and $|\alpha|_{v}=|\alpha_{1}|_{p},|\alpha^{\prime}|_{v}=|\alpha_{2}|_{p}$ to
obtain that $F_{\varphi_{p},\varrho_{p},v,s}\in
I(s,\chi_{v})I(-s,\chi_{v^{\prime}})$. ∎

Now, we will extend the matching result in Theorem 3.11 to standard sections.

######  Theorem 3.15.

In the setting of Theorem 3.11, suppose $p=vv^{\prime}$ splits and let
$\lambda_{\alpha,v,s}(\phi_{v})\in I_{v}(s,\chi_{v})$ denote the standard
section associated to $\lambda_{\alpha,v}(\phi_{v})\in I_{v}(0,\chi_{v})$ for
$\phi_{v}\in{\mathcal{S}}(W_{\alpha}(F_{v}))$. For any $r\in\mathbb{N}$, there
exists
$\varphi_{p}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{(G\cdot
T^{\Delta})(\mathbb{Z}_{p})}$ such that

(3.56)
$F_{\varphi_{p},\varrho_{p},v,s}(g)=\mathcal{L}(s)\lambda_{\alpha,v,s}(\phi_{v})(g_{v})\lambda_{\alpha,v,-s}(\phi_{v^{\prime}})(g_{v^{\prime}})+O(s^{r}).$

for all $g=(g_{v},g_{v^{\prime}})\in G_{0}(\mathbb{Q}_{p})$, where
$\mathcal{L}(s):=(1-p^{-2})|D_{E}|_{p}^{-1}L(1+s,\chi_{v})L(1-s,\chi_{v^{\prime}})$.

######  Remark 3.16.

If $\varrho_{p}$ is unramified and $\varphi_{p}$ is the characteristic
function of the maximal lattice in $V_{p}$, then

(3.57)
$F_{\varphi_{p},\varrho_{p},v,s}(1)=L(1+s,\chi_{v})L(1-s,\chi_{v^{\prime}})(1-p^{-2}).$

by a similar calculation as in Lemma 3.2, and
$F_{\varphi_{p},\varrho_{p},v,s}/(L(1+s,\chi_{v})L(1-s,\chi_{v^{\prime}})(1-p^{-2}))$
is already the standard section
$\lambda_{\alpha,v,s}(\phi_{v})(g_{v})\lambda_{\alpha,v,-s}(\phi_{v^{\prime}})(g_{v^{\prime}})$.

###### Proof.

For $\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})$ and $g\in
G_{0}(\mathbb{Q}_{p})$, we write

$\displaystyle F_{\varphi,\varrho_{p},v,s}(g)$ $\displaystyle=\sum_{n\geq
0}\frac{a_{n}(\varphi,g)}{n!}(-\log p\cdot s)^{n},$ $\displaystyle
a_{n}(\varphi,g)$ $\displaystyle:=(-\log
p)^{-n}\partial_{s}^{n}\left(F_{\varphi,\varrho_{p},v,s}(g)\right)\mid_{s=0}=\int_{H_{1}(\mathbb{Q}_{p})}\Phi_{0}({\mathscr{F}}(\varphi_{p}))(g_{p},h_{1})\varrho(h_{1})\operatorname{ord}_{v}(h_{1})^{n}dh_{1}.$

It is easy to check from definition that
$F_{\varphi,\varrho_{p},v,s}:G_{0}(\mathbb{Z}_{p})\to\mathbb{C}$ satisfies
(3.50), hence so does the function $a_{n}(\varphi,)$ for all $n\geq 0$. Now we
define

$\varphi^{(n)}:=\frac{1}{(-2)^{n}n!}(\omega((p,p^{-1})^{2})-1)^{n}\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})$

with $(p,p^{-1})\in H_{1}(\mathbb{Q}_{p})$. An easy induction shows that
$F_{\varphi^{(n)},\varrho_{p},v,s}=\tfrac{1}{n!}(\tfrac{1-p^{2s}}{2})^{n}F_{\varphi,\varrho_{p},v,s},$
which implies

(3.58) $a_{n^{\prime}}(\varphi^{(n)},g)=\begin{cases}0&\text{ if
}n^{\prime}<n,\\\ F_{\varphi,\varrho_{p}}&\text{ if }n^{\prime}=n.\end{cases}$

When $\varphi\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))$ is
$(G\cdot T^{\Delta})(\mathbb{Z}_{p})$-invariant, so is the function
$\varphi^{(n)}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))$.
Furthermore, the function
$a_{n}(\varphi,\cdot):G_{0}(\mathbb{Z}_{p})\to\mathbb{Q}(\zeta_{p^{\infty}})$
satisfies conditions (3.50) and (3.51). By Proposition 3.10, there exists
$\varphi_{n}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{(G\cdot
T^{\Delta})(\mathbb{Z}_{p})}$ such that

(3.59) $F_{\varphi_{n},\varrho_{p}}(k)=a_{n}(\varphi,k)$

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

Now we prove the Theorem by induction on $r$. The case $r=1$ is just the
content of Theorem 3.11. Note that $|D|_{p}=1$ when $p$ splits in $F$. Now
suppose we have $\varphi$ satisfying (3.56) for some $r\geq 1$. As

$\Phi_{s}:=\lambda_{\alpha,v,s}(\phi_{v})(g_{v})\lambda_{\alpha,v,-s}(\phi_{v^{\prime}})(g_{v^{\prime}})\in
I(s,\chi_{v})I(-s,\chi_{v^{\prime}})$

is a standard section, it satisfies $\Phi_{s}(k)=\Phi_{0}(k)$ when $k\in
G_{0}(\mathbb{Z}_{p})$. So in that case, we have

$\displaystyle
F_{\varphi,\varrho_{p},v,s}(k)-\mathcal{L}(s)\Phi_{s}(k)=\left(a_{r}(\varphi,k)-c_{r}\Phi_{0}(k)\right)\frac{(-\log
p\cdot s)^{r}}{r!}+O(s^{r+1}),$

with $c_{r}:=(-\log p)^{-r}\partial^{r}_{s}\mathcal{L}(s)\mid_{s=0}$ rational.
If we set

$\tilde{\varphi}:=\varphi-\varphi^{(r)}_{r}-c_{r}\varphi^{(r)}\in{\mathcal{S}}(V_{p},\mathbb{Q}(\zeta_{p^{\infty}}))^{(G\cdot
T^{\Delta})(\mathbb{Z}_{p})},$

then equations (3.58) and (3.59) give us

$\displaystyle\quad
F_{\tilde{\varphi},\varrho_{p},v,s}(k)-\mathcal{L}(s)\Phi_{s}(k)$
$\displaystyle=\left(a_{r}(\varphi,k)-F_{\varphi_{r},\varrho_{p}}(k)+c_{r}F_{\varphi,\varrho_{p}}(k)-c_{r}\Phi_{0}(k)\right)\frac{(-\log
p\cdot s)^{r}}{r!}+O(s^{r+1})=O(s^{r+1}).$

So $\tilde{\varphi}$ satisfies the claim for $r+1$. This completes the proof.
∎

Now, we can state a consequence of the matching result in Theorem 3.15.

######  Proposition 3.17.

For matching sections
$\varphi_{p}\in{\mathcal{S}}(V_{p};\mathbb{Q}(\zeta_{p^{\infty}}))^{G(\mathbb{Z}_{p})}$
and $\\{\phi_{v}\in{\mathcal{S}}(W_{\alpha}(F_{v})):v\mid p\\}$ as in Theorem
3.15 with $r=2$, we have

(3.60) $\prod_{v\mid
p}W^{*}_{t,v}(\phi_{v})=\sum_{n\in\mathbb{Z}}\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h){\mathscr{F}}_{1}(\varphi_{p})((0,p^{n}),t^{\prime}/p^{n},-h^{-1}t/p^{n})dh$

for all $t\in F_{p}^{\times}$. Furthermore when $p=vv^{\prime}$ is a split
prime and $W_{t,v}^{*}(\phi_{v})=0$, then we have

(3.61)
$\frac{W^{*,\prime}_{t,v}(\phi_{v})W^{*}_{t,v^{\prime}}(\phi_{v^{\prime}})}{\log
p}=\sum_{n\in\mathbb{Z}}\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h){\mathscr{F}}_{1}(\varphi_{p})((0,p^{n}),t^{\prime}/p^{n},-h^{-1}t/p^{n})o_{v}(h)dh.$

######  Remark 3.18.

Since $o_{v^{\prime}}(h)=-o_{v}(h)$ for all $h\in H_{1}(\mathbb{Q}_{p})$, the
left hand side of (3.61) gets a minus sign if $o_{v}(h)$ is replaced by
$o_{v^{\prime}}(h)$ on the right hand side.

###### Proof.

To prove (3.60). we apply the definition of $W^{*}_{t,v}$ in (2.55) and
Theorem 3.11 to obtain

$\displaystyle\prod_{v\mid p}W^{*}_{t,v}(\phi_{v})$
$\displaystyle=|D/D_{E}|_{p}^{1/2}\prod_{v\mid
p}L(1,\chi_{v})\int_{F_{v}}\lambda_{\alpha,v}(\phi_{v})(wn(b_{v}))\psi_{v}(-tb_{v})db_{v}$
$\displaystyle=(1-p^{-2})^{-1}\int_{F_{p}}F_{\varphi_{p},\varrho_{p}}(wn(b))\psi_{p}(-\operatorname{Tr}(tb))db$
$\displaystyle=(1-p^{-2})^{-1}\int_{H_{1}(\mathbb{Q}_{p})}\varrho(h_{1})\int_{F_{p}}{\mathscr{F}}(\omega_{p}((wn(b),h_{1}))\varphi_{p})(0)\psi_{p}(-\operatorname{Tr}(tb))dbdh_{1}.$

Now the right hand side of (3.60) can be rewritten as

Right hand side of
$\displaystyle\eqref{eq:FCmatch}=\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h_{1})\sum_{n\in\mathbb{Z}}{\mathscr{F}}_{1}(\omega_{p}(h_{1})\varphi_{p})((0,p^{n}),t^{\prime}/p^{n},-t/p^{n})dh_{1}$
$\displaystyle=\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h_{1})\sum_{n\in\mathbb{Z}}(1-p^{-1})^{-1}\int_{p^{n}\mathbb{Z}_{p}^{\times}}{\mathscr{F}}_{1}(\omega_{p}(h_{1})\varphi_{p})((0,u),t^{\prime}/u,-t/u)d^{\times}udh_{1}$
$\displaystyle=(1-p^{-1})^{-1}\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h_{1})\int_{\mathbb{Q}_{p}^{\times}}{\mathscr{F}}_{1}(\omega_{p}(h_{1})\varphi_{p})((0,u),t^{\prime}/u,-t/u)d^{\times}udh_{1}.$

For the second line, we have used $\omega(m(a))\varphi_{p}=\varphi_{p}$ for
all $a\in\mathbb{Z}_{p}^{\times}$ since $\varphi_{p}$ is
$G(\mathbb{Z}_{p})$-invariant. Equation (3.60) now follows from applying
Proposition 3.19 to $\varphi=\omega_{p}(h_{1})\varphi_{p}$.

We now prove (3.61). Let $\Phi_{s}\in I(s,\chi_{v})I(-s,\chi_{v^{\prime}})$ be
the standard sections extending
$\lambda_{\alpha,v}(\phi_{v})\lambda_{\alpha,v^{\prime}}(\phi_{v^{\prime}})$.
By Theorem 3.15 with $r=2$, we have

$F_{\varphi_{p},\varrho_{p},v,s}=(1-p^{-2})|D_{E}|_{p}^{-1}L(1,\chi_{v})L(1,\chi_{v^{\prime}})\lambda_{\alpha,v,s}(\phi_{v})\lambda_{\alpha,v^{\prime},-s}(\phi_{v^{\prime}})+O(s^{2}).$

Therefore using $W^{*}_{t,v}(\phi_{v})=0$, we have

$\displaystyle
W^{*,\prime}_{t,v}(\phi_{v})W^{*}_{t,v^{\prime}}(\phi_{v^{\prime}})=\partial_{s}\left(W^{*}_{t,v}(1,s,\phi_{v})W^{*}_{t,v^{\prime}}(1,-s,\phi_{v^{\prime}})\right)\mid_{s=0}$
$\displaystyle=|D_{E}|_{p}^{-1}L(1,\chi_{v})L(1,\chi_{v^{\prime}})\partial_{s}\left(\int_{F_{p}}\Phi_{s}((wn(b_{v}),wn(b_{v^{\prime}})))\psi_{v}(-tb_{v})\psi_{v^{\prime}}(-tb_{v^{\prime}})db_{v}db_{v^{\prime}}\right)\mid_{s=0}$
$\displaystyle=(1-p^{-2})^{-1}\partial_{s}\left(\int_{F_{p}}F_{\varphi_{p},\varrho_{p},v,s}(wn(b))\psi_{p}(-\operatorname{Tr}(tb))db\right)\mid_{s=0}$
$\displaystyle=(1-p^{-2})^{-1}\partial_{s}\left(\int_{H_{1}(\mathbb{Q}_{p})}\varrho(h_{1})|h_{1}|_{v}^{s}\int_{F_{p}}{\mathscr{F}}(\omega_{p}((wn(b),h_{1}))\varphi_{p})(0)\psi_{p}(-\operatorname{Tr}(tb))dbdh_{1}\right)\mid_{s=0}$
$\displaystyle=\log
p(1-p^{-2})^{-1}\int_{H_{1}(\mathbb{Q}_{p})}\varrho(h_{1})\operatorname{ord}_{v}(h_{1})\int_{F_{p}}{\mathscr{F}}(\omega_{p}((wn(b),h_{1}))\varphi_{p})(0)\psi_{p}(-\operatorname{Tr}(tb))dbdh_{1}.$

Applying Proposition 3.19 and continuing as in the second half of the proof of
(3.60) then proves (3.61). ∎

We end this section with the following technical result used in the proof of
the previous proposition.

######  Proposition 3.19.

For any $\varphi\in{\mathcal{S}}(V_{p};\mathbb{C})^{G(\mathbb{Z}_{p})}$ and
$t\in F_{p}^{\times}$, we have

(3.62)
$(1+1/p)^{-1}\int_{F_{p}}(\omega_{p}((wn(\beta)){\mathscr{F}}(\varphi))(0)\psi_{p}(-\operatorname{Tr}(t\beta))d\beta=\int_{\mathbb{Q}_{p}^{\times}}{\mathscr{F}}_{1}(\varphi)((0,u),t^{\prime}/u,-t/u)d^{\times}u.$

###### Proof.

Since the left hand side is essentially the Fourier transform of
$(\omega(wn(\beta)){\mathscr{F}}(\varphi_{p}))(0)$ as a function of $\beta\in
F_{p}$, it suffices to calculate the inverse Fourier transform of the right
hand side, though we need to be careful about the singularity of right hand
side when $t\not\in F_{p}^{\times}$. To take care of this, we define

(3.63)
$G_{\epsilon}(t,\varphi):=\begin{cases}\int_{\mathbb{Q}_{p}^{\times}}{\mathscr{F}}_{1}(\varphi)((0,u),t^{\prime}/u,-t/u)d^{\times}u,&\text{
if }|t|>\epsilon,\\\ 0,&\text{ otherwise.}\end{cases}$

for $\epsilon>0$ and $t\in F_{p}$. Note that $|t|:=\min\\{|t_{1}|,|t_{2}|\\}$
when $t=(t_{1},t_{2})\in F_{p}=\mathbb{Q}_{p}^{2}$, Given any fixed $t\in
F_{p}^{\times}$, the limit $\lim_{\epsilon\to 0}G_{\epsilon}(t,\varphi)$
exists and is the right hand side of (3.62). Also for any fixed $\epsilon>0$,
the function $G_{\epsilon}(t,\varphi)$ is a Schwartz function on $F_{p}$. Its
inverse Fourier transform is given by

$\displaystyle\hat{G}_{\epsilon}(\beta,\varphi)$
$\displaystyle:=\int_{F_{p}}G_{\epsilon}(t,\varphi)\psi_{p}(\operatorname{Tr}(t^{\prime}\beta^{\prime}))dt$
$\displaystyle=\int_{F_{p}\backslash
D_{\epsilon}}\int_{\mathbb{Q}_{p}^{\times}}{\mathscr{F}}_{1}(\varphi)((0,u),t^{\prime}/u,-t/u)d^{\times}u\psi_{p}(\operatorname{Tr}(t^{\prime}\beta^{\prime}))dt$
$\displaystyle=\int_{\mathbb{Q}_{p}^{\times}}\int_{F_{p}\backslash
D_{\epsilon/|u|}}{\mathscr{F}}_{1}(\varphi)((0,u),\tilde{t}^{\prime},-\tilde{t})\psi_{p}(\operatorname{Tr}(\tilde{t}^{\prime}u\beta^{\prime}))|u|^{2}d\tilde{t}{d^{\times}u},$

where $D_{\epsilon}\subset F_{p}$ is the $\epsilon$-neighborhood of $0$. Note
that

$\displaystyle{\mathscr{F}}_{1}(\varphi)((0,u),\tilde{t}^{\prime},-\tilde{t})\psi_{p}(\operatorname{Tr}(\tilde{t}^{\prime}u\beta^{\prime}))$
$\displaystyle={\mathscr{F}}_{1}(\varphi_{\beta})((0,u),-\tilde{t},-\tilde{t}),$

where $\varphi_{\beta}:=\omega(w_{2}n(\beta))\varphi$ and $w_{i}\in
H(\mathbb{Q})$ is defined in (3.6). Therefore,
$\hat{G}_{\epsilon}(\beta,\varphi)$ is given by

$\displaystyle\hat{G}_{\epsilon}(\beta,\varphi)$
$\displaystyle=\int_{\mathbb{Q}_{p}^{\times}}\int_{F_{p}\backslash
D_{\epsilon/|u|}}{\mathscr{F}}_{1}(\varphi_{\beta})((0,u),-\tilde{t},-\tilde{t})|u|^{2}d\tilde{t}{d^{\times}u}$
$\displaystyle=\int_{\mathbb{Q}_{p}^{\times}}\left({\mathscr{F}}(\varphi_{\beta})((0,u),0)-\int_{D_{\epsilon/|u|}}\phi_{\beta}((0,u),s,s)ds\right)|u|^{2}d^{\times}u$

with $s=-\tilde{t}$ and $ds=d\tilde{t}$. Using the
$G(\mathbb{Z}_{p})$-invariance of $\varphi$, we have

${\mathscr{F}}(\varphi_{\beta})((0,p^{n}u),0)={\mathscr{F}}(\varphi_{\beta})((0,p^{n}),0)={\mathscr{F}}(\varphi_{\beta})(a,0)$

for all $u\in\mathbb{Z}_{p}^{\times},n\in\mathbb{Z}$ and
$a\in(p^{n}\mathbb{Z}_{p})^{2}-(p^{n+1}\mathbb{Z}_{p})^{2}$. Applying this, we
can evaluate the first part as

$\displaystyle\int_{\mathbb{Q}_{p}^{\times}}{\mathscr{F}}(\varphi_{\beta})((0,u),0)|u|^{2}d^{\times}u=\sum_{n\in\mathbb{Z}}\int_{\mathbb{Z}_{p}^{\times}}{\mathscr{F}}(\varphi_{\beta})((0,p^{n}u),0)|p^{n}u|^{2}d^{\times}u$
$\displaystyle=\sum_{n\in\mathbb{Z}}{\mathscr{F}}(\varphi_{\beta})((0,p^{n}),0)p^{-2n}(1-1/p)=(1+1/p)^{-1}\sum_{n\in\mathbb{Z}}{\mathscr{F}}(\varphi_{\beta})((0,p^{n}),0)\int_{(p^{n}\mathbb{Z}_{p})^{2}-(p^{n+1}\mathbb{Z}_{p})^{2}}da$
$\displaystyle=(1+1/p)^{-1}\sum_{n\in\mathbb{Z}}\int_{(p^{n}\mathbb{Z}_{p})^{2}-(p^{n+1}\mathbb{Z}_{p})^{2}}{\mathscr{F}}(\varphi_{\beta})(a,0)da=(1+1/p)^{-1}\int_{\mathbb{Q}_{p}^{2}}{\mathscr{F}}(\varphi_{\beta})(a,0)da$
$\displaystyle=(1+1/p)^{-1}{\mathscr{F}}(\omega(w_{1})\varphi_{\beta})(0)=(1+1/p)^{-1}(\omega(wn(\beta)){\mathscr{F}}(\varphi))(0).$

Then for any fixed $t\in F_{p}^{\times}$, we have

$\displaystyle G_{\epsilon}(t,\varphi)$
$\displaystyle=\int_{F_{p}}G_{\epsilon}(\beta,\varphi)\psi_{p}(-\operatorname{Tr}(\beta
t))d\beta$
$\displaystyle=(1+1/p)^{-1}\int_{F_{p}}((\omega(wn(\beta)){\mathscr{F}})(\varphi))(0)\psi_{p}(-\operatorname{Tr}(\beta
t))d\beta-E_{\epsilon}(\varphi),$ $\displaystyle E_{\epsilon}(\varphi)$
$\displaystyle:=\int_{F_{p}}\psi_{p}(-\operatorname{Tr}(\beta
t))\int_{\mathbb{Q}_{p}^{\times}}|u|^{2}\int_{D_{\epsilon/|u|}}{\mathscr{F}}_{1}(\varphi)((0,u),-s^{\prime},s)\psi_{p}(-\operatorname{Tr}(su\beta))dsd^{\times}ud\beta$

Since ${\mathscr{F}}_{1}(\varphi)$ is a Schwartz function, we can replace the
domain $\mathbb{Q}_{p}^{\times}\times D_{\epsilon/|u|}$ with a compact open
subset independent of $\beta$, and interchange the order of integration to
compute the integral over $\beta$ first, which yields

$E_{\epsilon}(\varphi)=\int_{\mathbb{Q}_{p}^{\times}}|u|^{2}\int_{D_{\epsilon/|u|}}\int_{F_{p}}\psi_{p}(-\operatorname{Tr}(\beta(t+su)))d\beta{\mathscr{F}}_{1}(\varphi)((0,u),-s^{\prime},s)dsd^{\times}u.$

When $\epsilon$ is sufficiently small, we have $t\not\in D_{\epsilon}$ and
$t+su\neq 0$, in which case $E_{\epsilon}=0$. This finishes the proof. ∎

##  4\.  Doi-Naganuma Lift of the Deformed Theta Integral

In this section, we will define and study the properties of the function
$\tilde{\mathcal{I}}$ mentioned in the introduction. In particular, we will
calculate its Fourier coefficients and images under lowering differential
operators. The actions of differential operators follow from those on the
theta kernel, which are given in section 4.1. The Fourier expansion
computations are carried out in section 4.2, with the main result being
proposition 4.7. Section 4.3 contains rationality results about theta lifts
that will be needed to handle the error term mentioned in section 1.2.

Choose
$\varphi^{(1,1)}:=\varphi_{f}\varphi^{(1,1)}_{\infty}\in{\mathcal{S}}(V(\mathbb{A}))^{K_{\varrho}}$
with
$\varphi^{(1,1)}_{\infty}:=\varphi^{(1,1)}_{0,\infty}\otimes\varphi^{+}_{\infty}\in{\mathcal{S}}(V_{0}(\mathbb{R}))\otimes{\mathcal{S}}(V_{1}(\mathbb{R}))$
and

(4.1)
$\varphi^{(1,1)}_{0,\infty}(a,b,\nu,\nu^{\prime}):=-i(a-b+i(\nu+\nu^{\prime}))e^{-\pi(a^{2}+b^{2}+\nu^{2}+(\nu^{\prime})^{2})}\in{\mathcal{S}}(V_{0}(\mathbb{R})).$

For $\tilde{\varrho}_{C}$ as in (2.77), we can define

(4.2)
$\tilde{\mathcal{I}}(g_{0}):=\mathcal{I}(g_{0},\varphi^{(1,1)},\tilde{\varrho}_{C})=\int_{[H_{1}]}\int_{[G]}\theta(g,(g_{0},h_{1}),\varphi^{(1,1)})dg\tilde{\varrho}_{C}(h_{1})dh_{1}.$

We will now analyze various properties of this integral.

###  4.1. Lowering operator action

To calculate the action of differential operators on $\tilde{\mathcal{I}}$, it
suffices to understand the effect on $\varphi_{0}$ via the Weil
representation, which can be done in the Fock model. For this, we follow the
appendices in [FM06] and [Li22] (see also [KM90]).

We identify
$(V_{0}(\mathbb{R}),Q)=(M_{2}(\mathbb{R}),\det)\cong\mathbb{R}^{2,2}$ with the
basis

(4.3) $\begin{split}v_{1}&:=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix},v_{2}:=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},v_{3}:=\frac{1}{\sqrt{2}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},v_{4}:=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\end{split}$

which identifies
${\mathcal{S}}(\mathbb{R}^{2,2})\cong{\mathcal{S}}(V_{0}(\mathbb{R}))$ and
gives us

(4.4) $\begin{pmatrix}a&\nu_{1}\\\
\nu_{2}&b\end{pmatrix}=\frac{a+b}{\sqrt{2}}v_{1}+\frac{-a+b}{\sqrt{2}}v_{3}+\frac{-\nu_{1}+\nu_{2}}{\sqrt{2}}v_{2}+\frac{\nu_{1}+\nu_{2}}{\sqrt{2}}v_{4}.$

The polynomial Fock space is the subspace
$\mathbb{S}(\mathbb{R}^{2,2})\subset{\mathcal{S}}(\mathbb{R}^{2,2})$ spanned
by functions of the form $\prod_{1\leq j\leq 4}D_{j}^{r_{j}}\varphi^{\circ}$
for $r_{j}\in\mathbb{N}_{0}$, where
$\varphi^{\circ}\in{\mathcal{S}}(\mathbb{R}^{2,2})$ is the Gaussian

$\varphi^{\circ}(x_{1},x_{2},x_{3},x_{4}):=e^{-\pi(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})}.$

and $D_{r}$ are operators on ${\mathcal{S}}(\mathbb{R}^{2,2})$ defined by

(4.5) $D_{r}:=\partial_{x_{r}}-2\pi x_{r},~{}1\leq r\leq 4.$

There is an isomorphism
$\iota:\mathbb{S}(\mathbb{R}^{2,2})\to\mathscr{P}(\mathbb{C}^{4})=\mathbb{C}[\mathfrak{z}_{1},\mathfrak{z}_{2},\mathfrak{z}_{3},\mathfrak{z}_{4}]$
such that $\iota(\varphi^{\circ})=1$, $D_{r}$ acts as
$(-1)^{\lfloor(r-1)/2\rfloor}i\mathfrak{z}_{r}$. We now set

(4.6)
$\mathfrak{v}:=\mathfrak{z}_{1}+i\mathfrak{z}_{2},~{}\mathfrak{w}:=\mathfrak{z}_{3}-i\mathfrak{z}_{4}.$

Then using (4.4), the Schwartz functions
$\varphi^{(\epsilon,\epsilon^{\prime})}_{0,\infty}\in{\mathcal{S}}(V_{0}(\mathbb{R}))$
in (3.22) and (4.1) become

(4.7)
$\begin{split}\iota(\varphi^{(1,-1)}_{0,\infty})&=i\sqrt{2}\iota((x_{1}+ix_{2})\varphi^{\circ})=-\frac{i\sqrt{2}}{4\pi}\iota((D_{1}+iD_{2})\varphi^{\circ})=\frac{\sqrt{2}}{4\pi}\mathfrak{v},\\\
\iota(\varphi^{(-1,1)}_{0,\infty})&=-\frac{\sqrt{2}}{4\pi}\overline{\mathfrak{v}},~{}\iota(\varphi^{(1,1)}_{0,\infty})=-\frac{\sqrt{2}}{4\pi}\mathfrak{w}.\end{split}$

Let $(W,\langle,\rangle)$ be the $\mathbb{R}$-symplectic space of dimension 2,
and $\mathbb{W}:=V_{0}(\mathbb{R})\otimes W$ the symplectic space with the
skew-symmetric form $(,)\otimes\langle,\rangle$. The Lie algebra
${\mathfrak{sp}}(\mathbb{W}\otimes\mathbb{C})$ acts on ${\mathcal{S}}(V_{0})$
through the infinitesimal action induced by $\omega$, which we also denote by
$\omega$. In ${\mathfrak{sp}}(\mathbb{W}\otimes\mathbb{C})$, we have the
subalgebra
${\mathfrak{sp}}(W\otimes\mathbb{C})\times\mathfrak{o}(V_{0}\otimes\mathbb{C})$.
Through $\iota$, the elements
$L,R\in{\mathfrak{sl}}_{2}(\mathbb{C})\cong{\mathfrak{sp}}(W\otimes\mathbb{C})$
defined in (4.11) act on
$\mathbb{C}[\mathfrak{z}_{1},\mathfrak{z}_{2},\mathfrak{z}_{3},\mathfrak{z}_{4}]$
as (see [FM06, Lemma A.2])

(4.8)
$\omega(L)=-8\pi\partial_{\mathfrak{v}}\partial_{\overline{\mathfrak{v}}}+\frac{1}{8\pi}\mathfrak{w}\overline{\mathfrak{w}},~{}\omega(R)=-8\pi\partial_{\mathfrak{w}}\partial_{\overline{\mathfrak{w}}}+\frac{1}{8\pi}\mathfrak{v}\overline{\mathfrak{v}}.$

Using the isomorphism

$\displaystyle{\mathfrak{sl}}_{2}(\mathbb{C})^{2}$
$\displaystyle\to\mathfrak{o}(V_{0}\otimes\mathbb{C})$ $\displaystyle(A,B)$
$\displaystyle\mapsto(v\mapsto Av+vB^{t}),$

we see that the elements $L_{1}=(L,0),L_{2}=(0,L_{2}),R_{1}=(R,0),R_{2}=(0,R)$
in ${\mathfrak{sl}}_{2}(\mathbb{C})^{2}$ act on
$\mathbb{C}[\mathfrak{z}_{1},\mathfrak{z}_{2},\mathfrak{z}_{3},\mathfrak{z}_{4}]$
through $\iota$ as (see [FM06, Lemma A.1])

(4.9)
$\begin{split}\omega(L_{1})&=8\pi\partial_{\mathfrak{v}}\partial_{{\mathfrak{w}}}-\frac{1}{8\pi}\overline{\mathfrak{v}}\overline{\mathfrak{w}},~{}\omega(R_{1})=8\pi\partial_{\overline{\mathfrak{v}}}\partial_{\overline{\mathfrak{w}}}-\frac{1}{8\pi}\mathfrak{v}\mathfrak{w},\\\
\omega(L_{2})&=8\pi\partial_{\overline{\mathfrak{v}}}\partial_{{\mathfrak{w}}}-\frac{1}{8\pi}{\mathfrak{v}}\overline{\mathfrak{w}},~{}\omega(R_{2})=8\pi\partial_{{\mathfrak{v}}}\partial_{\overline{\mathfrak{w}}}-\frac{1}{8\pi}\overline{\mathfrak{v}}\mathfrak{w},\end{split}$

For convenience, we slightly abuse notation and write $L,R,L_{j},R_{j}$ for
their corresponding actions on $\mathscr{P}(\mathbb{C}^{4})$.

When we consider the decomposition $V_{0}=V_{00}\oplus U_{D}$ in (3.2), the
map $\iota$ induces
$\mathbb{S}(V_{00}(\mathbb{R}))\cong\mathscr{P}(\mathbb{C}^{3})=\mathbb{C}[\mathfrak{z}_{1},\mathfrak{z}_{3},\mathfrak{z}_{4}]$
and
$\mathbb{S}(U_{D}(\mathbb{R}))\cong\mathscr{P}(\mathbb{C})=\mathbb{C}[\mathfrak{z}_{2}]$.
For $a,b,c\in\mathbb{N}_{0}$, we also define
$\varphi^{(a,b)}_{00,\infty}\in\mathbb{S}(V_{00}(\mathbb{R}))$ and
$\varphi^{c}_{D,\infty}\in\mathbb{S}(\mathbb{R})$ by

(4.10)
$\varphi^{(a,b)}_{00,\infty}:=\left(-\frac{\sqrt{2}}{4\pi}\right)^{a+b}\iota^{-1}(\mathfrak{z}_{1}^{a}\mathfrak{w}^{b}),~{}\varphi^{(c)}_{D,\infty}:=\left(-\frac{\sqrt{2}i}{4\pi}\right)^{c}\iota^{-1}(\mathfrak{z}_{2}^{c}).$

For $r\in\mathbb{N}_{0}$, we have the operators
$\mathrm{RC}_{r},\widetilde{\mathrm{RC}}_{r}$ defined in (2.8) that also act
on $\mathscr{P}(\mathbb{C}^{4})$. They are related by the following lemma.

######  Lemma 4.1.

In the notations above, we have

(4.11)
$\begin{split}(L_{1}+L_{2})\mathrm{RC}_{r}(\mathfrak{w})&=-L(\widetilde{\mathrm{RC}}_{r}(\mathfrak{v}+(-1)^{r}\overline{\mathfrak{v}})),\\\
(\widetilde{\mathrm{RC}}_{r}-(R_{1}+R_{2})^{r})(\mathfrak{v})&=(-8\pi)^{1-2r}2^{r}R(\mathfrak{w}^{r}p_{r}(\mathfrak{v},\overline{\mathfrak{v}})),\\\
(\widetilde{\mathrm{RC}}_{r}-(R_{1}+R_{2})^{r})(\overline{\mathfrak{v}})&=(-8\pi)^{1-2r}2^{r}R(\mathfrak{w}^{r}p_{r}(\overline{\mathfrak{v}},\mathfrak{v})),~{}\end{split}$

where $p_{r}(X,Y):=-(\tilde{Q}_{r}(X,Y)-(X+Y)^{r})/Y\in\mathbb{Q}[X,Y]$ for
all $r\in\mathbb{N}_{0}$.

###### Proof.

It is easy to check that

(4.12)
$\begin{split}\mathrm{RC}_{r}(\mathfrak{w})&=(-8\pi)^{-2r}2^{r}\mathfrak{w}^{r+1}Q_{r}(\mathfrak{v},\overline{\mathfrak{v}}),\\\
\widetilde{\mathrm{RC}}_{r}(\mathfrak{v})&=(-8\pi)^{-2r}2^{r}\mathfrak{w}^{r}\tilde{Q}_{r}(\mathfrak{v},\overline{\mathfrak{v}})\mathfrak{v},\\\
\widetilde{\mathrm{RC}}_{r}(\overline{\mathfrak{v}})&=(-8\pi)^{-2r}2^{r}\mathfrak{w}^{r}\tilde{Q}_{r}(\mathfrak{v},\overline{\mathfrak{v}})\overline{\mathfrak{v}}.\end{split}$

This leads directly to the second equation in (4.11) from the definition. To
prove the first equation, it is enough to verify

$(L_{1}+L_{2})\mathfrak{w}^{r+1}Q_{r}(\mathfrak{v},\overline{\mathfrak{v}})=-L(Q_{r}(\mathfrak{v},\overline{\mathfrak{v}})(\mathfrak{v}+\overline{\mathfrak{v}})\mathfrak{w}^{r}),$

which follows from (2.6). ∎

######  Proposition 4.2.

Let $\phi_{f}\in{\mathcal{S}}(W_{\alpha}(\hat{F}))$ and
$\varphi_{f}\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})$ be matching
sections as in Theorem 3.3, and denote
$\epsilon:={\mathrm{sgn}}(\alpha_{1})=-{\mathrm{sgn}}(\alpha_{2})$. Then for
$\tilde{\mathcal{I}}$ defined in (4.2), we have

(4.13)
$\begin{split}\frac{\pi}{3}(L_{1}+L_{2})\mathrm{RC}_{r}&\tilde{\mathcal{I}}(g)=-(-4\pi)^{-r}(R_{1}+R_{2})^{r}(E^{*}(g,\phi^{(1,-1)})-(-1)^{r}E^{*}(g,\phi^{(-1,1)}))\\\
&-2\log\varepsilon_{\varrho}\frac{\pi}{3}\widetilde{\mathrm{RC}}_{r}\mathcal{I}_{f}(g,\varphi^{(1,-1)}-(-1)^{r}\varphi^{(-1,1)},\varrho),\end{split}$

where $\mathcal{I}_{f}$ is defined in (3.23) and $\varphi^{(\pm 1,\mp
1)}=\varphi_{f}\otimes\varphi^{(\pm 1,\mp
1)}_{\infty}\in{\mathcal{S}}(V(\mathbb{A}))$ is defined in (3.22).

###### Proof.

Suppose $\varphi_{f}=\varphi_{0,f}\otimes\varphi_{1,f}$ and denote
$\varphi^{\pm}_{1}=\varphi_{1,f}\varphi^{\pm}_{\infty},\varphi_{0}^{(k,k^{\prime})}=\varphi_{0,f}\varphi^{(k,k^{\prime})}_{0,\infty}$.
Then

$\displaystyle(L_{1}+L_{2})\mathrm{RC}_{r}(\tilde{\mathcal{I}}(g))=\int_{[G]}(L_{1}+L_{2})\mathrm{RC}_{r}(\theta_{0}(g^{\prime},g,\varphi^{(1,1)}_{0}))\vartheta_{1}(g^{\prime},\varphi_{1}^{+},\tilde{\varrho}_{C})dg^{\prime}$
$\displaystyle=\int_{[G]}L(\widetilde{\mathrm{RC}}_{r}(\theta_{0}(g^{\prime},g,\varphi^{(1,-1)}_{0})-(-1)^{r}\theta_{0}(g^{\prime},g,\varphi^{(-1,1)}_{0})))\vartheta_{1}(g^{\prime},\varphi_{1}^{+},\tilde{\varrho}_{C})dg^{\prime}$

by Lemma 4.1 and (4.7). Now applying Stokes’ theorem and Theorem 2.7 gives us

$\displaystyle\int_{[G]}$ $\displaystyle
L(\widetilde{\mathrm{RC}}_{r}(\theta_{0}(g^{\prime},g,\varphi_{0}^{\epsilon,-\epsilon})))\vartheta_{1}(g^{\prime},\varphi_{1}^{+},\tilde{\varrho}_{C})dg^{\prime}=-\int_{[G]}\widetilde{\mathrm{RC}}_{r}(\theta_{0}(g^{\prime},g,\varphi_{0}^{\epsilon,-\epsilon}))L\vartheta_{1}(g^{\prime},\varphi_{1}^{+},\tilde{\varrho}_{C})dg^{\prime}$
$\displaystyle=-\int_{[G]}\widetilde{\mathrm{RC}}_{r}(\theta_{0}(g^{\prime},g,\varphi_{0}^{\epsilon,-\epsilon}))\left(\vartheta_{1}(g^{\prime},\varphi_{1}^{-},\varrho)+2\log\varepsilon_{\varrho}\Theta_{1}(g^{\prime},\varphi_{1}^{-},\varrho)\right)dg^{\prime}.$

with $\epsilon=\pm 1$. Since
$R\vartheta_{1}(g^{\prime},\varphi_{1}^{-},\varrho)=0$, we can apply Stokes’
theorem, Lemma 4.1 and Theorem 3.3 to obtain

$\displaystyle\int_{[G]}$
$\displaystyle\widetilde{\mathrm{RC}}_{r}(\theta_{0}(g^{\prime},g,\varphi_{0}^{\epsilon,-\epsilon}))\vartheta_{1}(g^{\prime},\varphi_{1}^{-},\varrho)dg^{\prime}$
$\displaystyle=(-4\pi)^{-r}(R_{1}+R_{2})^{r}\int_{[G]}(\theta_{0}(g^{\prime},g,\varphi_{0}^{\epsilon,-\epsilon}))\vartheta_{1}(g^{\prime},\varphi_{1}^{-},\varrho)dg^{\prime}$
$\displaystyle=(-4\pi)^{-r}(R_{1}+R_{2})^{r}\mathcal{I}(g,\varphi^{(\epsilon,-\epsilon)},\varrho)=\frac{3}{\pi}(-4\pi)^{-r}(R_{1}+R_{2})^{r}E^{*}(g,\phi^{(\epsilon,-\epsilon)}).$

Putting these together finishes the proof. ∎

To understand the first term on the right hand side of (4.13), recall the
decomposition for $\theta_{0}(g^{\prime},g^{\Delta},\varphi_{0})$ in (3.18)
when $\varphi_{0,\infty}\in\mathbb{S}(V_{0})$. This allows us to define

(4.14)
$(\mathrm{RC}^{{}^{\prime},\Delta}_{r^{\prime},(k_{1},k_{2})}\theta_{0})(g^{\prime},g_{00}^{\Delta},\varphi_{0}):=(-4\pi)^{-r^{\prime}}Q_{r^{\prime},(k_{1},k_{2})}(R_{1}^{\prime},R_{2}^{\prime})\left(\theta_{00}(g_{1}^{\prime},g_{00},\varphi_{00})\theta_{D}(g_{2}^{\prime},\varphi_{D})\right)\mid_{g^{\prime}_{1}=g^{\prime}_{2}=g^{\prime}}$

for $\varphi_{0}=\varphi_{00}\otimes\varphi_{D}$ with
$\varphi_{00}\in{\mathcal{S}}(V_{00}(\mathbb{A})),\varphi_{D}\in{\mathcal{S}}(U_{D}(\mathbb{A}))$,
and $R^{\prime}_{j}$ the raising operator on $g^{\prime}_{j}$. In the Fock
model, $R^{\prime}_{1}$, resp. $R^{\prime}_{2}$, acts on
$\mathbb{C}[\mathfrak{z}_{1},\mathfrak{z}_{3},\mathfrak{z}_{4}]$, resp.
$\mathbb{C}[\mathfrak{z}_{2}]$, as

(4.15)
$\omega(R^{\prime}_{1})=-8\pi\partial_{\mathfrak{w}}\partial_{\overline{\mathfrak{w}}}+\frac{1}{8\pi}\mathfrak{z}_{1}^{2},~{}\omega(R^{\prime}_{2})=\frac{1}{8\pi}\mathfrak{z}_{2}^{2}.$

This definition also extends by linearity to all
$\varphi_{0}\in{\mathcal{S}}(V_{0}(\mathbb{A}))$ satisfying
$\varphi_{0,\infty}\in\mathbb{S}(V_{0}(\mathbb{R}))$. We now record the
following lemma.

######  Lemma 4.3.

For $r\in\mathbb{N}_{0}$, denote $r_{0}:=\lfloor r/2\rfloor$. Then

(4.16)
$\begin{split}(\widetilde{\mathrm{RC}}_{r}\theta_{0})&(g^{\prime},g_{00}^{\Delta},\varphi^{(1,-1)}_{0}+(-1)^{r}\varphi^{(-1,1)}_{0})\\\
&=-2^{2r_{0}-r+1}(\mathrm{RC}^{{}^{\prime},\Delta}_{r_{0},(-r+1/2,r-2r_{0}+1/2)}\theta_{0})(g^{\prime},g_{00}^{\Delta},\varphi_{0,f}(\varphi^{(1,r)}_{00,\infty}\otimes\varphi^{r-2r_{0}}_{D,\infty}))\end{split}$

###### Proof.

Suppose $\varphi_{0,f}=\varphi_{00,f}\otimes\varphi_{D,f}$. Then equations in
(4.7) implies

$(\widetilde{\mathrm{RC}}_{r}\theta_{0})(g^{\prime},g_{00}^{\Delta},\varphi^{(1,-1)}_{0}-(-1)^{r}\varphi^{(-1,1)}_{0})=\frac{\sqrt{2}}{4\pi}\theta_{0}(g^{\prime},g_{00}^{\Delta},\varphi_{0,f}\iota^{-1}\widetilde{\mathrm{RC}}_{r}(\mathfrak{v}+(-1)^{r}\overline{\mathfrak{v}})).$

From (4.12) and the definition of $P_{r}$ in (2.5), we have

$\displaystyle\widetilde{\mathrm{RC}}_{r}(\mathfrak{v}+(-1)^{r}\overline{\mathfrak{v}}))$
$\displaystyle=(-4\pi)^{-2r}2^{-r}\mathfrak{w}^{r}\tilde{Q}_{r}(\mathfrak{v},\overline{\mathfrak{v}})(\mathfrak{v}+(-1)^{r}\overline{\mathfrak{v}})=(-4\pi)^{-2r}2^{-r}\mathfrak{w}^{r}Q_{r}(\mathfrak{v},\overline{\mathfrak{v}})(\mathfrak{v}+\overline{\mathfrak{v}})$
$\displaystyle=(-4\pi)^{-2r}2\mathfrak{w}^{r}\mathfrak{z}_{1}^{r+1}P_{r}(i\mathfrak{z}_{2}/\mathfrak{z}_{1})$
$\displaystyle=(-4\pi)^{-2r}2\mathfrak{w}^{r}\mathfrak{z}_{1}(-1)^{r_{0}}\sum_{s=0}^{r_{0}}\binom{r_{0}-r-1/2}{r_{0}-s}\binom{r-r_{0}-1/2}{s}\mathfrak{z}_{1}^{2s}(i\mathfrak{z}_{2})^{r-2s}$
$\displaystyle=(-4\pi)^{r_{0}-2r}2i^{r}(-2)^{r_{0}}Q_{r_{0},(-r+1/2,r-2r_{0}+1/2)}(R_{1}^{\prime},R_{2}^{\prime})\mathfrak{z}_{1}\mathfrak{w}^{r}\mathfrak{z}_{2}^{r-2r_{0}}.$

Substituting the definition (4.10) finishes the proof. ∎

The following technical lemma about change of integral follows from the proof
of Lemma 5.4.3 in [Li22].

######  Lemma 4.4.

Given $\varphi_{i,f}\in{\mathcal{S}}(\hat{V}_{i})$ with $i=0,1$, let
$\Gamma\subset\operatorname{PSL}_{2}(\mathbb{Z})\subset G_{00}(\mathbb{Q})$ be
a congruence subgroup that acts trivially on $\varphi_{0,f}$. For any $a\geq
1,b\geq 0$ and $f\in M^{!}_{-2b}(\Gamma)$, we have

$\displaystyle\int^{\mathrm{reg}}_{\Gamma\backslash\mathbb{H}}y^{b}f(z)\int^{\mathrm{}}_{[\operatorname{SL}_{2}]}$
$\displaystyle\theta_{1}(g,\xi,\varphi_{1}^{-})\theta_{0}(g,g_{z}^{\Delta},\varphi_{0,f}(\varphi^{(a,b)}_{00,\infty}\otimes\varphi^{b-a+1}_{D,\infty}))dgd\mu(z)$
$\displaystyle=\int^{\mathrm{reg}}_{[\operatorname{SL}_{2}]}\theta_{1}(g,\xi,\varphi_{1}^{-})\int^{\mathrm{reg}}_{\Gamma\backslash\mathbb{H}}y^{b}f(z)\theta_{0}(g,g_{z}^{\Delta},\varphi_{0,f}(\varphi^{(a,b)}_{00,\infty}\otimes\varphi^{b-a+1}_{D,\infty}))d\mu(z)dg.$

###  4.2. Fourier Expansion of $\tilde{\mathcal{I}}$

To evaluate the Fourier expansion of $\tilde{\mathcal{I}}$ in (4.2), we change
to a mixed model of the Weil representation using the partial Fourier
transform ${\mathscr{F}}_{1}$ defined in (3.28).

Throughout the section, we write

(4.17) $g_{0}=(g_{z_{1}},g_{z_{2}})\in G_{0}(\mathbb{R})$

for $(z_{1},z_{2})\in\mathbb{H}^{2}$ with $z_{i}=x_{i}+iy_{i}$ and
$g_{\bm{\tau}}\in G(\mathbb{R})$ with
${\bm{\tau}}={\bm{u}}+i{\bm{v}}\in\mathbb{H}$, then (3.27) implies

(4.18)
$\begin{split}\omega_{0,\infty}(g_{0})&{\mathscr{F}}_{1}(\varphi_{0,\infty})((0,r),\nu)=\sqrt{y_{1}y_{2}}{\mathbf{e}}(r(x_{2}\nu+x_{1}\nu^{\prime})){\mathscr{F}}_{1}(\varphi_{0,\infty})((0,r\sqrt{y_{1}y_{2}}),\nu\sqrt{y_{2}/y_{1}}),\\\
\omega_{0,\infty}(g_{\bm{\tau}})&{\mathscr{F}}_{1}(\varphi_{0,\infty})((0,r),\nu)=\sqrt{{\bm{v}}}{\mathbf{e}}(-{\bm{u}}\nu\nu^{\prime}){\mathscr{F}}_{1}(\varphi_{0,\infty})((0,r/\sqrt{{\bm{v}}}),\sqrt{{\bm{v}}}\nu).\end{split}$

Also when $\varphi_{0,\infty}=\varphi^{(k,k^{\prime})}_{0,\infty}$ with
$k,k^{\prime}=\pm 1$ as given in (3.22) and (4.1), we have

(4.19)
${\mathscr{F}}_{1}(\varphi^{(k,k^{\prime})}_{0,\infty})((0,r),\nu)=\varphi^{(k,k^{\prime})}_{0,\infty}(ir,0,\nu)e^{-2\pi
r^{2}}$

with $r\in\mathbb{R},\nu=(\nu_{1},\nu_{2})\in\mathbb{R}^{2}$. After applying
Poisson summation and unfolding, we can rewrite the theta kernel
$\theta_{0}(g,g_{0},\varphi_{0})$ as

$\displaystyle\theta_{0}(g,g_{0},\varphi_{0})$ $\displaystyle=\sum_{\lambda\in
V_{0}(\mathbb{Q})}\omega_{0}(g,g_{0})\varphi_{0}(\lambda)=\sum_{\begin{subarray}{c}\nu\in
V_{-1}(\mathbb{Q})\\\
\eta\in\mathbb{Q}^{2}\end{subarray}}\omega_{-1}(g)\omega_{0}(g_{0}){\mathscr{F}}_{1}(\varphi_{0})(\eta
g,\nu)$ $\displaystyle=\sum_{\nu\in
V_{-1}(\mathbb{Q})}\omega_{-1}(g)\omega_{0}(g_{0})\left({\mathscr{F}}_{1}(\varphi_{0})((0,0),\nu)+\sum_{\begin{subarray}{c}r\in\mathbb{Q}^{\times}\\\
\gamma\in\Gamma_{\infty}\backslash\operatorname{SL}_{2}(\mathbb{Z})\end{subarray}}{\mathscr{F}}_{1}(\varphi_{0})((0,r)\gamma
g,\nu)\right).$

For a bounded, integrable function $f$ on $[G]$ such that
$\theta_{0}(g,g_{0},\varphi_{0})f(g)$ is right
$\operatorname{SL}_{2}(\hat{\mathbb{Z}}){\mathrm{SO}}_{2}(\mathbb{R})$-invariant,
we have

$I_{0}(g_{0},\varphi_{0},f)=\int_{[G]}\theta_{0}(g,g_{0},\varphi_{0})f(g)dg=\frac{3}{\pi}\int_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\theta_{0}(g_{\bm{\tau}},g_{0},\varphi_{0})f(g_{\bm{\tau}})d\mu({\bm{\tau}}),$

which can be written as
$I_{0}(g_{0},\varphi_{0},f)=I^{0}_{0}(g_{0},\varphi_{0},f)+I^{+}_{0}(g_{0},\varphi_{0},f)$
with (see e.g. equation (4.2) in [Kud16])

(4.20)
$\begin{split}I^{0}_{0}(g_{0},\varphi_{0},f)&:=\frac{3}{\pi}\sum_{\nu\in
V_{-1}(\mathbb{Q})}\int_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\omega_{-1}(g_{\bm{\tau}})\omega_{0}(g_{0}){\mathscr{F}}_{1}(\varphi_{0})((0,0),\nu)f(g_{\bm{\tau}})d\mu({\bm{\tau}}),\\\
I^{+}_{0}(g_{0},\varphi_{0},f)&=\frac{3}{\pi}\sum_{\begin{subarray}{c}\nu\in
V_{-1}(\mathbb{Q}),~{}r\in\mathbb{Q}^{\times}\\\
\gamma\in\Gamma_{\infty}\backslash\operatorname{SL}_{2}(\mathbb{Z})\end{subarray}}\int_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}f(g_{\gamma{\bm{\tau}}})\omega_{-1}(g_{\gamma{\bm{\tau}}})\omega_{0}(g_{0}){\mathscr{F}}_{1}(\varphi_{0})((0,r)g_{\gamma{\bm{\tau}}},\nu)d\mu({\bm{\tau}})\\\
&=\sum_{\nu\in
V_{-1}(\mathbb{Q}),~{}r\in\mathbb{Q}^{\times}}{\mathscr{F}}_{1}(\varphi_{0,f})((0,r),\nu)\mathfrak{F}_{r,\nu}(\varphi_{0,\infty})(z_{1},z_{2},f)\\\
\mathfrak{F}_{r,\nu}(\varphi)(z_{1},z_{2},f)&:=\frac{3}{\pi}\int_{\Gamma_{\infty}\backslash\mathbb{H}}(\omega_{0}(g_{\bm{\tau}},g_{0}){\mathscr{F}}_{1}(\varphi))((0,r),\nu)f(g_{\bm{\tau}})d\mu({\bm{\tau}}).\end{split}$

Using the $\operatorname{SL}_{2}(\hat{\mathbb{Z}})$-invariance of
$\theta_{0}(g,g_{0},\varphi_{0})f(g)$, we can rewrite for any $N\in\mathbb{N}$

(4.21)
$\mathfrak{F}_{r,\nu}(\varphi)(z_{1},z_{2},f):=N^{-1}\frac{3}{\pi}\int_{\Gamma_{\infty}^{N}\backslash\mathbb{H}}(\omega_{0}(g_{\bm{\tau}},g_{0}){\mathscr{F}}_{1}(\varphi))((0,r),\nu)f(g_{\bm{\tau}})d\mu({\bm{\tau}})$

with $\Gamma_{\infty}^{N}:=\\{n(Nb):b\in\mathbb{Z}\\}\subset\Gamma_{\infty}$.

For our purpose, we are interested in the case when
$f(g)=\vartheta_{1}(g,\varphi_{1},\rho)$ with
$\rho\in\\{\varrho,\tilde{\varrho}_{C}\\},\varphi_{1}=\varphi^{\pm}_{1}$ and
$\varphi_{0}=\varphi_{0}^{(k,k^{\prime})}$ for $k,k^{\prime}=\pm 1$. In that
case, we have $I_{0}(g_{0},\varphi_{0},f)=\mathcal{I}(g_{0},\varphi,\rho)$
with $\varphi=\varphi_{0}\otimes\varphi_{1}$, and denote

(4.22)
$\mathcal{I}^{+}(g_{0},\varphi,\rho):=I^{+}_{0}(g_{0},\varphi_{0},\vartheta(\cdot,\varphi_{1},\rho)),~{}\mathcal{I}^{0}(g_{0},\varphi,\rho):=I^{0}_{0}(g_{0},\varphi_{0},\vartheta(\cdot,\varphi_{1},\rho)).$

This can be extended by $\mathbb{Q}$-linearity to all
$\varphi=\varphi_{f}\varphi_{\infty}\in{\mathcal{S}}(V(\mathbb{A}))$ with
$\varphi_{f}\in{\mathcal{S}}(\hat{V})$ and
$\varphi_{\infty}\in\mathbb{S}(V(\mathbb{R}))$.

The constant term $I^{0}_{0}$ in the Fourier expansion of
$I_{0}(g_{0},\varphi_{0},f)$ is independent of $x_{1},x_{2}$ and can be
evaluated by the change of model in section 3.3. For our purpose, we will
state a decay result needed to prove Theorem 5.1

######  Lemma 4.5.

Suppose there is $s\in\mathbb{R}$ such that $|f(g_{\tau})|\ll v^{s}$ for all
$\tau$ in the standard fundamental domain of
$\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$. Then

$\lim_{y\to\infty}y^{-c}\left(\partial_{y_{1}}^{a}\partial_{y_{2}}^{b}\frac{I^{0}_{0}(g_{0},\varphi_{0},f)}{\sqrt{y_{1}y_{2}}}\right)\mid_{y_{1}=y_{2}=y}=0$

for any $a,b,c\in\mathbb{N}_{0}$ satisfying $a+b+c\geq 1$. When $a=b=c=0$, the
limit exists.

######  Remark 4.6.

It is easy to check that $f(g)=\vartheta(g,\varphi_{1},\tilde{\varrho}_{C})$
fulfills the condition in the lemma.

###### Proof.

Let ${\mathcal{F}}$ denote the fundamental domain. Then we can use (4.18) to
obtain

$\displaystyle\frac{I^{0}_{0}(g_{0},\varphi_{0},f)}{\sqrt{y_{1}y_{2}}}$
$\displaystyle=\frac{3}{\pi}\sum_{\nu\in
F}{\mathscr{F}}_{1}(\varphi_{0,f})((0,0),\nu)\left(\nu\sqrt{y_{2}/y_{1}}+\nu^{\prime}\sqrt{y_{1}/y_{2}}\right)$
$\displaystyle\times\int_{{\mathcal{F}}}{\mathbf{e}}(-{\bm{u}}\nu\nu^{\prime})e^{-\pi{\bm{v}}(\nu^{2}y_{2}^{2}+(\nu^{\prime})^{2}y_{1}^{2})/(y_{1}y_{2})}f(g_{\bm{\tau}})\frac{d{\bm{u}}d{\bm{v}}}{{\bm{v}}}.$

Since ${\mathscr{F}}_{1}(\varphi_{0,f})$ is a Schwartz function, we can
suppose the sum over $\lambda\in F$ is replaced by a sum over the translate of
a lattice. For the integral on the second line, we can trivially estimate it
by

$\int^{\infty}_{\sqrt{3}/2}e^{-\pi{\bm{v}}(\nu^{2}y_{2}^{2}+(\nu^{\prime})^{2}y_{1}^{2})/(y_{1}y_{2})}{\bm{v}}^{s}\frac{d{\bm{v}}}{{\bm{v}}}.$

From this, we see that $|y^{-1}I^{0}_{0}((g_{z},g_{z}),\varphi_{0},f)|$ is
bounded independent of $y$, and the second claim holds. This also gives the
first claim for $a=b=0$ and $c\geq 1$. The other cases follows from first
applying $\partial_{y_{1}}^{a}\partial_{y_{2}}^{b}$ to
$\frac{I^{0}_{0}(g_{0},\varphi_{0},f)}{\sqrt{y_{1}y_{2}}}$, then conducting
the same estimate. ∎

We will now evaluate the non-constant term $\mathcal{I}^{+}$. Let $\varrho$
and $\tilde{\varrho}_{C}$ be as in (2.77), and $K_{\varrho}\subset
H_{1}(\hat{\mathbb{Z}}),\Gamma_{\varrho}\subset H_{1}^{+}(\mathbb{Q})$ be as
in (2.14) and (2.68) respectively. For
$f(g)=\vartheta_{1}(g,\varphi^{-}_{1},\varrho)$ and
$\varphi_{0,\infty}=\varphi^{(\pm 1,\mp 1)}_{0,\infty}$ defined in (3.22), we
can apply (4.18), (4.19) and (4.21) to obtain

$\displaystyle\mathfrak{F}_{r,\nu}(\varphi^{(\pm 1,\mp
1)}_{0,\infty})(z_{1},z_{2},f)$
$\displaystyle={\mathrm{sgn}}(\nu)2\sqrt{y_{1}y_{2}}{\mathbf{e}}(r((x_{2}\mp
iy_{2})\nu+(x_{1}\pm iy_{1})\nu^{\prime}))$
$\displaystyle\times\mathrm{vol}(K_{\varrho})\sum_{\xi\in
C}\varrho(\xi)\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\
\beta\beta^{\prime}=\nu\nu^{\prime}<0\end{subarray}}{\mathrm{sgn}}(\beta)\varphi_{1,f}(\xi^{-1}\beta)$

if $\mp r\nu>0$, and zero otherwise. After the change of variable
$t=r\nu^{\prime}$, we have

$\displaystyle\mathcal{I}^{+}(g_{0},\varphi^{(1,-1)},\varrho)$
$\displaystyle=\frac{3}{\pi}\sqrt{y_{1}y_{2}}\sum_{t\in
F^{\times},~{}t_{1}>0>t_{2}}c_{t}(\varphi_{f},\varrho){\mathbf{e}}(t_{1}z_{1}+t_{2}\overline{z_{2}}),$
$\displaystyle\mathcal{I}^{+}(g_{0},\varphi^{(-1,1)},\varrho)$
$\displaystyle=\frac{3}{\pi}\sqrt{y_{1}y_{2}}\sum_{t\in
F^{\times},~{}t_{2}>0>t_{1}}c_{t}(\varphi_{f},\varrho){\mathbf{e}}(t_{1}\overline{z_{1}}+t_{2}{z_{2}}),$
$\displaystyle c_{t}(\varphi_{f},\varrho)$
$\displaystyle:=2\mathrm{vol}(K_{\varrho})\sum_{\xi\in
C}\varrho(\xi)\sum_{r\in\mathbb{Q}^{\times}}\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\
{\mathrm{Nm}}(\beta)={\mathrm{Nm}}(t)/r^{2}<0\end{subarray}}{\mathrm{sgn}}(\beta_{1}t_{2}/r){\mathscr{F}}_{1}(\varphi_{f})((0,r),t/r,\xi^{-1}\beta).$

In the case of $\mathcal{I}^{+}(g_{0},\varphi,\tilde{\varrho}_{C})$, we have
the following result.

######  Proposition 4.7.

Given $\phi_{f}\in{\mathcal{S}}(\hat{W}_{\alpha})$, let
$\varphi_{f}\in{\mathcal{S}}(\hat{V})^{G(\hat{\mathbb{Z}})T^{\Delta}(\hat{\mathbb{Z}})K_{\varrho}}$
be a matching section as in Theorem 3.3. For $\tilde{\varrho}_{C}$ as in
(2.77), we have

(4.23)
$\frac{\mathcal{I}^{+}(g_{0},\varphi_{f}\varphi^{(1,1)}_{\infty},\tilde{\varrho}_{C})}{\sqrt{y_{1}y_{2}}}=\frac{3}{\pi}\sum_{t\in
F^{\times},~{}t\gg
0}\tilde{c}_{t}(\varphi_{f},\varrho){\mathbf{e}}(t_{1}z_{1}+t_{2}{z_{2}})+\sum_{t\in
F^{\times}}e_{t}(\varphi_{f};y_{1},y_{2}){\mathbf{e}}(t_{1}z_{1}+t_{2}z_{2})$

where $\tilde{c}_{t}(\varphi_{f},\varrho)\in\mathbb{C}$ and
$e_{t}(\varphi_{f};\cdot):\mathbb{R}_{>0}^{2}\to\mathbb{C}$ are given in
(4.27) and (4.28) below. There is a constant $M\in\mathbb{N}$ such that
$\tilde{c}_{t}(\varphi_{f},\varrho)=0$ when $t\not\in M^{-1}{\mathcal{O}}$.

Let

(4.24) $S_{C}:=\\{v\text{ prime in }F:\operatorname{ord}_{v}(\xi)\neq 0\text{
for some }\xi\in C\\}$

be a finite set of primes, and Then there exists $\kappa\in\mathbb{N}$ and
$\beta(t,\phi_{f})\in F^{\times}$ such that
$\tilde{c}_{t}(\varphi_{f},\varrho)=-\frac{2}{\kappa}\log|\beta(t,\phi_{f})/\beta(t,\phi_{f})^{\prime}|$
and

(4.25)
$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}(\beta(t,\phi_{f})/\beta(t,\phi_{f})^{\prime})=\begin{cases}\tilde{W}_{t}(\phi_{f}),&\hbox{if
}\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}\\},\\\
-\tilde{W}_{t}(\phi_{f}),&\hbox{if
}\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}^{\prime}\\},\\\
0,&\hbox{otherwise, }\end{cases}$

when $\mathrm{Diff}(W_{\alpha},t)\subset S_{C}^{c}$ with $\tilde{W}_{t}$
defined in (2.63). Furthermore, the function $e_{t}$ satisfies

(4.26)
$\lim_{y\to\infty}y^{-c}\left(\partial^{a}_{y_{1}}\partial^{b}_{y_{2}}e_{t}(y_{1},y_{2})\right)\mid_{y_{1}=y_{2}=y}=0$

for all $a,b,c\in\mathbb{N}_{0}$.

######  Remark 4.8.

Note that we have

$\tilde{\mathcal{I}}(g_{0})=\mathcal{I}(g_{0},\varphi^{(1,1)},\tilde{\varrho}_{C})=\mathcal{I}^{0}(g_{0},\varphi^{(1,1)},\tilde{\varrho}_{C})+\mathcal{I}^{+}(g_{0},\varphi^{(1,1)},\tilde{\varrho}_{C}).$

from (4.2) and (4.22).

###### Proof.

Suppose $\varphi_{f}=\varphi_{0,f}\otimes\varphi_{1,f}$, as the general case
follows by linearity. We first prove (4.23). Using (4.20), it is enough to
evaluate
$\mathfrak{F}_{r,\nu}(\varphi^{(1,1)}_{0,\infty})(z_{1},z_{2},\vartheta_{1}(\cdot,\varphi^{+}_{1},\tilde{\varrho}_{C}))$.
If we set $t:=r\nu^{\prime}$, then we have by (4.18), (4.19), and Theorem 2.7

$\displaystyle\mathfrak{F}_{r,\nu^{\prime}}(\varphi^{(1,1)}_{0,\infty})(z_{1},z_{2},\vartheta_{1}(\cdot,\varphi^{+}_{1},\tilde{\varrho}_{C}))$
$\displaystyle=\left(e_{r,\nu^{\prime}}(\varphi_{1,f};y_{1},y_{2})+\tilde{c}_{r,\nu^{\prime}}(\varphi_{1,f},\varrho)\right)\sqrt{y_{1}y_{2}}{\mathbf{e}}(t_{1}z_{1}+t_{2}z_{2}))$

with $\tilde{c}_{r,\nu^{\prime}}(\varphi,\varrho)$ and
$e_{r,\nu}(\varphi;y,y^{\prime})$ given by

$\displaystyle\tilde{c}_{r,\nu^{\prime}}(\varphi,\varrho)$
$\displaystyle:=\begin{cases}4\mathrm{vol}(K_{\varrho})\displaystyle\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\ \beta\beta^{\prime}=\nu\nu^{\prime}>0\\\ \xi\in
C\end{subarray}}\varrho(\xi){\mathrm{sgn}}(r\beta)\varphi(\xi^{-1}\beta)\log(\varepsilon_{\varrho})\left\\{\log_{\varepsilon_{\varrho}}\sqrt{|\beta/\beta^{\prime}|}\right\\},&t\gg
0\\\ 0,&\text{ otherwise.}\end{cases}$ $\displaystyle
e_{r,\nu}(\varphi;y_{1},y_{2})$
$\displaystyle:=-\frac{\mathrm{vol}(K_{\varrho})}{\sqrt{y_{1}y_{2}}}\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\ \beta\beta^{\prime}=\nu\nu^{\prime}\\\ \xi\in
C\end{subarray}}\varrho(\xi)\varphi(\xi^{-1}\beta)\left(\delta_{\nu\nu^{\prime}<0}{\mathrm{sgn}}(\beta)e^{*}_{r,\nu}(y_{1},y_{2})+\frac{\log\varepsilon_{\varrho}}{\sqrt{\pi}}e^{\dagger}_{r,\nu,\beta}(y_{1},y_{2})\right).$

Here we have set

$\displaystyle e^{*}_{r,\nu}(y_{1},y_{2})$
$\displaystyle:=\int^{\infty}_{0}\Gamma(0,4\pi|\nu\nu^{\prime}|{\bm{v}})K_{r,\nu}({\bm{v}},y_{1},y_{2})\frac{d{\bm{v}}}{{\bm{v}}},$
$\displaystyle e^{\dagger}_{r,\nu,\beta}(y_{1},y_{2})$
$\displaystyle:=\int^{\infty}_{0}\sum_{\epsilon\in\Gamma_{1}}{\mathrm{sgn}}(\beta\epsilon-\beta^{\prime}\epsilon^{\prime})\Gamma(1/2,\pi(\beta\epsilon-\beta^{\prime}\epsilon^{\prime})^{2}{\bm{v}})\sqrt{{\bm{v}}}K_{r,\nu}({\bm{v}},y_{1},y_{2})\frac{d{\bm{v}}}{{\bm{v}}},$
$\displaystyle K_{r,\nu}({\bm{v}},y_{1},y_{2})$
$\displaystyle:=e^{-\pi\left(\frac{A}{\sqrt{{\bm{v}}}}-B\sqrt{{\bm{v}}}\right)^{2}}\left(\frac{A}{\sqrt{{\bm{v}}}}+B\sqrt{{\bm{v}}}\right),~{}A:=r\sqrt{y_{1}y_{2}},~{}B:=\frac{\nu
y_{1}+\nu^{\prime}y_{2}}{\sqrt{y_{1}y_{2}}}.$

So if we set

(4.27) $\displaystyle\tilde{c}_{t}(\varphi_{f},\varrho)$
$\displaystyle:=\sum_{r\in\mathbb{Q}^{\times}}{\mathscr{F}}_{1}(\varphi_{0,f})((0,r),t^{\prime}/r)\tilde{c}_{r,t^{\prime}/r}(\varphi_{1,f},\varrho),$
(4.28) $\displaystyle e_{t}(\varphi_{f};y_{1},y_{2})$
$\displaystyle:=\sum_{r\in\mathbb{Q}^{\times}}{\mathscr{F}}_{1}(\varphi_{0,f})((0,r),t^{\prime}/r)e_{r,t^{\prime}/r}(\varphi_{1,f};y_{1},y_{2}),$

then equation (4.23) holds by (4.20). Since $\varphi_{f}$ is a Schwartz
function, the sum defining $\tilde{c}_{t}$ is finite, and equals to 0 when
$t\not\in M^{-1}{\mathcal{O}}$ for some $M\in\mathbb{N}$ depending only on
$\varphi_{f}$.

To prove (4.25), notice that

$\displaystyle\tilde{c}_{t}(\varphi_{f},\varrho)$
$\displaystyle=4\mathrm{vol}(K_{\varrho})\sum_{r\in\mathbb{Q}^{\times}}\displaystyle\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\ 1\leq|\beta/\beta^{\prime}|\leq\varepsilon_{\varrho}^{2}\\\
\beta\beta^{\prime}=tt^{\prime}/r^{2}>0\\\ \xi\in
C\end{subarray}}\varrho(\xi){\mathscr{F}}_{1}(\varphi_{f})((0,r),t^{\prime}/r,\xi^{-1}\beta){\mathrm{sgn}}(r\beta)\log\sqrt{|\beta/\beta^{\prime}|}$

By Theorem 3.11 and Lemma 3.8, there exists $c\in\mathbb{N}$ such that
$2c{\mathscr{F}}_{1}(\varphi_{f})((0,r),\nu,\lambda)\in\mathbb{Z}$ for all
$r\in\hat{\mathbb{Q}}$ and $\nu,\lambda\in\hat{F}$. Then we can write

$\tilde{c}_{t}(\varphi_{f},\varrho)=-\frac{2}{\kappa}\log\left|\frac{\beta(t,\varphi_{f})}{\beta(t,\varphi_{f})^{\prime}}\right|,~{}\beta(t,\varphi_{f}):=\prod_{r\in\mathbb{Q}^{\times}}\displaystyle\prod_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\ 1\leq|\beta/\beta^{\prime}|\leq\varepsilon_{\varrho}^{2}\\\
\beta\beta^{\prime}=tt^{\prime}/r^{2}>0\\\ \xi\in
C\end{subarray}}(r\beta)^{-\mathrm{vol}(K_{\varrho})\varrho(\xi)\kappa{\mathscr{F}}_{1}(\varphi_{f})((0,r),t^{\prime}/r,\xi^{-1}\beta){\mathrm{sgn}}(r\beta)}$

For any split rational prime $p=\mathfrak{p}\mathfrak{p}^{\prime}$ with any
$\mathfrak{p}\not\in S_{C}$, we have

$\displaystyle\kappa^{-1}$
$\displaystyle\operatorname{ord}_{\mathfrak{p}}\beta(t,\varphi_{f})=-\mathrm{vol}(K_{\varrho})\sum_{r\in\mathbb{Q}^{\times}}\sum_{\begin{subarray}{c}\beta\in\Gamma_{\varrho}\backslash
F^{\times}\\\ \beta\beta^{\prime}=tt^{\prime}/r^{2}>0\\\ \xi\in
C\end{subarray}}\varrho(\xi){\mathscr{F}}_{1}(\varphi_{f})((0,r),t^{\prime}/r,\xi^{-1}\beta){\mathrm{sgn}}(r\beta)\operatorname{ord}_{\mathfrak{p}}(\xi^{-1}r\beta)$
$\displaystyle=-\mathrm{vol}(K_{\varrho})\sum_{r\in\mathbb{Q}^{\times}}\sum_{\begin{subarray}{c}h\in\Gamma_{\varrho}\backslash
H_{1}(\mathbb{Q})^{+}\\\ \xi\in
C\end{subarray}}\varrho(\xi){\mathscr{F}}_{1}(\varphi_{f}-\omega_{f}(-1)\varphi_{f})((0,r),t^{\prime}/r,\xi^{-1}h^{-1}t/r){\mathrm{sgn}}(h^{-1}t)\operatorname{ord}_{\mathfrak{p}}(\xi^{-1}h^{-1}t)$
$\displaystyle=2\sum_{r\in\mathbb{Q}^{\times}}\int_{H_{1}(\hat{\mathbb{Q}})}\varrho(h_{1}){\mathscr{F}}_{1}(\varphi_{f})((0,r),t^{\prime}/r,-h_{1}^{-1}t/r)\operatorname{ord}_{\mathfrak{p}}(h_{1}^{-1}t^{\prime})dh_{1}$

since $t\gg
0,\operatorname{ord}_{\mathfrak{p}}(\beta)=\operatorname{ord}_{\mathfrak{p}}(\xi^{-1}\beta)$
and $\varrho_{f}(h)={\mathrm{sgn}}(h)$, where $h_{1}=-\xi h\in
H_{1}(\hat{\mathbb{Q}})$ and $\varrho(h_{1})=-\varrho(\xi)$. The last step
follows from (2.72).

Notice that the quantity above factors as the following product of sums of
local integrals

$\displaystyle\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}\beta(t,\varphi_{f})$
$\displaystyle=2\prod_{\ell<\infty,~{}\ell\neq
p}\left(\sum_{n\in\mathbb{Z}}\int_{H_{1}(\mathbb{Q}_{\ell})}\varrho_{\ell}(h_{1,\ell}){\mathscr{F}}_{1}(\varphi_{f,\ell})((0,\ell^{n}),t^{\prime}/\ell^{n},-h_{1,\ell}^{-1}t/\ell^{n})dh_{1,\ell}\right)$
$\displaystyle\times\left(\sum_{n\in\mathbb{Z}}\int_{H_{1}(\mathbb{Q}_{p})}\varrho_{p}(h_{1,p}){\mathscr{F}}_{1}(\varphi_{f,p})((0,p^{n}),t^{\prime}/p^{n},-h_{1,p}^{-1}t/p^{n})\operatorname{ord}_{\mathfrak{p}}(h_{1,p}^{-1}t^{\prime})dh_{1,p}\right).$

Applying (3.60) turns the first line on the right hand side into
$2\prod_{v<\infty,~{}v\nmid p}W^{*}_{t,v}(\phi_{v})$. If this is non-zero,
then $\mathrm{Diff}(W_{\alpha},t)$ is either $\\{\mathfrak{p}\\}$ or
$\\{\mathfrak{p}^{\prime}\\}$ as it has odd size. If
$\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}\\}$, then
$W^{*}_{t,\mathfrak{p}}(\phi_{\mathfrak{p}})=0$ and Proposition 3.17 tell us
that the second line on the right hand side becomes

$\displaystyle\frac{W^{*,\prime}_{t,\mathfrak{p}}(\phi_{\mathfrak{p}})W^{*}_{t,\mathfrak{p}^{\prime}}(\phi_{\mathfrak{p}^{\prime\prime}})}{\log
p}+W^{*}_{t,\mathfrak{p}}(\phi_{\mathfrak{p}})W^{*}_{t,\mathfrak{p}^{\prime}}(\phi_{\mathfrak{p}^{\prime}})\operatorname{ord}_{\mathfrak{p}}(t^{\prime})=\frac{W^{*,\prime}_{t,\mathfrak{p}}(\phi_{\mathfrak{p}})W^{*}_{t,\mathfrak{p}^{\prime}}(\phi_{\mathfrak{p}^{\prime}})}{\log
p}$

as
$\operatorname{ord}_{\mathfrak{p}}(h_{1,p}^{-1}t^{\prime})=\operatorname{ord}_{\mathfrak{p}}(h_{1,p}^{-1})+\operatorname{ord}_{\mathfrak{p}}(t^{\prime})$.
This gives us

$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}\beta(t,\varphi_{f})=\tilde{W}_{t}(\phi_{f})/2.$

Repeat the above argument toegther with Remark 3.18, we obtain
$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}^{\prime}}\beta(t,\varphi_{f})=-\tilde{W}_{t}(\phi_{f})/2$.
Putting this together gives us (4.25), where the case with
$\mathrm{Diff}(W_{\alpha},t)=\\{\mathfrak{p}^{\prime}\\}$ is obtained
similarly.

Now we will prove (4.26). Since $\varphi$ has compact support, the summation
over $\xi$ and $\beta$ in $e_{r,\nu}$ and the summation over $r$ in (4.28) are
finite sums, it suffices to establish (4.26) with $e_{t}$ replaced by
$e^{*}_{r,\nu}$ and $e^{\dagger}_{r,\nu,\beta}$ with $r>0$. For any fixed
$C,\epsilon>0$, $s\in\mathbb{R}$ and $a,b,c\in\mathbb{N}_{0}$, we have

$\displaystyle\left|y^{c}\partial^{a}_{y}\partial^{b}_{y^{\prime}}\int^{A^{1-\epsilon}}_{0}e^{-C{\bm{v}}}{\bm{v}}^{s}K_{r,\nu}({\bm{v}},y,y^{\prime})\frac{d{\bm{v}}}{{\bm{v}}}\right|$
$\displaystyle\ll\int^{A^{1-\epsilon}}_{0}K_{r,\nu}({\bm{v}},y,y^{\prime})\frac{d{\bm{v}}}{{\bm{v}}}\ll
e^{-A^{\epsilon/2}},$
$\displaystyle\left|y^{c}\partial^{a}_{y}\partial^{b}_{y^{\prime}}\int^{\infty}_{A^{1-\epsilon}}e^{-C{\bm{v}}}{\bm{v}}^{s}K_{r,\nu}({\bm{v}},y,y^{\prime})\frac{d{\bm{v}}}{{\bm{v}}}\right|$
$\displaystyle\ll\int^{\infty}_{A^{1-\epsilon}}e^{-C{\bm{v}}/2}d{\bm{v}}\ll
e^{-A^{1/2-\epsilon/2}}$

when $B$ is in a compact subset of $\mathbb{R}$ and $A>0$ is sufficiently
large. Furthermore, it is easy to see that there exists $C>0$ such that

$|\Gamma(0,4\pi|\nu\nu^{\prime}|{\bm{v}})|\ll{\bm{v}}^{-1}e^{-C{\bm{v}}},~{}\left|\sum_{\varepsilon\in\Gamma_{1}}{\mathrm{sgn}}(\beta\varepsilon-\beta^{\prime}\varepsilon^{\prime})\Gamma(1/2,\pi(\beta\varepsilon-\beta^{\prime}\varepsilon^{\prime})^{2}{\bm{v}})\right|\ll{\bm{v}}^{-1/2}e^{-C{\bm{v}}}$

for all ${\bm{v}}>0$. Combining these then proves (4.26). ∎

###  4.3. Rationality of Theta Lifts

Recall that the rational quadratic space $V_{\alpha}$ is the restriction of
scalar of the $F$-quadratic space $W_{\alpha}$. The following result shows
that the Millson theta lift preserves rationality.

######  Proposition 4.9.

Let $f=\sum_{\mu\in L^{\vee}/L}f_{\mu}\mathfrak{e}_{\mu}\in
M^{!}_{-2r,\rho_{L}}$ be weakly holomorphic for some $r\in\mathbb{N}$ and
lattice $L\subset V_{\alpha}$. For
$\varphi^{(1,r)}=\varphi^{(1,r)}_{00,\infty}\varphi_{f}$ with
$\varphi_{f}\in{\mathcal{S}}(\hat{V}_{00};\mathbb{Q}^{\mathrm{ab}})^{T^{\Delta}(\hat{\mathbb{Z}})}$,
let $\Gamma(N)\subset\operatorname{SL}_{2}(\mathbb{Z})$ be a congruence
subgroup contained in $\ker(\rho_{L})$ that fixes $\varphi_{f}$. The following
regularized integral333The regularization is the same as in [BF04] or [ANS18].

(4.29)
$I^{M}(\tau,f_{\mu},\varphi_{f}):=\frac{1}{[\mathrm{SL}_{2}(\mathbb{Z}):\Gamma(N)]}\int^{\mathrm{reg}}_{\Gamma(N)\backslash\mathbb{H}}y^{r}f_{\mu}(z)\theta_{00}(g_{\tau},h_{z},\varphi^{(1,r)})d\mu(z)$

defines a weakly holomorphic modular form of weight $-r+1/2<0$. Suppose $f$
has rational Fourier coefficients at the cusp $\infty$, so does
$I^{M}(\tau,f_{\mu},\varphi_{f})$ for all $\mu\in L^{\vee}/L$.

######  Remark 4.10.

When $r=0$ and $f$ has vanishing constant term, the same proof shows that the
weakly holomorphic modular form $I^{M}(\tau,f_{\mu},\varphi_{f})$ has rational
Fourier coefficients up to algebraic multiples of weight $1/2$ unary theta
series.

###### Proof.

We will use the Fourier expansion of Millson theta lift calculated in Theorem
5.1 of [ANS18], which we now recall. Fix an orientation on
$V_{00}(\mathbb{R})$. For an isotropic line $\ell\subset V_{00}$, let
$G_{00,\ell}\subset G_{00}$ be its stabilizer and
$\gamma_{\ell}\in\operatorname{SL}_{2}(\mathbb{Z})$ such that
$\gamma_{\ell}^{-1}G_{00,\ell}\gamma_{\ell}=G_{00,\ell_{\infty}}$ with
$\ell_{\infty}=\mathbb{Q}v_{\infty}$ and
$v_{\infty}:=\left(\begin{smallmatrix}1&0\\\ 0&0\end{smallmatrix}\right)$.
Denote $c_{\ell}(m,\mu)$ the $m$-th Fourier coefficient of
$f_{\mu}\mid_{-2r}\gamma_{\ell}$. If $x\in V_{00}(\mathbb{Q})$ satisfies
$\sqrt{-Q(x)}=d\in\mathbb{Q}_{>0}$, then $x^{\perp}$ is a hyperbolic plane
spanned by two isotropic lines $\ell_{x}$ and $\ell_{-x}$ such that
$x,\gamma_{\ell_{x}}v_{\infty},\gamma_{\ell_{-x}}v_{\infty}$ is positively
oriented. We can then define $r_{x}\in\mathbb{Q}$ by

$\gamma_{\ell_{x}}^{-1}\cdot x=-d\begin{pmatrix}2r_{x}&1\\\ 1&0\end{pmatrix}.$

Suppose $r\geq 1$. From [ANS18, Theorem 5.1], we know that
$[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma(N)]\cdot I^{M}$ is weakly
holomorphic of weight $-r+1/2<0$ with principal part given by 444Up to a sign
$(-1)^{r+1}$ depending on the orientation of $V_{00}(\mathbb{R})$.

$\sum_{d>0}\frac{{\mathbf{e}}(-d^{2}\tau)}{2d^{1+r}}\sum_{\begin{subarray}{c}x\in\Gamma_{L}\backslash
V_{00,-d^{2}}(\mathbb{Q})\\\
w\in\mathbb{Q}_{<0}\end{subarray}}w^{k}\varphi_{f}(x)(c_{\ell_{x}}(w,\mu){\mathbf{e}}(r_{x}w)+(-1)^{r+1}c_{\ell_{-x}}(w,\mu){\mathbf{e}}(r_{-x}w))\in\mathbb{Q}^{\mathrm{ab}}.$

Note that the inner sum above vanishes for $d$ sufficiently large by
Proposition 4.7 in [BF04], and $I^{M}$ is uniquely determined by its principal
part because its weight is negative. Now we can enlarge $N$ such that $Nr_{\pm
x}w\in\mathbb{Z}$ whenever $c_{\ell_{\pm x}}(w,\mu)\neq 0$. Then given a prime
$p\nmid N$, for an element $x\in\Gamma_{L}\backslash
V_{00,-d^{2}}(\mathbb{Q})$ to have a representative $\tilde{x}\in V_{00}$ such
that both $t(p)\cdot\tilde{x}$ and $\tilde{x}$ are both $p$-integral is
equivalent to find a $p$-integral representative
$\left(\begin{smallmatrix}A&B\\\ B&C\end{smallmatrix}\right)$ with $p\nmid A$.
Note that the set

$S_{d}(\varphi):=\\{x\in\Gamma_{L}\backslash
V_{00,-d^{2}}(\mathbb{Q}):\varphi(x)\neq 0\\}$

is a finite set for any $\varphi\in{\mathcal{S}}(\hat{V}_{00})$.

For any $\sigma_{a}\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$
with $a\in\hat{\mathbb{Z}}^{\times}$, we have $t(a)\in
T(\hat{\mathbb{Z}})\subset{\mathrm{GL}}_{2}(\hat{\mathbb{Z}})$. Choose an odd
prime $p\nmid N$ such that $a\equiv p\bmod{N}$ and every $x\in
S_{d}(\varphi_{f})$ has a $p$-integral representative $\tilde{x}\in
V_{00}(\mathbb{Q})$ such that $t(p)\cdot\tilde{x}$ is $p$-integral. Let
$\tilde{S}_{d}(\varphi_{f})$ be such a set of representatives.

Denote $\tilde{x}^{\prime}:=t(p)\cdot\tilde{x}$ for
$\tilde{x}\in\tilde{S}_{d}(\varphi_{f})$, which is $p$-integral and satisfies
$\ell_{\pm\tilde{x}^{\prime}}=t(p)\cdot\ell_{\pm\tilde{x}}$ and

$\gamma_{\ell_{\pm\tilde{x}^{\prime}}}\equiv
t(p)\gamma_{\ell_{\pm\tilde{x}}}t(p)^{-1}\equiv
t(a)\gamma_{\ell_{\pm\tilde{x}}}t(a)^{-1}\bmod{N},~{}r_{\pm\tilde{x}^{\prime}}w-pr_{\pm\tilde{x}}w\in\mathbb{Z}.$

when $c_{\ell_{\tilde{x}}}(w)$ or $c_{\ell_{-\tilde{x}}}(w)$ is non-zero. By
equation (2.30), $\Gamma(N)\subset\ker(\rho_{L})$ and the fact that $f$ has
rational Fourier coefficients, we then have

$\sigma_{a}(f\mid_{-2r}\gamma_{\ell_{\pm\tilde{x}}})=\rho_{L}(t(a)\gamma_{\ell_{\pm\tilde{x}}})f=\rho_{L}(t(a)\gamma_{\ell_{\pm\tilde{x}}}t(a)^{-1})f=\rho_{L}(\gamma_{\ell_{\pm\tilde{x}^{\prime}}})f=f\mid_{-2r}\gamma_{\ell_{\pm\tilde{x}^{\prime}}}.$

These imply that

(4.30)
$\sigma_{a}(c_{\ell_{\pm\tilde{x}}}(w,\mu))=c_{\ell_{\pm\tilde{x}^{\prime}}}(w,\mu),~{}\sigma_{a}({\mathbf{e}}(r_{\tilde{x}}w))={\mathbf{e}}(pr_{\tilde{x}}w)={\mathbf{e}}(r_{\tilde{x}^{\prime}}w)$

for all $d>0$, $\tilde{x}\in\tilde{S}_{d}(\varphi_{f})$ and
$w\in\mathbb{Q}_{<0}$. Finally, we have
$\varphi_{f}(\tilde{x})=\varphi_{f}(t(p)^{-1}\tilde{x}^{\prime})$, which
implies

(4.31)
$\begin{split}\varphi_{f}(\tilde{x}))=\varphi_{f}(t(a)^{-1}\tilde{x}^{\prime})=\omega(\iota(t(a)))\varphi_{f})(\tilde{x}^{\prime}).\end{split}$

since $\varphi_{f}\in{\mathcal{S}}(L)$ and $p$ is co-prime to the level of
$L$. Here $\iota$ is the map defined in (3.10). The map
$\tilde{x}\mapsto\tilde{x}^{\prime}$ then gives a bijection between
$S_{d}(\varphi_{f})$ and $S_{d}(\omega(\iota(t(a)))\varphi_{f})$. From this,
we then obtain

(4.32)
$\sigma_{a}(I^{M}(\tau,f_{\mu},\varphi_{f}))=I^{M}(\tau,f_{\mu},\omega(\iota(t(a)))\sigma_{a}(\varphi_{f}))=I^{M}(\tau,f_{\mu},\omega(t(a),\iota(t(a)))\varphi_{f}).$

As $(t(a),\iota(t(a)))\in T^{\Delta}(\hat{\mathbb{Z}})$, and $\varphi_{f}$ is
$T^{\Delta}(\hat{\mathbb{Z}})$-invariant, the modular form
$I^{M}(\tau,f_{\mu},\varphi_{f})$ has rational Fourier coefficients. ∎

From Propositions 2.8 and 4.9, we can deduce the following result.

######  Proposition 4.11.

Let $r\in\mathbb{N}_{0}$ and $f\in M^{!}_{-2r,\rho_{L}}$ as in Proposition
4.9. For each $\mu\in L^{\vee}/L$, let
$\varphi_{\mu}\in{\mathcal{S}}(\hat{V};\mathbb{Q}^{\mathrm{ab}})^{(G\cdot
T^{\Delta})(\hat{\mathbb{Z}})}$ be a matching section with $\phi_{\mu}$ as in
Theorem 3.3. Then the regularized integral

(4.33)
$c_{\mu}(f):=\sqrt{D}^{-r}\frac{\pi}{3}\frac{1}{[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma(N)]}\int^{\mathrm{reg}}_{\Gamma(N)\backslash\mathbb{H}}v^{r}f_{\mu}(\tau)\widetilde{\mathrm{RC}}_{r}\mathcal{I}_{f}(g_{\tau}^{\Delta},\varrho,\varphi_{\mu}^{(1,-1)}-(-1)^{r}\varphi_{\mu}^{(-1,1)})d\mu(\tau)$

is a rational number for all $\mu\in L^{\vee}/L$ and congruence subgroup
$\Gamma(N)\subset\ker(\rho_{L})$ fixing $\varphi_{\mu}$.

###### Proof.

Since $\varphi_{\mu}$ is $G(\hat{\mathbb{Z}})$-invariant, we can rewrite the
constant $c_{\mu}:=c_{\mu}(f)$ as

$\displaystyle\sqrt{D}^{r}c_{\mu}=$
$\displaystyle\int^{\mathrm{reg}}_{\Gamma_{L}\backslash\mathbb{H}}v^{r}f_{\mu}(\tau)\lim_{T^{\prime}\to\infty}\int_{{\mathscr{F}}_{T^{\prime}}}\int_{H_{1}(\mathbb{Q})\backslash
H_{1}(\hat{\mathbb{Q}})}\theta(g_{\tau^{\prime}},(g^{\Delta},h_{1}),\varphi_{\mu}^{r})\varrho(h_{1})dh_{1}d\mu(\tau^{\prime})d\mu(\tau)$

with
$\varphi^{r}_{\mu}:=\widetilde{\mathrm{RC}}_{r}(\varphi_{\mu}^{(1,-1)}-(-1)^{r}\varphi_{\mu}^{(-1,1)})\in{\mathcal{S}}(V(\mathbb{A}))$.
Using Lemma 4.4, we can switch the regularized integral in $g$ with the limit
in $T^{\prime}$. Then by the rational decomposition $V=V_{00}\oplus
U_{D}\oplus V_{1}$ , we can write

$\varphi_{\mu}=\sum_{j\in
J}\varphi_{00,\mu,j}\otimes\varphi_{D,\mu,j}\otimes\varphi_{1,\mu,j},~{}$

with
$\varphi_{00,\mu,j}\in{\mathcal{S}}(\hat{V}_{00};\mathbb{Q}^{\mathrm{ab}})^{T^{\Delta}(\hat{\mathbb{Z}})},\varphi_{1,\mu,j}\in{\mathcal{S}}(\hat{V}_{1})$
and $\varphi_{D,\mu,j}\in{\mathcal{S}}(\hat{U}_{D})$. The constant $c_{\mu}$
can then be rewritten as

${c}_{\mu}=\sum_{j\in J}c_{\mu,j},$

where $c_{\mu,j}$ is defined by

$\displaystyle c_{\mu,j}$
$\displaystyle:=\sqrt{D}^{-r}\mathrm{vol}(K_{\varrho})\lim_{T^{\prime}\to\infty}\int_{{\mathscr{F}}_{T^{\prime}}}\Theta_{1,C}(g_{\tau^{\prime}},\varphi_{1,\mu,j}^{-},\varrho)$
$\displaystyle\times\int^{\mathrm{reg}}_{\Gamma(L)\backslash\mathbb{H}}v^{r}f_{\mu}(\tau)\theta_{0}(g_{\tau^{\prime}},g^{\Delta},\widetilde{\mathrm{RC}}_{r}(\varphi_{0,\mu,j}^{(1,-1)}-(-1)^{r}\varphi_{0,\mu,j}^{(-1,1)}))d\mu(\tau)d\mu(\tau^{\prime}).$

Now with Lemma 4.3, we obtain

$\displaystyle\frac{c_{\mu,j}}{2^{2r_{0}-r+1}}$
$\displaystyle=\mathrm{vol}(K_{\varrho})\lim_{T^{\prime}\to\infty}\int_{{\mathscr{F}}_{T^{\prime}}}(v^{\prime})^{-1/2}\Theta_{1,C}(g_{\tau^{\prime}},\varphi_{1,\mu,j}^{-},\varrho)G_{\mu,j}(\tau^{\prime})d\mu(\tau^{\prime}),$

where $G_{\mu,j}$ is a weakly holomorphic modular form of weight $-1$ defined
by

$G_{\mu,j}(\tau^{\prime}):=\sqrt{v^{\prime}}\sqrt{D}^{-3r}\mathrm{RC}^{{}^{\prime},\Delta}_{r_{0},(-r+1/2,r-2r_{0}+1/2)}\left(\theta_{D}(g_{\tau_{2}^{\prime}},\varphi^{r-2r_{0}}_{D,\mu,j})I^{M}(\tau_{1}^{\prime},f_{\mu},\varphi_{00,\mu,j})\right)\mid_{\tau_{1}^{\prime}=\tau_{2}^{\prime}=\tau^{\prime}},$

and has rational Fourier coefficient at the cusp $\infty$ by Proposition 4.11.
As $\varphi_{\mu}$ is $\operatorname{SL}_{2}(\hat{\mathbb{Z}})$-invariant, the
function $(v^{\prime})^{-1/2}\sum_{j\in
J}\Theta_{1,C}(g_{\tau^{\prime}},\varphi_{1,\mu,j}^{-},\varrho)G_{\mu,j}(\tau^{\prime})$
is $\operatorname{SL}_{2}(\mathbb{Z})$-invariant in $\tau^{\prime}$. Applying
Proposition 2.8 and Stokes’ Theorem, we then have

$\displaystyle\frac{c_{\mu}}{2^{2r_{0}-r+1}}$
$\displaystyle=\mathrm{vol}(K_{\varrho})\lim_{T^{\prime}\to\infty}\int_{{\mathscr{F}}_{T^{\prime}}}\sum_{j\in
J}L_{\tau^{\prime}}\left(\sqrt{v^{\prime}}\tilde{\Theta}_{1,C}(g_{\tau^{\prime}},\varphi_{1,\mu,j}^{-},\varrho)\right)G_{\mu,j}(\tau^{\prime},j)d\mu(\tau^{\prime})$
$\displaystyle=-\mathrm{vol}(K_{\varrho})\sum_{j\in
J}\mathrm{CT}\left(\sqrt{v^{\prime}}\tilde{\Theta}^{+}_{1,C}(g_{\tau^{\prime}},\varphi_{1,\mu,j}^{-},\varrho)G_{\mu,j}(\tau^{\prime},j)\right)\in\mathbb{Q}.$

This finishes the proof. ∎

##  5\. Proofs of Theorems

In this section, we will prove Theorem 1.2. First, we state and prove the case
for $\mathrm{O}(2,2)$.

######  Theorem 5.1.

Let $F$ be a real quadratic field and $W=W_{\alpha}$ an $F$-quadratic space
and $W_{{\alpha^{\vee}}}$ its neighborhood quadratic space as in section 2.4.
Suppose $\alpha_{1}<0<\alpha_{2}$. For $r\in\mathbb{N}_{0}$ and a lattice
$L\subset W_{\mathbb{Q}}$, suppose

$f=\sum_{m\in\mathbb{Q},~{}\mu\in
L^{\vee}/L}c(m,\mu)q^{m}\mathfrak{e}_{\mu}\in M_{-2r,\rho_{L}}^{!}$

is a weakly holomorphic modular form with rational Fourier coefficients.
Furthermore, suppose it has vanishing constant term when $r=0$. Then there
exists $\kappa,M\in\mathbb{N}$ depending on $f$ such that

(5.1)
$\begin{split}\kappa&\left(\Phi^{r}_{f}(Z(W_{\alpha}))-(-1)^{r}\Phi^{r}_{f}(Z(W_{{\alpha^{\vee}}}))\right)\\\
&=-\frac{\deg(Z(W))}{\Lambda(0,\chi)}\sum_{m>0,~{}\mu\in
L^{\vee}/L}c(-m,\mu)m^{r}\sum_{\begin{subarray}{c}\lambda\in F^{\times}\cap
M^{-1}{\mathcal{O}}\\\ \lambda\gg 0\\\
\operatorname{Tr}(\lambda)=m\end{subarray}}P_{r}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)\log\left|\frac{\beta(\lambda,\phi_{\mu})}{\beta(\lambda,\phi_{\mu})^{\prime}}\right|,\end{split}$

where $\beta(t,\phi_{f})\in F^{\times}$ is non-zero which has the property

(5.2)
$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}(\beta(t,\phi_{f})/\beta(t,\phi_{f})^{\prime})=\begin{cases}\tilde{W}_{t}(\phi_{f}),&\hbox{if
}\mathrm{Diff}(W,t)=\\{\mathfrak{p}\\},\\\ -\tilde{W}_{t}(\phi_{f}),&\hbox{if
}\mathrm{Diff}(W,t)=\\{\mathfrak{p}^{\prime}\\},\\\ 0,&\hbox{otherwise.
}\end{cases}$

###### Proof.

By the Siegel-Weil formula in (2.57), we have

$\Phi^{r}_{f}(Z(W))=\frac{C^{\prime}}{(-4\pi)^{r}}\cdot\int^{\mathrm{reg}}_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}v^{r}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)R^{r}_{\tau}E^{*}(g_{\tau}^{\Delta},\phi_{\mu}^{({\mathrm{sgn}}(\alpha),-{\mathrm{sgn}}(\alpha))})d\mu(\tau)$

with $C^{\prime}:=\deg(Z(W))/(2\Lambda(0,\chi))\in\mathbb{Q}^{\times}$. For
each $\mu\in L^{\vee}/L$, let $\varphi_{\mu}\in{\mathcal{S}}(\hat{V})^{(G\cdot
T^{\Delta})(\hat{\mathbb{Z}})}$ be a matching section of $\phi_{\mu}$ as in
Theorem 3.3. Then we can apply Proposition 4.2 to obtain

$\displaystyle\Phi^{r}_{f}$
$\displaystyle(Z(W_{{\alpha^{\vee}}}))-(-1)^{r}\Phi^{r}_{f}(Z(W_{\alpha}))$
$\displaystyle=\frac{C^{\prime}}{(-4\pi)^{r}}\cdot\int^{\mathrm{reg}}_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)R^{r}_{\tau}\left(E^{*}(g_{\tau}^{\Delta},\phi_{\mu}^{(1,-1)})-(-1)^{r}E^{*}(g_{\tau}^{\Delta},\phi_{\mu}^{(-1,1)})\right)d\mu(\tau)$
$\displaystyle=-{C^{\prime}}\cdot\frac{\pi}{3}\int^{\mathrm{reg}}_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)L_{\tau}(\mathrm{RC}_{r}\tilde{\mathcal{I}})(g_{\tau}^{\Delta},\tilde{\varrho}_{C},\varphi^{(1,1)}_{\mu})d\mu(\tau)$
$\displaystyle+\frac{C^{\prime}2\log\varepsilon_{\varrho}}{[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma(N)]}\sum_{\mu\in
L^{\vee}/L}\frac{\pi}{3}\int^{\mathrm{reg}}_{\Gamma(N)\backslash\mathbb{H}}f_{\mu}(\tau)\widetilde{\mathrm{RC}}_{r}\mathcal{I}_{f}(g_{\tau}^{\Delta},\varrho,\varphi^{(1,-1)}_{\mu}-(-1)^{r}\varphi^{(-1,1)}_{\mu})d\mu(\tau)$
$\displaystyle=-{C^{\prime}}\cdot\frac{\pi}{3}\int^{\mathrm{reg}}_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)L_{\tau}(\mathrm{RC}_{r}\tilde{\mathcal{I}})(g_{\tau}^{\Delta},\tilde{\varrho}_{C},\varphi^{(1,1)}_{\mu})d\mu(\tau)+{2C^{\prime}\sqrt{D}^{r}\log\varepsilon_{\varrho}\sum_{\mu\in
L^{\vee}/L}c_{\mu}(f)}.$

By Proposition 4.11, we know that $c_{\mu}(f)\in\mathbb{Q}$ for all $\mu\in
L^{\vee}/L$. For the other term, we can apply Stokes’ theorem to obtain

$\displaystyle{C^{\prime}}$
$\displaystyle\cdot\frac{\pi}{3}\lim_{T\to\infty}\int_{{\mathscr{F}}_{T}}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)L_{\tau}(\mathrm{RC}_{r}\tilde{\mathcal{I}})(g_{\tau}^{\Delta},\tilde{\varrho}_{C},\varphi^{(1,1)}_{\mu})d\mu(\tau)$
$\displaystyle={C^{\prime}}\cdot\frac{\pi}{3}\lim_{v\to\infty}\int_{0}^{1}\sum_{\mu\in
L^{\vee}/L}f_{\mu}(\tau)(\mathrm{RC}_{r}\tilde{\mathcal{I}})(g_{\tau}^{\Delta},\tilde{\varrho}_{C},\varphi^{(1,1)}_{\mu})du$
$\displaystyle=C^{\prime}\sum_{m>0,~{}\mu\in
L^{\vee}/L}c(-m,\mu)\sum_{\lambda\in F^{\times},~{}\lambda\gg
0,~{}\operatorname{Tr}(\lambda)=m}m^{r}P_{r}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)\tilde{c}_{\lambda}(\varphi_{\mu},\varrho)$

For the last step, we have applied Remark 4.8 to replace $\tilde{\mathcal{I}}$
with $\mathcal{I}^{0}+\mathcal{I}^{+}$, used Lemma 4.5 to see that
$\mathcal{I}^{0}$ contribute nothing, and substitute in the Fourier
coefficients of $\mathrm{RC}_{r}\mathcal{I}^{+}$ in terms of
$\tilde{c}_{\lambda}(\varphi_{\mu},\varrho)$, the $\lambda$-th Fourier
coefficient of
$\tilde{\mathcal{I}}(g_{\tau}^{\Delta},\varphi^{(1,1)}_{\mu},\tilde{\varrho}_{C})$.
Note that $m^{r}P_{r}((\lambda-\lambda^{\prime})/m)$ appears by (2.9) and is a
rational multiple of $\sqrt{D}^{r}$. As the sum above is finite, we can choose
$C$ such that $S_{C}^{c}$ contains $\mathrm{Diff}(W,t)$ for all the $t$ that
appears in this sum. Finally, the knowledge about the factorization of these
coefficients in Proposition 4.7 finishes the proof. ∎

######  Corollary 5.2.

In the setting of Theorem 5.1, suppose that $Z_{f}$ does not intersect with
$Z(W)$ when $r=0$. Then there exists $\kappa\in\mathbb{N}$ and
$\gamma(\lambda,\phi_{\mu})\in F^{\times}$ such that

(5.3)
$\begin{split}\kappa\Phi^{r}_{f}(Z(W))&=-2\frac{\deg(Z(W))}{\Lambda(0,\chi)}\sum_{m>0,~{}\mu\in
L^{\vee}/L}c(-m,\mu)m^{r}\sum_{\begin{subarray}{c}\lambda\in F^{\times}\cap
M^{-1}{\mathcal{O}}\\\ \lambda\gg 0\\\
\operatorname{Tr}(\lambda)=m\end{subarray}}P_{r}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)\log|\gamma(\lambda,\phi_{\mu})|\end{split}$

and

(5.4)
$\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}(\gamma(t,\phi_{f}))=\begin{cases}\tilde{W}_{t}(\phi_{f}),&\hbox{if
}\mathrm{Diff}(W,t)=\\{\mathfrak{p}\\},\\\ 0,&\hbox{otherwise. }\end{cases}$

######  Remark 5.3.

The constant $\frac{\deg(Z(W))}{\Lambda(0,\chi)}$ can be explicitly given when
$X_{K}=X_{0}(1)^{2}$ (see [Li21b, Remark 3.6]).

###### Proof.

By Theorem 5.10 in [BEY21], we have

$\displaystyle(\Phi^{r}_{f}(Z(W_{\alpha}))$
$\displaystyle+(-1)^{r}\Phi^{r}_{f}(Z(W_{{\alpha^{\vee}}})))$
$\displaystyle=\frac{\deg(Z(W))}{\Lambda(0,\chi)}\sum_{m>0,~{}\mu\in
L^{\vee}/L}c(-m,\mu)m^{r}\sum_{\begin{subarray}{c}\lambda\in F^{\times}\\\
\lambda\gg 0\\\
\operatorname{Tr}(\lambda)=m\end{subarray}}P_{r}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)a_{\lambda}(\phi_{\mu}),$

where $a_{t}(\phi_{\mu})$ is $t$-th the Fourier coefficient of the holomorphic
part of the incoherent Eisenstein series, and given in (2.62). Adding this to
(5.1) and applying (5.2) finishes the proof. ∎

Now we can prove Theorem 1.2.

###### Proof of Theorem 1.2.

Write $\mathrm{V}=\mathrm{V}_{\circ}\oplus W_{\mathbb{Q}}$ and $L_{W}:=L\cap
W_{\mathbb{Q}},~{}L_{\circ}:=L\cap\mathrm{V}_{\circ}$. Then $L_{\circ}\oplus
L_{W}\subset L$ is a full sublattice, and we can write

$\langle
f(\tau),\overline{\Theta_{L}(\tau,Z(W))}\rangle_{L}=\langle\tilde{f}(\tau,\tau),\overline{\Theta_{L_{W}}(\tau,Z(W))}\rangle_{L_{W}}$

with
$\tilde{f}(\tau_{1},\tau_{2}):=\langle\operatorname{Tr}^{L}_{L_{\circ}\oplus
L_{W}}(f(\tau_{1})),\overline{\Theta_{L_{\circ}}(\tau_{2})}\rangle_{L_{\circ}}$.
Using (2.10), we have

$\displaystyle(4\pi)^{-r}$ $\displaystyle\langle
R^{r}_{\tau}f(\tau),\overline{\Theta_{L}(\tau,Z(W))}\rangle_{L}=(4\pi)^{-r}(R_{\tau_{1}}^{r}\langle
f(\tau_{1}),\overline{\Theta_{L}(\tau,Z(W))}\rangle_{L}\mid_{\tau_{1}=\tau}$
$\displaystyle=\langle(4\pi)^{-r}(R_{\tau_{1}}^{r}(\tilde{f})^{\Delta},\overline{\Theta_{L_{W}}(\tau,Z(W))}\rangle_{L_{W}}$
$\displaystyle=\sum_{\ell=0}^{r}c^{(r;k_{1},k_{2})}_{\ell}(4\pi)^{-r+\ell}\langle
R_{\tau}^{r-\ell}\tilde{f}_{\ell},\overline{\Theta_{L_{W}}(\tau,Z(W))}\rangle_{L_{W}},$

where $k_{1}=-2r+1-\frac{n}{2},k_{2}=\frac{n}{2}-1$ and

$\tilde{f}_{\ell}:=\mathrm{RC}_{\ell,(k_{1},k_{2})}(\tilde{f})^{\Delta}\in
M^{!}_{-2r+2\ell,L_{W}}$

has rational Fourier coefficients. Therefore, we have

(5.5)
$\Phi^{r}_{f}(Z(W))=\sum_{\ell=0}^{r}c^{(r;k_{1},k_{2})}_{\ell}\Phi^{r-\ell}_{\tilde{f}_{\ell}}(Z(W)).$

If $f$ has the Fourier expansion

$f(\tau)=\sum_{\nu\in
L^{\vee}/L,~{}n\in\mathbb{Z}+Q(\nu)}c(n,\nu)q^{n}\mathfrak{e}_{\nu},$

then the $(m,\mu)$-th Fourier coefficient of $\tilde{f}_{\ell}$, denoted by
$c_{\ell}(m,\mu)$, can be expressed as

$c_{\ell}(m,\mu)=\sum_{\lambda_{\circ}\in
L_{\circ}^{\vee}}Q_{\ell,(k_{1},k_{2})}(m-Q(\lambda_{\circ}),Q(\lambda_{\circ}))c(m-Q(\lambda_{\circ}),(\lambda_{\circ},\mu)).$

with $Q_{\ell,(k_{1},k_{2})}(X,Y)\in\mathbb{Q}[X,Y]$ defined in (2.4). In
particular when $Z(W)\cap Z_{f}=\emptyset$, we have $c_{r}(0,\mu)=0$ for all
$0\leq\ell\leq r$ and $\mu\in L^{\vee}/L$ as
$c(-Q(\lambda_{\circ}),(\lambda_{\circ},\mu))=0$ for all $\lambda_{\circ}\in
L_{\circ}^{\vee}$ and $\mu\in L_{W}^{\vee}/L_{W}$ by (2.44).

By Corollary 5.2, we can write

$\displaystyle\kappa\Phi^{r}_{f}(Z(W))=-2\frac{\deg(Z(W))}{\Lambda(0,\chi)}$
$\displaystyle\sum_{\ell=0}^{r}c^{(r;k_{1},k_{2})}_{\ell}\sum_{m>0,~{}\mu\in
L_{W}^{\vee}/L_{W}}c_{\ell}(-m,\mu)m^{r-\ell}$
$\displaystyle\times\sum_{\begin{subarray}{c}\lambda\in F^{\times}\cap
M^{-1}{\mathcal{O}}\\\ \lambda\gg 0\\\
\operatorname{Tr}(\lambda)=m\end{subarray}}P_{r-\ell}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)\log|\gamma_{\ell}(\lambda,\phi_{\mu})|$

with $\gamma_{\ell}(\lambda,\phi_{\mu})\in F^{\times}$ having factorization as
in (5.4) independent of $\ell$, and express $\Phi^{r}_{f}(Z(W))$ as in (1.6)
such that

(5.6)
$\begin{split}\kappa^{-1}\operatorname{ord}_{\mathfrak{p}}(a_{j})=-2\frac{\deg(Z(W))}{\Lambda(1,\chi)}&\sum_{\begin{subarray}{c}m>0\\\
\mu\in L_{W}^{\vee}/L_{W}\\\ \lambda_{\circ}\in
L_{\circ}^{\vee}\end{subarray}}c(-m-Q(\lambda_{\circ}),(\lambda_{\circ},\mu))\\\
&\times\sum_{\begin{subarray}{c}0\leq\ell\leq r\\\ r-\ell\equiv
j\bmod{2}\end{subarray}}c^{(r;k_{1},k_{2})}_{\ell}Q_{\ell,(k_{1},k_{2})}(-m-Q(\lambda_{\circ}),Q(\lambda_{\circ}))\\\
&\times\sum_{\begin{subarray}{c}\lambda\in F^{\times}\cap
M^{-1}{\mathcal{O}}\\\ \lambda\gg 0\\\ \operatorname{Tr}(\lambda)=m\\\
\mathrm{Diff}(W,\lambda)=\\{\mathfrak{p}\\}\end{subarray}}\frac{1}{\sqrt{D}^{j\bmod{2}}}P_{r-\ell}\left(\frac{\lambda-\lambda^{\prime}}{m}\right)\tilde{W}_{\lambda}(\phi_{\mu})\end{split}$

for all prime $\mathfrak{p}$ of $F$. ∎

Finally, we prove Theorem 1.4.

###### Proof of Theorem 1.4.

By the main result in [Li21a], we have $\tilde{\kappa}\in\mathbb{N}$ and
Galois equivariant maps $\tilde{\alpha}_{j}:T_{W}(\hat{\mathbb{Q}})\to
E^{\mathrm{ab}}$ satisfying

(5.7)
$\Phi_{f}^{r}([z_{0},h])-\Phi_{f}^{r}([z_{0},h^{\prime}])=\frac{1}{\tilde{\kappa}}\left(\log\left|\frac{\tilde{\alpha}_{1}(h)}{\tilde{\alpha}_{1}(h^{\prime})}\right|+\sqrt{D}\log\left|\frac{\tilde{\alpha}_{2}(h)}{\tilde{\alpha}_{2}(h^{\prime})}\right|\right)$

for all $h,h^{\prime}\in T_{W}(\hat{\mathbb{Q}})$. Furthermore when $n=2$, we
have $\tilde{\alpha}_{j}=1$ for $j\equiv r\bmod{2}$. Setting
$\alpha_{j}(h):=a_{j}\prod_{[z_{0},h^{\prime}]\in
Z(W)\backslash[z_{0},h]}\frac{\tilde{\alpha}_{j}(h)}{\tilde{\alpha}_{j}(h^{\prime})}$
and $\kappa:=\tilde{\kappa}|Z(W)|$ and applying equations (5.7) and (1.6)
proves the first two claims. Combining with Corollary 2.4, we see that
Conjecture 1.1 holds. ∎

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