# Deformations of Theta Integrals and A Conjecture of Gross-Zagier

Jan H. Bruinier  ,  Yingkun Li  Fachbereich Mathematik, Technische Universität
Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Tonghai
Yang  Department of Mathematics, University of Wisconsin Madison, Van Vleck
Hall, Madison, WI 53706, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we complete the proof of the conjecture of Gross and Zagier
concerning algebraicity of higher Green functions at a single CM point on the
product of modular curves. The new ingredient is an analogue of the incoherent
Eisenstein series over a real quadratic field, which is constructed as the
Doi-Naganuma theta lift of a deformed theta integral on hyperbolic space.

###### 2020 Mathematics Subject Classification:

11F37, 11F41, 11G15.

###### Contents

  1. 1 Introduction
    1. 1.1 A Problem posed by Gross and Zagier
    2. 1.2 Proof Strategy
    3. 1.3 Outlook and Organization
  2. 2 Preliminaries
    1. 2.1 Differential operators
    2. 2.2 Quadratic Space associated to Real Quadratic Field
    3. 2.3 The Weil Representation and Theta Functions
    4. 2.4 CM Points and Higher Green Functions
    5. 2.5 Product of Modular Curves as a Shimura Variety
    6. 2.6 Eisenstein Series
    7. 2.7 Hecke’s cusp form
    8. 2.8 The deformed theta integral
  3. 3  Doi-Naganuma Lift of Hecke’s Cusp Form
    1. 3.1 Quadratic Spaces
    2. 3.2 Theta Integral
    3. 3.3 Fourier Transform and Siegel-Weil formula
    4. 3.4 Matching Global Sections
    5. 3.5 Matching Local Sections I
    6. 3.6 Matching Local Sections II
  4. 4  Doi-Naganuma Lift of the Deformed Theta Integral
    1. 4.1 Lowering operator action
    2. 4.2 Fourier Expansion of $\tilde{\mathcal{I}}$
    3. 4.3 Rationality of Theta Lifts
  5. 5 Proofs of Theorems

##  1\. Introduction

Just over half a century ago, Doi and Naganuma discovered a Hecke-equivariant
lifting map from a weight $k$ elliptic modular form to a weight $(k,k)$
Hilbert modular form for a real quadratic field $F$ [DN70]. This is a special
case of cyclic base change [JL70], which has now become a basic and useful
tool in the theory of automorphic forms and automorphic representations. By
the exceptional isogeny

(1.1) $\mathrm{O}(2,2)\sim\mathrm{Res}_{F/\mathbb{Q}}\operatorname{SL}_{2},$

the Doi-Naganuma lifting is also an instance of a theta lifting from
$\operatorname{SL}_{2}$ to $\mathrm{O}(2,2)$ [Kud78].

###  1.1. A Problem posed by Gross and Zagier

In the seminal paper [GZ86], Gross and Zagier proved their formula relating
the central derivative of some Rankin-Selberg $L$-functions attached to a
weight 2 level $N$ newform $f$ and the Néron-Tate height pairing of
$f$-isopytical components of Heegner points in the Jacobian of the modular
curve $X_{0}(N)$. This was extended in [GKZ87] to describe the positions of
these Heegner points in the Jacobian using Fourier coefficients of modular
forms. In the degenerate case $N=1$, the Gross-Zagier formula yields a
beautiful factorization formula of the norm of differences of singular moduli
[GZ85].

To calculate the archimedean contribution to the height pairings, one requires
the automorphic Green function

(1.2)
$\begin{split}G^{\Gamma_{0}(N)}_{s}(z_{1},z_{2})&:=-2\sum_{\gamma\in\Gamma_{0}(N)}Q_{s-1}\left(1+\frac{|z_{1}-\gamma
z_{2}|^{2}}{2\Im(z_{1})\Im(\gamma z_{2})}\right),~{}\Re(s)>1,\\\
Q_{s-1}(t)&:=\int^{\infty}_{0}(t+\sqrt{t^{2}-1}\cosh(u))^{-s}du\end{split}$

on $X_{0}(N)\times X_{0}(N)$. It is an eigenfunction with respect to the
Laplacians in $z_{1}$ and $z_{2}$ with eigenvalue $s(1-s)$. The function
vanishes when one of the $z_{i}$ approaches the cusps, and has a logarithmic
singularity along the diagonal. In fact, these properties characterize it
uniquely. Using Hecke operators acting on either $z_{1}$ or $z_{2}$, we can
define

(1.3)
$\begin{split}G^{\Gamma_{0}(N),m}_{s}(z_{1},z_{2})&:=\sum_{\gamma\in\Gamma_{0}(N)\backslash
R_{N},~{}\det(\gamma)=m}G_{s}^{\Gamma_{0}(N)}(z_{1},\gamma z_{2})\\\
&=G^{\Gamma_{0}(N)}_{s}(z_{1},z_{2})\mid
T_{m,z_{1}}=G^{\Gamma_{0}(N)}_{s}(z_{1},z_{2})\mid T_{m,z_{2}},\end{split}$

where $R_{N}:=\\{\left(\begin{smallmatrix}a&b\\\
Nc&d\end{smallmatrix}\right):a,b,c,d\in\mathbb{Z}\\}$. Then
$G^{\Gamma_{0}(N),m}_{s}$ has a logarithmic singularity along the $m$-th Hecke
correspondence $T_{m}\subset X_{0}(N)^{2}$ (see (1.2) in Chapter II of
[GZ86]).

For integral parameters $s=r+1\in\mathbb{N}_{\geq 2}$, these functions are
called higher Green functions. In Section V.1 of [GKZ87], two problems about
these functions were raised. The first one was to give an interpretation of
their values at Heegner points as archimedean contributions of certain higher
weight height pairings. This was answered by Zhang in [Zha97] (see also
[Xue10]), where the Néron-Tate height pairing of Heegner points is replaced by
the arithmetic intersection of Heegner cycles on Kuga-Sato varieties.

The second problem dealt with the algebraicity of higher Green functions at a
single CM point. Let $M^{!,\infty}_{-2r}(\Gamma_{0}(N))$ be the space of
weakly holomorphic modular forms for $\Gamma_{0}(N)$ of weight $-2r<0$ with
poles only at the cusp infinity (see (2.28)). Given
$f=\sum_{m\gg-\infty}c(m)q^{m}\in M^{!,\infty}_{-2r}(\Gamma_{0}(N))$, we call
the following linear combination of higher Green functions

(1.4)
$G^{\Gamma_{0}(N)}_{r+1,f}(z_{1},z_{2}):=\sum_{m\in\mathbb{N}}c(-m)m^{k-1}G^{\Gamma_{0}(N),m}_{r+1}(z_{1},z_{2})$

the principal higher Green function associated to $f$. Along the divisor

$Z_{f}:=\sum_{m\geq 1,~{}c(-m)\neq 0}T_{m}.$

the function $G_{r+1,f}^{\Gamma_{0}(N)}$ has a logarithmic singularity. Using
Serre duality, this function is the same as the higher Green function defined
via relations in Section V.4 of [GZ86] (see Remark 2.5). We say that it is
rational when $f$ has rational Fourier coefficients at the cusp infinity. Even
though the theory of complex multiplication does not directly apply as in the
case of automorphic Green functions, the value of a rational, principal higher
Green function $G^{\Gamma_{0}(N)}_{r+1,f}$ at a single CM point on
$X_{0}(N)\times X_{0}(N)$ should be algebraic in nature, predicted by the
following conjecture (see e.g. [Mel08] and [Via11]).

######  Conjecture 1.1.

Suppose $f\in M_{-2r}^{!,\infty}(\Gamma_{0}(N))$ has rational Fourier
coefficients at the cusp infinity. Then for any CM point $(z_{1},z_{2})\in
X_{0}(N)^{2}\backslash Z_{f}$ with $z_{j}$ having discriminant $d_{j}<0$,
there exists
$\alpha=\alpha(z_{1},z_{2})\in\overline{\mathbb{Q}}\subset\mathbb{C}$ such
that

$G^{\Gamma_{0}(N)}_{r+1,f}(z_{1},z_{2})=|d_{1}d_{2}|^{-r/2}\log|\alpha|.$

Over the years, there have been a lot of partial results towards this
conjecture. When $d_{1}d_{2}$ is a perfect square, this conjecture was proved
in [Zha97] conditional on the non-degeneracy of the height pairing of CM
cycles. Using regularized theta liftings, an analytic proof was given in
[Via11] with restrictions on $N,d_{j}$ and later in full generality in
[BEY21]. When $d_{1}d_{2}$ is not a perfect square, less was known before. For
$N=1,z_{1}=i$ and $r=1$, Mellit proved the conjecture in his thesis [Mel08]
using an algebraic approach. When one averages over the full Galois orbit of
the CM point $(z_{1},z_{2})$, the conjecture follows from [GKZ87] for $r$
even. More partial results are available when one averages over different
Galois orbits [Li22, BEY21] when $N=1$.

Motivated by Conjecture 1.1, the first and third author, together with S.
Ehlen, considered its generalization to the setting of orthogonal Shimura
varieties in [BEY21]. More precisely, let $\mathrm{V}$ be a rational quadratic
space of signature $(n,2)$ with $n\geq 1$, and $X_{K}$ be the Shimura variety
associated to ${\tilde{H}}_{\mathrm{V}}:=\operatorname{GSpin}(\mathrm{V})$ and
a compact open subgroup $K\subset{\tilde{H}}_{\mathrm{V}}(\hat{\mathbb{Q}})$.
For a non-negative integer $r$ and a vector-valued harmonic Maass form $f$ of
weight $1-n/2-2r$, denote by $\Phi_{f}^{r}$ its regularized theta lift (see
[Bru02] or equation (2.42)). This function is an eigenfunction of the
Laplacian on $X_{K}$ and has a logarithmic singularity along the special
divisor $Z_{f}$ associated to $f$ (see (2.43)). We call it a higher Green
function on $X_{K}$, and say that it is principal, resp. rational, if $f$ is
weakly holomorphic, resp. has rational principal part Fourier coefficients.
When $\mathrm{V}=M_{2}(\mathbb{Q})$ and $X_{K}=X_{0}(N)^{2}$, the function
$\Phi_{f}^{r}$ becomes $G^{\Gamma_{0}(N)}_{r+1,f}$ (see Corollary 2.4).

For a totally real field $F$ of degree $d$ and an $F$-quadratic space $W=E$
with $E/F$ a quadratic CM extension, suppose there is an isometry

(1.5) $W_{\mathbb{Q}}:=\mathrm{Res}_{F/\mathbb{Q}}W\hookrightarrow\mathrm{V},$

which in particular implies that $n+2\geq 2d$. Then we obtain a CM cycle
$Z(W)$ on $X_{K}$ from a torus $T_{W}$ in ${\tilde{H}}_{\mathrm{V}}$ (see
section 2.4 for details). Note that $Z(W)$ is defined over $F$, and is the big
CM cycle $Z(W,z_{0}^{\pm})$ in [BKY12]. We denote $Z(W)_{\mathbb{Q}}$ the
union of the $F$-conjugates of $Z(W)$. If $F$ is quadratic, we write
$Z(W)_{\mathbb{Q}}=Z(W)\cup Z(W)^{\prime}$.

In [Li21a], the second author studied the algebraicity of the difference of a
rational, principal $\Phi^{r}_{f}$ at two CM points in $Z(W)$, and was able to
verify the analogue of Conjecture 1.1 in that setting. This opens up the
possibility of proving Conjecture 1.1 when one proves an algebraicity result
for the averaged value $\Phi^{r}_{f}(Z(W))$. In this paper, we complete this
step by proving the following result complementary to [Li21a].

######  Theorem 1.2 (Algebraicity and Factorization).

Let $\Phi_{f}^{r}$ be a rational, principal higher Green function on $X_{K}$.
Suppose that $E/\mathbb{Q}$ is a biquadratic CM number field with the real
quadratic field $F=\mathbb{Q}(\sqrt{D})$, and $Z(W)\cap Z_{f}=\emptyset$. Then
there exists a positive integer $\kappa$ and $a_{1},a_{2}\in F^{\times}$ such
that

(1.6)
$\Phi_{f}^{r}(Z(W))=\frac{1}{\kappa}\left(\log\left|a_{1}\right|+\sqrt{D}\log\left|a_{2}\right|\right).$

For any prime $\mathfrak{p}$ of $F$, the value
$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}(a_{j})$ is given in (5.6). When
$n=2$, we have $a_{j}=1$ for $j\equiv r\bmod{2}$.

######  Remark 1.3.

The denominator $\kappa$ appears as a consequence of our matching of sections
(see Propositions 4.7 and 4.11), and only depends on $Z(W)$ and $r$ when $f$
has integral Fourier coefficients.

Theorem 1.2 also applies to the case $r=0$ when $f$ has zero constant term, in
which case we have $a_{2}=1$. The value of the regularized Borcherds lift
$\Phi_{f}^{0}=\Phi_{f}$ at the $\mathbb{Q}$-rational CM-cycle
$Z(W)_{\mathbb{Q}}$ has been extensively studied. When $n=2$, the CM-value
$\Phi_{f}(Z(W)_{\mathbb{Q}})$ was the subject of the classical work of Gross-
Zagier on singular moduli [GZ85], and generalizations by the first and third
author [BY06]. A crucial ingredient in these works is a real-analytic Hilbert
Eisenstein series $E^{*}$ of parallel weight $1$ over $F$. It is an incoherent
Eisenstein series in the sense of the Kudla program [Kud97]. The arithmetic
Siegel-Weil formula predicts that the Fourier coefficients of its derivative,
$E^{*,\prime}$, are given by arithmetic degrees of special cycles [HY11,
HY12].

For general $F$, one can express the value of a principal automorphic Green
function $\Phi_{f}$ at the CM cycle $Z(W)_{\mathbb{Q}}$ in terms of the
Fourier coefficients of the holomorphic part of an incoherent Eisenstein
series $E^{*,\prime}$ following the idea in [Kud03] (see [BKY12]). For a
harmonic Maass form $f$, the value $\Phi_{f}(Z(W)_{\mathbb{Q}})$ is the
archimedean contribution of the derivative of a Rankin-Selberg $L$-function
involving $\xi(f)$ at $s=0$ [BY09, BKY12, AGHMP18].

On the other hand, to understand CM-values of higher Green functions, it is
not enough to just have the incoherent Eisenstein series. When $r\geq 1$, the
value of $\Phi^{r}_{f}(Z(W)_{\mathbb{Q}})$ can be studied using incoherent
Eisenstein series only when $n\leq 2$ and $r$ is even, as a result of the
limitation of the Rankin-Cohen differential operator used in [GKZ87].
Furthermore, it is more difficult to study the value $\Phi^{r}_{f}$ at a
single CM point in $Z(W)_{\mathbb{Q}}$, even when $r=0$. For this to work, one
would ideally like to express the incoherent Eisenstein series as a sum of
real-analytic modular forms, one for each CM point in $Z(W)_{\mathbb{Q}}$.
When $d=1$, this is possible and the real-analytic modular forms are weight
one harmonic Maass forms that map to the holomorphic theta series attached to
each CM point under a suitable differential operator $\xi$. These harmonic
Maass forms have been extensively studied in [DL15, Ehl17, Via19] and the
appendix of [BEY21]. Using such forms, and the idea to view the higher Green
function on a product of two modular curves partially averaged in the first
variable as a higher Green function for the orthogonal group of signature
$(1,2)$, Ehlen and the first and third authors were able to obtain an
algebraicity result for certain partial averages of
$G_{r+1,f}^{\operatorname{SL}_{2}(\mathbb{Z})}$ at points in
$Z(W)_{\mathbb{Q}}$, leading to a proof of Conjecture 1.1 when $N=1$ and the
class group of the imaginary quadratic field $\mathbb{Q}(\sqrt{d_{1}})$ is an
elementary 2-group (see [BEY21, Theorem 7.13]).

When $d\geq 2$, such decomposition of the incoherent Eisenstein series is not
clear at all. Therefore, the techniques in the works mentioned above cannot be
used to study the CM-value $\Phi^{r}_{f}(Z(W))$. In this paper, we give a
candidate for such decomposition by constructing an analogue
$\tilde{\mathcal{I}}$ of $E^{*,\prime}$ when the CM field $E$ is biquadratic,
in particular $F$ is real quadratic. In particular, it needs to satisfy an
analogue of the differential equation (1.7). Furthermore, its Fourier
coefficients should be logarithms of numbers in $F$, with factorizations
mimicking that of $E^{*,\prime}$. The form $\tilde{\mathcal{I}}$ is a real-
analytic Hilbert modular form of weight $(1,1)$ and defined in Equation (4.2).
It is also the main innovation of this paper, and we will give more details
about it in section 1.2 below. In a nutshell, $\tilde{\mathcal{I}}$ is
constructed by pulling back a deformed theta integral for the dual pair
$(\operatorname{SL}_{2},{\mathrm{SO}}(3,3))$ to the subgroup
${\mathrm{SO}}(2,2)$, and viewing it as a Hilbert modular form via the isogeny
(1.1). Using $\tilde{\mathcal{I}}$ as a substitute for $E^{*,\prime}$, we are
able to prove Theorem 1.2, which is new even for $r=0$. In that case, the
algebraic nature of $a_{1}$ in clear, but its factorization in (5.6) is new.
When $n=2$, this factorization becomes simpler and is given in Corollary 5.2.
Furthermore in Theorem 1.2, we do not impose any restriction on the levels of
CM cycle $Z(W)$ and the Shimura variety $X_{K}$.

Combining Theorem 1.2 with the main result in [Li21a], we deduce the
algebraicity of a rational, principal higher Green function at a single CM
point when $E/\mathbb{Q}$ is biquadratic, hence Conjecture 1.1 in particular.

######  Theorem 1.4.

In the setting of Theorem 1.2, there exists $\kappa\in\mathbb{N}$ and Galois
equivariant maps $\alpha_{1},\alpha_{2}:T_{W}(\hat{\mathbb{Q}})\to
E^{\mathrm{ab}}$ such that

$\Phi_{f}^{r}([z_{0},h])=\frac{1}{\kappa}\left(\log|\alpha_{1}(h)|+\sqrt{D}\log|\alpha_{2}(h)|\right)$

for all $[z_{0},h]\in Z(W)$. Furthermore for $n=2$, we can choose
$\alpha_{j}(h)=1$ for $j\equiv r\bmod{2}$, i.e. there exists a Galois-
equivariant map $\alpha:T_{W}(\hat{\mathbb{Q}})\to E^{\mathrm{ab}}$ such that

$\Phi^{r}_{f}([z_{0},h])=\frac{1}{\kappa\sqrt{D}^{r}}\log|\alpha(h)|$

for all $h\in T_{W}(\hat{\mathbb{Q}})$. In particular Conjecture 1.1 is true.

It is an interesting question to ask for an algebraic interpretation of
Theorem 1.4 analogous to the one in [BY09, AGHMP18] in the case when $r=0$.
One might speculate that for a harmonic Maass form $f$ of weight $1-n/2-2r$,
there is a higher special cycle $Z^{r}(f)$ on a fiber product power of the
Kuga-Satake abelian scheme over the canonical integral model of $X_{K}$. Then
the evaluation of the higher Green function $\Phi^{r}_{f}([z_{0},h])$ should
have an interpretation as an archimedian height pairing of $Z^{r}(f)$ with a
higher CM cycle above $[z_{0},h]$. If $f$ is weakly holomorphic, the cycle
$Z^{r}(f)$ should be rationally equivalent to zero, and therefore the global
height pairing should vanish. Then $\Phi^{r}_{f}([z_{0},h])$ would be equal to
the negative of the non-archimedian intersection pairing and given by
logarithms of algebraic numbers. The results in [Zha97], [Xue10], and [BEY21]
in the case of modular curves support such a speculation. Making such
speculation concrete would require some non-trivial work (see [Mel08] for the
case $n=2$).

###  1.2. Proof Strategy

For simplicity, we focus on the case $n=2$, from which the general case is not
hard to derive (see section 5 for details). Applying the strategy in [Kud03]
and the Rankin-Cohen operator, one can express
$\Phi^{r}_{f}(Z(W))+(-1)^{r}\Phi^{r}_{f}(Z(W)^{\prime})$ as an $F$-linear
combination of Fourier coefficients of the holomorphic part of $E^{*,\prime}$,
which are logarithms of rational numbers. This is a standard procedure
involving the Siegel-Weil formula and Stokes’ Theorem (see e.g. the proof of
Theorem 3.5 in [Li21b]). A crucial property of the incoherent Eisenstein
series is the following differential equation (see [BKY12, Lemma 4.3])

(1.7)
$2(L_{1}+L_{2})E^{*,\prime}(g,0,\Phi^{(1,1)})=E^{*}(g,0,\Phi^{(1,-1)})+E^{*}(g,0,\Phi^{(-1,1)}).$

Here $L_{j}$ are lowering operators in the $j$-th variable, and
$\Phi^{(\epsilon_{1},\epsilon_{2})}=\Phi_{f}\otimes\Phi^{(\epsilon_{1},\epsilon_{2})}_{\infty}$
are Siegel-Weil sections in the degenerate principal series $I(0,\chi)$ with
$\chi=\chi_{E/F}$ being the quadratic Hecke character of $F$ associated to
$E/F$ (see section 2.6 for details). In particular,
$E^{*}(g,0,\Phi^{(\epsilon,-\epsilon)})$ is a coherent Eisenstein series of
weight $(\epsilon,-\epsilon)$ for $\epsilon=\pm 1$. To prove Theorem 1.2, it
suffices to understand
$\Phi^{r}_{f}(Z(W))-(-1)^{r}\Phi^{r}_{f}(Z(W)^{\prime})$, which means we need
a substitute of $E^{*,\prime}$ on the left hand side of (1.7) such that the
right hand side is $E^{*}(g,0,\Phi^{(1,-1)})-E^{*}(g,0,\Phi^{(-1,1)})$.

To obtain this minus sign, we apply the exceptional isogeny in (1.1) and view
the coherent Eisenstein series as modular forms on the group
$H_{0}:={\mathrm{SO}}(V_{0})$ for the quadratic space $V_{0}$ of signature
$(2,2)$ defined in section 3.1. Since $E/\mathbb{Q}$ is biquadratic, there is
an odd character $\varrho=\varrho_{f}\cdot{\mathrm{sgn}}$ of
$[F^{1}]=F^{1}\backslash\mathbb{A}_{F}^{1}$ such that
$\chi=\varrho\circ{\mathrm{Nm}}^{-}$ (see Remark 3.5). By viewing $\varrho$ as
an automorphic form on $H_{1}:={\mathrm{SO}}(V_{1})$ for the quadratic space
$V_{1}=(F,{\mathrm{Nm}})$, we can consider its theta lift following the
diagram

(1.8)
$H_{1}\stackrel{{\scriptstyle\theta_{1}}}{{\to}}G\stackrel{{\scriptstyle\theta_{0}}}{{\to}}H_{0},$

where $G=\operatorname{SL}_{2}$. The first map lifts $\varrho$ to a weight one
holomorphic cusp form $\vartheta(g^{\prime},\varphi^{-}_{1},\varrho)$ on $G$,
which was first studied by Hecke. Here $\varphi_{1}^{\pm}$ is a Schwartz
function on $V_{1}(\mathbb{A})$ whose archimedean component
$\varphi^{\pm}_{\infty}$ is the Schwartz function in $V_{1}(\mathbb{R})$
defined in (2.66). Then the second map lifts it to a coherent Eisenstein
series, and is an instance of the Rallis tower property ([Ral84])111By a
change of integration order, we can also rewrite the map as
$G\stackrel{{\scriptstyle\theta}}{{\to}}H\stackrel{{\scriptstyle\text{Pullback}}}{{\longrightarrow}}H_{0},$
where $H={\mathrm{SO}}(V)$ contains $H_{0}\times H_{1}$. . From this, we
obtain the equation

(1.9) $\begin{split}\mathcal{I}(g,\varphi^{(\pm 1,\mp
1)},\varrho)&:=\int_{G(\mathbb{Q}))\backslash
G(\mathbb{A})}\theta_{0}(g^{\prime},g,\varphi_{0}^{(\pm 1,\mp
1)})\vartheta(g^{\prime},\varphi^{-}_{1},\varrho)dg^{\prime}\\\
&=\frac{3}{\pi}E^{*}(g,0,\Phi^{(\pm 1,\mp 1)}),\end{split}$

where $\theta_{0}$ is the theta kernel for the quadratic space $V_{0}$,
$\varphi^{(\pm 1,\mp 1)}=\varphi_{0}^{(\pm 1,\mp 1)}\otimes\varphi_{1}^{-}$ is
a Schwartz function on $V(\mathbb{A})$ with $V:=V_{0}\oplus V_{1}$ and
$\Phi^{(\pm 1,\mp 1)}=F_{\varphi,\varrho}$ is the section defined in (3.37).
Our first main result is Theorem 3.3, which ensures that all coherent
Eisenstein series can be realized as such lifts. This is reduced to the
corresponding local problem and solved in section 3.5.

To construct $\tilde{\mathcal{I}}$, we first modify the character $\varrho$ to
the function $\tilde{\varrho}_{C}$ on $H_{1}(\mathbb{A})$ defined in (2.77).
It is a preimage of $\varrho$ under the first order invariant differential
operator $t\frac{d}{dt}$ on $H_{1}(\mathbb{R})\cong\mathbb{R}^{\times}$, and
hence not a classical automorphic form on $H_{1}$. We call its lift
$\vartheta(g^{\prime},\varphi^{+}_{1},\tilde{\varrho}_{C})$ to $G$ a deformed
theta integral, since the archimedean component of $\tilde{\varrho}_{C}$ is
essentially $\log t$ and comes from the first term in the Laurent expansion of
$t^{s}$ at $s=0$. This deformed theta integral was first studied in [CL20]. It
satisfies the following important property (see Theorem 2.7)

(1.10)
$L\vartheta(g^{\prime},\varphi_{1}^{+},\tilde{\varrho}_{C})=\vartheta(g^{\prime},\varphi_{1}^{-},\varrho)+\mathrm{error}.$

Here $L$ is the lowering operator on $G$, and $\mathrm{error}$ is the special
value of the theta kernel on $V_{1}$.

We now define
$\tilde{\mathcal{I}}(g):=\mathcal{I}(g,\varphi^{(1,1)},\tilde{\varrho}_{C})$
in (4.2) using the theta kernel $\theta_{0}(g^{\prime},g,\varphi_{0}^{(1,1)})$
with the archimedean component of $\varphi^{(1,1)}_{0}$ being the Schwartz
function $\varphi^{(1,1)}_{0,\infty}$ defined in (4.1) (the integral is
similar to (1.9)). A key observation is that there is an identity between the
actions of the universal enveloping algebras of $H_{0}$ and $G$, which gives
in this special case (see (4.7) and Lemma 4.1)

(1.11)
$(L_{1}+L_{2})\theta_{0}(g^{\prime},g,\varphi_{0}^{(1,1)})=L\theta_{0}(g^{\prime},g,\varphi_{0}^{(1,-1)})-L\theta_{0}(g^{\prime},g,\varphi_{0}^{(-1,1)}),$

where $L_{1},L_{2}$, resp. $L$, are differential operators for the variable
$g\in H_{0}$, resp. $g^{\prime}\in G$. Putting these together, we see that
$\tilde{\mathcal{I}}$ satisfies the following property (see the proof of
Proposition 4.2 with $r=0$ for details)

$\displaystyle-(L_{1}+L_{2})\tilde{\mathcal{I}}(g)$
$\displaystyle=E^{*}(g,0,\Phi^{(1,-1)})-E^{*}(g,0,\Phi^{(-1,1)})+\mathrm{error}^{\prime}.$

Up to this the term $\mathrm{error}^{\prime}$, which is a manifestation of the
error term in (1.10), we have constructed the Hilbert modular form satisfying
the desired analogue of the differential equation (1.7).

In addition to satisfying the differential equation, we still need to better
understand the Fourier coefficients of $\tilde{\mathcal{I}}$, and compare them
to those of $E^{*,\prime}$. This is done in section 4.2, where we show that
they are logarithms of algebraic numbers and give precise factorization
information. To achieve this, we introduce a new local section with an
$s$-variable in (3.54) and match it with the standard section involving the
$s$-variable up to an error of $O(s^{m})$ for any positive integer $m$. This
builds upon the results in section 3.5 and is accomplished in Theorem 3.15.
These new local sections are of independent interest, as they do not come from
pullback of the standard section on $H\cong{\mathrm{SO}}(3,3)$. In fact, they
do not even tensor together to form a global section with an $s$-variable.

Finally, we still need to handle the term arising from the error on the right
hand side in (1.10). This boils down to proving the rationality of a Millson
theta lift, which is given in Proposition 4.9. For this, we will need the
Fourier expansion of such a lift computed in [ANS18], and to choose the
matching section with a suitable invariance property. Proceeding essentially
as in [GKZ87] or [BEY21], with $E^{*,\prime}$ replaced by its sum with
$\tilde{\mathcal{I}}$, we complete the proof of Theorem 1.2.

###  1.3. Outlook and Organization

The factorization of the algebraic numbers appearing in the Fourier
coefficients of $\tilde{\mathcal{I}}$ are very closely related to the Fourier
coefficients of $E^{*,\prime}$, which suggests that they should reflect the
non-archimedean part of the arithmetic intersection between integral versions
of $Z_{f}$ and $Z(W)$ defined over the ring of integers of $F$. It would be
very interesting to relate this arithmetic intersection to special values of
derivatives of $L$-functions as in [BKY12] by applying and refining the
results in [AGHMP18].

It would be interesting to investigate the analogues of Theorems 1.2 and 1.4
for other CM, étale $\mathbb{Q}$-algebras $E/\mathbb{Q}$. When $E/\mathbb{Q}$
has degree 4, there are four cases

  1. (1)

$E/\mathbb{Q}$ is biquadratic,

  2. (2)

$E/\mathbb{Q}$ is a product of imaginary quadratic fields,

  3. (3)

$E/\mathbb{Q}$ is cyclic,

  4. (4)

$E/\mathbb{Q}$ is a non-Galois, quartic extension.

The CM points $Z(W)$ have a moduli interpretation as abelian surfaces with CM
by the reflex CM algebra $E^{\\#}$. The present paper treats case (1). In
cases (2) and (3), the reflex algebras $E^{\\#}$ are quartic, abelian field
extensions of $\mathbb{Q}$, and the CM cycle $Z(W)$ is already defined over
$\mathbb{Q}$. In the last case, $E^{\\#}/\mathbb{Q}$ is a quartic, non-Galois
field, and $Z(W)$ is defined over a real quadratic field. We plan to extend
the ideas and techniques in this paper to prove the analogue of Theorem 1.4 in
cases (2)-(4). When $E/\mathbb{Q}$ is a field of degree greater than 4, one
would need some new ideas.

In addition, there are other applications of these expected results. For cases
(2) and (3), by combining the analogue of Theorem 1.2 and the idea in [Li21b],
we hope to obtain non-existence result of genus 2 curves with CM Jacobian and
having everywhere good reduction in certain families, generalizing the main
result in [HP17]. In the last case, we expect a variation of our construction
to lead to a proof of the factorization conjecture of CM-values of twisted
Borcherds product in [BY07].

The paper is organized in the following way. Section 2 contains preliminaries.
Most of these are standard, except for section 2.8, which contains the adelic
version of the results in [CL20]. Section 3 matches the coherent Eisenstein
series with the Doi-Naganuma lift of Hecke’s cusp form. Section 4 defines
$\tilde{\mathcal{I}}$ and studies its various properties. Finally, we give the
proofs of Theorems 1.2 and 1.4 in the last section.

Acknowledgement: We thank Claudia Alfes-Neumann, Chao Li and Shaul Zemel for
helpful comments on an earlier version. J.B. and Y.L. were supported by the
LOEWE research unit USAG, and by the Deutsche Forschungsgemeinschaft (DFG)
through the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of
Uniformized Structures”, project number 444845124. T.Y. was partially
supported by the Dorothy Gollmar Chair’s Fund and Van Vleck research fund of
the department of mathematics, UW-Madison.

##  2\. Preliminaries

In this seciton, we introduce some preliminary notions, most of which are
standard from the literature. The only materials not easily found in the
literature are in sections 2.7 and 2.8 concerning the weght one cusp forms of
Hecke in the adelic language, which are translated from the results in [CL20]
in the classical language.

Let $\mathbb{N}$ denote the set of positive integers and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. For a number field $E$, let
$\mathbb{A}_{E}$ be its ring of adeles, $\hat{E}$ the finite adeles, and
$\mathbb{A}=\mathbb{A}_{\mathbb{Q}}$ with $\psi=\psi_{f}\psi_{\infty}$ its
usual additive character. For algebraic group $G$ over $E$, denote
$[G]=G(E)\backslash G(\mathbb{A}_{E})$. As usual, let
$G=\operatorname{SL}_{2}$ with standard Borel $B=MN\subset G$. Denote also

$m(a)=\left(\begin{smallmatrix}a&\\\ &a^{-1}\end{smallmatrix}\right)\in
M,n(b):=\left(\begin{smallmatrix}1&b\\\ &1\end{smallmatrix}\right)\in
N,~{}w=\left(\begin{smallmatrix}&-1\\\ 1&\end{smallmatrix}\right)\in G,$

and

$T(R)=\\{t(a):=\left(\begin{smallmatrix}a&\\\ &1\end{smallmatrix}\right):a\in
R^{\times}\\}\subset{\mathrm{GL}}_{2}(R).$

Throughout the paper, $F$ will be a real quadratic field (unless stated
otherwise). Let $\prime\in\operatorname{Gal}(F/\mathbb{Q})$ be the non-trivial
element. It induces an automorphism of
$\mathbb{A}_{F},\mathbb{A}_{F}^{\times}$ and $F_{p}:=F\otimes\mathbb{Q}_{p}$
for each prime $p\leq\infty$, If $p$ is a finite prime that splits in $F$
(resp. is the infinite place), then $F$ has two embeddings into
$\mathbb{Q}_{p}$ (resp. $\mathbb{R}$), and $F_{p}$ is isomorphic to
$\mathbb{Q}_{p}^{2}$ (resp. $\mathbb{R}^{2}$). For $\lambda\in F$, let
$\lambda_{1},\lambda_{2}$ denote the images under those embeddings. We will
also sometimes use $\lambda$ to represent the pair $(\lambda_{1},\lambda_{2})$
in $\mathbb{Q}_{p}^{2}$ (resp. $\mathbb{R}^{2}$), and $\lambda^{\prime}$ would
represent $(\lambda_{2},\lambda_{1})$. We have the incomplete Gamma function

$\Gamma(s,x)=\int^{\infty}_{x}t^{s-1}e^{-t}dt.$

###  2.1. Differential operators

For a real-analytic function $f$ on $G(\mathbb{R})$, the Lie algebra
${\mathfrak{sl}}_{2}(\mathbb{C})$ acts via

(2.1)
$A(f)(g):=\partial_{t}f(ge^{tA})\mid_{t=0},~{}A\in{\mathfrak{sl}}_{2}(\mathbb{C}).$

We define the raising and lowering operator

(2.2) $R:=\frac{1}{2}\begin{pmatrix}1&i\\\
i&-1\end{pmatrix},~{}L:=\frac{1}{2}\begin{pmatrix}1&-i\\\ -i&-1\end{pmatrix}.$

If $f$ is right $K_{\infty}$-equivariant of weight $k$, then we have

$\sqrt{v}^{-(k+2)}R(f)(g_{\tau})=R_{\tau,k}(\sqrt{v}^{-k}f(g_{\tau})),~{}\sqrt{v}^{-(k-2)}L(f)(g_{\tau})=L_{\tau,k}(\sqrt{v}^{-k}f(g_{\tau})),$

where $R_{\tau,k}$ and $L_{\tau,k}$ are the usual raising and lowering
operators given by

(2.3)
$R_{\tau,k}:=2i\partial_{\tau}+\frac{k}{v},~{}L_{\tau,k}:=-2iv^{2}\partial_{\overline{\tau}}.$

We say that $f$ is holomorphic, resp. anti-holomorphic, if $L(f)=0$, resp.
$R(f)=0$.

For $r\in\mathbb{N}_{0}$ and $k_{1},k_{2}\in{\tfrac{1}{2}}\mathbb{Z}$, define

(2.4)
$\begin{split}Q_{r,(k_{1},k_{2})}(X,Y)&:=\sum_{s=0}^{r}\binom{r+k_{1}-1}{s}\binom{r+k_{2}-1}{r-s}X^{r-s}(-Y)^{s},\\\
\tilde{Q}_{r}(X,Y)&:=\frac{Q_{r,(1,1)}(X,Y)(X+Y)}{X+(-1)^{r}Y}\end{split}$

in $\mathbb{Q}[X,Y]$. We omit $(k_{1},k_{2})$ from the notation when it is
$(1,1)$, in which case

$Q_{r}(X,Y)=(X+Y)^{r}P_{r}\left(\frac{X-Y}{X+Y}\right),$

with $P_{r}(x)$ the $r$-th Legendre polynomial given explicitly by

(2.5)
$P_{r}(x)=2^{-r}\sum_{s=0}^{r}\binom{r}{s}^{2}(x-1)^{r-s}(x+1)^{s}=(-1)^{r_{0}}\sum_{s=0}^{r_{0}}\binom{r_{0}-r-1/2}{r_{0}-s}\binom{r-r_{0}-1/2}{s}x^{r-2s},$

where $r_{0}:=\lfloor r/2\rfloor$. The second identity comes from (3.133) on
page 38 of [Gou72] and direct calculation. We thank Zhiwei Sun for pointing us
to this reference.

From the differential equation satisfied by $P_{r}$, we have

(2.6)
$(\partial_{X}\partial_{Y})(Q_{r}(X,Y)(X+Y))=(r+1)(\partial_{X}+\partial_{Y})Q_{r}(X,Y).$

For $A\in{\mathfrak{sl}}_{2}(\mathbb{C})$, denote

(2.7) $A_{1}=(A,0),~{}A_{2}=(0,A)$

in ${\mathfrak{sl}}_{2}(\mathbb{C})^{2}$. Then we have two operators

(2.8)
$\mathrm{RC}_{r,(k_{1},k_{2})}:=(-4\pi)^{-r}Q_{r,k_{1},k_{2}}(R_{1},R_{2}),~{}\widetilde{\mathrm{RC}}_{r}:=(-4\pi)^{-r}\tilde{Q}_{r}(R_{1},R_{2})$

on real-analytic functions on $G(\mathbb{R})^{2}$. If
$f:G(\mathbb{R})^{2}\to\mathbb{C}$ is holomorphic and right
$K_{\infty}^{2}$-equivariant of weight $(k_{1},k_{2})$, then
$\mathrm{RC}_{r,(k_{1},k_{2})}(f)^{\Delta}:G(\mathbb{R})\to\mathbb{C}$ is
holomorphic and right $K_{\infty}$-equivariant of weight $k_{1}+k_{2}+2r$, and
the operator $\mathrm{RC}$ is usually called the Rankin-Cohen operator. Here
$f^{\Delta}(g):=f(g^{\Delta})=f(g,g)$ is the restriction of $f$ to the
diagonal $G(\mathbb{R})\subset G(\mathbb{R})^{2}$. In fact, we have (see
[BvdGHZ08, Proposition 19])

$\mathrm{RC}_{r,(k_{1},k_{2})}(f)^{\Delta}(g_{z})=(2\pi
i)^{-r}\left(Q_{r}(\partial_{z_{1}},\partial_{z_{2}})f(g_{z_{1}},g_{z_{2}})\right)\mid_{z_{1}=z_{2}=z}.$

For example, if $f(g_{z_{1}},g_{z_{2}})={\mathbf{e}}(m_{1}z_{1}+m_{2}z_{2})$,
then

(2.9)
$\mathrm{RC}_{r,(k_{1},k_{2})}(f)^{\Delta}(g_{z})=Q_{r,(k_{1},k_{2})}(m_{1},m_{2}){\mathbf{e}}((m_{1}+m_{2})z).$

From Lemma 2.2 in [Li21a], we know that there are unique constants
$c^{(r;k_{1},k_{2})}_{\ell}\in\mathbb{Q}$ such that

(2.10)
$(4\pi)^{-r}(R_{1}^{r}f)^{\Delta}=\sum_{\ell=0}^{r}c^{(r;k_{1},k_{2})}_{\ell}(4\pi)^{-r+\ell}R^{r-\ell}\mathrm{RC}_{\ell,(k_{1},k_{2})}(f)^{\Delta}$

whenever $k_{1}+k_{2}+2r<2$.

###  2.2. Quadratic Space associated to Real Quadratic Field

Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field, which becomes a
$\mathbb{Q}$-quadratic space of signature $(1,1)$ with respect to the
quadratic form

$Q_{a}(\lambda):=a{\mathrm{Nm}}(\lambda),~{}\lambda\in F$

for any $a\in\mathbb{Q}^{\times}$. We denote this quadratic space by $V_{a}$
and identify

(2.11)
$\begin{split}\iota_{a}:V_{a}(\mathbb{R})=F\otimes_{\mathbb{Q}}\mathbb{R}&\cong\mathbb{R}^{2}\\\
(\lambda_{1},\lambda_{2})&\mapsto(\lambda_{1},{\mathrm{sgn}}(a)\lambda_{2})\sqrt{|a|}.\end{split}$

This is an isometry, where the quadratic form on $\mathbb{R}^{2}$ is given by
$Q((x_{1},x_{2}))=x_{1}x_{2}$. The special orthogonal group
$H_{a}:=\mathrm{SO}({V_{a}})$ satisfies

$H_{a}(\mathbb{Q})\cong F^{1}(\cong F^{\times}/\mathbb{Q}^{\times}),$

where $F^{1}:=\\{\mu\in F:{\mathrm{Nm}}(\mu)=1\\}$ acts on $V$ via
multiplication. Furthermore, we identify

$H_{a}(\mathbb{R})\cong\mathbb{R}^{\times},$

where $t\in\mathbb{R}^{\times}$ acts on
$V_{a}(\mathbb{R})=F\otimes_{\mathbb{Q}}\mathbb{R}\cong\mathbb{R}^{2}$ via
$(x_{1},x_{2})\mapsto(tx_{1},t^{-1}x_{2})$. Note that the invariant measure on
$H(\mathbb{R})\cong\mathbb{R}^{\times}$ is $\frac{dt}{|t|}$. We have a
character ${\mathrm{sgn}}:H_{a}(\mathbb{R})\cong\mathbb{R}^{\times}\to\\{\pm
1\\}$. Denote

$H_{a}(\mathbb{R})^{+}:=\ker({\mathrm{sgn}})\cong\mathbb{R}_{>0}$

its kernel, which is the connected component of $H_{a}(\mathbb{R})$ containing
the identity. We also denote

(2.12) $H_{a}(\mathbb{Q})^{+}:=H_{a}(\mathbb{R})^{+}\cap H_{a}(\mathbb{Q})$

which is an index 2 subgroup of $H_{a}(\mathbb{Q})$.

######  Remark 2.1.

Let $\chi=\chi_{E/F}$ be a Hecke character associated to a quadratic extension
$E/F$. Suppose $E/\mathbb{Q}$ is biquadratic. Then
$\chi\mid_{\mathbb{A}^{\times}}$ is trivial and $\chi$ factors through the map
${\mathrm{Nm}}^{-}:\mathbb{A}_{F}^{\times}\to H_{a}(\mathbb{A})$, i.e. there
exists $\varrho=\varrho_{E/F}:H_{a}(\mathbb{A})\to\\{\pm 1\\}$ such that

(2.13) $\varrho\circ{\mathrm{Nm}}^{-}=\chi.$

Note that $\varrho$ is odd if and only if $E$ is totally imaginary. We also
denote the compact subgroup

(2.14) $K_{\varrho}:=H_{a}(\hat{\mathbb{Z}})\cap\mathrm{ker}(\varrho).$

Note that $H_{a}(\mathbb{Q})\backslash H_{a}(\hat{\mathbb{Q}})/K_{\varrho}$ is
a finite set.

###  2.3. The Weil Representation and Theta Functions

Let $(V,Q)$ be a rational quadratic space of signature $(p,q)$, and
$H_{V}:={\mathrm{SO}}(V)$. For a subfield $E\subset\mathbb{C}$, we denote
${\mathcal{S}}(\hat{V};E)$, resp. ${\mathcal{S}}(V_{p};E)$, to denote the
space of Schwartz functions on $\hat{V}:=V(\hat{\mathbb{Q}})$, resp.
$V_{p}:=V(\mathbb{Q}_{p})$, with values in $E$, which is a $E$-vector space.
We omit $E$ from the notation if it is $\mathbb{Q}$. On the other hand, we
write ${\mathcal{S}}(V(\mathbb{R}))$ and ${\mathcal{S}}(V(\mathbb{A}))$ for
the space of Schwartz function on $V(\mathbb{R})$ and $V(\mathbb{A})$ valued
in $\mathbb{C}$ respectively.

For a lattice222Lattices will be even and integral throughout the paper.
$L\subset V$, we denote $L^{\vee}\subset V$ its dual lattice,
$\hat{L}:=L\otimes\hat{\mathbb{Z}}$,
$\hat{L}^{\vee}:=L^{\vee}\otimes\hat{\mathbb{Z}}$ and

(2.15) $L_{m,\mu}:=\\{\lambda\in L+\mu:Q(\lambda)=m\\}.$

for $m\in\mathbb{Q},\mu\in L^{\vee}/L$. The finite dimensional $E$-subspace

(2.16) ${\mathcal{S}}(L;E):=\\{\phi\in{\mathcal{S}}(\hat{V};E):\phi\text{ is
$\hat{L}$-invariant with support on
}\hat{L}^{\vee}\\}\subset{\mathcal{S}}(\hat{V};E),$

is spanned by $\\{\phi_{L+\mu}:\mu\in\hat{L}^{\vee}/\hat{L}\cong
L^{\vee}/L\\}$ with

(2.17)
$\phi_{L+\mu}:=\operatorname{Char}(\hat{L}+\mu)\in{\mathcal{S}}(\hat{V}).$

For full sublattices $M\subset L\subset V$, it is clear that

(2.18)
${\mathcal{S}}(L;E)\subset{\mathcal{S}}(M;E)\subset{\mathcal{S}}(\hat{V};E).$

As above, we also denote ${\mathcal{S}}(L):={\mathcal{S}}(L;\mathbb{Q})$.
Furthermore, since

${\mathcal{S}}(\hat{V})=\bigcup_{L\subset V\text{ lattice}}{\mathcal{S}}(L),$

we have ${\mathcal{S}}(\hat{V};E)={\mathcal{S}}(\hat{V})\otimes_{\mathbb{Q}}E$
for any subfield $E\subset\mathbb{C}$.

Suppose $V=V_{1}\oplus V_{2}$ is a decomposition of rational quadratic spaces.
For any lattice $L_{i}\subset V_{i}$, we have ${\mathcal{S}}(L_{1}\oplus
L_{2};E)={\mathcal{S}}(L_{1};E)\otimes{\mathcal{S}}(L_{2};E)\subset{\mathcal{S}}(\hat{V}_{1};E)\otimes{\mathcal{S}}(\hat{V}_{2};E)$
via the natural restriction map. This also gives us

(2.19)
${\mathcal{S}}(\hat{V};E)={\mathcal{S}}(\hat{V}_{1};E)\otimes{\mathcal{S}}(\hat{V}_{2};E)=\bigoplus_{L_{1}\subset
V_{1},~{}L_{2}\subset V_{2}\text{
lattices}}{\mathcal{S}}(L_{1};E)\otimes{\mathcal{S}}(L_{2};E).$

Combining with equation (2.18), we see that for any given
$\phi\in{\mathcal{S}}(\hat{V};E)$, we can find a lattice $L=L_{1}\oplus
L_{2}\subset V$ such that $L_{i}\subset V_{i}$ and

(2.20)
$\phi\in{\mathcal{S}}(L;E)={\mathcal{S}}(L_{1};E)\otimes{\mathcal{S}}(L_{2};E).$

The group $(G\times H_{V})(\mathbb{A})$ acts on ${\mathcal{S}}(V(\mathbb{A}))$
via the Weil representation $\omega=\omega_{V,\psi}$ given by [KR94]

(2.21)
$\begin{split}(\omega(m(a))\varphi)(x)&=\chi_{V}(a)|a|_{\mathbb{A}}^{\dim
V/2}\varphi(ax),~{}a\in\mathbb{A}^{\times},\\\
(\omega(n(b))\varphi)(x)&=\psi(bQ(x))\varphi(x),~{}b\in\mathbb{A},\\\
(\omega(w)\varphi)(x)&=\gamma(V)\int_{V(\mathbb{A})}\varphi(b)\psi(-(x,b))db,~{}w\in
G(\mathbb{A}),\\\ (\omega(h)\varphi)(x)&=\varphi(h^{-1}x),~{}h\in
H_{V}(\mathbb{A}),\\\ \end{split}$

for any $\varphi\in{\mathcal{S}}(V(\mathbb{A}))$ and $x\in V(\mathbb{A})$. For
each prime $p\leq\infty$, we also have the local Weil representation
$\omega_{p}$ of $G(\mathbb{Q}_{p})$ acting on
${\mathcal{S}}(V(\mathbb{Q}_{p});\mathbb{C})$. When $p<\infty$, the
representation descends to
${\mathcal{S}}(V(\mathbb{Q}_{p});\mathbb{Q}(\zeta_{p^{\infty}}))$ with

(2.22) $\mathbb{Q}(\zeta_{p^{\infty}}):=\bigcup_{n\geq
1}\mathbb{Q}(\zeta_{p^{n}})\subset\mathbb{Q}^{\mathrm{ab}}$

the maximal abelian extension of $\mathbb{Q}$ ramified only at $p$. Then
$\omega_{f}:=\otimes_{p<\infty}\omega_{p}$ gives a representation of
$G(\hat{\mathbb{Q}})$ on ${\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$.
For any given $\phi\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$, one
can choose a finite abelian extension $E/\mathbb{Q}$ such that
$G(\hat{\mathbb{Q}})$-module $\phi$ generates is contained in
${\mathcal{S}}(\hat{V};E)$.

For any lattice $L\subset V$, the subspace ${\mathcal{S}}(L;\mathbb{C})$
defined in (2.16) is a $K_{f}$-invariant subspace with
$K_{f}:=G(\hat{\mathbb{Z}})$. It has a unitary pairing $\langle,\rangle$ with
the vector space $\mathbb{C}[L^{\vee}/L]:=\oplus_{\mu\in
L^{\vee}/L}\mathbb{C}\mathfrak{e}_{\mu}$ given by

(2.23)
$\langle\mathfrak{e}_{\mu},\phi_{\mu^{\prime}}\rangle:=\langle\mathfrak{e}_{\mu},\phi_{\mu^{\prime}}\rangle_{L}:=\begin{cases}1,&\mu=\mu^{\prime},\\\
0,&\text{ otherwise.}\end{cases}$

More generally if $L=L_{1}\oplus L_{2}$, we have
$L^{\vee}/L=L_{1}^{\vee}/L_{1}\oplus L_{2}^{\vee}/L_{2}$ and
$\mathbb{C}[L^{\vee}/L]\cong\mathbb{C}[L_{1}^{\vee}/L_{1}]\otimes\mathbb{C}[L_{2}^{\vee}/L_{2}]$.
Therefore, we can extend the pairing above to

(2.24)
$\langle\cdot,\cdot\rangle_{L_{2}}:\mathbb{C}[L^{\vee}/L]\times{\mathcal{S}}(L_{2};\mathbb{C})\to\mathbb{C}[L_{1}^{\vee}/L_{1}],~{}(\mathfrak{v}_{1}\otimes\mathfrak{v}_{2},\phi)\mapsto\langle\mathfrak{v}_{2},\phi\rangle\mathfrak{v}_{1}$

with $\mathfrak{v}_{i}\in\mathbb{C}[L_{i}^{\vee}/L_{i}]$ and
$\phi\in{\mathcal{S}}(L_{2};\mathbb{C})$.

With respect to the perfect pairing in (2.23), the unitary dual of
$\omega_{f}$ is the representation $\rho_{L}$ on $\mathbb{C}[L^{\vee}/L]$
given by

(2.25)
$\begin{split}\rho_{L}(n(1))(\mathfrak{e}_{\mu})&:={\mathbf{e}}(Q(\mu))\mathfrak{e}_{\mu},\\\
\rho_{L}(w)(\mathfrak{e}_{\mu})&:=\frac{{\mathbf{e}}(-(p-q)/8)}{\sqrt{|L^{\vee}/L|}}\sum_{\mu\in
L^{\vee}/L}{\mathbf{e}}(-(\mu,\mu^{\prime}))\mathfrak{e}_{\mu^{\prime}},\end{split}$

where $(p,q)$ is the signature of $V(\mathbb{R})$. This is the Weil
representation on finite quadratic modules used by Borcherds in [Bor98]. If we
identify ${\mathcal{S}}(L;\mathbb{C})$ and $\mathbb{C}[L^{\vee}/L]$ with
$\mathbb{C}^{|L^{\vee}/L|}$ via the bases
$\\{\phi_{\mu}:\mu\in\hat{L}^{\vee}/\hat{L}\\}$ and
$\\{\mathfrak{e}_{\mu}:\mu\in L^{\vee}/L\\}$ respectively, then
$\omega_{f}=\overline{\rho_{L}}=\rho_{L}^{-1}$. For full sublattices $M\subset
L$, the following trace map

(2.26)
$\operatorname{Tr}^{L}_{M}:\mathbb{C}[L^{\vee}/L]\to\mathbb{C}[M^{\vee}/M],~{}\mathfrak{e}_{\mu}\mapsto\frac{1}{[L:M]}\sum_{h\in
M^{\vee}/M,~{}h\equiv\mu\bmod{L}}\mathfrak{e}_{h}$

intertwines the Weil representation and is compatible with the inclusion in
(2.18) in the sense that

(2.27)
$\langle\operatorname{Tr}^{L}_{M}(\mathfrak{v}),\phi\rangle_{M}=\langle\mathfrak{e},\phi\rangle_{L}$

for any
$\mathfrak{v}\in\mathbb{C}[L^{\vee}/L],\phi\in{\mathcal{S}}(L;\mathbb{C})$.
Note that all of these still hold when $\mathbb{C}$ is replaced by some
totally imaginary abelian extension $E_{L}/\mathbb{Q}$.

As usual, we let $H_{k,L}(\Gamma)$ denote the space of harmonic Maass forms
valued in $\mathbb{C}[L^{\vee}/L]$ of weight $k\in\frac{1}{2}\mathbb{Z}$ and
representation $\rho_{L}$ on a congruence subgroup
$\Gamma\subset\operatorname{SL}_{2}(\mathbb{Z})$. It contains the subspace
$M^{!}_{k,L}(\Gamma)$ of vector-valued weakly holomorphic modular forms. Post-
composing with $\operatorname{Tr}^{L}_{M}$ in (2.26) induces a map
$H_{k,L}(\Gamma)\to H_{k,M}(\Gamma)$, which preserves holomorphicity and
rationality of holomorphic part Fourier coefficients. If $L^{\vee}/L$ is
trivial (resp. $\Gamma=\operatorname{SL}_{2}(\mathbb{Z})$), then we drop $L$
(resp. $\Gamma$) from the above notations. Furthermore, we let

(2.28) $M^{!,\infty}_{k}(\Gamma):=\\{f\in M^{!}_{k}(\Gamma):f\text{ is
holomorphic away from the cusp }\infty\\}.$

and denote for $f(\tau)=\sum_{m\in\mathbb{Q},~{}\mu\in
L^{\vee}/L}c(m,\mu)q^{m}\mathfrak{e}_{\mu}\in M^{!}_{k,L}$

(2.29) $\mathrm{prin}(f):=\sum_{m\in\mathbb{Q}_{<0},~{}\mu\in
L^{\vee}/L}c(m,\mu)q^{m}\mathfrak{e}_{\mu}$

the principal part of $f$.

In [McG03, Theorem 4.3], McGraw extended the representation $\rho_{L}$ to an
action of ${\mathrm{GL}}_{2}(\mathbb{Z}/N\mathbb{Z})$ on
$\mathbb{C}[L^{\vee}/L]$, where $N$ is the level of $L$. We can phrase this
extension adelically in the following manner. Using the short exact sequence

$1\to\operatorname{SL}_{2}\to{\mathrm{GL}}_{2}\stackrel{{\scriptstyle\det}}{{\to}}\mathbb{G}_{m}\to
1,$

we can identify ${\mathrm{GL}}_{2}\cong\operatorname{SL}_{2}\rtimes T$. Then
$\omega_{f}$ extends to a $\mathbb{Q}$-linear action of
${\mathrm{GL}}_{2}(\hat{\mathbb{Q}})={\mathrm{GL}}_{2}(\mathbb{Q}){\mathrm{GL}}_{2}(\hat{\mathbb{Z}})$
on ${\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$ via

(2.30)
$(\omega_{f}(g,t(a))\phi)(x):=(\omega_{f}(g)\sigma_{a}(\phi))(x)=\sigma_{a}((\omega_{f}(t(a)^{-1}gt(a))(\phi))(x))$

for
$g\in\operatorname{SL}_{2}(\hat{\mathbb{Q}}),a\in\hat{\mathbb{Q}}^{\times},\phi\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$,
where $\sigma_{a}\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$
satisfies $\sigma_{a}(\psi_{f}(a^{\prime}))=\psi_{f}(aa^{\prime})$ for all
$a,a^{\prime}\in\hat{\mathbb{Q}}^{\times}$ and acts on
${\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$ via its action on
$\mathbb{Q}^{\mathrm{ab}}$. This gives us

(2.31)
${\mathcal{S}}(\hat{V})={\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})^{T(\hat{\mathbb{Z}})}.$

For each $p<\infty$, this gives an extension of $\omega_{p}$ to a
$\mathbb{Q}$-linear action of ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$ on
${\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))$, which satisfies

(2.32)
${\mathcal{S}}(V_{p})={\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{T(\mathbb{Z}_{p})}.$

For $\varphi\in{\mathcal{S}}(V(\mathbb{A}))$, we have the theta function

(2.33) $\theta_{V}(g,h,\varphi):=\sum_{x\in
V(\mathbb{Q})}(\omega(g)\varphi)(h^{-1}x)$

for $(g,h)\in(G\times H_{V})(\mathbb{A})$ as usual. For a lattice $L\subset
V$, we also denote

(2.34) $\Theta_{L}(\tau,h):=v^{-(p-q)/2}\sum_{\mu\in
L^{\vee}/L}\theta_{V}(g_{\tau},h,\phi_{\mu}\phi_{\infty})\phi_{\mu}$

the vector-valued theta function with $\phi_{\infty}$ the Gaussian.

###  2.4. CM Points and Higher Green Functions

We follow [BKY12] and [BEY21] to recall CM points and higher Green functions.
Let $(\mathrm{V},Q)$ be a rational quadratic space of signature $(n,2)$, and
${\tilde{H}}={\tilde{H}}_{\mathrm{V}}:=\mathrm{GSpin}(\mathrm{V})$. For an
open compact subgroup $K\subset{\tilde{H}}(\hat{\mathbb{Q}})$, we have the
associated Shimura variety $X_{K}$, whose $\mathbb{C}$-points are given by

$X_{K}(\mathbb{C})={\tilde{H}}(\mathbb{Q})\backslash(\mathbb{D}\times{\tilde{H}}(\hat{\mathbb{Q}})/K).$

Here $\mathbb{D}=\mathbb{D}^{+}\sqcup\mathbb{D}^{-}$ is the symmetric space
associated to $\mathrm{V}(\mathbb{R})$. For $m\in\mathbb{Q}$ and
$\varphi\in{\mathcal{S}}(\hat{\mathrm{V}};\mathbb{C})$, one can define the
special divisor $Z(m,\varphi)$ on $X_{K}$ (see e.g. [BEY21, section 2]).

The CM points on $X_{K}$ can be described as follows. For a totally real field
$F$ of degree $d$ with real embeddings $\sigma_{j},1\leq j\leq d$, denote
$\alpha_{j}:=\sigma_{j}(\alpha)$ for $\alpha\in F$. Suppose $\alpha_{j_{0}}<0$
for some $j_{0}$ and $\alpha_{j}>0$ when $j\neq j_{0}$. Then a CM quadratic
extension $E/F$ becomes an $F$-quadratic space $W=W_{\alpha}=E$ with respect
to the quadratic form $Q_{\alpha}:=\alpha{\mathrm{Nm}}_{E/F}$. Suppose there
is isometric embedding as in (1.5). Then the subspace
$W\otimes_{F,\sigma_{j_{0}}}\mathbb{R}\subset\mathrm{V}(\mathbb{R})$ is a
negative 2-plane, and determines a point $z_{0}^{\pm}\in\mathbb{D}^{\pm}$ with
a choice of orientation. For convenience, we denote

(2.35) $z_{0}:=z_{0}^{+}.$

The group $\mathrm{Res}_{F/\mathbb{Q}}({\mathrm{SO}}(W))$ is contained in
${\mathrm{SO}}(\mathrm{V})$, whose preimage in ${\tilde{H}}_{\mathrm{V}}$ is a
torus denoted by $T_{W}$. Note that $T_{W}(\mathbb{Q})=E^{\times}/F^{1}$. We
denote the CM cycle on $X_{K}$ associated to $T_{W}$ by

(2.36) $Z(W):=T_{W}(\mathbb{Q})\backslash(\\{z_{0}^{\pm}\\}\times
T_{W}(\hat{\mathbb{Q}})/K_{W})\subset X_{K}$

with $K_{W}:=K\cap T_{W}(\hat{\mathbb{Q}})$. It is defined over $F$, and its
Galois conjugates are the CM cycles $Z(W(j))$ with $1\leq j\leq d$, where
$W(j)$ is the neighborhood $F$-quadratic spaces at $\sigma_{j}$ of admissible
incoherent $\mathbb{A}_{F}$-quadratic space $\mathbb{W}$ associated to $W$
(see [BY11, BKY12] for details). Note that $W(j)=W_{\alpha(j)}$ for some
$\alpha(j)\in F^{\times}$ and $\alpha(j_{0})=\alpha$. When $F$ is real
quadratic (i.e. $d=2$), for $\alpha\in F^{\times}$ with
${\mathrm{Nm}}(\alpha)<0$, we set

(2.37) ${\alpha^{\vee}}:=\alpha(2)\in F^{\times}.$

Then $({\alpha^{\vee}})^{\vee}=\alpha$. Finally for totally positive $t\in F$,
we define the “Diff” set

(2.38) $\mathrm{Diff}(W,t):=\\{\mathfrak{p}:W_{\mathfrak{p}}\text{ does not
represent }t\\}$

following [Kud97]. Note that this set is finite and odd (see [YY19,
Proposition 2.7]).

######  Example 2.2.

Suppose $E_{1},E_{2}$ are imaginary quadratic fields such that $E=E_{1}E_{2}$
is biquadratic containing a real quadratic field $F$, i.e. $E_{1}\neq E_{2}$.
Then for any CM points $z_{j}\in E_{j}$, the point
$(z_{1},z_{2})\in\mathbb{H}^{2}$ is send to $Z(W_{\alpha})\cup
Z(W_{\alpha^{\vee}})\subset X_{K}$ under the isomorphism in (2.47) (see
section 3.2 in [YY19] for details).

Denote

(2.39) $\sigma_{0}:=\frac{n}{4}-\frac{1}{2}.$

Let $L\subset\mathrm{V}$ be an even integral lattice such that $K$ stabilizes
$\hat{L}$. For $\mu\in L^{\vee}/L$ and $m\in\mathbb{Z}+Q(\mu)$, we write
$Z(m,\mu):=Z(m,\phi_{\mu})$. The automorphic Green function on
$X_{K}\backslash Z(m,\mu)$ is defined by

(2.40)
$\Phi_{m,\mu}(z,h,s):=2\frac{\Gamma(s+\sigma_{0})}{\Gamma(2s)}\sum_{\lambda\in
h(L_{m,\mu})}\left(\frac{m}{Q(\lambda_{z^{\perp}})}\right)^{s+\sigma_{0}}F\left(s+\sigma_{0},s-\sigma_{0},2s;\frac{m}{Q(\lambda_{z^{\perp}})}\right),$

for $\mathrm{Re}(s)>\sigma_{0}+1$, where $F(a,b,c;z)$ is the Gauss
hypergeometric function [AS64, Chapter 15]. At $Z(m,\mu)$, the function
$\Phi_{m,\mu}$ has logarithmic singularity.

At $s=\sigma_{0}+1+r$ with $r\in\mathbb{N}$, the function
$\Phi_{m,\mu}(z,h,s)$ is called a higher Green function. For a harmonic Maass
form $f=\sum_{m,\mu}c(m,\mu)q^{-m}\phi_{\mu}+O(1)\in H_{k-2r,L}$ with
$k:=-2\sigma_{0}$, define

(2.41) $\Phi_{f}^{r}(z,h):=r!\sum_{m>0,~{}\mu\in
L^{\prime}/L}c(m,\mu)m^{r}\Phi_{m,\mu}(z,h,\sigma_{0}+1+r)$

to be the associated higher Green function. Following from the work of
Borcherds [Bor98] and generalization by Bruinier [Bru02] (also see [BEY21,
Proposition 4.7]), the function $\Phi_{f}^{r}$ has the following integral
representation

(2.42)
$\begin{split}\Phi_{f}^{r}(z,h)&=(4\pi)^{-r}\lim_{T\to\infty}\int_{{\mathcal{F}}_{T}}\langle
R^{r}_{\tau}f(\tau),\overline{\Theta_{L}(\tau,z,h)}\rangle d\mu(\tau)\\\
&=(-4\pi)^{-r}\lim_{T\to\infty}\int_{{\mathcal{F}}_{T}}\langle
f(\tau),\overline{R^{r}_{\tau}\Theta_{L}(\tau,z,h)}\rangle
d\mu(\tau),\end{split}$

where ${\mathcal{F}}_{T}$ is the truncated fundamental domain of
$\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ at height $T>1$. It
has logarithmic singularity along the special divisor

(2.43) $Z_{f}:=\sum_{m>0,~{}\mu\in L^{\vee}/L}c(-m,\mu)Z(m,\mu)$

on $X_{K}$. Note that $[z_{0},h]\in Z(W)\cap Z_{f}$ if and only if

(2.44) $h(L_{m,\mu})\cap z_{0}^{\perp}\neq\emptyset$

for some $m,\mu$ with $c(-m,\mu)\neq 0$.

###  2.5. Product of Modular Curves as a Shimura Variety

We follow and slightly modify [YY19, section 3] to express $X_{0}(N)\times
X_{0}(N)$ as $\mathrm{O}(2,2)$ orthogonal Shimura variety. Consider
$(\mathrm{V},Q)=(M_{2}(\mathbb{Q}),\det)$, and the lattice

$L:=\left\\{\begin{pmatrix}a&b\\\
Nc&d\end{pmatrix}:a,b,c,d\in\mathbb{Z}\\}\subset\mathrm{V}\right\\}$

for any $N\in\mathbb{N}$. Then the dual lattice $L^{\vee}$ is given by

$L^{\vee}:=\left\\{\begin{pmatrix}a&b/N\\\
c&d\end{pmatrix}:a,b,c,d\in\mathbb{Z}\\}\subset\mathrm{V}\right\\}$

and $L^{\vee}/L\cong(\mathbb{Z}/N\mathbb{Z})^{2}$ is isomorphic to that of a
scaled hyperbolic plane.

For $g_{j}\in\operatorname{SL}_{2}(\mathbb{Q})$ and
$\Lambda\in\mathrm{V}(\mathbb{Q})$, the map

(2.45) $\Lambda\mapsto g_{1}\Lambda g_{2}^{-1}$

gives
$\operatorname{SL}_{2}\times\operatorname{SL}_{2}\cong\operatorname{Spin}(\mathrm{V})$
and identifies $H_{\mathrm{V}}$ as a subgroup of
${\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2}$ [YY19, section 3.1]. Let
$K(N):=K(\Gamma_{0}(N))\subset{\mathrm{GL}}_{2}(\hat{\mathbb{Z}})$ be the open
compact subgroup in [YY19, section 3.1] and $K:=(K(N)\times K(N))\cap
H_{\mathrm{V}}(\hat{\mathbb{Q}})$. Then the map

(2.46)
$w:\mathbb{H}^{2}\to\mathbb{D}^{+},~{}(z_{1},z_{2})\mapsto\mathbb{R}\Re\left(\begin{smallmatrix}z_{1}&-z_{1}z_{2}\\\
1&-z_{2}\end{smallmatrix}\right)+\mathbb{R}\Im\left(\begin{smallmatrix}z_{1}&-z_{1}z_{2}\\\
1&-z_{2}\end{smallmatrix}\right)$

induces an isomorphism

(2.47) $X_{0}(N)\times X_{0}(N)\cong
X_{K},~{}(z_{1},z_{2})\mapsto[w(z_{1},z_{2}),1]$

with $X_{K}$ the Shimura variety for $H_{\mathrm{V}}$.

Under the map (2.45), the inverse images of the discriminant kernel
$\Gamma_{L}\subset{\mathrm{SO}}(L)$ are

$\Gamma^{\Delta}_{0}(N):=\left\\{(g_{1},g_{2})\in\Gamma_{0}(N)^{2}:g_{1}g_{2}\in\Gamma_{1}(N)\right\\}\subset\Gamma_{0}(N)^{2},$

which contains $\Gamma_{1}(N)\times\Gamma_{1}(N)$ and is a normal subgroup of
$\Gamma_{0}(N)^{2}$ satisfying

$\Gamma_{0}(N)^{2}/\Gamma^{\Delta}_{0}(N)\cong(\mathbb{Z}/N\mathbb{Z})^{\times},~{}(\left(\begin{smallmatrix}a_{1}&*\\\
*&*\end{smallmatrix}\right),\left(\begin{smallmatrix}a_{2}&*\\\
*&*\end{smallmatrix}\right))\mapsto a_{1}a_{2}\bmod{N}.$

The group
${\mathrm{SO}}(L^{\vee}/L):={\mathrm{SO}}(L)/\Gamma_{L}\cong\Gamma_{0}(N)^{2}/\Gamma^{\Delta}_{0}(N)\cong(\mathbb{Z}/N\mathbb{Z})^{\times}$
acts on $L^{\vee}/L\cong(\mathbb{Z}/N\mathbb{Z})^{2}$ via

$\alpha\cdot(b,c):=(\alpha b,\alpha^{-1}c),$

and the induced linear map on $\mathbb{C}[L^{\vee}/L]$ intertwines the Weil
representation $\rho_{L}$.

Now given $f\in M^{!}_{k}(\Gamma_{0}(N))$ for $k\in 2\mathbb{Z}$, we can lift
it to a vector-valued modular form in $M^{!}_{k,\rho_{L}}$ via the following
map

(2.48) $\mathrm{vv}:M^{!}_{k}(\Gamma_{0}(N))\to
M^{!}_{k,\rho_{L}},~{}f\mapsto\sum_{\gamma\in\operatorname{SL}_{2}(\mathbb{Z})/\Gamma_{0}(N)}(f\mid_{k}\gamma)\rho_{L}(\gamma)^{-1}\cdot\mathfrak{e}_{0}.$

This map and its generalizations are well-studied (see e.g. [Sch09]), whose
properties are summarized in the following result.

######  Lemma 2.3.

When $k<0$, we have

(2.49) $\mathrm{prin}(\mathrm{vv}(f))=\mathrm{prin}(f)\mathfrak{e}_{0}.$

for all $f\in M^{!,\infty}_{k}(\Gamma_{0}(N))$, on which space the map
$\mathrm{vv}$ is an isomorphism with the inverse given by

(2.50) $g=\sum_{\mu\in L^{\vee}/L}g_{\mu}\mathfrak{e}_{\mu}\mapsto g_{0},$

Furthermore, it preserves the rationality of the Fourier expansion at the cusp
infinity.

###### Proof.

See Proposition 4.2 in [Sch09] and Proposition 6.12, Corollary 6.14 in
[BHK+20]. ∎

As a consequence, we can relate the higher Green function
$G^{\Gamma_{0}(N)}_{r+1,f}$ from the introduction to one on the Shimura
variety $X_{K}$.

######  Corollary 2.4.

Under the isomorphism (2.47), we have

(2.51)
$G^{\Gamma_{0}(N)}_{r+1,f}(z_{1},z_{2})=-\Phi_{\mathrm{vv}(f)}^{r}([w(z_{1},z_{2}),1])$

with $G^{\Gamma_{0}(N)}_{r+1,f}$ the higher Green function defined in (1.4)
for $f\in M^{!,\infty}_{-2r}(\Gamma_{0}(N))$ with $r>0$.

###### Proof.

Under (2.47), the divisor

(2.52)
$Z(m,0)=\Gamma_{0}(N)^{2}\backslash\\{(z_{1},z_{2})\in\mathbb{H}^{2}:(\left(\begin{smallmatrix}z_{1}&-z_{1}z_{2}\\\
1&-z_{2}\end{smallmatrix}\right),x)=0\text{ for some }x\in L_{m,0}\\}$

on $X_{K}$ is simply the $m$-th Hecke correspondence $T_{m}$ on
$X_{0}(N)\times X_{0}(N)$. Therefore, the two sides of (2.51) have logarithmic
singularity along the same divisor. Using Corollary 4.2 and Theorem 4.4 of
[BEY21], we see that their difference is a smooth function in
$L^{2}(X_{0}(N)^{2})$ and an eigenfunction of the Laplacians in $z_{1}$ and
$z_{2}$ with eigenvalue $r(1-r)<0$. By fixing $z_{2}$, this difference is an
eigenfunction of the Laplacian on $X_{0}(N)$ with negative eigenvalue, which
vanishes identically. This holds for any $z_{2}\in X_{0}(N)$, and we obtain
(2.51). ∎

######  Remark 2.5.

Following section V.4 of [GZ86], we call a set of integers
$\\{\lambda_{m}:m\in\mathbb{N}\\}$ a relation for $S_{2-k}(\Gamma_{0}(N))$ if
only finitely many $\lambda_{m}$ are non-zero and

$\sum_{m\geq 1}\lambda_{m}a_{m}=0$

for all $\sum_{m\geq 1}a_{m}q^{m}\in S_{2-k}(\Gamma_{0}(N))$. Since $g_{0}\in
S_{2-k}(\Gamma_{0}(N))$ for all $g\in S_{2-k,L^{-}}$, we have

$\\{f_{P},g\\}:=\mathrm{CT}\left(\sum_{\mu\in
L^{\vee}/L}f_{P,\mu}g_{\mu}\right)=0$

for $f_{P}=\sum_{m\geq 1}\lambda_{m}q^{-m}\mathfrak{e}_{0}$. By Serre duality
[Bor99], there exists $f\in M^{!}_{k,L}$ with $\mathrm{prin}(f)=f_{P}$.

###  2.6. Eisenstein Series

We recall coherent and incoherent Eisenstein series for the group
$G=\operatorname{SL}_{2}$ following [BKY12]. Let $F$ be a totally real field
of degree $d$ and discriminant $D_{F}$, $E/F$ be a quadratic CM extension with
absolute discriminant $D_{E}$ and
$\chi=\chi_{E/F}=\otimes_{v\leq\infty}\chi_{v}$ the associated Hecke
character. For a standard section $\Phi\in I(s,\chi)$ with

(2.53)
$I(s,\chi):=\operatorname{Ind}^{G(\mathbb{A}_{F})}_{B(\mathbb{A}_{F})}(|\cdot|^{s}\chi)=\bigotimes_{v\leq\infty}I_{v}(s,\chi_{v}),~{}I_{v}(s,\chi_{v}):=\operatorname{Ind}^{G(F_{v})}_{B(F_{v})}(|\cdot|_{v}^{s}\chi_{v}),$

we can form the Eisenstein series

$E^{*}(g,s,\Phi):=\Lambda(s+1,\chi)E(g,s,\Phi),~{}E(g,s,\Phi):=\sum_{\gamma\in
B\backslash\operatorname{SL}_{2}(F)}\Phi(\gamma g,s).$

where $\Lambda(s,\chi)$ is the completed $L$-function for $\chi$ (see equation
(4.6) in [BKY12]). When $\Phi=\otimes_{v}\Phi_{v}$, the Eisenstein series
$E(g,s,\Phi)$ has the Fourier expansion

$E^{*}(g,s,\Phi)=E^{*}_{0}(g,s,\Phi)+\sum_{{t}\in
F^{\times}}E^{*}_{t}(g,s,\Phi),~{}$

and for $t\in F^{\times}$,

$E^{*}_{t}(g,s,\Phi)=\prod_{v}W^{*}_{{t},v}(g,s,\Phi_{v}),$

where $W^{*}_{t,v}$ is the normalized local Whittaker function defined by

(2.54)
$\begin{split}W^{*}_{{t},v}(g_{v},s,\Phi_{v})&:=|D_{E}/D_{F}|_{v}^{-(s+1)/2}L(s+1,\chi_{v})\int_{F_{v}}\Phi_{v}(wn(b)g_{v},s)\psi_{v}(-{t}b)db.\end{split}$

For simplicity, we denote

(2.55)
$W^{*}_{t,v}(\Phi_{v}):=W^{*}_{t,v}(1,0,\Phi_{v}),~{}W^{*,\prime}_{t,v}(\Phi_{v}):=\partial_{s}W^{*}_{t,v}(1,s,\Phi_{v})\mid_{s=0}.$

We will be interested in the case when $\Phi$ is a Siegel-Weil section.

Given $\alpha\in F^{\times}$, we view $W_{\alpha}=E$ as an $F$-quadratic space
with quadratic form $Q_{\alpha}(z):=\alpha z\bar{z}$. Denote $\omega_{\alpha}$
the associated Weil representation. We have an
$\operatorname{SL}_{2}$-equivariant map
$\lambda_{\alpha}:{\mathcal{S}}(W_{\alpha}(\mathbb{A}_{F}))\to I(0,\chi)$
given by

(2.56) $\lambda_{\alpha}(\phi)(g)=(\omega_{\alpha}(g)\phi)(0).$

At each place $v$, there are local versions of $\omega_{\alpha}$ and
$\lambda_{\alpha}$ as well, denoted by $\omega_{\alpha,v}$ and
$\lambda_{\alpha,v}$. When $\Phi=\lambda_{\alpha}(\phi)$, resp.
$\Phi_{v}=\lambda_{\alpha,v}(\phi_{v})$, we replace $\Phi$, resp. $\Phi_{v}$,
from the notations above by $\phi$, resp. $\phi_{v}$.

Let $Z(W_{\alpha})$ be the CM points on $X_{K}$ as in Section 2.4 and
$W_{\alpha,\mathbb{Q}}\subset\mathrm{V}$ the rational quadratic space as in
(1.5). A special case of the Siegel-Weil formula (see [BKY12, Theorem 4.5])
gives us

(2.57) $\theta(g,Z(W_{\alpha}),\phi)=C\cdot E(g^{\Delta},0,\phi)$

for any
$\phi\in{\mathcal{S}}(W_{\alpha,\mathbb{Q}}(\mathbb{A}))={\mathcal{S}}(W_{\alpha}(\mathbb{A}_{F}))$.
Here $C=\deg(Z(W_{\alpha}))/2$.

Suppose only the $j$-th real embedding of $\alpha$ is negative. Denote
$\mathds{1}:=(1,\dots,1)$ and $\mathds{1}(j):=(1,\dots,-1,\dots,1)$ with $-1$
at the $j$-th slot. The sections
$\Phi(j)=\lambda_{\alpha,f}(\phi)\otimes\Phi^{\mathds{1}(j)}_{\infty}$ and
$\Phi=\lambda_{\alpha,f}(\phi)\otimes\Phi^{\mathds{1}}_{\infty}$ are coherent
and incoherent respectively. For all
$\phi\in{\mathcal{S}}(W_{\alpha}(\hat{F});\mathbb{C})$, the Eisenstein series
$E^{*}(g,s,\Phi(j))$ is holomorphic of weight $\mathds{1}(j)$ at $s=0$ and

(2.58)
$E^{*}(\tau,\phi,\mathds{1}(j)):={\mathrm{Nm}}(v)^{-1/2}E^{*}(g_{\tau},0,\Phi)$

is called a coherent Eisenstein series. On the other hand, the Eisenstein
series $E^{*}(g,s,\Phi)$ vanishes at $s=0$ and its derivative

(2.59)
$E^{*,\prime}(\tau,\phi):=\partial_{s}{\mathrm{Nm}}(v)^{-1/2}E^{*}(g_{\tau},s,\Phi)\mid_{s=0}$

is called an incoherent Eisenstein series, which is related to the coherent
Eisenstein series via the differential equation [BKY12, Lemma 4.3]

(2.60) $2L_{\tau_{j}}E^{*,\prime}(\tau,\phi)=E^{*}(\tau,\phi,\mathds{1}(j))$

for all $1\leq j\leq d$. Furthermore, it has the Fourier expansion

(2.61)
$E^{*,\prime}(\tau,\phi)=\mathcal{E}(\tau,\phi)+\phi(0)\Lambda(0,\chi)\log{\mathrm{Nm}}(v)+\mathcal{E}^{*}(\tau,\phi),$

where $\mathcal{E}^{*}(\tau,\phi)$ has exponential decay near the cusp
infinity and

$\mathcal{E}(\tau,\phi)=a_{0}(\phi)+\sum_{t\in F,~{}t\gg
0}a_{t}(\phi){\mathbf{e}}(\operatorname{Tr}(t\tau)).$

Here, $a_{0}(\phi)$ is an explicit constant (see (2.24) in [YY19]) and

(2.62)
$\begin{split}a_{t}(\phi)&:=\begin{cases}(-i)^{d}\tilde{W}_{t}(\phi)\log{\mathrm{Nm}}(\mathfrak{p}),&\text{
if }\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}\\},\\\ 0,&\text{
otherwise.}\end{cases}\end{split}$

The coefficient $\tilde{W}_{t}(\phi)$ is given by (see [YY19, Proposition
2.7])

(2.63)
$\tilde{W}_{t}(\phi)=2^{d}\frac{W^{*,\prime}_{t,\mathfrak{p}}(\phi_{\mathfrak{p}})}{\log{\mathrm{Nm}}(\mathfrak{p})}\prod_{v\nmid\mathfrak{p}\infty}W^{*}_{t,v}(\phi_{v})\in\mathbb{Q}(\phi)$

when $\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}\\}$

###  2.7. Hecke’s cusp form

Denote $\omega_{a}=\omega_{V_{a},\psi}$ the Weil representation and
$\theta_{a}:=\theta_{V_{a}}$ the theta function as in (2.33). For a bounded,
integrable function $\rho:H_{a}(\mathbb{Q})\backslash
H_{a}(\mathbb{A})/K\to\mathbb{C}$, consider the following theta lift

(2.64)
$\vartheta_{a}(g,\varphi,\rho):=\int_{[H_{a}]}\theta_{a}(g,h,\varphi)\rho(h)dh,$

Such integral was first considered by Hecke in [Hec27] when $\rho=\varrho$ is
an odd, continuous character, i.e.

(2.65) $\varrho(h)={\mathrm{sgn}}(h_{\infty})\varrho_{f}(h_{f})$

with $\varrho_{f}$ a continuous character on $H_{a}(\hat{\mathbb{Q}})$ and
$\varphi=\varphi^{\pm}_{\infty}\varphi_{f}$ with

(2.66) $\varphi^{\pm}_{\infty}(x_{1},x_{2}):=(x_{1}\pm
x_{2})e^{-\pi(x_{1}^{2}+x_{2}^{2})}\in{\mathcal{S}}(\mathbb{R}^{2}).$

Notice that $\varphi^{\pm}_{\infty}$ satisfies
$\varphi^{\pm}_{\infty}(-x_{1},-x_{2})=-\varphi^{\pm}_{\infty}(x_{1},x_{2})$.

In this case, the $m$-th Fourier coefficient of $\vartheta_{a}$ is given by

$W_{m}(\varphi,\varrho)(g):=\int_{[N]}\vartheta_{a}(ng,\varphi,\varrho){\psi(-mn)}dn.$

for $m\in\mathbb{Q}$. To evaluate it, we apply the usual unfolding trick

$\displaystyle W_{m}(\varphi,\varrho)(g)$
$\displaystyle=\int_{[N]}\int_{[H_{a}]}\sum_{\lambda\in
V_{a}(\mathbb{Q})}(\omega_{a}(ng)\varphi)(h^{-1}\lambda)\varrho(h)dh{\psi(-mn)}dn$
$\displaystyle=\int_{[H_{a}]}\sum_{\lambda\in
V_{a,m}(\mathbb{Q})}(\omega_{a}(g)\varphi)(h^{-1}\lambda)\varrho(h)dh$
$\displaystyle=\sum_{\lambda\in H_{a}(\mathbb{Q})\backslash
V_{a,m}(\mathbb{Q})}\int_{H_{a,\lambda}(\mathbb{Q})\backslash
H_{a}(\mathbb{A})}(\omega_{a}(g)\varphi)(h^{-1}\lambda)\varrho(h)dh.$

When $m=0$, we have $\lambda=0$ since $V_{a}$ is anistropic and
$\varphi(0)=\varphi^{\pm}_{\infty}(0)\varphi_{f}(0)=0$. When
$Q_{a}(\lambda)=m\neq 0$, the group $H_{a,\lambda}$ is trivial. We can then
write $g=g_{\tau}g_{f}$ with $g_{\tau}=n(u)m(\sqrt{v})\in G(\mathbb{R})$ and
$g_{f}\in G(\hat{\mathbb{Q}})$ and obtain

$W_{m}(\varphi,\varrho)(g)=\sum_{\lambda\in H_{a}(\mathbb{Q})\backslash
V_{a,m}(\mathbb{Q})}\int_{\mathbb{R}^{\times}}(\omega_{a}(g_{\tau})\varphi^{\pm}_{\infty})(t^{-1}(\iota_{a}(\lambda)))\frac{dt}{t}\int_{H_{a}(\hat{\mathbb{Q}})}\varrho_{f}(h)(\omega_{a}(g_{f})\varphi_{f})(h^{-1}\lambda)dh.$

The group $H_{a}(\mathbb{Q})$ acts on $V_{a,m}(\mathbb{Q})$ transitively. The
archimedean integral can be evaluated as

$\displaystyle\int_{\mathbb{R}^{\times}}(\omega_{a}(g_{\tau})\varphi^{\pm}_{\infty})(t^{-1}(\iota_{a}(\lambda)))\frac{dt}{t}$
$\displaystyle=2{\mathbf{e}}(mu){\frac{v}{\sqrt{a}}}\int_{0}^{\infty}(t^{-1}\lambda_{1}\pm{\mathrm{sgn}}(a)t\lambda_{2})e^{-\pi\frac{v}{|a|}(t^{-2}\lambda_{1}^{2}+t^{2}\lambda_{2}^{2})}\frac{dt}{t}$
$\displaystyle=2v{\mathbf{e}}(mu){\mathrm{sgn}}(\lambda_{1})\sqrt{|m|}\int^{\infty}_{0}(t^{-1}\pm{\mathrm{sgn}}(m)t)e^{-\pi
v|m|(t^{-2}+t^{2})}\frac{dt}{t}.$

This is 0 if $\pm{\mathrm{sgn}}(m)<0$ by the change of variable $t\mapsto
1/t$, and can be otherwise evaluated using the lemma below.

######  Lemma 2.6.

For any $\beta>0$, we have

$\int^{\infty}_{0}(t^{-1}+t)e^{-\beta\pi(t^{-2}+t^{2})}\frac{dt}{t}=\frac{e^{-2\pi\beta}}{\sqrt{\beta}}.$

Therefore, we have

(2.67)
$W_{m}(\varphi^{\pm},\varrho)((g_{\tau},g_{f}))=2\sqrt{v}{\mathbf{e}}(mu){\mathbf{e}}(|m|iv){\mathrm{sgn}}(\lambda_{1})\int_{H_{a}(\hat{\mathbb{Q}})}\varrho_{f}(h)(\omega_{a}(g_{f})\varphi_{f})(h^{-1}\lambda)dh,$

when $\pm m>0$ and $\lambda\in V_{a,m}(\mathbb{Q})$. Otherwise, it is 0. The
integral in (2.67) can be evaluated locally. Notice that it always converges
as $\varphi_{f}(h^{-1}\lambda)$ has compact support as a function of $h\in
H_{a}(\hat{\mathbb{Q}})$.

###  2.8. The deformed theta integral

We recall the real-analytic modular form of weight one constructed in [CL20]
using the notations of section 2.7. Let $\varrho$ be an odd, continuous
character as in (2.65), and $K_{\varrho}\subset H_{a}(\hat{\mathbb{Z}})$ the
open compact subgroup defined in (2.14). The intersection
$H_{a}(\mathbb{Q})^{+}\cap K_{\varrho}$, where
$H_{a}(\mathbb{Q})^{+}:=H_{a}(\mathbb{Q})\cap H_{a}(\mathbb{R})^{+}$, is a
cyclic subgroup

(2.68)
$\Gamma_{\varrho}=\langle\varepsilon_{\varrho}\rangle,~{}\varepsilon_{\varrho}>1>\varepsilon_{\varrho}^{\prime}>0$

of the totally positive units in ${\mathcal{O}}$. Then we have

(2.69) $H_{a}(\mathbb{A})=\coprod_{\xi\in
C}H_{a}(\mathbb{Q})H_{a}(\mathbb{R})^{+}K_{\varrho}\xi,$

where $C\subset H_{a}(\hat{\mathbb{Q}})$ is a finite subset of elements
representing $H_{a}(\mathbb{Q})^{+}\backslash
H_{a}(\hat{\mathbb{Q}})/K_{\varrho}$. So given $h=(h_{f},h_{\infty})$, we can
find $\alpha\in H_{a}(\mathbb{Q}),t\in H_{a}(\mathbb{R})^{+},k_{1}\in
K_{\varrho},\xi\in C$ all depending on $h$ such that

(2.70) $h=(\alpha k_{1}\xi,\alpha t),$

though the choice is not unique. This gives us the identification

(2.71) $H_{a}(\mathbb{Q})\backslash
H_{a}(\mathbb{A})/K_{\varrho}\cong\coprod_{\xi\in C}\Gamma_{\varrho}\backslash
H_{a}(\mathbb{R})^{+}\xi$

by sending $h=(\alpha k_{1}\xi,\alpha t)\in H_{a}(\mathbb{A})$ as in (2.70) to
$t\in H_{a}(\mathbb{R})^{+}$ in the $\xi$-component. Just like the
decomposition (2.69), this isomorphism depends on the choice of the set of
representatives $C$. Similarly, we have

(2.72) $H_{a}(\hat{\mathbb{Q}})/K_{\varrho}\cong\coprod_{\xi\in
C}\Gamma_{\varrho}\backslash H_{a}(\mathbb{Q})^{+}\xi.$

Using the Fourier coefficient $W_{m}$ in (2.67) and the decomposition in
(2.72), we can write

(2.73)
$\vartheta_{a}(g,\varphi^{\pm},\varrho)=\mathrm{vol}(K_{\varrho})\sum_{\xi\in
C}\varrho(\xi)\sum_{\beta\in\Gamma_{\varrho}\backslash
V_{a}(\mathbb{Q}),~{}\pm
Q_{a}(\beta)>0}(\omega_{a}(g)\varphi^{0,\pm}))(\xi^{-1}\beta).$

where $\varphi^{\pm}=\varphi_{f}\varphi_{\infty}^{\pm}$ and

(2.74)
$\varphi^{0,\pm}=\varphi_{f}\varphi_{\infty}^{0,\pm},~{}\varphi^{0,\pm}_{\infty}(x_{1},x_{2}):={\mathrm{sgn}}(x_{1})e^{\mp
2\pi x_{1}x_{2}}.$

Although $\varphi^{0,\pm}_{\infty}$ is not a Schwartz function on
$\mathbb{R}^{2}$, the sum above still converges absolutely. The subgroup
$B(\mathbb{R})\subset G(\mathbb{R})$ acts on it via $\omega_{a}$ as in (2.21),
whereas we impose a weight $\pm 1$ action by ${\mathrm{SO}}_{2}(\mathbb{R})$.
We also have a left $G(\mathbb{Q})$-invariant function

(2.75)
$\begin{split}\Theta_{a,C}(g,\varphi^{\pm},\varrho)&:=\mathrm{vol}({K_{\varrho}})\sum_{\xi\in
C}\varrho(\xi)\theta_{a}(g,\xi,\varphi^{\pm})\end{split}$

on $G(\mathbb{A})$ for $\varphi\in{\mathcal{S}}(V_{a}(\mathbb{A}))$. Note that
$\Theta_{a,C}$ depends on the choice of $C$.

We now define a function $\lg_{C}:H_{a}(\mathbb{Q})\backslash
H_{a}(\mathbb{A})/{K_{\varrho}}\to[0,1)$ by

(2.76) $\begin{split}\lg_{C}((\alpha k_{1}\xi,\alpha
t))&:=2\log\varepsilon_{\varrho}\cdot\\{\log
t/\log\varepsilon_{\varrho}\\},\\\ \\{a\\}&:=a-\lim_{\epsilon\to
0}\frac{1}{2}\left(\lfloor a+\epsilon\rfloor+\lfloor
a-\epsilon\rfloor\right).\end{split}$

Note that $\\{0\\}=\frac{1}{2}$. Unlike the function considered by Hecke,
$\lg_{C}$ cannot be written as the product of functions on
$H_{a}(\hat{\mathbb{Q}})$ and $H_{a}(\mathbb{R})$. Denote

(2.77) $\tilde{\varrho}_{C}:=\lg_{C}\cdot\varrho:H_{a}(\mathbb{Q})\backslash
H_{a}(\mathbb{A})/{K_{\varrho}}\to\mathbb{C}.$

Given $\varphi=\varphi_{f}\varphi_{\infty}\in{\mathcal{S}}(V_{a}(\mathbb{A}))$
for some ${K_{\varrho}}$-invariant
$\varphi_{f}\in{\mathcal{S}}(\hat{V}_{a};\mathbb{C})$, the deformed theta
integral $\vartheta_{a}(g,\varphi,\tilde{\varrho}_{C})$, where $\vartheta_{a}$
is defined in (2.64), was studied in [CL20], To describe its Fourier
expansion, denote

$\displaystyle\vartheta^{*}_{a}(g,\varphi_{f},\varrho)$
$\displaystyle:=\int_{H_{a}(\mathbb{Q})^{+}\backslash
H_{a}(\hat{\mathbb{Q}})}\varrho(h)\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
V_{a}(\mathbb{Q})\\\
Q_{a}(\beta)<0\end{subarray}}(\omega_{a}(g)\varphi^{0,*})(h^{-1}\beta)dh$
$\displaystyle=-\mathrm{vol}({K_{\varrho}})\sqrt{v}\sum_{\xi\in
C}\varrho(\xi)\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
V_{a}(\mathbb{Q})\\\
Q_{a}(\beta)<0\end{subarray}}(\omega_{a}(g_{f})\varphi_{f})(\xi^{-1}\beta){\mathrm{sgn}}(\beta){\mathbf{e}}(Q_{a}(\beta)\tau)\Gamma(0,4\pi|Q_{a}(\beta)|v).$

for $g=(g_{f},g_{\tau})\in G(\mathbb{A})$ with
$\varphi^{0,*}=\varphi_{f}\varphi^{0,*}_{\infty}$ and

(2.78) $\varphi^{0,*}_{\infty}(x_{1},x_{2}):=-{\mathrm{sgn}}(x_{1})e^{-2\pi
x_{1}x_{2}}\Gamma(0,4\pi|x_{1}x_{2}|).$

Note that $\vartheta^{*}_{a}$ is not necessarily left-$G(\mathbb{Q})$
invariant. But it is modular after applying the lowering operator as

$L\left(\vartheta^{*}_{a}(g,\varphi_{f},\varrho)\right)=\vartheta_{a}(g,\varphi^{-},\varrho).$

Similarly for $\xi\in H_{a}(\hat{\mathbb{Q}})$, define

(2.79) $\begin{split}\theta^{*}_{a}&(g,\xi,\varphi_{f}):=\sum_{\lambda\in
V_{a}(\mathbb{Q})}(\omega_{a}(g)\varphi^{*})(\xi^{-1}\lambda)=\sum_{\lambda\in
V_{a}(\mathbb{Q})}(\omega_{a}(g_{f})\varphi_{f})(\xi^{-1}\lambda)(\omega_{a}(g_{\tau})\varphi^{*})(\lambda)\\\
&=-v\sum_{\lambda\in
V_{a}(\mathbb{Q})}(\omega_{a}(g_{f})\varphi_{f})(\xi^{-1}\lambda)\frac{{\mathrm{sgn}}(\lambda_{1}-{\mathrm{sgn}}(a)\lambda_{2})}{\sqrt{\pi}}{\mathbf{e}}(Q_{a}(\lambda)\tau)\Gamma\left(\frac{1}{2},\frac{\pi
v}{|a|}(\lambda_{1}-{\mathrm{sgn}}(a)\lambda_{2})^{2}\right).\end{split}$

Here, we have employed the rapidly decaying function

(2.80) $\varphi^{*}(x_{1},x_{2}):=-e^{-2\pi
x_{1}x_{2}}{\mathrm{sgn}}(x_{1}-x_{2})\Gamma\left(\frac{1}{2},\pi(x_{1}-x_{2})^{2}\right),$

where $B(\mathbb{R})\subset G(\mathbb{R})$ acts via $\omega_{a}$ and
${\mathrm{SO}}_{2}(\mathbb{R})$ acts with weight 1. Also, we denote

(2.81)
$\Theta^{*}_{a,C}(g,\varphi_{f},\varrho):=\mathrm{vol}({K_{\varrho}})\sum_{\xi\in
C}\varrho(\xi)\theta^{*}_{a}(g,\xi,\varphi_{f}),$

which satisfies

(2.82)
$L\Theta^{*}_{a,C}(g,\varphi_{f},\varrho)=\Theta_{a,C}(g,\varphi^{-},\varrho).$

We recall some results.

######  Theorem 2.7.

Let $\tilde{\varrho}_{C}$ be as in (2.77) and $\varphi_{f}\in
V_{a}(\hat{\mathbb{Q}})$ a right-${K_{\varrho}}$ invariant function. Then the
integral $\vartheta_{a}(g,\varphi^{+},\tilde{\varrho}_{C})$ defines a
$G(\mathbb{Q})$-invariant function in $g\in G(\mathbb{A})$ of weight 1 with
respect to ${\mathrm{SO}}_{2}(\mathbb{R})$. Furthermore, it has the Fourier
expansion

(2.83)
$\begin{split}\vartheta_{a}(g,\varphi^{+},\tilde{\varrho}_{C})&={\vartheta^{*}_{a}(g,\varphi_{f},\varrho)}+\log\varepsilon_{\varrho}\Theta^{*}_{a,C}(g,\varphi_{f},\varrho)\\\
&\quad+\mathrm{vol}({K_{\varrho}})\sum_{\begin{subarray}{c}\xi\in C\\\
\beta\in\Gamma_{\varrho}\backslash V_{a}(\mathbb{Q})\\\
Q_{a}(\beta)>0\end{subarray}}\tilde{\varrho}_{C}((\xi,\sqrt{|\beta/\beta^{\prime}|}))(\omega_{a}(g)\varphi^{0,+})(\xi^{-1}\beta),\end{split}$

where $\varphi^{0,+}=\varphi_{f}\varphi^{0,+}_{\infty}$ is defined in (2.74),
and satisfies

(2.84)
$L\vartheta_{a}(g,\varphi^{+},\tilde{\varrho}_{C})=\vartheta_{a}(g,\varphi^{-},\varrho)+\log\varepsilon_{\varrho}\Theta_{a,C}(g,\varphi^{-},\varrho).$

###### Proof.

This follows essentially from Proposition 5.5 in [CL20]. For completeness, we
include a different (and slightly shorter) proof here. As in the evaluation of
$\vartheta_{a}(g,\varphi,\varrho)$ in (2.73), we have

$\displaystyle W_{m}(\varphi,\tilde{\varrho}_{C})(g_{\tau})$
$\displaystyle=\mathrm{vol}({K_{\varrho}}){\mathbf{e}}(mu)\sqrt{v}\sum_{\xi\in
C}\sum_{\beta\in\Gamma_{\varrho}\backslash
V_{a,m}(\mathbb{Q})}\varphi_{f}(\xi^{-1}\beta)J(\beta,v),$ $\displaystyle
J(\beta,v)$
$\displaystyle:=2\log\varepsilon_{\varrho}\int^{\infty}_{0}\varphi^{+}_{\infty}(t^{-1}\cdot(\iota_{a}(\beta)\sqrt{v}))\left\\{\frac{\log
t}{\log\varepsilon_{\varrho}}\right\\}\frac{dt}{t}$
$\displaystyle=2\log\varepsilon_{\varrho}{\mathrm{sgn}}(m\beta)\int^{\infty}_{0}\varphi^{{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\left\\{\frac{\log
t}{\log\varepsilon_{\varrho}}+\frac{1}{2}\frac{\log|\beta/\beta^{\prime}|}{\log\varepsilon_{\varrho}}\right\\}\frac{dt}{t}.$

To verify (2.84), we start with

$\displaystyle L_{\tau}$
$\displaystyle{\mathbf{e}}(mu)J(\beta,v)={\mathbf{e}}(m\tau)v^{2}\partial_{v}\left(e^{2\pi
mv}J(\beta,v)\right)$
$\displaystyle={\mathbf{e}}(m\tau)2\log\varepsilon_{\varrho}{\mathrm{sgn}}(m\beta)\int^{\infty}_{0}v^{2}\partial_{v}e^{2\pi
mv}\varphi^{{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\left\\{\frac{\log
t}{\log\varepsilon_{\varrho}}+\frac{1}{2}\frac{\log|\beta/\beta^{\prime}|}{\log\varepsilon_{\varrho}}\right\\}\frac{dt}{t}$
$\displaystyle=v{\mathbf{e}}(mu){\mathrm{sgn}}(m\beta)\log\varepsilon_{\varrho}\int^{\infty}_{0}\partial_{t}\varphi^{-{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\left\\{\frac{\log
t}{\log\varepsilon_{\varrho}}+\frac{1}{2}\frac{\log|\beta/\beta^{\prime}|}{\log\varepsilon_{\varrho}}\right\\}dt$
$\displaystyle=-v{\mathbf{e}}(mu)\left({\mathrm{sgn}}(m\beta)\int^{\infty}_{0}\varphi^{-{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\frac{dt}{t}-\log\varepsilon_{\varrho}\sum_{\varepsilon\in\Gamma_{\varrho}}\varphi^{+}_{\infty}(\iota_{a}(\beta\varepsilon))\right)$
$\displaystyle=v{\mathbf{e}}(mu)\left(\delta_{m<0}{\mathrm{sgn}}(\beta)+\log\varepsilon_{\varrho}\sum_{\varepsilon\in\Gamma_{\varrho}}\varphi^{+}_{\infty}(\iota_{a}(\beta\varepsilon))\right).$

Substitute this into the left hand side of (2.84) proves it.

Now to calculate the Fourier expansion, it suffices prove the claim

(2.85) $\lim_{v\to\infty}e^{2\pi
mv}J(\beta,v)={\mathrm{sgn}}(\beta)\lg_{C}((1,\sqrt{|\beta/\beta^{\prime}|}).$

For each $\beta\in\Gamma_{\varrho}\backslash V_{a}(\mathbb{Q})$, we choose the
unique representative $\beta_{0}\in\Gamma_{\varrho}\beta$ such that
$|\beta_{0}/\beta_{0}^{\prime}|\in[1,\varepsilon_{\varrho}^{2})$. We can then
write $J(\beta,v)=J_{1}(\beta_{0},v)+J_{2}(\beta_{0},v)$, where

$\displaystyle J_{1}(\beta_{0},v)$
$\displaystyle:=2\int^{\infty}_{0}\varphi^{+}_{\infty}(t^{-1}\cdot(\iota_{a}(\beta_{0})\sqrt{v}))\log
t\frac{dt}{t},$ $\displaystyle J_{2}(\beta_{0},v)$
$\displaystyle:=-2\log\varepsilon_{\varrho}\int^{\infty}_{0}\varphi^{+}_{\infty}(t^{-1}\cdot(\iota_{a}(\beta_{0})\sqrt{v}))\left\lfloor\frac{\log
t}{\log\varepsilon_{\varrho}}\right\rfloor\frac{dt}{t}.$

For $J_{1}$, we have

$\displaystyle\lim_{v\to\infty}e^{2\pi mv}J_{1}(\beta_{0},v)$
$\displaystyle=\log|\beta_{0}/\beta_{0}^{\prime}|{\mathrm{sgn}}(m\beta_{0})\lim_{v\to\infty}e^{2\pi
mv}\int^{\infty}_{0}\varphi^{{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\frac{dt}{t}$
$\displaystyle=\log|\beta_{0}/\beta_{0}^{\prime}|{\mathrm{sgn}}(\beta_{0})\delta_{m>0}.$

For $J_{2}$, the limit vanishes unless $|\beta_{0}/\beta_{0}^{\prime}|=1$, in
which case

$\displaystyle\lim_{v\to\infty}e^{2\pi mv}J_{2}(\beta_{0},v)$
$\displaystyle=2\log\varepsilon_{\varrho}{\mathrm{sgn}}(m\beta_{0})\lim_{v\to\infty}e^{2\pi
mv}\int^{1}_{\varepsilon_{\varrho}^{-1}}\varphi^{{\mathrm{sgn}}(m)}_{\infty}(\sqrt{|m|v}(t,t^{-1}))\frac{dt}{t}$
$\displaystyle=\log\varepsilon_{\varrho}{\mathrm{sgn}}(\beta_{0})\delta_{m>0}.$

Putting these together proves claim (2.85). ∎

Finally, we record a result as a direct consequence of Theorem 4.5 in [CL20]
(see also section 5 in [LS22]).

######  Proposition 2.8.

For any $\varphi_{f}\in{\mathcal{S}}(\hat{V}_{1};\mathbb{C})$, there exists a
real-analytic modular form
$\tilde{\Theta}_{a,C}(g,\varphi^{-},\varrho)=\tilde{\Theta}^{+}_{a,C}(g,\varphi^{-},\varrho)+\tilde{\Theta}^{*}_{a,C}(g,\varphi^{-},\varrho)$
such that $L\tilde{\Theta}_{a,C}=\Theta_{a,C}$ and
$\sqrt{v}\tilde{\Theta}^{+}_{a,C}(g_{\tau},\varphi^{-},\varrho)$ is
holomorphic in $\tau$ with Fourier coefficients in $\mathbb{Q}(\varphi_{f})$.

##  3\.  Doi-Naganuma Lift of Hecke’s Cusp Form

In this section, we are interested in computing the $\mathrm{O}(2,2)$ theta
lift of Hecke’s cusp form from section 2.7, and realize them as coherent
Hilbert Eisenstein series from 2.6 over real quadratic fields. The main result
of this section is the global matching Theorem 3.3, where we show that any
coherent Eisenstein series can be realized as such theta lifts. This global
statement follows from its local counterpart in Theorem 3.11, which is
improved further in Theorem 3.15 to allow matching deformed local sections.
This last result will be crucial for us in proving the factorization result in
proposition 4.7 later.

###  3.1. Quadratic Spaces

Let $V_{\pm 1}$ be as in Section 2.2, $\ell^{+},\ell^{-}$ be isotropic lines
such that $\ell^{+}\oplus\ell^{-}$ is a hyperbolic plane and denote

(3.1) $V:=V_{0}\oplus V_{1},~{}V_{0}:=\ell^{+}\oplus\ell^{-}\oplus V_{-1}.$

We can realize

$\displaystyle V_{0}(\mathbb{Q})$ $\displaystyle\cong\left\\{\Lambda\in
M_{2}(F):\Lambda^{t}=\Lambda^{\prime}\right\\}$ $\displaystyle(a,b,\lambda)$
$\displaystyle\mapsto\begin{pmatrix}a&\lambda\\\
\lambda^{\prime}&b\end{pmatrix}$

with $\det$ as the quadratic form, and furthermore write

(3.2) $V_{0}=V_{00}\oplus U_{D},~{}V_{00}:=V_{0}\cap
M_{2}(\mathbb{Q}),~{}U_{D}:=\sqrt{D}\mathbb{Q}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\cong(\mathbb{Q},Q_{D}),$

where $Q_{D}(x)=Dx^{2}$. So $V$ has Witt rank 3 and admits the isotropic
decomposition

(3.3)
$V=V^{+}+V^{-},~{}V^{\pm}:=\ell^{\pm}+(V_{-1}+V_{1})^{\pm},~{}(V_{-1}+V_{1})^{\pm}(\mathbb{Q}):=\\{(\lambda,\pm\lambda):\lambda\in
F\\}$

with $V^{\pm}$ maximal totally isotropic subspaces. For a $\mathbb{Q}$-algebra
$R$, e.g.
$R\in\\{\mathbb{Q},\mathbb{Q}_{p},\mathbb{R},\hat{\mathbb{Q}},\mathbb{A}\\}$,
We will use

(3.4) $(a,b,\lambda,\mu)\in V(R),~{}a,b\in R,\lambda\in R\otimes F\cong
V_{-1}(R),\mu\in R\otimes F\cong V_{1}(R)$

to represent elements in $V(R)$. Define elements $f_{j}^{\pm}\in V$ by

(3.5)
$\begin{split}f_{1}^{+}&:=(1,0,0,0),~{}f_{1}^{-}:=(0,1,0,0),~{}f_{2}^{+}:=(0,0,1/2,1/2),~{}f_{2}^{-}:=(0,0,1/2,-1/2),\\\
f_{3}^{+}&:=(0,0,\sqrt{D}/2,\sqrt{D}/2),~{}f_{3}^{-}:=(0,0,1/(2\sqrt{D}),-1/(2\sqrt{D})).\end{split}$

Then $\\{f_{j}^{\pm}:j=1,2,3\\}\subset V^{\pm}$ is a $\mathbb{Q}$-basis of
$V^{\pm}$. With respect to the ordered basis
$(f_{1}^{+},f_{2}^{+},f_{3}^{+},f_{1}^{-},f_{2}^{-},f_{3}^{-})$, the Gram
matrix of $Q$ is $\left(\begin{smallmatrix}0&I_{3}\\\
I_{3}&0\end{smallmatrix}\right)$. For $i=1,2,3$, the following linear
transformations

(3.6) $w_{i}(f_{j}^{\pm}):=\begin{cases}f_{j}^{\mp},&\text{ if }i=j,\\\
f_{j}^{\pm},&\text{ otherwise.}\end{cases}$

are easily checked to be in $\mathrm{O}(V)$. The unimodular lattice

(3.7) $V_{\mathbb{Z}}:=\\{(a,b,\lambda,\mu)\in
V(\mathbb{Q})\cap(\mathbb{Z}^{2}\times(\mathfrak{d}^{-1})^{2}):\lambda-\mu\in{\mathcal{O}}_{F}\\}\subset
V$

provides $V$ with an integral structure. Similarly for $?\in\\{00,0,1\\}$, the
lattice $V_{?,\mathbb{Z}}:=V_{\mathbb{Z}}\cap V_{?}$ in $V_{?}$ gives it with
an integral structure.

For $?\in\\{00,0,1,-1,\emptyset\\}$. we write

(3.8)
${\tilde{H}}_{?}:=\operatorname{GSpin}(V_{?}),~{}H_{?}:={\mathrm{SO}}(V_{?}),$

which are subgroups of ${\tilde{H}}$ and $H$ respectively by acting trivially
on $V^{\perp}_{?}$, and have the following exact sequences

(3.9) $1\to\mathrm{G}_{m}\to{\tilde{H}}_{?}\to H_{?}\to 1.$

For any commutative ring $R$, we have explicitly

(3.10)
$\iota:\operatorname{GSpin}(V_{0,\mathbb{Z}})(R)\cong\\{\gamma\in{\mathrm{GL}}_{2}({\mathcal{O}}\otimes_{\mathbb{Z}}R):\det(\gamma)\in
R^{\times}\\},$

via the action of
$\gamma\in{\mathrm{GL}}_{2}({\mathcal{O}}\otimes_{\mathbb{Z}}R)$ on
$V_{\mathbb{Z},0}(R)$

(3.11) $\Lambda\mapsto\det(\gamma)^{-1}\gamma\Lambda(\gamma^{\prime})^{t}.$

For any $\mathbb{Q}$-algebra $R$, we also have
${\tilde{H}}_{?}(R)=\operatorname{GSpin}(V_{?,\mathbb{Z}})(R)$ for
$?\in\\{00,0,1,\emptyset\\}$. Therefore through $\iota$, we have

(3.12) $G_{0}:=\mathrm{Spin}(V_{0})\cong
G_{F}:=\mathrm{Res}_{F/\mathbb{Q}}(G),~{}G_{00}:=\mathrm{Spin}(V_{00})\cong
G,~{}H_{00}\cong\mathrm{PGL}_{2},$

and will represent elements in $H_{0}$ by their preimages in $G_{F}$. Denote
$T_{0}:=\iota^{-1}(T)\subset{\tilde{H}}_{0}$ Then the relations among these
groups can be visualized in the following diagram

(3.13)
${G_{00}}$${G_{0}}$${{\tilde{H}}_{0}}$${T_{0}}$${\operatorname{SL}_{2}}$${G_{F}}$${{\mathrm{GL}}_{2/F}}$${T}$${H_{00}}$${H_{0}}$${H_{0}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Here the horizontal and vertical arrows are natural inclusions and surjections
of algebraic groups respectively, and the diagonal arrows are induced by
$\iota$. Let $B_{F}\subset G_{F}$ be the standard parabolic subgroup, and
$B_{0}:=\iota^{-1}(B_{F})\subset G_{0}$. They can be visualized as

(3.14)
${B_{0}}$${G_{0}}$${B_{F}}$${G_{F}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

which gives us

(3.15) $B_{0}(\mathbb{Q})\backslash G_{0}(\mathbb{Q})\cong
B_{F}(\mathbb{Q})\backslash G_{F}(\mathbb{Q})=B(F)\backslash G(F)$

via $\iota$. We also denote

(3.16) $T^{\Delta}\subset T\times
T_{0}\subset{\mathrm{GL}}_{2}\times{\tilde{H}}_{0}$

the diagonal, which will play a crucial role in the local matching result in
section 3.5.

Now let $P\subset H$ be the Siegel parabolic stabilizing $V^{+}$, whose Levi
factor is isomorphic to ${\mathrm{GL}}(V^{+})$. Then $P_{0}:=P\cap
H_{0}\subset H$ is the subgroup stabilizing the line $\ell^{+}$ and acting
trivially on $V_{1}$. The preimage of $P_{0}H_{-1}\subset H_{0}$ in
${\tilde{H}}_{0}$ is given by $B_{0}T_{0}$. Combining with (3.15), we obtain

(3.17) $(P_{0}H_{-1})(\mathbb{Q})\backslash
H_{0}(\mathbb{Q})=(B_{0}T_{0})(\mathbb{Q})\backslash{\tilde{H}}_{0}(\mathbb{Q})=B_{0}(\mathbb{Q})\backslash
G_{0}(\mathbb{Q})\cong B(F)\backslash G(F).$

For $\alpha\in F^{\times},\beta\in F$, we then have $m(\alpha),n(\beta)\in
G_{0}(\mathbb{Q})\subset{\tilde{H}}(\mathbb{Q})$. It is easy to check that

$\displaystyle(\omega(m(\alpha))\varphi)(a,b,\nu,\lambda)$
$\displaystyle=\varphi(a/{\alpha\alpha^{\prime}},\alpha\alpha^{\prime}b,\alpha^{\prime}\nu/\alpha,\lambda),$
$\displaystyle(\omega(n(\beta))\varphi)(a,b,\nu,\lambda)$
$\displaystyle=\varphi(a-\beta\nu-\beta^{\prime}\nu^{\prime}+\beta\beta^{\prime}b,\beta,\nu-\beta
b,\lambda)$

for a Schwartz function $\varphi\in{\mathcal{S}}(V(\mathbb{A}))$.

###  3.2. Theta Integral

Let $\theta_{0}(g,g_{1},\varphi_{0})$ denote the theta function on $[G\times
H_{0}]$ associated to $\varphi_{0}\in{\mathcal{S}}(V_{0}(\mathbb{A}))$.
Suppose $\varphi_{0,\infty}$ is in the polynomial Fock space
$\mathbb{S}(V_{0}(\mathbb{R}))$ (see section 4.1). We can then restrict
$\theta_{0}$ to the subgroup $[G\times H_{00}]$ and view it as a function on
$[G\times G_{00}]$, and write

(3.18) $\theta_{0}(g,g_{00},\varphi_{0})=\sum_{j\in
J}\theta_{00}(g,g_{00},\varphi_{00,j})\theta_{D}(g,\varphi_{D,j}),$

where $\varphi_{0}=\sum_{j\in J}\varphi_{00,j}\varphi_{D,j}$ with
$\varphi_{00,j}\in{\mathcal{S}}(V_{00}(\mathbb{A}))$ and
$\varphi_{D,j}\in{\mathcal{S}}(U_{D}(\mathbb{A}))$.

We now define

(3.19)
$I_{0}(h_{0},\varphi_{0},f):=\int_{[G]}\theta_{0}(g,h_{0},\varphi_{0})f(g)dg$

for $f$ a bounded, integrable function on $[G]$. Note that the measure $dg$ is
normalized so that $[G]$ has volume 1. In particular for a right
$G(\hat{\mathbb{Z}})$-invariant function $\phi$ on $[G]$, we have

(3.20)
$\int_{[G]}\phi(g)dg=\frac{3}{\pi}\int_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\phi(g_{\tau})d\mu(\tau),~{}d\mu(\tau):=\frac{dudv}{v^{2}}.$

When $f(g)=\vartheta_{1}(g,\varphi_{1},\rho)$ for a bounded, integrable
function $\rho$ on $H_{1}(\mathbb{Q})\backslash H_{1}(\mathbb{A})$, the
integral $I_{0}$ above becomes

(3.21)
$\begin{split}\mathcal{I}(h_{0},\varphi,\rho)&:=\int_{[H_{1}]}I((h_{0},h_{1}),\varphi)\rho(h_{1})dh_{1}=I_{0}(h_{0},\varphi_{0},\vartheta_{1}(\cdot,\varphi_{1},\rho)),\\\
I(h,\varphi)&:=\int_{[G]}\theta(g,h,\varphi)dg\end{split}$

with $\varphi=\varphi_{0}\otimes\varphi_{1}$.

For our purpose, $\rho=\varrho$ will be an odd, continuous character as in
(2.65), and $\varphi=\varphi_{f}\varphi^{(\pm 1,\mp 1)}_{\infty}$ with

(3.22) $\begin{split}\varphi^{(\pm 1,\mp 1)}_{\infty}&:=\varphi^{(\pm 1,\mp
1)}_{0,\infty}\otimes\varphi^{-}_{\infty},\\\ \varphi^{(\pm 1,\mp
1)}_{0,\infty}(a,b,\nu_{1},\nu_{2})&:=({\pm}i(a+b)+(\nu_{1}-\nu_{2}))e^{-\pi(a^{2}+b^{2}+\nu_{1}^{2}+\nu_{2}^{2})},\end{split}$

and $\varphi^{\pm}_{\infty}$ defined in (2.66). Here we have identified
$V(\mathbb{R})=\mathbb{R}^{2}\oplus V_{1}(\mathbb{R})\oplus
V_{-1}(\mathbb{R})\cong(\mathbb{R}^{2})^{\oplus 3}$ via (2.11). We have

$\omega(w)\varphi^{(\pm 1,\mp 1)}_{\infty}=\varphi^{(\pm 1,\mp 1)}_{\infty}$

where $\omega$ is the Weil representation on $V(\mathbb{R})$. Therefore,
$\varphi^{(\pm 1,\mp 1)}_{\infty}$ is a
$K_{\infty}(=\mathrm{SO}_{2}(\mathbb{R}))$-invariant vector. On the other
hand, for $h=(w,1),h^{\prime}=(1,w)\in H_{0}(\mathbb{R})$, it is easy to see
that

$\omega(h)\varphi^{(\pm 1,\mp 1)}_{\infty}=\mp i\varphi^{(\pm 1,\mp
1)}_{\infty},~{}\omega(h^{\prime})\varphi^{(\pm 1,\mp 1)}_{\infty}=\pm
i\varphi^{(\pm 1,\mp 1)}_{\infty}.$

So $\varphi^{(\pm 1,\mp 1)}_{\infty}$ is equivariant of weight $(\pm 1,\mp 1)$
with respect to the maximal compact of $H_{0}(\mathbb{R})$. Later, we will
also consider the following integral

(3.23)
$\mathcal{I}_{f}(h_{0},\varphi,\varrho):=\int_{H_{1}(\mathbb{Q})\backslash
H_{1}(\hat{\mathbb{Q}})}\varrho(h_{1})\int_{[G]}\theta(g,(h_{0},h_{1}),\varphi)dgdh_{1},$

which is similar to $\mathcal{I}(h_{0},\varphi,\varrho)$ and well-defined as

$\varrho(-h_{1})\theta(g,(h_{0},-h_{1}),\varphi)=\varrho(h_{1})\theta(g,(h_{0},h_{1}),\varphi)$

for all $g,h_{0},h_{1}$ and $\varphi_{f}\in{\mathcal{S}}(\hat{V})$. When
$\varphi=\varphi_{0}\otimes\varphi_{1}$, we have

(3.24)
$\mathcal{I}_{f}(h_{0},\varphi,\varrho)=I_{0}(h_{0},\varphi_{0},\Theta_{1}(\cdot,\varphi_{1},\varrho)).$

###  3.3. Fourier Transform and Siegel-Weil formula

We recall the Siegel-Weil formula needed for our purpose following [GQT14].
The range we need is in the 1st term range, and was originally proved in
[KR94]. Let
$\varphi=\varphi_{\infty}\varphi_{f}\in{\mathcal{S}}(V(\mathbb{A}))$ with
$\varphi_{\infty}$ as in (3.22) above. For $(g,h)\in G(\mathbb{A})\times
H(\mathbb{A})$, we have the theta function $\theta(g,h,\varphi)$, and are
interested in the value of convergent integral $I(h,\varphi)$ defined in
(3.21).

For a rational quadratic space $(V,(,)_{V})$, suppose
$V=U^{+}+U^{-}+V_{\circ}$ with $U^{+},U^{-}$ complementary totally isotropic
subspaces and $V_{\circ}=(U^{+}+U^{-})^{\perp}$. Let $W=X+Y$ denote the
symplectic space of rank 2 over $\mathbb{Q}$ with the symplectic pairing
$\langle,\rangle_{W}$. The rational vector space $\mathbb{W}:=V\otimes W$ is
then a symplectic space with respect to the pairing

(3.25) $\langle v_{1}\otimes w_{1},v_{2}\otimes
w_{2}\rangle_{\mathbb{W}}:=(v_{1},v_{2})_{V}\langle w_{1},w_{2}\rangle_{W}.$

From this, we have the Fourier transform
${\mathscr{F}}_{U^{+}}:{\mathcal{S}}(V(\mathbb{A}))\to{\mathcal{S}}(((U^{-}\otimes
W)+V_{\circ})(\mathbb{A}))$ defined by

(3.26)
${\mathscr{F}}_{U^{+}}(\varphi)(\eta,v_{\circ}):=\int_{U^{+}(\mathbb{A})}\varphi(u^{+},\eta_{1},v_{\circ})\psi((u^{+},\eta_{2})_{V})du^{+}$

with $\eta=(\eta_{1},\eta_{2})\in(U^{-})^{2}(\mathbb{A})\cong(U^{-}\otimes
W)(\mathbb{A})$ and $\eta_{i}\in U^{-}(\mathbb{A})$. Note that we have
$(u^{+},\eta_{1},v_{\circ})\in(U^{+}\otimes X+U^{-}\otimes
X+V_{\circ})(\mathbb{A})=V(\mathbb{A})$. Note that on
${\mathcal{S}}((U^{-}\otimes W+V_{\circ})(\mathbb{A}))$, the Weil
representation $\omega$ acts as

(3.27)
$\begin{split}(\omega(g,1)\phi)(\eta,v_{\circ})&=\omega_{V_{\circ}}(g)\phi(\eta
g,v_{\circ}),g\in G(\mathbb{A}),\\\
(\omega(1,a)\phi)(\eta,v_{\circ})&=|\det(a)|\phi(a^{-1}\eta,v_{\circ}),a\in{\mathrm{GL}}(U^{+})(\mathbb{A}),\\\
(\omega(1,u)\phi)(\eta,v_{\circ})&=\psi(\langle
u(\eta),\eta\rangle/2)\phi(\eta,v_{\circ}),u\in
N(U^{+})(\mathbb{A})\subset\mathrm{Hom}_{\mathbb{Q}}(U^{-},U^{+})(\mathbb{A}),\end{split}$

which makes ${\mathscr{F}}_{U^{+}}$ an intertwining map.

For $V$ in (3.1), we can take $U^{\pm}=V^{\pm}$ and $V_{\circ}$ trivial with
$V^{\pm}$ defined in (3.3). Another possibility is to take
$U^{\pm}=\ell^{\pm}$ and $V_{\circ}=V_{1}\oplus V_{-1}$, which will be used in
calculating the Fourier expansion of the theta integral $I_{0}$ in (3.19). To
simplify notations, we write

(3.28)
${\mathscr{F}}:={\mathscr{F}}_{V^{+}},~{}{\mathscr{F}}_{1}:={\mathscr{F}}_{\ell^{+}},$

and use them to represent the Fourier transform at the finite and infinite
places as well. For example, ${\mathscr{F}}_{1}$ is given by

(3.29)
${\mathscr{F}}_{1}(\varphi)((\eta_{1},\eta_{2}),\nu,\lambda)=\int_{\mathbb{A}}\varphi(b,\eta_{1},\nu,\lambda)\psi(b\eta_{2})db$

for $\varphi\in{\mathcal{S}}(V(\mathbb{A}))$. As ${\mathscr{F}}_{1}$ acts
trivially on
${\mathcal{S}}(V_{1}(\mathbb{A}))\subset{\mathcal{S}}(V(\mathbb{A}))={\mathcal{S}}(V_{0}(\mathbb{A}))\otimes{\mathcal{S}}(V_{1}(\mathbb{A}))$,
we will denote its restriction to
${\mathcal{S}}(V_{0}(\mathbb{A}))\subset{\mathcal{S}}(V(\mathbb{A}))$ by
${\mathscr{F}}_{1}$ as well.

For a place $v\leq\infty$ of $\mathbb{Q}$ and corresponding local field
$k=\mathbb{Q}_{v}$, recall we have the Siegel-Weil section

$\displaystyle\Phi_{v}:{\mathcal{S}}((V^{-}\otimes W)(k))$ $\displaystyle\to
I^{H}_{v}(0)$ $\displaystyle\phi_{v}$
$\displaystyle\mapsto(h\mapsto(\omega_{v}(h)\phi_{v})(0)),$

where $I^{H}_{v}(s)=\operatorname{Ind}^{H(k)}_{P(k)}(|\cdot|^{s})$ is the
degenerate principal series. The image of $\Phi_{v}$ is a submodule of
$I^{H}_{v}(0)$ denoted by $R_{v}(W)$. When $v<\infty$, it is known that (see
[GQT14, Proposition 5.2(ii)])

$I^{H}_{v}(0)=R_{v}(W)\oplus(R_{v}(W)\otimes{\det}_{H}).$

It is clear that

(3.30) $\Phi_{v}(\omega(g)\phi_{v})=\Phi_{v}(\phi_{v})$

for any $g\in G(k)$.

Given any $\phi=\otimes_{v}\phi_{v}\in{\mathcal{S}}((V^{-}\otimes
W)(\mathbb{A}))$, we denote $\Phi_{s}(\phi)\in I^{H}(s)$ the standard section
satisfying $\Phi_{0}(\phi)=\otimes_{v}\Phi_{v}(\phi_{v})$. We can then form
the Eisenstein series

$E^{H}_{P}(s,\phi)(h):=\sum_{\gamma\in P(\mathbb{Q})\backslash
H(\mathbb{Q})}\Phi_{s}(\phi)(\gamma h),$

which has meromorphic continuation to $s\in\mathbb{C}$ and is holomorphic at
$s=0$. The regularized Siegel-Weil formula by Kudla-Rallis gives then the
following equality (see [GQT14, Theorem 7.3(ii)])

(3.31) $2I(h,\varphi)=E^{H}_{P}(0,{\mathscr{F}}(\varphi))(h).$

As a special case of the proposition in section 2 of [Mœg97], following an
argument in [GPSR87], we have the following lemma.

######  Lemma 3.1.

For any $h\in H(\mathbb{A})$, we have

(3.32) $E^{H}_{P}(s,\phi)(h)=\sum_{\gamma_{0}\in B(F)\backslash
G(F),~{}\gamma_{1}\in
H_{1}(\mathbb{Q})}\Phi_{s}(\phi)((\gamma_{0},\gamma_{1})h).$

###### Proof.

We will show that $B(F)\backslash G(F)\cong(P(\mathbb{Q})\backslash
H(\mathbb{Q}))\times H_{1}(\mathbb{Q})$ with the map induced by (3.11). First,
we have $P(\mathbb{Q})\backslash H(\mathbb{Q})=(P\cap(H_{0}\times
H_{1}))(\mathbb{Q})\backslash(H_{0}\times H_{1})(\mathbb{Q})$. Let
$H_{-1}\subset H_{0}$ denote the image of ${\mathrm{SO}}(V_{-1})$, which is
isomorphic to $H_{1}$, and $P_{0}:=P\cap H_{0}$. Then $P\cap(H_{0}\times
H_{1})=P_{0}P_{1}^{\Delta}$ with $P_{1}^{\Delta}\cong H_{1}$ the image of the
diagonal embedding of $H_{1}$ into $H_{-1}\times H_{1}$. From this, we obtain

$(P\cap(H_{0}\times H_{1}))(\mathbb{Q})\backslash(H_{0}\times
H_{1})(\mathbb{Q})=(P_{0}P_{1}^{\Delta})(\mathbb{Q})\backslash(H_{0}\times
H_{1})(\mathbb{Q})=(((P_{0}H_{-1})\backslash H_{0})\times H_{1})(\mathbb{Q}).$

Equation (3.17) then finishes the proof. ∎

Suppose $\varrho=\otimes_{p\leq\infty}\varrho_{p}$ is an odd character of
$H_{1}(\mathbb{A})/H_{1}(\mathbb{Q})$ and

(3.33) $\chi:=\varrho\circ{\mathrm{Nm}}^{-}=\otimes_{v\leq\infty}\chi_{v}$

a totally odd character of $\mathbb{A}_{F}^{\times}/F^{\times}$, which can be
viewed as a character on $B_{0}(\mathbb{A})$. Denote

(3.34)
$I^{G_{0}}(\chi):=\mathrm{Ind}_{B_{0}(\mathbb{A})}^{G_{0}(\mathbb{A})}\chi,~{}I^{G_{0}}_{p}(\chi_{p}):=\mathrm{Ind}_{B_{0}(\mathbb{Q}_{p})}^{G_{0}(\mathbb{Q}_{p})}\chi_{p},~{}\chi_{p}:=\bigotimes_{v\mid
p}\chi_{v}.$

From (3.14), we see that

(3.35) $I^{G_{0}}(\chi)=I(0,\chi),~{}I^{G_{0}}_{p}(\chi_{p})=\bigotimes_{v\mid
p}I_{v}(0,\chi_{v})$

with $I(s,\chi)$ and $I_{v}(s,\chi_{v})$ defined in (2.53). Using the formula
(3.31) and Lemma 3.1, we can rewrite the function
$\mathcal{I}(g_{0},\varphi,\varrho)$ in (3.21) as

$\displaystyle 2\mathcal{I}(g_{0},\varphi,\varrho)$
$\displaystyle=\mathrm{CT}_{s=0}\int_{[H_{1}]}\varrho(h_{1})E^{H}_{P}(s,{\mathscr{F}}(\varphi))(g_{0},h_{1})dh_{1}=E^{G_{0}}_{B_{0}}(0,F_{\varphi,\varrho})(g_{0}),$

for $g_{0}\in G_{0}(\mathbb{A})$, where
$E^{G_{0}}_{B_{0}}(s^{\prime},F_{\varphi,\varrho})$ is the Eisenstein series
for the standard section associated to

(3.36)
$\begin{split}F_{\varphi,\varrho}(g_{0})&:=F_{\varphi,\varrho,0}(g_{0})\in\operatorname{Ind}^{G_{0}}_{B_{0}}\chi,\\\
F_{\varphi,\varrho,s}(g_{0})&:=\int_{H_{1}(\mathbb{A})}\Phi_{s}({\mathscr{F}}(\varphi))(g_{0},h_{1})\varrho(h_{1})dh_{1}.\end{split}$

Note that $F_{\varphi,\varrho,s}$ is not a standard section.

If $\varrho=\otimes_{p\leq\infty}\varrho_{p}$ and
$\varphi=\otimes_{p\leq\infty}\varphi_{p}$, then $F_{\varphi,\varrho,s}$ is a
product of local integrals.

(3.37)
$F_{\varphi_{p},\varrho_{p},s}(g_{0,p}):=\int_{H_{1}(\mathbb{Q}_{p})}\Phi_{s}({\mathscr{F}}(\varphi_{p}))(g_{0,p},h_{1})\varrho(h_{1})dh_{1},~{}F_{\varphi_{p},\varrho_{p}}:=F_{\varphi_{p},\varrho_{p},0}\in
I_{0,p}(\chi_{p}).$

We have explicitly

(3.38) $F_{\varphi_{p},\varrho_{p}}(g)=\int_{H_{1}(\mathbb{Q}_{p})\times
F_{p}\times\mathbb{Q}_{p}}\varphi_{p}((g,h_{1})^{-1}(x,0,\lambda,\lambda))\varrho_{p}(h_{1})dx\,d\lambda\,dh_{1}$

with $d\lambda$ the self-dual measure on $F_{p}$ such that
$\int_{{\mathcal{O}}_{F_{p}}}d\lambda=|D|_{p}^{1/2}$. From this, we see that

(3.39)
$\sigma_{a}(|D|_{p}^{1/2}F_{\varphi_{p},\varrho_{p}}(g))=|D|_{p}^{1/2}F_{\sigma_{a}(\varphi_{p}),\varrho_{p}}(g),~{}F_{\varphi_{p},\varrho_{p}}(t_{0}g)=F_{\varphi_{p},\varrho_{p}}(g)$

for all $a\in\mathbb{Z}_{p}^{\times}$ and $t_{0}\in T_{0}(\mathbb{Z}_{p})$. At
all but finitely many cases, the function $F_{\varphi_{p},\varrho_{p},s}$ is
given explicitly as follows.

######  Lemma 3.2.

Suppose $p$ is unramified in $E$ and $\varphi_{p}$ is the characteristic
function of the maximal lattice $V_{\mathbb{Z}}\otimes\mathbb{Z}_{p}\subset
V_{p}$. Then

(3.40) $F_{\varphi_{p},\varrho_{p},s}(g_{p})=(1-p^{-2-2s})\prod_{v\mid
p}L(1+s,\chi_{v})$

for all $g_{p}\in G_{0}(\mathbb{Z}_{p})$.

###### Proof.

Since $\varphi_{p}$ is $G_{0}(\mathbb{Z}_{p})$-invariant, we can suppose
$g_{p}=1$.

If $p$ is inert in $F$, then

$\Phi_{s}({\mathscr{F}}(\varphi_{p}))(1,h_{1})=\Phi_{0}({\mathscr{F}}(\varphi_{p}))(1,h_{1})=1=\varrho(h_{1})$

for all $h_{1}\in
H_{1}(\mathbb{Q}_{p})=H_{1}(\mathbb{Z}_{p})={\mathcal{O}}_{F_{p}}^{1}\subset{\mathcal{O}}_{F_{p}}^{\times}$,
and
$F_{\varphi_{p},\varrho_{p},s}(g_{p})=\int_{H_{1}(\mathbb{Z}_{p})}dh_{1}=1$.

If $p$ is split in $F$, we have $F_{p}\cong\mathbb{Q}_{p}^{2}$,
$H_{1}(\mathbb{Q}_{p})=\\{(\alpha,\alpha^{-1})\in
F_{p}:\alpha\in\mathbb{Q}_{p}^{\times}\\}\cong\mathbb{Q}_{p}^{\times}$ and and
$\chi_{v}=\chi_{v^{\prime}}$ is a character of $\mathbb{Q}_{p}^{\times}$.
Straightforward (though involved) calculations show that

$\Phi_{s}({\mathscr{F}}(\varphi_{p}))(1,h_{1})=\Phi_{0}({\mathscr{F}}(\varphi_{p}))(1,h_{1})\min\\{|h_{1}|_{v},|h_{1}|_{v^{\prime}}\\}^{s}.$

For $h_{1}=(\alpha,\alpha^{-1})$ with $o(\alpha)=m$, we have

(3.41)
$\begin{split}\Phi_{0}({\mathscr{F}}(\varphi_{p}))(1,h_{1})&=\int_{\mathbb{Q}_{p}^{3}}\operatorname{Char}(\mathbb{Z}_{p}^{6})(a,0,\lambda_{1},\lambda_{2},\alpha^{-1}\lambda_{1},\alpha\lambda_{2})dad\lambda_{1}d\lambda_{2}\\\
&=\int_{p^{\max\\{0,m\\}}\mathbb{Z}_{p}}d\lambda_{1}\int_{p^{\max\\{0,-m\\}}\mathbb{Z}_{p}}d\lambda_{2}=p^{-|m|}.\end{split}$

Since $p$ is unramified in $E$, we have
$\varrho_{p}((\alpha,\alpha^{-1}))=\epsilon^{o(\alpha)}$ with
$\epsilon:=\varrho_{p}((p,p^{-1}))=\chi_{v}(p)=\chi_{v^{\prime}}(p)$. Putting
these together then gives us

(3.42)
$\begin{split}F_{\varphi_{p},\varrho_{p},v,s}(1)&=\int_{\mathbb{Q}_{p}^{\times}}\min\\{|\alpha|_{p},|\alpha^{-1}|_{p}\\}^{1+s}\epsilon^{o(\alpha)}|\alpha|_{p}^{s}d^{\times}\alpha\\\
&=\sum_{m\in\mathbb{Z}}\epsilon^{m}p^{-|m|(1+s)}=L(1+s,\chi_{v})L(1+s,\chi_{v^{\prime}})(1-p^{-2-2s}).\end{split}$

This finishes the proof. ∎

###  3.4. Matching Global Sections

The function $\mathcal{I}(g_{0},\varphi^{(k,k^{\prime})},\varrho)$ is a
Hilbert modular form of weight $(k,k^{\prime})$. We want to suitably choose
$\varrho$ and $\varphi_{f}$ and compare this function to a coherent Eisenstein
series.

Let $\chi=\chi_{E/F}$ be a Hecke character associated to a quadratic extension
$E/F$ with $E/\mathbb{Q}$ biquadratic. Let $\alpha\in F^{\times},W_{\alpha}$
be the same as in section 2.6. For our purpose, we will choose
$\phi^{(k,k^{\prime})}_{\infty}\in{\mathcal{S}}(W_{\alpha}(F\otimes\mathbb{R}))$
to be eigenfunctions of $K_{\infty}=\operatorname{SL}_{2}(\mathbb{R})^{2}$
with weight $(k,k^{\prime})$ and normalized to have

$\phi^{(k,k^{\prime})}_{\infty}(0)=1.$

The matching result we will prove is the following.

######  Theorem 3.3.

Let $\varrho:\mathbb{A}_{F}^{\times}/F^{\times}\to\mathbb{C}^{\times}$ be a
character satisfying (2.13) whose kernel in $H_{1}(\hat{\mathbb{Z}})$ is
$K_{\varrho}$. For $\alpha\in F^{\times}$ with ${\mathrm{Nm}}(\alpha)<0$,
given any $\phi_{f}\in{\mathcal{S}}(\hat{W}_{\alpha})$, there exists
$\varphi_{f}\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$ such that
$\omega_{f}(-1)\varphi_{f}=-\varphi_{f}$ for $-1\in H_{1}(\hat{\mathbb{Q}})$,
it is invariant with respect to the compact subgroup
$G(\hat{\mathbb{Z}})T^{\Delta}(\hat{\mathbb{Z}})K_{\varrho}\subset
G(\mathbb{A})\times H(\mathbb{A})$, and satisfies

(3.43)
$\frac{\pi}{3}F_{\varphi,\varrho}=2\Lambda(1,\chi)\lambda_{\alpha}(\phi)\in
I(0,\chi),$

where $\varphi=\varphi_{f}\varphi^{(\epsilon,-\epsilon)}_{\infty}$ with
$\epsilon:={\mathrm{sgn}}(\alpha_{1})=-{\mathrm{sgn}}(\alpha_{2})$ and
$\varphi^{(\pm 1,\mp 1)}_{\infty}$ defined in (3.22), and
$\phi=\phi_{f}\phi^{(\epsilon,-\epsilon)}_{\infty}$. In particular, we have
the equality

(3.44) $\frac{\pi}{3}\mathcal{I}(g,\varphi,\varrho)=E^{*}(g,\phi).$

######  Remark 3.4.

The constants
$\Lambda(0,\chi)=\Lambda(1,\chi)=\frac{\sqrt{D_{E}/D}}{\pi^{2}}L(1,\chi)$ and
$\sqrt{D_{E}}$ are in $\mathbb{Q}^{\times}$.

######  Remark 3.5.

The condition $\varrho\circ{\mathrm{Nm}}^{-}=\chi$ is equivalent to
$\chi\mid_{\mathbb{A}^{\times}}$ being trivial, which is equivalent to
$E/\mathbb{Q}$ being a biquadratic Galois extension.

######  Remark 3.6.

For $L\subset W_{\alpha}(\hat{\mathbb{Q}})$ a lattice and $\mu\in L^{\vee}/L$,
suppose $\varphi_{\mu}\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$
satisfies (3.43) with $\phi_{f}=\phi_{L+\mu}$. Then it is easy to see that

$\sum_{\mu\in
L^{\vee}/L}\mathcal{I}(g^{\Delta}_{\tau},\varphi_{\mu},\varrho)\mathfrak{e}_{\mu}:\mathbb{H}\to\mathbb{C}[L^{\vee}/L]$

is a (non-holomorphic) vector-valued modular form of weight 0 on
$\operatorname{SL}_{2}(\mathbb{Z})$ with representation $\overline{\rho_{L}}$.

######  Remark 3.7.

If we decompose $V_{0}=U\oplus U^{\perp}$ with $U=\ell^{+}+\ell^{-}$ the
hyperbolic plane, then it is easy to see that
$T_{0}\subset{\mathrm{SO}}(U)\subset H_{0}$. Therefore for any
$\varphi\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})^{T^{\Delta}(\hat{\mathbb{Z}})}$,
we can write it as

$\varphi=\sum_{j\in J}\varphi_{U,j}\otimes\varphi_{U^{\perp},j}$

such that
$\varphi_{U,j}\in{\mathcal{S}}(\hat{U};\mathbb{Q}^{\mathrm{ab}})^{T^{\Delta}(\hat{\mathbb{Z}})}$
and
$\varphi_{U^{\perp},j}\in{\mathcal{S}}(\hat{U}^{\perp};\mathbb{Q}^{\mathrm{ab}})^{T^{\Delta}(\hat{\mathbb{Z}})}$
for all $j\in J$. This in particular implies that $\varphi_{U^{\perp},j}$ is
$T(\hat{\mathbb{Z}})$-invariant, i.e. it is $\mathbb{Q}$-valued by (2.31).

###### Proof of Theorem 3.3.

Suppose $\phi=\otimes_{v\leq\infty}\phi_{v}$. By Theorem 3.11, there exists
$\varphi_{p}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))$ invariant
with respect to $G(\mathbb{Z}_{p})T^{\Delta}(\mathbb{Z}_{p})$ and satisfying
(3.52). Furthermore, $\varphi_{p}$ is the characteristic function of the
maximal lattice in $V_{p}$ for all but finitely many $p$. Therefore
$\varphi_{f}:=\bigotimes_{p<\infty}\varphi_{p}$ is in
${\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})^{(G\cdot
T^{\Delta})(\hat{\mathbb{Z}})}$, and satisfies

$F_{\varphi_{f},\varrho_{f}}=\zeta(2)^{-1}{L(1,\chi)}{\sqrt{D_{E}/D}}\lambda_{\alpha}(\phi_{f})=6\Lambda(1,\chi)\lambda_{\alpha}(\phi_{f}).$

Since $\varrho_{f}(-1)={\mathrm{sgn}}(-1)=-1$, the function
$\omega_{f}(-1)\varphi_{f}$ with $-1\in H_{1}(\hat{\mathbb{Q}})$ also satisfy
these conditions, we can replace $\varphi_{f}$ by
$(\varphi_{f}-\omega_{f}(-1)\varphi_{f})/2$ so that
$\omega_{f}(-1)\varphi_{f}=-\varphi_{f}$. Furthermore, we have
$F_{\omega_{f}(h)\varphi_{f},\varrho_{f}}=F_{\varphi_{f},\varrho_{f}}$ for all
$h\in K_{\varrho}$, and can therefore average over $K_{\varrho}$ to ensure
that $\varphi_{f}$ is $K_{\varrho}$-invariant.

To prove (3.44), it suffices to check that
$F_{\varphi^{(\epsilon,-\epsilon)}_{\infty},\varrho_{\infty}}(g)=\pi^{-1}\lambda_{\alpha}(\phi_{\infty}^{(\epsilon,-\epsilon)})(g)$
for $g=(g_{\tau_{1}},g_{\tau_{2}})$. Using

$\displaystyle\Phi_{\infty}({\mathscr{F}}(\varphi^{(\epsilon,-\epsilon)}_{\infty}))(g,t)$
$\displaystyle={\mathscr{F}}(\omega(g,t)\varphi^{(\epsilon,-\epsilon)}_{\infty})(0)=\int_{\mathbb{R}^{3}}(\omega(g,t)\varphi^{(\epsilon,-\epsilon)}_{\infty})(a,0,\lambda_{1},\lambda_{2},\lambda_{1},\lambda_{2})dad\lambda_{1}d\lambda_{2}$
$\displaystyle=\int_{\mathbb{R}^{3}}\varphi^{(\epsilon,-\epsilon)}_{\infty}\left(\frac{a-\lambda_{1}u_{1}-\lambda_{2}u_{2}}{\sqrt{v_{1}v_{2}}},0,v_{1}\lambda_{1}/\sqrt{v_{1}v_{2}},v_{2}\lambda_{2}/\sqrt{v_{1}v_{2}},t^{-1}\lambda_{1},t\lambda_{2}\right)dad\lambda_{1}d\lambda_{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}(v_{1}\lambda_{1}-v_{2}\lambda_{2})(t^{-1}\lambda_{1}-t\lambda_{2})e^{-\frac{\pi}{v_{1}v_{2}}((v_{1}\lambda_{1}-v_{2}\lambda_{2})^{2}+v_{1}v_{2}((t^{-1}\lambda_{1}+t\lambda_{2})^{2}))}d\lambda_{1}d\lambda_{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}x\frac{2x+(t^{-1}v_{2}-tv_{1})y}{v_{1}t+v_{2}t^{-1}}e^{-\frac{\pi}{v_{1}v_{2}}(x^{2}+v_{1}v_{2}y^{2}))}\frac{dxdy}{v_{1}t+v_{2}t^{-1}}$
$\displaystyle=\pi^{-1}2{(v_{1}v_{2})^{3/2}}{}\frac{t^{2}}{(v_{1}t^{2}+v_{2})^{2}},$

where we have used the change of variable
$x=v_{1}\lambda_{1}-v_{2}\lambda_{2},y=t^{-1}\lambda_{1}+t\lambda_{2}$, we
obtain

$\displaystyle
F_{\varphi^{(\epsilon,-\epsilon)}_{\infty},\varrho_{\infty}}(g)$
$\displaystyle=\int_{0}^{\infty}\Phi_{\infty}({\mathscr{F}}(\varphi_{\infty}))(g,t)\frac{dt}{t}=\pi^{-1}2{(v_{1}v_{2})^{3/2}}{}\int_{0}^{\infty}\frac{tdt}{(v_{1}t^{2}+v_{2})^{2}}=\pi^{-1}\sqrt{v_{1}v_{2}}.$

On the other hand, we have

$\lambda_{\alpha}(\phi^{(\epsilon,-\epsilon)}_{\infty})(g)=\sqrt{v_{1}v_{2}}.$

This finishes the proof. ∎

The requirement that $\varphi_{f}$ in Theorem 3.3 is invariant with respect to
$T^{\Delta}(\hat{\mathbb{Z}})$ will be important to deduce important
rationality results in section 4.3. We give a taste of such results in the
following lemma.

######  Lemma 3.8.

If $\varphi_{0}\in{\mathcal{S}}(V_{0};\mathbb{Q}^{\mathrm{ab}})$ is invariant
with respect to $T^{\Delta}(\hat{\mathbb{Z}})\subset({\mathrm{GL}}_{2}\times
H_{0})(\hat{\mathbb{Z}})$, then
${\mathscr{F}}_{1}(\varphi_{0})\in{\mathcal{S}}(((\ell^{-}\otimes
W)+V_{-1})(\hat{\mathbb{Q}});\mathbb{Q}^{\mathrm{ab}})$ satisfies

(3.45)
$\sigma_{a}\left({\mathscr{F}}_{1}(\varphi_{0})((\eta_{1},\eta_{2}),\nu)\right)={\mathscr{F}}_{1}(\varphi_{0})((a^{-1}\eta_{1},\eta_{2}),\nu)$

for any $\sigma_{a}\in\mathrm{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$
associated to $a\in\hat{\mathbb{Z}}^{\times}$ as in section 2.3. In
particular, we have

(3.46) ${\mathscr{F}}_{1}(\varphi_{0})((0,r),\nu)\in\mathbb{Q}$

for all $r\in\hat{\mathbb{Q}},\nu\in\hat{F}$.

###### Proof.

Using the expression for ${\mathscr{F}}_{1}$ in (3.29), we can write

$\displaystyle\sigma_{a}$
$\displaystyle\left({\mathscr{F}}_{1}(\varphi_{0})((\eta_{1},\eta_{2}),\nu)\right)=\sigma_{a}\left(\int_{\hat{\mathbb{Q}}}\varphi_{0}(b,\eta_{1},\nu)\psi_{f}(b\eta_{2})db\right)=\int_{\hat{\mathbb{Q}}}\sigma_{a}(\varphi_{0}(b,\eta_{1},\nu))\psi_{f}(ab\eta_{2})db$
$\displaystyle=\int_{\hat{\mathbb{Q}}}\omega((t(a),1))(\varphi_{0})(b,\eta_{1},\nu)\psi_{f}(ab\eta_{2})db=\int_{\hat{\mathbb{Q}}}\omega((1,\iota(t(a^{-1}))))(\varphi_{0})(b,\eta_{1},\nu)\psi_{f}(ab\eta_{2})db$
$\displaystyle=\int_{\hat{\mathbb{Q}}}\varphi_{0}(ab,a^{-1}\eta_{1},\nu)\psi_{f}(ab\eta_{2})db={\mathscr{F}}_{1}(\varphi_{0})((a^{-1}\eta_{1},\eta_{2}),\nu).$

For the second step, we moved $\sigma_{a}$ inside the integral as
$\varphi_{0}$ is a Schwartz function and the integral is a finite sum. The
third and fourth steps used (2.30) and the invariance of $\varphi_{0}$ under
$(t,\iota(t))\in T^{\Delta}(\hat{\mathbb{Z}})$ respectively. Equation (3.46)
now follows from (3.45) via (2.30). ∎

###  3.5. Matching Local Sections I

The goal of this section is to prove Theorem 3.11, the non-archimedean local
counterpart of the matching result 3.3. For this purpose, we fix a prime
$p<\infty$ throughout this section. The main input to Theorem 3.11 is the
following surjectivity result.

######  Proposition 3.9.

Let $\varrho_{p}$ and $\chi_{p}$ be as in (3.33). Then the following map

(3.47)
$\begin{split}\beta:{\mathcal{S}}(V_{p};\mathbb{C})^{G(\mathbb{Z}_{p})}\subset{\mathcal{S}}(V_{p};\mathbb{C})&\to
I_{p}^{G_{0}}(\chi_{p})\\\ \varphi&\mapsto F_{\varphi,\varrho_{p}}\end{split}$

is surjective. Furthermore if $\Phi\in I_{p}^{G_{0}}(\chi_{p})$ is valued in
$\mathbb{Q}(\zeta_{p^{\infty}})$, then there exists
$\varphi\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$
satisfying $\beta(\varphi)=\Phi$. Here
$\mathbb{Q}(\zeta_{p^{\infty}})\subset\mathbb{Q}^{\mathrm{ab}}$ is the
subfield defined in (2.22).

###### Proof.

Using (3.35), we can suppose $\Phi=\otimes_{v\mid p}\Phi_{v}$ with
$\Phi_{v}\in I_{v}(0,\chi_{v})$. Since
$F_{\omega(g)\varphi,\varrho_{p}}=F_{\varphi,\varrho_{p}}$ for all $g\in
G(\mathbb{Z}_{p})$ and $\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})$, it
suffices to prove the surjectivity of $\beta$ on
${\mathcal{S}}(V_{p};\mathbb{C})$. To do this, we will use the $m$-th Fourier
coefficient of $\Phi_{v}\in I(0,\chi_{v})$ for $m\in F_{v}$, which is defined
by

(3.48) $W_{m}(\Phi_{v}):=\int_{F_{v}}\Phi_{v}(wn(b))\psi_{v}(-mb)db$

with $\psi_{v}$ an additive character of $F_{v}$. For $m=(m_{v})_{v\mid p}\in
F_{p}$ and $\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})$, we denote

(3.49) $\begin{split}W_{m}(\varphi)&:=\prod_{v\mid
p}W_{m_{v}}((F_{\varphi,\varrho_{p}})_{v})\\\ &=\int_{F_{p}\times
H_{1}(\mathbb{Q}_{p})}(\omega_{p}(wn(b),h_{1}){\mathscr{F}}(\varphi))(0)\psi_{p}(-mb)\varrho_{p}(h_{1})dh_{1}\,db\end{split}$

with $\psi_{p}:=\prod_{v\mid p}\psi_{v}$. Now, the $G(F_{v})$-module
$I(0,\chi_{v})$ can be written as

$I(0,\chi_{v})=\oplus_{\alpha\in E_{v}^{\times}/F_{v}^{\times}}R(W_{\alpha})$

with $R(W_{\alpha})$ the image of $\lambda_{\alpha,v}$ and irreducible. So
$I(0,\chi_{v})$ is irreducible if and only if $\chi_{v}$ is trivial.
Otherwise, it is the direct sum of two irreducible submodules. Furthermore for
$\Phi\in R(W_{\alpha})$, the coefficient $W_{m}(\Phi)$ is zero unless
$m/\alpha\in\operatorname{N}_{E_{v}/F_{v}}E_{v}^{\times}$. We then have two
cases to consider, depending on whether $\chi_{v}=\chi_{v^{\prime}}$ is
trivial or not.

When $\chi_{v}=\chi_{v^{\prime}}$ is trivial, Lemma 3.13 gives us $\varphi$
such that $W_{m}(\varphi)\neq 0$ for some $m\in F_{p}^{\times}$. So for $v\mid
p$, the projection of $\mathrm{im}(\beta)\subset I_{p}^{G_{0}}(\chi_{p})$ to
$I(0,\chi_{v})$ is a non-trivial, irreducible sub $G(F_{v})$-module. As
$I(0,\chi_{v})$ is irreducible, the map $\beta$ is surjective. When
$\chi_{v}=\chi_{v^{\prime}}$ is non-trivial, we again apply Lemma 3.13 and
similar argument proves surjectivity.

When $\Phi=\otimes_{v\mid p}\Phi_{v}$ has value in
$\mathbb{Q}(\zeta_{p^{\infty}})$, we apply the surjectivity of $\beta$ and the
discussion in section 2.3 to choose
$\varphi_{j}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$
and $c_{j}\in\mathbb{C}$ such that

$\varphi:=\sum_{j=1}^{J}c_{j}\varphi_{j}\in{\mathcal{S}}(V_{p};\mathbb{C})$

satisfies $\beta(\varphi)=\Phi$ and $J$ is minimal. Therefore,
$F_{\varphi,\varrho_{p}}=\sum_{j=1}^{J}c_{j}F_{\varphi_{j},\varrho_{p}}$ is
valued in $\mathbb{Q}(\zeta_{p^{\infty}})$. By the minimality of $J$, the
section $F_{\varphi_{j},\varrho_{p}}$ is not identically zero for all $j$.
Therefore, the set $\\{1,c_{1},\cdots,c_{J}\\}\subset\mathbb{C}$ is linearly
dependent over $\mathbb{Q}(\zeta_{p^{\infty}})$. The minimality of $J$ then
implies that $J=1$ and $c_{1}\in\mathbb{Q}(\zeta_{p^{\infty}})$, hence
$\varphi\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$.
∎

Using this proposition, we can use the map $\beta$ to extend any continuous
function on $G_{0}(\mathbb{Z}_{p})$. Furthermore, we can incorporate Galois
action to obtain the following result.

######  Proposition 3.10.

In the setting of Proposition 3.9, given any continuous function
$\Phi:G_{0}(\mathbb{Z}_{p})\to\mathbb{C}$ satisfying

(3.50) $\Phi(m(a)n(b)k)=\chi(a)\Phi(k),~{}$

for all $m(a),n(b)\in B_{0}(\mathbb{Z}_{p}),k\in G_{0}(\mathbb{Z}_{p})$, there
exists $\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})^{G(\mathbb{Z}_{p})}$ such
that $F_{\varphi,\varrho_{p}}(g)=\Phi(g)$ for all $g\in
G_{0}(\mathbb{Z}_{p})$. Furthermore, if $\Phi$ has value in
$\mathbb{Q}(\zeta_{p^{\infty}})$ and satisfies

(3.51)
$\sigma_{a}(|D|_{p}^{-1/2}\Phi(t_{0}^{-1}gt_{0}))=|D|_{p}^{-1/2}\Phi(g),~{}$

with
$t_{0}=\iota(t(a))\in{\tilde{H}}_{0}(\mathbb{Z}_{p}),t(a)=\left(\begin{smallmatrix}a&\\\
&1\end{smallmatrix}\right)\in T\subset{\mathrm{GL}}_{2}(\mathbb{Z}_{p})$ for
all $a\in\mathbb{Z}_{p}^{\times}$ and $g\in G_{0}(\mathbb{Z}_{p})$, then
$\varphi\in{\mathcal{S}}(V;\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$
can be chosen to be $T^{\Delta}(\mathbb{Z}_{p})$-invariant.

###### Proof.

A continuous function $\Phi$ on $G_{0}(\mathbb{Z}_{p})$ satisfying (3.50) can
be uniquely extended to a section $\tilde{\Phi}\in I^{G_{0}}_{p}(\chi_{p})$ by
setting

$\tilde{\Phi}(g):=\chi_{p}(a)\Phi(k)$

with $g=m(a)n(b)k$ the Iwasawa decomposition of $g$. Therefore the first claim
is a direct consequence of Proposition 3.9.

For the second claim, we take any
$\varphi\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$
and observe that

$\displaystyle\frac{F_{\omega_{p}(t,t_{0})\varphi_{p},\varrho_{p}}(g)}{|D|_{p}^{1/2}}$
$\displaystyle=\frac{F_{\omega_{p}(t)\varphi_{p},\varrho_{p}}(gt_{0})}{|D|_{p}^{1/2}}=\frac{F_{\sigma_{a}(\varphi_{p}),\varrho_{p}}(t^{-1}gt_{0})}{|D|_{p}^{1/2}}=\sigma_{a}\left(\frac{F_{\varphi_{p},\varrho_{p}}(t^{-1}gt)}{|D|_{p}^{1/2}}\right)$
$\displaystyle=\sigma_{a}(|D|_{p}^{-1/2}\Phi(t_{0}^{-1}gt_{0}))=|D|_{p}^{-1/2}\Phi(g)$

for any $(t,t_{0})\in T^{\Delta}(\mathbb{Z}_{p})$ with
$t=t(a),t_{0}=\iota(t(a))$ and $g\in G_{0}(\mathbb{Z}_{p})$. Here we used
(3.39) for the first line. By averaging $\varphi$ over
$T^{\Delta}(\mathbb{Z}_{p})$, we can suppose that it is
$T^{\Delta}(\mathbb{Z}_{p})$-invariant. This finishes the proof. ∎

We are now ready to state and prove the local matching result. This is just
the Kudla matching principle [Kud03] in some sense.

######  Theorem 3.11.

For any $\phi_{v}\in{\mathcal{S}}(W_{\alpha}(F_{v}))$ with $v\mid p$, there
exists
$\varphi_{p}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{(G\cdot
T^{\Delta})(\mathbb{Z}_{p})}$ such that

(3.52) $F_{\varphi_{p},\varrho_{p}}=(1-p^{-2})|D/D_{E}|_{p}^{1/2}\prod_{v\mid
p}L(1,\chi_{v})\lambda_{\alpha,v}(\phi_{v}).$

In addition, if $p$ is unramified in $E$ and co-prime to $\alpha$, and
$\phi_{v}$ is the characteristic function of the maximal lattice in
$W_{\alpha}(F_{v})$, then we can choose $\varphi_{p}$ to be the characteristic
function of the maximal lattice in $V_{p}$.

###### Proof.

Suppose $p$ and $\alpha$ are co-prime, $E_{p}/\mathbb{Q}_{p}$ is unramified
and
$\phi_{v}=\operatorname{Char}({\mathcal{O}}_{E_{v}}),\varphi_{p}=\operatorname{Char}({\mathcal{O}}_{F_{p}}\times\mathbb{Z}_{p}^{2}\times{\mathcal{O}}_{F_{p}})$.
Then it is easy to check that $F_{\varphi_{p},\varrho_{p}}$ and $\prod_{v\mid
p}\lambda_{\alpha,v}(\phi_{v})$ are both right
$G({\mathcal{O}}_{F_{p}})$-invariant. Since they are both in $\prod_{v\mid
p}I(0,\chi_{v})$, we only need to check that

$F_{\varphi_{p},\varrho_{p}}(1)=(1-p^{-2})\prod_{v\mid
p}L(1,\chi_{v})\lambda_{\alpha,v}(\phi_{v})(1)$

by the Iwasawa decomposition of $G(F_{p})$. This is given precisely by Lemma
3.2, and proves (3.52) for all but finitely many places.

When $\phi_{v}$ is $\mathbb{Q}$-valued, we can use (2.30) to check that

$\displaystyle\sigma_{a}\left(\prod_{v\mid
p}\lambda_{\alpha,v}(\phi_{v})(t^{-1}gt)\right)$ $\displaystyle=\prod_{v\mid
p}\sigma_{a}\left(\omega_{\alpha,v}(t^{-1}gt)(\phi_{v})(0)\right)=\prod_{v\mid
p}\left(\omega_{\alpha,v}(g)(\sigma_{a}(\phi_{v}))(0)\right)$
$\displaystyle=\prod_{v\mid
p}\left(\omega_{\alpha,v}(g)(\phi_{v})(0)\right)=\prod_{v\mid
p}\lambda_{\alpha,v}(\phi_{v})(g)$

for any $(t,t_{0})\in T^{\Delta}(\mathbb{Z}_{p})$ with
$t=t(a),t_{0}=\iota(t(a))$ and $g\in G_{0}(\mathbb{Z}_{p})$. Proposition 3.10
combined with Remark 3.4 then completes the proof. ∎

Finally, we record the two local calculation lemmas used in proving
Proposition 3.9.

######  Lemma 3.12.

Suppose $F_{p}/\mathbb{Q}_{p}$ is non-split with valuation ring
${\mathcal{O}}_{p}$, uniformizer $\varpi$, residue field size $q$, and a non-
trivial additive character $\psi$. For a character $\varrho$ of
$H_{1}(\mathbb{Q}_{p})=F_{p}^{1}\subset{\mathcal{O}}_{p}^{\times}$, let

$n(\psi):=\min\\{n:\psi(\varpi^{n}{\mathcal{O}}_{p})=1\\},~{}n(\varrho):=\min\\{n\geq
0:\varrho(K_{n})=1\\},$

be the conductors of $\psi$ and $\varrho$ respectively, where
$K_{n}:=F_{p}^{1}\cap(1+\varpi^{n}{\mathcal{O}}_{p})$. Then

$\int_{F_{p}^{1}}\varrho(x)\psi(mx)dx\neq 0$

for some $m\in{\mathcal{O}}_{p}^{\times}$ if and only if $n(\varrho)\leq
n(\psi)$.

###### Proof.

Let

$f(m)=\begin{cases}\int_{F_{p}^{1}}\varrho(x)\psi(mx)dx&\operatorname{if}m\in{\mathcal{O}}_{p}^{\times}\\\
0&\hbox{otherwise}.\end{cases}$

Then $f\in{\mathcal{S}}(F_{p})$ and its Fourier transformation with respect to
$\psi$ is

$\displaystyle\hat{f}(m)$
$\displaystyle=\int_{F_{p}}f(n)\psi(-nm)dn=\int_{F_{p}^{1}}\varrho(x)\int_{{\mathcal{O}}_{p}^{\times}}\psi(n(x-m)dndx$
$\displaystyle=\int_{F_{p}^{1}}\varrho(x)(\operatorname{Char}(m+\varpi^{n(\psi)}{\mathcal{O}}_{p})(x)-q^{-1}\operatorname{Char}(m+\varpi^{n(\psi)-1}{\mathcal{O}}_{p})(x))dx.$

First assume that there is some $h_{0}\in F_{p}^{1}$ such that
$h_{0}-m\in\varpi^{n(\psi)}{\mathcal{O}}_{p}$. Then

$\displaystyle\hat{f}(m)$
$\displaystyle=\varrho(h_{0})(\int_{K_{n(\psi)}}\varrho(x)dx-q^{-1}\int_{K_{n(\psi)-1}}\varrho(x)dx)$
$\displaystyle=\begin{cases}0&\operatorname{if}n(\varrho)>n(\psi),\\\
\varrho(h_{0})\mathrm{vol}(K_{n(\psi)})&\operatorname{if}n(\varrho)=n(\psi),\\\
\varrho(h_{0})(\mathrm{vol}(K_{n(\psi)})-q^{-1}K_{n(\psi)-1})&\operatorname{if}n(\varrho)<n(\psi).\end{cases}$

Next we assume that there no $h_{0}\in F_{p}^{1}$ such that
$h_{0}-m\in\varpi^{n(\psi)}{\mathcal{O}}_{p}$ but some $h_{0}\in F_{p}^{1}$
with $h_{0}-m\in\varpi^{n(\psi)-1}{\mathcal{O}}_{p}$. Then

$\hat{f}(m)=\begin{cases}0&\operatorname{if}n(\varrho)\geq n(\psi),\\\
-q^{-1}\varrho(h_{0})\mathrm{vol}(K_{n(\psi)-1})&\operatorname{if}n(\varrho)<n(\psi).\end{cases}$

Finally, if there is no $h_{0}\in F_{p}^{1}$ with
$h_{0}-m\in\varpi^{n(\psi)-1}{\mathcal{O}}_{p}$, then $\hat{f}(m)=0$. Now the
lemma is clear. ∎

######  Lemma 3.13.

When $\chi_{v}$ is trivial, there exists $\phi\in{\mathcal{S}}(V_{p})$ such
that $F_{\phi,\varrho_{p}}$ is non-trivial. When $\chi_{v}$ is non-trivial,
then for any $\epsilon=(\epsilon_{v})_{v\mid p}$ with $\epsilon_{v}=\pm 1$
there exists $\phi^{\epsilon}\in{\mathcal{S}}(V_{p})$ and $m^{\epsilon}\in
F_{p}^{\times}$ such that $W_{m^{\epsilon}}(\phi^{\epsilon})\neq 0$ and
$m^{\epsilon}=(m^{\epsilon_{v}}_{v})_{v\mid p}$ with
$\chi_{v}(m^{\epsilon_{v}}_{v})=\epsilon_{v}$.

###### Proof.

When $\chi_{v}$ is trivial, the character $\varrho_{p}$ of
$H_{1}(\mathbb{Q}_{p})$ is also trivial. Suppose $\phi$ is the characteristic
function of the maximal lattice in $V_{p}$, then integral in (3.38) is
positive at $g=1$, which means $F_{\phi,\varrho_{p}}$ is non-trivial.

Suppose now that $\chi_{v}$, hence $\varrho_{p}$, is non-trivial. We can
suppose that $n(\psi)=0$. When $\phi=\phi_{0}\otimes\phi_{1}$ with
$\phi_{i}\in{\mathcal{S}}(V_{i,p})$, we can apply (3.38) to write

$W_{m}(\phi)=\int_{F_{p}\times H_{1}(\mathbb{Q}_{p})\times
F_{p}\times\mathbb{Q}_{p}}\phi_{0}((wn(b))^{-1}\cdot\left(\begin{smallmatrix}x&\lambda\\\
\lambda^{\prime}&0\end{smallmatrix}\right))\phi_{1}(h_{1}^{-1}\lambda)\psi_{p}(-mb)\varrho_{p}(h_{1})dx\,d\lambda\,dh_{1}\,db.$

We first assume that $p$ is non-split and use the notation in Lemma 3.12. In
this case, there is a unique place $v$ of $F$ above $p$, and $\epsilon=\pm 1$.
For $n\geq\max\\{n(\varrho)+1,1\\}$ and $\beta\in{\mathcal{O}}_{p}^{\times}$,
let $\phi_{1}=\phi_{1,\beta}=\operatorname{Char}(\beta+p^{n}\mathcal{O}_{p})$
and

$\phi_{0}=\operatorname{Char}\left(\begin{smallmatrix}\mathbb{Z}_{p}&\varpi^{n}\mathcal{O}_{p}\\\
\varpi^{n}\mathcal{O}_{p}&1+p^{n}\mathbb{Z}_{p}\end{smallmatrix}\right).$

Then

$\displaystyle W_{m}(\phi_{0}\otimes\phi_{1})=\int_{F_{p}\times
F_{p}^{1}\times
F_{p}\times\mathbb{Q}_{p}}\phi_{0}\left(\begin{smallmatrix}b\lambda+b^{\prime}\lambda^{\prime}+bb^{\prime}x&-\lambda^{\prime}-bx\\\
-\lambda-b^{\prime}x&x\end{smallmatrix}\right)\phi_{1}(h^{-1}\lambda)\varrho(h)\psi(-mb)dx\,d\lambda\,dh\,db$
$\displaystyle=\int_{F_{p}^{1}}c(h)\varrho(h)dh,$

where

$\displaystyle c(h)$ $\displaystyle:=\int_{\beta
h+\varpi^{n}\mathcal{O}_{p}}\int_{-\lambda^{\prime}+\varpi^{n}\mathcal{O}_{\mathfrak{p}}}\int_{1+p^{n}\mathbb{Z}_{p}}\psi(-mb)dx\,db\,d\lambda$
$\displaystyle=p^{-n(1+f)}\operatorname{Char}(\varpi^{-n}\mathcal{O}_{p})(m)\int_{\beta
h+\varpi^{n}\mathcal{O}_{p}}\psi(m\lambda^{\prime})d\lambda=C\operatorname{Char}(\varpi^{-n}\mathcal{O}_{p})(m)\psi(m\beta^{\prime}h^{-1})$

for some nonzero constant $C$. Here $f=1$ or $2$ depending on whether
$F/\mathbb{Q}$ is ramified or inert at $p$. Using
$\varrho(h)=\varrho(h^{-1})$, we have

(3.53)
$W_{m}(\phi_{0}\otimes\phi_{1})=C\operatorname{Char}(\varpi^{-n}\mathcal{O}_{p})(m)\int_{F_{p}^{1}}\varrho(h)\psi(m\beta^{\prime}h)dh.$

If $F_{p}/\mathbb{Q}_{p}$ is inert, then $\chi_{p}$ is ramified and non-
trivial when restricted to ${\mathcal{O}}_{p}^{\times}$. By Lemma 3.12, we can
find $m_{0}\in\varpi^{-n}{\mathcal{O}}_{p}^{\times}$ such that

$W_{m_{0}/\beta^{\prime}}(\phi_{0}\otimes\phi_{1,\beta})=C\int_{F_{p}^{1}}\varrho(h)\psi(m_{0}h)dh\neq
0$

as $n(\psi(\varpi^{-n}\cdot))=n\geq n(\varrho)$. We can choose
$\beta=\beta^{\pm}$ such that $\chi(m_{0}/(\beta^{\pm})^{\prime})=\pm 1$. Then
taking $\phi^{\pm}=\phi_{0}\otimes\phi_{1,\beta^{\pm}}$ proves the Lemma. If
$F_{p}/Q_{p}$ is ramified, then $E_{v}/F_{v}$ is inert and
$\chi_{v}(\varpi)=-1,\chi_{v}\mid_{{\mathcal{O}}_{p}^{\times}}=1$. Again by
Lemma 3.12, we can find $m_{j}\in\varpi^{-n+j}{\mathcal{O}}_{p}^{\times}$ for
$j=0,1$ such that

$W_{m_{j}}(\phi_{0}\otimes\phi_{1,1})=C\int_{F_{p}^{1}}\varrho(h)\psi(m_{j}h)dh\neq
0$

as $n(\psi(m_{j}\cdot))=n-j\geq n(\varrho)$. Therefore,
$\phi^{\pm}=\phi_{0}\otimes\phi_{1,1}$ satisfies the Lemma.

Finally, we come to the case when $p=v_{1}v_{2}$ splits and
$\eta:=\chi_{v_{1}}=\chi_{v_{2}}$ is non-trivial. In this case,
$F_{p}=F_{v_{1}}\times F_{v_{2}}=\mathbb{Q}_{p}^{2}$ and $\eta=\varrho_{p}$ is
a character of $\mathbb{Q}_{p}^{\times}\cong H_{1}(\mathbb{Q}_{p})$. For $m\in
F_{p}$, we write $m=(m_{1},m_{2})$ with $m_{j}\in\mathbb{Q}_{p}$ and

$\displaystyle V_{0,p}$ $\displaystyle\cong M_{2}(\mathbb{Q}_{p})$
$\displaystyle\left(\begin{smallmatrix}a&\lambda\\\
\lambda^{\prime}&b\end{smallmatrix}\right)$
$\displaystyle\mapsto\left(\begin{smallmatrix}a&\lambda_{1}\\\
\lambda_{2}&b\end{smallmatrix}\right)$

So we take $\phi_{1}=\phi_{1,1}\otimes\phi_{1,2}$ with
$\phi_{1,j}\in{\mathcal{S}}(\mathbb{Q}_{p})$ and
$\phi_{0}\in{\mathcal{S}}(M_{2}(\mathbb{Q}_{p}))$. Simple calculation gives us

$\displaystyle W_{m}(\phi_{0}\otimes\phi_{1})=$
$\displaystyle\int_{\mathbb{Q}_{p}^{2}\times\mathbb{Q}_{p}^{\times}\times\mathbb{Q}_{p}^{2}\times\mathbb{Q}_{p}}\phi_{1,1}(h^{-1}\lambda_{1})\phi_{1,2}(h\lambda_{2})\phi_{0}\left(\begin{smallmatrix}b_{1}\lambda_{1}+b_{2}\lambda_{2}+b_{1}b_{2}x&-\lambda_{2}-b_{1}x\\\
-\lambda_{1}-b_{2}x&x\end{smallmatrix}\right)$
$\displaystyle\quad\cdot\eta(h)\psi(-m_{1}b_{1}-m_{2}b_{2})dx\,d\lambda_{1}\,d\lambda_{2}\,{d^{\times}h}\,db_{1}\,db_{2}.$

Taking $\phi_{1,j}=\operatorname{Char}(1+p^{n}\mathbb{Z}_{p})$,
$n\geq\max\\{1,n(\eta)\\}$, and

$\phi_{0}=\operatorname{Char}(\left(\begin{smallmatrix}\mathbb{Z}_{p}&p^{n}\mathbb{Z}_{p}\\\
p^{n}\mathbb{Z}_{p}&1+p^{n}\mathbb{Z}_{p}\end{smallmatrix}\right)),$

The same calculation as above gives

$\displaystyle W_{m}(\phi_{1}\otimes\phi_{0})$
$\displaystyle=C\int_{\mathbb{Q}_{p}^{\times}}\eta(x)\psi(m_{2}x+m_{1}x^{-1})\operatorname{Char}(p^{-n}\mathbb{Z}_{p}^{2})(m_{1},m_{2})\operatorname{Char}(p^{-n}\mathbb{Z}_{p}^{2})(m_{1}x^{-1},m_{2}x)\frac{dx}{|x|}$

for some nonzero constant $C$.

When $\eta$ is ramified, we take $n=n(\rho)$, $m_{1}=p^{l}$ with $l\geq n$ and
$m_{2}=m_{0}p^{-n}\in p^{-n}\mathbb{Z}_{p}^{\times}$ and obtain (write
$o(x)=\hbox{ord}_{p}x$)

$\displaystyle W_{m}(\phi_{1}\otimes\phi_{0})$ $\displaystyle=C\int_{0\leq
o(x)\leq n+l}\eta(x)\psi(m_{2}x)\psi(m_{1}x^{-1})\frac{dx}{|x|}$
$\displaystyle=C\sum_{0\leq i\leq
n}\int_{p^{i}\mathbb{Z}_{p}^{\times}}\eta(x)\psi(m_{2}x)\frac{dx}{|x|}+C\sum_{1\leq
i\leq
l}\int_{p^{-n-i}\mathbb{Z}_{p}^{\times}}\eta(x)^{-1}\psi(m_{1}x)\frac{dx}{|x|}$
$\displaystyle=C\left[\eta(m_{0})^{-1}\int_{\mathbb{Z}_{p}^{\times}}\eta(x)\psi(p^{-n}x)dx+\eta(p)^{n+l}\int_{\mathbb{Z}_{p}^{\times}}\eta^{-1}(x)\psi(p^{-n}x)dx\right]$
$\displaystyle\neq 0,$

for some $m_{2}$.

When $\eta$ is unramified, we have $n(\eta)=0$ and take $n=1$. For $m_{j}\in
p^{-1}\mathbb{Z}_{p}^{\times}$, $j=1,2$, we have

$\displaystyle
W_{(m_{1},m_{2})}(\phi_{0}\otimes\phi_{1})=C\int_{\mathbb{Z}_{p}^{\times}}\psi(m_{2}x+m_{1}x^{-1})dx.$

If we sum this over $m_{j}\in p^{-1}\mathbb{Z}_{p}^{\times}$, then the result
is non-zero. So there exists $m_{j}\in p^{-1}\mathbb{Z}_{p}^{\times}$ such
that $W_{(m_{1},m_{2})}(\phi_{0}\otimes\phi_{1})\neq 0$. For
$m_{j}\in\mathbb{Z}_{p}^{\times}$, $j=1,2$, we have

$\displaystyle W_{(m_{1},m_{2})}(\phi_{0}\otimes\phi_{1})$
$\displaystyle=C\left[\int_{\mathbb{Z}_{p}^{\times}}dx+\eta(p)\int_{p\mathbb{Z}_{P}^{\times}}\psi(m_{1}x^{-1})\frac{dx}{|x|}+\eta(p)^{-1}\int_{p^{-1}\mathbb{Z}_{p}^{\times}}\psi(m_{2}x)\frac{dx}{|x|}\right]$
$\displaystyle=C(1-p^{-1}+p^{-1}+p^{-1})\neq 0,$

as $\eta(p)=-1$. Replacing $\phi_{1}$ by
$\phi_{1}^{\prime}=\operatorname{Char}(1+p^{n}\mathbb{Z}_{p},p+p^{n}\mathbb{Z}_{p})$,
the same calculation gives

$W_{(m_{1},m_{2})}(\phi_{0}\otimes\phi_{1}^{\prime})\neq 0$

when $m_{j}\in\mathbb{Z}_{p}^{\times}$ and $m_{3-j}\in
p^{-1}\mathbb{Z}_{p}^{\times}$. This completes the proof. ∎

###  3.6. Matching Local Sections II

In order to give the factorization result, we also need a matching result
involving the following local sections with the $s$ parameter. When
$p=vv^{\prime}$ splits in $F$, we define a slightly modified section

(3.54)
$F_{\varphi_{p},\varrho_{p},v,s}(g_{p}):=\int_{H_{1}(\mathbb{Q}_{p})}\Phi_{0}({\mathscr{F}}(\varphi_{p}))(g_{p},h_{1})\varrho(h_{1})|h_{1}|_{v}^{s}dh_{1}.$

This function on $G_{0}(\mathbb{Q}_{p})$ depends on the choice of $v\mid p$,
and has the following property.

######  Lemma 3.14.

When $|s|<1$, the integral in (3.54) converges absolutely and defines a
rational function in $p^{s}$ defined over
$\mathbb{Q}(\zeta_{p^{\infty}})(\varphi_{p})$. Furthermore, it defines a
section in $I(s,\chi_{v})I(-s,\chi_{v^{\prime}})$.

###### Proof.

For the first claim, one can suppose
$\omega(g_{p}){\mathscr{F}}_{1}(\varphi_{p})$ is the characteristic function
of

$C_{1}\times(p^{a_{1}}\mathbb{Z}_{p}+r_{1})\times(p^{a_{2}}\mathbb{Z}_{p}+r_{2})\times(p^{b_{1}}\mathbb{Z}_{p}+t_{1})\times(p^{b_{2}}\mathbb{Z}_{p}+t_{2}),$

with $C_{1}\subset\mathbb{Q}_{p}^{2}$ a compact subset and
$a_{j},b_{j}\in\mathbb{Z},r_{j},t_{j}\in\mathbb{Q}_{p}$. As in Lemma 3.2, the
integral defining $F_{\varphi_{p},\varrho_{p},v,s}(g_{p})$ is given by

$\displaystyle F_{\varphi_{p},\varrho_{p},v,s}(g_{p})$
$\displaystyle=\int_{\mathbb{Q}_{p}^{\times}}\int_{\mathbb{Q}_{p}^{4}}(\omega(g_{p}){\mathscr{F}}_{1}(\varphi_{p}))(0,0,\lambda_{1},\lambda_{2},\alpha\lambda_{1},\alpha^{-1}\lambda_{2})d\lambda_{1}d\lambda_{2}\varrho((\alpha,\alpha^{-1}))|\alpha|_{p}^{s}d^{\times}\alpha.$

Suppose $(0,0)\in C_{1}$, otherwise the integral vanishes identically. When
$|\alpha|_{p}\geq p^{N}$ for $N$ sufficiently large, we have
$|\alpha^{-1}\lambda_{2}|_{p}$ very small for all $\lambda_{2}\in
p^{a_{2}}\mathbb{Z}_{p}+r_{2}$. Therefore, when $t_{2}\not\in
p^{b_{2}}\mathbb{Z}_{p}$ or $r_{1}\not\in p^{a_{1}}\mathbb{Z}_{p}$, the
integral over those $\alpha$ with $|\alpha|_{p}\geq p^{N}$ is zero. When
$t_{2}\in p^{b_{2}}\mathbb{Z}_{p}$ and $r_{1}\in p^{a_{1}}\mathbb{Z}_{p}$, we
have

$\displaystyle\quad\int_{|\alpha|_{p}\geq
p^{N}}\int_{\mathbb{Q}_{p}^{4}}(\omega(g_{p}){\mathscr{F}}_{1}(\varphi_{p}))(0,0,\lambda_{1},\lambda_{2},\alpha\lambda_{1},\alpha^{-1}\lambda_{2})d\lambda_{1}d\lambda_{2}\varrho((\alpha,\alpha^{-1}))|\alpha|_{p}^{s}d^{\times}\alpha$
$\displaystyle=\mathrm{vol}(C_{1})\sum_{n\geq
N}\varrho((p,p^{-1}))^{-n}p^{ns}\int_{\mathbb{Z}_{p}^{\times}}\varrho((a,a^{-1}))$
$\displaystyle\quad\times\mathrm{vol}(p^{a_{1}}\mathbb{Z}_{p}\cap
p^{n}(p^{b_{1}}+a^{-1}t_{1}))\mathrm{vol}((p^{a_{2}}\mathbb{Z}_{p}+r_{2})\cap
p^{-n+b_{2}}\mathbb{Z}_{p})d^{\times}a$
$\displaystyle=\mathrm{vol}(C_{1})\mathrm{vol}(p^{a_{2}}\mathbb{Z}_{p}+r_{2})\mathrm{vol}(p^{b_{1}}+a^{-1}t_{1})\int_{\mathbb{Z}_{p}^{\times}}\varrho((a,a^{-1}))d^{\times}a\sum_{n\geq
N}(\varrho((p,p^{-1}))p^{-1+s})^{n},$

which converges when $|s|<1$ and defines a rational function in $p^{s}$. The
same argument takes care of the case when $|\alpha|_{p}$ is sufficiently
small. This proves the first claim.

For the second claim, it is easily checked that

(3.55)
$(\omega(m(\alpha)g,h){\mathscr{F}}(\varphi_{p}))(0)=|\alpha\alpha^{\prime}|_{v}(\omega(g,h(\alpha^{\prime}/\alpha)){\mathscr{F}}(\varphi))(0)$

for $\alpha=(\alpha_{1},\alpha_{2})\in F_{p}=\mathbb{Q}_{p}^{2}$. We can then
use (3.55),
$\varrho(\alpha/\alpha^{\prime})=\chi_{v}(\alpha)\chi_{v^{\prime}}(\alpha)$