, we measured the magnetic susceptibility as a function of magnetic
fields and temperatures (Fig. S2). No divergence in susceptibility was
observed in the zero field limit. We note that the compressible BKT phase for
the TFIM, that is characterized by a divergent correlation length and quasi-
long-range order in a temperature window above the true long-range ordering
transition and describes the long-distance and low-energy properties of the
TFIM, is fragile to inter-layer coupling ($-0.3$K for the dipole-dipole
interaction between the neighboring layer sites) and/or weak disorder.

In addition, Li et al. suggested that there would be a magnetic peak at M
points at finite temperature due to the proliferated vortex-antivortex pairs
based on Monte-Carlo calculations sTMGO_WLi . We performed polarized elastic
neutron scattering measurement across M2 points at different temperatures and
no magnetic peaks were found (Fig. S3).

##  V More inelastic neutron scattering data.

The inelastic neutron scattering measurements were performed in different
external longitudinal fields, 0, 0.5, 1.5, 2, 2.5, 3, 5, 8.5 T for experiments
at LET and 0, 1, 1.7, 2.2, 4 T for experiments at AMATERAS. For the
measurements at LET, the 8.5 T data can be taken as an ideal background since
the spin gap is high enough to disentangle the low-energy background and the
high-energy signals. Thus, in Fig. 4b, for the data measured between 0 and 5 T
we show the background subtracted results while the raw 8.5 T data is
presented. By integrating the scattering function along the energy axis, we
can determine the total local moment, $\langle m^{2}\rangle$, which are shown
as All in the inset of Fig. 4b. The elastic contributions are evaluated by
fitting the elastic lines with Gaussian profiles. The difference between the
total moment and elastic channel gives the inelastic spectra. For the
experiments at AMATERAS, no high-field data are available for background
subtraction.

In the main text, we have shown the dispersion under several representative
external fields (Fig. 3), along with the spin wave calculation using the spinw
program sSPINW . Here, we present the rest of the data, including the LET data
under 0.5, 2 and 8.5 T (Fig. S4) and AMATERAS data under 1, 1.7, 2.2 and 4 T
(Fig. S5).

## References

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# On quaternion algebras over some extensions of quadratic number fields

Vincenzo Acciaro  Vincenzo Acciaro, Dipartimento di Economia, Università di
Chieti–Pescara, Viale della Pineta, 4, 65127 Pescara, Italy
<EMAIL_ADDRESS>,  Diana Savin  Diana Savin, Faculty of Mathematics and
Computer Science, Ovidius University, Bd. Mamaia 124, 900527, Constanta,
Romania<EMAIL_ADDRESS>,  Mohammed Taous  Mohammed Taous,
Department of Mathematics, Faculty of Sciences and Technology, Moulay Ismail
University, Errachidia, Morocco<EMAIL_ADDRESS>and  Abdelkader Zekhnini
Abdelkader Zekhnini, Mohammed First University, Pluridisciplinary Faculty of
Nador, Department Mathematics and Informatics, Nador, Morocco<EMAIL_ADDRESS>
###### Abstract.

Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime
integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such
that $K$ is a dihedral extension of ${\mathbb{Q}}$ of degree $\ell$ over $F$
or $K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$
divides the class number of $F$. In this paper, we obtain a complete
characterization of division quaternion algebras $H_{K}(p,q)$ over $K$.

###### Key words and phrases:

Quaternion algebras, quadratic fields, dihedral extension, Hilbert class field

###### 2000 Mathematics Subject Classification:

Primary 11R11; 11R21 ; 11R32, 11R52; 11S15; Secondary 11R37, 11R29, 11A41,
11R04, 11F85

##  1\. Introduction

Let $F$ be a field with $char(F)\neq 2$ and let $a,b\in F\backslash\\{0\\}$.
The generalized quaternion algebra $H_{F}(a,b)$ is the associative algebra
generated over the field $F$ by two elements $i$ and $j$, subject to the
relations $i^{2}=a$, $j^{2}=b$ and $ij=-ji$.

Quaternion algebras turn out to be central simple algebras of dimension $4$
over $F$. A basis for $H_{F}(a,b)$ over $F$ is given by $\\{1,i,j,ij\\}$.

It can be shown that every four dimensional central simple algebra over a
field of characteristic $\neq 2$ is a quaternion algebra.

If $x=x_{1}1+x_{2}i+x_{3}j+x_{4}k\in H_{F}(a,b),$ with $x_{i}\in F$, the
conjugate $\overline{x}$ of $x$ is defined as
$\overline{x}=x_{1}1-x_{2}i-x_{3}j-x_{4}k$, and the norm of $x$ as
$\boldsymbol{n}(x)=x\overline{x}=x_{1}^{2}-ax_{2}^{2}-bx_{3}^{2}+abx_{4}^{2}.$

If the equations $ax=b,\ ya=b$ have unique solutions for all $a,b\in A,\ a\neq
0$, then the algebra $A$ is called a division algebra. If $A$ is a finite-
dimensional algebra, then $A$ is a division algebra if and only if $A$ has no
zero divisors. In the case of generalized quaternion algebras there is a
simple criterion that guarantees them to be division algebras: $H_{F}(a,b)$ is
a division algebra if and only if there is a unique element of zero norm,
namely $x=0$.

Let $L$ be an extension field of $F$, and let $A$ be a central simple algebra
over $F$. We recall that $A$ is said to split over $L$, and $L$ is called a
splitting field for $A$, if $A\otimes_{F}L$ is isomorphic to a full matrix
algebra over $L$.

Several results are known about the splitting behavior of quaternion algebras
and symbol algebras over specific fields [4, 5, 7, 11, 12].

Explicit conditions which guarantee that a quaternion algebra splits over the
field of rationals numbers, or else is a division algebra, were studied in
[3].

In [1] we studied the splitting behavior of some quaternion algebras over
quadratic fields. Then, in [2] we extended the previous results to the
composite of quadratic number fields.

Traditionally, in order to decide whether a quaternion algebra is a division
algebra or it splits, one either looks for the primes which ramify in the
algebra or otherwise one has to appeal to Hasse’s norm theorem which allows to
reduce the problem to computations of local Hilbert symbols [13, 14]. Very
recently, Goldstein [8] tried to combine these two techniques.

In this paper, as we did in our previous papers [1, 2], we adopt the former
approach, i.e we study the ramification of certain integral primes, and we
obtain a nice characterization of quaternion division algebras $H_{K}(p,q)$
solely in terms of quadratic residues, assuming that $p$ and $q$ are positive
primes and $K$ is a dihedral extension of ${\mathbb{Q}}$ of prime degree over
an imaginary quadratic field. The layout of the paper is the following. In
Section 2 we state some preliminary results which we will need later. In
Section 3 we apply these results to study quaternion algebras over dihedral
extensions $K$ of ${\mathbb{Q}}$ of prime degree $l$ over an imaginary
quadratic field.

##  2\.  Preliminary results

In this section we recall some basic results concerning quaternion algebras.
Unless otherwise stated, when we say “prime integer” we mean “positive prime
integer”.

Let $K$ be a number field and let $\mathcal{O}_{K}$ be its ring of integers.
If $v$ is a place of $K$, let us denote by $K_{v}$ the completion of $K$ at
$v$. We recall that a quaternion algebra $H_{K}(a,b)$ is said to ramify at a
place $v$ of $K$ \- or $v$ is said to ramify in $H_{K}(a,b)$ \- if the
quaternion $K_{v}$-algebra $H_{v}=K_{v}\otimes H_{K}(a,b)$ is a division
algebra. This happens exactly when the Hilbert symbol $(a,b)_{v}$ is equal to
$-1$, i.e. when the equation $ax^{2}+by^{2}=1$ has no solutions in $K_{v}$. We
also recall that the reduced discriminant $D_{H_{K}(a,b)}$ of the quaternion
algebra $H_{K}(a,b)$ is defined as the product of those prime ideals of the
ring of integers $\mathcal{O}_{K}$ of $K$ which ramify in $H_{K}(a,b)$. The
following splitting criterion for a quaternion algebras is well known [3,
Corollary 1.10]:

######  Proposition 2.1.

Let $K$ be a number field. Then, the quaternion algebra ${H}_{K}(a,b)$ is
split if and only if its discriminant $D_{H_{K}(a,b)}$ is equal to the ring of
integers $\mathcal{O}_{K}$ of $K$.

If $\mathcal{O}_{K}$ is a principal ideal domain, then we may identify the
ideals of $\mathcal{O}_{K}$ with their generators, up to units. Thus, in a
quaternion algebra $H$ over $\mathbb{Q}$, the element $D_{H}$ turns out to be
an integer, and $H$ is split if and only if $D_{H}=1$.

The next proposition gives us a geometric interpretation of splitting [7,
Proposition 1.3.2]:

######  Proposition 2.2.

Let $K$ be a field. Then, the quaternion algebra ${H}_{K}(a,b)$ is split if
and only if the conic $C(a,b):$ $ax^{2}+by^{2}=z^{2}$ has a rational point in
$K$, i.e. there are $x_{0},y_{0},z_{0}\in K$ such that
$ax_{0}^{2}+by_{0}^{2}=z_{0}^{2}$.

The next proposition due to Hasse relates the norm group of extensions of the
base field to the splitting behavior of a quaternion algebra [7, Proposition
1.1.7]:

######  Proposition 2.3.

Let $F$ be a field. Then, the quaternion algebra ${H}_{F}(a,b)$ is split if
and only if $a$ is the norm of an element of $F(\sqrt{b})$.

For quaternion algebras it is true the following [7, Proposition 1.1.7]:

######  Proposition 2.4.

Let $K$ be a field with char $K\neq 2$ and let $a,b\in K\backslash\\{0\\}$.
Then the quaternion algebra $H_{K}(a,b)$ is either split or a division
algebra.

In particular, this tells us that a quaternion algebra $H_{\mathbb{Q}}(a,b)$
is a division algebra if and only if there is a prime $p$ such that
$p|D_{H_{\mathbb{Q}}(a,b)}$.

It is known [10] that if a prime integer $p$ divides $D_{H(a,b)}$ then it must
divide $2ab$, hence we may restrict our attention to these primes. In other
words, in order to obtain a sufficient condition for a quaternion algebra
$H_{\mathbb{Q}(\sqrt{d})}(p,q)$ to be a division algebra, it is important to
study the ramification of the primes $2,p,q$ in the algebra
$H_{\mathbb{Q}}(p,q)$. The following lemma from the classical book by Alsina
[3, Lemma 1.21] gives us a hint:

######  Lemma 2.5.

Let $p$ and $q$ be two primes, and let $H_{\mathbb{Q}}(p,q)$ be a quaternion
algebra of discriminant $D_{H}$.

  1. i.

if $p\equiv q\equiv 3\pmod{4}$ and $(\frac{q}{p})\neq 1$, then $D_{H}=2p$;

  2. ii.

if $q=2$ and $p\equiv 3\pmod{8}$, then $D_{H}=pq=2p$;

  3. iii.

if $p$ or $q\equiv 1\pmod{4}$ , with $p\neq q$ and $(\frac{p}{q})=-1$, then
$D_{H}=pq$.

We recall that a small ramified $\mathbb{Q}$-algebra is a rational quaternion
algebra having the discriminant equal to the product of two distinct prime
numbers. The following necessary and sufficient explicit condition for a small
ramified $\mathbb{Q}$-algebra $H_{\mathbb{Q}}(p,q)$ to be a division algebra
over a quadratic field $\mathbb{Q}(\sqrt{d})$ was proved in [1]:

######  Proposition 2.6.

Let $p$ and $q$ be two distinct odd primes, with $p$ or $q\equiv 1\pmod{4}$
and $(\frac{p}{q})=-1.$ Let $K=\mathbb{Q}(\sqrt{d})$ and let $\Delta_{K}$ be
the discriminant of $K.$ Then the quaternion algebra
$H_{\mathbb{Q}(\sqrt{d})}(p,q)$ is a division algebra if and only if
$(\frac{\Delta_{K}}{p})=1$ or $(\frac{\Delta_{K}}{q})=1$.

When $q=2$ and $p$ is a prime such that $p\equiv 3$ (mod 8), then, according
to Lemma 2.5 the discriminant $D_{H_{\mathbb{Q}(p,q)}}$ is equal to $2p$, so
$H_{\mathbb{Q}}(p,q)$ is a division algebra. The next proposition, which was
proved in [1], shows what happens when we extend the field of scalars from
${\mathbb{Q}}$ to ${\mathbb{Q}}(\sqrt{d})$:

######  Proposition 2.7.

Let $p$ be an odd prime, with $p\equiv 3\pmod{8}$. Let
$K=\mathbb{Q}(\sqrt{d})$ and let $\Delta_{K}$ be the discriminant of $K.$ Then
$H_{\mathbb{Q}(\sqrt{d})}(p,2)$ is a division algebra if and only if
$(\frac{\Delta_{K}}{p})=1$ or $d$ $\equiv 1\pmod{8}$.

When $p$ and $q$ are primes both congruent to $3$ modulo $4$ and
$(\frac{q}{p})\neq 1$ then, according to Lemma 2.5(i) the discriminant
$D_{H_{\mathbb{Q}}(p,q)}$ is equal to $2p$, so $H_{\mathbb{Q}}(p,q)$ is a
division algebra. The next proposition which also was proved in [1], tells us
when the quaternion algebra $H_{\mathbb{Q}(\sqrt{d})}(p,q)$ is still a
division algebra:

######  Proposition 2.8.

Let $p$ and $q$ be two odd prime integers, with $p\equiv q\equiv 3\pmod{4}$
and $(\frac{q}{p})\neq 1$. Let $K=\mathbb{Q}(\sqrt{d})$ and let $\Delta_{K}$
be the discriminant of $K.$ Then the quaternion algebra
$H_{\mathbb{Q}(\sqrt{d})}(p,q)$ is a division algebra if and only if
$(\frac{\Delta_{K}}{p})=1$ or $d$ $\equiv 1\pmod{8}$.

##  3\.  Main results

Let’s ask ourselves now what happens when we consider a quaternion algebra
over a Galois extension $K$ of $\mathbb{Q}$, with nonabelian Galois group of
order $2l$, where $l$ is an odd prime integer. For this purpose, we recall the
following result, which can be found as an exercise in [11, p. 77]:

######  Remark 3.1.

Let $K/F$ be a finite extension of fields of odd degree, and let $a,b\in
F\backslash\\{0\\}$. Then the quaternion algebra $H_{K}(a,b)$ splits if and
only if $H_{F}(a,b)$ splits.

Let us first consider the case $Gal(K/\mathbb{Q})\cong S_{3}$, i.e. the
dihedral group $D_{3}$. The following three propositions will deal with this
case:

######  Proposition 3.2.

Let $\epsilon$ be a primitive third root of unity, and put
$F=\mathbb{Q}\left(\epsilon\right)$. Let $\alpha\in K\backslash\\{0,1\\}$ be a
cubicfree integer, put $K=F\left(\sqrt[3]{\alpha}\right)$ and let $p$, $q$ be
two distinct odd prime integers such that $(\frac{p}{q})=-1$ and $p$ or
$q\equiv 1\pmod{4}$. Then the quaternion algebra $H_{K}(p,q)$ is a division
algebra if and only if $(\frac{-3}{p})=1$ or $(\frac{-3}{q})=1.$

###### Proof.

Clearly $F=\mathbb{Q}\left(\epsilon\right)=\mathbb{Q}\left(i\sqrt{3}\right)$
is an imaginary quadratic number field and $\left[K:F\right]=3$. Moreover,
$K/\mathbb{Q}$ is Galois and Gal$(K/\mathbb{Q})\cong S_{3}$. According to
Remark 3.1 and Proposition 2.4, $H_{K}(p,q)$ is a division algebra if and only
if $H_{F}(p,q)$ is a division algebra. By Proposition 2.6, this can happen if
and only if $(\frac{-3}{p})=1$ or $(\frac{-3}{q})=1.$ ∎

######  Proposition 3.3.

Let $\epsilon$ be a primitive third root of unity, and put
$F=\mathbb{Q}\left(\epsilon\right)$. Let also $\alpha\in K\backslash\\{0,1\\}$
be a cubicfree integer, put $K=F\left(\sqrt[3]{\alpha}\right)$ and let $p$ be
an odd prime integer such that $p\equiv 3\pmod{8}$. Then the quaternion
algebra $H_{K}(p,2)$ is a division algebra if and only if $(\frac{-3}{p})=1.$

###### Proof.

The proof is obtained from the proof of Proposition 3.2 by replacing
Proposition 2.6 with Proposition 2.7. ∎

######  Proposition 3.4.

Let $\epsilon$ be a primitive third root of unity, and put
$F=\mathbb{Q}\left(\epsilon\right)$. Let $\alpha\in K\backslash\\{0,1\\}$ be a
cubicfree integer, put $K=F\left(\sqrt[3]{\alpha}\right)$ and let $p$ and $q$
be distinct odd prime integers with $(\frac{q}{p})\neq 1$ and
$p$$\equiv$$q$$\equiv 3$ (mod $4$). Then the quaternion algebra $H_{K}(p,q)$
is a division algebra if and only if $(\frac{-3}{p})=1.$

###### Proof.

The proof is obtained from the proof of Proposition 3.2 after replacing
Proposition 2.6 with Proposition 2.8. ∎

In what follows, let $\ell$ be an odd positive prime integer and $F$ a
quadratic number field. Let $K$ be an extension of $F$ defined as follows: $K$
is a dihedral extension of ${\mathbb{Q}}$ of prime degree $\ell$ over $F$, or
$K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$ divides
the class number of $F$ (so $[K:F]=\ell^{n}$ which is odd, with
$n\in\mathds{N}^{*}$). The existence of such a $K$ containing $F$ is
guaranteed, in the second case, by class field theory, but the first one is a
tipical problem in inverse Galois theory. The following result from [9, p.
352-353] guarantees us that such a $K$ indeed exists (for the first case):

######  Theorem 3.5.

For any prime $\ell$ and any quadratic field $F=\mathbb{Q}(\sqrt{d})$ there
exist infinitely many dihedral fields $K$ of degree $2\ell$ containing $F$.

If the quaternion algebra $H_{F}(p,q)$ is a division algebra, we would like to
know when $H_{K}(p,q)$ is still a division algebra. The following three
propositions will allow us to achieve this task:

######  Proposition 3.6.

Let $F$ be a quadratic field and $\Delta_{F}$ its discriminant. Let $K$ be an
extension of $F$ defined as above. Let $p$ and $q$ be distinct odd prime
integers, with $(\frac{p}{q})=-1$, and $p$ or $q\equiv 1\pmod{4}$. Then the
quaternion algebra $H_{K}(p,q)$ is a division algebra if and only if
$\left(\frac{\Delta_{F}}{p}\right)=1$ or
$\left(\frac{\Delta_{F}}{q}\right)=1.$

###### Proof.

Note first that the degree $\left[K:F\right]$ is odd, since it is equal to
$\ell$ or $\ell^{n}$ with $n\in\mathds{N}^{*}$. According to Remark 3.1 and
Proposition 2.4, $H_{K}(p,q)$ is a division algebra if and only if
$H_{F}(p,q)$ is a division algebra. By Proposition 2.6, this can happen if and
only if $\left(\frac{\Delta_{F}}{p}\right)=1$ or
$\left(\frac{\Delta_{F}}{q}\right)=1.$ ∎

######  Proposition 3.7.

Let $d\neq 1$ be a squarefree integer and let $F=\mathbb{Q}(\sqrt{d})$ be a
quadratic number field and $\Delta_{F}$ its discriminant. Let $K$ be an
extension of $F$ defined as above. Let $p\equiv 3\pmod{8}$ be an odd prime
integer. Then the quaternion algebra $H_{K}(p,q)$ is a division algebra if and
only if $\left(\frac{\Delta_{F}}{p}\right)=1$ or $d\equiv 1\pmod{8}$.

###### Proof.

The proof is obtained from the proof of Proposition 3.6, after replacing
Proposition 2.6 with Proposition 2.7. ∎

######  Proposition 3.8.

Let $d\neq 1$ be a squarefree integer and let $F=\mathbb{Q}(\sqrt{d})$ be a
quadratic number field and $\Delta_{F}$ its discriminant. Let $K$ be an
extension of $F$ defined as above. Let $p$ and $q$ be distinct odd prime
integers with $(\frac{q}{p})\neq 1$ and $p\equiv q\equiv 3\pmod{4}$. Then the
quaternion algebra $H_{K}(p,q)$ is a division algebra if and only if
$\left(\frac{\Delta_{F}}{p}\right)=1$ or $d\equiv 1\pmod{8}$.

###### Proof.

The proof is obtained from the proof of Proposition 3.6 after replacing
Proposition 2.6 with Proposition 2.8. ∎

We conclude our discussion with the main theorem of the paper:

######  Theorem 3.9.

Let $d\neq 1$ be a squarefree integer and $\ell$ an odd positive prime
integer. Let $F=\mathbb{Q}(\sqrt{d})$ be a quadratic number field and
$\Delta_{F}$ its discriminant. Let $K$ be a dihedral extension of
${\mathbb{Q}}$ of prime degree $\ell$ over $F$ or $K$ is an abelian
$\ell$-extension unramified over $F$ whenever $\ell$ divides the class number
of $F$. Let $p$ and $q$ be two distinct odd prime integers. Then the
quaternion algebra $H_{K}(p,q)$ is a division algebra if and only if one of
the following conditions is verified:

  1. 1.

$p$ or $q\equiv 1\pmod{4}$, $(\frac{p}{q})=-1$, and
$\left(\frac{\Delta_{F}}{p}\right)=1$ or
$\left(\frac{\Delta_{F}}{q}\right)=1$;

  2. 2.

$p\equiv 3\pmod{8}$, and $\left(\frac{\Delta_{F}}{p}\right)=1$ or $d\equiv
1\pmod{8}$;

  3. 3.

$p\equiv q\equiv 3\pmod{4}$, $(\frac{q}{p})\neq 1$, and
$\left(\frac{\Delta_{F}}{p}\right)=1$ or $d\equiv 1\pmod{8}$.

## References

  * [1] V. Acciaro, D. Savin, M. Taous and A. Zekhnini, On quaternion algebras over quadratic number fields, preprint, 2019\.
  * [2] V. Acciaro, D. Savin, M. Taous and A. Zekhnini, On quaternion algebras over the composite of quadratic number fields, preprint, 2019\.
  * [3] M. Alsina and P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves, CRM Monograph Series, 22, American Mathematical Society, 2004.
  * [4] A. Chapman, D. J.Grynkiewiczb, E. Matzri, L. H.Rowen, U. Vishne, Kummer spaces in symbol algebras of prime degree, J. of Pure and Applied Algebra, vol. 220, issue 10 (2016): 3363-3371.
  * [5] A. Chapman, Symbol length of p-algebras of prime exponent, J. of Algebra and its Applications, vol. 16, No. 7 (2017), 1750136 (9 p.).
  * [6] T. Chinburg and E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. (2), 60(1):33–44, 1999.
  * [7] P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, 2006.
  * [8] D. Goldstein and M. Schacher, Norms in central simple algebras, Pacific Journal of Mathematics, Vol. 292, No. 2, 2018, pp. 373-388.
  * [9] C.U. Jensen, N. Yui, Polynomials with $D_{p}$ as Galois Group, Journal of Number Theory 15, (1982) pp. 347-375.
  * [10] D.R. Kohel, Quaternion algebras, available online at http://www.i2m.univ-amu.fr/perso/david.kohel/alg/doc/AlgQuat.pdf.
  * [11] T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, 2004.
  * [12] L. Rowen, D. J. Saltman, Tensor products of division algebras and fields, Journal of Algebra, 394 (2013), pp. 296-309.
  * [13] J. P. Serre, Local fields, Graduate Texts in Mathematics 67, Springer, New York, 1979.
  * [14] S.V. Vostokov, Explicit formulas for the Hilbert symbol, Geometry and Topology Monographs, Vol.3: Invitation to higher local fields, Part I, section 8, pp 81-89.


# Representations of almost regular $m$-gonal forms $I$

Dayoon Park  Department of Mathematics, The University of Hong Kong, Hong Kong
<EMAIL_ADDRESS>
###### Abstract.

It is known that any $m$-gonal form of $\operatorname{rank}n\geq 5$ is almost
regular. In this article, we study the sufficiently large integers which are
represented by (almost regular) $m$-gonal forms of $\operatorname{rank}n\geq
6$.

##  1\. Introduction

The $m$-gonal number is defined as a total number of dots to constitute a
regular $m$-gon. We especially call the total number of dots to constitute a
regular $m$-gon with $x$ dots for each side

(1.1) $P_{m}(x)=\frac{m-2}{2}(x^{2}-x)+x$

as $x$-th $m$-gonal number. The $m$-gonal number which was designed in 2nd
century BC has been an interesting target in number theory for a long time. A
Fermat’s conjecture which states that every positive integer written as a sum
of at most $m$ $m$-gonal numbers was resolved by Lagrange, Gauss, and Cauchy
for $m=4$, $m=3$, and for all $m\geq 3$, respectively.

By definition of $m$-gonal number, only positive integer $x$ would be admitted
in (1.1). But we may naturally generalize the $m$-gonal number by admitting
every rational integer $x$ in (1.1). Very recently, as a general version of
Fermat’s conjecture, the author completed the optimal $\ell_{m}$ for which
every positive integer is written as a sum of $\ell_{m}$ generalized $m$-gonal
numbers for all $m\geq 3$ in [12]. We call a weighted sum of $m$-gonal numbers

(1.2) $F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$

as $m$-gonal form for $(a_{1},\cdots,a_{n})\in{\mathbb{N}}^{n}$. For a
positive integer $N\in{\mathbb{N}}$, if the diophantine equation

$F_{m}(\mathbf{x})=N$

has an integer solution $\mathbf{x}\in{\mathbb{Z}}^{n}$, then we say that
$F_{m}(\mathbf{x})$ represents $N$. If an $m$-gonal form $F_{m}(\mathbf{x})$
represents every positive integer, then we say that $F_{m}(\mathbf{x})$ is
universal. When

$F_{m}(\mathbf{x})\equiv N\pmod{r}$

has an integer solution $\mathbf{x}\in{\mathbb{Z}}^{n}$ for all
$r\in{\mathbb{Z}}$ (which is equivalent with that $F_{m}(\mathbf{x})=N$ has a
$p$-adic integer solution $\mathbf{x}\in{\mathbb{Z}}_{p}^{n}$ for every prime
$p$), we say that $F_{m}(\mathbf{x})$ locally represents $N$. If an $m$-gonal
form $F_{m}(\mathbf{x})$ represents all positive integers which are locally
represented, then we call $F_{m}(\mathbf{x})$ is regular. If an $m$-gonal form
$F_{m}(\mathbf{x})$ represents all positive integers which are locally
represented but finitely many, then we call $F_{m}(\mathbf{x})$ is almost
regular. The author and Kim determined every type of regular $m$-gonal form of
$\operatorname{rank}\geq 4$ for all $m\geq 14$ with $m\not\equiv 0\pmod{4}$
and $m\geq 28$ with $m\equiv 0\pmod{4}$ in [4]. Chan and Oh showed that every
quadratic polynomial of $\operatorname{rank}\geq 5$ is almost regular in [1].
So any $m$-gonal form $F_{m}(\mathbf{x})$ of $\operatorname{rank}\geq 5$ would
represent every sufficiently large integer which is locally represented by
$F_{m}(\mathbf{x})$. On the other words, there would be a constant
$N_{m;(a_{1},\cdots,a_{n})}>0$ for which
$F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ with $n\geq 5$
represents $N$ provided that

$\begin{cases}N\geq N_{m;(a_{1},\cdots,a_{n})}\\\ N\text{ is locally
represented by $F_{m}(\mathbf{x})$}.\end{cases}$

On the other hand, there are some results on the sufficiently large integers
which are represented by $a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ for some
fixed $(a_{1},\cdots,a_{n})\in{\mathbb{N}}^{n}$ and for all $m\geq 3$. In
1830, Legendre claimed that for $m\geq 5$, every positive integer $N\geq
28(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+P_{m}(x_{3})+P_{m}(x_{4})+\delta_{m}(N)$

for some $(x_{1},x_{2},x_{3},x_{4})\in{\mathbb{N}}^{4}$, $\delta_{m}(N)=0$ if
$m\not\equiv 0\pmod{2}$, and $\delta_{m}(N)\in\\{0,1\\}$ if $m\equiv
0\pmod{2}$. Nathanson gave a simple proof of the above Legendre’s claim and
Cauchy’s polygonal number theorem in [9] and [10]. In [8], Meng and Sun showed
that when $m-2\equiv 0\pmod{4}$, every sufficiently large integer $N\geq
28(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+P_{m}(x_{3})+P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{N}}^{4}$, when $m\not\equiv
0\pmod{4}$, every positive integer $N\geq 1628(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+2P_{m}(x_{3})+2P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{N}}^{4}$, every positive integer
$N\geq 924(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+P_{m}(x_{3})+3P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{N}}^{4}$, every positive integer
$N\geq 1056(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+2P_{m}(x_{3})+4P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{N}}^{4}$. And Kim [5] showed that
when $m\not\equiv 0\pmod{4}$, every sufficiently large integer
$N\geq\frac{1}{8}(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+P_{m}(x_{3})+P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{Z}}^{4}$, every sufficiently large
integer $N\geq\frac{1}{10}(m-2)^{3}$ is written as

$P_{m}(x_{1})+P_{m}(x_{2})+P_{m}(x_{3})+2P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{Z}}^{4}$, when $m\not\equiv
0\pmod{4}$, every sufficiently large integer $N\geq\frac{1}{3}(m-2)^{3}$ is
written as

$P_{m}(x_{1})+P_{m}(x_{2})+2P_{m}(x_{3})+2P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{Z}}^{4}$, and every sufficiently
large integer $N\geq\frac{7}{8}(m-2)^{3}$ is written as

$P_{m}(x_{1})+2P_{m}(x_{2})+2P_{m}(x_{3})+2P_{m}(x_{4})$

for some $(x_{1},\cdots,x_{4})\in{\mathbb{Z}}^{4}$.

In this article, we more generally consider the sufficiently large integers
which are represented by an $m$-gonal form
$a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ with fixed
$(a_{1},\cdots,a_{n})\in{\mathbb{N}}^{n}$. The following theorem is the main
theorem in this article.

######  Theorem 1.1.

Let $F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ be an
$m$-gonal form of rank $n\geq 6$. Then $F_{m}(\mathbf{x})$ represents any
integer $N$ provided that

(1.3) $\begin{cases}N\geq N(a_{1},\cdots,a_{n})\cdot(m-2)^{3}\\\ N\text{ is
locally represented by }F_{m}(\mathbf{x})\end{cases}$

where $N(a_{1},\cdots,a_{n})$ is a constant which is dependent only on
$a_{1},\cdots,a_{n}$. In this sense, the cubic with respect to $m$ is optimal
in (1.3).

In order to prove Theorem 1.1, we use the arithmetic theory of quadratic form.

In [2], Kane and Liu showed that the asymptotically increasing $\gamma_{m}$
($\gamma_{m}$ is the smallest positive integer for which an $m$-gonal form is
universal if the $m$-gonal form represents every positive integer up to
$\gamma_{m}$) is bounded by $C_{\epsilon}\cdot m^{7+\epsilon}$ for any
$\epsilon>0$ and an absolute constant $C_{\epsilon}>0$ which is dependent on
$\epsilon$ by observing the sufficiently large integers which are represented
by the $m$-gonal forms of escalator tree of $m$-gonal forms of fixed depth
$6$. Theorem 1.1 may directly improve the upper bound for $\gamma_{m}$ as
$C\cdot(m-2)^{3}$ for some absolute constant $C$ since every $m$-gonal form of
escalator tree of $m$-gonal forms of depth $6$ is locally universal. Although,
the exact growth of $\gamma_{m}$ on $m$ was already revealed as linear on $m$
by the author and Kim [3].

##  2\. preliminaries

Before we move on, we set our languages and notations which are used
throughout this article. Basically, our languages and notations follow
conventional that in the arithmetic theory of quadratic form. A quadratic form
over ${\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$) is a ${\mathbb{Z}}$ (or
${\mathbb{Z}}_{p}$)-module $L$ equipped with a quadratic form
$Q:L\rightarrow{\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$) and the corresponding
symmetric bilinear map $B(x,y):=\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$. For a
quadratic form $L={\mathbb{Z}}v_{1}+\cdots+{\mathbb{Z}}v_{n}$ of
$\operatorname{rank}n$, we usually identify $L$ with its Gram-matrix
$\begin{pmatrix}B(v_{i},v_{j})\end{pmatrix}_{n\times n}$. When $L$ is a binary
(i.e., $n=2$) quadratic form over ${\mathbb{Z}}$, we adopt the notation
$[Q(v_{1}),B(v_{1},v_{2}),Q(v_{2})]$ instead of
$\begin{pmatrix}B(v_{i},v_{j})\end{pmatrix}_{2\times 2}$. We write an
$m$-gonal form

(2.1) $a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$

simply as $\left<a_{1},\cdots,a_{n}\right>_{m}$. When $m=4$, we conventionally
denote the square form (i.e., diagonal quadratic form) of (2.1) by
$\left<a_{1},\cdots,a_{n}\right>$ instead of
$\left<a_{1},\cdots,a_{n}\right>_{4}$.

Note that an integer $A(m-2)+B$ is represented by an $m$-gonal form

$\sum_{i=1}^{n}a_{i}P_{m}(x_{i})=\frac{m-2}{2}\\{(a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2})-(a_{1}x_{1}+\cdots+a_{n}x_{n})\\}+(a_{1}x_{1}+\cdots+a_{n}x_{n})$

over ${\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$) if and only if the diophantine
system

(2.2) $\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable for some $k\in{\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$) over
${\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$). The system (2.2) is solvable over
${\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$) if and only if

(2.3)
$\left(B+k(m-2)-\left(\sum\limits_{i=2}^{n}a_{i}x_{i}\right)\right)^{2}+\sum\limits_{i=2}^{n}a_{1}a_{i}x_{i}^{2}=2Aa_{1}+Ba_{1}+k(m-4)a_{1}$

is solvable over ${\mathbb{Z}}$ (or ${\mathbb{Z}}_{p}$). The left hand side of
the equation (2.3) may be written as

$Q_{a_{1};\mathbf{a}}((x_{2},\cdots,x_{n})-(B+k(m-2))(r_{2},\cdots,r_{n}))+(B+k(m-2))^{2}\cdot\left(1-\sum\limits_{i=2}^{n}a_{i}r_{i}\right)$

where
$Q_{a_{1};\mathbf{a}}(x_{2},\cdots,x_{n}):=\sum_{i=2}^{n}(a_{1}a_{i}+a_{i}^{2})x_{i}^{2}+\sum_{2\leq
i<j\leq n}2a_{i}a_{j}x_{i}x_{j}$ is a quadratic form and
$r_{1},\cdots,r_{n}\in{\mathbb{Q}}$ are the solution for

$\begin{cases}(a_{1}a_{2}+a_{2}^{2})r_{2}+a_{2}a_{3}r_{3}+\cdots+a_{2}a_{n}r_{n}=a_{2}\\\
a_{2}a_{3}r_{2}+(a_{1}a_{3}+a_{3}^{2})r_{3}+\cdots+a_{3}a_{n}r_{n}=a_{3}\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\vdots\\\
a_{2}a_{n}r_{2}+a_{3}a_{n}r_{3}+\cdots+(a_{1}a_{n}+a_{n}^{2})r_{n}=a_{n}.\end{cases}$

Note that since

(2.4)
$Q_{a_{1};\mathbf{a}}=\begin{pmatrix}a_{1}a_{2}+a_{2}^{2}&a_{2}a_{3}&\cdots&a_{2}a_{n}\\\
a_{2}a_{3}&a_{1}a_{3}+a_{3}^{2}&\cdots&a_{3}a_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
a_{2}a_{n}&a_{3}a_{n}&\cdots&a_{1}a_{n}+a_{n}^{2}\end{pmatrix}$

is a sub-quadratic form of rank $n-1$ of the positive definite diagonal
quadratic form $\sum_{i=1}^{n}a_{1}a_{i}x_{i}^{2}$ satisfying
$a_{1}x_{1}+\cdots+a_{n}x_{n}=0$, we have that the determinant of
$Q_{a_{1};\mathbf{a}}$ is positive. We denote the non-zero determinant of
$Q_{a_{1};\mathbf{a}}$ as $d_{a_{1};\mathbf{a}}$.

By Theorem 4.9 in [1], since every quadratic polynomial of rank greater than
or equal to $5$ is almost regular, when $n\geq 6$, we may get an absolute
constant

(2.5) $C(a_{1},\cdots,a_{n})$

which is dependent on $a_{1},\cdots,a_{n},r_{1},\cdots,r_{n}$ (so it is
essentially dependent only on $a_{1},\cdots,a_{n}$) for which

(2.6)
$Q_{a_{1};\mathbf{a}}((x_{2},\cdots,x_{n})-(B+k(m-2))(r_{2},\cdots,r_{n}))=c$

has an integer solution $(x_{2},\cdots,x_{n})\in{\mathbb{Z}}^{n-1}$ if the
equation (2.6) is locally solvable and $c\geq C(a_{1},\cdots,a_{n})$.

And then we may observe that an integer $A(m-2)+B$ with $0\leq B\leq m-3$
satisfying that

the system (2.2) with $k\in{\mathbb{Z}}$ is locally solvable

is represented by the $m$-gonal form $\left<a_{1},\cdots,a_{n}\right>_{m}$ if

$A\geq\frac{1}{2a_{1}}\left(C(a_{1},\cdots,a_{n})+(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)\right)-B-k(m-2).$

Throughout this article, we show that for any integer $A(m-2)+B$ with $0\leq
B\leq m-3$ which is locally represented by
$\left<a_{1},\cdots,a_{n}\right>_{m}$ with $n\geq 6$, we may choose an integer
$k$ in a bounded scope $[0,K(a_{1},\cdots,a_{n})-1]$ for which the system
(2.2) is locally solvable where $K(a_{1},\cdots,a_{n})$ is an absolute
constant which is dependent only on $a_{1},\cdots,a_{n}$. Which may induce the
first argument in Theorem 1.1.

In this article, to avoid the appearance of superfluous duplications, we
always assume that an $m$-gonal form $\left<a_{1},\cdots,a_{n}\right>_{m}$ is
primitive, i.e., $\gcd(a_{1},\cdots,a_{n})=1$. Any unexplained terminology and
notation can be found in [7].

##  3\. Local representation

######  Proposition 3.1.

Let $F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ be a
primitive $m$-gonal form.

  * (1)

When $p$ is an odd prime with $p|m-2$, $F_{m}(\mathbf{x})$ is universal over
${\mathbb{Z}}_{p}$.

  * (2)

When $m\not\equiv 0\pmod{4}$, $F_{m}(\mathbf{x})$ is universal over
${\mathbb{Z}}_{2}$.

  * (3)

When $p$ is an odd prime with $(p,m-2)=1$, an integer $N$ is represented by
$F_{m}(\mathbf{x})$ over ${\mathbb{Z}}_{p}$ if and only if the integer
$8(m-2)N+(a_{1}+\cdots+a_{n})(m-4)^{2}$ is represented by the diagonal
quadratic form $\left<a_{1},\cdots,a_{n}\right>$ over ${\mathbb{Z}}_{p}$.

  * (4)

When $m\equiv 0\pmod{4}$, an integer $N$ is represented by $F_{m}(\mathbf{x})$
over ${\mathbb{Z}}_{2}$ if and only if the integer
$2(m-2)N+(a_{1}+\cdots+a_{n})\left(\frac{m-4}{2}\right)^{2}$ is represented by
the diagonal quadratic form $\left<a_{1},\cdots,a_{n}\right>$ over
${\mathbb{Z}}_{2}$.

###### Proof.

(1) One may easily show that for any $N\in{\mathbb{Z}}_{p}$, $P_{m}(x)=N$ is
solovable over ${\mathbb{Z}}_{p}$ by using Hensel’s Lemma (13:9 in [7]). Which
yields the claim from the assumption that $F_{m}(\mathbf{x})$ is primitive.
(2) For any $N\in{\mathbb{Z}}_{2}$, the equation $P_{m}(x)=N$ has a $2$-adic
integer solution
$\frac{m-4-\sqrt{(m-4)^{2}-8N(m-2)}}{2(m-2)}\in{\mathbb{Z}}_{2}$. Which yields
the claim from the assumption that $F_{m}(\mathbf{x})$ is primitive.
(3) Note that

$N=\sum_{i=1}^{n}a_{i}P_{m}(x_{i})$

if and only if

$8(m-2)N+(a_{1}+\cdots+a_{n})(m-4)^{2}=\sum_{i=1}^{n}a_{i}(2(m-2)x_{i}-(m-4))^{2}$

where $2(m-2)\in{\mathbb{Z}}_{p}^{\times}$. It completes the proof.
(4) Note that

$N=\sum_{i=1}^{n}a_{i}P_{m}(x_{i})$

if and only if

$2(m-2)N+(a_{1}+\cdots+a_{n})\left(\frac{m-4}{2}\right)^{2}=\sum_{i=1}^{n}a_{i}\left((m-2)x_{i}-\frac{m-4}{2}\right)^{2}$

where $m-2\in 2{\mathbb{Z}}_{2}^{\times}$ and $\frac{m-4}{2}\in
2{\mathbb{Z}}_{2}$. It completes the proof. ∎

######  Proposition 3.2.

For an odd prime $p$, if there are more than four units of ${\mathbb{Z}}_{p}$
in $\\{a_{1},\cdots,a_{n}\\}$ with $n\geq 6$ by admitting a recursion, then

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{p}$ for any $A,B,k\in{\mathbb{Z}}$.

###### Proof.

First assume that $a_{1}+\cdots+a_{n}\not\equiv 0\pmod{p}$. Then by using
82:15 and 91:9 in [7], we may obtain that for any $A,B,k$, the binary
quadratic ${\mathbb{Z}}_{p}$-lattice
$[a_{1}+\cdots+a_{n},B+k(m-2),2A+B+k(m-4)]\otimes{\mathbb{Z}}_{p}$ is
represented as

$\begin{pmatrix}1&1&\cdots&1\\\
x_{1}&x_{2}&\cdots&x_{n}\end{pmatrix}\begin{pmatrix}a_{1}&0&\cdots&0\\\
0&a_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&a_{n}\end{pmatrix}\begin{pmatrix}1&x_{1}\\\ 1&x_{2}\\\
\vdots&\vdots\\\ 1&x_{n}\end{pmatrix}$

for some $(x_{1},\cdots,x_{n})\in{\mathbb{Z}}_{p}^{n}$ since every quaternary
unimodular ${\mathbb{Z}}_{p}$-lattice is universal. Which yields the claim.

Now assume that $a_{1}+\cdots+a_{n}\equiv 0\pmod{p}$. Without loss of
generality, let $0=\text{ord}_{p}(a_{1})$. Then through same arguments with
the above, one may induce that

$\begin{cases}a_{2}x_{2}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{2}x_{2}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{p}$ for any $A,B,k$ since
$a_{2}+\cdots+a_{n}\not\equiv 0\pmod{p}$ and every ternary unimodular
${\mathbb{Z}}_{p}$-lattice is universal. This completes the proof. ∎

######  Proposition 3.3.

Let $p$ be an odd prime with $(p,m-2)=1$. If an integer $A(m-2)+B$ is
represented by $\left<a_{1},\cdots,a_{n}\right>_{m}$ of
$\operatorname{rank}n\geq 6$ over ${\mathbb{Z}}_{p}$, then there is a residue
$k_{m;\mathbf{a},A,B}(p)$ in
${\mathbb{Z}}/p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}$ for which

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{p}$ for any $k\equiv
k_{m;\mathbf{a},A,B}(p)\pmod{p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}}$.

###### Proof.

It would be enough to show that if

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{p}$, then

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k^{\prime}(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k^{\prime}(m-2)\end{cases}$

is also solvable over ${\mathbb{Z}}_{p}$ for any $k^{\prime}\in
k+p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$. In other ways,
we show that if the equation

$Q_{a_{1};\mathbf{a}}(\mathbf{x}-(B+k(m-2))\mathbf{r})=a_{1}(2A+B+k(m-4))-(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)$

is solvable over ${\mathbb{Z}}_{p}$, then the equation

$Q_{a_{1};\mathbf{a}}(\mathbf{x}-(B+k^{\prime}(m-2))\mathbf{r})=a_{1}(2A+B+k^{\prime}(m-4))-(B+k^{\prime}(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)$

is also solvable over ${\mathbb{Z}}_{p}$ for any $k^{\prime}\in
k+p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$. In virtue of the
Jordan decomposition (see 91 in [7]), we may take $T\in
GL_{n-1}({\mathbb{Z}}_{p})$ for which

$T^{t}\cdot\begin{pmatrix}a_{1}a_{2}+a_{2}^{2}&a_{2}a_{3}&\cdots&a_{2}a_{n}\\\
a_{2}a_{3}&a_{1}a_{3}+a_{3}^{2}&\cdots&a_{3}a_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
a_{2}a_{n}&a_{3}a_{n}&\cdots&a_{1}a_{n}+a_{n}^{2}\end{pmatrix}\cdot
T=\begin{pmatrix}a_{2}^{\prime}&0&\cdots&0\\\ 0&a_{3}^{\prime}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&a_{n}^{\prime}\end{pmatrix}.$

Which allows us to consider the simple diagonal form

$\sum_{i=2}^{n}a_{i}^{\prime}(x_{i}-(B+k(m-2))r_{i}^{\prime})^{2}$

instead of $Q_{a_{1};\mathbf{a}}(\mathbf{x}-(B+k(m-2))\mathbf{r})$ where
$\mathbf{r}^{\prime}=T\cdot\mathbf{r}$. Note that

$\left({\mathbb{Z}}_{p}-(B+k^{\prime}(m-2))r_{i}^{\prime}\right)^{2}=((B+k(m-2))r_{i}^{\prime})\cdot{\mathbb{Z}}_{p}+((B+k(m-2))r_{i}^{\prime})^{2}$

for any $k^{\prime}\in
k+p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$ if
$\text{ord}_{p}((B+k(m-2))r_{i}^{\prime})<0$. So if there is an $i$ such that
$\text{ord}_{p}((B+k(m-2))r_{i}^{\prime})<0$, then the statement would hold
because

$\begin{cases}\text{ord}_{p}(r_{i}^{\prime})\geq-\text{ord}_{p}(d_{a_{1};\mathbf{a}})\\\
\text{ord}_{p}(a_{i}^{\prime})\leq\text{ord}_{p}(d_{a_{1};\mathbf{a}})\end{cases}$

for all $i$.

Now assume that $\text{ord}_{p}((B+k(m-2))r_{i}^{\prime})\geq 0$ (i.e.,
$(B+k(m-2))r_{i}^{\prime}\in{\mathbb{Z}}_{p}$) for all $i$. If

$a_{1}(2A+B+k(m-4))+(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)<\text{ord}_{p}(d_{a_{1};\mathbf{a}}),$

then
$a_{1}(2A+B+k^{\prime}(m-4))+(B+k^{\prime}(m-2))^{2}\cdot(1-\sum_{i=2}^{n}a_{i}r_{i})$
is equivalent with
$a_{1}(2A+B+k(m-4))+(B+k(m-2))^{2}\cdot(1-\sum_{i=2}^{n}a_{i}r_{i})$ up to
${\mathbb{Z}}_{p}$-unit square for any $k^{\prime}\in
k+p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$. So we obtain the
claim. If

$a_{1}(2A+B+k(m-4))+(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)\geq\text{ord}_{p}(d_{a_{1};\mathbf{a}}),$

then
$a_{1}(2A+B+k^{\prime}(m-4))+(B+k^{\prime}(m-2))^{2}\cdot(1-\sum_{i=2}^{n}a_{i}r_{i})\in
p^{\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$ for any
$k^{\prime}\in k+p^{2\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$.
On the other hand, the diagonal quadratic ${\mathbb{Z}}_{p}$-lattice

$\left<a_{2}^{\prime},\cdots,a_{n}^{\prime}\right>\otimes{\mathbb{Z}}_{p}$

of $\operatorname{rank}n-1\geq 5$ is
$p^{\text{ord}_{p}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{p}$-universal since
$\max\\{\text{ord}_{p}(a_{2}^{\prime}),\cdots,\text{ord}_{p}(a_{n}^{\prime})\\}\leq\text{ord}_{p}(d_{a_{1};\mathbf{a}})$.
This completes the proof. ∎

######  Proposition 3.4.

Let $p$ be an odd prime with $p|m-2$. Then for any integer $A(m-2)+B$, there
is a residue $k_{m;\mathbf{a},A,B}(p)$ in ${\mathbb{Z}}/p^{\max\limits_{1\leq
i\leq n}\\{\text{ord}_{p}(a_{i})\\}}{\mathbb{Z}}$ for which

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

with $n\geq 6$ is solvable over ${\mathbb{Z}}_{p}$ for any $k\equiv
k_{m;\mathbf{a},A,B}(p)\pmod{p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}}$.

###### Proof.

First assume that $a_{1}+\cdots+a_{n}\not\equiv 0\pmod{p}$. According to
Hensel’s Lemma (13:9 in [7]), we may take a residue $k_{m;\mathbf{a},A,B}(p)$
in ${\mathbb{Z}}/p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}{\mathbb{Z}}$ satisfying

$d_{m;\mathbf{a},A,B,k}\equiv 0\pmod{p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}}$

for any $k\equiv k_{m;\mathbf{a},A,B}(p)\pmod{p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}}$. By the uniqueness of Jordan decomposition up
to Jordan type (see 91:9 in [7]), the orthogonal complement of the unary
unimodular $\left<a_{1}+\cdots+a_{n}\right>\otimes{\mathbb{Z}}_{p}$ in
$\left<a_{1},\cdots,a_{n}\right>\otimes{\mathbb{Z}}_{p}$ of
$\operatorname{rank}n\geq 6$ would be $p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}{\mathbb{Z}}_{p}$-universal. By using 82:15 and
91:9 in [7], we may see that for any integer $k\equiv
k_{m;\mathbf{a},A,B}(p)\pmod{p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}}$, the binary quadratic
${\mathbb{Z}}_{p}$-lattice
$[a_{1}+\cdots+a_{n},B+k(m-2),2A+B+k(m-4)]\otimes{\mathbb{Z}}_{p}$ (which is
equivalent with

$\left<a_{1}+\cdots+a_{n},(a_{1}+\cdots+a_{n})^{-1}\cdot
d_{m;\mathbf{a},A,B,k}\right>\otimes{\mathbb{Z}}_{p}$

with the discriminant $d_{m;\mathbf{a},A,B,k}\equiv
0\pmod{p^{\max\limits_{1\leq i\leq n}\\{\text{ord}_{p}(a_{i})\\}}}$) has a
representation by $\left<a_{1},\cdots,a_{n}\right>\otimes{\mathbb{Z}}_{p}$ of
the type

$\begin{pmatrix}1&1&\cdots&1\\\
x_{1}&x_{2}&\cdots&x_{n}\end{pmatrix}\begin{pmatrix}a_{1}&0&\cdots&0\\\
0&a_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&a_{n}\end{pmatrix}\begin{pmatrix}1&x_{1}\\\ 1&x_{2}\\\
\vdots&\vdots\\\ 1&x_{n}\end{pmatrix}$

for some $(x_{1},\cdots,x_{n})\in{\mathbb{Z}}_{p}^{n}$. This completes the
claim.

When $a_{1}+\cdots+a_{n}\equiv 0\pmod{p}$, without loss of generality, let
$0=\text{ord}_{p}(a_{1})$. Then one may show that

$\begin{cases}a_{2}x_{2}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{2}x_{2}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{p}$ for any $k\equiv
k_{m;\mathbf{a},A,B}(p)\pmod{p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}}$ where $k_{m;\mathbf{a},A,B}(p)$ is the
residue in ${\mathbb{Z}}/p^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{p}(a_{i})\\}}{\mathbb{Z}}$ satisfying

$d_{m;\mathbf{a},A,B,k_{m,\mathbf{a},A,B}(p)}\equiv
0\pmod{p^{\max\limits_{1\leq i\leq n}\\{\text{ord}_{p}(a_{i})\\}}}$

through same arguments with the above. This completes the proof. ∎

######  Proposition 3.5.

Let $m\equiv 0\pmod{4}$. If an integer $A(m-2)+B$ is represented by an
$m$-gonal form $\left<a_{1},\cdots,a_{n}\right>_{m}$ of
$\operatorname{rank}n\geq 6$ over ${\mathbb{Z}}_{2}$, then there is a residue
$k_{m;\mathbf{a},A,B}(2)$ in ${\mathbb{Z}}/8\cdot
2^{2\text{ord}_{2}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}$ for which for any
$k\equiv k_{m;\mathbf{a},A,B}(2)\pmod{8\cdot
2^{\text{ord}_{2}(d_{a_{1},\mathbf{a}})}}$,

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{2}$.

###### Proof.

After Jordan decomposing (see 91 in [7])

$\begin{pmatrix}a_{1}a_{2}+a_{2}^{2}&a_{2}a_{3}&\cdots&a_{2}a_{n}\\\
a_{2}a_{3}&a_{1}a_{3}+a_{3}^{2}&\cdots&a_{3}a_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
a_{2}a_{n}&a_{3}a_{n}&\cdots&a_{1}a_{n}+a_{n}^{2}\end{pmatrix}\otimes{\mathbb{Z}}_{2},$

through similar arguments with the proof of Proposition 3.3, one may prove
this by showing that if the equation

$Q_{a_{1};\mathbf{a}}(\mathbf{x}-(B+k(m-2))\mathbf{r})=a_{1}(2A+B+k(m-2))-(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)$

is solvable over ${\mathbb{Z}}_{2}$, then the equation

$Q_{a_{1};\mathbf{a}}(\mathbf{x}-(B+k^{\prime}(m-2))\mathbf{r})=a_{1}(2A+B+k^{\prime}(m-2))-(B+k^{\prime}(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)$

is also solvable over ${\mathbb{Z}}_{2}$ for any $k^{\prime}\in k+8\cdot
2^{2\text{ord}_{2}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{2}$. One may use that
$Q_{a_{1};\mathbf{a}}\otimes{\mathbb{Z}}_{2}$ is $2\cdot
2^{\text{ord}_{2}(d_{a_{1};\mathbf{a}})}{\mathbb{Z}}_{2}$-universal, for
$u,u^{\prime}\in{\mathbb{Z}}_{2}^{\times}$,

$u^{\prime}\in u({\mathbb{Z}}_{2})^{2}\text{ if and only if }u\equiv
u^{\prime}\pmod{8},$

for $q\in{\mathbb{Q}}_{2}$,

$\begin{cases}\left({\mathbb{Z}}_{2}-q\right)^{2}=4q{\mathbb{Z}}_{2}+q^{2}&\text{if
}\text{ord}_{2}(q)=-1\\\
\left({\mathbb{Z}}_{2}-q\right)^{2}=2q{\mathbb{Z}}_{2}+q^{2}&\text{if
}\text{ord}_{2}(q)<-1,\end{cases}$

and

$\begin{cases}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\par\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&-\frac{1}{2}\end{pmatrix}^{t}\par\begin{pmatrix}2&0\\\
0&-2\end{pmatrix}\par\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&-\frac{1}{2}\end{pmatrix}\par\\\ \begin{pmatrix}2&1\\\
1&2\end{pmatrix}=\begin{pmatrix}1&\frac{1}{2}\\\
0&\frac{1}{2}\end{pmatrix}^{t}\par\begin{pmatrix}2&0\\\
0&6\end{pmatrix}\par\begin{pmatrix}1&\frac{1}{2}\\\
0&\frac{1}{2}\end{pmatrix}.\par\end{cases}$

∎

######  Proposition 3.6.

Let $m\not\equiv 0\pmod{4}$. For any integer $A(m-2)+B$, there is a residue
$k_{m;\mathbf{a},A,B}(2)$ in ${\mathbb{Z}}/2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}{\mathbb{Z}}$ for which for any $k\equiv
k_{m;\mathbf{a},A,B}(2)\pmod{2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}}$,

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

with $n\geq 6$ is solvable over ${\mathbb{Z}}_{2}$.

###### Proof.

Without loss of generality, let
$0=\text{ord}_{2}(a_{1})\leq\text{ord}_{2}(a_{2})\leq\cdots\leq\text{ord}_{2}(a_{n})$.
First, assume that $a_{1}+\cdots+a_{n}$ is odd. For $m\not\equiv 0\pmod{4}$
and $a_{1}+\cdots+a_{n}$ is odd, we may take $k_{2}\in{\mathbb{Z}}_{2}$ for
which

$d_{m;\mathbf{a},A,B,k_{2}}:=(a_{1}+\cdots+a_{n})(2A+B+k_{2}(m-4))-(B+k_{2}(m-2))^{2}=0.$

Put the residue $k_{m;\mathbf{a},A,B}(2)$ in ${\mathbb{Z}}/2\cdot
2^{\max\limits_{1\leq i\leq n}\\{\text{ord}_{2}(a_{i})\\}}{\mathbb{Z}}$ as
$k_{m;\mathbf{a},A,B}(2)\equiv k_{2}\pmod{2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}{\mathbb{Z}}_{2}}$. For $k\equiv
k_{m;\mathbf{a},A,B}(2)\pmod{2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}}$, unary unimodular
$\left<a_{1}+\cdots+a_{n}\right>\otimes{\mathbb{Z}}_{2}$ splits the binary
quadratic ${\mathbb{Z}}_{2}$-lattice

$\begin{pmatrix}a_{1}+\cdots+a_{n}&B+k(m-2)\\\
B+k(m-2)&2A+B+k(m-4)\end{pmatrix}\otimes{\mathbb{Z}}_{2}$

with discriminant $d_{m;A,B,k}\equiv 0\pmod{2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}}$. On the other words,
$[a_{1}+\cdots+a_{n},B+k(m-2),2A+B+k(m-4)]\otimes{\mathbb{Z}}_{2}$ is
isometric to diagonal $\left<a_{1}+\cdots+a_{n},(a_{1}+\cdots+a_{n})^{-1}\cdot
d_{m;A,B,k}\right>\otimes{\mathbb{Z}}_{2}$. The orthogonal complement of unary
unimodular $\left<a_{1}+\cdots+a_{n}\right>\otimes{\mathbb{Z}}_{2}$ in
$\left<a_{1},\cdots,a_{n}\right>\otimes{\mathbb{Z}}_{2}$ of
$\operatorname{rank}n\geq 6$ would represent every $2$-adic integer in $2\cdot
2^{\max\limits_{1\leq i\leq n}\\{\text{ord}_{2}(a_{i})\\}}{\mathbb{Z}}_{2}$.
From 82:15 in [7], we may see that

$\begin{pmatrix}a_{1}+\cdots+a_{n}&B+k(m-2)\\\
B+k(m-2)&2A+B+k(m-4)\end{pmatrix}\otimes{\mathbb{Z}}_{2}$

has a representation by
$\left<a_{1},\cdots,a_{n}\right>\otimes{\mathbb{Z}}_{2}$ of the type

$\begin{pmatrix}1&1&\cdots&1\\\
x_{1}&x_{2}&\cdots&x_{n}\end{pmatrix}\begin{pmatrix}a_{1}&0&\cdots&0\\\
0&a_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&a_{n}\end{pmatrix}\begin{pmatrix}1&x_{1}\\\ 1&x_{2}\\\
\vdots&\vdots\\\ 1&x_{n}\end{pmatrix}$

for some $(x_{1},\cdots,x_{n})\in{\mathbb{Z}}_{2}^{n}$. It completes the proof
when $a_{1}+\cdots+a_{n}$ is odd.

When $a_{1}+\cdots+a_{n}$ is even, one may show that

$\begin{cases}a_{2}x_{2}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{2}x_{2}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is solvable over ${\mathbb{Z}}_{2}$ for any integer $k\equiv
k_{m;\mathbf{a},A,B}(2)\pmod{2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}}$ where $k_{m;\mathbf{a},A,B}(2)$ is the
residue in ${\mathbb{Z}}/2\cdot 2^{\max\limits_{1\leq i\leq
n}\\{\text{ord}_{2}(a_{i})\\}}{\mathbb{Z}}$ satisfying

$d_{m;(a_{2},\cdots,a_{n}),A,B,k_{m;\mathbf{a},A,B}(2)}\equiv 0\pmod{2\cdot
2^{\max\limits_{1\leq i\leq n}\\{\text{ord}_{2}(a_{i})\\}}}$

similarly with the above. ∎

##  4\. Representing integer which is locally represented

######  Lemma 4.1.

There is a constant $K(a_{1},\cdots,a_{n})>0$ which is dependent only on
$a_{1},\cdots,a_{n}$ for which for any integer $A(m-2)+B$ with $0\leq B\leq
m-3$ which is locally represented by an $m$-gonal form
$\left<a_{1},\cdots,a_{n}\right>_{m}$ of $\operatorname{rank}n\geq 6$, there
is $0\leq k\leq K(a_{1},\cdots,a_{n})-1$ such that

$\begin{cases}a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}=2A+B+k(m-4)\\\
a_{1}x_{1}+\cdots+a_{n}x_{n}=B+k(m-2)\end{cases}$

is locally solvable.

###### Proof.

Define the finite set $T$ as the set of all odd primes $p$ for which the
number of units of ${\mathbb{Z}}_{p}$ in $\\{a_{1},\cdots,a_{n}\\}$ by
admitting a recursion is less than $5$. Then the lemma holds for

$K(a_{1},\cdots,a_{n}):=\prod\limits_{p\in\\{2\\}\cup
T}p^{\max\\{\text{ord}_{p}(8\cdot(d_{a_{1};\mathbf{a}})^{2}),\text{ord}_{p}(2a_{1}),\cdots,\text{ord}_{p}(2a_{n})\\}}$

by Proposition 3.1, Proposition 3.2, Proposition 3.3, Proposition 3.4,
Proposition 3.5, and Proposition 3.6. ∎

###### proof of Theorem 1.1.

Recall that for an integer $A(m-2)+B$ with $0\leq B\leq m-3$ such that the
system (2.2) is locally solvable, if

$A\geq\frac{1}{2a_{1}}\left(C(a_{1},\cdots,a_{n})+(B+k(m-2))^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)\right)-B-k(m-2)$

where $C(a_{1},\cdots,a_{n})$ is the constant in (2.5), then $A(m-2)+B$ is
represented by the $m$-gonal form $\left<a_{1},\cdots,a_{n}\right>_{m}$. Now
we define $N(a_{1},\cdots,a_{n})$ as

$\frac{1}{2a_{1}}\left(C(a_{1},\cdots,a_{n})+K(a_{1},\cdots,a_{n})^{2}\cdot\left(1-\sum_{i=2}^{n}a_{i}r_{i}\right)\right).$

Then by Lemma 4.1, we complete the first claim.

One may notice that

$(a_{1}+\cdots+a_{n})(a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2})\geq(a_{1}x_{1}+\cdots+a_{n}x_{n})^{2}$

and

$a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})\equiv
a_{1}x_{1}+\cdots+a_{n}x_{n}\pmod{m-2}.$

Therefore an integer $A(m-2)+B$ with $0\leq B\leq m-2$ would be never
represented by $\left<a_{1},\cdots,a_{n}\right>_{m}$ if

$A<\begin{cases}\frac{B^{2}}{2(a_{1}+\cdots+a_{n})}+\frac{B}{2}&\text{when
}B\leq\frac{m-2}{2}\\\
\frac{(B-(m-2))^{2}}{2(a_{1}+\cdots+a_{n})}+\frac{(B-(m-2))}{2}&\text{when
}B>\frac{m-2}{2}.\end{cases}$

Which implies that the cubic with respect to $m$ in (1.3) is optimal. ∎

######  Remark 4.2.

Recall that every $m$-gonal form of $\operatorname{rank}\geq 5$ is almost
regular. Indeed, the $5$ is optimal, i.e., there is a non-almost regular
$m$-gonal form of $\operatorname{rank}4$. By Proposition 3.1, the $m$-gonal
$\left<1,1,1,1\right>_{m}$ is locally universal for all $m\geq 3$, but
$\left<1,1,1,1\right>_{m}$ with $m\equiv 0\pmod{4}$ and $m>8$ is not almost
universal (see Theorem 1.2 in [5]). Theorem 1.1 is missing the sufficiently
large integers which are represented by (almost regular) $m$-gonal form of
$\operatorname{rank}5$. Filling out the remaining answer on sufficiently large
integers which are represented by $m$-gonal form of $\operatorname{rank}5$
could be a challenging problem.

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