Quantum atmosphere effective radii for different spin fields from quantum
gravity inspired black holes

Douglas M. Gingrich

Department of Physics, University of Alberta, Edmonton, AB T6G 2E1 Canada
TRIUMF, Vancouver, BC V6T 2A3 Canada

e-mail<EMAIL_ADDRESS>
###### Abstract

Quantum atmosphere effective radii for the emission of spin-0, -1/2, -1, and
-2 massless fields from Schwarzschild, Tangherlini, non-commutative geometry
inspired, and polymeric black holes are calculated. The power observed from
the black hole at spatial infinity taking greybody factors into account is
compared to an equal-power black-body radiator of the same temperature but
different effective radius. A large range of different radii are obtained for
different spin fields and black holes. The equal-power black-body effective
radius is not, in general, a good proxy for the location of the quantum
atmosphere.

##  1 Introduction

The Hawking radiation from evaporating black holes is thought to originate
from quantum excitations near the horizon [1]. Giddings [2] has argued that
the radiation originates from an effective radius $r_{\mathrm{A}}$ outside the
horizon radius $r_{\mathrm{H}}$ call the quantum atmosphere:
$r_{\mathrm{A}}-r_{\mathrm{H}}\sim r_{\mathrm{H}}$. It is of interest to test
the validity of Giddings’ claim.

The quantum atmosphere is the location where most of the Hawking radiation
comes from. A few different arguments have been given for the location of the
quantum atmosphere. The thermal wavelength of typical Hawking radiation is
much larger than the horizon size. Heuristic arguments using a gravitational
version of the Schwinger effect for particle production by tidal forces
outside the horizon have been made [3, 4]. Another reasoning uses the $(1+1)$
dimensional renormalized stress-energy tensor [2, 3, 5]. In addition, the
radius can be given by an effective black-body emission surface [2, 6]. In
this paper, we examine the later of these definitions.

Ref. [3, 5] have corroborated Giddings’ conclusion by obtaining
$r_{\mathrm{A}}-r_{\mathrm{H}}\approx r_{\mathrm{H}}$ for the Schwarzschild
black hole using gravitational Schwinger effect arguments and a more precise
calculation using the stress-energy tensor. While the different arguments
agree that the location of the quantum atmosphere is some distance from the
horizon, they do not all give a common estimate for the numerical value.

It is of interest to examine if Giddings’ arguments are applicable to other
types of black holes. Hod [6] showed that the quantum atmosphere radius for a
massless scalar field from a Tangherlini black hole emitting radiation in the
bulk is a decreasing function of the number of space dimensions; Hod finds
$r_{\mathrm{A}}-r_{\mathrm{H}}\ll r_{\mathrm{H}}$ for high number of extra
dimensions. The Reissner-Nordström black hole has also been considered in Ref.
[4]. These metrics give contradicting conclusions to Ref. [2, 3, 5].

In this paper, we calculate exact greybody factors numerically for all spin
fields. The results are used to calculate the double-differential frequency
spectrum which is then integrated over all frequencies to obtain the power. By
equating the power to that of a black body, we determine an effective emission
surface of the quantum atmosphere. The potentials seen by different spin
fields are different so we could expect the quantum atmosphere to depend on
the emitted field’s spin. We find that the apparent radius should not be used,
in general, as a proxy for the location of the quantum atmosphere. For
example, $r_{\mathrm{A}}$ can not be used as a definition for the location of
the quantum atmosphere for gravitons for most black hole metrics we consider.

##  2 Effective radius calculation

The effective potential barrier around a black hole is commonly encoded in a
set of transmission coefficients, greybody factors, that depend on the
properties of the black hole, the properties of the emitted radiation,
frequency and modes of the emitted radiation. A physical observable that can
be formed from the transmission coefficients is the absorption cross section
which is a sum of the transmission coefficients over all radiation modes
divided by the frequency squared. Weighting the absorption cross section by a
temperature-dependent statistical factor corresponding to the spin-statistics
of the emitted radiation gives the radiation flux or power per unit frequency.
By integrating over all energies, the total radiated power or luminosity is
obtained. In the absence of absorption – step-function transmission
coefficients – the Stefan-Boltzmann law is obtained. For black holes, one can
convert the temperature dependence into a dependence on the horizon radius,
and in principle a dependence on the black hole parameters. The power thus
allows a determination of the effects of the transmission coefficients
integrated over all frequencies. By comparing the power generated by a black
hole with the equivalent power from a black body seen at spatial infinity, one
obtains and effective area for the black hole, or in the case of spherically
symmetric black holes, an effective radius. The method of equal-power infers
the size of the radiating body.

The calculated power emitted from a black hole seen by an observe at spatial
infinity is compared to the equivalent power $P_{\mathrm{B}}$ from an idealize
black-body radiator in flat space using the generalized Stefan-Boltzmann
relation (see for example Ref. [7])

$P_{\mathrm{B}}=\sigma A_{n+2}({r_{\mathrm{A}}})T_{\mathrm{B}}^{n+4}\,,$ (1)

where $T_{\mathrm{B}}$ is the black- body temperature,
$A_{n+2}(r_{\mathrm{A}})$ is the surface area of a $(n+4)$-dimensional
emitting body, and $\sigma$ is the appropriate Stefan-Boltzmann constant for
bosons or fermions in $n$ extra dimensions. We use units of
$G=c=\hbar=k_{\mathrm{B}}=1$. Although Eq. (1) is written in the general form
to allow comparison with higher-dimensional black holes, it reduces to the
more familiar form of the Stefan-Boltzmann law when $n=0$.

For a black hole, once the greybody factors $\Gamma_{s,\ell}(\omega)$ for
massless spin field $s$ emitted with spheroidal harmonic mode $\ell$ with
frequency $\omega$ have been calculated, the absorption cross section in four
spacetime dimensions is obtained:

$\sigma_{s}(\omega)=\frac{\pi}{\omega^{2}}\sum_{\ell\geq
s}(2\ell+1)\Gamma_{s,\ell}(\omega)\,.$ (2)

The $(2\ell+1)$ factor is the degeneracy of the axial quantum number or
angular momentum $m$ modes.

The total power in four spacetime dimensions is then given by

$P=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\frac{\omega^{3}\sigma_{s}(\omega)}{\exp(\omega/T)-(-1)^{2s}}\mathrm{d}\omega\,,$
(3)

where $T$ is the Hawking temperature as measured at spatial infinity.

We define the effective radius $r_{\mathrm{A}}$ of the black hole quantum
atmospheres by equating the Hawking radiation power from the black hole Eq.
(3) with the corresponding Stefan-Boltzmann radiation power of a flat space
perfect black-body emitter Eq. (1):

$P(r_{\mathrm{H}},T)=P_{\mathrm{B}}(r_{A},T_{\mathrm{B}})\,.$ (4)

This equation determines the effective radius assuming equal temperature:
$T=T_{\mathrm{B}}$. One could likewise determine the effective temperature of
the filtered radiation by assuming equal radii [8].

Using $A_{n+2}\propto R^{n+2}$, we obtain the effective radius using

$\frac{r_{\mathrm{A}}}{r_{\mathrm{H}}}=\left[\frac{P(r_{\mathrm{H}},T)}{P_{\mathrm{B}}(r_{\mathrm{H}},T)}\right]^{\frac{1}{n+2}}\,,$
(5)

where $P$ depends on the emitted field’s spin and $P_{\mathrm{B}}$ is
different for bosons and fermions. The dimensionless radii
$r_{\mathrm{A}}/r_{\mathrm{H}}$ characterizes the black hole quantum
atmospheres.

As in Ref. [6], it is beneficial to characterize the effective quantum
atmosphere using

$\bar{r}_{\mathrm{A}}=\frac{r_{\mathrm{A}}-r_{\mathrm{H}}}{r_{\mathrm{H}}}\,.$
(6)

Values of $\bar{r}_{\mathrm{A}}\gtrsim 1$ validate Giddings’ argument and
negative values imply the quantum atmosphere is behind the horizon.

##  3 Black holes thermodynamics

In this section, we write down the black-body power for the different metrics
consider. The formula contain only a single polarization for each spin field.
We make no claim about the validity of the two quantum inspired black hole
metrics considered here. They are partly chosen for their different black-body
features and the ease of greybody calculation.

###  3.1 Schwarzschild-Tangherlini black holes

For the Schwarzschild-Tangherlini [9] black hole radiating into the bulk, the
higher dimensional $(n+4)$ black-body power is [7, 6]

$P_{\mathrm{B}}=\sigma A_{n+2}(R)T^{n+4}\,,$ (7)

where the higher-dimensional Stefan-Boltzmann constant is

$\sigma=\frac{(n+3)\Gamma((n+3)/2)\zeta(n+4)}{2\pi^{(n+3)/2+1}}\,,$ (8)

and $\Gamma$ is the gamma function and $\zeta$ is the Riemann zeta function.
The higher-dimensional surface area of the emitting body of radius $R$ is

$A_{n+2}(R)=\frac{2\pi^{(n+3)/2}}{\Gamma((n+3)/2)}R^{n+2}\,.$ (9)

We will also need the black hole temperature

$T=\frac{n+1}{4\pi r_{\mathrm{H}}}\,,$ (10)

where

$r_{\mathrm{H}}=\frac{1}{\sqrt{\pi}M_{*}}\left(\frac{M}{M_{*}}\right)^{1/(n+1)}\left[\frac{8\Gamma((n+3)/2)}{n+2}\right]^{1/(n+1)}\,.$
(11)

The above equations reduce to the familiar Stefan-Boltzmann law and
Schwarzschild black hole when $n=0$, and $M_{*}=\sqrt{\hbar c/G}$ is the
Planck mass.

###  3.2 Non-commutative geometry inspired black holes

Non-commutative geometry inspired black holes are interesting in that the form
of the black-body area of the Schwarzschild-Tangherlini remains unchanged but
the temperature dependence is different [10, 11]. The temperature is given by

$T=\frac{n+1}{4\pi
r_{\mathrm{H}}}\left[1-\frac{2}{n+1}\left(\frac{r_{\mathrm{H}}}{2\sqrt{\theta}}\right)^{n+3}\frac{\mathrm{e}^{-r_{\mathrm{H}}/(4\theta)}}{\gamma\left(\frac{n+3}{2},\frac{r_{\mathrm{H}}^{2}}{4\theta}\right)}\right]\,,$
(12)

where $\gamma$ is the upper incomplete gamma function. The horizon radius is
obtained by solving

$\frac{M}{M_{*}}=\frac{n+2}{8\gamma\left(\frac{n+3}{2},\frac{r^{2}}{4\theta}\right)}(\sqrt{\pi}M_{*}r_{\mathrm{H}})^{n+1}\,.$
(13)

The minimum length parameter $\sqrt{\theta}$ is take to be a free parameter
and could be well above the Planck length. As $\theta\to 0$, the radius and
temperature approach the Tangherlini values. The metric give one, two, or no
horizon. For a single horizon the temperature vanishes and a black hole
remnant is expected to form. The temperature has a maximum but vanishes at the
remnant radius. The non-commutative black hole is similar to the Tangherlini
black hole for large masses.

To model the effects of an effective ultra-violet cut-off in the frequency
$\omega$ of the emitted quanta an additional factor [12] of
$\exp(-\theta\omega^{2}/2)$ should multiply Eq. (3). Although we have included
this factor, it has a small effect.

###  3.3 Polymeric black holes

In loop quantum gravity, semi-classical corrections due to the effects of
quantum gravity have been derived to give a so-called polymer Schwarzschild
black hole [13, 14]. The model has two free parameters $\epsilon$ and $a_{0}$.
The parameter $a_{0}=8\pi A_{\mathrm{min}}$ is related to the minimum area of
loop quantum gravity and is expected to be of the Planck scale.

A positive deformation parameter $\epsilon$ represents the typical scale of
the geometry fluctuations in the Hamiltonian constraints of the theory as they
get renormalized from the Planck scale to the astrophysical scales. It’s
thought that $\epsilon\ll 1$, and values of $\epsilon\lesssim 0.8$ will have
little effect on what follows. For large $\epsilon$, deviations from the
Schwarzschild metric are apparent for astronomical size black holes.

The horizon area is not the usual form but is given by

$A=4\pi(2m)^{2}\left[1+\left(\frac{\sqrt{a_{0}}}{2m}\right)^{4}\right]\,.$
(14)

The temperature is given by

$T=\frac{1}{4\pi(2m)}(1-P(\epsilon)^{2})\left[1+\left(\frac{\sqrt{a_{0}}}{2m}\right)^{4}\right]^{-1}\,,$
(15)

where the polymerization function is

$P(\epsilon)=\frac{\sqrt{1+\epsilon^{2}}-1}{\sqrt{1+\epsilon^{2}}+1}\,.$ (16)

The total integrated power given by the Stefan-Boltzmann law is

$P=\frac{\sigma}{256\pi^{3}}m^{-2}(1-P(\epsilon)^{2})^{4}\left[1+\left(\frac{\sqrt{a_{0}}}{2m}\right)^{4}\right]^{-3}\,,$
(17)

where $\sigma=\pi^{2}/120$ for bosons and $\sigma=7\pi^{2}/960$ for fermions.
In the above equations $m$ is a parameter that is related to the ADM mass $M$
by $M=m(1+P)^{2}$.

##  4 Results

We calculate the quantum atmosphere effective radius for spin-0, -1/2, -1, and
-2 massless fields from two quantum inspired black holes. Our calculations are
numerical and follow the procedures used in Ref [15] which are based on the
general potentials in Ref [16] and the path-ordered matrix exponentials in Ref
[17]. The procedure enables previously rather difficult calculations.

###  4.1 Schwarzschild black hole

We consider the Schwarzschild black holes as a warm-up. Table 1 shows
dimensionless effective radii for all spin fields from a Schwarzschild black
hole. Our numerical calculations reproduce the results of Page [18] for
spin-1, -1/2, -2, and Elster [19] for scalars. In terms of the quantum
atmosphere, the case of spin-1 was first discussed in Ref. [2] and the spin-0
in Ref. [6]. The case of spin-2 shows a breakdown of Gidding’s principle (Ref.
[2] restricted the discussion to $s\leq 1$).

Table 1: Dimensionless radii $\bar{r}_{\mathrm{A}}$ for massless fields of spin $s$ from a Schwarzschild black hole.  $s$ | 0 | 1/2 | 1 | 2
---|---|---|---|---
$\bar{r}_{\mathrm{A}}$ | 1.68 | 1.13 | 0.27 | $-0.57$

For the Schwarzschild black hole, the black-body power is well known to have a
$P\sim M^{-2}$ dependence. We find that including graybody factors, this mass
dependence is maintained, i.e. $\Gamma$ does not introduce any additional $M$
dependence.

###  4.2 Tangherlini black hole

To help validate our procedure, we reproduce a previous result in Ref. [6].
Table 2 shows dimensionless effective radii for scalars from a Tangherlini
black hole radiating in the bulk. We have taken $M=M_{*}=1$. To obtain these
results, we have calculated the emission on the brane and used the bulk-to-
brane emission ratios obtained in Ref. [20]. Our results agree with Ref. [6]
to within the numerical accuracy of the calculations.

Table 2: Dimensionless radii $\bar{r}_{\mathrm{A}}$ for a massless scalar field from a $(n+4)$-dimensional Tangherlini black hole radiating in the bulk.  $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
$\bar{r}_{\mathrm{A}}$ | 0.99 | 0.71 | 0.59 | 0.50 | 0.44 | 0.39 | 0.33

We are now equipped to calculate something new. Table 3 shows dimensionless
effective radii for all spin fields from a Tangherlini black hole radiating on
the brane. Looking at the large values of $\bar{r}_{\mathrm{A}}$ for brane
emission, we reach a different conclusion from bulk emission, and support
Giddings’ argument much better.

Table 3: Dimensionless radii $\bar{r}_{\mathrm{A}}$ for massless fields of spin $s$ from a $(n+4)$-dimensional Tangherlini black hole radiating on the brane.  | $n$
---|---
$s$ | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 3.44 | 4.98 | 5.83 | 5.86 | 5.21 | 4.16 | 2.78
1/2 | 3.44 | 5.10 | 5.90 | 5.84 | 5.13 | 4.04 | 2.71
1 | 2.68 | 4.70 | 5.82 | 5.99 | 5.39 | 4.32 | 2.95
2 | 0.96 | 2.67 | 3.89 | 4.37 | 4.14 | 3.43 | 2.34

###  4.3 Non-commutative geometry inspired black hole

The non-commutative geometry inspired black hole we consider has a minimum
horizon radius at a finite mass (a black hole remnant), and a temperature that
has a maximum before the temperature vanishes. Thus the power does not follow
the $M^{-2}$ dependence near the end of the black hole’s lifetime and
$\bar{r}_{\mathrm{A}}$ depends on the black hole mass. For high
$M\sqrt{\theta}$, we reproduce the Schwarzschild results. For the black-body
case, below about $M\sqrt{\theta}<6$, the power dependence deviates from a
pure $M^{-2}$ dependence and vanishes as $M\to 1.9/\sqrt{\theta}$. The black
hole power falls faster than the black-body power with $M$ except for the
spin-0 field.

Table 4 shows dimensionless effective radii for all spin fields from a non-
commutative geometry inspired black hole in higher dimensions radiating on the
brane at the maximum temperature; we have taken $\sqrt{\theta}=1$.

Table 4: Dimensionless radii $\bar{r}_{\mathrm{A}}$ for massless fields of spin $s$ from a $(n+4)$-dimensional non-commutative geometry inspired black hole radiating on the brane with the maximum temperature and $\sqrt{\theta}=1$.  | $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | $1.70$ | $-0.98$ | 5.85 | 7.35 | 7.93 | 7.62 | 6.60 | 5.21
1/2 | $1.03$ | $-0.98$ | 6.03 | 7.51 | 7.99 | 7.57 | 6.47 | 5.04
1 | $0.12$ | $-0.99$ | 5.33 | 7.22 | 8.04 | 7.83 | 6.83 | 5.42
2 | $-0.68$ | $-0.99$ | 2.75 | 4.50 | 5.55 | 5.77 | 5.25 | 4.28

###  4.4 Polymeric black hole

The polymeric black hole also has a maximum temperature but the temperature
vanishes at zero mass. Combined with the non-trivial area dependence, the
power does not follow a $M^{-2}$ dependence and $\bar{r}_{\mathrm{A}}$ depends
on the mass of the black hole. We have taken $\epsilon=0.01$ and $a_{0}=1$.
For this value of $\epsilon$, $P=2.5\times 10^{-5}$ and gives a negligible
contribution to the power, and causes $m\approx M$. For high
$2M/\sqrt{a_{0}}$, we reproduce the Schwarzschild results. For the black-body
case, below about $2M/\sqrt{a_{0}}<2$, the power dependence deviates from a
pure $M^{-2}$ dependence and vanishes as $M\to 0$. The black hole power falls
faster than the black-body power with $M$ except for the spin-0 field.

Table 5 shows dimensionless effective radii for all spin fields from a
polymeric black hole at the maximum temperature.

Table 5: Dimensionless radii $\bar{r}_{\mathrm{A}}$ for massless fields of spin $s$ from a polymeric black hole with the maximum temperature, and $\epsilon=0.01$ and $a_{0}=1$.  $s$ | 0 | 1/2 | 1 | 2
---|---|---|---|---
$\bar{r}_{\mathrm{A}}$ | 1.57 | 0.62 | $-0.22$ | $-0.90$

##  5 Discussion

We have calculated the quantum atmosphere for all massless spin fields for the
first time. Two quantum gravity inspired metrics posing different black-body
power formula have been compared with exact numerical calculations of the
total power from the black hole including greybody factors.

Giddings’ argument of $\bar{r}_{\mathrm{A}}\sim 1$ clearly depends on the spin
of the emitted radiation, decreasing by a factor of about six when going from
scalars to vectors, and in general does not apply to gravitons.

Hod’s [6] result $\bar{r}_{\mathrm{A}}<1$ for higher-dimensional black holes
is reproduced, but if the radiation is confined to our brane, the conclusion
is very different. Values of $\bar{r}_{\mathrm{A}}\sim 5$ for most spins and
extra dimensions are obtained. The higher-dimensional form of the black-body
formula plays a significant role beyond just the greybody factors.

We have examined two quantum gravity inspired black holes in the regime were
quantum effects are important and the radiation will have its maximum
intensity. The quantum atmosphere for scalar fields in four space-time
dimensions appears similar regardless of the quantum inspired metric and is
similar to Schwarzschild black holes.

The power in the spin-0 field always has a quantum atmosphere radius of about
1.7 times the horizon radius in four space-time dimensions. We can see that,
in general, the effective radius of an equivalent black-body radiator is not a
good proxy for the quantum atmosphere. On the other hand, the effective radius
$\bar{r}_{\mathrm{A}}$ could be considered an intuitive measure of greybody
effects on the total power received by an observer.

The greybody factors themselves are of little interest until they are used to
calculate physical observables. It is common to calculate the absorption cross
section and compare the high-frequency limit against the geometric cross
section and the low-frequency limit against the surface area. These limits
allow an easily quantifiable measure of the greybody effects of different
metrics. Perhaps a more measurement observable, someday, will be the total
particle fluxes and energy spectra measured by a distant observer. First
measurements of these quantities are likely to be integrated over the
detecting instrument’s acceptance and resolution to obtain single numbers for
the number of particles per unit time and energy per unit time (or power),
before full spectra are measured. Expressing these measurements in terms of an
effective black-body radius could prove to be a useful mnemonic for
elucidating quantum gravity effects.

##  6 Acknowledgments

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC). Nous remercions le Conseil de recherches en
sciences naturelles et en génie du Canada (CRSNG) de son soutien.

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Hyper-Algebras of Vector-Valued Modular Forms

Hyper-Algebras of Vector-Valued Modular Forms††This paper is a contribution to
the Special Issue on Modular Forms and String Theory in honor of Noriko Yui.
The full collection is available at http://www.emis.de/journals/SIGMA/modular-
forms.html

Martin RAUM

M. Raum

Chalmers tekniska högskola och Göteborgs Universitet,
Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden
<EMAIL_ADDRESS>http://raum-brothers.eu/martin/

Received May 07, 2018, in final form September 30, 2018; Published online
October 04, 2018

We define graded hyper-algebras of vector-valued Siegel modular forms, which
allow us to study tensor products of the latter. We also define vector-valued
Hecke operators for Siegel modular forms at all places of ${\mathbb{Q}}$,
acting on these hyper-algebras. These definitions bridge the classical and
representation theoretic approach to Siegel modular forms. Combining both the
product structure and the action of Hecke operators, we prove in the case of
elliptic modular forms that all cusp forms of sufficiently large weight can be
obtained from products involving only two fixed Eisenstein series. As a
byproduct, we obtain inclusions of cuspidal automorphic representations into
the tensor product of global principal series.

Siegel modular forms; vector-valued Hecke operators; automorphic
representations

11F40; 11F60; 11F70

##  1 Introduction

Products of scalar-valued modular forms can be used to construct new ones, and
in special situations they reveal deep arithmetic information. For example,
Rankin showed in [14] that the Petersson scalar product of an elliptic cusp
form $f$ and the product $E_{k}E_{l}$ of two Eisenstein series, can be
expressed in terms of special $L$-values attached to $f$. This very relation
reappeared in [8], where Kohnen and Zagier define a ${\mathbb{Q}}$-structure
on elliptic modular forms that is different from the one originating in
Fourier expansions. The product of scalar-valued modular forms naturally leads
to the definition of the graded algebra ${\mathrm{M}}(\bullet)$ of modular
forms.

Vector-valued modular forms in the elliptic case can be associated with any
representation $\rho$ of the modular group ${\mathrm{SL}_{2}}({\mathbb{Z}})$.
As opposed to scalar-valued ones, their product structure has mostly been
neglected. Instead, one considers graded modules
${\mathrm{M}}(\bullet\otimes\rho)$ over ${\mathrm{M}}(\bullet)$. For example,
Marks and Mason proved in [11] that ${\mathrm{M}}(\bullet\otimes\rho)$ is free
over ${\mathrm{M}}(\bullet)$ and that its rank relates directly to the
eigenvalues of $\rho\big{(}{-}I^{(2)})$ for $I^{(2)}$ the $2\times 2$ identity
matrix.

It is possible to define tensor products of vector-valued modular forms by
means of the tensor product of smooth functions from the Poincaré upper half
plane
$\mathbb{H}=\\{\tau\in{\mathbb{C}}\colon\operatorname{\mathfrak{Im}}\tau>0\\}$
to the representation spaces $V(\rho)$ and $V(\rho^{\prime})$ of $\rho$ and
$\rho^{\prime}$

$\displaystyle\otimes\colon\ {\mathrm{C}}^{\infty}\big{(}\mathbb{H}\rightarrow
V(\rho)\big{)}\times{\mathrm{C}}^{\infty}\big{(}\mathbb{H}\rightarrow
V(\rho^{\prime})\big{)}\longrightarrow{\mathrm{C}}^{\infty}\big{(}\mathbb{H}\rightarrow
V(\rho\otimes\rho^{\prime})\big{)},$ $\displaystyle\hphantom{\otimes\colon}{}\
(f\otimes g)(\tau)\longmapsto f(\tau)\otimes g(\tau).$

Tensor products of vector-valued modular forms have seldom been studied. In
[23], we employed them to express elliptic cusp forms of any level as tensor
products of at most two Eisenstein series. Tensor products can be conveniently
subsumed under the concept of hyper-algebras. Our description of tensor
products of and differential operators on almost holomorphic Siegel modular
forms that we gave in [6], for example, employed hyper-algebras to classify
almost holomorphic Siegel modular forms. In this work, we review the concept
of hyper-algebras in the context of modular forms.

Hyper-algebras mimic algebras but they have multivalued multiplication. In
particular, they are natural analogues of hyper-groups [21]. Note that we
suppress Wall’s assumption of dimensionality of hyper-groups, which does not
hold for examples that we treat. Hyper-groups arise naturally in the context
of double cosets in group theory. The definition of a group $G$ includes the
product $a\cdot b\in G$ of $a,b\in G$. Given a sufficiently nice subgroup $H$
– that is, if $(G,H)$ is a Gelfand pair – it is standard in modular forms to
define a (commutative) Hecke algebra. Specifically, Hecke algebras assign
meaning to the product of double cosets $HaH\cdot HbH$. For instance, if
$G={\mathrm{GSp}_{g}}({\mathbb{Q}})$ and $K={\mathrm{GSp}_{g}}({\mathbb{Z}})$,
then the Hecke algebra with integral coefficients is free as a
${\mathbb{Z}}$-module and has basis $HcH$ for certain diagonal $c\in G$. Every
product of double cosets can be written as a sum $\sum_{c\in G}m_{c}HcH$,
where $0\leq m_{c}\in{\mathbb{Z}}$ and $m_{c}\neq 0$ for only finitely many
$c$. The corresponding notion of hyper-groups, to the author’s knowledge, has
not yet appeared in the context of modular forms, but is standard in other
areas of mathematics. Rephrasing the product in Hecke algebras, we arrive at a
notion of multivalued multiplication of double cosets. One can decompose
$HaH=\bigcup_{c\in{\mathrm{L}}(H,a)}Hc$ and
$HbH=\bigcup_{c\in{\mathrm{L}}(H,b)}Hc$ as a disjoint union of left cosets
with coset representatives ${\mathrm{L}}(H,a),{\mathrm{L}}(H,b)\subseteq G$.
By definition of the multiplication in Hecke algebras, the multiset
$\\{\\!\\{H(d_{a}\cdot d_{b})\colon
d_{a}\in{\mathrm{L}}(H,a),d_{b}\in{\mathrm{L}}(H,b)\\}\\!\\}$ has a
decomposition into the disjoint union of $\\{\\!\\{Hd_{c}\colon
d_{c}\in{\mathrm{L}}(H,c)\\}\\!\\}$ where each ${\mathrm{L}}(H,c)$ occurs with
multiplicity $m_{c}$ that occurred before. A more detailed, explicit
explanation can be found in [9]. Summarizing, we obtain a multiplication of
double cosets that takes values in multisets of double cosets:

$\displaystyle H\backslash G/H\times H\backslash
G/H\longrightarrow\mathrm{Multiset}(H\backslash G/H).$

In this way, we obtain a commutative hyper-group. Note that every group can be
naturally viewed as a hyper-group. Vice versa, a hyper-group whose
multiplication takes values in multisets of size $1$ is a group.

It is possible to extend the definition of hyper-groups to hyper-algebras.
Addition is axiomatized as in the case of algebras. Multiplication takes
values in the set of all subspaces and compatibility relations that mimic
those for algebras are imposed on it. We give a precise definition of hyper-
algebras in Section 3. Given an algebra $A$ we obtain a hyper-algebra by
assigning to two elements the module spanned by their product. In general, it
is not possible to recover the algebra from this, because the hyper-algebra
product $a\underline{\,\cdot\,}b$ associated with $a,b\in A$ and
$(ra)\underline{\,\cdot\,}(r^{\prime}b)$ is the same for all units $r$,
$r^{\prime}$ in the base ring $R$.

Siegel modular forms are assigned to a weight $\sigma$ and a type $\rho$,
which are representations of ${\mathrm{GL}_{g}}({\mathbb{C}})$ and
${\mathrm{Sp}_{g}}({\mathbb{Z}})$, respectively. Given Siegel modular forms
$f$ and $g$ of weights and types $\sigma_{f}$, $\rho_{f}$, $\sigma_{g}$, and
$\rho_{g}$, then their tensor product $f\otimes g$ has weight
$\sigma_{f}\otimes\sigma_{g}$ and type $\rho_{f}\otimes\rho_{g}$. Even if the
weights and types of $f$ and $g$ are irreducible, their tensor products can be
reducible. On the other hand, to avoid redundancies, graded modules of Siegel
modular forms are preferably build from irreducible weights and types. As a
consequence, tensor products do not yield an algebra structure on such
modules.

Since weights and types behave similarly with respect to the construction that
we discuss now, we focus on the former. To remedy the lack of algebra
structures on the graded module of Siegel modular forms, we have previously
suggested to attach to $f$ and $g$ the span

$\displaystyle
f\,{\mathop{\underline{\otimes}}}\,g=\operatorname{span}\big{\\{}\phi\circ(f\otimes
g)\colon\phi\in\operatorname{\mathrm{Hom}}(\sigma_{f}\otimes\sigma_{g},\sigma)\big{\\}}\subset\bigoplus_{\sigma}{{\mathrm{M}}^{(g)}}(\sigma),$

where $\sigma$ runs through a fixed set of representatives of isomorphism
classes of irreducible weights, and ${{\mathrm{M}}^{(g)}}(\sigma)$ denotes the
corresponding space of Siegel modular forms. In [6], we identified the
resulting structure as a hyper-algebra. It is a special case of hyper-algebras
of Siegel modular forms, defined in Section 3.1.

In [6], we also defined the concept of differential hyper-algebras to subsume
the action of covariant differential operators. In Section 3.4, we revisit
this definition and reinterpret it in terms of a Hecke operator
${\mathrm{T}}_{\infty}$ at the infinite place of ${\mathbb{Q}}$. Returning to
the case of general weights and types, we also study vector-valued Hecke
operators ${\mathrm{T}}_{v}$ at all places $v$ of ${\mathbb{Q}}$. If $v=p$ is
a prime, then ${\mathrm{T}}_{p}$ generalizes classical Hecke operators for
Siegel modular forms. It extends the vector-valued Hecke operators for
elliptic modular forms that we have already defined in [23]. The polynomial
algebra ${\mathbb{T}}$ generated by formal elements ${\mathrm{T}}_{v}$ acts by
means of these Hecke operators on the hyper-algebra of Siegel modular forms.
Specifically, at the infinite place, the Hecke operator acts as a hyper-
derivation. At the finite places, Hecke operators respect the hyper-product
structure. A precise description can be found in Section 3.6.

Section 4 contains a more detailed study of tensor products and the action of
${\mathbb{T}}$ on modular forms is studied in more detail in the genus $1$
case. Let ${\mathrm{S}}^{(1)}(k)$ be the space of genus $1$ cusp forms of
weight $k$ and level $1$. Write $E^{(1)}(l)$ for the level $1$ Eisenstein
series of weight $l$. The following is a slight modification of the main
result in [23].

######  Theorem 1.1.

Let $l,l^{\prime}\geq 4$ be even integers and $\rho$ a finite-dimensional
representation of ${\mathrm{SL}_{2}}({\mathbb{Z}})$ whose kernel is a
congruence subgroup. Then for every $k\geq l+l^{\prime}$, we have

$\displaystyle{\mathrm{S}}^{(1)}(k\otimes\rho)\subset{\mathbb{T}}\big{(}{\mathbb{T}}E^{(1)}(l)\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}E^{(1)}(l^{\prime})\big{)}.$
(1.1)

The right-hand side of (1.1) can be efficiently computed and its left-hand
side is genuinely interesting. We argue that the proof can serve as a
blueprint to a whole family of analogous statements. Thus Theorem 1.1 sets the
path towards a general method of computing Siegel modular forms. Section 4
contains a specific conjecture in the case of genus $2$.

Next, we discuss a connection between Theorem 1.1 and automorphic
representations for ${\mathrm{PGL}_{2}}$. To this end, we restrict to
representations $\rho$ whose kernel is a congruence subgroup. Vector-valued
Hecke operators defined in this paper can be related to adelic automorphic
representations. Covariant differential operators correspond to the action of
an appropriate Lie algebra on Harish-Chandra modules. Hecke operators at
finite places arise directly from a representation of a group over
${{{\mathbb{Q}}_{p}}}$. It is natural to ask how much Theorem 1.1 relates to
results from automorphic representation theory. We make such a relation
precise, and determine some constituents in the tensor product of principal
series. Specifically, we consider automorphic representations for
${\mathrm{PGL}_{2}}$. We let $\varpi(k-1,{\mathds{1}})$ be the principal
series that is unramified at the finite places and has Harish-Chandra
parameter $k-1$ at infinity. Let $\varpi_{\infty}(k-1)$ be the discrete series
with Harish-Chandra parameter $k-1$.

######  Theorem 1.2.

Given even integers $l,l^{\prime}\geq 4$, then

$\displaystyle\varpi\lhook\joinrel\relbar\joinrel\rightarrow\varpi(l-1,{\mathds{1}})\otimes\varpi(l^{\prime}-1,{\mathds{1}})$

for every cuspidal automorphic representation
$\varpi=\varpi_{\infty}(k-1)\otimes\varpi_{\mathrm{f}}$ with $k\geq
l+l^{\prime}$ and $\varpi_{\mathrm{f}}$ a representation of
${\mathrm{PSL}_{2}}({\mathbb{A}}_{\mathrm{f}})$.

When phrased in this language, similarity to the Gross–Prasad conjectures for
the inclusion
${\mathrm{O}_{1,2}}\cong{\mathrm{SL}_{2}}\hookrightarrow{\mathrm{SL}_{2}}\times{\mathrm{SL}_{2}}\cong{\mathrm{O}_{2,2}}$
becomes apparent.

In Section 2, we collect preliminaries on Siegel modular forms. In Section 3,
we define hyper-algebras of Siegel modular forms. In Section 4, we show that
products of certain Eisenstein series yield cusp forms. In Section 5, we give
an interpretation of our results in terms of adelic automorphic
representations.

##  2 Modular forms

###  2.1 The classical setup

The Siegel upper half space
$\big{\\{}\tau\in{\mathrm{Mat}_{g}}({\mathbb{C}})\colon{\,{}^{{\mathrm{t}}}\\!}\tau=\tau,\,\operatorname{\mathfrak{Im}}\tau>0\big{\\}}$
of genus $g$ is denoted by $\mathbb{H}^{(g)}$. It carries an action of the
real symplectic group

$\displaystyle{\mathrm{Sp}_{g}}({\mathbb{R}})=\big{\\{}\gamma\in{\mathrm{Mat}_{2g}}({\mathbb{R}})\colon{\,{}^{{\mathrm{t}}}\\!}\gamma
J^{(g)}\gamma=J^{(g)}\big{\\}},\qquad
J^{(g)}=\left(\begin{smallmatrix}0&-I^{(g)}\\\
I^{(g)}&0\end{smallmatrix}\right),$

where $I^{(g)}$ is the $g\times g$ identity matrix. This action is explicitly
given by $\gamma\tau=(a\tau+b)(c\tau+d)^{-1}$, where we employ the block
decomposition of $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$. The subgroup
${\Gamma^{(g)}}={\mathrm{Sp}_{g}}({\mathbb{Z}})$ of symplectic transformation
matrices with integral entries is called the Siegel modular group of genus
$g$.

Generally, a complex representation $\sigma$ of
${\mathrm{GL}_{g}}({\mathbb{C}})$ is called a weight. For the purpose of this
paper, we restrict to representations that factor through
${\mathrm{GL}_{g}}({\mathbb{C}})/\big{\\{}{\pm}I^{(g)}\big{\\}}$. A finite-
dimensional representation $\rho$ of ${\mathrm{Sp}_{g}}({\mathbb{Z}})$ is
called a type. Throughout this note, we focus on types whose kernel is a
congruence subgroup that contains $-I^{(g)}$. The representation spaces of
$\sigma$ and $\rho$ are denoted by $V(\sigma)$ and $V(\rho)$. A weight and a
type together determine a slash action

$\displaystyle\big{(}f\big{|}_{\sigma,\rho}\gamma\big{)}(\tau)=\sigma(c\tau+d)^{-1}\rho(\gamma)^{-1}f(\gamma\tau)$

on functions $f\colon\mathbb{H}^{(g)}\rightarrow V(\sigma)\otimes V(\rho)$.

The space of genus $g$ Siegel modular forms of weight $\sigma$ and type $\rho$
is defined as the space of holomorphic functions
$f\colon\mathbb{H}^{(g)}\rightarrow V(\sigma)\otimes V(\rho)$ such that
$f|_{\sigma,\rho}\gamma=f$ for all $\gamma\in{\Gamma^{(g)}}$ and which satisfy
$f(x+{\rm i}y)=O(1)$ as $y\rightarrow\infty$, if $g=1$. We write

$\displaystyle{{\mathrm{M}}^{(g)}}(\sigma\otimes\rho)={\mathrm{E}}^{(g)}(\sigma\otimes\rho)\oplus{\mathrm{S}}^{(g)}(\sigma\otimes\rho)$

for this space, the space of Eisenstein series, and the space of cusp forms,
respectively.

###  2.2 The symplectic group and its Lie algebra

We write ${\mathrm{G}}={\mathrm{PGSp}_{g}}$ for the ${\mathbb{Q}}$-split
algebraic group of projective symplectic similitudes. Throughout we work with
the model

$\displaystyle{\mathrm{PGSp}_{g}}({\mathbb{Z}})=\big{\\{}\gamma\in{\mathrm{Mat}_{2g}}({\mathbb{Z}})\colon{\,{}^{{\mathrm{t}}}\\!}\gamma
J^{(g)}\gamma=s(\gamma)J^{(g)},\,s(\gamma)\in{\mathbb{Z}}\big{\\}}\big{/}\big{\\{}sI^{(g)}\colon
s\in{\mathbb{Z}}\big{\\}}.$

Compact subgroups ${K^{(g)}_{\infty}}$ and ${K^{(g)}_{p}}$ of
${\mathrm{G}}({\mathbb{R}})$ and ${\mathrm{G}}({{{\mathbb{Q}}_{p}}})$ are
${\mathrm{U}_{g}}({\mathbb{R}})/\big{\\{}{\pm}I^{(g)}\big{\\}}$ and
${\mathrm{PGSp}_{g}}({{\mathbb{Z}}_{p}})$, where the former is embedded into
${\mathrm{G}}({\mathbb{R}})$ by $a{\rm
i}+b\mapsto\left(\begin{smallmatrix}a&-b\\\ b&a\end{smallmatrix}\right)$.
Weights correspond to complex representations of ${K^{(g)}_{\infty}}$ by means
of the restriction along
${\mathrm{U}_{g}}({\mathbb{R}})/\big{\\{}{\pm}I^{(g)}\big{\\}}\rightarrow{\mathrm{GL}_{g}}({\mathbb{C}})/\big{\\{}{\pm}I^{(g)}\big{\\}}$.

We let ${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}$ be a fixed
set of representatives of isomorphism classes of finite-dimensional, complex
representations of ${K^{(g)}_{\infty}}$. For all $g$, the representation
$\det^{k}\colon{\mathrm{U}_{g}}({\mathbb{R}})\rightarrow{\mathbb{C}}$,
$k\mapsto\det(k)$ with $k\in{\mathbb{Z}}$, $2\,|\,gk$ yields an irreducible
representation of ${K^{(g)}_{\infty}}$ via the above isomorphism of
${K^{(g)}_{\infty}}$ and
${\mathrm{U}_{g}}({\mathbb{R}})/\big{\\{}{\pm}I^{(g)}\big{\\}}$. In the
context of spaces of modular forms, we will occasionally write $k$ instead of
$\det^{k}$ to match more closely the classical notation. For example,

$\displaystyle{{\mathrm{M}}^{(g)}}(k\otimes\rho)={{\mathrm{M}}^{(g)}}\big{(}{\det}^{k}\otimes\rho\big{)}.$

We denote the $l$-th symmetric power of the standard representation by
${\mathrm{sym}}^{l}$. The dual of a representation is indicated by a
superscript $\vee$: $\sigma^{\vee}$ is the dual of $\sigma$.

We let ${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{p}}\big{)}$ be a fixed set
of representatives of isomorphism classes of finite-dimensional, complex
representations of ${K^{(g)}_{p}}$. To simplify notation, we let
${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}$ be the set of
restricted tensor products of the representations in
${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{p}}\big{)}$. That is, it consists
of exterior tensor products of representations of ${K^{(g)}_{p}}$ which are
trivial for all but finitely many $p$. There is a correspondence between
${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}$ and finite-
dimensional, complex representations of ${\Gamma^{(g)}}$ with a congruence
subgroup and the matrix $-I_{2g}$ in their kernel.

Subsets of irreducible representations are denoted by

$\displaystyle{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}\subset{\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}\qquad\text{and}\qquad{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}\subset{\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}.$

###  2.3 Covariant differential operators

We call a differential operator ${\mathrm{D}}$ covariant from
$\big{|}_{\sigma,\rho}$ to $\big{|}_{\sigma^{\prime},\rho}$ if for all
$g\in{\mathrm{G}}({\mathbb{R}})$ and all smooth functions
$f\colon\mathbb{H}^{(g)}\rightarrow V(\sigma)\otimes V(\rho)$ , we have

$\displaystyle{\mathrm{D}}\big{(}f\big{|}_{\sigma,\rho}g\big{)}={\mathrm{D}}(f)\big{|}_{\sigma^{\prime},\rho}g.$

If ${\mathrm{D}}$ is covariant for the trivial type then it yields a
differential operator that is covariant for all types.

A theorem of Helgason [3], allows us to classify order $1$ differential
operators. For each $\sigma$ there is a lowering operator
${\mathrm{L}}={\mathrm{L}}_{\sigma}$ that is covariant from $\sigma$ to
${\mathrm{L}}\sigma={\mathrm{sym}}^{2\vee}\sigma$ and a raising operator
${\mathrm{R}}={\mathrm{R}}_{\sigma}$ that is covariant from $\sigma$ to
${\mathrm{R}}\sigma={\mathrm{sym}}^{2}\sigma$. We only fix a normalization in
the case $g=1$, setting

$\displaystyle{\mathrm{L}}=-2{\rm
i}y^{2}\partial_{\overline{\tau}}\qquad\text{and}\qquad{\mathrm{R}}=2{\rm
i}\partial_{\tau}+ky^{-1}.$

In [6], we gave explicit expressions for all $g$. The normalization employed
there, however, does not coincide with the one we choose here.

Covariant differential operators allow us to define almost holomorphic Siegel
modular forms. A function $f\colon\mathbb{H}^{(g)}\rightarrow V(\sigma)\otimes
V(\rho)$ that vanishes under the $(d+1)$-th tensor power of the lowering
operator, i.e., ${\mathrm{L}}^{d}f=0$, that is invariant of weight $\sigma$
and type $\rho$, i.e., $f\big{|}_{\sigma,\rho}\gamma=f$ for all
$\gamma\in{\Gamma^{(g)}}$, and that satisfies $f(\tau)=O(1)$ as
$y\rightarrow\infty$, if $g=1$, is called an almost holomorphic Siegel modular
form of weight $\sigma$, type $\rho$, and depth $d$. The space of such
functions is denoted by

$\displaystyle{\sideset{{}^{[d]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\sigma\otimes\rho).$

###  2.4 Hecke operators

We extend the definition of vector-valued Hecke operators in [23] for elliptic
modular forms to Siegel modular forms. Many of the proofs in op. cit. apply to
the general case word by word. For this reason, we skip several arguments in
this subsection.

#### Hecke operators on representations

For a positive integer $M$, we let

$\displaystyle\Delta_{M}=\Big{\\{}\left(\begin{smallmatrix}a&b\\\
0&d\end{smallmatrix}\right)\in{\mathrm{GSp}_{g}}({\mathbb{Q}})\colon{\,{}^{{\mathrm{t}}}\\!}ad=MI^{(g)},\,a,b,d\in{\mathrm{Mat}_{g}}({\mathbb{Z}}),\,d\
\text{upper triangular},$
$\displaystyle\hphantom{\Delta_{M}=\Big{\\{}}{}\forall\,i<j\colon 0\leq
d_{i,j}<d_{j,j},\,\forall\,i\colon,0\leq b_{i,j}<d_{j,j}\Big{\\}},$

where ${\mathrm{GSp}_{g}}$ is the group of symplectic similitude
transformations. We have a right action of ${\mathrm{Sp}_{g}}({\mathbb{Z}})$
on $\Delta_{M}$ defined by $(m,\gamma)\mapsto\overline{m\gamma}$ with
$\gamma^{\prime}\overline{m\gamma}=m\gamma$ for some
$\gamma^{\prime}\in{\mathrm{Sp}_{g}}({\mathbb{Z}})$. This, in particular,
defines a cocycle $I_{m}(\gamma)=\gamma^{\prime}$; That is, we have
$I_{m}(\gamma_{1}\gamma_{2})=I_{m}(\gamma_{1})I_{m\gamma_{1}}(\gamma_{2})$.

We denote the natural basis of ${\mathbb{C}}[\Delta_{M}]$ by
${\mathfrak{e}}_{m}$, $m\in\Delta_{M}$. To every type $\rho$ we associated the
type $T_{M}\rho$ defined by

$\displaystyle
V(T_{M}\rho):=V(\rho)\otimes{\mathbb{C}}[\Delta_{M}]\qquad\text{and}\qquad(T_{M}\rho)(\gamma)(v\otimes{\mathfrak{e}}_{m}):=\rho\big{(}I_{m}^{-1}\big{(}\gamma^{-1}\big{)}\big{)}(v)\otimes{\mathfrak{e}}_{m\gamma^{-1}}.$

The cocycle property of $I_{m}(\gamma)$ implies that it is a representation of
${\mathrm{Sp}_{g}}({\mathbb{Z}})$. Given a scalar product
$\langle\,\cdot\,,\,\cdot\,\rangle_{\rho}$ on $V(\rho)$, we obtain one on
$V({\mathrm{T}}_{M}\rho)$ by

$\displaystyle\langle
v\otimes{\mathfrak{e}}_{m_{v}},w\otimes{\mathfrak{e}}_{m_{w}}\rangle=\begin{cases}\langle
v,w\rangle_{\rho},&\text{if $m_{v}=m_{w}$},\\\
0,&\text{otherwise}.\end{cases}$

Hecke operators on representations are compatible with homomorphisms between
types and with tensor products. Specifically, the following are homomorphism
of ${\mathrm{Sp}_{g}}({\mathbb{Z}})$-representations:

$\displaystyle T_{M}\phi\colon\ T_{M}\rho\longrightarrow
T_{M}\rho^{\prime},\qquad
v\otimes{\mathfrak{e}}_{m}\longmapsto\phi(v)\otimes{\mathfrak{e}}_{m},$
$\displaystyle\hskip
49.79231pt{\mathds{1}}\lhook\joinrel\relbar\joinrel\rightarrow
T_{M}{\mathds{1}},\hskip 48.36967ptc\longmapsto
c\sum_{m\in\Delta_{M}}{\mathfrak{e}}_{m},$
$\displaystyle(T_{M}\rho)\otimes(T_{M}\sigma)\relbar\joinrel\twoheadrightarrow
T_{M}(\rho\otimes\sigma),$
$\displaystyle(v\otimes{\mathfrak{e}}_{m})\otimes(w\otimes{\mathfrak{e}}_{m^{\prime}})\longmapsto\begin{cases}(v\otimes
w)\otimes{\mathfrak{e}}_{m},&\text{if $m=m^{\prime}$},\\\
0,&\text{otherwise},\end{cases}$ (2.1)

where $\phi\in\operatorname{\mathrm{Hom}}(\rho,\rho^{\prime})$. If
$\psi\colon\rho\rightarrow\rho^{\prime\prime}$ is a further homomorphism, then
$T_{M}(\psi\circ\phi)=(T_{M}\psi)\circ(T_{M}\phi)$. For later reference, we
denote the morphism in (2.1) by $\pi_{M,\rho,\rho^{\prime}}$.

#### Hecke operators on modular forms

For $m=\left(\begin{smallmatrix}a&b\\\
0&d\end{smallmatrix}\right)\in{\mathrm{GSp}_{g}}({\mathbb{R}})$ of similitude
$M$, and for $f\colon\mathbb{H}\rightarrow V(\sigma)$, we define

$\displaystyle\big{(}f\big{|}_{\sigma}m\big{)}(\tau)=\sigma\big{(}d/\sqrt{M}\big{)}^{-1}f\big{(}(a\tau+b)d^{-1}\big{)}.$

Fixing a type $\rho$ and a positive integer $M$, we define the vector-valued
Hecke operator ${\mathrm{T}}_{M}$ acting on
$f\in{{\mathrm{M}}^{(g)}}\big{(}\sigma\otimes\rho\big{)}$ by

$\displaystyle\big{(}T_{M}f\big{)}(\tau)=\sum_{m\in\Delta_{M}}\big{(}f\big{|}_{\sigma}m\big{)}(\tau)\otimes{\mathfrak{e}}_{m}\in{{\mathrm{M}}^{(g)}}\big{(}\sigma\otimes{\mathrm{T}}_{M}\rho\big{)}.$

The above homomorphism $\pi_{M,\rho,\rho^{\prime}}$ is compatible with Hecke
operators. Specifically, we have

$\displaystyle\pi_{M,\rho,\rho^{\prime}}\big{(}({\mathrm{T}}_{M}f)\otimes({\mathrm{T}}_{M}g)\big{)}={\mathrm{T}}_{M}\big{(}f\otimes
g\big{)}.$

##  3 Hyper-algebras

We start with the formal definition of hyper-groups, which is the blueprint to
our Definition 3.2 of hyper-algebras. Given a set $S$, let
$\mathrm{Multiset}(S)\cong\\{f\colon S\rightarrow{\mathbb{Z}}_{\geq 0}\\}$ be
the set of all multisubsets of $S$. Multisets are throughout denoted by double
curly brackets $\\{\\!\\{\cdots\\}\\!\\}$. The union of multisets corresponds
to the sum of functions $S\rightarrow{\mathbb{Z}}_{\geq 0}$. As a shorthand
notation, for the binary operator appearing in the next definition, we set
$a{\mathop{\underline{\,\cdot\,}}}\\{\\!\\{b_{i}\\}\\!\\}_{i}=\bigcup_{b_{i}}a{\mathop{\underline{\,\cdot\,}}}b_{i}$
and similarly
$\\{\\!\\{a_{i}\\}\\!\\}_{i}{\mathop{\underline{\,\cdot\,}}}b=\bigcup_{a_{i}}a_{i}{\mathop{\underline{\,\cdot\,}}}b$.

######  Definition 3.1.

A pair $(G,{\mathop{\underline{\,\cdot\,}}})$ of a set $G$ and a binary
operator ${\mathop{\underline{\,\cdot\,}}}\colon G\times
G\rightarrow\mathrm{Multiset}(G)$ is called a hyper-group if the following
axioms are satisfied:

  1. (i)

(Finite image) For any $a,b\in G$, the multiset
$a{\mathop{\underline{\,\cdot\,}}}b$ is finite.

  2. (ii)

(Associativity) For any $a,b,c\in G$, we have
$(a{\mathop{\underline{\,\cdot\,}}}b){\mathop{\underline{\,\cdot\,}}}c=a{\mathop{\underline{\,\cdot\,}}}(b{\mathop{\underline{\,\cdot\,}}}c)$,
that is,

$\displaystyle\bigcup_{h\in
a{\mathop{\underline{\,\cdot\,}}}b}h{\mathop{\underline{\,\cdot\,}}}c=\bigcup_{h\in
b{\mathop{\underline{\,\cdot\,}}}c}a{\mathop{\underline{\,\cdot\,}}}h.$

  3. (iii)

(Identity) There is an element $e\in G$ such that for every $a\in G$ we have
$a\in a{\mathop{\underline{\,\cdot\,}}}e$ and $a\in
e{\mathop{\underline{\,\cdot\,}}}a$.

  4. (iv)

(Inverse) For every element $a\in G$ there is element $a^{-1}\in G$ such that
$e\in a{\mathop{\underline{\,\cdot\,}}}a^{-1}$ and $e\in
a^{-1}{\mathop{\underline{\,\cdot\,}}}a$.

Extending this notion to hyper-algebras is straightforward. However, the
reader should be warned that the word “hyper-algebra” is used in the context
of algebraic groups [18], too, and these two notions should not be confused.

Given a commutative ring $R$ (with identity), and an $R$-module $M$, we let
$\mathrm{SubMod}_{R}(M)$ be the set of all $R$-submodules of $M$. In analogy
with the definition of multisets, we extend the binary operator
${\mathop{\underline{\,\cdot\,}}}$ in the following definition to submodules
by $a{\mathop{\underline{\,\cdot\,}}}N=\bigcup_{b\in
N}a{\mathop{\underline{\,\cdot\,}}}b$ and
$N{\mathop{\underline{\,\cdot\,}}}b=\bigcup_{a\in
N}a{\mathop{\underline{\,\cdot\,}}}b$ for $N\in\mathrm{SubMod}_{R}(M)$.

######  Definition 3.2.

Let $R$ be a ring. A triple $(A,+,\,{\mathop{\underline{\,\cdot\,}}}\,)$ with
$(A,+)$ an $R$-module and binary operator

$\displaystyle{\mathop{\underline{\,\cdot\,}}}\colon\ A\times
A\longrightarrow\mathrm{SubMod}_{R}(A)$

is called a hyper-algebra (with identity) if

  1. (i)

(Linearity) Given $a,b\in A$ and $r\in R$, we have
$(ra){\mathop{\underline{\,\cdot\,}}}b=r\big{(}a{\mathop{\underline{\,\cdot\,}}}b\big{)}=a{\mathop{\underline{\,\cdot\,}}}(rb)$.

  2. (ii)

(Identity) There is $e\in A$ such that we have
$e{\mathop{\underline{\,\cdot\,}}}a=a{\mathop{\underline{\,\cdot\,}}}e=\operatorname{span}_{R}a$
for all $a\in A$.

  3. (iii)

(Associativity) Given $a,b,c\in A$, we have
$a{\mathop{\underline{\,\cdot\,}}}(b{\mathop{\underline{\,\cdot\,}}}c)=(a{\mathop{\underline{\,\cdot\,}}}b){\mathop{\underline{\,\cdot\,}}}c$.

  4. (iv)

(Distributivity) for $a,b,c\in A$, we have
$a{\mathop{\underline{\,\cdot\,}}}(b+c)\subseteq
a{\mathop{\underline{\,\cdot\,}}}b+a{\mathop{\underline{\,\cdot\,}}}c$.

Concepts like commutativity, grading, and derivation extend to hyper-algebras.
Fixing a hyper-algebra $A$, we call it commutative if for all $a,b\in A$, we
have $ab=ba$. We say that it is graded by a hyper-group $G$ if
$A=\bigoplus_{g}A_{g}$ as an $R$-module and
$A_{g}A_{g^{\prime}}\subseteq\bigcup_{h\in gg^{\prime}}A_{h}$. A hyper-
derivation $d$ on $A$ is an $R$-module endomorphism such that
$d(a{\mathop{\underline{\,\cdot\,}}}b)\subseteq(da){\mathop{\underline{\,\cdot\,}}}b+a{\mathop{\underline{\,\cdot\,}}}(db)$.

Note that for a hyper-algebra $A$, we can define a (left) hyper-module $M$ as
an $R$-module with binary operator ${\mathop{\underline{\,\cdot\,}}}\colon
A\times M\rightarrow\mathrm{SubMod}_{R}(M)$ satisfying the analogue of the
axioms in Definition 3.2.

###  3.1 Hyper-algebras of modular forms

Recall the various sets of representations defined in Section 2.2. The hyper-
algebra of holomorphic Siegel modular forms as a vector space is

$\displaystyle{{{\mathrm{M}}^{(g)}}(\bullet\otimes\bullet)}=\bigoplus_{\sigma,\rho}{{\mathrm{M}}^{(g)}}(\sigma\otimes\rho),\qquad\sigma\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}},\qquad\rho\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}}$

with product
$fg=f\,{\mathop{\underline{\otimes}}}\,g\subset{{{\mathrm{M}}^{(g)}}(\bullet\otimes\bullet)}$
defined as

$\displaystyle\operatorname{span}\Big{\\{}(\phi_{\sigma}\otimes\phi_{\rho})\circ(f\otimes
g)\colon\sigma\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}},\,\phi_{\sigma}\colon\sigma_{f}\otimes\sigma_{g}\rightarrow\sigma,$
$\displaystyle\hphantom{\operatorname{span}\Big{\\{}(\phi_{\sigma}\otimes\phi_{\rho})\circ(f\otimes
g)\colon}{}\rho\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}},\,\phi_{\rho}\colon\rho_{f}\otimes\rho_{g}\rightarrow\rho\Big{\\}}.$
(3.1)

Note that both
${{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}}$ and
${{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}}$ are
hyper-groups, and the hyper-algebra of Siegel modular forms is graded by both.
Our notation for submodules, for example
${{\mathrm{M}}^{(g)}}(\bullet\otimes\rho)$, has the obvious meaning.

We will also work with the hyper-algebras of almost holomorphic Siegel modular
forms

$\displaystyle{{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}=\bigcup_{0\leq
d}\bigoplus_{\sigma,\rho}{\sideset{{}^{[d]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\sigma\otimes\rho),\qquad\sigma\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}},\qquad\rho\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}}$

with product as in (3.1).

###  3.2 Computing the hyper-algebra product in Sage

In the case of elliptic modular forms, all irreducible weights are
$1$-dimensional. Types can, however, be arbitrary dimensional. We adopt an
example from [23], which illustrates how to compute products in
${{{\mathrm{M}}^{(g)}}(\bullet\otimes\bullet)}$ of modular forms with
nontrivial type. For convenience, we give Sage code [16]111Computations make
implicit use of the libraries [2, 12] and possibly further ones that are less
obvious from the source code of Sage., which the reader can modify to perform
his or her own computation – the code is not optimized for performance, but
for clarity.

We consider the representation $\rho_{3}$, which we realize as a matrix
representation by

$\displaystyle\rho_{3}(T)=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
-1&-1&-1\end{pmatrix},\qquad\rho_{3}(S)=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
-1&-1&-1\end{pmatrix}.$

Its kernel contains the congruence subgroup $\Gamma(3)$. The space of
Eisenstein series of weight $12$ and type $\rho_{3}$ has dimension one. A
basis element $E_{12,\rho_{3}}$ can be obtained from vector-valued Hecke
operators that are discussed in Section 2.4. We let $\zeta$ be a third root of
unity. The image of ${\mathrm{T}}_{3}$ on the level $1$ Eisenstein series of
weight $12$ is

⬇

K.<zeta> = CyclotomicField(3)

R.<q3> = K[[]]

E12 = EisensteinForms(1,12).basis()[0]

E12T3 = lambda n: vector([

3** 6 * E12.qexp(n//3+1).subs(q=q3**9).add_bigoh(3*n)

, 3**-6 * E12.qexp(3*n).subs(q=q3)

, 3**-6 * E12.qexp(3*n).subs(q=zeta*q3)

, 3**-6 * E12.qexp(3*n).subs(q=zeta**2*q3)

])

Fourier expansions are computed in terms of $\text{q3}=e(\tau/3)$. The
components of an Eisenstein series of type $\rho_{3}$ can be found by applying
a homomorphism from ${\mathrm{T}}_{3}{\mathds{1}}$ to $\rho_{3}$, which can be
computed by the method that we describe below when decomposing
$\rho_{3}\rho_{3}$,

⬇

E12rho3 = lambda n: matrix(3, [ 1,-1/3,-1/3,-1/3

, -1/3,1,-1/3,-1/3

, -1/3,-1/3,1,-1/3

]) * E12T3(n)

To compute $E_{12,\rho_{3}}\,{\mathop{\underline{\otimes}}}\,E_{12,\rho_{3}}$,
we have to decompose the $9$-dimensional representation
$\rho_{3}\otimes\rho_{3}$. Since the kernel of $\rho_{3}$ has finite index in
${\mathrm{SL}_{2}}({\mathbb{Z}})$, character theory for finite groups is one
way to achieve this. We use a more general method that applies to all
representation: By exhibiting the trivial representation in
$(\rho_{3}\otimes\rho_{3})^{\vee}\otimes\rho$ for various representations
$\rho$, we compute the space of homomorphisms
$\operatorname{\mathrm{Hom}}(\rho_{3}\otimes\rho_{3},\rho)$. For a systematic
decomposition of $\rho_{3}\otimes\rho_{3}$, one could apply the MeatAxe
algorithm [4] in conjunction with a multimodular approach.

We start by defining the representation matrices of the trivial representation
${\mathds{1}}$, $\rho_{3}$, and $\rho_{3}\otimes\rho_{3}$,

⬇

s_triv = t_triv = identity_matrix(QQ, 1)

s3 = matrix(QQ, 3, [0,1,0, 1,0,0, -1,-1,-1])

t3 = matrix(QQ, 3, [1,0,0, 0,0,1, -1,-1,-1])

s33 = s3.tensor_product(s3)

t33 = t3.tensor_product(t3)

We compute homomorphisms between representations by employing the isomorphism
of vector spaces
$\operatorname{\mathrm{Hom}}(\rho,\rho^{\prime})\cong\big{(}\rho^{\vee}\otimes\rho^{\prime}\big{)}({\mathds{1}})$.
Representation matrices of the dual representation $\rho^{\vee}$ are given by
the transpose inverses of those of $\rho$. We can immediately compute
homomorphisms from $\rho_{3}\rho_{3}$ to ${\mathds{1}}$ and $\rho_{3}$,

⬇

dual = lambda m: m.transpose().inverse()

hom = lambda s1,t1,s2,t2: \

(dual(s1).tensor_product(s2)-1).right_kernel() \

.intersection((dual(t1).tensor_product(t2)-1).right_kernel())

hom_triv = hom(s33,t33, s_triv,t_triv)

hom_rho3 = hom(s33,t33, s3,t3)

There is one copy of ${\mathds{1}}$ and two copies of $\rho_{3}$ in
$\rho_{3}\rho_{3}$. We are facing the problem of decomposing their complement.
In our case, it turns out that it consists of two inequivalent one-dimensional
representations with representation matrices

$\displaystyle\rho_{\zeta}(S)=\begin{pmatrix}1\end{pmatrix},\qquad\rho_{\zeta}(T)=\begin{pmatrix}\zeta\end{pmatrix}\qquad\text{and}\qquad\rho_{\zeta^{2}}(S)=\begin{pmatrix}1\end{pmatrix},\qquad\rho_{\zeta^{2}}(T)=\begin{pmatrix}\zeta^{2}\end{pmatrix}.$

In Sage, we implement them by means of

⬇

K.<zeta> = CyclotomicField(3)

szeta = identity_matrix(K,1)

tzeta = matrix(K,1,[zeta])

szeta2 = identity_matrix(K,1)

tzeta2 = matrix(K,1,[zeta**2])

First, we determine the kernel of the homomorphism that we have determined so
far

⬇

bm = reduce( lambda s,l: s.intersection(matrix([l]).right_kernel())

, [ hom_triv.basis()[0] ]

+ [b[ix::3] for b in hom_rho3.basis() for ix in range(3)]

, VectorSpace(QQ,9) ) \

.basis_matrix().transpose()

srest = bm.solve_right(s33*bm)

trest = bm.solve_right(t33*bm)

Second, we observe that $S$ acts trivially on that kernel, so that it suffices
to decompose the action of $T$ into eigenspaces. In the present case they are
defined over a third order cyclotomic extension of the rationals. In fact,
they are isomorphic to $\rho_{\zeta}$ and $\rho_{\zeta^{2}}$ given above

⬇

hom_zeta = hom(s33,t33, szeta,tzeta)

hom_zeta2 = hom(s33,t33, szeta2,tzeta2)

We construct matrices from the homomorphism spaces that we previously
determined

⬇

phi_zeta = matrix(hom_zeta.basis()[0])

phi_zeta2 = matrix(hom_zeta2.basis()[0])

phi_rho3_1 = matrix([hom_rho3.basis()[0][ix::3] for ix in range(3)])

phi_rho3_2 = matrix([hom_rho3.basis()[1][ix::3] for ix in range(3)])

Assembling the results that we have computed via Sage, we find that with
respect to the given bases, we have homomorphisms

$\displaystyle\phi_{{\mathds{1}}}\colon\ $
$\displaystyle\rho_{3}\otimes\rho_{3}\longrightarrow{\mathds{1}},\qquad$
$\displaystyle\begin{pmatrix}1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&1\end{pmatrix},$
$\displaystyle\phi_{\zeta}\colon\ $
$\displaystyle\rho_{3}\otimes\rho_{3}\longrightarrow\rho_{\zeta},\qquad$
$\displaystyle\begin{pmatrix}1&\zeta+1&-\zeta&\zeta+1&\zeta&-1&-\zeta&-1&-\zeta-1\end{pmatrix},$
$\displaystyle\phi_{\zeta^{2}}\colon\ $
$\displaystyle\rho_{3}\otimes\rho_{3}\longrightarrow\rho_{\zeta^{2}},\qquad$
$\displaystyle\begin{pmatrix}1&-\zeta&\zeta+1&-\zeta&-\zeta-1&-1&\zeta+1&-1&\zeta\end{pmatrix},$
$\displaystyle\phi_{\rho_{3},1}\colon\ $
$\displaystyle\rho_{3}\otimes\rho_{3}\longrightarrow\rho_{3},\qquad$
$\displaystyle\begin{pmatrix}1&0&-1&-1&-1&-2&0&1&-1\\\
-1&0&1&-1&1&0&-2&-1&-1\\\ -1&-2&-1&1&-1&0&0&-1&1\end{pmatrix},$
$\displaystyle\phi_{\rho_{3},2}\colon\ $
$\displaystyle\rho_{3}\otimes\rho_{3}\longrightarrow\rho_{3},\qquad$
$\displaystyle\begin{pmatrix}0&1&-1&-1&0&-3&1&3&0\\\ 0&1&3&-1&0&1&-3&-1&0\\\
0&-3&-1&3&0&1&1&-1&0\end{pmatrix}.$

Summarizing, we find that
$E_{12,\rho_{3}}\,{\mathop{\underline{\otimes}}}\,E_{12,\rho_{3}}$ is
supported on ${\mathds{1}}$, $\rho_{\zeta}$, $\rho_{\zeta^{2}}$, and
$\rho_{3}$. The corresponding subspaces are spanned by elements whose Fourier
coefficients are too large to display them all. We confine ourselves to the
trivial type:

$\displaystyle\frac{564856947200}{1594323}-\frac{1894333004462080000}{84584326707}q-\frac{1261863434802833408000}{28194775569}q^{2}+O\big{(}q^{3}\big{)}$
$\displaystyle\qquad{}\in
E_{12,\rho_{3}}\,{\mathop{\underline{\otimes}}}\,E_{12,\rho_{3}}\cap{\mathrm{M}}\big{(}24\otimes{\mathds{1}}\big{)}.$

Beyond this, we illustrate how to compute the remaining Fourier expansions
with Sage. The tensor square of $E_{12,\rho_{3}}$ with precision at least $n$
is given by the following function $\text{Esq}(n)$

⬇

Esq = lambda n: vector(R,[c1*c2 for c1 in E12rho3(n)

for c2 in E12rho3(n)])

print phi_triv * Esq(3)

print phi_zeta * Esq(3)

print phi_zeta2 * Esq(3)

print (phi_rho3_1 * Esq(3), phi_rho3_2 * Esq(3))

###  3.3 From weights to isomorphism classes of irreducible weights

We have defined weights as complex, finite-dimensional representations of
${K^{(g)}_{\infty}}$ without any further restriction. If we wished to
construct a graded algebra of modular forms whose grading includes all these
weights, this would not be possible. All representations of
${K^{(g)}_{\infty}}$ together do not constitute a set, but rather they are
objects in a category $\operatorname{Rep}\big{(}{K^{(g)}_{\infty}}\big{)}$.
Given this fact, one might be inclined to pass from weights to isomorphism
classes of weights – the skeleton of
$\operatorname{Rep}\big{(}{K^{(g)}_{\infty}}\big{)}$. Alternatively, we can
and will focus on a fixed set
${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}$ of representatives
of isomorphism class. For every
$\sigma\in{\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}$ the space
of modular forms ${{\mathrm{M}}^{(g)}}(\sigma)$ of weight $\sigma$ is a vector
space.

Let us consider to what extent we can define a multiplication. Given
$f\in{{\mathrm{M}}^{(g)}}(\sigma_{f})$ and
$g\in{{\mathrm{M}}^{(g)}}(\sigma_{g})$ with
$\sigma_{f},\sigma_{g}\in{\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}$,
we find that

$\displaystyle fg=f\otimes
g\in{{\mathrm{M}}^{(g)}}(\sigma_{f}\otimes\sigma_{g})$

for their tensor product. Its weight, in general, is not in
${\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}$. Denote by
$\overline{\sigma_{f}\sigma_{g}}$ the corresponding representative. A priori,
$fg$ lies in ${\mathrm{C}}^{\infty}({\mathbb{H}^{(g)}}\rightarrow
V(\sigma_{f}\sigma_{g}))$. We have to obtain from it an element in
${\mathrm{C}}^{\infty}\big{(}{\mathbb{H}^{(g)}}\rightarrow
V(\overline{\sigma_{f}\sigma_{g}})\big{)}$. The most naive way is to choose an
endomorphism $\phi$ in
$\operatorname{\mathrm{Hom}}(\sigma_{f}\sigma_{g},\overline{\sigma_{f}\sigma_{g}})$
and compose it with $fg$. The resulting function $\phi\circ fg$ depends on
$\phi$. More precisely, it depends on $f$ and $g$ up to automorphisms of
$\overline{\sigma_{f}\sigma_{g}}$.

Except in the case $g=1$, there seems to be no natural choice of endomorphisms
for all tensor products. For instance,
${\mathrm{sym}}^{l}{\mathrm{sym}}^{l^{\prime}}$ contains
${\mathrm{sym}}^{l+l^{\prime}}$ in a natural way by realizing
${\mathrm{sym}}^{l}$ and ${\mathrm{sym}}^{l^{\prime}}$ as a representation on
polynomials of degree $l$ and $l^{\prime}$ in $g$ variables. But, for example
if $g=2$, there is no natural inclusion of ${\mathrm{sym}}^{l+l^{\prime}-2}$
into ${\mathrm{sym}}^{l}{\mathrm{sym}}^{l^{\prime}}$ for $l,l^{\prime}\geq 2$.
The situation becomes even more difficult for $g\geq 3$, because then
irreducible representations can occur with multiplicities greater than $1$ in
tensor products of irreducible representations.

Since there is no natural choice of a single homomorphism from
$\sigma_{f}\sigma_{g}$ to $\overline{\sigma_{f}\sigma_{g}}$, we resort to
taking the span over all of them. The product of $f$ and $g$ in this setting
is defined as the space

$\displaystyle\operatorname{span}\big{\\{}\phi\circ
fg\colon\phi\in\operatorname{\mathrm{Hom}}\big{(}\sigma_{f}\sigma_{g},\,\overline{\sigma_{f}\sigma_{g}}\big{)}\big{\\}}\subset\bigoplus_{\sigma\in{\overline{\mathrm{Rep}}}\big{(}{K^{(g)}_{\infty}}\big{)}}{{\mathrm{M}}^{(g)}}(\sigma).$

This product yields a hyper-algebra of modular forms whose weights run through
all isomorphism classes of representations. Since at this point, we have
already lost the algebra structure and arrived at hyper-algebras, there is no
additional harm in considering only irreducible weights. This motivates our
definition in Section 3.1.

###  3.4 Covariant differential operators on hyper-algebras

As in the case of tensor products the image of lowering and raising operators
has, in general, reducible weight. For example, in the case $g=2$ and if
$l\geq 2$, the weight
${\mathrm{R}}\det^{k}{\mathrm{sym}}^{l}=\det^{k}{\mathrm{sym}}^{2}{\mathrm{sym}}^{l}$
allows for a decomposition into irreducible weights
$\det^{k-2}{\mathrm{sym}}^{l-2}$, $\det^{k-1}{\mathrm{sym}}^{l}$, and
$\det^{k}{\mathrm{sym}}^{l+2}$. It is therefore natural to define a vector
space valued action of covariant differential operators.

Let ${\mathbb{C}}[{\mathrm{L}},{\mathrm{R}}]$ be the polynomial algebra in two
formal variables ${\mathrm{L}}$ and ${\mathrm{R}}$. Notation overlaps with the
one for lowering and raising operators, but it will be clear from the context
to what we refer when writing ${\mathrm{L}}f$ or ${\mathrm{R}}f$. The action
of ${\mathrm{L}}$ and ${\mathrm{R}}$ on
${{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}$
is given by

$\displaystyle{\mathrm{L}}f=\operatorname{span}\big{\\{}\phi\circ{\mathrm{L}}_{\sigma_{f}}f\colon\sigma\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}},\,\phi\colon\sigma\rightarrow{\mathrm{L}}\sigma_{f}\big{\\}}\qquad\text{and}$
$\displaystyle{\mathrm{R}}f=\operatorname{span}\big{\\{}\phi\circ{\mathrm{R}}_{\sigma_{f}}f\colon\sigma\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\infty}}\big{)}},\,\phi\colon\sigma\rightarrow{\mathrm{R}}\sigma_{f}\big{\\}}.$

Viewing the differential operators ${\mathrm{L}}$ and ${\mathrm{R}}$ jointly
as a vector-valued Hecke operator at the infinite place acting on almost
holomorphic Siegel modular forms, we set for
$f\in{{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}$

$\displaystyle{\mathbb{T}}_{\infty}={\mathbb{C}}[{\mathrm{T}}_{\infty}],\qquad{\mathrm{T}}_{\infty}f={\mathrm{L}}f+{\mathrm{R}}f\subset{{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}.$

One readily verifies by employing the defining formulas of lowering and
raising operators in [6] that ${\mathrm{T}}_{\infty}$ acts as a hyper-
derivation.

###  3.5 Hecke actions on hyper-algebras

Based on the vector-valued Hecke operators that we have introduced in Section
2.4, we get an additional hyper-module structure on (almost) holomorphic
Siegel modular forms. Let

$\displaystyle{\mathbb{T}}_{\mathrm{f}}={\mathbb{C}}[{\mathrm{T}}_{p}\colon p\
\text{prime}]$

be the polynomial ring in infinitely many formal variables ${\mathrm{T}}_{p}$.
As in the case of covariant differential operators, notation coincides with
the one for actual Hecke operators, but this should not lead to confusion. The
action of ${\mathrm{T}}_{p}$ on
${{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}$
is defined by

$\displaystyle{\mathrm{T}}_{p}f=\operatorname{span}\big{\\{}\phi\circ{\mathrm{T}}_{p}f\colon\rho\in{{\overline{\mathrm{IrrRep}}}\big{(}{K^{(g)}_{\mathrm{f}}}\big{)}},\,\phi\colon{\mathrm{T}}_{p}\rho_{f}\rightarrow\rho\big{\\}}\subset{{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}.$

This equips
${{\sideset{{}^{[\bullet]}}{{}^{(g)}}{\mathop{{\mathrm{M}}}}}(\bullet\otimes\bullet)}$
with the structure of a ${\mathbb{T}}_{\mathrm{f}}$-hyper-module.

###  3.6 The formal Hecke algebra

We combine both formal Hecke algebras into one

$\displaystyle{\mathbb{T}}={\mathbb{T}}_{\infty}{\mathbb{T}}_{{{{\mathbb{A}}_{\mathrm{f}}}}}={\mathbb{C}}[{\mathrm{T}}_{\infty},{\mathrm{T}}_{p}\colon
p\ \text{prime}].$

Recall that ${\mathbb{T}}_{\infty}\subset{\mathbb{T}}$ acts on almost
holomorphic Siegel modular forms by hyper-derivations, and
${\mathbb{T}}_{\mathrm{f}}$ acts by endomorphisms.

##  4 Essential surjectivity of tensor products of modular forms

The hyper-algebra structure and the action of the formal Hecke algebra make it
rather easy to formulate essential surjectivity for products of modular forms.
The blueprint for such results are the ones in [6, 23]. The first of them,
phrased in the language that we have developed reads as follows. For all
$k\geq 8$, all $l,k-l\geq 4$, and all congruence representations $\rho$, we
have

$\displaystyle{\mathrm{M}}^{(1)}(k\otimes\bullet)={\mathrm{E}}^{(1)}(k\otimes\bullet)+{\mathbb{T}}_{\mathrm{f}}E^{(1)}(l)\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\mathrm{f}}E^{(1)}(k-l).$
(4.1)

It is now straightforward to ask for analogues.

###  4.1 A conjecture in the case of genus 2

We start with a conjecture, that we will not prove in this paper. There is
experimental evidence in [15] that

$\displaystyle{\mathrm{M}}^{(2)}(k\otimes{\mathds{1}})={\mathrm{S}}{\mathrm{K}}^{(2)}(k\otimes{\mathds{1}})+\sum_{4\leq
l\leq
k-4}{\mathrm{S}}{\mathrm{K}}^{(2)}(l\otimes{\mathds{1}})\cdot{\mathrm{S}}{\mathrm{K}}^{(2)}(k-l\otimes{\mathds{1}}),$
(4.2)

where ${\mathrm{S}}{\mathrm{K}}$ denotes the space of (holomorphic)
Saito–Kurokawa lifts. In light of this and of formula (4.1) an extension to
all $\rho$ seems possible. Could it be that for sufficiently large $k$, $l$,
and $k-l$, and for any congruence type $\rho$ of genus $2$ Siegel modular
forms, we have

$\displaystyle{\mathrm{M}}^{(2)}(k\otimes\rho)={\mathrm{S}}{\mathrm{K}}^{(2)}(k\otimes\rho)+{\mathbb{T}}_{\mathrm{f}}{\mathrm{S}}{\mathrm{K}}^{(2)}(l\otimes{\mathds{1}})\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\mathrm{f}}{\mathrm{S}}{\mathrm{K}}^{(2)}(k-l\otimes{\mathds{1}})\quad\text{?}$
(4.3)

Note that we have restricted to scalar weights in the above. To cover vector-
valued weights, we have to first study the effect of covariant differential
operators on tensor products.

On inspection of the proofs in [23], we observe that applying vector-valued
Hecke operators on the right-hand side of (4.3) reduces the proof to a certain
unfolding of Petersson scalar products and a nonvanishing of twisted
${\mathrm{L}}$-values. In the case of Siegel modular forms, unfolding to
twisted spinor ${\mathrm{L}}$-series was studied in [10] following older ideas
from [7]. A sufficient nonvanishing result for the purpose of [23] follows
from Waldspurger’s treatment of half-integral weight modular forms [8, 20].
Since the untwisted analogue in the case of genus $2$ Siegel modular forms has
now been established [1], there is hope to prove the following, stronger
statement:

$\displaystyle{\mathrm{M}}^{(2)}(k\otimes\rho)={\mathrm{S}}{\mathrm{K}}^{(2)}(k\otimes\rho)+{\mathbb{T}}_{\mathrm{f}}{\mathrm{E}}^{(2)}(l\otimes{\mathds{1}})\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\mathrm{f}}{\mathrm{S}}{\mathrm{K}}^{(2)}(k-l\otimes{\mathds{1}})\quad\text{?}$

###  4.2 Hecke operators at all places in genus 1

Making use of not only Hecke operators at the finite places, but of the full
formal Hecke algebra ${\mathbb{T}}$, one can strengthen surjectivity results
such as the one in (4.1).

######  Theorem 4.1.

Let $l,l^{\prime}\geq 4$ be even integers, and $\rho$ a congruence type. Then
for every even $k\geq l+l^{\prime}$ we have

$\displaystyle{\mathrm{S}}^{(1)}(k\otimes\rho)\subset{\mathbb{T}}_{\infty}\big{(}{\mathbb{T}}E^{(1)}(l)\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}E^{(1)}(l^{\prime})\big{)}.$
(4.4)

###### Proof.

To ease notation, we suppress the superscript $(1)$ for Eisenstein series
throughout this proof. The notation in this proof is adopted from [23]: We let
${\mathds{1}}_{|D|}$ be the square of the Kronecker character $\epsilon_{D}$
for a negative fundamental discriminant $D$. Note hat ${\mathds{1}}_{|D|}$ is
a trivial non-primitive Dirichlet character. The modular form
$f_{\epsilon_{D}}$ is the twist of $f$ by $\epsilon_{D}$. The induction of
modular forms for $\Gamma_{0}(N)$ to vector-valued modular forms for
${\mathrm{SL}_{2}}({\mathbb{Z}})$ is denoted by ${\mathrm{Ind}}$. The
induction of a Dirichlet character $\chi$ of modulus $N$ from $\Gamma_{0}(N)$
to ${\mathrm{SL}_{2}}({\mathbb{Z}})$ is denoted by $\rho_{\chi}$.

Without loss of generality, we can assume that $l\leq l^{\prime}$. In complete
analogy with [23], we note that the right-hand side of (4.4) is a
${\mathbb{T}}_{{\mathrm{f}}}$-module. It therefore suffices to show the
following: Any scalar multiple of a newform $f$ for the congruence subgroup
$\Gamma_{0}(N)$ vanishes, if for $t=(l+l^{\prime}-k)/2\in{\mathbb{Z}}_{\geq
0}$ and all $D$, the (holomorphic) modular form $f_{\epsilon_{D}}$ is
orthogonal to the almost holomorphic modular form

$\displaystyle{\mathrm{Ind}}\big{(}E_{l,{\mathds{1}}_{|D|}}\big{)}\otimes{\mathrm{R}}^{t}\big{(}E_{l^{\prime},{\mathds{1}}_{|D|},\infty}\big{)}.$

We define the following two vector-valued Eisenstein series:

$\displaystyle
E_{l,{\mathds{1}}_{|D|}}(\tau)=\sum_{n=1}^{\infty}\sigma_{l-1,{\mathds{1}}_{|D|}}(n)q^{n},$
$\displaystyle
E_{l,{\mathds{1}}_{|D|},\infty,s}=\sum_{\gamma\in\Gamma_{\infty}\backslash{\mathrm{SL}_{2}}({\mathbb{Z}})}{\mathfrak{e}}_{\Gamma_{0}(N)}\otimes{\mathfrak{e}}_{\Gamma_{0}(N)}y^{s}\Big{|}_{k,\rho_{{\mathds{1}}_{|D|}}\otimes\rho_{{\mathds{1}}_{|D|}}}\gamma.$

Real-analytic vector-valued Eisenstein series have, for example, appeared in
[19].

We establish the described vanishing condition by relating the Petersson
scalar product to special $L$-values, as in [5, 8, 23]. Combining
regularization and unfolding as is described in detail in [23], we find that

$\displaystyle\big{\langle}{\mathrm{Ind}}f_{\epsilon_{D}},{\mathrm{Ind}}\big{(}E_{l,{\mathds{1}}_{|D|}}\big{)}\otimes{\mathrm{R}}^{t}\big{(}E_{l^{\prime},{\mathds{1}}_{|D|},\infty,s}\big{)}\big{\rangle}$
$\displaystyle\qquad{}=\int_{{\Gamma^{(1)}}\backslash\mathbb{H}^{(1)}}\pi\big{(}f\cdot
E_{l,{\mathds{1}}_{|D|}}(\tau)\cdot{\mathfrak{e}}_{\Gamma_{0}(N)}\otimes{\mathfrak{e}}_{\Gamma_{0}(N)}{\mathrm{R}}^{t}y^{s}\big{)}\frac{{\rm
d}x{\rm d}y}{y^{2-k}},$ (4.5)

where $\pi$ is the projection adjoint to the inclusion

$\displaystyle{\mathds{1}}\longrightarrow\rho_{|D|}\otimes\big{(}\rho_{|D|}\otimes\rho_{N|D|^{2}}\big{)}\otimes\rho_{N|D|^{2}},\qquad
1\longmapsto\sum_{\begin{subarray}{c}\gamma\colon\Gamma_{0}(|D|)\backslash{\mathrm{SL}_{2}}({\mathbb{Z}})\\\
\gamma^{\prime}\colon\Gamma_{0}(|D|N)\backslash{\mathrm{SL}_{2}}({\mathbb{Z}})\end{subarray}}{\mathfrak{e}}_{\gamma}\otimes\big{(}{\mathfrak{e}}_{\gamma}\otimes{\mathfrak{e}}_{\gamma^{\prime}}\big{)}\otimes{\mathfrak{e}}_{\gamma^{\prime}}.$

We will next evaluate the integral, employ analytic continuation, and then
insert $s=0$, to obtain a product of special values of Dirichlet series.

A direct verification shows that $\lim\limits_{s\rightarrow
0}{\mathrm{R}}^{t}_{l^{\prime}}y^{s}=(l^{\prime})^{\uparrow}_{t}y^{-t}$, where
$(a)^{\uparrow}_{n}=a(a+1)\cdots(a+n-1)$ is the upper factorial. The above
scalar product (4.5) therefore equals

$\displaystyle(l^{\prime})^{\uparrow}_{t}\left(\sum_{n=0}^{\infty}\int_{0}^{\infty}c(f_{\epsilon_{D}};n)e(n{\rm
i}y)\sigma_{l-1,{\mathds{1}}_{|D|}}(n)e(n{\rm i}y)y^{-t+s}\frac{{\rm
d}y}{y^{2-k}}\right)_{s=0}$
$\displaystyle\qquad{}=(l^{\prime})^{\uparrow}_{t}\left(\frac{\Gamma(k-1-t)}{(4\pi)^{k-1-t}}\sum_{n=0}^{\infty}\frac{\sigma_{l}(n)c(f_{\epsilon_{D}};n)}{n^{k-1-t+s}}\right)_{s=0}.$

By the extension of Rankin’s result [14] in [23], we find that it does
converge absolutely, if $l>l^{\prime}$ or allows for a suitable analytic
continuation of $l=l^{\prime}$. It can be expressed in terms of special
$L$-values as follows

$\displaystyle\frac{L\big{(}f_{\epsilon_{D}},k-1-t\big{)}L\big{(}f_{\epsilon_{D}}\times{\mathds{1}}_{|D|},k-l-t\big{)}}{L\big{(}{\mathds{1}}_{|D|}{\mathds{1}}_{|D|}{\mathds{1}},k-1-2t-l)\big{)}}.$

The denominator equals $L\big{(}{\mathds{1}}_{|D|},l^{\prime}-1)\big{)}$ by
the relation $k=2t+l+l^{\prime}$. The first factor in the numerator is a
special value of an Euler product, since $t=(k-l-l^{\prime})/2<\frac{k}{2}-2$
and thus $k-1-t>\frac{k}{2}+1$. In order to inspect the second factor in the
numerator, note that $l+t\leq\frac{k}{2}$, since we have assumed that $l\leq
l^{\prime}$. If $l+t<\frac{k}{2}$ then $k-t-l\geq\frac{k}{2}+1$, so that
$L(f_{\epsilon_{D}},k-t-l)$ is the special value of a convergent Euler
product. Otherwise, we infer that the central value
$L\big{(}f_{\epsilon_{D}},\frac{k}{2}\big{)}$ vanishes for all negative
fundamental discriminants $D$. Using Waldspurger’s and Kohnen–Zagier’s results
[8, 20], we can argue as in [23] to finish the proof. More precisely, the
vanishing of $L\big{(}f_{\epsilon_{D}},\frac{k}{2}\big{)}$ implies the
vanishing of the Shintani lift of $f$, because $f$ is a scalar multiple of a
newform. Since the Shintani lift is injective, we find that $f=0$ as desired.
∎

Theorem 4.1 does not cover the case of small weights $k$. The next statement
clarifies that $k<l+l^{\prime}$ cannot appear on the right-hand side of (4.4).

######  Proposition 4.2.

Suppose that $k<l_{1}+l_{2}$ for positive even $l_{1},l_{2}\geq 4$ and
positive even $k$. Then any weight $k$ cusp form $f$ is orthogonal with
respect to the Petersson scalar product to
${\mathrm{R}}^{t_{1}}g_{1}\cdot{\mathrm{R}}^{t_{2}}g_{2}$ for $g_{1}$ and
$g_{2}$ modular forms of weight $l_{1}$ and $l_{2}$, and
$t_{1},t_{2}\in{\mathbb{Z}}_{\geq 0}$ such that $k=l_{1}+2t_{1}+l_{2}+2t_{2}$.

###### Proof.

The almost holomorphic modular form
${\mathrm{R}}^{t_{1}}g_{1}\cdot{\mathrm{R}}^{t_{2}}g_{2}$ has depth
$t_{1}+t_{2}$. That is, it allows for a decomposition

$\displaystyle{\mathrm{R}}^{t_{1}}g_{1}\cdot{\mathrm{R}}^{t_{2}}g_{2}=\sum_{t=0}^{t_{1}+t_{2}}{\mathrm{R}}^{t}h_{t}$

for holomorphic modular forms $h_{t}$ of weight $k-2t$. Since $d>t_{1}+t_{2}$,
each term in this sum is orthogonal to $f$ by Shimura’s orthogonality
relations [17] – also confer [13]. ∎

##  5 Adelic automorphic representations

Automorphic representation theory in most modern settings focuses on adelic
representations. That is, one investigates the right regular representation of
${\mathrm{G}}({\mathbb{A}})$ on
$L^{2}({\mathrm{G}}({\mathbb{Q}})\backslash{\mathrm{G}}({\mathbb{A}}))$, where
${\mathrm{G}}={\mathrm{PGSp}_{g}}$. It is an important aspect of the theory
that one can split any automorphic representation into a restricted tensor
product of local components. The local theory is over $p$-adic fields and the
theory over the infinite places.

In the classical theory of modular forms these two aspects of automorphic
representation theory are reflected by Hecke operators and covariant
differential operators. To support our claim that hyper-algebras of modular
forms together with the action of ${\mathbb{T}}$ are mitigating between the
classical language and the representation theoretic one, we show how to pass
from classical modular forms to the attached local representations and how
some of their aspects can be interpreted in terms of hyper-algebras. In
particular, we rephrase our result in Section 4 in terms of tensor products of
automorphic representations.

###  5.1 Harish-Chandra modules

It is formally correct to work with the group $G={\mathrm{PGSp}_{g}}$ in this
subsection. However, as is common, we will instead work with
$G={\mathrm{Sp}_{g}}$, i.e., $G({\mathbb{R}})={\mathrm{Sp}_{g}}({\mathbb{R}})$
for which all aspects that we discuss here are the same. Further, we adopt
notation from [24] to shorten this exposition.

Given any almost holomorphic Siegel modular form $f$, we can associate to it
an irreducible Harish-Chandra module. Recall that a Harish-Chandra module is
an admissible $({\mathfrak{g}},{\mathrm{K}})$-module. A
$({\mathfrak{g}},{\mathrm{K}})$-module is a representation of
${\mathrm{K}}={K^{(g)}_{\infty}}$ that is simultaneously a
${\mathfrak{g}}$-module with ${\mathfrak{g}}={\mathfrak{sp}_{g}}$, and for
which these two structures are compatible. A detailed definition can be found
in [22, Section 3.3.1]. A $({\mathfrak{g}},{\mathrm{K}})$-module $M$ is called
admissible if it is a unitarizable $K$-representation and if
$\operatorname{\mathrm{Hom}}_{K}(\sigma,M)$ is finite-dimensional for every
finite-dimensional $K$-representation $\sigma$.

Starting with $f$, we produce a function ${\mathrm{A}}_{\infty}(f)$ on $G$,
${\mathrm{A}}_{\infty}$ stands for adelization at the infinite place, which
here is isomorphic to ${\mathbb{R}}$:

$\displaystyle{\mathrm{A}}_{\infty}(f)(g)=\big{(}f\big{|}_{\sigma,\rho}g\big{)}\big{(}iI^{(g)}\big{)}.$

It takes values in $V(\sigma)\otimes V(\rho)$, and when contracting with
$V(\sigma)^{\vee}$, we obtain a space of functions that take values in
$V(\rho)$ and which is a $K$-module isomorphic to $\sigma^{\vee}\otimes
V(\rho)$. We denote the contraction by ${\mathrm{A}}_{\infty}(f)\cdot
V(\sigma)^{\vee}$, and the $({\mathfrak{g}},{\mathrm{K}})$-module generated by
it will be denoted by

$\displaystyle\overline{{\mathrm{A}}}_{\infty}(f)=({\mathfrak{g}},{\mathrm{K}})\big{(}{\mathrm{A}}_{\infty}(f)\cdot
V(\sigma)^{\vee}\big{)}.$

Let ${\mathfrak{k}}\subset{\mathfrak{g}}$ be the Lie algebra of $K$. It was
established in [24] that application of covariant differential operators and
the action of the ${\mathfrak{k}}$-complement
${\mathfrak{m}}\subseteq{\mathfrak{g}}$ commute with passing back and forth
between modular forms and $K$-types in Harish-Chandra modules. Concretely, we
established commutativity of the following diagram, featuring the hyper-
derivations ${\mathrm{L}}$ and ${\mathrm{R}}$:

${f}$${{\mathrm{A}}_{\infty}(f)}$${{\mathrm{R}}f}$${{\mathfrak{m}}^{+}{\mathrm{A}}_{\infty}(f),}$${\mathrm{A}}_{\infty}$${\mathrm{A}}_{\infty}$${\mathrm{R}}$${\mathfrak{m}}^{+}$${f}$${{\mathrm{A}}_{\infty}(f)}$${{\mathrm{L}}f}$${{\mathfrak{m}}^{-}{\mathrm{A}}_{\infty}(f).}$${\mathrm{A}}_{\infty}$${\mathrm{A}}_{\infty}$${\mathrm{L}}$${\mathfrak{m}}^{-}$

The product of vector-valued modular forms then contains information about the
tensor product of Harish-Chandra modules. And thus it yields a lower bound on
smooth functions in the tensor product of $G({\mathbb{R}})$-representations.
Given a Harish-Chandra module we can obtain from it a representation of
${\mathrm{G}}({\mathbb{R}})$. In our case, we denote this representation by
${\mathrm{G}}({\mathbb{R}})\overline{A}_{\infty}$.

######  Corollary 5.1.

Given two almost holomorphic Siegel modular forms $f$ and $g$, we have the
following inclusion of $({\mathfrak{g}},{\mathrm{K}})$-modules:

$\displaystyle\overline{{\mathrm{A}}}_{\infty}\big{(}{\mathbb{T}}_{\infty}f\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\infty}g\big{)}\subseteq\overline{{\mathrm{A}}}_{\infty}(f)\otimes\overline{{\mathrm{A}}}_{\infty}(g).$

By passing back to representations of ${\mathrm{Sp}_{g}}({\mathbb{R}})$, we
find that

$\displaystyle{\mathrm{G}}({\mathbb{R}})\overline{{\mathrm{A}}}_{\infty}\big{(}{\mathbb{T}}_{\infty}f\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\infty}g\big{)}\subseteq{\mathrm{G}}({\mathbb{R}})\overline{{\mathrm{A}}}_{\infty}(f)\otimes{\mathrm{G}}({\mathbb{R}})\overline{{\mathrm{A}}}_{\infty}(g).$

###### Proof.

This follows from the commutative diagrams above, except that we have to
verify that the hyper-product
${\mathbb{T}}_{\infty}f\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{\infty}g$
yields an admissible $({\mathfrak{g}},{\mathrm{K}})$-module. This becomes
clear when inspecting the action of the center of ${\mathfrak{k}}$. ∎

###  5.2 The finite places

As opposed to the infinite place it is important to insist on the group
${\mathrm{G}}={\mathrm{PGSp}_{g}}$ when treating the finite places. Recall
that as a maximal compact group over ${{{\mathbb{Q}}_{p}}}$, we choose
${K^{(g)}_{p}}={\mathrm{G}}({{\mathbb{Z}}_{p}})$. Also recall that any type in
this paper is a congruence type, which means that it gives rise to a
representation over ${{{\mathbb{A}}_{\mathrm{f}}}}$.

To approximate at $p$, we need the relation
${\mathrm{G}}({{{\mathbb{Q}}_{p}}})={\mathrm{G}}({\mathbb{Q}}){K^{(g)}_{p}}$.
Consider a vector-valued almost holomorphic Siegel modular form $f$ of type
$\rho=\otimes^{\prime}\rho_{p}$. Given
$gk\in{\mathrm{G}}({{{\mathbb{Q}}_{p}}})$ we set

$\displaystyle{\mathrm{A}}_{p}(f)(gk)=\rho_{p}^{-1}(k)\big{(}f\big{|}_{\sigma,\rho}g\big{)}\big{(}iI^{(g)}\big{)}.$

As a first remark note that $f$ is constant on
$\ker\rho_{p}\subseteq{K^{(g)}_{p}}$. The contraction of ${\mathrm{A}}_{p}(f)$
with $V(\rho_{p})^{\vee}$ will be denoted by $\overline{{\mathrm{A}}}_{p}(f)$.
It is a space of functions on ${\mathrm{G}}({{{\mathbb{Q}}_{p}}})$ taking
values in $V(\sigma)\otimes V(\otimes^{\prime}_{p\neq
p^{\prime}}\rho_{p^{\prime}})$. As a ${K^{(g)}_{p}}$-module it has isomorphism
type $\rho_{p}^{\vee}\otimes\big{(}V(\sigma)\otimes V(\otimes^{\prime}_{p\neq
p^{\prime}}\rho_{p^{\prime}})\big{)}$.

The commutative diagram corresponding to the one in Section 5.1 is

${f}$${{\mathrm{A}}_{p}(f)}$${{\mathrm{T}}_{p}f}$${\Delta_{p}{\mathrm{A}}_{p}(f),}$${\mathrm{A}}_{p}$${\mathrm{A}}_{p}$${\mathrm{T}}_{p}$$\Delta_{p}$

where $\Delta_{p}$ is the vector space with basis consisting of determinant
$p$ matrices in ${K^{(g)}_{p}}$. In analogy with the infinite case, we write
${\mathrm{G}}({{{\mathbb{Q}}_{p}}})\overline{{\mathrm{A}}}_{p}(f)$ for the
representation over ${{{\mathbb{Q}}_{p}}}$ that corresponds to $f$.

######  Corollary 5.2.

Given Siegel modular forms $f$ and $g$, we have the following inclusion of
${\mathrm{G}}({{{\mathbb{Q}}_{p}}})$-representations:

$\displaystyle{\mathrm{G}}({{{\mathbb{Q}}_{p}}}){\mathrm{A}}_{p}\big{(}{\mathbb{T}}_{p}f\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}_{p}g\big{)}\subseteq{\mathrm{G}}({{{\mathbb{Q}}_{p}}}){\mathrm{A}}_{p}(f)\otimes{\mathrm{G}}({{{\mathbb{Q}}_{p}}}){\mathrm{A}}_{p}(g).$

###  5.3 Global tensor products

We now transfer the statements of Corollaries 5.1 and 5.2 to the global
setting.

######  Theorem 5.3.

Given almost holomorphic Siegel modular forms $f$ and $g$, with automorphic
representations $\varpi(f)$ and $\varpi(g)$ associated with them. Suppose that
for a newform $h$ we have

$\displaystyle{\mathrm{Ind}}(h)\in{\mathbb{T}}f\,{\mathop{\underline{\otimes}}}\,{\mathbb{T}}g.$

Then the associated automorphic representation $\varpi(h)$ is contained in the
tensor product of those associated with $f$ and $g$:

$\displaystyle\varpi(h)\lhook\joinrel\relbar\joinrel\rightarrow\varpi(f)\otimes\varpi(g).$

###### Proof.

Using strong approximation, we obtain a map from holomorphic Siegel modular
forms $f$ to functions

$\displaystyle{\mathrm{A}}(f)={\mathrm{A}}_{\infty}(f)\otimes\Big{(}\bigotimes_{p}{\mathrm{A}}_{p}(f)\Big{)}$

on ${\mathrm{G}}({\mathbb{A}})$, which generate an automorphic representation
$\varpi(f)=\varpi_{\infty}(f)\otimes\otimes^{\prime}_{p}\varpi_{p}(f)$
associated with $f$. Note that the Harish-Chandra module attached to a
representation of ${\mathrm{G}}({\mathbb{R}})$ consists of the subspace of
smooth vectors. In particular, it is a subspace of $V(\varpi_{\infty}(f))$.
The commutative diagrams in Sections 5.1 and 5.2 show that
${\mathrm{A}}_{\infty}({\mathbb{T}}_{\infty}f)\subset V(\varpi_{\infty})$ and
${\mathrm{A}}_{p}({\mathbb{T}}_{p}f)\subset V(\varpi_{p})$. ∎

As a consequence of Theorem 5.3, we can reinterpret (4.1) and (4.4) in terms
of tensor products of global representations. This provides a proof of Theorem
1.2.

### Acknowledgements

The author was partially supported by Vetenskapsrådet Grant 2015-04139. The
author is grateful to the referees for various helpful comments.

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