# Electrical detectability of magnon-mediated spin current shot noise

Luise Siegl1 Michaela Lammel1 Akashdeep Kamra2 Hans Huebl3,4,5 Wolfgang
Belzig1 Sebastian T. B. Goennenwein1 1Department of Physics, University of
Konstanz, 78457 Konstanz, Germany
2Condensed Matter Physics Center (IFIMAC) and Departamento de Física Teórica
de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid,
Spain
3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748
Garching, Germany
4TUM School of Natural Sciences, Technische Universität München, 85748
Garching, Germany
5Munich Center for Quantum Science and Technology (MCQST), 80799 München,
Germany

###### Abstract

A magnonic spin current crossing a ferromagnet-metal interface is accompanied
by spin current shot noise arising from the discrete quanta of spin carried by
magnons. In thin films, e.g., the spin of so-called squeezed magnons have been
shown to deviate from the common value $\hbar$, with corresponding changes in
the spin noise. In experiments, spin currents are typically converted to
charge currents via the inverse spin Hall effect. We here analyze the
magnitude of the spin current shot noise in the charge channel for a typical
electrically detected spin pumping experiment, and find that the voltage noise
originating from the spin current shot noise is much smaller than the
inevitable Johnson-Nyquist noise. Furthermore, we find that due to the local
nature of the spin-charge conversion, the ratio of spin current shot noise and
Johnson-Nyquist noise cannot be systematically enhanced by tuning the sample
geometry, in contrast to the linear increase in dc spin pumping voltage with
sample length. Instead, the ratio depends sensitively on material-specific
transport properties. Our analysis thus provides guidance for the experimental
detection of squeezed magnons through spin pumping shot noise.

The power spectral density of charge current fluctuations contains fundamental
information about the underlying transport and dynamics [1]. For example, the
discrete nature of electric charge results in shot noise [2]. Vice-versa, shot
noise experiments allow to quantify the quantum of charge relevant for
transport. In diodes and related structures, the existence of shot noise shows
that the electrical current is carried by elementary charges, while in
fractional quantum Hall systems, composite Fermions with fractional charge are
the relevant electrical transport quanta [3, 4, 5]. In superconducting
contacts, multiple charge quanta have been predicted [6, 7, 8, 9] and observed
[10, 11, 12]. On the other hand, thermal fluctuations of charge carriers
inside an electrical conductor at equilibrium lead to the Johnson-Nyquist
noise [13, 14].

In recent years, pure spin transport has attracted considerable interest. In
particular, spin pumping has emerged as a powerful method for the generation
of pure spin currents in ferromagnetic/normal metal (FM/N) heterostructures
[15, 16, 17, 18, 19, 20, 21, 22]. There, a magnon mode in the FM is populated
using a coherent microwave drive, i.e., in ferromagnetic resonance (FMR). The
resulting nonequilibrium magnonic spin is partially absorbed by the electrons
in N causing a pure spin current flow across the FM/N interface. Taking
advantage of the (inverse) spin Hall effect which interconverts pure spin and
charge currents in a metal with strong spin-orbit coupling [23, 24, 25], such
spin currents can be detected in N as an electrical current or voltage signal.

Thermal fluctuations of a pure spin current have been detected experimentally
in a yttrium iron garnet/platinum bilayer employing a magnetic field
orientation-dependent measurement of the voltage noise power spectral density
[26]. The observed thermal spin current noise was theoretically shown to be
related to the spin Hall magnetoresistance (SMR) effect via the fluctuation-
dissipation theorem [26]. The SMR effect is governed by spin current flow
across a magnetic insulator/metal interface [27, 28, 29, 30, 31]. More
precisely, the resistance of the metal layer changes as a function of the
magnetization orientation due to spin current flow across the interface.
However, while serving as a proof-of-concept for spin current noise
measurements [26], the observed thermal voltage noise in platinum did not
provide deeper insights into the microscopic mechanisms of spin transport.

The situation is different for fluctuations of a non-equilibrium current [32,
33], such as spin shot noise arising from a pure spin current $I_{\mathrm{s}}$
flowing across the FM/N interface [34, 35], or non-equilibrium spin
accumulations [36]. In analogy to electrical shot noise, this spin current
shot noise can be used to experimentally detect and quantify the spin
transport quantum [34, 35]. For spin transport arising from a coherently
driven magnon mode, as is the case in a typical ferromagnetic resonance
scenario, the non-integer character of the magnon spin transport quantum, e.g.
due to squeezing effects, could become experimentally accessible. The same
information is harder to infer from a thermally driven spin current shot
noise, for example via the spin Seebeck effect [37], as it involves multiple
magnon modes with different effective spins.

However, previous theoretical works [34, 35] on spin current shot noise have
restricted themselves to examine spin currents, with only a preliminary
discussion on spin to charge current noise conversion [37, 38, 39]. Moreover,
to assess the experimental detectability, the additional noise contributions
arising from charge fluctuations in the normal metal must be taken into
account. Here, we calculate the magnitude of the spin current shot noise power
spectral density upon conversion to the charge channel, and consider how the
spin to charge conversion impacts the detectability of spin current shot
noise. A key prerequisite for this consideration is that the spatial
correlations in the electronic spin current injected into N are short ranged
comparable to the electronic wavelengths. This implies that the conversion
factor for spin to charge dc currents, widely used in spin-Hall effect based
studies, does not provide a complete picture for the conversion of the
interfacial spin current to the electrically measured voltage fluctuations. In
consequence, we find that the voltage noise resulting from the spin current
shot noise is substantially smaller than the purely charge based Johnson-
Nyquist (JN) noise in the normal metal.

We consider the shot noise associated with the spin pumping current in a
system at a finite temperature $T$, driven by a coherent microwave magnetic
field at driving frequency $\omega$ [35] which corresponds to the
ferromagnetic resonance frequency. The ensuing power spectral density in the
spin channel for a total spin current $I_{\mathrm{s}}$ traversing the FM/N
interface is given by
$S_{I_{\mathrm{s}}I_{\mathrm{s}}}=2\hbar^{*}I_{\mathrm{s}}\frac{2k_{\rm{B}}T}{\hbar\omega}$
[35], where $k_{\rm{B}}$ is the Boltzmann constant, $\hbar$ is the reduced
Planck constant and $\hbar^{*}$ the effective spin. Interestingly, spin shot
noise thus reflects magnon squeezing effects, since the quantum of angular
momentum relevant for the noise is no longer $\hbar$, but
$\hbar^{*}=\hbar(1+\delta)$ [34], where $\delta$ is a material and sample
geometry dependent factor. These results suggest that spin current shot noise
can be used to experimentally detect and quantify magnon squeezing by the
effective spin $\hbar^{*}$ via spin pumping experiments.

Now, we consider the experimental detection scheme to measure spin current
shot noise electrically via the inverse spin Hall effect (ISHE) in the N
layer. Within the FM layer the magnetisation $\bm{M}$ is coherently driven out
of its equilibrium position by FMR [40, 41, 42]. Thus, a pure spin current
$I_{\mathrm{s}}=j_{\rm{s}}wl$ propagating along z-direction is pumped over the
interface of width $w$ and length $l$ into the metal layer, as depicted in
Fig. 1. Note that for this expression of the total spin pumping current, we
have assumed a spatially uniform spin pumping current density $j_{\rm{s}}$,
which is valid for a dc current or the expectation value of an ac current.
Since a direct experimental detection of spin is difficult, the spin current
density is converted into a charge current density
$\bm{j}_{\rm{c}}=\frac{2e}{\hbar}\theta_{\rm{SH}}\,\bm{j}_{\rm{s}}\times\bm{s}$
via the inverse spin Hall effect in N [23, 19, 43, 25], as sketched in Fig. 1.
Here, $e$ is the elementary charge, and $\theta_{\rm{SH}}$ the spin Hall
angle. In most experiments, open circuit electrical boundary conditions are
implemented, such that an open-circuit dc voltage $V$ is detected instead of
the charge current.

Figure 1: Schematic of a typical spin pumping experiment with electrical
detection via the inverse spin Hall effect. The resonantly driven coherent
precession of the magnetization $\bm{M}$ in the ferromagnetic layer (FM)
drives a pure spin current density $\bm{j}_{\rm{s}}$ propagating in the
z-direction, across the interface into the normal metal (N) layer. In the
latter, it is converted into a charge current density $\bm{j}_{\rm{c}}$ owing
to the ISHE. For open circuit electrical boundary conditions, a dc voltage $V$
proportional to $\bm{j}_{\rm{s}}$ arises.

Applying such an ISHE-based electrical detection scheme to spin current shot
noise, one obtains a voltage noise power spectral density $S_{VV}\propto
S_{I_{\rm{s}}I_{\rm{s}}}$ in the charge channel inside the normal metal.
Generally, the spin shot noise can be detected as an electrical voltage noise
power spectral density $S_{VV}$ or current noise power spectral density
$S_{II}$. Since the current and voltage power spectral densities can be
transformed into each other by
$\left(\frac{S_{VV}}{V^{2}}\right)_{I=\rm{const}}=\left(\frac{S_{II}}{I^{2}}\right)_{U=\rm{const}}$
[44], we here focus on $S_{VV}$.

The typical theoretical analysis [45] exploits the spatial homogeneity of the
spin pumping current density $\bm{j}_{\rm{s}}$ over the FM/N interface to
relate $\bm{j}_{\rm{s}}$ with the experimentally measurable quantity, i.e. the
total charge current $I$ (or voltage $V$) through the normal metal. This, in
turn, is a consequence of the spatial invariance of the coherent microwave
drive causing FMR and thus, the magnetization precession in the FM layer. On
the other hand, the spin pumping current noise or fluctuations are expected to
have short ranged correlations determined by the wavelength of the electrons
that absorb and carry the spin current in N. Thus, we need to go beyond the
typical relation for dc currents [45, 37], as discussed below, to relate the
power spectral density of the total spin pumping current noise
$S_{I_{\mathrm{s}}I_{\mathrm{s}}}$ [34, 35] to the total charge current $I$
through the normal metal. The spin current shot noise
$S_{I_{\mathrm{s}}I_{\mathrm{s}}}$ scales with the system temperature and is
largest under the condition of ferromagnetic resonance $\omega=\omega_{0}$
[35]. We thus exploit the result from Ref. [35] in the high temperature limit
$k_{\rm{B}}T\gg\hbar\omega$ for sufficiently low driving frequencies $\omega$,
as it is the experimentally relevant limit. The low temperature limit
$k_{\rm{B}}T\ll\hbar\omega$ is briefly discussed below. Assuming a y-polarized
spin current density $j_{\rm{s}}$, we obtain the spatially and temporally
resolved spin pumping current density correlator

$\langle
j_{\rm{s}}(t,\bm{\rho})j_{\rm{s}}(t^{\prime},\bm{\rho^{\prime}})\rangle=2\hbar^{*}j_{\rm{s}}\frac{2k_{\rm{B}}T}{\hbar\omega_{0}}\delta(t-t^{\prime})\delta(\bm{\rho}-\bm{\rho^{\prime}})$
(1)

local in time $t$ and space, where $\bm{\rho}$ is the two-dimensional position
vector in the interfacial plane. Equation (1) captures the low-frequency and
frequency-independent part of the spin current noise power spectral density.
It has been derived starting from the correlator of the total spin current
across the interface [35], and considering a coherent region. Assuming
additionally that the coherence length is much smaller than the sample
dimensions, a delta function in space is obtained. Taking this spatio-temporal
correlation for the interfacial spin current density and following an analysis
similar to that in Ref. [26], we evaluate the voltage noise power spectral
density $S_{VV}^{\rm{shot}}$ of the spin pumping current shot noise in the
charge channel

$S_{VV}^{\rm{shot}}=16\frac{\theta_{\rm{SH}}^{2}\lambda_{\rm{sd}}^{2}\rho_{\rm{N}}^{2}l}{wt_{\rm{N}}^{2}}j_{\rm{s}}e^{2}\frac{\hbar^{*}}{\hbar^{2}}\frac{2k_{\rm{B}}T}{\hbar\omega_{0}}\tanh^{2}\left(\frac{t_{\rm{N}}}{2\lambda_{\rm{sd}}}\right).$
(2)

Here, $\omega_{0}$ is the FMR frequency, and $\lambda_{\rm{sd}}$,
$\rho_{\rm{N}}$ and $t_{\rm{N}}$ are the spin diffusion length, the
resistivity and the thickness of the metal layer N, respectively.

We now interpret the shot noise enhancement with temperature [35]. We begin by
recognizing that the dc spin current expression contains an integral of the
form $\sim\int d\epsilon[f(\epsilon)-f(\epsilon+\hbar\omega)]$, where
$\epsilon$ denotes energy, $\omega$ is the magnon (microwave drive) frequency,
and $f(\epsilon)$ is the Fermi function [35]. The physical content is that the
rate of spin transfer is determined by the difference between available
electrons and holes (unoccupied electronic states) within $\hbar\omega$. This
is reasonable since a magnon is absorbed by raising an electron from an
occupied to an unoccupied state or by creating a pair of electron and hole
excitations. On the other hand, the expression for the shot noise contains an
integral $\sim\int d\epsilon f(\epsilon)[1-f(\epsilon+\hbar\omega)]$ which
shows that the coherently driven and populated magnon mode is causing
correlations between existing electron and hole excitations [35]. At zero
temperature, since there are no pre-existing excitations, the only way to
create a correlation is by exciting them at the same time via the absorption
of a magnon. However, at finite temperatures, the macroscopically occupied
magnon mode may induce correlations in the thermally excited electron-hole
pairs resulting in stronger correlations that are still proportional to the
number and spin of magnons. In mathematical terms, at zero temperature, the
integrands in both the integrals above are unity and nonzero only over a
$\hbar\omega$-wide energy window resulting in their proportionality. While
finite temperatures do not affect the integral for the dc current, they
significantly enhance the window over which electron and hole excitations
coexist and the integrand is nonzero. The shot noise induced correlations are
thus enhanced with temperature. To conclude, the simple picture that shot
noise results solely from the quantized nature of transport is only valid at
zero temperature.

In addition to the voltage noise, Eq. (2), arising from the spin pumping
current traversing the FM/N interface, the normal metal harbors a thermal
charge fluctuations-based Johnson-Nyquist (JN) noise with a power density

$S_{VV}^{\rm{JN}}=4k_{\rm{B}}TR.$ (3)

Here, $R=\rho_{\rm{N}}l/(t_{\rm{N}}w)$ is the resistance of N. Note, that the
noise represented by Eq. (3) is different from the contribution of the thermal
magnonic spin current fluctuations [35, 37], which can be considered a
magnonic spin transport analogue of the JN noise. As the thermal magnonic spin
current fluctuations has been theoretically shown to be smaller than the shot
noise in a wide range of parameters [35, 37] we disregard this magnonic
contribution in our analysis here. The voltage noise in N thus will have (at
least) two contributions, $S_{VV}=S_{VV}^{\rm{shot}}+S_{VV}^{\rm{JN}}$. Since
both are frequency independent (white) at low frequencies, we consider and
compare their absolute magnitudes. The ratio

$\frac{S_{VV}^{\rm{shot}}}{S_{VV}^{\rm{JN}}}=8\frac{\theta^{2}_{\rm{SH}}\lambda^{2}_{\rm{sd}}\rho_{\rm{N}}}{t_{\rm{N}}}j_{\rm{s}}\frac{\hbar^{*}e^{2}}{\hbar^{3}\omega_{0}}\tanh^{2}\left(\frac{t_{\rm{N}}}{2\lambda_{\rm{sd}}}\right)$
(4)

should be maximized for the detection of spin current shot noise. Since
$\delta$ is on the order of $1$ in thin ferromagnetic films, we use
$\hbar^{*}=2\hbar$ to estimate the ratio [34]. Therefore, the adjustable
parameters for maximizing the spin noise voltage signal are the magnitude of
the spin current density $j_{\rm{s}}$, the FMR frequency $\omega_{0}$, and the
thickness $t_{\rm{N}}$ of the metal layer together with its material-specific
properties $\theta_{\rm{SH}}$, $\lambda_{\rm{sd}}$ and $\rho_{\rm{N}}$.
Similar conclusions have been drawn by Luo et al. regarding the magnitude of
magnon shot noise in spin Seebeck effect measurements [46].

In finding the optimal sample design for electrical spin current shot noise
experiments, we first note that only the ratio of spin diffusion length
$\lambda_{\rm{sd}}$ and metal layer thickness $t_{\rm{N}}$ enters in Eq. (4).
Since $t_{\rm{N}}$ can be straightforwardly chosen by appropriate sample
design, we numerically optimize the expression

$p=\frac{\lambda_{\rm{sd}}}{t_{\rm{N}}}\tanh^{2}\left(\frac{t_{\rm{N}}}{2\lambda_{\rm{sd}}}\right)$
(5)

from Eq. (4). Using Newton’s method we find numerically that Eq. (5) has a
global maximum at $p\approx 0.29$ for $t_{\rm{N}}\approx
2.18\lambda_{\rm{sd}}$.

Next, we consider the material properties of the normal metal N in the
combination $\theta^{2}_{\rm{SH}}\lambda_{\rm{sd}}\rho_{\rm{N}}$ in Eq. (4).
Figure 2 shows a compilation of values from literature for a set of different
spin Hall active metals [25, 47, 48, 49, 50, 31]. The symbols hereby indicate
different materials, the colors the references from which the values were
taken. As evident from Fig. 2, the values of
$\theta^{2}_{\rm{SH}}\lambda_{\rm{sd}}\rho_{\rm{N}}$ span 5 orders of
magnitude. This large variation reflects the broad scatter in spin Hall angles
and spin diffusion lengths reported even for the same material [25, 47, 48,
49, 50, 31].

Figure 2: The combination of material parameters
$\theta^{2}_{\rm{SH}}\lambda_{\rm{sd}}\rho_{\rm{N}}$ relevant for Eq. (4) for
different normal metals N. The data were taken from Sinova et al. [25] (blue),
Hong et al. [47] (grey), Ma et al. [48] (dark green), Isasa et al. [49] (light
green), Sagasta et al. [50] (purple) and Weiler et al. [31] (yellow). The
symbols correspond to different materials.

Finally, we turn to the magnitude of the spin current density $j_{\rm{s}}$. In
spin pumping experiments, values of
$$1.4\text{\times}{10}^{-11}\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$<j_{\rm{s}}<$8.8\text{\times}{10}^{-9}\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$
have been reported [31, 51]. Note that Weiler et al. found that the spin
Seebeck effect allows generating larger $j_{\rm{s}}$ than those obtained from
spin pumping [31], presumably due to the contribution of magnons with a broad
range of wave vectors and frequencies. However, the non-integer effective spin
$\hbar^{*}$ of the squeezed magnon is largest for the $\bm{k}=0$ eigenmode
(Kittel mode) and $\hbar^{*}\to\hbar$ with increasing $\bm{k}$ [52].
Therefore, experiments based on the spin Seebeck effect appear less suitable
for investigating the basic mechanisms behind spin current shot noise and
magnon squeezing effects. We therefore here focus on pure spin currents
generated via spin pumping.

Taking together the previous results we can extract the ratio of the spin
current shot noise and Johnson-Nyquist noise for different parameter
combinations. Figure 3 shows this ratio in the voltage channel,
$S_{VV}^{\rm{shot}}/S_{VV}^{\rm{JN}}$, as given by Eq. (4). We hereby assumed
a FMR frequency of
$\omega_{0}/2\pi=$10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ as typical for
measurements with a X band microwave cavity, and $\hbar^{*}=2\hbar$ as
mentioned above. Based on the parameter values discussed in the preceding
paragraphs, the range of experimentally achievable noise power ratios is
indicated as a semi-transparent rectangle in the figure. Notably, even for the
best possible combination of material parameters and spin current densities
reported in the literature the ratio $S_{VV}^{\rm{shot}}/S_{VV}^{\rm{JN}}$ is
smaller than $10^{-4}$. This upper experimental boundary is marked by the
black line in Fig. 3. In previous experiments addressing the voltage noise due
to thermal spin current fluctuations, changes in the noise magnitude of
$\approx$1\text{\times}{10}^{-3}$$ of the Johnson-Nyquist noise [26] could be
resolved. We furthermore include data from Weiler et al. (yellow symbols) [31]
in Fig. 3, as a typical example for electrically detected spin pumping data
recorded using yttrium iron garnet/Pt thin film bilayers and a X band
microwave cavity. In these experiments, the ratio of the spin current shot
noise and Johnson-Nyquist noise is smaller than $10^{-6}$. Note that, while in
cavity-based FMR a frequency of $10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ is
widely used, it would be beneficial for spin current shot noise experiments to
reduce the microwave frequency as much as possible to increase the shot noise.
In the low temperature limit where $k_{\rm{B}}T\ll\hbar\omega$, the Johnson-
Nyquist voltage noise power spectral density is suppressed exponentially[53,
54]. However, in this regime other noise sources, e.g. originating from
quantum fluctuations might dominate [55, 56]. This warrants a quantitative
analysis of fluctuations associated with quantum effects, which is beyond the
scope of this work.

Figure 3: False color plot showing the voltage noise power density due to spin
current shot noise in relation to the Johnson-Nyquist voltage noise density,
$S_{VV}^{\rm{shot}}/S_{VV}^{\rm{JN}}$, as given by Eq. (4) for a FMR frequency
$\omega_{0}/2\pi=$10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
$\hbar^{*}=2\hbar$. The semi-transparent rectangle highlights the accessible
range of $S_{VV}^{\rm{shot}}/S_{VV}^{\rm{JN}}$ values considering literature
values [25, 47, 48, 49, 50, 31, 51] for the spin current density $j_{\rm{s}}$
achievable in spin pumping experiments and the combination of spin Hall angle
$\theta_{\mathrm{SH}}$, spin diffusion length $\lambda_{\mathrm{sd}}$ and
resistivity $\rho_{\mathrm{N}}$ for different spin Hall active normal metals
of thickness $t_{\mathrm{N}}$ (see main text). The yellow symbols correspond
to the experimental data taken from Weiler et al. [31].

Our analysis thus indicates that for the detection of the spin current shot
noise in an electrical experiment, both the sample properties and the spin
current drive need to be carefully optimized. Furthermore, to resolve the
effective spin $\hbar^{*}$ of the magnon from spin pumping driven experiments
[34], even higher experimental precision will be required.

In summary, we derived the correlator of the spin pumping current density
including its time and spatial dependence. Considering the spin-to-charge
conversion process typically used in electrically detected spin current
experiments, we find that in the voltage channel, the spin current shot noise
is small compared to the ubiquitous Johnson-Nyquist noise. More precisely, the
ratio of the spin current shot noise to Johnson-Nyquist noise is estimated to
be at most $10^{-3}$ using parameters from literature and assuming a driving
frequency of $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. We thus conclude that
a careful choice of materials is of key importance for the measurement of the
spin pumping current shot noise and thus the effective spin of squeezed
magnon. Hence, our work offers important guidance regarding sample design and
optimization for the experimental detection of spin current shot noise.

###### Acknowledgements.

We acknowledge financial support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) via the SFB 1432 – Project-ID 425217212. H.
Huebl acknowledges financial support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) via Germany’s Excellence Strategy
EXC-2111-390814868. A. Kamra acknowledges financial support from the Spanish
Ministry for Science and InnovationAEI Grant CEX2018-000805-M (through the
“Maria de Maeztu” Programme for Units of Excellence in R&D) and grant
RYC2021-031063-I funded by MCIN/AEI/10.13039/501100011033 and “European Union
Next Generation EU/PRTR”.

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On Riemann type relations for theta functions on bounded symmetric domains of
type $I$ Atsuhira Nagano

###### Abstract

We provide a practical technique to obtain plenty of algebraic relations for
theta functions on the bounded symmetric domains of type $I$. In our
framework, each theta relation is controlled by combinatorial properties of a
pair $(T,P)$ of a regular matrix $T$ over an imaginary quadratic field and a
positive-definite Hermitian matrix $P$ over the complex number field.

000Keywords: theta functions ; symmetric domains 000Mathematics Subject
Classification 2020: Primary 11F27 ; Secondary 32M15, 14K25

## Introduction

Theta functions are special functions which play significant roles in number
theory, geometry, combinatorics and mathematical physics. The
$g^{2}$-dimensional bounded symmetric domain $\mathbb{H}_{I}^{(g)}$ of type
$I$ is the homogeneous space of the unitary group $U(g,g)$ (see (0.5)). In
this paper, we will present a practical technique to obtain plenty of
algebraic relations of theta functions on $\mathbb{H}_{I}^{(g)}$. Relations
obtained by our technique can be recognized as natural counterparts of the
classical Riemann theta relations for the Jacobi theta functions (see (0.1)
and (Introduction)). For this reason, in this paper, we will call our results
the Riemann type relations for the theta functions on $\mathbb{H}_{I}^{(g)}$.

Let us recall classical Riemann theta relations. The Jacobi theta functions
are defined as

$\displaystyle\mathbb{C}\times\mathbb{H}\ni(z,\tau)\mapsto\vartheta_{ab}(z,\tau)=\sum_{n\in\mathbb{Z}}\exp\left(\pi\sqrt{-1}\left(n+\frac{a}{2}\right)^{2}\tau+2\pi\sqrt{-1}\left(n+\frac{a}{2}\right)\left(z+\frac{b}{2}\right)\right)\in\mathbb{C},$
(0.1)

where $\mathbb{H}=\\{\tau\in\mathbb{C}\mid{\rm Im}(\tau)>0\\}$ and
$a,b\in\\{0,1\\}$. Classical Riemann theta relations are non-trivial quartic
relations which the theta functions (0.1) satisfy. One can find them in [Mu]
Chap.I Sect.5 (see also Section 1.2 of this paper). Here, let us give a simple
example:

$\displaystyle
2\vartheta_{00}(x^{\prime},\tau)\vartheta_{00}(y^{\prime},\tau)\vartheta_{00}(u^{\prime},\tau)\vartheta_{00}(v^{\prime},\tau)$
$\displaystyle=\vartheta_{00}(x,\tau)\vartheta_{00}(y,\tau)\vartheta_{00}(u,\tau)\vartheta_{00}(v,\tau)+\vartheta_{10}(x,\tau)\vartheta_{10}(y,\tau)\vartheta_{10}(u,\tau)\vartheta_{10}(v,\tau)$
$\displaystyle+\vartheta_{01}(x,\tau)\vartheta_{01}(y,\tau)\vartheta_{01}(u,\tau)\vartheta_{01}(v,\tau)+\vartheta_{11}(x,\tau)\vartheta_{11}(y,\tau)\vartheta_{11}(u,\tau)\vartheta_{11}(v,\tau),$
(0.2)

where $x^{\prime},y^{\prime},u^{\prime},v^{\prime}$ are complex variables
related to $x,y,u,v$ by the orthogonal matrix

$\displaystyle T_{0}=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&-1&1&-1\\\
1&1&-1&-1\\\ 1&-1&-1&1\end{pmatrix}$ (0.3)

as ${}^{t}(x^{\prime},y^{\prime},u^{\prime},v^{\prime})=T_{0}{}^{t}(x,y,u,v)$.
This example of classical Riemann theta relations has various applications.

  * [a]

Let us recall a simple and notable example. Just putting $x=y=u=v=0$ to
(Introduction), we immediately obtain

$\vartheta_{00}^{4}(\tau)=\vartheta_{10}^{4}(\tau)+\vartheta_{01}^{4}(\tau),$

where $\vartheta_{ab}(\tau)=\vartheta_{ab}(0,\tau).$ This relation is called
the Jacobi theta identity. This is important to study elliptic modular forms
for the principal congruence subgroup in ${\rm SL}(2,\mathbb{Z})$ of level
$2$.

The Jacobi theta functions satisfy other interesting formulas.

  * [b]

There are “half formulas” for the Jacobi theta functions (0.1). For detail,
see (2.4). They play an important role in the study of the arithmetic-
geometric mean by Gauss (see [C] ; see also the beginning of Section 2.1).

For a ring $R$, let ${\rm Mat}(g,h;R)$ be the abelian group of $g\times h$
matrices over $R$, and $R^{g}$ be the $R$-module of column vectors of size $g$
over $R$. The well-known Riemann theta function on the
$\frac{g(g+1)}{2}$-dimensional Siegel upper half plane
$\mathfrak{S}_{g}=\\{\Omega\in{\rm
Mat}(g,g;\mathbb{C})\mid{}^{t}\Omega=\Omega,{\rm Im}(\Omega)\text{ is positive
definite}\\}$ is defined as

$\displaystyle\mathfrak{S}_{g}\ni\Omega\mapsto\vartheta(\Omega)=\sum_{n\in\mathbb{Z}^{g}}\exp\left(\pi\sqrt{-1}{}^{t}n\Omega
n\right)\in\mathbb{C}.$ (0.4)

  * [c]

One can define Riemann theta functions with characteristics (see (1.14)). They
satisfy various non-trivial homogeneous relations. For instance, let us refer
to the relation (2.3), which is important in algebraic geometry and number
theory. This is essential in studies of abelian varieties, because Riemann
theta functions are associated with moduli spaces of abelian varieties in
nature (for example, see [I] and [K]).

There are various proofs of the classical Riemann relations such as
(Introduction). To the best of the author’s knowledge, each proof is based on
combinatorial properties derived from the orthogonal matrix $T_{0}$ of (0.3).
Additionally, one can prove the above results [b] and [c] on the basis of
combinatorial properties of suitable matrices $T$, which are taken instead of
$T_{0}$, and the orthogonality of characters (for example, see [Mu] Chap.II).
Here, we remark that the replaced matrices $T$ are not necessarily orthogonal
matrices. In the present paper, we will call the algebraic relations suggested
in [b] and [c] Riemann type relations, following the naming of (Introduction).

The main purpose of this paper is to obtain generalizations of the above
classical theta relations. We study theta functions on the $g^{2}$-dimensional
bounded symmetric domain

$\displaystyle\mathbb{H}_{I}^{(g)}=\left\\{W\in{\rm Mat}(g,g;\mathbb{C})\hskip
2.84526pt\Bigg{|}\hskip 2.84526pt\frac{W-{}^{t}\overline{W}}{2\sqrt{-1}}\text{
is a positive-definite Hermitian matrix}\right\\}$ (0.5)

of type $I$. This space naturally contains the Siegel upper half plane
$\mathfrak{S}_{g}$. Moreover, the unitary group $U(g,g)$ acts on
$\mathbb{H}_{I}^{(g)}$ in a canonical way. Thus, we have motives to study
theta functions on $\mathbb{H}_{I}^{(g)}$. In fact, by using the lattice
$\mathfrak{O}_{K}^{g}$ given by the ring $\mathfrak{O}_{K}$ of integers of an
imaginary quadratic field $K$, we have the theta function

$\displaystyle\mathbb{H}_{I}^{(g)}\ni
W\mapsto\Theta(W)=\sum_{n\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}{}^{t}\overline{n}Wn\right)\in\mathbb{C}.$
(0.6)

Such a theta function is studied in [F], [DK] or [Ma2] (see also (1.9) and
Remark 1.2). The main theorem (Theorem 1.1) of this paper gives a systematic
way to obtain various relations for the theta function (0.6). We can regard
our results as generalizations of the above mentioned classical results [a],
[b] and [c]. Therefore, we call our results Riemann type relations for the
theta function (0.6). More precisely, in our framework, a pair $(T,P)$
determines an algebraic theta relation. Here, $T$ is a matrix over $K$, which
is a counterpart of $T_{0}$ of (0.3). Note that we do not need to assume that
$T$ is an orthogonal matrix or a unitary matrix. Also, $P$ is a positive-
definite Hermitian matrix over $K$. For detail, see Theorem 1.1.

Probably, the biggest difference between the theta functions (0.6) and (0.4)
is whether we need an imaginary quadratic field $K$ or not. In fact, the theta
function (0.6) depends on the imaginary quadratic field $K$. However, such a
substitutability of $K$ is necessary, when we practically study the theta
functions. Actually, it is well known that the Riemann theta function (0.4) is
useful to study the moduli space of $\mathscr{K}_{\rm Kum}$, where
$\mathscr{K}_{\rm Kum}$ is the family of Kummer surfaces coming from
principally polarized abelian surfaces (see [I]). [MSY] constructs a family
$\mathscr{K}_{{\rm MSY}}$ of $K3$ surfaces which is an extension of
$\mathscr{K}_{\rm Kum}$. By virtue of the theta function (0.6) associated with
$K=\mathbb{Q}\left(\sqrt{-1}\right)$, we are able to study the moduli space of
$\mathscr{K}_{{\rm MSY}}$. On the other hand, [NS] studies another family
$\mathscr{K}_{NS}$ of $K3$ surfaces. Although $\mathscr{K}_{\rm NS}$ also
contains $\mathscr{K}_{\rm Kum}$ as a subfamily, this is different from
$\mathscr{K}_{{\rm MSY}}$. In order to study the moduli space of
$\mathscr{K}_{NS}$, we need the theta function (0.6) for
$K=\mathbb{Q}\left(\sqrt{-3}\right)$ (see Table 1).

$\mathscr{K}_{\rm Kum}$ | $\mathscr{K}_{{\rm MSY}}$ | $\mathscr{K}_{{\rm NS}}$
---|---|---
Riemann theta of (0.4) | Theta of (0.6) for $K=\mathbb{Q}\left(\sqrt{-1}\right)$ | Theta of (0.6) for $K=\mathbb{Q}\left(\sqrt{-3}\right)$
Table 1: Families of $K3$ surfaces and related theta functions

Algebraic relations of theta functions enable us to obtain interesting results
in number theory and geometry. To the best of the author’s knowledge, there
are several deep research related with $\mathscr{K}_{\rm MSY}$ since the
publication of [MSY] and [Ma1]. In this case, the theta function (0.6) is
defined by the use of the ring of Gaussian integers. Theta relations for them
lead a study of an arithmetic hyperbolic $3$-manifold (a generalization of the
arithmetic-geometric mean, resp.) in [MNY] ([MT], resp.).

Thus, there is a necessity to study algebraic relations for the theta function
(0.6) on $\mathbb{H}_{I}^{(g)}$. Our main theorem provides a handy tool to
obtain non-trivial theta relations. It is valid for arbitrary imaginary
quadratic fields $K$. The author expects that our technique is beneficial to
obtain plenty of theta relations and it accelerates further research of moduli
spaces of families of algebraic varieties like $\mathscr{K}_{\rm MSY}$ or
$\mathscr{K}_{{\rm NS}}$.

The contents of this paper are as follows. In Section 1, we introduce a new
theta function $\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$,
where $A_{0},B_{0}\in{\rm Mat}(g,h;K)$ and $P\in{\rm Mat}(h,h;\mathbb{C})$ is
a positive-definite Hermitian matrix (see (1.2)). We prove an algebraic
relation for $\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$
(Theorem 1.1). This is a natural counterpart of the formula of [Mu] Theorem
6.3 in Chap.II (see also Section 1.2). If $P$ is a matrix over $\mathbb{Q}$,
according to Theorem 1.2, $\Theta^{P}$ is related to the theta function of
(0.6). We will recognize theta relations derived from Theorem 1.1 as
generalizations of the classical Riemann type relations such as
(Introduction), [a], [b] and [c]. By taking suitable pairs $(T,P)$ and
applying Theorem 1.1, we obtain non-trivial theta relations by relatively
simple calculations. In order to see practical effects of our main result, we
will apply it in Section 2 for appropriately chosen $(T,P)$.

##  1 Relations for theta functions on $\mathbb{H}_{I}^{(g)}$

###  1.1 Main result

Throughout this paper, $K$ denotes the imaginary quadratic field
$\mathbb{Q}\left(\sqrt{-d}\right)$ for a positive square-free integer $d$, and
$\mathfrak{O}_{K}=\mathbb{Z}+\delta\mathbb{Z}$ denotes the ring of integers of
$K$, where

$\displaystyle\delta=\begin{cases}\sqrt{-d}\quad&(-d\not\equiv 1(\text{mod
}4)),\\\ \frac{1+\sqrt{-d}}{2}\quad&(-d\equiv 1(\text{mod }4)).\end{cases}$

Note that it holds $|\delta|^{2}\in\mathbb{Z}$.

As mentioned in Introduction, $\mathbb{H}_{I}^{(g)}$ in (0.5) stands for the
$g^{2}$-dimensional bounded symmetric domain of type $I$. Here, we remark that
every $W\in\mathbb{H}_{I}^{(g)}$ can be decomposed into the sum of a Hermitian
matrix and $\sqrt{-1}$ times a positive-definite Hermitian matrix:

$\displaystyle
W=\frac{W+{}^{t}\overline{W}}{2}+\sqrt{-1}\frac{W-{}^{t}\overline{W}}{2\sqrt{-1}}.$
(1.1)

Let $P\in{\rm Mat}(h,h;\mathbb{C})$ be a positive-definite Hermitian matrix.
Let us define a theta function

$\displaystyle\Theta^{P}(W)=\sum_{N\in{\rm
Mat}(g,h;\mathfrak{O}_{K})}\exp\left(\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{N}WNP\right)\right)$

on $\mathbb{H}_{I}^{(g)}$ associated with the imaginary quadratic field $K$.
The right hand side is an absolutely and uniformly convergent series on any
compact set in $\mathbb{H}_{I}^{(g)}$.

Additionally, we define the theta function with characteristics
$A_{0},B_{0}\in{\rm Mat}(g,h;K)$

$\displaystyle\Theta^{P}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(W)=\sum_{N\in{\rm
Mat}(g,h;\mathfrak{O}_{K})}\exp\left(\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{(N+A_{0})}W(N+A_{0})P\right)+2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{(N+A_{0})}B_{0}\right)\right)$ (1.2)

on $\mathbb{H}_{I}^{(g)}$ associated with $K$.

Here, let us briefly see the convergence of (1.2) just in case. Since $P$ is a
positive-definite Hermitian matrix, there is a unitary matrix $U$ such that
$\overline{U}P{}^{t}U=\text{diag}(\gamma_{1},\ldots,\gamma_{h})$ with
$\gamma_{j}>0$, where $\text{diag}(\gamma_{1},\ldots,\gamma_{h})$ is the
diagonal matrix with diagonal entries $\gamma_{1},\ldots,\gamma_{h}.$ Then,
putting $N^{\prime}=(\nu_{1},\ldots,\nu_{h})=(N+A_{0}){}^{t}U$, we have ${\rm
Tr}\left({}^{t}\overline{(N+A_{0})}W(N+A_{0})P\right)=\sum_{j=1}^{h}\gamma_{j}{}^{t}\overline{\nu_{j}}W\nu_{j},$
where ${\rm Tr}$ is the matrix trace. Thereby, from (1.1), we obtain

$\displaystyle\sum_{N\in{\rm
Mat}(g,h;\mathfrak{O}_{K})}\Big{|}\exp\left(\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{(N+A_{0})}W(N+A_{0})P\right)+2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{(N+A_{0})}B_{0}\right)\right)\Big{|}$
$\displaystyle=\prod_{j=1}^{h}\sum_{\nu_{j}}\exp\left(-\gamma_{j}{}^{t}\overline{\nu_{j}}\frac{W-{}^{t}\overline{W}}{2\sqrt{-1}}\nu_{j}\right).$

According to (0.5), we can compare the right hand side of (1.2) with a product
of the Gaussian integrals $\int_{\mathbb{R}}e^{-X^{2}}dX<\infty$. Thus, we can
check the convergence of (1.2).

We set

$\displaystyle\widehat{B}=\begin{cases}B\quad\quad&(-d\not\equiv 1\hskip
2.84526pt(\text{mod }4)),\\\ 2B\quad\quad&(-d\equiv 1\hskip
2.84526pt(\text{mod }4)),\\\ \end{cases}$

for any $B\in{\rm Mat}(g,h;K)$.

Using the above notations, we have the following theorem.

######  Theorem 1.1.

(Riemann type relation of theta functions on $\mathbb{H}_{I}^{(g)}$) Let $K$
be an imaginary quadratic field. Let $A_{0},B_{0}\in{\rm Mat}(g,h;K)$. Suppose
$(T,P)$ is a pair of matrices satisfying

  * (i)

$T\in{\rm GL}_{h}(K)$,

  * (ii)

$P\in{\rm Mat}(h,h;\mathbb{C})$ is a positive-definite Hermitian matrix.

Also, let $Q$ be a positive-definite Hermitian matrix given by
$Q={}^{t}\overline{T}PT$. Then, the theta function (1.2) associated with $K$
satisfies the relation

$\displaystyle\Theta^{Q}\begin{bmatrix}A_{0}{}^{t}\overline{T}^{-1}\\\
B_{0}T\end{bmatrix}(W)=\frac{1}{{}^{\sharp}G_{2}}\sum_{B\in G_{2}}\sum_{A\in
G_{1}}\exp\left(-2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{A_{0}}\widehat{B}\right)\right)\cdot\Theta^{P}\begin{bmatrix}A_{0}+A\\\
B_{0}+\widehat{B}\end{bmatrix}(W).$ (1.3)

Here, $G_{1}$ and $G_{2}$ are finite abelian groups defined as

  * •

$G_{1}={\rm Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}/\left({\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}\cap{\rm
Mat}(g,h;\mathfrak{O}_{K})\right)$,

  * •

$G_{2}={\rm Mat}(g,h;\mathfrak{O}_{K})T^{-1}/\left({\rm
Mat}(g,h;\mathfrak{O}_{K})T^{-1}\cap{\rm Mat}(g,h;\mathfrak{O}_{K})\right)$,

and ${}^{\sharp}G_{2}$ denotes the order of $G_{2}$.

###### Proof.

By the definition, we have

$\displaystyle\Theta^{Q}\begin{bmatrix}A_{0}{}^{t}\overline{T}^{-1}\\\
B_{0}T\end{bmatrix}(W)$ $\displaystyle=\sum_{N\in{\rm
Mat}(g,h;\mathfrak{O}_{K})}\exp\bigg{(}\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{\left(N+A_{0}{}^{t}\overline{T}^{-1}\right)}W\left(N+A_{0}{}^{t}\overline{T}^{-1}\right)\left({}^{t}\overline{T}PT\right)\right)$
$\displaystyle\hskip 142.26378pt+2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{\left(N+A_{0}{}^{t}\overline{T}^{-1}\right)}B_{0}T\right)\bigg{)}$
$\displaystyle=\sum_{N\in{\rm
Mat}(g,h;\mathfrak{O}_{K})}\exp\Big{(}\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{(N{}^{t}\overline{T}+A_{0})}W(N{}^{t}\overline{T}+A_{0})P\right)+2\pi\sqrt{-1}{\rm
Re}{\rm
Tr}\left({}^{t}\overline{\left(N{}^{t}\overline{T}+A_{0}\right)}B_{0}\right)\Big{)}$
$\displaystyle=\sum_{M\in{\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}}\exp\Big{(}\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}W\left(M+A_{0}\right)P\right)+2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}B_{0}\right)\Big{)}.$
(1.4)

On the other hand, since $A\in G_{1}$, we have

$\displaystyle\sum_{B\in G_{2}}\sum_{A\in G_{1}}\exp\left(-2\pi\sqrt{-1}{\rm
Re}{\rm
Tr}\left({}^{t}\overline{A_{0}}\widehat{B}\right)\right)\cdot\Theta^{P}\begin{bmatrix}A_{0}+A\\\
B_{0}+\widehat{B}\end{bmatrix}(W)$ $\displaystyle=\sum_{B\in G_{2}}\sum_{A\in
G_{1}}\sum_{N\in{\rm Mat}(g,h;\mathfrak{O}_{K})}\exp\Big{(}\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{(N+A_{0}+A)}W(N+A_{0}+A)P\right)$
$\displaystyle\hskip 142.26378pt+2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{(N+A_{0}+A)}\left(B_{0}+\widehat{B}\right)\right)-2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{A_{0}}\widehat{B}\right)\Big{)}$
$\displaystyle=\sum_{B\in G_{2}}\hskip 8.53581pt\sum_{M\in{\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}+{\rm
Mat}(g,h;\mathfrak{O}_{K})}\exp\Big{(}\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}W\left(M+A_{0}\right)P\right)$
$\displaystyle\hskip 196.32384pt+2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}B_{0}\right)+2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{M}\widehat{B}\right)\Big{)}.$ (1.5)

Now, let $M\in{\rm Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}+{\rm
Mat}(g,h;\mathfrak{O}_{K})$. Then, ${\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)\in\mathbb{Z}$ for every $B\in
G_{2}$ if and only if $M\in{\rm Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}$
(see Lemma 1.1 below). We note that the correspondence

$B\mapsto\exp\left(2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)\right)$

gives a character of the finite abelian group $G_{2}$. In fact, letting
$C=LT^{-1}\in{\rm Mat}(g,h;\mathfrak{O}_{K})T^{-1}\cap{\rm
Mat}(g,h;\mathfrak{O}_{K})$ with $L\in{\rm Mat}(g,h;\mathfrak{O}_{K})$, we
have

$\displaystyle\exp\left(2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left({}^{t}\overline{\left(M_{1}{}^{t}\overline{T}+M_{2}\right)}\widehat{C}\right)\right)=\exp\left(2\pi\sqrt{-1}\left({\rm
Re}{\rm Tr}\left({}^{t}\overline{M_{1}}\widehat{L}\right)+{\rm Re}{\rm
Tr}\left({}^{t}\overline{M_{2}}\widehat{C}\right)\right)\right)=1\cdot 1=1$

for any $M_{1},M_{2}\in{\rm Mat}(g,h;\mathfrak{O}_{K}).$ Therefore, it holds

$\displaystyle\sum_{B\in G_{2}}\exp\left(2\pi\sqrt{-1}{\rm Re}{\rm
Tr}\left(M\widehat{B}\right)\right)=\begin{cases}{}^{\sharp}G_{2}\quad&(M\in{\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}),\\\ \hskip
5.69054pt0\quad&(\text{otherwise}).\end{cases}$ (1.6)

Hence, by (1.1) and (1.6), we obtain

$\displaystyle\sum_{B\in G_{2}}\sum_{A\in G_{1}}\exp\left(-2\pi\sqrt{-1}{\rm
Re}{\rm
Tr}({}^{t}\overline{A_{0}}\widehat{B})\right)\Theta^{P}\begin{bmatrix}A_{0}+A\\\
B_{0}+\widehat{B}\end{bmatrix}(W)$
$\displaystyle={}^{\sharp}G_{2}\sum_{M\in{\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}}\exp\left(\pi\sqrt{-1}{\rm
Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}W\left(M+A_{0}\right)P\right)+2\pi\sqrt{-1}{\rm
Re}{\rm Tr}\left({}^{t}\overline{\left(M+A_{0}\right)}B_{0}\right)\right).$
(1.7)

By comparing (1.1) with (1.1), we have (1.3). ∎

######  Lemma 1.1.

Suppose that $M\in{\rm Mat}(g,h;K)$. Then, ${\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)\in\mathbb{Z}$ for every
$B\in{\rm Mat}(g,h;\mathfrak{O}_{K})T^{-1}$ if and only if $M\in{\rm
Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}$.

###### Proof.

Suppose that ${\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)\in\mathbb{Z}$ for all $B\in{\rm
Mat}(g,h;\mathfrak{O}_{K})T^{-1}$. Let us take $B=\ell E_{ab}T^{-1}$, where
$\ell\in\mathfrak{O}_{K}$ and $E_{ab}$ $(1\leq a\leq g,1\leq b\leq h)$ is the
matrix unit. Setting $S=M{}^{t}\overline{T}^{-1}=(s_{jk})\in{\rm Mat}(g,h;K)$,
we have

$\displaystyle{\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)=\begin{cases}{\rm
Re}(\ell\overline{s_{ab}})\quad&(-d\not\equiv 1\hskip 2.84526pt(\text{mod
}4)),\\\ 2{\rm Re}(\ell\overline{s_{ab}})\quad&(-d\equiv 1\hskip
2.84526pt(\text{mod }4)).\end{cases}$ (1.8)

By the assumption, together with (1.8), it follows that
$s_{ab}\in\mathfrak{O}_{K}$ for all $a$ and $b$. Hence,
$M=S{}^{t}\overline{T}\in{\rm Mat}(g,h;\mathfrak{O}_{K}){}^{t}\overline{T}$.

Conversely, by setting $M=S{}^{t}\overline{T}$ and $B=LT^{-1}$ for $S,L\in{\rm
Mat}(g,h;\mathfrak{O}_{K})$, we obtain

$\displaystyle{\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)=\begin{cases}{\rm Re}{\rm
Tr}\left({}^{t}\overline{S}L\right)\quad&(-d\not\equiv 1\hskip
2.84526pt(\text{mod }4)),\\\ 2{\rm Re}{\rm
Tr}\left({}^{t}\overline{S}L\right)\quad&(-d\equiv 1\hskip 2.84526pt(\text{mod
}4)).\end{cases}$

Since $\mathfrak{O}_{K}$ is a ring, ${\rm Re}{\rm
Tr}\left({}^{t}\overline{M}\widehat{B}\right)$ is a rational integer. ∎

######  Remark 1.1.

The groups $G_{1}$ and $G_{2}$ of the theorem are finite, because every entry
$t$ of $T\in{\rm Mat}(h,h;K)$ has an expression $t=\frac{\nu}{n}$ with
$\nu\in\mathfrak{O}_{K}$ and $n\in\mathbb{Z}$. This is why we assume that $T$
is a matrix over $K$.

###  1.2 Backgrounds

For $a,b\in K^{g}$, we have the holomorphic theta function

$\displaystyle\Theta\begin{bmatrix}a\\\
b\end{bmatrix}(W)=\sum_{n\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}\left({}^{t}\overline{(n+a)}W(n+a)\right)+2\pi\sqrt{-1}{\rm
Re}\left({}^{t}\overline{(n+a)}b\right)\right)$ (1.9)

on $\mathbb{H}_{I}^{(g)}$ associated with $K$. By the definition, this
function satisfies

$\displaystyle\Theta\begin{bmatrix}ua\\\
ub\end{bmatrix}(W)=\Theta\begin{bmatrix}a+\nu\\\
b\end{bmatrix}(W)=\Theta\begin{bmatrix}a\\\ b\end{bmatrix}(W),$ (1.10)

for every $\nu\in\mathfrak{O}_{K}^{g}$ and every unit
$u\in\mathfrak{O}_{K}^{\times}$.

######  Remark 1.2.

The theta function $\Theta\begin{bmatrix}a\\\ b\end{bmatrix}(W)$ of (1.9) is
essentially equal to the theta function defined in [Ma2] as

$\displaystyle\widecheck{\Theta}\begin{bmatrix}a\\\
b\end{bmatrix}(W)=\begin{cases}\sum_{n\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}\left({}^{t}\overline{(n+a)}W(n+a)\right)+2\pi\sqrt{-1}{\rm
Re}\left({}^{t}\overline{n}b\right)\right)&\quad(-d\not\equiv 1\hskip
2.84526pt({\rm mod}\hskip 2.84526pt4)),\\\
\sum_{n\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}\left({}^{t}\overline{(n+a)}2W(n+a)\right)+4\pi\sqrt{-1}{\rm
Re}\left({}^{t}\overline{n}b\right)\right)&\quad(-d\equiv 1\hskip
2.84526pt({\rm mod}\hskip 2.84526pt4)).\end{cases}$ (1.11)

If $-d\not\equiv 1\hskip 2.84526pt({\rm mod}\hskip 2.84526pt4)$ ($-d\equiv
1\hskip 2.84526pt({\rm mod}\hskip 2.84526pt4)$, resp.), the difference between
$\widecheck{\Theta}\begin{bmatrix}a\\\ b\end{bmatrix}(W)$ and our theta
function $\Theta\begin{bmatrix}a\\\ b\end{bmatrix}(W)$
($\Theta\begin{bmatrix}a\\\ 2b\end{bmatrix}(2W)$, resp.) is the factor
$e^{2\pi\sqrt{-1}ab}$ ($e^{4\pi\sqrt{-1}ab}$, resp.).

In the following, let us see the reason why we call the equation (1.3) the
Riemann type relation while retracing the origin. Let $I_{h}$ be the identity
matrix of size $h$.

  * •

Let us start with the simplest case. If $h=1$ and $P=I_{1}$, then
$\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$ of (1.2) is the same
as the theta function of (1.9).

  * •

If $P=I_{h}$, then $\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$
has an expression of a product of $h$ theta functions of (1.9). In practice,
if we suppose $A_{0}=(a_{1},\ldots,a_{h})$ and $B_{0}=(b_{1},\ldots,b_{h})$
with $a_{j},b_{j}\in K^{g}$, by the definition, we have

$\displaystyle\Theta^{I_{h}}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(W)=\sum_{n_{1},\ldots,n_{h}\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}\left(\sum_{j=1}^{h}{}^{t}\overline{(n_{j}+a_{j})}W(n_{j}+a_{j})\right)+2\pi\sqrt{-1}{\rm
Re}\left(\sum_{j=1}^{h}{}^{t}\overline{(n_{j}+a_{j})}b_{j}\right)\right)$
$\displaystyle=\prod_{j=1}^{h}\sum_{n_{j}\in\mathfrak{O}_{K}^{g}}\exp\left(\pi\sqrt{-1}\left({}^{t}\overline{(n_{j}+a_{j})}W(n_{j}+a_{j})\right)+2\pi\sqrt{-1}{\rm
Re}\left({}^{t}\overline{(n_{j}+a_{j})}b_{j}\right)\right)=\prod_{j=1}^{h}\Theta\begin{bmatrix}a_{j}\\\
b_{j}\end{bmatrix}(W).$ (1.12)

Similarly, if $P$ is a diagonal matrix
$\text{diag}(\alpha_{1},\ldots,\alpha_{h})$ with positive rational entries
$\alpha_{j}$, we obtain

$\displaystyle\Theta^{\text{diag}(\alpha_{1},\ldots,\alpha_{h})}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(W)=\prod_{j=1}^{h}\Theta\begin{bmatrix}a_{j}\\\
b_{j}\end{bmatrix}(\alpha_{j}W).$ (1.13)

  * •

Let $\Omega\in\mathfrak{S}_{g}$ and $z\in\mathbb{C}^{g}$. Set

$\displaystyle\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}(z,\Omega)=\sum_{n\in\mathbb{Z}^{g}}\exp\left(\pi\sqrt{-1}\left({}^{t}(n+a)\Omega(n+a)\right)+2\pi\sqrt{-1}\left({}^{t}(n+a)(z+b)\right)\right),$
(1.14)

where $a,b\in\mathbb{Q}^{g}$ (recall (Introduction)). This is the well-known
Riemann theta function with characteristics $a,b$. The theta function (1.9) is
a natural counterpart of (1.14) for $z=0\in\mathbb{C}^{g}$. One can define the
theta functions $\vartheta^{P}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(Z,\Omega)$ for any positive-definite symmetric matrix
$P\in{\rm Mat}(h,h)(\mathbb{R})$ and $Z\in{\rm Mat}(g,h;\mathbb{C})$. For
$(T,P)$ and $Q$, where $P$ and $Q$ are positive-definite symmetric matrices
satisfying $Q={}^{t}TPT$ with $T\in{\rm GL}_{h}(\mathbb{Q})$, one can obtain a
non-trivial relation for $\vartheta^{P}$ and $\vartheta^{Q}$ (see [Mu] Chap.II
Sect.6). In fact, our Theorem 1.1 is a counterpart of that to theta functions
on $\mathbb{H}_{I}$.

  * •

When $g=1$, the Riemann theta function (1.14) is equal to the classical Jacobi
theta function (0.1). In this simple case, the pair $(T_{0},I_{4})$ in the
above sense induces quartic relations of Jacobi theta functions, where $T_{0}$
is the orthogonal matrix of (0.3). They coincide with the classical Riemann
theta relations mentioned in Introduction, like (Introduction).

###  1.3  $\Theta^{P}$ for $P$ over $\mathbb{Q}$

If $P\in{\rm Mat}(h,h;\mathbb{Q})$ is a positive-definite symmetric matrix,
the theta function $\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$
of (1.2) has an expression in terms of $\Theta\begin{bmatrix}a\\\
b\end{bmatrix}(W)$ of (1.9) as the following theorem. This result gives a
natural extension of the facts (• ‣ 1.2) and (1.13).

######  Theorem 1.2.

Let $K$ be an arbitrary imaginary quadratic field. Suppose $P\in{\rm
Mat}(h,h;\mathbb{Q})$ be a positive-definite symmetric matrix. Let
$A_{0},B_{0}\in{\rm Mat}(g,h;K)$. Then, there exist positive rational numbers
$\lambda_{1},\ldots,\lambda_{h}$ such that $\Theta^{P}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(W)$ of (1.2) associated with $K$ is expressed by a
homogeneous polynomial in $\Theta\begin{bmatrix}a\\\
b\end{bmatrix}(\lambda_{j}W)$ ($j\in\\{1,\ldots,h\\}$) of degree $h$ over
$\mathbb{Q}$. Here, $\Theta\begin{bmatrix}a\\\ b\end{bmatrix}(W)$ is the theta
function of (1.9).

###### Proof.

We suppose that the positive-definite symmetric matrix $P$ is given by

$P=\begin{pmatrix}\mu_{1}&R\\\ {}^{t}R&P_{1}\end{pmatrix},$

where $\mu_{1}\in\mathbb{Q},R\in{\rm Mat}(1,h-1;\mathbb{Q})$ and $P_{1}\in{\rm
Mat}(h-1,h-1;\mathbb{Q})$. Since $P$ is positive definite, it follows that
$\mu_{1}>0$ and $P_{1}$ is also positive definite. In fact, we have
${}^{t}\overline{x}Px>0$ for any non-zero $x\in\mathbb{R}^{h}$. So, if we take
$x={}^{t}(x_{1},0,\ldots,0)$ ($x={}^{t}(0,x_{2},\ldots,x_{h})$, resp.), we can
see that $\mu_{1}>0$ ($P_{1}$ is positive definite, resp.) In particular,
$P_{1}$ is a nonsingular matrix. Set

$M=\begin{pmatrix}1&0\\\ -P_{1}^{-1}{}^{t}R&I_{h-1}\end{pmatrix}$

We remark $M\in{\rm Mat}(h,h;\mathbb{Q})$, because $P$ is a matrix over
$\mathbb{Q}$. Then, it holds

$\displaystyle{}^{t}\overline{M}PM=\begin{pmatrix}\mu_{1}-RP_{1}^{-1}{}^{t}R&0\\\
0&P_{1}\end{pmatrix}=:S.$ (1.15)

Obviously, $\det(M)=1$ and $M$ is nonsingular. Setting $K=M^{-1}$, we have

$\displaystyle{}^{t}\overline{K}SK=P.$ (1.16)

Put $\lambda_{1}=\mu_{1}-RP_{1}^{-1}{}^{t}R\in\mathbb{Q}.$ Then, it holds
$\det(S)=\lambda_{1}\det(P_{1})=\det(P)$, for $\det(K)=1$. Since $P_{1}$ is
positive definite, we have $\lambda_{1}>0$. Also, the symmetric matrix $S$ is
positive definite. Thus, especially according to (1.16), the matrices $K,S$
and $P$ satisfy the condition of Theorem 1.1. By virtue of that theorem, we
can obtain an expression of $\Theta^{P}\begin{bmatrix}A_{0}\\\
B_{0}\end{bmatrix}(W)$ given by a linear combination of theta functions
$\Theta^{S}\begin{bmatrix}A\\\ B\end{bmatrix}(W)$ over $\mathbb{Q}$ for
appropriate $A,B\in{\rm Mat}(g,h;K)$. From (1.15) and the definition (1.2) of
our theta functions, each $\Theta^{S}\begin{bmatrix}A\\\ B\end{bmatrix}(W)$ is
equal to a product of $\Theta\begin{bmatrix}a\\\ b\end{bmatrix}(\lambda_{1}W)$
and $\Theta^{P_{1}}\begin{bmatrix}A_{1}\\\ B_{1}\end{bmatrix}(W)$ for some
$a,b\in K^{g}$ and $A_{1},B_{1}\in{\rm Mat}(g,h-1;K)$. From the above,
$\Theta^{P}\begin{bmatrix}A_{0}\\\ B_{0}\end{bmatrix}(W)$ is given by a linear
combination of $\Theta\begin{bmatrix}a\\\
b\end{bmatrix}(\lambda_{1}W)\cdot\Theta^{P_{1}}\begin{bmatrix}A_{1}\\\
B_{1}\end{bmatrix}(W)$ over $\mathbb{Q}$. Therefore, by mathematical
induction, we have the assertion. ∎

######  Remark 1.3.

If we consider cases for general positive-definite Hermitian matrices
$P\in{\rm Mat}(h,h;K)$, instead of symmetric matrices over $\mathbb{Q}$, the
statement of the above theorem is not always true. In fact, the counterpart of
the relation (1.15) does not always hold if $P$ is not a matrix over
$\mathbb{Q}$.

The above theorem guarantees that the theta function
$\Theta^{P}\begin{bmatrix}A\\\ B\end{bmatrix}(W)$ of (1.2) has a polynomial
expression in terms of the ordinary theta functions of (1.9), if $P$ is a
positive-definite matrix over $\mathbb{Q}$. Therefore, such a theta function
$\Theta^{P}\begin{bmatrix}A\\\ B\end{bmatrix}(W)$ is useful to obtain non-
trivial relations for theta functions on $\mathbb{H}_{I}^{(g)}$.

##  2 Examples of applications

Recall that the classical Riemann theta relation (Introduction) can be derived
from the combinatorial properties associated with the particular matrix
$T_{0}$ of (0.3). Also, by using appropriate matrices $T$ instead of $T_{0}$,
we can systematically obtain the Riemann type relations such as [b] and [c] of
Introduction. In the author’s opinion, the most interesting aspect of Theorem
1.1 is the fact that the theorem enables us to obtain explicit algebraic theta
relations by just taking suitable pairs $(T,P)$. In this section, we will see
how to apply our main theorem to obtain explicit relations for the theta
functions on $\mathbb{H}_{I}^{(g)}$ by virtue of appropriate $(T,P)$.

Matsumoto [Ma2] obtains non-trivial algebraic relations of theta functions on
$\mathbb{H}_{I}^{(g)}$ for the quadratic fields
$K=\mathbb{Q}\left(\sqrt{-1}\right),\mathbb{Q}\left(\sqrt{-2}\right)$ and
$\mathbb{Q}\left(\sqrt{-3}\right)$. He gives formulas in the form

$\displaystyle\left(\text{a monomial in
}\widecheck{\Theta}\begin{bmatrix}a_{j}\\\
b_{j}\end{bmatrix}(W)\right)=\left(\text{a polynomial in
}\widecheck{\Theta}\begin{bmatrix}a_{k}^{\prime}\\\
b_{k}^{\prime}\end{bmatrix}(W)\right).$ (2.1)

In fact, our main theorem helps us to comprehend the results of [Ma2]. Namely,
by applying Theorem 1.1 to an appropriately chosen unitary matrix $T$ and
$P=I_{h}$, together with (1.13), we can acquire each result of [Ma2]. For
example, let $\widecheck{\Theta}\begin{bmatrix}a\\\ b\end{bmatrix}(W)$ be the
theta function of (1.11) associated with $K=\mathbb{Q}\left(\sqrt{-1}\right)$.
Then, Theorem 1 of [Ma2] is stated as

$\displaystyle 2^{g}\widecheck{\Theta}\begin{bmatrix}a_{1}+a_{2}\\\
b_{1}+b_{2}\end{bmatrix}(W)\cdot\widecheck{\Theta}\begin{bmatrix}a_{1}-a_{2}\\\
b_{1}-b_{2}\end{bmatrix}(W)$
$\displaystyle=\sum_{e,f\in(\frac{1}{1+\sqrt{-1}}\mathfrak{O}_{K}/\mathfrak{O}_{K})^{g}}\exp\left(2\pi\sqrt{-1}{\rm
Re}\left(\left(1+\sqrt{-1}\right){}^{t}(b_{1}+b_{2})\overline{f}\right)\right)$
$\displaystyle\hskip
142.26378pt\cdot\widecheck{\Theta}\begin{bmatrix}e+(1+\sqrt{-1})a_{1}\\\
f+(1+\sqrt{-1})b_{1}\end{bmatrix}(W)\cdot\widecheck{\Theta}\begin{bmatrix}e+(1+\sqrt{-1})a_{2}\\\
f+(1+\sqrt{-1})b_{2}\end{bmatrix}(W),$ (2.2)

where $a_{1},a_{2},b_{1},b_{2}\in K^{g}$. Now, let us consider the matrices
$T,P$ and $Q$ given by

$T=\frac{1-\sqrt{-1}}{2}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},\quad
P=Q=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$

Also, we set $A_{0}=(a_{1}+a_{2},a_{1}-a_{2})$ and
$B_{0}=(b_{1}+b_{2},b_{1}-b_{2})$. Now, $T$ is a unitary matrix:
${}^{t}\overline{T}=T^{-1}=\frac{1+\sqrt{-1}}{2}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}.$ Then, we can see that
$G_{1}=G_{2}=\left(\frac{1}{1+\sqrt{-1}}\mathfrak{O}_{K}/\mathfrak{O}_{K}\right)^{g}.$
Regarding the correspondence between $\widecheck{\Theta}$ and $\Theta$ stated
in Remark 1.2, we can see that our Theorem 1.1 derives (2).

In this section, in order to demonstrate availabilities of our Theorem 1.1, we
will obtain theta relations which are different from (2.1). In particular, the
matrices $T$ need not to be unitary matrices. This section is organized as
follows.

  * •

In Subsection 2.1, we generalize the Riemann type relations suggested in [b],
which induce “half formulas” for the Jacobi or Riemann theta functions. The
results obtained in this subsection are valid for any imaginary quadratic
fields $K$.

  * •

The Cartan matrices of root systems are positive-definite matrices which play
significant roles in various areas in mathematics. In Subsection 2.2, we give
a simple expression of the theta function $\Theta^{A_{h}}$ defined by the use
of the Cartan matrix $A_{h}$ of rank $h$ of the root system of type $A$. The
results obtained in this subsection are valid for any imaginary quadratic
fields $K$ also.

We remark that $\mathbb{Q}\left(\sqrt{-3}\right)$
($\mathbb{Q}\left(\sqrt{-1}\right)$, resp.) is the imaginary quadratic field
with the smallest (second smallest, resp.) discriminant. These imaginary
quadratic fields are characterized by the property that those rings of
integers have non-trivial units. Furthermore, theta functions (1.9) for these
fields are closely related to the moduli spaces of the concrete families of
$K3$ surfaces as in Table 1 in Introduction. Thus, theta functions for
$\mathbb{Q}\left(\sqrt{-3}\right)$ and $\mathbb{Q}\left(\sqrt{-1}\right)$ are
supported by solid motivations to study.

  * •

In Subsection 2.3, we give examples of cubic relations of theta functions for
$K=\mathbb{Q}\left(\sqrt{-3}\right)$.

  * •

In Subsection 2.4, we give examples of quartic relations of theta functions
for $K=\mathbb{Q}\left(\sqrt{-1}\right)$.

In this section, in order to make the expressions of our results simple, we
will study formulas for the cases $B_{0}=O$. It should be reasonable for us to
focus on such cases. It is known that many practically useful relations are
given in terms of theta functions for $B_{0}=O$, like (2.3), (2.4) or (2.5)
below. Also, we are able to know how to apply Theorem 1.1 by considering such
cases.

###  2.1 Particular quadratic theta relations

For the Riemann theta function of (1.14), we set $\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}(\Omega)=\vartheta\begin{bmatrix}a\\\ b\end{bmatrix}(0,\Omega)$
for $\Omega\in\mathfrak{S}_{g}$. It is known that they satisfy a quadratic
equation

$\displaystyle\vartheta\begin{bmatrix}a_{1}\\\
0\end{bmatrix}(\Omega)\cdot\vartheta\begin{bmatrix}a_{2}\\\
0\end{bmatrix}(\Omega)=\sum_{d\in(\mathbb{Z}/2\mathbb{Z})^{g}}\vartheta\begin{bmatrix}\frac{1}{2}(d+a_{1}+a_{2})\\\
0\end{bmatrix}(2\Omega)\cdot\vartheta\begin{bmatrix}\frac{1}{2}(d+a_{1}-a_{2})\\\
0\end{bmatrix}(2\Omega)$ (2.3)

(see [Mu] Chap.II Sect.6). There are various interesting applications of
(2.3). For example, let us consider the case of $g=1$ and let
$\tau\in\mathbb{H}$ stands for $\Omega$. Then, $\vartheta\begin{bmatrix}0\\\
0\end{bmatrix}(\tau)$ ($\vartheta\begin{bmatrix}\frac{1}{2}\\\
0\end{bmatrix}(\tau)$, resp.) is traditionally denoted by
$\vartheta_{00}(\tau)$ ($\vartheta_{10}(\tau)$, resp.). The relation (2.3)
derives “half formulas”

$\displaystyle\vartheta_{00}^{2}(\tau)=\vartheta_{00}^{2}(2\tau)+\vartheta_{10}^{2}(2\tau),\quad\vartheta_{10}^{2}(\tau)=2\vartheta_{00}(2\tau)\vartheta_{10}(2\tau).$
(2.4)

They are useful. For instance, by transformation formulas for
$\tau\mapsto-\frac{1}{\tau}$ in [Mu] Table V in Chap.I Sect.9, the above
formulas induce “double formulas”

$\displaystyle\vartheta_{00}^{2}(2\tau)=\frac{1}{2}(\vartheta_{00}^{2}(\tau)+\vartheta_{01}^{2}(\tau)),\quad\vartheta_{01}^{2}(2\tau)=\vartheta_{00}(\tau)\vartheta_{01}(\tau).$
(2.5)

We remark that the formulas (2.5) are very important to understand Gauss’s
approach to the arithmetic-geometric means and elliptic integrals (for
example, see [C]).

In this subsection, we will give natural extensions of the above useful
results for the Riemann theta functions.

We can regard the following proposition as a natural extension of (2.3).

######  Proposition 2.1.

For $a\in K^{g}$, let $\Theta\begin{bmatrix}a\\\ 0\end{bmatrix}(W)$ be the
theta function (1.9) associated with $K$. Let $\alpha_{1},\alpha_{2}\in
K^{g}$. Then, one has

$\displaystyle\Theta\begin{bmatrix}\alpha_{1}\\\
0\end{bmatrix}(|\delta|^{2}W)\cdot\Theta\begin{bmatrix}\alpha_{2}\\\
0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}u\in(\mathbb{Z}/2|\delta|^{2}\mathbb{Z})^{g}\\\
v\in(\mathbb{Z}/2\mathbb{Z})^{g}\\\
w\in(\mathbb{Z}/|\delta|^{2}\mathbb{Z})^{g}\end{matrix}}}\Theta\begin{bmatrix}\frac{1}{2\delta}\left(\alpha_{1}\delta+\alpha_{2}+u+v\delta+2w\right)\\\
0\end{bmatrix}(2|\delta|^{2}W)\cdot\Theta\begin{bmatrix}\frac{1}{2\delta}\left(\alpha_{1}\delta-\alpha_{2}+u+v\delta\right)\\\
0\end{bmatrix}(2|\delta|^{2}W).$

###### Proof.

We note that the values of the theta functions in the right hand side of the
formula are independent of the choice of representatives of $u,v,w$. Let us
consider the matrices

$T=\frac{1}{2\overline{\delta}}\begin{pmatrix}\overline{\delta}&1\\\
\overline{\delta}&-1\end{pmatrix},\quad P=\begin{pmatrix}2|\delta|^{2}&0\\\
0&2|\delta|^{2}\end{pmatrix},\quad Q=\begin{pmatrix}|\delta|^{2}&0\\\
0&1\end{pmatrix}.$

Then, we have $T^{-1}=\begin{pmatrix}1&1\\\
\overline{\delta}&-\overline{\delta}\end{pmatrix}\in{\rm
Mat}(2,2;\mathfrak{O}_{K})$. Therefore, due to the definition of the group
$G_{2}$ of Theorem 1.1, it follows that $G_{2}$ is the unit group:
$G_{2}=\\{0\\}.$ From Theorem 1.1, we obtain

$\displaystyle\Theta^{Q}\begin{bmatrix}A_{0}{}^{t}\overline{T}^{-1}\\\
O\end{bmatrix}(W)=\sum_{A\in G_{1}}\Theta^{P}\begin{bmatrix}A_{0}+A\\\
O\end{bmatrix}(W).$ (2.6)

Now, for $m_{1},m_{2}\in\mathfrak{O}_{K}^{g}$, we have

$(m_{1},m_{2}){}^{t}\overline{T}=\left(\frac{m_{1}\delta+m_{2}}{2\delta},\frac{m_{1}\delta-
m_{2}}{2\delta}\right)=\left(\frac{r+2m_{2}}{2\delta},\frac{r}{2\delta}\right),$

where $r=m_{1}\delta-m_{2}$. Since $r$ and $m_{2}$ attain arbitrary elements
of $\mathfrak{O}_{K}^{g}=\left(\mathbb{Z}+\delta\mathbb{Z}\right)^{g}$, we
obtain

$G_{1}=\left\\{\left(\frac{u+v\delta+2w}{2\delta},\frac{u+v\delta}{2\delta}\right)\hskip
2.84526pt\bigg{|}\hskip
2.84526ptu\in(\mathbb{Z}/2|\delta|^{2}\mathbb{Z})^{g},v\in(\mathbb{Z}/2\mathbb{Z})^{g},w\in(\mathbb{Z}/|\delta|\mathbb{Z})^{g}\right\\}.$

If $A_{0}{}^{t}\overline{T}^{-1}=(\alpha_{1},\alpha_{2})$, then
$A_{0}=\left(\frac{\alpha_{1}\delta+\alpha_{2}}{2\delta},\frac{\alpha_{1}\delta-\alpha_{2}}{2\delta}\right).$
Hence, by (2.6), together with (• ‣ 1.2) and (1.13), we have the assertion. ∎

The next proposition gives another extension of (2.3).

######  Proposition 2.2.

One has

$\displaystyle\Theta\begin{bmatrix}\alpha_{1}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{2}\\\ 0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}u\in(\mathbb{Z}/2|\delta|^{2}\mathbb{Z})^{g}\\\
v\in(\mathbb{Z}/2\mathbb{Z})^{g}\\\
w\in(\mathbb{Z}/|\delta|^{2}\mathbb{Z})^{g}\end{matrix}}}\Theta\begin{bmatrix}\frac{1}{2\delta}\left(\alpha_{1}+\alpha_{2}+u+v\delta+2w\right)\\\
0\end{bmatrix}(2|\delta|^{2}W)\cdot\Theta\begin{bmatrix}\frac{1}{2\delta}\left(\alpha_{1}-\alpha_{2}+u+v\delta\right)\\\
0\end{bmatrix}(2|\delta|^{2}W).$

###### Proof.

Let us consider the matrices

$T=\frac{1}{2\overline{\delta}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},\quad
P=\begin{pmatrix}2|\delta|^{2}&0\\\ 0&2|\delta|^{2}\end{pmatrix},\quad
Q=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$

Then, we have $T^{-1}=\overline{\delta}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\in{\rm Mat}(2,2;\mathfrak{O}_{K})$. By an argument similar
to the proof of Proposition 2.1 (1), we have the formula in question. In
particular, we can see that the finite groups $G_{1}$ and $G_{2}$ are the same
as those of Proposition 2.1. ∎

###  2.2 Expressions of theta functions related with the root system $A_{h}$

Let us denote the Cartan matrix of the root lattice of type $A_{h}$ by the
letter expressing the type, i.e.,

$A_{h}=\begin{pmatrix}2&-1&\cdots&0&0\\\ -1&2&\cdots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&2&-1\\\
0&0&\cdots&-1&2\end{pmatrix}.$

In this subsection, we will give an expression of the theta function of (1.2)
coming from the real positive-definite matrix $P=A_{h}$.

######  Proposition 2.3.

For $\alpha_{1},\ldots,\alpha_{h}\in K^{g}$, let
$\Theta^{A_{h}}\begin{bmatrix}(\alpha_{1},\alpha_{2},\ldots,\alpha_{h})\\\
O\end{bmatrix}(W)$ be the theta function (1.2) for $P=A_{h}$ associated with
$K$. Then, one has the expression

$\displaystyle\Theta^{A_{h}}\begin{bmatrix}(\alpha_{1},\alpha_{2},\ldots,\alpha_{h})\\\
O\end{bmatrix}(W)$
$\displaystyle=\sum_{u_{1},v_{1}}\cdots\sum_{u_{h},v_{h}}\Theta\begin{bmatrix}\left(\frac{\alpha_{1}+u_{1}+v_{1}\delta}{h\delta}\right)\\\
0\end{bmatrix}((h+1)h|\delta|^{2}W)\cdot\Theta\begin{bmatrix}\left(-\frac{\alpha_{1}+u_{1}+v_{1}\delta}{h\delta}+\frac{\alpha_{2}+u_{2}+v_{2}\delta}{(h-1)\delta}\right)\\\
0\end{bmatrix}(h(h-1)|\delta|^{2}W)$ $\displaystyle\hskip
142.26378pt\cdots\Theta\begin{bmatrix}\left(-\frac{\alpha_{h-1}+u_{h-1}+v_{h-1}\delta}{2\delta}+\frac{\alpha_{h}+u_{h}}{\delta}\right)\\\
0\end{bmatrix}(2|\delta|^{2}W).$

Here, the summation $\sum_{u_{1},v_{1}}\cdots\sum_{u_{h},v_{h}}$ extends over
all $u_{j}\in(\mathbb{Z}/|\delta|^{2}(h+1-j)\mathbb{Z})^{g}$ and
$v_{j}\in(\mathbb{Z}/(h+1-j)\mathbb{Z})^{g}$ for $j\in\\{1,\ldots,h\\}$.

###### Proof.

Set

$\displaystyle T_{h}=\begin{pmatrix}\frac{1}{h}&0&0&\cdots&&0&0\\\
-\frac{1}{h}&\frac{1}{h-1}&0&\cdots&&0&0\\\
0&-\frac{1}{h-1}&\frac{1}{h-2}&\cdots&&0&0\\\
0&0&-\frac{1}{h-2}&\ddots&&0&0\\\ \vdots&\vdots&&&&\vdots&\vdots\\\
0&0&0&\cdots&&\frac{1}{2}&0\\\ 0&0&0&\cdots&&-\frac{1}{2}&1\\\
\end{pmatrix},\quad S_{h}=\begin{pmatrix}h&0&0&\cdots&0&0\\\
h-1&h-1&0&\cdots&0&0\\\ h-2&h-2&h-2&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 2&2&2&\cdots&2&0\\\
1&1&1&\cdots&1&1\end{pmatrix}$

and

$\displaystyle P_{h}=\begin{pmatrix}(h+1)h&0&\cdots&0\\\ 0&h(h-1)&\cdots&0\\\
\vdots&\vdots&\ddots&0\\\ 0&0&\cdots&2\cdot 1\end{pmatrix}.$

Let us study the matrices

$T=\frac{1}{\overline{\delta}}T_{h},\quad P=|\delta|^{2}P_{h},\quad Q=A_{h}.$

Then, we have $T^{-1}=\overline{\delta}S_{h}\in{\rm
Mat}(2,2;\mathfrak{O}_{K})$. So, the group $G_{2}$ of Theorem 1.1 is equal to
$\\{0\\}$. Also, if we take $(m_{1},\ldots,m_{h})\in{\rm
Mat}(g,h;\mathfrak{O}_{K})$, we have

$\displaystyle(m_{1},m_{2},\ldots,m_{h}){}^{t}\overline{T}=\frac{1}{\delta}\left(\frac{m_{1}}{h},-\frac{m_{1}}{h}+\frac{m_{2}}{h-1},-\frac{m_{2}}{h-1}+\frac{m_{3}}{h-2},\ldots,-\frac{m_{h-1}}{2}+m_{h}\right).$

Recall that $\mathfrak{O}_{K}=\mathbb{Z}+\delta\mathbb{Z}$. Considering the
fact that integers $k$ and $k+1$ are coprime if $k\geq 2$, the group $G_{1}$
of Theorem 1.1 is given by

$\displaystyle G_{1}$
$\displaystyle=\bigg{\\{}\frac{1}{\delta}\left(\frac{u_{1}+v_{1}\delta}{h},-\frac{u_{1}+v_{1}\delta}{h}+\frac{u_{2}+v_{2}\delta}{h-1},\ldots,-\frac{u_{h-1}+v_{h-1}\delta}{2}+u_{h}\right)$
$\displaystyle\hskip 142.26378pt\Big{|}\hskip
2.84526ptu_{j}\in\mathbb{Z}/(|\delta|^{2}(h+1-j)\mathbb{Z})^{g},v_{j}\in(\mathbb{Z}/(h+1-j)\mathbb{Z})^{g}\bigg{\\}}.$

Moreover, if we set
$A_{0}{}^{t}\overline{T}^{-1}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{h})$, then
$A_{0}=\frac{1}{\delta}\left(\frac{\alpha_{1}}{h},-\frac{\alpha_{1}}{h}+\frac{\alpha_{2}}{h-1},\ldots,-\frac{\alpha_{h-1}}{2}+\alpha_{h}\right)$.
Hence, we have the equation in question. ∎

###  2.3 Examples of cubic theta relations for
$\mathbb{Q}\left(\sqrt{-3}\right)$

In this subsection, we set $d=3$, $K=\mathbb{Q}\left(\sqrt{-3}\right)$,
$\mathfrak{O}_{K}=\mathbb{Z}[\omega]$, where $\omega=\frac{-1+\sqrt{-3}}{2}$.
Dern and Krieg [DK] study theta functions on $\mathbb{H}_{I}^{(2)}$ associated
with the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-3}\right)$. Those
theta functions are closely related to the period mapping for a family of
lattice polarized $K3$ surfaces (see [NS]). Thus, theta functions (1.9)
associated with $K=\mathbb{Q}\left(\sqrt{-3}\right)$ are interesting objects.

######  Proposition 2.4.

Let $K=\mathbb{Q}\left(\sqrt{-3}\right)$. Take
$\alpha_{1},\alpha_{2},\alpha_{3}\in K^{g}$. For $a\in K^{g}$, let
$\Theta\begin{bmatrix}a\\\ 0\end{bmatrix}(W)$ be the theta function (1.9)
associated with $K$. Then, it holds

$\displaystyle\Theta\begin{bmatrix}\alpha_{1}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{2}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{3}\\\ 0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}r\in(\mathfrak{O}_{K}/3\mathfrak{O}_{K})^{g}\\\
s\in(\sqrt{-3}\mathfrak{O}_{K}/3\mathfrak{O}_{K})^{g}\end{matrix}}}\Theta\begin{bmatrix}\frac{\alpha_{1}+\omega^{2}\alpha_{2}+\omega\alpha_{3}+r}{3}\\\
0\end{bmatrix}(3W)\cdot\Theta\begin{bmatrix}\frac{\alpha_{1}+\omega\alpha_{2}+\omega^{2}\alpha_{3}+r+s}{3}\\\
0\end{bmatrix}(3W)\cdot\Theta\begin{bmatrix}\frac{\alpha_{1}+\alpha_{2}+\alpha_{3}+r-s}{3}\\\
0\end{bmatrix}(3W).$ (2.7)

###### Proof.

Take the matrices

$T=\frac{1}{3}\begin{pmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\\\ \end{pmatrix},\quad P=3I_{3},\quad Q=I_{3}.$

Since $T^{-1}=3{}^{t}\overline{T}=\begin{pmatrix}1&1&1\\\
1&\omega^{2}&\omega\\\ 1&\omega&\omega^{2}\\\ \end{pmatrix}\in{\rm
Mat}(3,3;\mathfrak{O}_{K})$, the finite group $G_{2}$ of Theorem 1.1 is equal
to $\\{0\\}$. Also, for $(m_{1},m_{2},m_{3})\in{\rm
Mat}(g,3;\mathfrak{O}_{K}),$ we have
$(m_{1},m_{2},m_{3}){}^{t}\overline{T}=\frac{1}{3}(m_{1}+m_{2}+m_{3},m_{1}+\omega^{2}m_{2}+\omega
m_{3},m_{1}+\omega m_{2}+\omega^{2}m_{3}).$ Now, $1+\omega+\omega^{2}=0$.
Then, setting $r=m_{1}+\omega^{2}m_{2}+\omega_{1}m_{3}$ and
$r^{\prime}=m_{1}+\omega m_{2}+\omega^{2}m_{3}$, we obtain

$\displaystyle m_{1}+m_{2}+m_{3}=3m_{1}+(-m_{1}-\omega^{2}m_{2}-\omega
m_{3})+(-m_{1}-\omega m_{2}-\omega^{2}m_{3})\equiv-
r-r^{\prime}\quad\quad(\text{mod }3\mathfrak{O}_{K}^{g}).$

Moreover, set $s=r^{\prime}-r=\sqrt{-3}(m_{2}-m_{3})$. Note that $m_{2}-m_{3}$
can take any values of $\mathfrak{O}_{K}^{g}$. Therefore, the finite group
$G_{1}$ of Theorem 1.1 becomes

$G_{1}=\left\\{\frac{1}{3}(-2r-s,r,r+s)\hskip 2.84526pt\Big{|}\hskip
2.84526ptr\in(\mathfrak{O}_{K}/3\mathfrak{O}_{K})^{g},s\in(\sqrt{-3}\mathfrak{O}_{K}/3\mathfrak{O}_{K})^{g}\right\\}.$

If we set
$A_{0}{}^{t}\overline{T}^{-1}=(\alpha_{1},\alpha_{2},\alpha_{3})\in{\rm
Mat}(g,3;\frac{1}{3}\mathfrak{O}_{K}/\mathfrak{O}_{K})$, we obtain
$A_{0}=\frac{1}{3}(\alpha_{1}+\alpha_{2}+\alpha_{3},\alpha_{1}+\omega^{2}\alpha_{2}+\omega\alpha_{3},\alpha_{1}+\omega\alpha_{2}+\omega^{2}\alpha_{3}).$
Since $-2r\equiv r$ $(\text{mod }3\mathfrak{O}_{K})$ for any
$r\in\mathfrak{O}_{K}$, we have the formula in question. ∎

If we take particular $\alpha_{1},\alpha_{2},\alpha_{3}\in K^{g}$, the
equation (2.4) becomes simple. For example, by putting
$\alpha_{1}=\alpha_{2}=\alpha_{3}={}^{t}(v_{1},\ldots,v_{g})\in K^{g}$, we
have the following equation:

$\displaystyle\Theta^{3}\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}\\\ 0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}r_{1},\ldots,r_{g}\in\mathfrak{O}_{K}/3\mathfrak{O}_{K}\\\
s_{1},\ldots,s_{g}\in\sqrt{-3}\mathfrak{O}_{K}/3\mathfrak{O}_{K}\end{matrix}}}\Theta\begin{bmatrix}\frac{1}{3}\begin{pmatrix}r_{1}\\\
\vdots\\\ r_{g}\end{pmatrix}\\\
0\end{bmatrix}(3W)\cdot\Theta\begin{bmatrix}\frac{1}{3}\begin{pmatrix}r_{1}+s_{1}\\\
\vdots\\\ r_{g}+s_{g}\end{pmatrix}\\\
0\end{bmatrix}(3W)\cdot\Theta\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}+\frac{1}{3}\begin{pmatrix}r_{1}-s_{1}\\\ \vdots\\\
r_{g}-s_{g}\end{pmatrix}\\\ 0\end{bmatrix}(3W).$ (2.8)

Additionally, by substituting $\alpha_{1}={}^{t}(v_{1},\ldots,v_{g}),$
$\alpha_{2}={}^{t}(v_{1},\ldots,v_{g})+\frac{1}{\sqrt{-3}}{}^{t}(1,\ldots,1)$
and
$\alpha_{3}={}^{t}(v_{1},\ldots,v_{g})-\frac{1}{\sqrt{-3}}{}^{t}(1,\ldots,1)$,
we have the following equation:

$\displaystyle\Theta\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}+\frac{1}{\sqrt{-3}}\begin{pmatrix}1\\\ \vdots\\\
1\end{pmatrix}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}-\frac{1}{\sqrt{-3}}\begin{pmatrix}1\\\ \vdots\\\
1\end{pmatrix}\\\ 0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}r_{1},\ldots,r_{g}\in\mathfrak{O}_{K}/3\mathfrak{O}_{K}\\\
s_{1},\ldots,s_{g}\in\sqrt{-3}\mathfrak{O}_{K}/3\mathfrak{O}_{K}\end{matrix}}}\Theta\begin{bmatrix}\frac{1}{3}\begin{pmatrix}-1\\\
\vdots\\\ -1\end{pmatrix}+\frac{1}{3}\begin{pmatrix}r_{1}\\\ \cdots\\\
r_{g}\end{pmatrix}\\\
0\end{bmatrix}(3W)\cdot\Theta\begin{bmatrix}\frac{1}{3}\begin{pmatrix}1\\\
\vdots\\\ 1\end{pmatrix}+\frac{1}{3}\begin{pmatrix}r_{1}+s_{1}\\\ \vdots\\\
r_{g}+s_{g}\end{pmatrix}\\\ 0\end{bmatrix}(3W)$ $\displaystyle\hskip
284.52756pt\cdot\Theta\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \vdots\\\
v_{g}\end{pmatrix}+\frac{1}{3}\begin{pmatrix}r_{1}-s_{1}\\\ \vdots\\\
r_{g}-s_{g}\end{pmatrix}\\\ 0\end{bmatrix}(3W).$ (2.9)

As mentioned previously, the Riemann theta functions (1.14) whose
characteristics are in $\frac{1}{2}\mathbb{Z}/\mathbb{Z}$ have ample amount of
applications (for example, see the beginning of Section 2.1). Similarly, it
seems natural to study theta function of Proposition 2.4 whose characteristics
are in $\frac{1}{3}\mathfrak{O}_{K}/\mathfrak{O}_{K}.$ Remark that
$\mathfrak{O}_{K}/3\mathfrak{O}_{K}=\left\\{0,\pm 1,\pm\sqrt{-3},\pm
1\pm\sqrt{-3}\right\\}\simeq(\mathbb{Z}/3\mathbb{Z})\oplus(\mathbb{Z}/3\mathbb{Z})$
and
$\sqrt{-3}\mathfrak{O}_{K}/3\mathfrak{O}_{K}=\left\\{0,\pm\sqrt{-3}\right\\}\simeq(\mathbb{Z}/3\mathbb{Z}).$
Now, let $\rho_{j,k}\in\mathbb{Z}/3\mathbb{Z}=\\{0,1,-1\\}$ for
$j\in\\{1,\ldots,g\\},k\in\\{1,2\\}$. We introduce a theta function labeled by
$\frac{1}{3}\mathbb{Z}/\mathbb{Z}$:

$\displaystyle\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}\\\ \vdots&\vdots\\\
\rho_{g1},&\rho_{g2}\end{Bmatrix}(W)=\Theta\begin{bmatrix}\begin{pmatrix}\frac{1}{3}\rho_{11}+\frac{1}{\sqrt{-3}}\rho_{12}\\\
\vdots\\\ \frac{1}{3}\rho_{g1}+\frac{1}{\sqrt{-3}}\rho_{g2}\end{pmatrix}\\\
0\end{bmatrix}(W).$ (2.10)

The function (2.10) simplifies the relations (2.4), if
$\alpha_{1},\alpha_{2},\alpha_{3}\in\frac{1}{3}\mathfrak{O}_{K}^{g}$. In fact,
the all characteristics appearing in (2.4) are closed in
$\frac{1}{3}\mathfrak{O}_{K}/\mathfrak{O}_{K}$. For example, in the simplest
case such that $v_{1}=\cdots=v_{g}=0$, the equation (2.3) becomes

$\displaystyle\left(\Theta\begin{Bmatrix}0,&0\\\ \vdots&\vdots\\\
0,&0\end{Bmatrix}(W)\right)^{3}=\sum_{\rho_{11},\rho_{12},\ldots,\rho_{g1},\rho_{g2}\in\\{0,1,-1\\}}\sum_{\sigma_{12},\ldots,\sigma_{g2}\in\\{0,1,-1\\}}$
$\displaystyle\hskip 142.26378pt\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}\\\
\vdots&\vdots\\\
\rho_{g1},&\rho_{g2}\end{Bmatrix}(3W)\cdot\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}+\sigma_{12}\\\
\vdots&\vdots\\\
\rho_{g1},&\rho_{g2}+\sigma_{g2}\end{Bmatrix}(3W)\cdot\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}-\sigma_{12}\\\
\vdots&\vdots\\\ \rho_{g1},&\rho_{g2}-\sigma_{g2}\end{Bmatrix}(3W).$

Similarly, if we put $v_{1}=\cdots=v_{g}=0$, the equation (2.3) becomes

$\displaystyle\Theta\begin{Bmatrix}0,&0\\\ \vdots&\vdots\\\
0,&0\end{Bmatrix}(W)\cdot\Theta\begin{Bmatrix}0,&1\\\ \vdots&\vdots\\\
0,&1\end{Bmatrix}(W)\cdot\Theta\begin{Bmatrix}0,&-1\\\ \vdots&\vdots\\\
0,&-1\end{Bmatrix}(W)$
$\displaystyle=\sum_{\rho_{11},\rho_{12},\ldots,\rho_{g1},\rho_{g2}\in\\{0,1,-1\\}}\sum_{\sigma_{12},\ldots,\sigma_{g2}\in\\{0,1,-1\\}}$
$\displaystyle\hskip 71.13188pt\Theta\begin{Bmatrix}\rho_{11}-1,&\rho_{12}\\\
\vdots&\vdots\\\
\rho_{g1}-1,&\rho_{g2}\end{Bmatrix}(3W)\cdot\Theta\begin{Bmatrix}\rho_{11}+1,&\rho_{12}+\sigma_{12}\\\
\vdots&\vdots\\\
\rho_{g1}+1,&\rho_{g2}+\sigma_{g2}\end{Bmatrix}(3W)\cdot\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}-\sigma_{12}\\\
\vdots&\vdots\\\ \rho_{g1},&\rho_{g2}-\sigma_{g2}\end{Bmatrix}(3W).$

Although the above relations can be expressed in simpler forms via (1.10), we
will not pursue them in this paper.

###  2.4 Examples of quartic theta relations for
$\mathbb{Q}\left(\sqrt{-1}\right)$

In this subsection, we set $d=1$, $K=\mathbb{Q}\left(\sqrt{-1}\right)$,
$\mathfrak{O}_{K}=\mathbb{Z}[i]$, where $i=\sqrt{-1}$. Let us study the theta
functions (1.2) associated with $\mathbb{Q}\left(\sqrt{-1}\right)$. Since
$\mathfrak{O}_{K}$ is just the ring of Gaussian integers, there are already
various research for such theta functions (see [F], [Ma1], [MT]). We give the
following theta relation as an application of Theorem 1.1.

######  Proposition 2.5.

Let $K=\mathbb{Q}\left(\sqrt{-1}\right)$. Take
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in K^{g}$. For $a\in K^{g}$, let
$\Theta\begin{bmatrix}a\\\ 0\end{bmatrix}(W)$ be the theta function (1.9)
associated with $K$. Then, it holds

$\displaystyle\Theta\begin{bmatrix}\alpha_{1}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{2}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{3}\\\
0\end{bmatrix}(W)\cdot\Theta\begin{bmatrix}\alpha_{4}\\\ 0\end{bmatrix}(W)$
$\displaystyle=\sum_{\scriptsize{\begin{matrix}r\in(\mathfrak{O}_{K}/4\mathfrak{O}_{K})^{g}\\\
s_{1},s_{2}\in(2\mathfrak{O}_{K}/4\mathfrak{O}_{K})^{g}\end{matrix}}}\Theta\begin{bmatrix}\frac{\alpha_{1}-i\alpha_{2}-i\alpha_{3}-\alpha_{4}+r}{4}\\\
0\end{bmatrix}(4W)\cdot\Theta\begin{bmatrix}\frac{-i\alpha_{1}+\alpha_{2}-\alpha_{3}-i\alpha_{4}-ir+s_{1}}{4}\\\
0\end{bmatrix}(4W)$ $\displaystyle\hskip
142.26378pt\cdot\Theta\begin{bmatrix}\frac{i\alpha_{1}+\alpha_{2}-\alpha_{3}+i\alpha_{4}-ir+s_{2}}{4}\\\
0\end{bmatrix}(4W)\cdot\Theta\begin{bmatrix}\frac{\alpha_{1}+i\alpha_{2}+i\alpha_{3}-\alpha_{4}-r-is_{1}+is_{2}}{4}\\\
0\end{bmatrix}(4W).$

###### Proof.

Let us consider the matrices

$T=\frac{1}{4}\begin{pmatrix}1&i&i&-1\\\ i&1&-1&i\\\ -i&1&-1&-i\\\
1&-i&-i&-1\end{pmatrix},\quad P=4I_{4},\quad Q=I_{4}.$

Then, we have $T^{-1}=4{}^{t}\overline{T}=\begin{pmatrix}1&-i&i&1\\\
-i&1&1&i\\\ -i&-1&-1&i\\\ -1&-i&i&-1\end{pmatrix}\in{\rm
Mat}(4,4;\mathfrak{O}_{K}).$ Thereby, the group $G_{2}$ of Theorem 1.1 is
equal to $\\{0\\}.$ Letting $(m_{1},m_{2},m_{3},m_{4})\in{\rm
Mat}(g,4;\mathfrak{O}_{K})$, we have
$(m_{1},m_{2},m_{3},m_{4}){}^{t}\overline{T}=\frac{1}{4}(r,r^{\prime},r^{\prime\prime},r^{\prime\prime\prime}),$
where

$\displaystyle\begin{cases}&r=m_{1}-im_{2}-im_{3}-m_{4},\\\
&r^{\prime}=-im_{1}+m_{2}-m_{3}-im_{4},\\\
&r^{\prime\prime}=im_{1}+m_{2}-m_{3}+im_{4},\\\
&r^{\prime\prime\prime}=m_{1}+im_{2}+im_{3}-m_{4}.\end{cases}$

Then, $ir^{\prime}-r=2(im_{2}+m_{4})$ and
$-ir^{\prime\prime}+r=2(m_{1}-im_{2})$ hold. Here, $im_{2}+m_{4}$ and
$m_{1}-im_{2}$ can take any values of $\mathfrak{O}_{K}^{g}$ respectively.
Also,

$r+ir^{\prime}-ir^{\prime\prime}+r^{\prime\prime\prime}=4m_{1}\equiv
0\quad(\text{mod }4\mathfrak{O}_{K}^{g}).$

Therefore, the finite group $G_{1}$ of Theorem 1.1 has an expression

$G_{1}=\left\\{\frac{1}{4}(r,-ir+s_{1},-ir+s_{2},-r-s_{1}i+s_{2}i)\hskip
2.84526pt\bigg{|}\hskip
2.84526ptr\in(\mathfrak{O}_{K}/4\mathfrak{O}_{K})^{g},s_{1},s_{2}\in(2\mathfrak{O}_{K}/4\mathfrak{O}_{K})^{g}\right\\}.$

Setting
$A_{0}{}^{t}\overline{T}^{-1}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$,
we obtain
$A_{0}=\frac{1}{4}(\alpha_{1}-i\alpha_{2}-i\alpha_{3}-\alpha_{4},-i\alpha_{1}+\alpha_{2}-\alpha_{3}-i\alpha_{4},i\alpha_{1}+\alpha_{2}-\alpha_{3}+i\alpha_{4},\alpha_{1}+i\alpha_{2}+i\alpha_{3}-\alpha_{4})$.
By summarizing the above, we have the formula in question. ∎

Note that
$\mathfrak{O}_{K}/4\mathfrak{O}_{K}=\\{\rho_{1}+\rho_{2}i\mid\rho_{1},\rho_{2}\in\mathbb{Z}/4\mathbb{Z}\\}.$
Now, we set

$\displaystyle\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}\\\ \vdots&\vdots\\\
\rho_{g1},&\rho_{g2}\\\
\end{Bmatrix}(W)=\Theta\begin{bmatrix}\begin{pmatrix}\frac{1}{4}\rho_{11}+\frac{i}{4}\rho_{12}\\\
\vdots\\\ \frac{1}{4}\rho_{g1}+\frac{i}{4}\rho_{g2}\\\ \end{pmatrix}\\\
0\end{bmatrix}(W),$ (2.11)

for $\rho_{j,k}\in\mathbb{Z}/4\mathbb{Z}=\\{0,1,2,3\\}$ for
$j\in\\{1,\ldots,g\\},k\in\\{1,2\\}$. The expression (2.11) enables us to
obtain simple forms of the equation of the Proposition 2.5, if
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\frac{1}{4}\mathfrak{O}_{K}^{g}$.
For example, if we put $\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=0$, we
obtain an equation

$\displaystyle\left(\Theta\begin{Bmatrix}0,&0\\\ \vdots&\vdots\\\
0,&0\end{Bmatrix}(W)\right)^{4}$
$\displaystyle=\sum_{\rho_{11},\rho_{12},\ldots,\rho_{g1},\rho_{g2}\in\\{0,\pm
1,2\\}}\sum_{\scriptsize\begin{matrix}\sigma_{11},\sigma_{12},\ldots,\sigma_{g1},\sigma_{g2}\in\\{0,2\\}\\\
\sigma^{\prime}_{11},\sigma^{\prime}_{12},\ldots,\sigma^{\prime}_{g1},\sigma^{\prime}_{g2}\in\\{0,2\\}\end{matrix}}$
$\displaystyle\hskip 42.67912pt\Theta\begin{Bmatrix}\rho_{11},&\rho_{12}\\\
\vdots&\vdots\\\
\rho_{g1},&\rho_{g2}\end{Bmatrix}(4W)\cdot\Theta\begin{Bmatrix}\rho_{12}+\sigma_{11},&-\rho_{11}+\sigma_{12}\\\
\vdots&\vdots\\\
\rho_{g2}+\sigma_{g1},&-\rho_{g1}+\sigma_{g2}\end{Bmatrix}(4W)\cdot\Theta\begin{Bmatrix}\rho_{12}+\sigma^{\prime}_{11},&-\rho_{11}+\sigma^{\prime}_{12}\\\
\vdots&\vdots\\\
\rho_{g2}+\sigma^{\prime}_{g1},&-\rho_{g1}+\sigma^{\prime}_{g2}\end{Bmatrix}(4W)$
$\displaystyle\hskip
250.38434pt\cdot\Theta\begin{Bmatrix}-\rho_{11}+\sigma_{12}-\sigma_{12}^{\prime},&-\rho_{12}-\sigma_{11}+\sigma_{11}^{\prime}\\\
\vdots&\vdots\\\
-\rho_{g1}+\sigma_{g2}-\sigma_{g2}^{\prime},&-\rho_{g2}-\sigma_{g1}+\sigma_{g1}^{\prime}\end{Bmatrix}(4W).$

## Acknowledgment

The author is supported by JSPS Grant-in-Aid for Scientific Research
(22K03226), JST FOREST Program (JPMJFR2235) and MEXT LEADER. He is thankful to
Professor Hironori Shiga for valuable discussions. He also appreciates the
reviewer’s accurate suggestions for improvement of the manuscript.

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Atsuhira Nagano
Faculty of Mathematics and Physics,
Institute of Science and Engineering,
Kanazawa University,
Kakuma, Kanazawa, Ishikawa
920-1192, Japan
(E-mail<EMAIL_ADDRESS>