††footnotetext: 2020 Mathematics Subject Classification: Primary 11F46, 11F66,
11F72
Key words and phrases. Low-lying zeros, spinor $L$-function, standard
$L$-function, Petersson formula, non-vanishing

# Weighted Low-lying Zeros of L-functions Attached to Siegel Modular Forms

Shifan Zhao

###### Abstract.

In this paper, we study weighted low-lying zeros of spinor and standard
$L$-functions attached to degree 2 Siegel modular forms. We show the symmetry
type of weighted low-lying zeros of spinor $L$-functions is symplectic, for
test functions whose Fourier transform have support in $(-1,1)$, extending the
previous range $(-\frac{4}{15},\frac{4}{15})$ in [KST12]. We then show the
symmetry type of weighted low-lying zeros of standard $L$-functions is also
symplectic. We further extend the range of support by performing an average
over weight. As an application, we discuss non-vanishing of central values of
those $L$-functions.

###### Contents

  1. 1 Introduction
  2. 2 Spinor and Standard L-functions
    1. 2.1 The Spinor L-function
    2. 2.2 The Standard L-function
    3. 2.3 Combinatorial Relations
  3. 3 The Petersson Formula
  4. 4 Proof of Main Theorems
    1. 4.1 Proof of Theorem 1.1
    2. 4.2 Proof of Theorem 1.2
    3. 4.3 Proof of Theorem 1.3
  5. 5 Application to Non-vanishing of Central Values
    1. 5.1 Proof of Corollary 1.1
    2. 5.2 Further Discussion

##  1\. Introduction

D. Hilbert and G. Pólya suggested that non-trivial zeros of the Riemann zeta
function $\zeta(s)$ correspond to eigenvalues of a self-adjoint operator on
some Hilbert space. The first evidence of such a connection was found by H. L.
Montgomery [Mon73], who investigated the pair correlation of non-trivial zeros
of $\zeta(s)$ and conjectured that it is, as pointed out by F. J. Dyson, the
same as the pair correlation of eigenvalues of random Hermitian or unitary
matrices of large order, also known as the Gaussian Unitary Ensemble (GUE)
model. This conjecture of Montgomery was later supported by numerical results
by A. M. Odlyzko [Odl87], based on values for the first $10^{5}$ zeros and for
zeros number $10^{12}+1$ to $10^{12}+10^{5}$. The local spacing between these
sample zeros matches prediction by the GUE model quite well.

Z. Rudnick and P. Sarnak [RS96] extended Montgomery’s work by computing the
general $n$-level correlation function of zeros of any principal $L$-function
$L(s,\pi)$ attached to a cuspidal automorphic representation $\pi$ of
GL${}_{m}(\mathbb{A}_{\mathbb{Q}})$ (in a restricted range). Their answer is
universal and is precisely the one predicted by the GUE model. Numerical
evidence were found by R. Rumely [Rum93] for primitive Dirichlet
$L$-functions, and by M. O. Rubinstein [Rub98] for Hasse-Weil $L$-functions of
3 distinct elliptic curves and for the Hecke $L$-function associated to
Ramanujan’s $\tau$-function.

Although the $n$-level correlation statistic of zeros of any fixed principal
automorphic $L$-function obeys the universal GUE law, there is another
statistic, called the $n$-level density of low-lying zeros, that is sensitive
to families. N. Katz and Sarnak [KS99] studied low-lying zeros of zeta
functions of varieties over finite fields (the ”function field” analogue). For
these they indicated that a spectral interpretation exists in terms of
eigenvalues of Frobenius on cohomology groups. On the number field side,
although many results concerning low-lying zeros have been proved, it is still
not clear where their spectral nature comes from. See also [KS99a] for a nice
survey on these topics.

Before stating our results, we first describe the problem in general terms.
Let $\mathcal{F}_{Q}$ be a family of automorphic forms, ordered by certain
conductor $Q\geq 1$. To each $f\in\mathcal{F}_{Q}$ one associates an
$L$-function

(1.1) $L(s,f)=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}},$

which converges absolutely for $s\in\mathbb{C}$ in certain right half-plane.
We assume that $L(s,f)$ continues analytically to the whole complex plane
$\mathbb{C}$ with possibly finitely many poles and satisfies a functional
equation

(1.2) $\Lambda(s,f)=L_{\infty}(s,f)L(s,f)=\varepsilon_{f}\Lambda(1-s,f),$

where $\varepsilon_{f}=\pm 1$ is the root number.

We also assume Generalized Riemann Hypothesis (GRH) for $L(s,f)$. That is,
non-trivial zeros of $L(s,f)$ all lie on the critical line. Thus we may denote
those zeros by

(1.3) $\rho_{f}=\frac{1}{2}+i\gamma_{f},\hskip
8.53581pt\gamma_{f}\in\mathbb{R}.$

Let $\Phi\in\mathcal{S}(\mathbb{R})$ be an even Schwartz function whose
Fourier transform $\hat{\Phi}$ has compact support. We call such a function a
”test function” throughout this paper. To this end we define the 1-level
density of low-lying zeros of $L(s,f)$, with respect to the test function
$\Phi$, to be

(1.4) $D(f;\Phi)=\sum_{\rho_{f}}\Phi\left(\frac{\gamma_{f}}{2\pi}\log
c_{f}\right),$

where $\rho_{f}$ runs through non-trivial zeros of $L(s,f)$, counted with
multiplicity, and $c_{f}$ is a parameter associated with
$f\in\mathcal{F}_{Q}$, comparable to the analytic conductor of $f$ (to be
specified later). The Density Conjecture for low-lying zeros of $L(s,f)$
asserts the following:

######  Conjecture 1.1 (Density Conjecture).

For any even Schwartz function $\Phi$ whose Fourier transform $\hat{\Phi}$ has
compact support, we have

(1.5)
$\lim_{Q\to\infty}\frac{1}{|\mathcal{F}_{Q}|}\sum_{f\in\mathcal{F}_{Q}}D(f;\Phi)=\int_{-\infty}^{\infty}\Phi(x)W(\mathcal{F})(x)dx$

for some distribution $W(\mathcal{F})$ depending only on $\mathcal{F}$.

Many observations and results in [KS99] suggest that the distribution
$W(\mathcal{F})$ depends on the family $\mathcal{F}$ through a symmetry group
$G(\mathcal{F})$. Those symmetry types are orthogonal O, special orthogonal
even SO(even), special orthogonal odd SO(odd), symplectic Sp and unitary U,
with corresponding distributions and Fourier transforms

(1.6) $\displaystyle W(O)(x)=1+\frac{1}{2}\delta_{0}(x)$ $\displaystyle,\hskip
8.53581pt\hat{W}(O)(y)=\delta_{0}(y)+\frac{1}{2},$ (1.7) $\displaystyle
W(SO(\text{even}))(x)=1+\frac{\sin 2\pi x}{2\pi x}$ $\displaystyle,\hskip
8.53581pt\hat{W}(SO(\text{even}))(y)=\delta_{0}(y)+\frac{1}{2}\eta(y),$ (1.8)
$\displaystyle W(SO(\text{odd}))(x)=1-\frac{\sin 2\pi x}{2\pi
x}+\delta_{0}(x)$ $\displaystyle,\hskip
8.53581pt\hat{W}(SO(\text{odd}))(y)=\delta_{0}(y)-\frac{1}{2}\eta(y)+1,$ (1.9)
$\displaystyle W(Sp)(x)=1-\frac{\sin 2\pi x}{2\pi x}$ $\displaystyle,\hskip
8.53581pt\hat{W}(Sp)(y)=\delta_{0}(y)-\frac{1}{2}\eta(y),$ (1.10)
$\displaystyle W(U)(x)=1$ $\displaystyle,\hskip
8.53581pt\hat{W}(U)(y)=\delta_{0}(y),$

where $\delta_{0}$ is the Dirac distribution at $0$, and
$\eta(y)=1,\frac{1}{2},0$ for $|y|<1,|y|=1$ and $|y|>1$ respectively. The
first three distributions of different orthogonal symmetry type have
indistinguishable Fourier transforms within $(-1,1)$, while the symplectic and
unitary symmetry types are distinguishable from the orthogonal ones.

The Density Conjecture 1.1 has been verified for many families (in restrict
ranges). See [ILS00, Rub01, FI03, Gul05, You06, DM06, DM06, GZ11, CK15, ST16,
LM17, KWY20], to name a few. In almost all results towards this direction, the
support of Fourier transform of the test function $\Phi$ is restricted within
certain range. One important question in this topic is how to extend the range
as large as possible, for the full Density Conjecture 1.1 does not require any
condition on the compact support of $\hat{\Phi}$.

One can also consider ”weighted” distribution of low-lying zeros by allowing
certain weight $\omega_{f}$. The weighted average density under consideration
is

(1.11)
$\left(\sum_{f\in\mathcal{F}_{Q}}\omega_{f}\right)^{-1}\sum_{f\in\mathcal{F}_{Q}}\omega_{f}D(f;\Phi).$

Often these weights $\omega_{f}$ contain important arithmetic information such
as central values of $L$-functions, and including them may possibly change the
symmetry type. Recent results in this direction include [KST12, KR19, SS22,
Faz24].

In this article we study weighted low-lying zeros of spinor and standard
$L$-functions attached to degree 2 Siegel modular forms. For a general
introduction on Siegel modular forms, we refer readers to [Kli90, Pit19]

We proceed to describe our results. Let $k\geq 6$ be an even integer. Let
$S_{k}(\Gamma_{2})$ be the space of degree 2 holomorphic Siegel cusp forms of
weight $k$ for the symplectic group $\Gamma_{2}=\text{Sp}_{4}(\mathbb{Z})$.
Each form $F\in S_{k}(\Gamma_{2})$ is a holomorphic function on the Siegel
upper half-plane

(1.12) $\mathbb{H}_{2}=\\{Z=X+iY\in M_{2}(\mathbb{C}):Z=Z^{T},Y>0\\},$

which satisfies the automorphic condition

(1.13) $F((AZ+B)(CZ+D)^{-1})=\det(CZ+D)^{k}F(Z),\hskip
8.53581pt\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in\Gamma_{2},\hskip
8.53581ptZ\in\mathbb{H}_{2}.$

Here and after we use $M_{n}(R)$ to denote the ring of $n\times n$ matrices
over a ring $R$.

The Fourier expansion of $F$ is

(1.14) $F(Z)=\sum_{T\in\mathcal{T}}a_{F}(T)(\det
T)^{\frac{k}{2}-\frac{3}{4}}e(\text{Tr}(TZ)),\hskip
8.53581ptZ\in\mathbb{H}_{2},$

where the summation is taken over the set

(1.15) $\mathcal{T}=\\{T=(t_{ij})\in
M_{2}(\mathbb{R}):T>0,t_{11}\in\mathbb{Z},t_{22}\in\mathbb{Z},2t_{12}=2t_{21}\in\mathbb{Z}\\}.$

We call $a_{F}(T)$ the (normalized) Fourier coefficient of $F$ at $T$. It is
known that $a_{F}(T)\in\mathbb{R}$.

We use $I$ to denote the $2\times 2$ identity matrix. For $F\in
S_{k}(\Gamma_{2})$ we set

(1.16)
$\omega_{F}=\frac{\sqrt{\pi}}{4}(4\pi)^{3-2k}\Gamma(k-3/2)\Gamma(k-2)\frac{a_{F}(I)^{2}}{||F||^{2}}$

to be the ”harmonic” weight attached to $F$, where $||F||$ is the Petersson
norm of $F$ defined by

(1.17) $||F||=\left(\int_{\Gamma_{2}\backslash\mathbb{H}_{2}}|F(Z)|^{2}(\det
Y)^{k}\frac{dXdY}{(\det Y)^{3}}\right)^{\frac{1}{2}}.$

We now choose a basis $H_{k}(\Gamma_{2})$ of $S_{k}(\Gamma_{2})$ consisting of
eigenforms for all Hecke operators (we call such a form a Hecke eigenform). It
is known (see, for example, equation (1.8) in [Blo19]) that

(1.18) $\sum_{F\in H_{k}(\Gamma_{2})}\omega_{F}=1+O(e^{-k}).$

To each form $F\in H_{k}(\Gamma_{2})$ one can attach a degree 4 spinor
$L$-function $L(s,F;\text{spin})$ and a degree 5 standard $L$-function
$L(s,F;\text{std})$, both normalized so that the central point is $s=1/2$. The
analytic conductors of those $L$-functions are of size $k^{2}$ and $k^{4}$,
respectively. Further properties of these $L$-functions are discussed in
section 2.

We assume GRH for both spinor and standard $L$-functions, and denote their
non-trivial zeros on the critical line by

(1.19) $\rho_{F,\text{spin}}=\frac{1}{2}+i\gamma_{F,\text{spin}},\hskip
8.53581pt\rho_{F,\text{std}}=\frac{1}{2}+i\gamma_{F,\text{std}}.$

The corresponding density functions with respect to a test function $\Phi$ are

(1.20) $\displaystyle D(F;\Phi;\text{spin})$
$\displaystyle=\sum_{\rho_{F,\text{spin}}}\Phi\left(\frac{\gamma_{F,\text{spin}}}{2\pi}\log
c_{F;\text{spin}}\right),$ (1.21) $\displaystyle D(F;\Phi;\text{std})$
$\displaystyle=\sum_{\rho_{F,\text{std}}}\Phi\left(\frac{\gamma_{F,\text{std}}}{2\pi}\log
c_{F;\text{std}}\right).$

Our first result concerning low-lying zeros of spinor $L$-functions is as
follows:

######  Theorem 1.1.

Let $\Phi$ be an even Schwartz function whose Fourier transform has support in
$(-1,1)$. For $F\in H_{k}(\Gamma_{2})$, define $D(F;\Phi;\text{spin})$ as in
1.20 with $c_{F;\text{spin}}=k^{2}$ and $\omega_{F}$ as in 1.16. Assume GRH
for $L(s,F;\text{spin})$. Then we have

(1.22) $\lim_{k\to\infty}\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}D(F;\Phi;\text{spin})=\hat{\Phi}(0)-\frac{\Phi(0)}{2}=\int_{-\infty}^{\infty}\Phi(x)W(Sp)(x)dx.$

######  Remark 1.1.

The result above has been obtained in [KST12], but only for test functions
$\Phi$ with supp$(\hat{\Phi})\subset(-\frac{4}{15},\frac{4}{15})$, as an
application of their quantitative local equidistribution result. Here we
extend the range of support to $(-1,1)$. This improvement is crucial in
application to non-vanishing problems, as we will explain in section 5.

Let $H_{k}^{*}(\Gamma_{2})\subset H_{k}(\Gamma_{2})$ denote a Hecke basis of
the space of Saito-Kurokawa lifts (these concepts will be discussed in section
2). Then as a direct corollary of Theorem 1.1 we can establish the following
non-vanishing result:

######  Corollary 1.1.

Assume GRH for $L(s,F;\text{spin})$. Then we have

(1.23) $\liminf_{k\to\infty}\sum_{\begin{subarray}{c}F\in
H_{k}(\Gamma_{2})\backslash H_{k}^{*}(\Gamma_{2})\\\ L(1/2,F;\text{spin})\neq
0\end{subarray}}\omega_{F}\geq\frac{3}{4}.$

######  Remark 1.2.

For comparison, it is shown in [Blo19] that

(1.24) $\sum_{\begin{subarray}{c}F\in H_{k}(\Gamma_{2})\backslash
H_{k}^{*}(\Gamma_{2})\\\ L(1/2,F;\text{spin})\neq
0\end{subarray}}\omega_{F}\gg\frac{1}{\log k},$

unconditionally for large $k$. This follows from asymptotic formulas for the
first and second moments of central values. Although it is not surprising that
GRH would yield much stronger result, one still needs the range of support in
Theorem 1.1 not to be too small to carry out the argument.

For low-lying zeros of standard $L$-functions, we have the following result:

######  Theorem 1.2.

Let $\Phi$ be an even Schwartz function whose Fourier transform has support in
$(-\frac{1}{4},\frac{1}{4})$. For $F\in H_{k}(\Gamma_{2})$, define
$D(F;\Phi;\text{std})$ as in 1.21 with $c_{F;\text{std}}=k^{4}$ and
$\omega_{F}$ as in 1.16. Assume GRH for $L(s,F;\text{std})$. Then we have

(1.25) $\lim_{k\to\infty}\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}D(F;\Phi;\text{std})=\hat{\Phi}(0)-\frac{\Phi(0)}{2}=\int_{-\infty}^{\infty}\Phi(x)W(Sp)(x)dx.$

######  Remark 1.3.

An unweighted version of Theorem 1.2 was established in [KWY20], for test
functions $\Phi$ whose Fourier transform have sufficiently small support (for
a precise range of support, see Proposition 9.3 in [KWY20]). The (unweighted)
symmetry type is also symplectic. For comparison, the symmetry type of low-
lying zeros of spinor $L$-functions changes from orthogonal to symplectic when
weighted by $\omega_{F}$.

We may further extend the range of support in Theorem 1.2 from
$(-\frac{1}{4},\frac{1}{4})$ to $(-\frac{5}{18},\frac{5}{18})$ by performing
an extra (smooth) average over weight $k$. Our result is the following:

######  Theorem 1.3.

Let $\Omega\in C_{c}^{\infty}(0,\infty)$ be such that $\Omega\geq 0$, not
identically 0. Let $\Phi$ be an even Schwartz function whose Fourier transform
has support in $(-\frac{5}{18},\frac{5}{18})$. For $F\in H_{k}(\Gamma_{2})$
and large parameter $K>0$, define $D(F;\Phi;\text{std})$ as in 1.21 with
$c_{F;\text{std}}=K^{4}$ and $\omega_{F}$ as in 1.16. Assume GRH for
$L(s,F;\text{std})$. Then we have

(1.26)
$\lim_{K\to\infty}\left(\sum_{k}\Omega\left(\frac{k}{K}\right)\right)^{-1}\sum_{k}\Omega\left(\frac{k}{K}\right)\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}D(F;\Phi;\text{std})=\hat{\Phi}(0)-\frac{\Phi(0)}{2}=\int_{-\infty}^{\infty}\Phi(x)W(Sp)(x)dx.$

where the summation in $k$ is over even integers.

This article is organized as follows: In section 2, we first review some facts
about spinor and standard $L$-functions. We then work out the combinatorial
relations between certain functions in Satake parameters of a form $F\in
H_{k}(\Gamma_{2})$ and its Fourier coefficients at scalar matrices. These
relations allow us to apply the Petersson formula, which we state in section
3. In section 3 we also take average over weight $k$ in the Petersson formula
and give an upper bound for the off-diagonal term. In section 4 we apply the
results established in previous sections, as well as the explicit formula to
prove Theorems 1.1-1.3. In section 5 we prove Corollary 1.1 and discuss some
other issues concerning non-vanishing of central $L$-values.

##  2\. Spinor and Standard L-functions

Let $F\in H_{k}(\Gamma_{2})$ be a Hecke eigenform. It is known that for each
prime $p$ there are 3 complex numbers
$\alpha_{F,0}(p),\alpha_{F,1}(p),\alpha_{F,2}(p)$, called the Satake
parameters of $F$ at $p$, with certain prescribed properties. See chapter 3 of
[Pit19] for a detailed discussion. In particular, these Satake parameters
satisfy the relation

(2.1) $\alpha_{F,0}(p)^{2}\alpha_{F,1}(p)\alpha_{F,2}(p)=1.$

Let $S_{2k-2}(\Gamma_{1})$ denote the space of holomorphic cusp forms of
weight $2k-2$ for the full modular group $\Gamma_{1}=$ SL${}_{2}(\mathbb{Z})$.
There is an injective Hecke-equivariant linear map

(2.2) $SK:S_{2k-2}(\Gamma_{1})\to S_{k}(\Gamma_{2}),\hskip 8.53581ptf\mapsto
F_{f},$

called the Saito-Kurokawa lifting. We denote the image of $SK$ by
$S_{k}^{*}(\Gamma_{2})$ and call forms in $S_{k}^{*}(\Gamma_{2})$ Saito-
Kurokawa lifts. We also use $H_{k}^{*}(\Gamma_{2})$ for a basis of
$S_{k}^{*}(\Gamma_{2})$ consisting of Hecke eigenforms. There are various ways
to construct such a lifting map. For a construction using half-integral weight
modular forms, see section 2.1.3 of [Pit19] and references there.

The Generalized Ramanujan Conjecture (GRC) asserts that all Satake parameters
$\alpha_{F,i}(p)$ ($i=0,1,2$) have absolute value 1. For elliptic modular
forms, the Ramanujan Conjecture is true due to Deligne’s work [Del74]. For
degree 2 Siegel modular forms $F\in H_{k}(\Gamma_{2})$, GRC is proved to be
true only for non-Saito-Kurokawa lifts, due to R. Weissauer’s result [Wei09].
For Saito-Kurokawa lifts $F_{f}\in H_{k}^{*}(\Gamma_{2})$, GRC is false, as we
will see in Andrianov’s explicit formula 2.8 below.

###  2.1. The Spinor L-function

The spinor $L$-function attached to a Hecke eigenform $F\in H_{k}(\Gamma_{2})$
is defined by a degree 4 Euler product

(2.3)
$\begin{split}L(s,F;\text{spin})&=\prod_{p}\left(1-\frac{\alpha_{F,0}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,0}(p)\alpha_{F,1}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,0}(p)\alpha_{F,2}(p)}{p^{s}}\right)^{-1}\\\
&\times\left(1-\frac{\alpha_{F,0}(p)\alpha_{F,1}(p)\alpha_{F,2}(p)}{p^{s}}\right)^{-1},\end{split}$

which converges absolutely in some right half-plane. By setting

(2.4) $\alpha_{F}(p)=\alpha_{F,0}(p),\hskip
8.53581pt\beta_{F}(p)=\alpha_{F,0}(p)\alpha_{F,1}(p),$

we may rewrite the above Euler product as

(2.5)
$L(s,F;\text{spin})=\prod_{p}\left(1-\frac{\alpha_{F}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\beta_{F}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F}(p)^{-1}}{p^{s}}\right)^{-1}\left(1-\frac{\beta_{F}(p)^{-1}}{p^{s}}\right)^{-1},$

in view of the relation 2.1.

It is proved by A. N. Andrianov [And74] that $L(s,F;\text{spin})$ extends to a
meromorphic function on $\mathbb{C}$, which has a simple pole at
$s=\frac{3}{2}$ if $F$ is a Saito-Kurokawa lift, and is entire otherwise. Its
functional equation takes the following form:

(2.6)
$\Lambda(s,F;\text{spin})=\Gamma_{\mathbb{C}}(s+1/2)\Gamma_{\mathbb{C}}(s+k-3/2)L(s,F;\text{spin})=\Lambda(1-s,F;\text{spin}),$

where $\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$. For $F_{f}\in
H_{k}^{*}(\Gamma_{2})$ a Saito-Kurokawa lift, its spinor $L$-function
decomposes as follows:

(2.7) $L(s,F_{f};\text{spin})=\zeta(s+1/2)\zeta(s-1/2)L(s,f),$

where $L(s,f)$ is the Hecke $L$-function of the elliptic cusp form $f$.

For $F\in H_{k}(\Gamma_{2})$ we have the following Andrianov’s explict formula
[And74]:

(2.8)
$a_{F}(I)L(s,F;\text{spin})=\zeta(s+1/2)L(s+1/2,\chi_{-4})\sum_{n=1}^{\infty}\frac{a_{F}(nI)}{n^{s}},$

where $\chi_{-4}$ is the non-trivial Dirichlet character modulo 4. From this
formula it follows

(2.9) $a_{F}(I)=0\implies a_{F}(nI)=0,\hskip 8.53581ptn\geq 1.$

###  2.2. The Standard L-function

The standard $L$-function attached to a Hecke eigenform $F\in
H_{k}(\Gamma_{2})$ is defined by a degree 5 Euler product

(2.10)
$L(s,F;\text{std})=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,1}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,1}(p)^{-1}}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,2}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F,2}(p)^{-1}}{p^{s}}\right)^{-1},$

which converges absolutely in some right half-plane. Using 2.1, we rewrite
this Euler product as

(2.11)
$\begin{split}L(s,F;\text{std})&=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F}(p)\beta_{F}(p)}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F}(p)^{-1}\beta_{F}(p)}{p^{s}}\right)^{-1}\\\
&\times\left(1-\frac{\alpha_{F}(p)\beta_{F}(p)^{-1}}{p^{s}}\right)^{-1}\left(1-\frac{\alpha_{F}(p)^{-1}\beta_{F}(p)^{-1}}{p^{s}}\right)^{-1}.\end{split}$

The analytic continuation and functional equation of standard $L$-functions
were worked out by S. Böcherer [Böc85]. He proved that $L(s,F;\text{std})$
extends to an entire function and satisfies a functional equation

(2.12)
$\Lambda(s,F;\text{std})=\Gamma_{\mathbb{R}}(s)\Gamma_{\mathbb{C}}(s+k-1)\Gamma_{\mathbb{C}}(s+k-2)L(s,F;\text{std})=\Lambda(1-s,F;\text{std}),$

where $\Gamma_{\mathbb{R}}=\pi^{-\frac{s}{2}}\Gamma(s/2)$ and
$\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$.

###  2.3. Combinatorial Relations

For $F\in H_{k}(\Gamma_{2})$, $m\geq 1$ and prime $p$, we set

(2.13) $\displaystyle c_{m}(p;F)$
$\displaystyle=\alpha_{F}(p)^{m}+\alpha_{F}(p)^{-m}+\beta_{F}(p)^{m}+\beta_{F}(p)^{-m},$
(2.14) $\displaystyle\tau_{2m}(p;F)$
$\displaystyle=1+\alpha_{F}(p)^{m}\beta_{F}(p)^{m}+\alpha_{F}(p)^{m}\beta_{F}(p)^{-m}+\alpha_{F}(p)^{-m}\beta_{F}(p)^{m}+\alpha_{F}(p)^{-m}\beta_{F}(p)^{-m}$

to be the $m^{th}$-power sum of local parameters of $L(s,F;\text{spin})$ and
$L(s,F;\text{std})$ at $p$ respectively.

The main goal of this section is to find expressions of these power sums in
terms of Fourier coefficients of $F$ at scalar matrices, for $m=1,2$, under
the assumption that $a_{F}(I)\neq 0$. Note that the condition $a_{F}(I)\neq 0$
is not a direct consequence of $F\neq 0$, unlike in the elliptic case, where a
primitive form $f$ vanishes if and only if its first Fourier coefficient
vanishes. In fact, determining whether $a_{F}(I)=0$ or not is a difficult
problem because $\omega_{F}$ is intimately connected to central values of
spinor $L$-functions (Böcherer’s conjecture). However, as we shall see later
in section 4, making this assumption here does no harm to our argument. Our
result is as follows:

######  Lemma 2.1.

Let $F\in H_{k}(\Gamma_{2})$ be a Hecke eigenform. For any prime $p$ and
$m\geq 1$, define $c_{m}(p;F)$ and $\tau_{2m}(p;F)$ as in 2.13 and 2.14.
Assume that $a_{F}(I)\neq 0$, and set
$U_{m}(p;F)=\frac{a_{F}(p^{m}I)}{a_{F}(I)}$. Also set
$\lambda_{p}=1+\chi_{-4}(p)$ and $\mu_{p}=\chi_{-4}(p)$, where $\chi_{-4}$ is
the non-trivial Dirichlet character modulo 4. Then we have

$\displaystyle c_{1}(p;F)$
$\displaystyle=U_{1}(p;F)+\frac{\lambda_{p}}{\sqrt{p}},$ $\displaystyle
c_{2}(p;F)$
$\displaystyle=-U_{1}(p;F)^{2}+2U_{2}(p;F)+\frac{\lambda_{p}^{2}-2\mu_{p}}{p},$
$\displaystyle\tau_{2}(p;F)$
$\displaystyle=U_{1}(p;F)^{2}-U_{2}(p;F)+\frac{\lambda_{p}}{\sqrt{p}}U_{1}(p;F)+\frac{\mu_{p}}{p}-1,$
$\displaystyle\tau_{4}(p;F)$
$\displaystyle=-U_{3}(p;F)U_{1}(p;F)+U_{2}(p;F)^{2}+\frac{\lambda_{p}}{\sqrt{p}}U_{2}(p;F)U_{1}(p;F)+\left(\frac{\mu_{p}}{p}-1\right)U_{1}(p;F)^{2}$
$\displaystyle-\frac{\lambda_{p}}{\sqrt{p}}U_{3}(p;F)+\left(\frac{\lambda_{p}^{2}-2\mu_{p}}{p}\right)U_{2}(p;F)+\left(\frac{\lambda_{p}\mu_{p}}{p^{\frac{3}{2}}}-2\frac{\lambda_{p}}{\sqrt{p}}\right)U_{1}(p;F)+\frac{\mu_{p}^{2}}{p^{2}}-\frac{\lambda_{p}^{2}}{p}+1.$

######  Remark 2.1.

The key feature of Lemma 2.1 is that we are able to express $\tau_{4}(p;F)$
using polynomials of $U_{m}(p;F)$ of degree 2 (and not of higher degree). This
is essential when we deal with weighted low-lying zeros of standard
$L$-functions using the Petersson formula.

###### Proof.

Throughout the proof, $F$ and $p$ are fixed. To save notations we use
$c_{m},\tau_{2m},U_{m},\alpha_{p},\beta_{p}$ to denote
$c_{m}(p;F),\tau_{2m}(p;F),U_{m}(p;F),\alpha_{F}(p),\beta_{F}(p)$
respectively, with the understanding that they depend on $F$ and $p$.

We start with Andrianov’s explicit formula 2.8:

$a_{F}(I)L(s,F;\text{spin})=\zeta(s+1/2)L(s+1/2,\chi_{-4})\sum_{n=1}^{\infty}\frac{a_{F}(nI)}{n^{s}}.$

Using Euler product expansions for the $L$-functions involved, we see that the
two Dirichlet series

(2.15)
$\prod_{p}\left(1-\frac{\alpha_{p}}{p^{s}}\right)\left(1-\frac{\beta_{p}}{p^{s}}\right)\left(1-\frac{\alpha_{p}^{-1}}{p^{s}}\right)\left(1-\frac{\beta_{p}^{-1}}{p^{s}}\right)\sum_{n=1}^{\infty}\frac{a_{F}(nI)a_{F}(I)^{-1}}{n^{s}}$

and

(2.16)
$\prod_{p}\left(1-\frac{1}{p^{s+1/2}}\right)\left(1-\frac{\chi_{-4}(p)}{p^{s+1/2}}\right)$

both converge absolutely in some right half-plane and are equal. Comparing
coefficients of $p^{-as}$, $a=1,2,3,4$, we obtain

(2.17) $\displaystyle-\frac{\lambda_{p}}{\sqrt{p}}$
$\displaystyle=U_{1}-c_{1},$ (2.18) $\displaystyle\frac{\mu_{p}}{p}$
$\displaystyle=U_{2}-U_{1}c_{1}+\tau_{2}+1,$ (2.19) $\displaystyle 0$
$\displaystyle=U_{3}-U_{2}c_{1}+U_{1}(\tau_{2}+1)-c_{1},$ (2.20)
$\displaystyle 0$
$\displaystyle=U_{4}-U_{3}c_{1}+U_{2}(\tau_{2}+1)-U_{1}c_{1}+1.$

We also have elementary relations

(2.21) $\displaystyle c_{1}^{2}$ $\displaystyle=c_{2}+2(\tau_{2}+1),$ (2.22)
$\displaystyle(\tau_{2}+1)^{2}$ $\displaystyle=3+\tau_{4}+4\tau_{2}+2c_{2}.$

From equations 2.17 and 2.18 we obtain directly

(2.23) $\displaystyle c_{1}$
$\displaystyle=U_{1}+\frac{\lambda_{p}}{\sqrt{p}},$ (2.24)
$\displaystyle\tau_{2}$
$\displaystyle=U_{1}^{2}-U_{2}+\frac{\lambda_{p}}{\sqrt{p}}U_{1}+\frac{\mu_{p}}{p}-1.$

Combining equations 2.21, 2.23 and 2.24 we have

(2.25) $c_{2}=-U_{1}^{2}+2U_{2}+\frac{\lambda_{p}^{2}-2\mu_{p}}{p}.$

Using equations 2.22, 2.24 and 2.25 we express $\tau_{4}$ as

(2.26)
$\begin{split}\tau_{4}&=U_{1}^{4}-2U_{2}U_{1}^{2}+2\frac{\lambda_{p}}{\sqrt{p}}U_{1}^{3}+U_{2}^{2}-2\frac{\lambda_{p}}{\sqrt{p}}U_{2}U_{1}+\left(\frac{\lambda_{p}^{2}+2\mu_{p}}{p}-2\right)U_{1}^{2}\\\
&-2\frac{\mu_{p}}{p}U_{2}+\left(2\frac{\lambda_{p}\mu_{p}}{p^{\frac{3}{2}}}-4\frac{\lambda_{p}}{\sqrt{p}}\right)U_{1}+\frac{\mu_{p}^{2}}{p^{2}}-2\frac{\lambda_{p}^{2}}{p}+1.\end{split}$

However, to get the final form of $\tau_{4}$, we must express $U_{1}^{4}$,
$U_{2}U_{1}^{2}$ and $U_{1}^{3}$ using degree 2 polynomials in
$U_{a}(a=1,2,3,4).$ Combining equations 2.19, 2.23 and 2.24, we have

(2.27)
$U_{1}^{3}=2U_{2}U_{1}-\frac{\lambda_{p}}{\sqrt{p}}U_{1}^{2}-U_{3}+\frac{\lambda_{p}}{\sqrt{p}}U_{2}+\left(1-\frac{\mu_{p}}{p}\right)U_{1}+\frac{\lambda_{p}}{\sqrt{p}}.$

Likely, equations 2.20, 2.23 and 2.24 give us

(2.28)
$U_{2}U_{1}^{2}=U_{3}U_{1}+U_{2}^{2}-\frac{\lambda_{p}}{\sqrt{p}}U_{2}U_{1}+U_{1}^{2}-U_{4}+\frac{\lambda_{p}}{\sqrt{p}}U_{3}-\frac{\mu_{p}}{p}U_{2}+\frac{\lambda_{p}}{\sqrt{p}}U_{1}-1.$

Further, we multiply 2.27 by $U_{1}$ and apply 2.27, 2.28 to get

(2.29)
$\begin{split}U_{1}^{4}&=U_{3}U_{1}+2U_{2}^{2}-3\frac{\lambda_{p}}{\sqrt{p}}U_{2}U_{1}+\left(3+\frac{\lambda_{p}^{2}-\mu_{p}}{p}\right)U_{1}^{2}-2U_{4}+3\frac{\lambda_{p}}{\sqrt{p}}U_{3}\\\
&-\left(\frac{\lambda_{p}^{2}+2\mu_{p}}{p}\right)U_{2}+\left(2\frac{\lambda_{p}}{\sqrt{p}}+\frac{\lambda_{p}\mu_{p}}{p^{\frac{3}{2}}}\right)U_{1}-\left(2+\frac{\lambda_{p}^{2}}{p}\right).\end{split}$

Finally, we insert 2.27, 2.28 and 2.29 into 2.26 to get

(2.30)
$\begin{split}\tau_{4}&=-U_{3}U_{1}+U_{2}^{2}+\frac{\lambda_{p}}{\sqrt{p}}U_{2}U_{1}+\left(\frac{\mu_{p}}{p}-1\right)U_{1}^{2}-\frac{\lambda_{p}}{\sqrt{p}}U_{3}+\left(\frac{\lambda_{p}^{2}-2\mu_{p}}{p}\right)U_{2}\\\
&+\left(\frac{\lambda_{p}\mu_{p}}{p^{\frac{3}{2}}}-2\frac{\lambda_{p}}{\sqrt{p}}\right)U_{1}+\frac{\mu_{p}^{2}}{p^{2}}-\frac{\lambda_{p}^{2}}{p}+1.\end{split}$

∎

##  3\. The Petersson Formula

The main tool used in this paper is a spectral summation formula of Petersson
type. This formula was first worked by Y. Kitaoka [Kit84] by computing Fourier
coefficients of Siegel Poincaré series. In this section we introduce this
formula and consider an averaged (over weight) version of it.

We begin by introducing some notations. For $k\geq 6$ even, we set

(3.1) $c_{k}=\frac{\sqrt{\pi}}{4}(4\pi)^{3-2k}\Gamma(k-3/2)\Gamma(k-2).$

For $T,Q\in\mathcal{T}$ we define

(3.2) $\Delta_{k}(T,Q)=\sum_{F\in
H_{k}(\Gamma_{2})}c_{k}\frac{a_{F}(T)a_{F}(Q)}{||F||^{2}}.$

For a matrix $C\in M_{2}(\mathbb{Z})$ with $\det C\neq 0$ (we denote the set
of such matrices by $\mathcal{C}$) and $Q,T\in\mathcal{T}$, define the
symplectic Kloosterman sum to be

(3.3) $K(Q,T;C)=\sum_{D}e(\text{Tr}(AC^{-1}Q+C^{-1}DT)),$

where $D$ runs through the set

(3.4) $\\{D\in M_{2}(\mathbb{Z})\bmod C\Lambda:\begin{pmatrix}A&*\\\
C&D\end{pmatrix}\in\Gamma_{2}\\},$

and $\Lambda$ is the set of $2\times 2$ symmetric integral matrices. By
elementary divisor theory one has the following estimate [Kit84]:

(3.5) $|K(Q,T;C)|\leq|\det C|^{3/2}.$

For $P=\begin{pmatrix}p_{1}&p_{2}/2\\\
p_{2}/2&p_{4}\end{pmatrix},S=\begin{pmatrix}s_{1}&s_{2}/2\\\
s_{2}/2&s_{4}\end{pmatrix}\in\mathcal{T}$ and $c\geq 1$, we define another
exponential sum

(3.6) $H^{\pm}(P,S;c)=\delta_{s_{4}=p_{4}}\sideset{}{{}^{*}}{\sum}_{d_{1}\bmod
c}\sum_{d_{2}\bmod
c}e\left(\frac{\overline{d_{1}}s_{4}d_{2}^{2}\mp\overline{d_{1}}p_{2}d_{2}+s_{2}d_{2}+\overline{d_{1}}p_{1}+d_{1}s_{1}}{c}\mp\frac{p_{2}s_{2}}{2cs_{4}}\right).$

For these we have the trivial bound

(3.7) $|H^{\pm}(P,S;c)|\leq c^{2}.$

For $P\in M_{2}(\mathbb{R})$ with positive eigenvalues
$\lambda_{1},\lambda_{2}>0$ we set

(3.8)
$\mathcal{J}_{k-3/2}(P)=\int_{0}^{\frac{\pi}{2}}J_{k-3/2}(4\pi\sqrt{\lambda_{1}}\sin\theta)J_{k-3/2}(4\pi\sqrt{\lambda_{2}}\sin\theta)\sin\theta
d\theta,$

where $J_{k-3/2}$ is the usual $J$-Bessel function of half-integral order
$k-\frac{3}{2}$. With these notations, we can now state the Petersson formula.

######  Lemma 3.1.

For $T,Q\in\mathcal{T}$ and $k\geq 6$ even, define $\Delta_{k}(T,Q)$ as in
3.2. Then we have

(3.9) $\Delta_{k}(T,Q)=\frac{1}{8}|\text{Aut}(T)|\left(\frac{\det Q}{\det
T}\right)^{\frac{k}{2}-\frac{3}{4}}\delta_{Q\sim
T}+\frac{\sqrt{2}\pi}{8}G_{1,k}(T,Q)+\pi^{2}G_{2,k}(T,Q),$

where $\text{Aut}(T),G_{1,k}(T,Q)$ and $G_{2,k}(T,Q)$ are defined by

(3.10) $\displaystyle\text{Aut}(T)$
$\displaystyle=\\{U\in\text{GL}_{2}(\mathbb{Z}):U^{T}TU=T\\},$ (3.11)
$\displaystyle G_{1.k}(T,Q)$
$\displaystyle=\sum_{\pm}\sum_{s=1}^{\infty}\sum_{c=1}^{\infty}\sum_{U,V}\frac{(-1)^{\frac{k}{2}}}{c^{\frac{3}{2}}s^{\frac{1}{2}}}H^{\pm}(UQU^{T},V^{-1}TV^{-T};c)J_{k-3/2}\left(\frac{4\pi\sqrt{\det(TQ)}}{cs}\right),$
(3.12) $\displaystyle G_{2,k}(T,Q)$
$\displaystyle=\sum_{C\in\mathcal{C}}\frac{K(Q,T;C)}{|\det
C|^{\frac{3}{2}}}\mathcal{J}_{k-3/2}(TC^{-1}QC^{-T}).$

Here the summation $\sum_{U,V}$ in 3.11 is over $U=(u_{ij})/\\{\pm
I\\},V=(v_{ij})\in$ GL${}_{2}(\mathbb{Z})$ such that

(3.13)
$(u_{21},u_{22})Q(u_{21},u_{22})^{T}=(-v_{21},v_{11})T(-v_{21},v_{11})^{T}=s.$

The delta symbol $\delta_{Q\sim T}$ is equal to 1 if $Q$ and $T$ are
equivalent in the sense of quadratic forms, and is equal to 0 otherwise.

######  Remark 3.1.

Following Kitaoka [Kit84], we call the three terms in 3.9 containing
$\delta_{Q\sim T}$, $G_{1,k}(T,Q)$ and $G_{2,k}(T,Q)$ the diagonal term, the
rank 1 term and the rank 2 term respectively. Note that the classical
Petersson formula for elliptic modular forms contains only a diagonal term and
an off-diagonal term.

######  Remark 3.2.

As pointed out by V. Blomer (see Remark 1 in [Blo19]), there are some
numerical errors in Kitaoka’s original derivation of the Petersson formula.
The version that we present here is based on Lemma 1 in [Blo19]. However, our
results do not depend on exact values of those constants.

The main purpose of this section is to establish the following averaged
Petersson formula:

######  Lemma 3.2.

Let $m,n\geq 1$ be positive integers such that $m|n$. For $k\geq 6$ even,
define $\Delta_{k}(mI,nI)$ as in 3.2. Let $\Omega\in C_{c}^{\infty}(0,\infty)$
be such that $\Omega\geq 0$, not identically 0. Then for large $K>0$ we have

(3.14)
$\left(\sum_{k}\Omega\left(\frac{k}{K}\right)\right)^{-1}\sum_{k}\Omega\left(\frac{k}{K}\right)\Delta_{k}(mI,nI)=\delta_{m=n}+O_{j,\epsilon,\Omega}\left(\left(\frac{m^{\frac{3}{2}-\epsilon}n^{-\frac{1}{2}+\epsilon}}{K^{4}}\right)+\left(\frac{(mn)^{2+\epsilon}}{K^{5+2\epsilon}}\right)+\left(\frac{(mn)^{\frac{j}{2}+1}}{K^{2j+3}}\right)\right)$

for any $j\geq 3$ and $\epsilon>0$ small. Here the summation $\sum_{k}$ is
over positive even integers $k\geq 6$.

######  Remark 3.3.

Our Lemma 3.2 can be viewed as a GSp4 analogue of the classical averaged
Petersson formula on GL2. See equation (5.81) in [Iwa97]. The main difficulty
is the presence of a product of two Bessel functions (instead of a single
Bessel function), each of half-integral order (instead of integral order). As
we shall see in the proof below, this can be overcome by applying an integral
representation 3.33 of a product of two Bessel functions.

###### Proof.

After applying the Petersson formula 3.9, we divide the left side of 3.14 into
three terms. We also set $g(x)=\Omega(\frac{x}{K})$ and $\ell=k-3/2$ to save
notations.

We denote the contribution of the diagonal term by $R_{0}$. Thus

(3.15)
$R_{0}=\frac{1}{8}\left(\sum_{k}g(k)\right)^{-1}\sum_{k}g(k)|\text{Aut}(mI)|\left(\frac{n}{m}\right)^{\ell}\delta_{mI\sim
nI}.$

Note that $mI$ and $nI$ define the same quadratic form if and only if $m=n$,
and that $|\text{Aut}(mI)|=8$. Thus the above expression reduces to
$R_{0}=\delta_{m=n}$.

Denote by $R_{1}$ the sum of the rank 1 term over $k$. We have

(3.16)
$R_{1}=\sum_{k}g(k)\sum_{\pm}\sum_{s=1}^{\infty}\sum_{c=1}^{\infty}\sum_{U,V}\frac{(-1)^{\frac{k}{2}}}{c^{\frac{3}{2}}s^{\frac{1}{2}}}H^{\pm}(nUU^{T},mV^{-1}V^{-T};c)J_{\ell}\left(\frac{4\pi
mn}{cs}\right),$

where the sum $\sum_{U,V}$ is over

(3.17) $n(u_{21}^{2}+u_{22}^{2})=m(v_{11}^{2}+v_{21}^{2})=s.$

So in particular $n|s$. Making change of variable $s\mapsto ns$, we may
rewrite $R_{1}$ as

(3.18)
$R_{1}=\sum_{k}g(k)\sum_{\pm}\sum_{s=1}^{\infty}\sum_{c=1}^{\infty}\sum_{U,V}\frac{(-1)^{\frac{k}{2}}}{c^{\frac{3}{2}}(ns)^{\frac{1}{2}}}H^{\pm}(nUU^{T},mV^{-1}V^{-T};c)J_{\ell}\left(\frac{4\pi
m}{cs}\right),$

where $\sum_{U,V}$ is over

(3.19) $u_{21}^{2}+u_{22}^{2}=s,\hskip
8.53581ptv_{11}^{2}+v_{22}^{2}=\frac{n}{m}s.$

These equations have $O(s^{\epsilon})$ and $O((\frac{n}{m}s)^{\epsilon})$
integral solutions, respectively, for any $\epsilon>0$, by the fact

(3.20) $|\\{(x,y)\in\mathbb{Z}^{2}:x^{2}+y^{2}=s\\}|=O(s^{\epsilon}).$

In view of the estimate [GR15]

(3.21) $J_{\ell}(x)\ll\left(\frac{x}{\ell}\right)^{\ell},\hskip
8.53581ptx>0,\ell>\frac{1}{2},$

we may cut-off the sum in 3.18 by $sc\ll\frac{m}{K}$ up to a negligible error.
In this range we change summation order and deal with the inner sum

(3.22) $\sum_{k}g(k)(-1)^{\frac{k}{2}}J_{\ell}\left(\frac{4\pi m}{cs}\right)$

by applying a lemma due to Blomer and A. Corbett (Lemma 20 in [BC22]) to
obtain

(3.23) $\sum_{k}g(k)(-1)^{\frac{k}{2}}J_{\ell}\left(\frac{4\pi
m}{cs}\right)=\omega_{0}\left(\frac{4\pi m}{cs}\right)+e^{\frac{4\pi
im}{cs}}\omega_{+}\left(\frac{4\pi m}{cs}\right)+e^{-\frac{4\pi
im}{cs}}\omega_{-}\left(\frac{4\pi m}{cs}\right),$

where $\omega_{0}(x),\omega_{\pm}(x)$ are some smooth functions on
$(0,\infty)$ satisfying

(3.24) $\displaystyle\omega_{0}(x)$ $\displaystyle\ll_{A}K^{-A},$ (3.25)
$\displaystyle\omega_{\pm}(x)$
$\displaystyle\ll_{A}\left(1+\frac{K^{2}}{x}\right)^{-A},$

for any $A>0$. The contribution of the $\omega_{0}$ term is negligible, while
the contribution of $\omega_{\pm}$ term depends on the size of $x=\frac{4\pi
m}{cs}$. For example, for $x\leq K^{2}$ (i.e. $cs\gg\frac{m}{K^{2}}$), we have

(3.26)
$\omega_{+}(x)\ll_{A}\left(1+\frac{K^{2}}{x}\right)^{-A}\leq\left(\frac{K^{2}}{x}\right)^{-A}\ll_{A}K^{-2A}m^{A}(cs)^{-A},$

for any $A>0$. This estimate, together with 3.7 and 3.20, give rise to

(3.27)
$\begin{split}R_{1}^{+,cs\gg\frac{m}{K^{2}}}&=\sum_{\pm}\sum_{\frac{m}{K^{2}}\ll
sc\ll\frac{m}{K}}\sum_{U,V}\frac{1}{c^{\frac{3}{2}}(ns)^{\frac{1}{2}}}H^{\pm}(nUU^{T},mV^{-1}V^{-T};c)\omega_{+}\left(\frac{4\pi
m}{cs}\right)\\\ &\ll_{\epsilon,A}\sum_{\frac{m}{K^{2}}\ll
sc\ll\frac{m}{K}}c^{-\frac{3}{2}}(ns)^{-\frac{1}{2}}s^{\epsilon}\left(\frac{n}{m}s\right)^{\epsilon}c^{2}K^{-2A}m^{A}c^{-A}s^{-A}\\\
&\ll_{\epsilon}m^{\frac{3}{2}-\epsilon}n^{-\frac{1}{2}+\epsilon}K^{-3},\end{split}$

for any small $\epsilon>0$ if one fixes some $A>3/2$. The case where
$cs\ll\frac{m}{K^{2}}$ is analyzed similarly, and its contribution
$R_{1}^{+,cs\ll\frac{m}{K^{2}}}$ to is again at most
$m^{\frac{3}{2}-\epsilon}n^{-\frac{1}{2}+\epsilon}K^{-3}$. Therefore, we have
obtained that

(3.28)
$R_{1}\ll_{\epsilon}m^{\frac{3}{2}-\epsilon}n^{-\frac{1}{2}+\epsilon}K^{-3},$

for any small $\epsilon>0$.

Denote by $R_{2}$ the sum of the rank 2 term over $k$. Explicitly,

(3.29) $R_{2}=\sum_{k}g(k)\sum_{C\in\mathcal{C}}\frac{K(nI,mI;C)}{|\det
C|^{\frac{3}{2}}}\int_{0}^{\frac{\pi}{2}}J_{\ell}(4\pi\sqrt{\lambda_{1}}\sin\theta)J_{\ell}(4\pi\sqrt{\lambda_{2}}\sin\theta)\sin\theta
d\theta,$

where $\lambda_{1},\lambda_{2}$ are eigenvalues of the matrix
$mnC^{-1}C^{-T}$. We set $\lambda_{\min}$ and $\lambda_{\max}$ to be the
smaller and the larger eigenvalue of $mnC^{-1}C^{-T}$ respectively. Denote by
$||\cdot||_{F}$ the Frobenius matrix norm. Then by Lemma 2 in [Blo19] we have

(3.30) $\lambda_{\min}\ll\frac{mn}{||C||_{F}^{2}}.$

Applying this estimate and 3.21 to
$J_{\ell}(4\pi\sqrt{\lambda_{\min}}\sin\theta)$, and applying the simple
estimate [GR15]

(3.31) $J_{\ell}(x)\ll 1,\hskip 8.53581ptx>0,\ell>\frac{1}{2}$

to $J_{\ell}(4\pi\sqrt{\lambda_{\max}}\sin\theta)$, we may cut off the sum in
$R_{2}$ by $||C||_{F}\ll\frac{\sqrt{mn}}{K}$ up to an negligible error. In
this range we change the summation order and deal with the inner sum

(3.32)
$\sum_{k}g(k)J_{\ell}(4\pi\sqrt{\lambda_{1}}\sin\theta)J_{\ell}(4\pi\sqrt{\lambda_{2}}\sin\theta)$

by making use of the following integral representation of product of two
Bessel functions (See equation (8) on page 47 of [Erd+81]):

(3.33)
$J_{v}(z)J_{v}(\zeta)=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\cos((z-\zeta)\cos\alpha)J_{2v}(2\sqrt{z\zeta}\sin\alpha)d\alpha,\hskip
8.53581pt\Re(v)>-\frac{1}{2},z>0,\zeta>0.$

By choosing $v=\ell$, $z=4\pi\sqrt{\lambda_{1}}\sin\theta$,
$\zeta=4\pi\sqrt{\lambda_{2}}\sin\theta$, and setting

(3.34) $\xi=2\sqrt{z\zeta}\sin\alpha=\frac{8\pi\sqrt{mn}}{\sqrt{|\det
C|}}\sin\theta\sin\alpha,$

we obtain

(3.35)
$\sum_{k}g(k)J_{\ell}(z)J_{\ell}(\zeta)=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\cos((z-\zeta)\cos\alpha)\left(\sum_{k}g(k)J_{2k-3}(\xi)\right)d\alpha.$

Let $r=2k-3$. We have $r\equiv 1\bmod 4$, since $k$ is even. Setting
$g_{1}(x)=g\left(\frac{x+3}{2}\right)$, we have

(3.36) $\sum_{k}g(k)J_{2k-3}(\xi)=\sum_{r\equiv 1\bmod 4}g_{1}(r)J_{r}(\xi).$

From here the method of Newmann series can be applied, in view of the
following integral representation of Bessel functions of integral order
[GR15]:

(3.37) $J_{r}(x)=\int_{-\frac{1}{2}}^{\frac{1}{2}}e(rt)e^{-ix\sin 2\pi t}dt.$

We quote the following result (Lemma 5.8 in [Iwa97]):

(3.38) $4\sum_{r\equiv 1\bmod 4}g_{1}(r)J_{r}(\xi)=g_{1}(\xi)+h(\xi)+O(\xi
c_{3}(g_{1})),$

where $h(\xi)$ and $c_{3}(g_{1})$ are defined by

(3.39) $\displaystyle h(\xi)$ $\displaystyle=\int_{0}^{\infty}g_{1}(\sqrt{2\xi
y})\sin(\xi+y-\frac{\pi}{4})(\pi y)^{-\frac{1}{2}}dy,$ (3.40) $\displaystyle
c_{3}(g_{1})$ $\displaystyle=\int_{-\infty}^{\infty}|\hat{g_{1}}(t)t^{3}|dt.$

We refer readers to section 5.5 of [Iwa97] for a proof of 3.38. Recall that
for $g_{1}$ we have

(3.41) $g_{1}^{(j)}(x)\ll_{j}K^{-j}$

for any $j\geq 0$. Thus by repeated partial integration we have

(3.42) $\displaystyle h(\xi)$ $\displaystyle\ll_{j}(\xi K^{-2})^{j},$ (3.43)
$\displaystyle c_{3}(g_{1})$ $\displaystyle\ll K^{-3}.$

See also (5.73) and (5.74) in [Iwa97].

The contribution of $g_{1}(\xi)$ to $R_{2}$ is

(3.44)
$R_{2}^{g_{1}(\xi)}=\sum_{||C||_{F}\ll\frac{\sqrt{mn}}{K}}\frac{K(nI,mI;C)}{|\det
C|^{\frac{3}{2}}}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\cos((z-\zeta)\cos\alpha)g_{1}(\xi)d\alpha\sin\theta
d\theta.$

In view of the support of $g_{1}$, the sum in 3.44 is confined in the range

(3.45) $\xi=\frac{8\pi\sqrt{mn}}{\sqrt{|\det C|}}\sin\theta\sin\alpha\gg K.$

Thus we have $\det(C)\ll\frac{mn}{K^{2}}$. By the estimate 3.5, we obtain

(3.46) $R_{2}^{g_{1}(\xi)}\ll\sum_{\begin{subarray}{c}0\neq|\det
C|\ll\frac{mn}{K^{2}}\\\
||C||_{F}\ll\frac{\sqrt{mn}}{K}\end{subarray}}1=\sum_{0\neq|d|\ll\frac{mn}{K^{2}}}\sum_{\begin{subarray}{c}\det
C=d\\\
||C||\ll\frac{\sqrt{mn}}{K}\end{subarray}}1=\sum_{0\neq|d|\ll\frac{mn}{K^{2}}}P_{d}\left(C\cdot\frac{mn}{K^{2}}\right),$

where $P_{d}(X)$ is the hyperbolic lattice counting function

(3.47)
$P_{d}(X)=|\\{(\alpha,\beta,\gamma,\delta)\in\mathbb{Z}^{4}:\alpha\delta-\beta\gamma=d,\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\leq
X\\}|,$

and $C>0$ is some constant. For $1\leq d\leq X$ we have the following
asymptotic formula (see Theorem 12.4 in [Iwa02]):

(3.48)
$P_{d}(X)=6\left(\sum_{\tau|d}\tau^{-1}\right)(X+O(d^{\frac{1}{3}}X^{\frac{2}{3}}))\ll
X\log|d|.$

This estimate also applies to $-X\leq d\leq-1$ by symmetry. Thus we have

(3.49)
$R_{2}^{g_{1}(\xi)}\leq\frac{mn}{K^{2}}\sum_{0\neq|d|\ll\frac{mn}{K^{2}}}\log|d|\ll\frac{m^{2}n^{2}}{K^{4}}\log\frac{mn}{K^{2}}\ll_{\epsilon}\left(\frac{(mn)^{2+\epsilon}}{K^{4+2\epsilon}}\right).$

The contributions of $h(\xi)$ and $O(\xi c_{3}(g_{1}))$ are analyzed
similarly, making use the bounds 3.42 and 3.43. We have

(3.50) $\displaystyle R_{2}^{h(\xi)}$
$\displaystyle\ll_{j}\frac{(mn)^{\frac{j}{2}+1}}{K^{2j+2}},$ (3.51)
$\displaystyle R_{2}^{O(\xi c_{3}(g_{1}))}$
$\displaystyle\ll_{\epsilon}\frac{(mn)^{2+\epsilon}}{K^{6+2\epsilon}},$

for any $j\geq 3$ and small $\epsilon>0$. Thus we obtain

(3.52)
$R_{2}\ll_{j,\epsilon}\frac{(mn)^{2+\epsilon}}{K^{4+2\epsilon}}+\frac{(mn)^{\frac{j}{2}+1}}{K^{2j+2}}.$

Combining the estimates of $R_{1}$ and $R_{2}$ above, and that

(3.53) $\sum_{k}g(k)=\sum_{k}\Omega\left(\frac{k}{K}\right)\gg K,$

by our choice of $\Omega$, the proof is now complete. ∎

##  4\. Proof of Main Theorems

In this section we prove Theorems 1.1-1.3. We assume $\hat{\Phi}$ is supported
in $(-\alpha,\alpha)$. We also set $\ell=k-3/2$ to save notations.

###  4.1. Proof of Theorem 1.1

By a standard explicit formula argument (see, for example, [ILS00]), we can
write the density function $D(F;\Phi;\text{spin})$ as

(4.1) $D(F;\Phi;\text{spin})=\frac{2}{2\pi
i}\int_{(2)}\Phi\left(\frac{s-\frac{1}{2}}{2\pi i}\log
k^{2}\right)\frac{\Lambda^{\prime}}{\Lambda}(s,F;\text{spin})ds+2\Phi\left(\frac{\log
k^{2}}{2\pi i}\right)\delta_{F\in H_{k}^{*}(\Gamma_{2})},$

where $\delta_{F\in H_{k}^{*}(\Gamma_{2})}=1$ if $F\in H_{k}^{*}(\Gamma_{2})$
is a Saito-Kurokawa lift (in which case $L(s,F;\text{spin})$ has a pole at
$s=3/2$) and is 0 otherwise. By 2.6 and 2.5 we may further write

(4.2) $\begin{split}D(F;\Phi;\text{spin})&=\frac{2}{\log
k^{2}}\int_{\mathbb{R}}\Phi(x)\left(-\log(2\pi)^{2}+\frac{\Gamma^{\prime}}{\Gamma}\left(1+\frac{2\pi
ix}{\log k^{2}}\right)+\frac{\Gamma^{\prime}}{\Gamma}\left(k-1+\frac{2\pi
ix}{\log k^{2}}\right)\right)dx\\\ &-\frac{2}{\log
k^{2}}\sum_{m=1}^{\infty}\sum_{p}c_{m}(p;F)\frac{\log
p}{p^{m/2}}\hat{\Phi}\left(\frac{m\log p}{\log k^{2}}\right)\\\
&+2\Phi\left(\frac{\log k^{2}}{2\pi i}\right)\delta_{F\in
H_{k}^{*}(\Gamma_{2})}\end{split}$

by shifting contour from $\sigma=2$ to $\sigma=1/2$.

For the integral involving gamma factors, we use the following estimates
[GR15]:

(4.3)
$\displaystyle\frac{\Gamma^{\prime}}{\Gamma}(a+bi)+\frac{\Gamma^{\prime}}{\Gamma}(a-bi)$
$\displaystyle=2\frac{\Gamma^{\prime}}{\Gamma}(a)+O\left(\frac{b^{2}}{a^{2}}\right),\hskip
8.53581pta>0,b\in\mathbb{R},$ (4.4)
$\displaystyle\frac{\Gamma^{\prime}}{\Gamma}(k-1)$ $\displaystyle=\log
k+O\left(\frac{1}{k}\right)$

to get

(4.5) $\frac{2}{\log
k^{2}}\int_{\mathbb{R}}\Phi(x)\left(-\log(2\pi)^{2}+\frac{\Gamma^{\prime}}{\Gamma}\left(1+\frac{2\pi
ix}{\log k^{2}}\right)+\frac{\Gamma^{\prime}}{\Gamma}\left(k-1+\frac{2\pi
ix}{\log k^{2}}\right)\right)dx=\hat{\Phi}(0)+o(1).$

For $c_{1}(p;F)$ and $c_{2}(p;F)$ we sum over $F$ against the weight
$\omega_{F}$. Using 1.16, Lemma 2.1 and the observation 2.9 we obtain

(4.6) $\displaystyle\sum_{F\in H_{k}(\Gamma_{2})}\omega_{F}c_{1}(p;F)$
$\displaystyle=\Delta_{k}(pI,I)+\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(I,I),$
(4.7) $\displaystyle\sum_{F\in H_{k}(\Gamma_{2})}\omega_{F}c_{2}(p;F)$
$\displaystyle=-\Delta_{k}(pI,pI)+2\Delta_{k}(p^{2}I,I)+O\left(\frac{1}{p}\right)\Delta_{k}(I,I).$

Collecting these we have the ”explicit formula”:

(4.8) $\begin{split}\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}D(F;\Phi;\text{spin})&=\hat{\Phi}(0)+o(1)\\\
&-\frac{2}{\log
k^{2}}\sum_{p}\left(\Delta_{k}(pI,I)+\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(I,I)\right)\frac{\log
p}{\sqrt{p}}\hat{\Phi}\left(\frac{\log p}{\log k^{2}}\right)\\\
&-\frac{2}{\log
k^{2}}\sum_{p}\left(-\Delta_{k}(pI,pI)+2\Delta_{k}(p^{2}I,I)+O\left(\frac{1}{p}\right)\Delta_{k}(I,I)\right)\frac{\log
p}{p}\hat{\Phi}\left(\frac{2\log p}{\log k^{2}}\right)\\\ &-\frac{2}{\log
k^{2}}\sum_{m=3}^{\infty}\sum_{p}\left(\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}c_{m}(p;F)\right)\frac{\log
p}{p^{m/2}}\hat{\Phi}\left(\frac{m\log p}{\log k^{2}}\right)\\\
&+2\Phi\left(\frac{\log k^{2}}{2\pi i}\right)\sum_{F_{f}\in
H_{k}^{*}(\Gamma_{2})}\omega_{F_{f}}\end{split}$

We treat the terms $\Delta_{k}(pI,I),\Delta_{k}(pI,pI)$ and
$\Delta_{k}(p^{2}I,I)$ using the Petersson formula 3.9. Take the term
$\Delta_{k}(pI,I)$ for example:

(4.9)
$\Delta_{k}(pI,I)=\frac{\sqrt{2}\pi}{8}G_{1,k}(pI,I)+\pi^{2}G_{2,k}(pI,I).$

The rank 1 term $G_{1,k}(pI,I)$ is

(4.10)
$G_{1,k}(pI,I)=\sum_{\pm}\sum_{s=1}^{\infty}\sum_{c=1}^{\infty}\sum_{U,V}\frac{(-1)^{\frac{k}{2}}}{c^{\frac{3}{2}}(ps)^{\frac{1}{2}}}H^{\pm}(UU^{T},pV^{-1}V^{-T};c)J_{\ell}\left(\frac{4\pi}{cs}\right),$

after a change of variable $s\mapsto ps$, where the summation $\sum_{U,V}$ is
over

(4.11) $u_{21}^{2}+u_{22}^{2}=ps,\hskip 8.53581ptv_{11}^{2}+v_{21}^{2}=s.$

By the estimates 3.7, 3.21 and 3.20, we bound $G_{1,k}(pI,I)$ as

(4.12)
$G_{1,k}(pI,I)\ll\sum_{s=1}^{\infty}\sum_{c=1}^{\infty}(ps)^{\epsilon}s^{\epsilon}c^{-\frac{3}{2}}p^{-\frac{1}{2}}s^{-\frac{1}{2}}c^{2}\left(\frac{4\pi}{cs\ell}\right)^{\ell}\ll
p^{-\frac{1}{2}+\epsilon}\left(\frac{4\pi}{\ell}\right)^{\ell},$

for $k$ sufficiently large. Thus its contribution to 4.8 is at most

(4.13) $\begin{split}\frac{1}{\log
k}\sum_{p}p^{-\frac{1}{2}+\epsilon}\left(\frac{4\pi}{\ell}\right)^{\ell}\frac{\log
p}{\sqrt{p}}\hat{\Phi}\left(\frac{\log p}{\log k^{2}}\right)&\ll\frac{1}{\log
k}\left(\frac{4\pi}{\ell}\right)^{\ell}\sum_{p\leq
k^{2\alpha}}p^{-1+\epsilon}\log p\\\ &\ll\frac{1}{\log
k}\left(\frac{4\pi}{\ell}\right)^{\ell}k^{2\alpha\epsilon}=o(1)\end{split}$

for any $\alpha>0$ as $k\to\infty$.

The rank 2 term $G_{2,k}(pI,I)$ is

(4.14) $G_{2,k}(pI,I)=\sum_{C\in\mathcal{C}}\frac{K(I,pI;C)}{|\det
C|^{3/2}}\int_{0}^{\frac{\pi}{2}}J_{\ell}(4\pi\sqrt{\lambda_{\min}}\sin\theta)J_{\ell}(4\pi\sqrt{\lambda_{\max}}\sin\theta)\sin\theta
d\theta,$

where $\lambda_{\min}$ and $\lambda_{\max}$ are the smaller and larger
eigenvalues of $pC^{-1}C^{-T}$ respectively. By estimates 3.5, 3.21, 3.31 and
3.30, we have

(4.15)
$G_{2,k}(pI,I)\ll\sum_{C\in\mathcal{C}}\left(\frac{4\pi\sqrt{p}\sin\theta}{\ell||C||_{F}}\right)^{\ell}\ll
p^{\frac{\ell}{2}}\left(\frac{4\pi}{\ell}\right)^{\ell},$

for $k$ sufficiently large. Thus its contribution to 4.8 is at most

(4.16) $\frac{1}{\log
k}\sum_{p}p^{\frac{\ell}{2}}\left(\frac{4\pi}{\ell}\right)^{\ell}\frac{\log
p}{\sqrt{p}}\hat{\Phi}\left(\frac{\log p}{\log k^{2}}\right)\ll\frac{1}{\log
k}\left(\frac{4\pi}{\ell}\right)^{\ell}\sum_{p\leq
k^{2\alpha}}p^{\frac{\ell-1}{2}}\log p\ll k^{\alpha}\left(\frac{4\pi
k^{\alpha}}{\ell}\right)^{\ell},$

which goes to 0 as $k\to\infty$ when $\alpha<1$. Thus we have proved the
contribution of $\Delta_{k}(I,pI)$ to 4.8 is small when $\alpha<1$. Other off-
diagonal contributions are estimated similarly, and are all small when
$\alpha<1$. We skip the details here.

The diagonal contribution of $\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(I,I)$ to
4.8 from the $m=1$ term is

(4.17) $\begin{split}-\frac{2}{\log k^{2}}\sum_{p}\frac{\lambda_{p}\log
p}{p}\hat{\Phi}\left(\frac{\log p}{\log k^{2}}\right)&=-\frac{2}{\log
k^{2}}\int_{1}^{\infty}\frac{\log x}{x}\hat{\Phi}\left(\frac{\log x}{\log
k^{2}}\right)d\pi(x)\\\ &=-\frac{2}{\log k^{2}}\int_{1}^{\infty}\frac{\log
x}{x}\hat{\Phi}\left(\frac{\log x}{\log k^{2}}\right)\frac{1}{\log
x}dx+o(1)\\\ &=-2\int_{0}^{\infty}\hat{\Phi}(y)dy+o(1)\\\
&=-\Phi(0)+o(1).\end{split}$

Here we have used the Prime Number Theorem (PNT) for the prime counting
function $\pi(x)$ and the fact the $\lambda_{p}=1+\chi_{-4}(p)$ takes values 0
and 2 for primes $p$ with density $\frac{1}{2}$ each.

The diagonal contribution of $-\Delta_{k}(pI,pI)$ from the $m=2$ term is

(4.18) $\frac{2}{\log k^{2}}\sum_{p}\frac{\log
p}{p}\hat{\Phi}\left(\frac{2\log p}{\log
k^{2}}\right)=\frac{\hat{\Phi}(0)}{2}+o(1).$

It is shown in [KST12] that the $m\geq 3$ terms in 4.8 contribute at most
$O\left(\frac{1}{\log k}\right)$. For non-Saito-Kurokawa lifts $F$ this
follows from the Ramanujan bound $|c_{m}(p;F)|\leq 4$ and the fact

(4.19) $\sum_{m=3}^{\infty}\sum_{p}\frac{\log p}{p^{m/2}}<\infty.$

For a treatment of Saito-Kurokawa lifts, we refer readers to section 5 of
[KST12].

For the last term in 4.8 we use the fact that (see for example page 1754 of
[Blo19])

(4.20)
$\omega_{F_{f}}\ll\frac{1}{k^{3}}\frac{L(1/2,f\times\chi_{-4})}{L(1,\text{sym}^{2}f)}.$

This, combined with the convexity bound for $L(1/2,f\times\chi_{-4})$ and the
lower bound [HL94]

(4.21) $L(1,\text{sym}^{2}f)\gg k^{-\epsilon}$

give us

(4.22) $\sum_{F_{f}\in H_{k}^{*}(\Gamma_{2})}\omega_{F_{f}}=o(1).$

Combining all results above, we finally have

(4.23) $\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}D(F;\Phi;\text{spin})=\hat{\Phi}(0)-\Phi(0)+\frac{\Phi(0)}{2}+o(1)=\hat{\Phi}(0)-\frac{\Phi(0)}{2}+o(1).$

for $\alpha<1$, as $k\to\infty$. This completes the proof of Theorem 1.1.

###  4.2. Proof of Theorem 1.2

The proof is similar to that of Theorem 1.1. We have seen in the proof there
that the symmetry type is determined by diagonal contributions. So we shall
concentrate on those terms and be brief about the rest.

By Lemma 2.1, the $m=1$ term is

(4.24) $\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}\tau_{2}(p;F)=\Delta_{k}(pI,pI)-\Delta_{k}(p^{2}I,I)+\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(pI,I)+\left(\frac{\mu_{p}}{p}-1\right)\Delta_{k}(I,I),$

in which the diagonal contribution of $\Delta_{k}(pI,pI)$ and
$-\Delta_{k}(I,I)$ cancel each other.

The $m=2$ term is

(4.25) $\begin{split}\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}\tau_{4}(p;F)&=-\Delta_{k}(p^{3}I,pI)+\Delta_{k}(p^{2}I,p^{2}I)+\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(p^{2}I,pI)+\left(\frac{\mu_{p}}{p}-1\right)\Delta_{k}(pI,pI)\\\
&-\frac{\lambda_{p}}{\sqrt{p}}\Delta_{k}(p^{3}I,I)+O\left(\frac{1}{p}\right)\Delta_{k}(p^{2}I,I)+O\left(\frac{1}{\sqrt{p}}\right)\Delta_{k}(pI,I)+\left(O\left(\frac{1}{p}\right)+1\right)\Delta_{k}(I,I).\end{split}$

The diagonal contribution from $\Delta_{k}(p^{2}I,p^{2}I),-\Delta_{k}(pI,pI)$
and $\Delta_{k}(I,I)$ is

(4.26) $-\frac{2}{\log k^{4}}\sum_{p}\frac{\log
p}{p}\hat{\Phi}\left(\frac{2\log p}{\log
k^{4}}\right)=-\frac{\hat{\Phi}(0)}{2}+o(1).$

To illustrate why the range of support is restricted to
$(-\frac{1}{4},\frac{1}{4})$, we analyze the contribution of the rank 2 term
$G_{2,k}(pI,pI)$:

(4.27) $G_{2,k}(pI,pI)=\sum_{C\in\mathcal{C}}\frac{K(pI,pI;C)}{|\det
C|^{3/2}}\int_{0}^{\frac{\pi}{2}}J_{\ell}(4\pi\sqrt{\lambda_{\min}}\sin\theta)J_{\ell}(4\pi\sqrt{\lambda_{\max}}\sin\theta)\sin\theta
d\theta,$

where $\lambda_{\min}$ and $\lambda_{\max}$ are eigenvalues of
$p^{2}C^{-1}C^{-T}$. We estimate as before to get

(4.28) $G_{2,k}(pI,pI)\ll\sum_{C\in\mathcal{C}}\left(\frac{4\pi
p\sin\theta}{\ell||C||_{F}}\right)^{\ell}\ll
p^{\ell}\left(\frac{4\pi}{\ell}\right)^{\ell}.$

Thus it contributes at most

(4.29) $\frac{1}{\log
k}\sum_{p}p^{\ell}\left(\frac{4\pi}{\ell}\right)^{\ell}\frac{\log
p}{\sqrt{p}}\hat{\Phi}\left(\frac{\log p}{\log k^{4}}\right)\ll\frac{1}{\log
k}\left(\frac{4\pi}{\ell}\right)^{\ell}\sum_{p\leq
k^{4\alpha}}p^{\ell-\frac{1}{2}}\log p\ll k^{2\alpha}\left(\frac{4\pi
k^{4\alpha}}{\ell}\right)^{\ell},$

which is $o(1)$ as $k\to\infty$ if $\alpha<\frac{1}{4}$. Other off-diagonal
terms are estimated similarly.

###  4.3. Proof of Theorem 1.3

The contribution of gamma factors and the diagonal contribution do not change
upon averaging over $k$ with respect to $\Omega$. To illustrate how we may
extend the range of support form $(-\frac{1}{4},\frac{1}{4})$ to
$(-\frac{5}{18},\frac{5}{18})$, we take the term $\Delta_{k}(pI,pI)$ for
example.

By Lemma 3.2, the off-diagonal part of

(4.30)
$\left(\sum_{k}\Omega\left(\frac{k}{K}\right)\right)^{-1}\sum_{k}\Omega\left(\frac{k}{K}\right)\Delta_{k}(pI,pI)$

is at most

(4.31)
$\frac{p}{K^{4}}+\frac{p^{4+2\epsilon}}{K^{5+2\epsilon}}+\frac{p^{j+2}}{K^{2j+3}}.$

for any $j\geq 3$ and small $\epsilon>0$. It contributes at most

(4.32) $\frac{1}{\log K}\sum_{p\leq
K^{4\alpha}}\left(\frac{p}{K^{4}}+\frac{p^{4+2\epsilon}}{K^{5+2\epsilon}}+\frac{p^{j+2}}{K^{2j+3}}\right)\frac{\log
p}{\sqrt{p}}\ll
K^{6\alpha-4}+K^{18\alpha-5+6\epsilon}+K^{(4j+10)\alpha-(2j+3)},$

which is $o(1)$ if $\alpha<\frac{5}{18}$, by taking $j$ sufficiently large.

Finally, to see

(4.33)
$\hat{\Phi}(0)-\frac{\Phi(0)}{2}=\int_{-\infty}^{\infty}\Phi(x)W(Sp)(x)dx,$

we use the Plancheral formula

(4.34)
$\int_{-\infty}^{\infty}\Phi(x)W(Sp)(x)dx=\int_{-\infty}^{\infty}\hat{\Phi}(y)\hat{W}(Sp)(y)dy$

and the Fourier pair 1.9. The proof is now complete.

##  5\. Application to Non-vanishing of Central Values

###  5.1. Proof of Corollary 1.1

By Theorem 1.1 we have

(5.1) $\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}\sum_{\rho_{F,\text{spin}}}\Phi\left(\frac{\gamma_{F,\text{spin}}}{2\pi}\log
k^{2}\right)<\hat{\Phi}(0)-\frac{\Phi(0)}{2}+\epsilon,$

for any $\epsilon>0$ and $k$ large enough. We further assume

(5.2) $\Phi(x)\geq 0,\hskip 8.53581pt\Phi(0)=1.$

By these conditions we may pick up only the zeros $\rho_{F,\text{spin}}=1/2$
to get

(5.3) $\begin{split}\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}\sum_{\rho_{F,\text{spin}}}\Phi\left(\frac{\gamma_{F,\text{spin}}}{2\pi}\log
k^{2}\right)&\geq\sum_{F\in
H_{k}(\Gamma_{2})}\omega_{F}\cdot\text{ord}_{s=1/2}L(s,F;\text{spin})\\\
&=\sum_{m=2}^{\infty}m\sum_{\text{ord}_{s=1/2}L(s,F;\text{spin})=m}\omega_{F}\\\
&\geq 2\sum_{\text{ord}_{s=1/2}L(s,F;\text{spin})\geq
2}\omega_{F}.\end{split}$

Here we have used the fact that the root number of $L(s,F;\text{spin})$ is
always $+1$. Thus the vanishing order of $L(s,F;\text{spin})$ at $s=1/2$ is
even. These inequalities, together with 1.18, give us

(5.4) $\sum_{L(1/2,F;\text{spin})\neq
0}\omega_{F}>1-\frac{1}{2}\left(\hat{\Phi}(0)-\frac{\Phi(0)}{2}\right)-\epsilon.$

It is discussed in [ILS00] that the Fourier pair

(5.5) $\Phi(x)=\left(\frac{\sin\pi vx}{\pi vx}\right)^{2};\hskip
8.53581pt\hat{\Phi}(y)=\frac{1}{v}\left(1-\frac{|y|}{v}\right),\hskip
8.53581pt|y|<v\hskip 8.53581pt(v>0).$

gives essentially the optimal bound. With this choice we have

(5.6) $\sum_{L(1/2,F;\text{spin})\neq
0}\omega_{F}>\frac{5}{4}-\frac{1}{2v}-\epsilon,$

for any $0<v<1$. Taking liminf in $k$ and $v\to 1$, we have

(5.7) $\liminf_{k\to\infty}\sum_{L(1/2,F;\text{spin})\neq
0}\omega_{F}\geq\frac{3}{4}.$

We can further ignore the contribution of Saito-Kurokawa lifts, in view of
4.22. This completes the proof of Corollary 1.1.

###  5.2. Further Discussion

From the proof of Corollary 1.1 we see that in order to obtain any result on
non-vanishing of central values of $L(s,F;\text{spin})$ or
$L(s,F;\text{std})$, the range of support in the corresponding Density Theorem
must go beyond $(-\frac{2}{5},\frac{2}{5})$. This range is by setting

(5.8) $\frac{5}{4}-\frac{1}{2v}=0.$

The previous range of support $(-\frac{4}{15},\frac{4}{15})$ obtained in
[KST12] for spinor $L$-functions is not large enough, for
$\frac{4}{15}<\frac{2}{5}$. Thus our extension to $(-1,1)$ is significant for
the purpose of non-vanishing.

Unfortunately, for standard $L$-functions, our range of support is still not
large enough to obtain a non-vanishing result, even after performing an
average over weight ($\frac{5}{18}<\frac{2}{5}$). The author would like to
address this problem by establishing a more refined version of Lemma 3.2 in
the future.

## Acknowledgement

The author would like to thank Prof. Wenzhi Luo for his suggestion on the
topic and his valuable comments.

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Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA.

E-mail address<EMAIL_ADDRESS>


utf8

# Arabic Word-level Readability Visualization
for Assisted Text Simplification

Reem Hazim, Hind Saddiki, Bashar Alhafni, Muhamed Al Khalil, Nizar Habash
Computational Approaches to Modeling Language Lab
New York University Abu Dhabi
<EMAIL_ADDRESS>
###### Abstract

utf8

##  1 Introduction

Models for automatic readability assessment and automatic text simplification
are relevant to many natural language processing (NLP) tasks such as
developing pedagogical language technologies that assist students with
learning languages, or teachers with curriculum design and writing assessment,
as well as personalized paraphrasing of NLP systems’ outputs to target
different users with different readability levels. Developing robust models
for readability assessment and simplification requires the creation of large-
scale lexical and annotated resources for training and evaluation. For
example, parallel texts with different paired readability levels can be used
to train readability models as well as simplification models. Figure 1
presents an example from an Arabic novel paired with a simplified version
targeting the fifth-grade readability level. Properly identifying which words
and phrases need to be rewritten in a simplified manner for a specific target
readability level and audience requires word-level readability annotation in a
framework that enables easy editing of the original text, as well as easy
checking on the updated text. To our knowledge, most of the available tools
for readability assessment work on the document or the sentence levels.

Figure 1:  An example original sentence from the Arabic novel “The Knight of
Bani Hamdan” Al-Jarim (1945). The red-marked words are all of readability
level 4 and 5 (difficult) per Al Khalil et al. (2018)’s readability lexicon.
The simplified text rewrites those words into lower (easier) levels (green-
marked words). The English translations are a best attempt to convey the
complexity level of the Arabic word choices to non-Arabic readers.

The system presented in this paper addresses this limitation by focusing on
word-level readability visualization to assist human annotators working on
identifying text readability levels and adjusting texts to simplify them in a
controlled setting. This effort is part of a project on the Simplification of
Arabic Masterpieces for Extensive Reading (SAMER) Al Khalil et al. (2017,
2018, 2020); Jiang et al. (2020). The project goals include the creation of a
lemma-based graded readability lexicon for Arabic and a corpus of parallel
original and simplified texts from Arabic novels (such as those presented in
Figure 1). The project plans to target two different simplified readability
levels: Grades 4-5 (Level III) and Grades 6-8 (Level IV). While our focus is
on Arabic, a language with limited annotated resources for text
simplification, the components we developed can be easily extended to other
languages. The demo system is a Google Docs add-on that includes morphological
analysis and light disambiguation of Arabic text, visualization of word
readability levels, and access to substitution options with their own explicit
readability levels. We make our add-on and its code publicly
available.LABEL:demo,LABEL:api Next, we present some relevant Arabic
linguistic facts (§2), and discuss related work (§3). We then present our
design and implementation decisions (§4). In §5 we discuss some examples and
use cases.

##  2 Relevant Arabic Linguistic Facts

Modern Standard Arabic (MSA) poses many challenges for NLP tasks Habash
(2010). Two in particular are directly relevant to the task at hand, and
affect many of our design decisions: morphological richness and orthographic
ambiguity.333We do not handle dialectal variants in this effort, although we
acknowledge that dialectal differences from MSA are an important factor in
readability assessment, since MSA is not the native variant of Arabic learned
at home Ferguson (1959); Holes (2004); Carroll et al. (2017).

#### Morphological Richness

Arabic employs a combination of templatic, affixational, and cliticization
morphological operations to realize a large number of features such as gender,
number, person, case, state, aspect, voice, and mood, in addition to a number
of attachable pronominal, preposition and determiner clitics. This leads to a
very large number of words to model. To address this aspect, we utilize a
morphological analysis component that is optimized for efficient
representation Graff et al. (2009); Taji et al. (2018).

#### Orthographic Ambiguity

Arabic is commonly written with optional diacritical marks – which are often
omitted – leading to rampant ambiguity. Orthographic ambiguity and
morphological richness interact heavily with each other. For example the word
فردها> frdhA444Arabic HSB transliteration Habash et al. (2007). has four core
lemmas Jiang et al. (2020): the verbs فَرَّد> far$\sim$ad ‘individualize,
separate in units’, and رَدّ> rad$\sim$ ‘answer, return’; and the nouns فَرْد>
fard ‘individual, unit’ and رَدّ> rad$\sim$ ‘response, return’.

Level | Grade | Age | Examples
---|---|---|---
I | 1 | 6 |  بَيْت، كَبير، أكَلَ، عَلى>
|  |  | house, big, to eat, on
II | 2-3 | 7-8 |  ذَهَب، أُسْطُواني، خَدَعَ، إذا >
|  |  | gold, cylindrical, to cheat, if
III | 4-5 | 9-10 |  رِئة، مُعادَلة، مُوَحَّد، أَغْرى>
|  |  | lung, equation, united, to entice
IV | 6-8 | 11-14 | اِقْتِصاد، طُمَأنينة، راقِي، نَكَثَ>
|  |  | economy, tranquility,
|  |  | sophisticated, to breach
V | 9+ | 15 - |  أَدَمة، مِطْياف، لَوْذَع، شُعَبيّ>
|  |  | epidermis, spectroscope,
|  |  | witty, bronchial
Table 1: The five readability levels, their grade equivalencies, and lemma and
English gloss examples, abridged from Al Khalil et al. (2020).

##  3 Related Work

#### Readability Resources

Text readability leveling is relevant to a wide range of NLP applications such
as text simplification and automatic readability assessment. Most research on
readability leveling has focused on English, leading to the development of
many resources Collins-Thompson and Callan (2004); Pitler and Nenkova (2008);
Feng et al. (2010); Vajjala and Meurers (2012); Xia et al. (2016); Nadeem and
Ostendorf (2018); Vajjala and Lučić (2018); Deutsch et al. (2020); Lee et al.
(2021). Specifically for MSA, datasets and modeling approaches have been
created and developed by leveraging text targeted towards L1 readers (native
speakers) Al-Khalifa and Al-Ajlan (2010); Al Tamimi et al. (2014); El-Haj and
Rayson (2016); Khalil et al. (2018) and L2 learners (non-native speakers)
Forsyth (2014); Saddiki et al. (2018). More recently, Al Khalil et al. (2020)
developed a 26,578-lemma lexicon (later extended to over 40,000 lemmas) with a
five-level readability scale. Examples of vocabulary from the different
readability levels and their corresponding grades and ages are shown in Table
1. This lexicon anchors readability at the lemma representation of the words.
We use this lexicon as our reference for readability levels. Jiang et al.
(2020) developed the online Readability Leveled Arabic Thesaurus interface
that leverages Al Khalil et al. (2020)’s lexicon, and extends its
coverage.555http://samer.camel-lab.com/ For a given user input word, this
interface provides the word’s possible lemmas, roots, English glosses, related
Arabic words and phrases from the Arabic WordNet Black et al. (2006), and
readability on a five-level readability scale. We make use of many components
of Jiang et al. (2020)’s interface in our add-on.

#### Readability Visualization

To the best of our knowledge, there has not been much work on developing web-
based visualization tools for word-level readability assessment, neither for
Arabic nor for other languages. Most of the existing tools work on the
document or the sentence levels. Such tools include
Readable666https://readable.com/ and datayze’s Readability
Analyzer777https://datayze.com/readability-analyzer.php for English, and the
recently proposed FABRA for French Wilkens et al.
(2022).888https://cental.uclouvain.be/fabra/ The lack of word-level tools for
Arabic has motivated us to create an easy-to-use Google Docs add-on for word-
level readability visualization.

#### Arabic Morphological Analysis and Disambiguation

There are a number of tools that support Arabic morphological analysis and
disambiguation and specifically lemmatization Pasha et al. (2014); Darwish and
Mubarak (2016); Obeid et al. (2020, 2022). Inspired by the JavaScript Chrome
extension developed by Khalifa et al. (2016) to assist Arabic learners in
understanding text written in MSA or dialectal Arabic (DA), we implement a
version of the Buckwalter core morphological analysis algorithm Buckwalter
(2002) in JavaScript as part of our add-on.

##  4 Design and Implementation

###  4.1 Design Considerations

We designed our interface with the following considerations in mind.

#### Openness and Ease-of-use

The system needs to be powerful and provides additive or complementary value
to existing text editors, so that simplifications and changes can be evaluated
on the fly and with minimal overhead. This needs to be accomplished with
minimal usability tradeoffs.

#### Handling Arabic Ambiguity and Rich Morphology

The system needs to be able to analyze fully inflected words and relate them
to their lemmas and part-of-speech (POS) tags. The lemmas and POS tags will be
used to identify the readability levels from Al Khalil et al. (2020)’s lexicon
and to link with the Arabic WordNet databases Black et al. (2006).
Additionally, the interface needs to provide the users with access to all the
analyses of a given word.

#### Visualizing Readability

The interface needs to provide summary readability statistics in word-token
and word-type spaces over full documents or arbitrary text selections. It
should highlight the words in context in a clear way to indicate intuitively
which words are easier and which are harder. And finally, the interface needs
to provide access to the readability levels of other unchosen analyses of any
word.

#### Access to Word Substitutions

The system should support the text simplification process by displaying
suggestions for related words and phrases, e.g., synonyms, antonyms,
hypernyms, and hyponyms, with different readability levels. We build on the
work of Jiang et al. (2020) who used the Arabic Wordnet to accomplish the
same.

Figure 2: A flowchart depicting the steps that our tool takes to process user
input. First, the user input is pre-processed and tokenized. Next, the lemma
and part-of-speech of each token are determined using a morphological
analyzer. Then, the tool looks up each lemma in the readability database to
identify its readability level. The tool then highlights individual words
accordingly and produces summary statistics describing the overall text
readability.

#### Explicit/Implicit Word Readability Markup

The system should allow the recording of explicit readability levels such that
when the automatic processes make mistakes, users can overwrite them. We want
those corrections and annotations to be persistent across different future
versions of the analyzer and lexicon. At the same time, unnecessary over-
specification can be distracting to the reader or annotator and should be
minimized. The system should support the ability to import and export text
files that could be marked for readability using external tools.

###  4.2 Implementation

#### Google Docs Add-on

We opted to implement our interface as a Google Docs add-on, which allows us
to use one of the world’s most used editing frameworks, without sacrificing
any of Google Docs’ advantages such as multi-author editing and other familiar
word-editing supports. We implemented the tool’s front-end in HTML, CSS and
JavaScript. The back-end was implemented in JavaScript, and it also utilizes
the Apps Script Document Service, which is a JavaScript API used to read and
modify Google Docs programmatically.

#### Readability Analysis and Visualization

The tool analyzes user input in four main steps that are summarized in Figure
2. First, in the back-end, the text is pre-processed and tokenized, and non-
word tokens are discarded. Second, the tokens are fed into the morphological
analysis algorithm, which produces the most likely lemma and POS pair for each
word. Third, we look up the lemmas in the readability database to identify
their readability levels.999We treat Proper nouns (Names) as a separate level.
Finally, we use the Apps Script Document API to highlight words with different
colors according to their readability levels. The tool also presents a summary
of the text’s readability distribution levels in a bar chart colored
consistently with the readability level word highlights.

#### Morphological and Lexical Analyses

Inspired by Khalifa et al. (2016)’s Chrome extension and Obeid et al. (2020)’s
out-of-context MLE disambiguation mode, we implemented a version of the
Buckwalter core morphological analysis algorithm Buckwalter (2002) in
JavaScript as part of our add-on. Besides being used to determine readability
levels, all lemma analyses are presented in a side bar to allow investigating
and reassigning readability levels if needed. It is worth noting that the
readability lexicon we use does not handle lexical polysemy. This is mainly
due to the lexical representation that is used in the lexicon, which follows
the representation of the Standard Arabic Morphological Analyzer (SAMA) Graff
et al. (2009). However, the design of our tool is independent of the
granularity level of lexical representation and therefore, any updates to
these components in the future can be easily integrated in our tool. Figure 3
presents an instance of the SAMER Google Docs add-on with marked up text.

Figure 3: The SAMER Google Docs add-on visualizing word-level and document-
level readability. Figure 4: The different word-level markup modes that are
supported by our tool.

#### Explicit/Implicit Word Readability Markup

By default, the system deterministically specifies a readability level for any
specific word based on its morphological and lexical readability resources.
When disagreement with the automatic levels happen, as in automatic errors or
importing text that was leveled externally, we ensure that the differences
from the deterministic readability levels are not lost. To accomplish this, a
prefix #<i># is explicitly added in front of the word in question forcing the
tool to interpret the word as having readability level of value <i>. For
example, the word كتب> ktb has a readability level of 1. However, the user can
manually assign it a level of 5 by adding #٥># (Indo-Arabic digit 5) in front
of the word, like so: كتب>#٥># #5#ktb . We also provide an interface button as
part of the morphological side bar discussed above to make such assignment.
The add-on also provides multiple markup visualization modes to navigate
between explicit and implicit readability level markup.

  1. (a)

Show: Explicitly mark all words with their readability levels.

  2. (b)

Minimize: Minimize all the markups by setting their font size to 1pt.

  3. (c)

Hide: Remove any markup whose readability level matches the internal level
chosen by the analyzer and only keep the disagreeing markups. By default the
Hide mode also minimizes the markup; however, the user can easily select the
full text and resize it to a preferred font size (Hide+Resize).

  4. (d)

Delete: Delete all markup from the text.

Figure 4 shows the supported markup modes.

Figure 5: An example of selecting a specific word and identifying all of its
analyses with their readability levels.

##  5 SAMER Add-on: Examples and Use Cases

We present some examples of how the SAMER project Google Docs add-on can be
used to analyze the readability of a literary text. We also discuss potential
use cases of our tool across a variety of tasks and how it can be extended to
other languages.

#### Examples

Figure 3 shows the result of using the tool to analyze a short segment of a
novel. After clicking on the Doc Level button at the top, the tool highlights
each word according to its readability level using different colors, and
presents a summary distribution of words in each readability level.Figure 5
shows the result of selecting a specific word (انحلت> AnHlt ‘be disbanded’)
and clicking on the Word Level button at the top. A side bar appears showing
the different lemma analyses by readability level. Various word substitution
alternatives are presented to the user including synonyms, hypernyms and
hyponyms, with their associated readability levels. If the user decides to
change the word, they can simply rewrite it and rerun the readability
analysis. If the user decides to change the automatically assigned readability
level, they can either change it directly manually, or by clicking on the
Assign button to change that specific word’s readability level markup or the
Assign All button to change all of its occurrences in the document.

#### Use Cases

Our goal behind creating an easy-to-use Google Docs add-on tool for Arabic
word-level readability analysis is to enable users to edit texts easily based
on a specific target readability level. We intend for our tool to be used by
human annotators to identify text readability levels and to simplify texts in
a controlled setting. However, we envision that our tool can be used to assist
writers in either making texts more sophisticated (harder readability) or in
providing alternatives for specific words that have the same readability
level.

#### Extending to Other Languages

Although our work focuses on Arabic, the SAMER add-on tool is designed in a
modular way and it can be easily extended to other languages. More concretely,
the following core components are needed to make such an extension possible:
(1) a readability level lexicon that relates lemmas to their readability
levels; (2) a morphological analysis database that specifies prefixes,
suffixes, stems and lemmas, and their co-occurrence compatibilities; (3) a
statistical lemma-based disambiguation model; and (4) synonym, hypernym,
hyponym and antonym lexical databases, such as those found in WordNet Fellbaum
(2010).

##  6 Conclusion

We presented a Google Docs add-on for automatic Arabic word-level readability
visualization. Our add-on includes a lemmatization component that is connected
to a five-level readability lexicon and Arabic WordNet-based substitution
suggestions. The add-on can be used for assessing the reading difficulty of a
text and identifying difficult words as part of the task of manual text
simplification. In future work, we plan on enhancing our tool’s readability
analysis by leveraging additional morphosyntatic features Saddiki et al.
(2018). We will use the add-on to annotate a corpus of parallel original and
simplified texts from Arabic novels.

## Limitations and Ethical Considerations

We acknowledge that the add-on we developed could be used maliciously to: (a)
modify texts under false pretenses, (b) plagiarize, or (c) profile people in a
biased way using their writing style. We also acknowledge that automatic
errors in readability analysis can lead to harmful results even when used with
good intent. We further recognize that the use of highlighting as a
visualization mechanism limits the conventional use of highlighting in text
editing. Another limitation of our work is the lack of extrinsic and intrinsic
evaluation. However, we are not aware of any manually annotated Arabic word-
level readability datasets . We plan to develop such datasets using our tool.
Finally, we acknowledge that further user studies are needed to confirm the
effectiveness of our tool in aiding annotators to perform tasks such as text
simplification.

## Acknowledgements

This project is funded by a New York University Abu Dhabi Research Enhancement
Fund grant. We thank Zhengyang Jiang, Go Inoue, and Ossama Obeid for helpful
discussions.

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