2.1 Intrinsic data

Let F be a number field, and let $\chi = \otimes_{v} \chi_v$ be a unitary Hecke character of $F^{\times} \backslash \mathbb{A}_F^{\times}$ . Let $\pi = \otimes_{v} \pi_v$ be a unitary generic automorphic representation of $G(\mathbb{A}_F)$ with central character $\omega= \otimes_{v} \omega_v$ .

Recall that our main objective is to establish a hybrid bound for $L(1/2, \pi \times \chi)$ as stated in Theorem 1.6.

2.1.2 Types of Archimedean parameters.

Let $v \mid \infty$ . For an admissible irreducible representation $\Pi_v$ of $G(F_v)$ , we denote its associated L-factor by

\[L_v(s, \Pi_v) = \Gamma_v(s + \gamma_{1,v}) \Gamma_v(s + \gamma_{2,v}). \]

Definition 2.1. Let $ 0 \lt \varepsilon_1 \lt 1 $ be a fixed parameter. We say that $ \Pi_v $ is of type I relative to $\varepsilon_1$ if its parameters $\gamma_{1,v}$ and $\gamma_{2,v}$ satisfy

\[\varepsilon_1 |\gamma_{2,v}| \leqslant |\gamma_{1,v}| \leqslant \varepsilon_1^{-1} |\gamma_{2,v}|. \]

Otherwise, we say that $ \Pi_v $ is of type II relative to $\varepsilon_1$ . Since $\varepsilon_1$ is a fixed small number, we will simply say that $ \Pi_v $ is of type I or type II, omitting ‘relative to $\varepsilon_1$ .’

Let $C_v(\pi \otimes \chi)$ be the local conductor defined analogously to $C_v(\pi)$ . Let $\varepsilon_0\gt0$ be a fixed constant, chosen sufficiently small. Define

(2.1) \begin{equation}T_v :=\begin{cases}100 C_v(\pi \otimes \chi)^{\frac{1}{2}} & \text{if } \pi_v \otimes \chi_v \text{ is of type I}, \\\varepsilon_0^{-100}C_v(\pi \otimes \chi) & \text{if } \pi_v \otimes \chi_v \text{ is of type II},\end{cases}\end{equation}

and

(2.2) \begin{equation}\iota_v:=\begin{cases}0 & \text{if } \pi_v \otimes \chi_v \text{ is of type I}, \\1/2 & \text{if } \pi_v \otimes \chi_v \text{ is of type II}.\end{cases}\end{equation}

Remark 2.2.

1. (i) The representation $\pi_v \otimes \chi_v$ is of type I in many favorable situations, such as when $\pi_v$ is fixed while $\chi_v$ varies, as in Theorem 1.1, or when $\pi_v$ varies with a fixed central character while $\chi_v$ remains constant.

2. (ii) If $\pi_v\otimes\chi_v$ is of type II, then $C_v(\chi)\leqslant 100\varepsilon_1C_v(\pi).$

2.1.3 Ramification parameters.

For $v\in\Sigma_{F,\operatorname{fin}},$ let $r_{\chi_v}$ (respectively, $r_{\omega_v}$ ) be the conductor exponent of $\chi_v$ (respectively, $\omega_v$ ). Let $r_{\pi_v}$ be the conductor exponent of $\pi_v.$ We set

(2.3) \begin{equation}m_v:=\max\{r_{\pi_v},n_v+r_{\omega_v}\}\geqslant n_v:=r_{\chi_v}.\end{equation}

This choice of $m_v$ is adopted from [Reference Blomer and HarcosBH14] and offers simplifications in certain local calculations when $\omega_v$ is nontrivial.

Define $\mathfrak{Q}=\prod_{v\lt\infty}\mathfrak{p}_v^{n_v}$ . For simplicity, we write $Q=C_{{\rm fin}}(\chi),$ $M=C_{{\rm fin}}(\pi),$ and $M'=C_{{\rm fin}}(\omega).$ Then $M'\mid M.$

2.1.5 Hecke operators.

Let $\pi=\otimes_{v}\pi_v$ be an automorphic representation of $G(\mathbb{A}_F)$ , with central character $\omega=\otimes_{v}\omega_v$ . Let $\mathfrak{n}$ be an integral ideal in F with primary decomposition $\mathfrak{n}=\prod_{v\mid\mathfrak{n}}\mathfrak{p}_v^{l_v}$ . The $\mathfrak{p}_v^{l_v}$ th Hecke operator is defined by

\begin{align*}T_{\mathfrak{p}_v^{l_v}}(\varphi):=q_v^{-{l_v}/{2}}\sum_{\substack{i+j=l_v\\i\geqslant j\geqslant0}}\int_{K_v{\operatorname{diag}}(\varpi_v^{i},\varpi_v^j)K_v}\pi_v(y_v)\varphi\, dy_v,\quad \varphi\in \pi.\end{align*}

Let $T_{\mathfrak{n}}(\varphi) := \prod_{v \mid \mathfrak{n}} T_{\mathfrak{p}_v^{l_v}}(\varphi)$ . When $\varphi$ is right-invariant under $\prod_{v \mid \mathfrak{n}} K_v$ , it becomes an eigenform for the operators $T_{\mathfrak{n}}$ and $T_{\mathfrak{p}_v^{l_v}}$ for each $v \mid \mathfrak{n}$ , with eigenvalues $\lambda_{\pi}(\mathfrak{n})$ and $\lambda_{\pi}(\mathfrak{p}_v^{l_v})$ , respectively. Moreover, we have $\lambda_{\pi}(\mathfrak{n}) = \prod_{v \mid\mathfrak{n}}\lambda_{\pi}(\mathfrak{p}_v^{l_v})$ .

2.1.6 Hecke relations.

Let $v\lt\infty$ , $l_v\geqslant 0$ and $l_v'\geqslant 0$ . We have

(2.4) \begin{equation}\overline{\lambda_{\pi}(\mathfrak{p}_v^{l_v})}=\omega_v^{-1}(\varpi_v^{l_v})\cdot \lambda_{\pi}(\mathfrak{p}_v^{l_v}),\end{equation}

and the Hecke relation

(2.5) \begin{equation}\lambda_{\pi}(\mathfrak{p}_v^{l_v})\lambda_{\pi}(\mathfrak{p}_v^{l_v'})=\sum_{j=0}^{\min\{l_v,l_v'\}}\omega_v(\varpi_v^{j})\lambda_{\pi}(\mathfrak{p}_v^{l_v+l_v'-2j}).\end{equation}

2.2 Construction of test functions

We shall construct a test function $f=\otimes_{v}f_v$ on $G(\mathbb{A}_F)$ in the following way.

• – At Archimedean places $v\mid\infty$ , we construct $f_v$ according to the types of $\pi_{v}\otimes \chi_{v}$ (see § 2.1.2).

  • * If $\pi_v \otimes \chi_v$ is of type I, the function $f_v$ will be constructed in § 2.2.1 based on Nelson’s work [Reference NelsonNel23] (see §§ 1.5.2 and 14 on p. 80) following the approach in [Reference Nelson and VenkateshNV21, § 1.10] (see also [Reference NelsonNel21, Part 2]). It will sieve out microlocal lifted vectors.

  • * If $\pi_v \otimes \chi_v$ is of type II, the function $f_v$ will be constructed in § 2.2.2 following the work of Michel and Venkatesh [Reference Michel and VenkateshMV10, § 3.6]. It will sieve out analytic newvectors.

Precise constructions will be given in §§ 2.2.1 and 2.2.2.

• – At the auxiliary fixed finite places $ v \in \{v_0, v_1\} $ constructed in § 2.1.4, we define $ f_v $ to simplify the regularized relative trace formula:

  • * The local test function $ f_{v_0} $ defined in § 2.2.3 simplifies the spectral side by annihilating the residues from the continuous spectrum during the regularization.

  • * The local test function $ f_{v_1} $ defined in § 2.2.4 simplifies the geometric side by annihilating certain orbital integrals.

Detailed discussions will be provided in §§ 3.4 and 2.4.2.

• – At ramified finite places, we construct the local test functions via a double average over unipotent translations weighted by characters (see § 2.2.6). This step is essential in handling the level aspect and also simplifies the regularized relative trace formula (see § 2.4.1).

• – At the remaining places, let $\mathfrak{n}$ be an integral ideal such that $\pi_v$ and $\chi_v$ are unramified for all $v \mid \mathfrak{n}$ , with $v_0 \nmid \mathfrak{n}$ and $v_1 \nmid \mathfrak{n}$ . Define $f_v$ as follows:

  • * For $v \mid \mathfrak{n}$ , we introduce normalized Hecke operators (following Duke, Friedlander, and Iwaniec [Reference Duke, Friedlander and IwaniecDFI02]) to facilitate further arithmetic amplification. The associated local test functions $f_v$ , constructed in § 2.2.5, play a crucial role in § 7 to prove Theorem 1.6.

  • * For $v \nmid \mathfrak{n}$ , we take $f_v$ to be the standard spherical vector.

2.2.1 Construction of ${f}_{\infty}$ : the type I case

Let $v\mid\infty.$ Suppose that $\pi_v\otimes\chi_v$ is of type I. Let $T_v=100 C_v(\pi \otimes \chi)^{\frac{1}{2}}$ be the parameter defined by (2.1).

By [Reference NelsonNel23, § 14] and [Reference Nelson and VenkateshNV21, § 2.5], there exists a function $\tilde{f}_{v} \in C_c^{\infty}(G(F_v))$ such that

(2.6) \begin{equation} \textrm{supp} \tilde{f}_{v} \subseteq \{ g \in G(F_v) : g = I_{n+1} + O(T_v^{-\varepsilon}),\ \operatorname{Ad}^*(g)\tau = \tau + O(T_v^{-\frac{1}{2} + \varepsilon}) \},\end{equation}

where the operator $\pi_{v}(\tilde{f}_{v})$ is approximately a rank-one projector with range spanned by a unit vector microlocalized at some chosen element $\tau_v$ in the coadjoint orbit of $\pi_v \otimes \chi_v$ .

An explicit construction of $\tilde{f}_{v}$ can be found in [Reference NelsonNel23, § 14]. Here, we summarize some basic properties of $\tilde{f}_{v}$ as follows. For $g \in G(F_v)$ , we may write

\[g = \begin{pmatrix}a & \;\;b \\c & \;\;d\end{pmatrix} \in G(F_v), \quad g^{-1} = \begin{pmatrix}a' &\;\; b' \\c' &\;\; d'\end{pmatrix} \in G(F_v). \]

Define

\[d_{v}(g) :=\begin{cases}\min\big\{ 1, |d^{-1}b|_v + |d^{-1}c|_v + |d'^{-1}b'|_v + |d'^{-1}c'|_v \big\} & \text{if $dd' \neq 0,$} \\1 & \text{if $dd' = 0$}.\end{cases} \]

As a consequence of [Reference NelsonNel21, (14.13)] and [Reference NelsonNel23, Theorem 15.1], along with the support constraint (2.6), we obtain

(2.7) \begin{equation}|\tilde{f}_v(g)| \ll T_v^{1+\varepsilon} \cdot\mathbf{1}_{|\operatorname{Ad}^*(g)\tau - \tau| \ll T_v^{-1/2+\varepsilon}} \cdot \mathbf{1}_{|g - I_2| \ll T_v^{-\varepsilon}} \cdot \min\bigg\{\! 1, \frac{T_v^{-\frac{1}{2}+\varepsilon}}{d_v(g)} \bigg\}.\end{equation}

We thus define the local test function $f_v$ by

(2.8) \begin{equation}f_{v}(g) := (f_{v}^{\dagger} * (f_{v}^{\dagger})^*)(g),\end{equation}

where $v \mid \infty$ , $g \in G(F_v)$ , and

(2.9) \begin{equation}f_{v}^{\dagger}(g) := \chi_v(\det g) \int_{F_v^{\times}} \tilde{f}_{v}(zg)\omega_{v}(z) \, d^{\times}z.\end{equation}

Remark 2.3. We do not recall the precise construction of $\tilde{f}_{v}$ from [Reference NelsonNel23, 14]; instead, we will only use its relevant properties, with appropriate references to the source.

2.2.2 Construction of ${f}_{\infty}$ : the type II case

Let $v\mid\infty$ . Suppose that $\pi_v\otimes\chi_v$ is of type II. Let $T_v=\varepsilon_0^{-100} C_v(\pi \otimes \chi)$ be the parameter defined by (2.1).

Let $ h_{v}^{\varepsilon_0} = (h_{1,v}, h_{2,v}, h_{3,v}) \in C_c^{\infty}(F_v^{\times} \times F_v \times F_v) $ be a triple of nonnegative Schwartz functions with compact support, satisfying the following conditions:

• – $ \int_{F_v^{\times}} h_{1,v}(a) \, d^{\times}a = 1 $ , $ h_{1,v}(a) = 0 $ unless $ |a - 1|_v \leqslant \varepsilon_0 $ , and

  \[h_{1,v}^{(j)}(a) \ll \varepsilon_0^{-j-1} \quad \text{for all } a \in F_v^{\times} \text{ and } 0 \leqslant j \leqslant 2; \]

• – $ |\int_{F_v} h_{2,v}(b) \psi_v(b)\, db| = 1 $ , and $ h_{2,v}(b) = 0 $ unless $ |b_v|_v \leqslant 10^{-3} $ ; and

• – $ \int_{F_v} h_{3,v}(c) \,dc = 1 $ , and $ h_{3,v}(c) = 0 $ unless $ |c|_v \leqslant \varepsilon_0 $ .

Let $h_{2,v}'$ be the derivative of $h_{2,v}$ . Define

(2.10) \begin{equation}\tilde{f}_v(g,h_v^{\varepsilon_0}):=T_v\chi_v(\det g)h_{1,v}(z)h_{1,v}(a)h_{2,v}'(b)h_{3,v}(T_v^{1+\varepsilon}c)\end{equation}

if $g=z\big(\!\begin{smallmatrix}a&\;\; b\\& 1\end{smallmatrix}\!\big)\big(\!\begin{smallmatrix}1& \\c &\;\; 1\end{smallmatrix}\!\big)\in G(F_v)$ , and set $\tilde{f}_v(g,h_v^{\varepsilon_0})\equiv 0$ otherwise. Define

(2.11) \begin{equation}f_{v}^{\dagger}(g):=\chi_v(\det g)T_v^{-\frac{1}{4}}\int_{F_v^{\times}}\tilde{f}_v(zg,h_{v}^{\varepsilon_0})\omega_{v}(z) \, d^{\times}z.\end{equation}

By the Bruhat decomposition, the set

\begin{align*}\bigg\{g=z\begin{pmatrix}a& \;\;b\\& 1\end{pmatrix}\begin{pmatrix}1& \\c &\;\; 1\end{pmatrix}\in G(F_v):\ a, z\in F_v^{\times},\ b, c\in F_v\!\bigg\}\end{align*}

is open in $G(F_v)$ . Consequently, for sufficiently small $\varepsilon_0$ , the function $f_{v}^{\dagger}(g)$ is smooth and supported in a small neighborhood of the identity matrix $I_2$ .

The local test function is then defined by

(2.12) \begin{equation}f_v(g)=(f_{v}^{\dagger}*(f_{v}^{\dagger})^*)(g),\quad g\in G(F_v).\end{equation}

2.2.3 Non-Archimedean test function $f_{v_0}$

Let $v_0$ be the fixed unramified place as in § 2.1.4. For $v=v_0$ , we define

(2.13) \begin{equation}\tilde{h}_v:=\sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\mathbf{1}_{u_{\beta}K_v}+\mathbf{1}_{\mathbf{d}K_v},\end{equation}

where $u_{\beta}=\big(\!\begin{smallmatrix}1&\;\; \beta\varpi_v^{-1}\\&1\end{smallmatrix}\!\big)$ , and $\mathbf{d}:=\operatorname{diag}(\varpi_v^{-1},\varpi_v).$ We define the local test function by

(2.14) \begin{equation}f_v(g):=(f_{v}^{\dagger}*(f_{v}^{\dagger})^*)(g),\quad g\in G(F_v),\end{equation}

where

(2.15) \begin{equation}f_{v}^{\dagger}(g):=\int_{F_v^{\times}}\tilde{h}_v(gz)\omega_v(z) \, d^{\times}z.\end{equation}

The test function $f_v(g_v)$ is introduced to eliminate contributions from the residues of period integrals on the spectral side when shifting contours during the regularization of the relative trace formula (see Proposition 3.1).

2.2.4 Non-Archimedean test function $f_{v_1}$

Let $v=v_1$ be the fixed unramified place as in § 2.1.4. We define the test function by

(2.16) \begin{equation}f_v(g_v)=f_{v}^{\dagger}(g):=\begin{cases}\displaystyle\frac{\omega_v(z_v)^{-1}\omega_v(E_{2,2}(k_v))^{-1}}{\operatorname{Vol}(\overline{K_{v}[1]})}&\ \text{if $g_v=z_vk_v\in Z(F_v)K_{v}[1]$,}\\0 &\ \text{otherwise,}\end{cases}\end{equation}

where $\overline{K_{0,v}[1]}$ is the image of $K_{0,v}[1]$ in $\overline{G}(F_v),$ and $E_{2,2}(k_v)$ is the (2,2)th entry of $k_v\in K_{0,v}[1]$ . We have $f_{v}(g)=(f_{v}^{\dagger}*(f_{v}^{\dagger})^*)(g)$ .

2.2.5 Non-Archimedean test function $f_v$ at $v\mid\mathfrak{n}$

Let $\mathfrak{n}$ be an integral ideal such that $\pi_v$ and $\chi_v$ are unramified for all $v \mid \mathfrak{n}$ , and $v_0 \nmid \mathfrak{n}$ and $v_1 \nmid \mathfrak{n}$ . For $v\mid\mathfrak{n}$ , we define

(2.17) \begin{equation}f_v(g):=q_v^{-({e_v(\mathfrak{n})})/{2}}\sum_{\substack{i+j=e_v(\mathfrak{n})\\i\geqslant j\geqslant0}}\int_{F_v^{\times}}\mathbf{1}_{K_v{\operatorname{diag}}(\varpi_v^i,\varpi_v^j)K_v}(zg)\omega_v(z) \, d^{\times}z,\quad g\in G(F_v).\end{equation}

2.2.6 Non-Archimedean test function $f_v$ : the remaining places

For $v\in\Sigma_{F,\operatorname{fin}}-\{v_0,v_1\}$ such that $v\nmid\mathfrak{n}$ , we define

(2.18) \begin{equation}f_{v}^{\dagger}(g):=\frac{1}{\operatorname{Vol}(\overline{K_{0,v}[m_v]})}\int_{F_v^{\times}}\mathbf{1}_{K_{0,v}[m_v]}(zg)\omega_v(z) \, d^{\times}z,\end{equation}

where $\overline{K_{0,v}[m_v]}$ is the image of $K_{0,v}[m_v]$ in $\overline{G}(F_v)$ . For $g\in G(F_v),$ define by

(2.19) \begin{equation}f_v(g)=\frac{1}{|\tau(\chi_v)|^{2}}\sum_{\substack{\alpha\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})\\\beta\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}}\chi_v(\alpha)\overline{\chi}_v(\beta)f_{v}^{\dagger}(g_{\alpha,\beta}),\end{equation}

where

$$\tau(\chi_v)=\sum_{\alpha\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\psi_v(\alpha\varpi_v^{-n_v})\chi_v(\alpha)$$

is the Gauss sum relative to the additive character $\psi_v$ , and

\begin{align*} g_{\alpha,\beta}:=\begin{pmatrix} 1&\;\;\alpha \varpi_v^{-n_v}\\ &1 \end{pmatrix}g\begin{pmatrix} 1&\;\;\beta \varpi_v^{-n_v}\\ &1 \end{pmatrix}\!.\end{align*}

Here, we identify $\alpha, \beta \in \mathcal{O}_v^{\times}/(1 + \varpi_v^{n_v} \mathcal{O}_v^{\times})$ with their representatives in $\mathcal{O}_v^{\times}$ to ensure that $g_{\alpha, \beta} \in G(F_v)$ .

Note that $n_v = 0$ for almost all $v \in \Sigma_{F, \mathrm{fin}}$ . Consequently, for all but finitely many $v \in \Sigma_{F, \mathrm{fin}}$ , we have $f_v=f_{v}^{\dagger}=f_{v}^{\dagger}*(f_{v}^{\dagger})^*$ , supported on $Z(F_v) K_{0,v}[m_v]$ .

By definition (2.3), we have $ m_v = 0 $ for $ v \in \{v_0, v_1\} $ and $v\mid\mathfrak{n}$ .

2.3 Decomposition of the kernel function

Let f be the test function defined in § 2.2.7. Then f defines an integral operator

(2.20) \begin{equation} R(f)\phi(g) = \int_{\overline{G}(\mathbb{A}_F)} f(g')\phi(gg') \, dg'\end{equation}

on the space $L^2([G], \omega )$ of functions on [G] that transform under $Z(\mathbb{A}_F)$ by $\omega$ and are square-integrable on $[\overline{G}]$ .

By making the change of variable $g' \mapsto g^{-1}g'$ in (2.20), and using the fact that $\phi$ is left-G(F)-invariant, we obtain

\[R(f)\phi(g) = \int_{[\overline{G}]} \operatorname{K}(g, g')\phi(g') \, dg', \]

where for $g_1, g_2 \in G(\mathbb{A}_F)$ ,

(2.21) \begin{equation}\operatorname{K}(g_1, g_2) := \sum_{\gamma \in \overline{G}(F)} f(g_1^{-1} \gamma g_2)\end{equation}

is the kernel function of R(f). Since f has compact support modulo the center, the sum in (2.21) converges absolutely.

It is well-known that $L^2([G],\omega)$ decomposes into the direct sum of the space $L_0^2([G],\omega)$ of cusp forms and spaces $L_{{\rm Eis}}^2([G],\omega)$ and $L_{{\rm Res}}^2([G],\omega)$ defined in terms of Eisenstein series and residues of Eisenstein series, respectively.

Let R be the right regular representation of $G(\mathbb{A}_F)$ on $L^2([G],\omega)$ . The spectral decomposition of the $L^2([G],\omega)$ (see [Reference Gelbart and JacquetGJ79]) gives that

(2.22) \begin{equation}R=\bigoplus_{\text{$\sigma$ cuspidal}}\sigma \oplus \int_{i\mathbb{R}}\bigoplus_{\eta\in\widehat{F^{\times}\backslash \mathbb{A}_F^{(1)}}}\sigma_{\lambda,\eta}\frac{d\lambda}{4\pi i} \oplus \bigoplus_{\substack{\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}\\ \eta^2=\omega}}\eta\circ \det,\end{equation}

where $\sigma_{\lambda,\eta}=\operatorname{Ind}(\eta|\cdot|^{\lambda},\eta^{-1}\omega|\cdot|^{-\lambda})$ denotes the normalized parabolic induction of the characters $\eta|\cdot|^{\lambda}$ and $\eta^{-1}\omega|\cdot|^{-\lambda}$ .

Then the kernel function $\operatorname{K}(g_1,g_2)$ decomposes as the pre-trace formula:

(2.23) \begin{equation} \operatorname{K}_0(g_1,g_2)+\operatorname{K}_{{\rm ER}}(g_1,g_2)=\operatorname{K}(g_1,g_2)=\sum_{\gamma\in \overline{G}(F)}f(g_1^{-1}\gamma g_2),\end{equation}

where $\operatorname{K}_0$ is the contribution from the cuspidal spectrum, and $\operatorname{K}_{{\rm ER}}(g_1,g_2)$ is the contribution from the noncuspidal spectrum. The subscript ‘ $\operatorname{ER}$ ’ refers to Eisenstein series and their residues. Explicit expansions of $\operatorname{K}_0(g_1,g_2)$ and $\operatorname{K}_{{\rm ER}}(g_1,g_2)$ will be given in § 3.2.

Substituting the Bruhat decomposition

\begin{align*}\overline{G}(F)=B_0(F)\bigsqcup B_0(F)wN(F),\quad w=\begin{pmatrix}&\;\; 1\\1\end{pmatrix}\end{align*}

into (2.21), we obtain $\operatorname{K}(g_1,g_2)=\operatorname{K}_{{\rm Small}}(g_1,g_2)+\operatorname{K}_{{\rm Big}}(g_1, g_2),$ where

\begin{align*} \operatorname{K}_{{\rm Small}}(g_1,g_2)=\sum_{\gamma\in B_0(F)}f(g_1^{-1}\gamma g_2),\quad \operatorname{K}_{{\rm Big}}(g_1,g_2)=\sum_{\gamma\in B_0(F)wN(F)}f(g_1^{-1}\gamma g_2). \end{align*}

Let $\mathcal{K}(\cdot,\cdot)\in \{\operatorname{K}(\cdot,\cdot), \operatorname{K}_0(\cdot,\cdot), \operatorname{K}_{{\rm ER}}(\cdot,\cdot), \operatorname{K}_{{\rm Small}}(\cdot,\cdot),\operatorname{K}_{{\rm Big}}(\cdot,\cdot)\}.$ Define

\begin{align*}\mathcal{F}_0\mathcal{F}_1\mathcal{K}(g_1,g_2)&:=\int_{[N]}\mathcal{K}(g_1,u_2g_2) \, du_2,\quad \mathcal{F}_1\mathcal{F}_0\mathcal{K}(g_1,g_2):=\int_{[N]}\mathcal{K}(u_1g_1,g_2) \, du_1,\\\mathcal{F}_1\mathcal{F}_1\mathcal{K}(g_1,g_2)&:=\int_{[N]}\int_{[N]}\mathcal{K}(u_1g_1,u_2g_2) \, du_2 \, du_1,\\\mathcal{F}_2\mathcal{F}_2\mathcal{K}(g_1,g_2)&:=\sum_{\alpha\in A(F)}\sum_{\beta\in A(F)}\int_{[N]}\int_{[N]}\mathcal{K}(u_1\alpha g_1,u_2\beta g_2)\theta(u_1)\overline{\theta}(u_2) \, du_2 \, du_1.\end{align*}

Using the Poisson summation twice yields

(2.24) \begin{align}\mathcal{F}_2\mathcal{F}_2\mathcal{K}(g_1,g_2)=\mathcal{K}(g_1,g_2)-\mathcal{F}_0\mathcal{F}_1\mathcal{K}(g_1,g_2)-\mathcal{F}_1\mathcal{F}_0\mathcal{K}(g_1,g_2)+\mathcal{F}_1\mathcal{F}_1\mathcal{K}(g_1,g_2).\end{align}

Since f has compact support modulo the center, (2.24) converges absolutely.

Lemma 2.4. For $x, y\in A(\mathbb{A}_F)$ , we have

\begin{align*}\mathcal{F}_0\mathcal{F}_1\operatorname{K}_{{\rm Big}}(x,y)=\mathcal{F}_1\mathcal{F}_0\operatorname{K}_{{\rm Big}}(x,y)=\mathcal{F}_1\mathcal{F}_1\operatorname{K}_{{\rm Big}}(x,y)\equiv 0.\end{align*}

Proof. By the definition of $\operatorname{K}_{{\rm Big}}(\cdot,\cdot)$ ,

\begin{align*}\mathcal{F}_0\mathcal{F}_1\operatorname{K}_{{\rm Big}}(x,y)=\sum_{b\in B_0(F)}\int_{N(\mathbb{A})}f(x^{-1}bwu_2y) \, du_2.\end{align*}

Now let $v\mid\mathfrak{Q}$ , so that $n_v\geqslant 1.$ By the definition of $f_v$ , we have

(2.25) \begin{align}&|\tau(\chi_v)|^2 \int_{N(F_v)} f_v(x_{v}^{-1} b w u_{v} y_{v}) \,du_{v}\nonumber\\&\quad = \sum_{\alpha, \beta \in \mathcal{O}_v^{\times} / (1 + \varpi_v^{n_v} \mathcal{O}_v^{\times})}\chi_v(\alpha) \overline{\chi}_v(\beta)\int_{N(F_v)}f_v^{\dagger} \bigg(\!\!\begin{pmatrix} 1 &\;\; \alpha \varpi_v^{n_v} \\ & 1\end{pmatrix}x_{v}^{-1} b w u_{v} y_{v}\begin{pmatrix} 1 & \;\;\beta \varpi_v^{n_v} \\ & 1\end{pmatrix}\!\!\bigg) \, du_{v},\end{align}

where the local test function $f_{v}^{\dagger}(\cdot)$ is defined by (2.18).

Since $y_{v} \in A(F_v)$ , after the change of variables

\[u_{v} \mapsto u_{v} y_{v}\begin{pmatrix} 1 &\;\; -\beta q_v^{n_v} \\ & 1\end{pmatrix}y_{v}^{-1}, \]

the right-hand side of (2.25) becomes

\begin{align*}\sum_{\beta \in \mathcal{O}_v^{\times} / (1 + \varpi_v^{n_v} \mathcal{O}_v^{\times})}\overline{\chi}_v(\beta)\sum_{\alpha \in \mathcal{O}_v^{\times} / (1 + \varpi_v^{n_v} \mathcal{O}_v^{\times})}\chi_v(\alpha)\int f_v^{\dagger} \bigg(\!\!\begin{pmatrix} 1 &\;\; \alpha q_v^{n_v} \\ & 1\end{pmatrix}x_{v}^{-1} b w u_{v} y_{v}\!\bigg) \, du_{v},\end{align*}

which vanishes as a consequence of the character orthogonality

\[\sum_{\beta \in \mathcal{O}_v^{\times} / (1 + \varpi_v^{n_v} \mathcal{O}_v^{\times})}\overline{\chi}_v(\beta) = 0. \]

Hence (2.25) vanishes, which implies that $\mathcal{F}_0\mathcal{F}_1\operatorname{K}_{{\rm Big}}(x,y)=0.$ Likewise, we have $\mathcal{F}_1\mathcal{F}_0\operatorname{K}_{{\rm Big}}(x,y)=\mathcal{F}_1\mathcal{F}_1\operatorname{K}_{{\rm Big}}(x,y)=0.$

2.4 The regularized relative trace formula

Let $\textbf{s}=(s_1, s_2)\in\mathbb{C}^2,$ with ${\rm Re}(s_1)\ggg 1$ and ${\rm Re}(s_2)\ggg 1.$ Consider the integral

(2.26) \begin{equation}\int_{[A]}\int_{[A]}\mathcal{F}_2\mathcal{F}_2\operatorname{K}(x,y)|\det x|^{s_1}|\det y|^{s_2}\chi(x)\overline{\chi}(y) \, d^{\times}x \, d^{\times}y.\end{equation}

It follows from [Reference YangYan23, § 6.2] that the above integral converges absolutely when ${\rm Re}(s_1)$ and ${\rm Re}(s_2)$ are sufficiently large.

We denote by $J_{\textrm{Spec}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ the integral (2.26) obtained by substituting the spectral expansion $\operatorname{K}(g_1,g_2)= \operatorname{K}_0(g_1,g_2)+\operatorname{K}_{{\rm ER}}(g_1,g_2)$ into it. Likewise, we denote by $J_{{\rm Geo}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ the integral obtained by substituting the geometric expansion (2.21) into (2.26). As a consequence, we have the coarse relative trace formula

(2.27) \begin{equation}J_{\textrm{Spec}}^{{\rm Reg}}(f,\mathbf{s},\chi)=J_{{\rm Geo}}^{{\rm Reg}}(f,\mathbf{s},\chi),\end{equation}

where ${\rm Re}(s_1)$ and ${\rm Re}(s_2)$ are sufficiently large.

For convenience, we refer to $J_{\textrm{Spec}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ as the spectral side of the relative trace formula and $J_{{\rm Geo}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ as the geometric side of the relative trace formula.

2.4.1 The spectral side.

Let $\textbf{s}=(s_1, s_2)\in\mathbb{C}^2,$ with ${\rm Re}(s_1)\ggg 1$ and ${\rm Re}(s_2)\ggg 1.$ Note that $\operatorname{K}_0(x,y)=\mathcal{F}_2\mathcal{F}_2\operatorname{K}_0(x,y).$ By the definition (2.26), we have

(2.28) \begin{equation}J_{\textrm{Spec}}^{{\rm Reg}}(f,\textbf{s},\chi)=J_{0}^{{\rm Reg}}(f,\textbf{s},\chi)+J_{{\rm Eis}}^{{\rm Reg}}(f,\textbf{s},\chi),\end{equation}

where

\begin{align*}J_{0}^{{\rm Reg}}(f,\textbf{s},\chi)&:=\int_{[A]}\int_{[A]}\operatorname{K}_0(x,y)|\det x|^{s_1}|\det y|^{s_2}\chi(x)\overline{\chi}(y) \, d^{\times}x \, d^{\times}y,\\J_{{\rm Eis}}^{{\rm Reg}}(f,\textbf{s},\chi)&:=\int_{[A]}\int_{[A]}\mathcal{F}_2\mathcal{F}_2\operatorname{K}_{{\rm ER}}(x,y)|\det x|^{s_1}|\det y|^{s_2}\chi(x)\overline{\chi}(y) \, d^{\times}x \, d^{\times}y.\end{align*}

In § 3 we will show that the spectral side $J_{\textrm{Spec}}^{{\rm Reg}}(f,\textbf{s},\chi)$ admits a holomorphic continuation $J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)$ to the region $-1/4\lt{\rm Re}(s_1),\ {\rm Re}(s_2)\lt1/4$ .

2.4.2 The geometric side.

Let ${\rm Re}(s_1)\ggg 1$ and ${\rm Re}(s_2)\ggg 1$ . By (2.24), Lemma 2.4, and the Bruhat decomposition $\operatorname{K}(x,y)=\operatorname{K}_{{\rm Small}}(x,y)+\operatorname{K}_{{\rm Big}}(x,y),$ we have

(2.29) \begin{equation}J_{{\rm Geo}}^{{\rm Reg}}(f,\textbf{s},\chi)=J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\textbf{s},\chi)+J^{{\rm Reg}}_{{\rm Geo},{\rm Big}}(f,\textbf{s},\chi),\end{equation}

where

\begin{align*}J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\textbf{s},\chi)&:=\int_{[A]}\int_{[A]}\mathcal{F}_2\mathcal{F}_2\operatorname{K}_{{\rm Small}}(x,y)|\det x|^{s_1}|\det y|^{s_2}\chi(x)\overline{\chi}(y) \, d^{\times}x \, d^{\times}y,\\J^{{\rm Reg}}_{{\rm Geo},{\rm Big}}(f,\textbf{s},\chi)&:=\int_{[A]}\int_{[A]}\operatorname{K}_{{\rm Big}}(x,y)|\det x|^{s_1}|\det y|^{s_2}\chi(x)\overline{\chi}(y) \, d^{\times}x \, d^{\times}y.\end{align*}

In view of the construction of $f_{v_1}$ in (2.16), we deduce from [Reference YangYan23, § 5] (or from the proof in [Reference Ramakrishnan and RogawskiRR05]) that

(2.30) \begin{equation}J^{{\rm Reg}}_{{\rm Geo},{\rm Big}}(f,\textbf{s},\chi)=J^{{\rm Reg}}_{{\rm Geo},{\rm Dual}}(f,\textbf{s},\chi)+J^{{\rm Reg},II}_{{\rm Geo},{\rm Big}}(f,\textbf{s},\chi),\end{equation}

where $J_{{\rm Geo},{\rm Dual}}^{{\rm Big}}(f,\textbf{s},\chi)$ is defined by

(2.31) \begin{equation}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F^{\times}}f\bigg(\!\!\begin{pmatrix} 1\\ x&\;\;1\end{pmatrix}\begin{pmatrix} y\\ &\;\;1\end{pmatrix}\!\!\bigg)|x|^{s_1+s_2}|y|^{s_2}\overline{\chi}(y) \, d^{\times}y \, d^{\times}x,\end{equation}

and the regular orbital $J^{{\rm Reg},II}_{{\rm Geo},{\rm Big}}(f,\textbf{s},\chi)$ is defined by

(2.32) \begin{equation}\sum_{t\in F-\{0,1\}}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F^{\times}}f\bigg(\!\!\begin{pmatrix} y&\;\;x^{-1}t\\ xy&1\end{pmatrix}\!\!\bigg)|x|^{s_1+s_2}|y|^{s_2}\overline{\chi}(y) \, d^{\times}y \, d^{\times}x.\end{equation}

Remark 2.5. Without the component $f_{v_1}$ , the decomposition (2.29) may include additional terms. For instance, if $\pi$ and $\chi$ are both unramified at all finite places (i.e., $m_v = 0$ for all $v \lt \infty$ , e.g., $\chi=|\cdot|^{it}$ ), then there are two extra terms in (2.30) (see [Reference Wei, Yang and ZhaoWYZ24, § 2.2.3]).

Since f is a Schwartz–Bruhat function, Tate’s thesis implies that $J_{{\rm Geo},{\rm Dual}}^{{\rm Big}}(f,\mathbf{s},\chi)$ converges absolutely for ${\rm Re}(s_1 + s_2) \gt 1$ , and it represents the complete Dedekind $\zeta$ -function $\Lambda(s_1 + s_2)$ . Likewise, $J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\textbf{s},\chi)$ is a Tate integral representing $\Lambda(1+s_1 + s_2)$ . Moreover, by Theorem 5.6 in [Reference YangYan23, § 5.4], the integral $J^{{\rm Reg},II}_{{\rm Geo},{\rm Big}}(f,\mathbf{s},\chi)$ converges absolutely for all $\mathbf{s}\in\mathbb{C}^2$ . In particular, the sum over $t \in F \setminus \{0,1\}$ is finite, a property referred to as stability of the regular orbital integrals (see [Reference Feigon and WhitehouseFW09, Reference Michel and RamakrishnanMR12]).

Throughout §§ 4–6, we give detailed quantitative descriptions of the functions $J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\mathbf{s},\chi)$ , $J^{{\rm Reg}}_{{\rm Geo},{\rm Dual}}(f,\mathbf{s},\chi)$ , and $J^{{\rm Reg},II}_{{\rm Geo},{\rm Big}}(f,\mathbf{s},\chi)$ . As a result, we obtain a holomorphic continuation $J_{{\rm Geo}}^{{\rm Reg},\heartsuit}(f,\mathbf{s},\chi)$ of $J_{{\rm Geo}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ to $\mathbf{s}\in\mathbb{C}^2.$

2.4.3 The relative trace formula.

Let $J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\mathbf{s},\chi)$ and $J_{{\rm Geo}}^{{\rm Reg},\heartsuit}(f,\mathbf{s},\chi)$ be the holomorphic continuations of $J_{\textrm{Spec}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ and $J_{{\rm Geo}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ , respectively. As a consequence of (2.27), we obtain the following.

Theorem 2.6. Let $\varepsilon\gt0$ be a small constant. Let $\mathcal{C}:=\{s\in\mathbb{C}:\ |s|=\varepsilon\}$ be the circle of radius $\varepsilon$ . Let $\mathbf{s}=(s,0)$ . Then

(2.33) \begin{align}J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\mathbf{0},\chi)&=\frac{1}{2\pi i}\int_{\mathcal{C}}\frac{J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\mathbf{s},\chi)}{s} \, ds\nonumber\\&\quad +\frac{1}{2\pi i}\int_{\mathcal{C}}\frac{J^{{\rm Reg}}_{{\rm Geo},{\rm Dual}}(f,\mathbf{s},\chi)}{s} \, ds+J^{{\rm Reg},II}_{{\rm Geo},{\rm Big}}(f,\textbf{0},\chi).\end{align}

Here, $J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\mathbf{s},\chi)$ and $J^{{\rm Reg}}_{{\rm Geo},{\rm Dual}}(f,\mathbf{s},\chi)$ are understood via their meromorphic continuation, as established by Tate’s thesis.

A more precise form of the formula (2.33) will be developed in §§ 3–6.

3.1 Eisenstein series

Define $H_T: T(\mathbb{A}_F)\rightarrow\mathbb{R}$ by setting $H\big(\!\big(\begin{smallmatrix} a\\ &\;\;d\end{smallmatrix}\big)\!\big)=\log |ad^{-1}|_{\mathbb{A}_F}.$ Extend $H_T$ to a map $H: G(\mathbb{A}_F)\rightarrow\mathbb{R}$ by using the Iwasawa decomposition $G(\mathbb{A}_F)=N(\mathbb{A}_F)T(\mathbb{A}_F)K,$ and defining $H(utk)=H_T(t),$ $u\in N(\mathbb{A}_F),$ $t\in T(\mathbb{A}_F),$ and $k\in K.$

For $\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}$ and $\lambda\in \mathbb{C}$ , let $\sigma_{\lambda,\eta}=\operatorname{Ind}(\eta|\cdot|^{\lambda},\eta^{-1}\omega|\cdot|^{-\lambda})$ be the parabolic induction. Let $\phi\in\sigma_{0,\eta}$ be a section. Define the Eisenstein series

\begin{align*}E(g,\phi,\lambda)=\sum_{\delta\in B(F)\backslash G(F)}\phi(\delta g)e^{(\frac{1}{2}+\lambda)H(\delta g)},\quad {\rm Re}(\lambda)\gt1.\end{align*}

Then $E(g,\phi,\lambda)$ converges absolutely and admits a meromorphic continuation to $\lambda\in\mathbb{C}$ with possible simple poles at $\lambda\in \{0,1\}$ , as well as a functional equation.

Let f be constructed in § 2.2. For $g\in G(\mathbb{A}_F),$ define

\begin{align*}E(g,\mathcal{I}(\lambda, f)\phi,\lambda)=\sigma_{0,\eta}(f)E(g,\phi,\lambda):=\int_{\overline{G}(\mathbb{A}_F)}f(y)E(gy,\phi,\lambda) \, dy,\quad {\rm Re}(\lambda)\gt1.\end{align*}

To simplify the notation, we also use $E(g,\phi,\lambda)$ and $E(g,\mathcal{I}(\lambda, f)\phi,\lambda)$ to denote their meromorphic continuations.

3.2 Spectral expansion of the kernel functions

Let the notation be as in § 2.2. Define $\operatorname{K}_0(x, y)$ and $\operatorname{K}_{{\rm ER}}(x, y)$ by (2.23). Then, by the decomposition (2.22),

\begin{align*}\operatorname{K}_0(x, y) &= \sum_{\sigma \in \mathcal{A}_0([G], \omega)} \sum_{\phi \in \mathfrak{B}_{\sigma}} \sigma(f) \phi(x) \overline{\phi(y)},\\\operatorname{K}_{{\rm ER}}(x, y) &= \frac{1}{4\pi} \sum_{\eta \in \widehat{F^{\times} \backslash \mathbb{A}_F^{(1)}}} \int_{i\mathbb{R}}\sum_{\phi \in \mathfrak{B}_{\sigma_{0,\eta}}}E(x, \mathcal{I}(\lambda, f)\phi, \lambda)\overline{E(y, \phi, \lambda)} \, d\lambda,\end{align*}

where $\mathfrak{B}_{\sigma}$ and $\mathfrak{B}_{\sigma_{0,\eta}}$ are orthonormal bases of $\sigma$ and $\sigma_{0,\eta} = \operatorname{Ind}(\eta, \eta^{-1} \omega)$ with respect to their associated inner products, respectively.

According to the construction in § 2.2, the test function f is right invariant under

(3.1) \begin{equation}K[\ast]:=K_{v_1}[1]\bigotimes \prod_{v\lt\infty,\ v\neq v_1}K_{0,v}[m_v].\end{equation}

Hence, we may take $\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}$ and $\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}$ in the above definitions of $\operatorname{K}_0(x,y)$ and $\operatorname{K}_{{\rm ER}}(x,y)$ , respectively. Here, $\mathfrak{B}_{\sigma}^{K[\ast]}$ (respectively, $\mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}$ ) consists of vectors in $\mathfrak{B}_{\sigma}$ that are right- $K[\ast]$ -invariant modulo the center.

3.3 Rankin–Selberg periods

Let $\varphi$ be a generic automorphic form on $G(\mathbb{A}_F).$ Define the associated Whittaker function by

(3.2) \begin{equation}W_{\varphi}(g):=\int_{[N]}\varphi(ug)\overline{\theta(u)} \, du,\quad g\in G(\mathbb{A}_F),\end{equation}

where $\theta=\otimes_v\theta_v$ is the generic additive character of $N(\mathbb{A}_F)$ .

By multiplicity one, $W_{\varphi}$ factors into local integrals, i.e., $W_{\varphi}(g)=\prod_{v}W_{\varphi,v}(g_v),$ where $g=\otimes_{v}g_v\in G(\mathbb{A}_F)$ and $W_{\varphi,v}$ is a local Whittaker function associated with the local generic character $\theta_v$ . Moreover, $W_{\varphi,v}$ is spherical for all but finitely many places $v\in\Sigma_F.$ Define the Rankin–Selberg integral

(3.3) \begin{equation}\Psi(s,\varphi,\chi):=\int_{\mathbb{A}_F^{\times}}W_{\varphi}\bigg(\!\!\begin{pmatrix} x\\ &\;\;1\end{pmatrix}\!\!\bigg)|x|^s\chi(x) \, d^{\times}x=\prod_{v}\Psi_v(s,\varphi,\chi),\end{equation}

where the local integral is defined by

(3.4) \begin{equation}\Psi_v(s,\varphi,\chi)=\int_{F_v^{\times}}W_{\varphi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)|x_v|_v^s\chi_v(x_v) \, d^{\times}x_v.\end{equation}

By standard estimates of $W_{\varphi,v}$ , the integral $\Psi(s,\varphi,\chi)$ converges absolutely for ${\rm Re}(s)\gt1.$ Furthermore, it is related to L-functions as follows.

If $\varphi\in\mathfrak{B}_{\sigma},$ where $\sigma\in \mathcal{A}_0([G],\omega),$ then by the Fourier expansion

\begin{align*}\varphi(g)=\sum_{\alpha\in F^{\times}}W_{\varphi}\bigg(\!\!\begin{pmatrix}\alpha\\&\;\; 1\end{pmatrix}g\!\bigg), \quad g\in G(\mathbb{A}_F),\end{align*}

we have

\begin{align*}\Psi(s,\varphi,\chi)=\int_{F^{\times}\backslash\mathbb{A}_F^{\times}}\varphi\bigg(\!\!\begin{pmatrix} x\\ &\;\;1\end{pmatrix}\!\!\bigg)|x|^s\chi(x) \, d^{\times}x,\end{align*}

which, by the rapid decay of $\varphi,$ converges absolutely for all $s\in\mathbb{C}.$ Hence, the function $\Psi(s,\varphi,\chi)$ is entire. By Hecke’s theory, $\Psi(s,\varphi,\chi)$ is the integral representation for the complete L-function $\Lambda(s+1/2,\sigma\times\chi).$

If $\varphi\in\mathfrak{B}_{0,\eta}$ for some $\eta\in\widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}},$ then each local integral

$$\int_{F_v^{\times}}W_{\varphi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)|x_v|_v^s\chi_v(x_v) \, d^{\times}x_v$$

represents the local L-function $L_v(s+1/2,\eta_v\chi_v)L_v(s+1/2,\eta_v^{-1}\omega_v\chi_v).$ Therefore, the function $\Psi(s,\varphi,\chi)$ admits a meromorphic continuation to $s\in\mathbb{C},$ which represents the complete L-function $\Lambda(s+1/2,\eta\chi)\Lambda(s+1/2,\eta^{-1}\omega\chi).$

3.4 The spectral side: meromorphic continuation

Let ${\rm Re}(s_1)\ggg 1$ and ${\rm Re}(s_2)\ggg 1$ . Substituting the Rankin–Selberg periods (see § 3.3) into the decomposition (2.28), we then obtain

\begin{align*}J_{\textrm{Spec}}^{{\rm Reg}}(f,\textbf{s},\chi)=J_{0}^{{\rm Reg}}(f,\textbf{s},\chi)+J_{{\rm Eis}}^{{\rm Reg}}(f,\textbf{s},\chi),\end{align*}

where

\begin{align*}J_{0}^{{\rm Reg}}(f,\textbf{s},\chi)&=\sum_{\sigma\in\mathcal{A}_0([G],\omega)}\sum_{\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}}\Psi(s_1,\sigma(f)\phi)\Psi(s_2,\overline{\phi},\overline{\chi}),\end{align*}

with $K[\ast]$ being defined by (3.1), and $J_{{\rm Eis}}^{{\rm Reg}}(f,\textbf{s},\chi)$ is defined by

\begin{align*}\frac{1}{4\pi}\sum_{\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}}\int_{i\mathbb{R}}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi}) \, d\lambda.\end{align*}

Proposition 3.1. The function $J_{{\rm Eis}}^{{\rm Reg}}(f,\textbf{s},\chi)$ extends to a holomorphic function $J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)$ in $-1/4\lt{\rm Re}(s_1),\ {\rm Re}(s_2)\lt1/4$ with $J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)$ given by

(3.5) \begin{equation}\frac{1}{4\pi}\sum_{\eta}\int_{i\mathbb{R}}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi}) \, d\lambda,\end{equation}

where the integrand $\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi})$ is identified with its meromorphic continuation. In particular,

(3.6) \begin{equation}J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi):=J_{0}^{{\rm Reg}}(f,\textbf{s},\chi)+J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)\end{equation}

is holomorphic in the region $-1/4\lt{\rm Re}(s_1),\ {\rm Re}(s_2)\lt1/4.$

We now consider the case that $\eta\in\{\chi^{-1},\chi\omega\}.$ For ${\rm Re}(s_1), {\rm Re}(s_2)\ggg 1,$ let

(3.7) \begin{equation}J_{\eta}(f,\textbf{s},\chi):=\frac{1}{4\pi}\int_{i\mathbb{R}}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi}) \, d\lambda.\end{equation}

By [Reference YangYan23, § 6.3.3], we have a meromorphic continuation

(3.8) \begin{equation}\tilde{J}_{\eta}(f,\textbf{s},\chi)=J_{\eta}^{\heartsuit}(f,\textbf{s},\chi)-\mathcal{G}_{\eta}(f,\mathbf{s},\chi)+\widetilde{\mathcal{G}}_{\eta}(f,\mathbf{s},\chi),\end{equation}

where $-1/4\lt{\rm Re}(s_1),\ {\rm Re}(s_2)\lt1/4,$ the function $J_{\eta}^{\heartsuit}(f,\textbf{s},\chi)$ is defined by (3.7) with the integrand $\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda))\Psi(s_2,\overline{E(\cdot, \phi,\lambda)})$ understood as its meromorphic continuation, and

\begin{align*}\mathcal{G}_{\eta}(f,\mathbf{s},\chi)&:=\frac{1}{4\pi}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\underset{\lambda=s_1-1/2}{\operatorname{Res}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi}),\\\widetilde{\mathcal{G}}_{\eta}(f,\mathbf{s},\chi)&:= \frac{1}{4\pi}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\underset{\lambda=1/2-s_2}{\operatorname{Res}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot, \phi,\lambda)},\overline{\chi}).\end{align*}

When ${\rm Re}(s_1+s_2)\gt1$ , by [Reference YangYan23, Theorem 8.1] the function $\mathcal{G}_{\eta}(f,\mathbf{s},\chi)$ is represented by the convergent integral

\begin{align*}-\frac{1}{4\pi}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F}\int_{\mathbb{A}_F}f\bigg(\!\!\begin{pmatrix} 1&\\ c&\;\;1\end{pmatrix}\begin{pmatrix} y&\;\;b\\ &1\end{pmatrix}\!\!\bigg)\psi(bx) \, db \, dc|x|^{s_1+s_2}\,d^{\times}x\overline{\chi}(y)|y|^{s_2} \, d^{\times}y,\end{align*}

and the function $\widetilde{\mathcal{G}}_{\eta}(f,\mathbf{s},\chi)$ is expressed by the convergent integral

\begin{align*}\frac{1}{4\pi}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F^{\times}}\int_{\mathbb{A}_F}\int_{\mathbb{A}_F}f\bigg(\!\!\begin{pmatrix} 1&\;\;b\\ &1\end{pmatrix}\begin{pmatrix} y&\\ cy&\;\;1\end{pmatrix}\!\!\bigg)\psi(bx) \, db \, dc|x|_{\mathbb{A}_F}^{s_1+s_2}\, d^{\times}x\overline{\chi}(y)|y|_{\mathbb{A}_F}^{s_2} \, d^{\times}y.\end{align*}

Utilizing the integral representations discussed above, we proceed to demonstrate that $\mathcal{G}_{\eta}(f,\mathbf{s},\chi)\equiv \widetilde{\mathcal{G}}_{\eta}(f,\mathbf{s},\chi)\equiv 0$ for the test function f constructed in § 2.2.

Let $v=v_0$ be the fixed unramified prime defined as in § 2.1.4. Consider the local integral

\begin{align*}\alpha_v(x_v,y_v):=\int_{F_v}\int_{F_v}f_v\bigg(\!\!\begin{pmatrix} 1&\\ c_v&\;\;1\end{pmatrix}\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\!\!\bigg)\psi_v(b_vx_v) \, db_v \, dc_v,\end{align*}

which is the inner integral of the local integral of $\mathcal{G}_{\eta}(f,\mathbf{s},\chi)$ .

Recall the construction of $f_v$ in § 2.2.3:

\begin{align*}f_v(g):=\int_{F_v^{\times}}(\tilde{h}_v*\tilde{h}_v^*)(gz)\omega_v(z) \, d^{\times}z,\quad g\in G(F_v).\tag{\ref{f2.12}}\end{align*}

By the definition of $\tilde{h}_v$ in (2.13), we have $\tilde{h}_v*\tilde{h}_v^*=\tilde{f}_v^{(1,1)}+\tilde{f}_v^{(1,2)}+\tilde{f}_v^{(2,1)}+\tilde{f}_v^{(2,2)},$ where

\begin{align*}\tilde{f}_v^{(1,1)}&:=\sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\sum_{\substack{\beta'\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}}\overline{\psi}_{v}(\beta'\varpi_v^{-1})\mathbf{1}_{u_{\beta}K_vu_{-\beta'}},\\\tilde{f}_v^{(1,2)}&:=\sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\mathbf{1}_{u_{\beta}K_v\mathbf{d}^{-1}},\\\tilde{f}_v^{(2,1)}&:=\sum_{\beta'\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\overline{\psi}_v(\beta'\varpi_v^{-1})\mathbf{1}_{\mathbf{d}K_vu_{-\beta'}},\quad \tilde{f}_v^{(2,2)}:=\mathbf{1}_{\mathbf{d}K_v\mathbf{d}^{-1}}.\end{align*}

Let $\beta\in \mathbb{F}_{v}^{\times}$ and $g\in G(F_v)$ . From the identity

\begin{align*}\begin{pmatrix}1& \;\;\beta\varpi_v^{-1}\\&1\end{pmatrix}=\begin{pmatrix}1& \\\beta^{-1}\varpi_v &\;\;1\end{pmatrix}\begin{pmatrix}\varpi_v^{-1}& \\ &\;\;\varpi_v\end{pmatrix}\begin{pmatrix}\varpi_v &\;\;\beta\\-\beta^{-1}&\end{pmatrix}\!,\end{align*}

it follows that

(3.9) \begin{equation}\tilde{f}_v^{(1,i)}(g) = \sum_{\substack{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}}\psi_{v}(\beta\varpi_v^{-1})\tilde{f}_v^{(2,i)}\bigg(\!\!\begin{pmatrix}1& \\-\beta^{-1}\varpi_v &\;\;1\end{pmatrix} g\! \bigg),\quad i\in\{1,2\}.\end{equation}

Substituting $g=\big(\!\begin{smallmatrix} 1&\\ c_v&\;1 \end{smallmatrix}\!\big)\big(\!\begin{smallmatrix} y_v&\;b_v\\ &1 \end{smallmatrix}\!\big)$ into (3.9), and applying the change of variable $c_v\mapsto c_v+\beta^{-1}\varpi_v^{-1}$ , the integral

\begin{align*}\int_{F_v}\int_{F_v}\tilde{f}_v^{(1,i)}\bigg(\!\!\begin{pmatrix}1&\\ c_v&\;\;1\end{pmatrix} \begin{pmatrix}y_v&b_v\\ &\;\;1\end{pmatrix} \!\!\bigg) \psi_v(b_vx_v) \, db_v \, dc_v\end{align*}

reduces to

(3.10) \begin{equation}\sum_{\substack{\beta}}\psi_{v}(\beta\varpi_v^{-1})\int_{F_v}\int_{F_v}\tilde{f}_v^{(2,i)}\bigg(\!\!\begin{pmatrix}1&\\ c_v&\;\;1\end{pmatrix} \begin{pmatrix}y_v&\;\;b_v\\ &1\end{pmatrix}\!\! \bigg) \psi_v(b_vx_v) \, db_v \, dc_v,\end{equation}

where $\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)$ . Since

\[\sum_{\substack{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}}\psi_{v}(\beta\varpi_v^{-1}) = -1, \]

it follows from (3.10) that

\begin{align*}\int_{F_v}\int_{F_v}\sum_{j=1}^2\tilde{f}_v^{(j,i)}\bigg(\!\!\begin{pmatrix}1&\\ c_v&\;\;1\end{pmatrix} \begin{pmatrix}y_v&\;\;b_v\\ &1\end{pmatrix}\!\! \bigg) \psi_v(b_vx_v) \, db_v \, dc_v \equiv 0,\quad i\in \{1,2\}.\end{align*}

Together with (2.14), we deduce that $\alpha_v(x_v,y_v) \equiv 0$ , which implies $\mathcal{G}_{\eta}(f,\mathbf{s},\chi) \equiv 0$ in the region ${\rm Re}(s_1) \ggg 1$ and ${\rm Re}(s_2) \ggg 1$ . By meromorphic continuation, it follows that $\mathcal{G}_{\eta}(f,\mathbf{s},\chi) \equiv 0$ . Similarly, we obtain $\widetilde{\mathcal{G}}_{\eta}(f,\mathbf{s},\chi) \equiv 0$ .

In conjunction with (3.8), we deduce the meromorphic continuation $J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\mathbf{s},\chi)$ , as defined in (3.5), of the function $J_{{\rm Eis}}^{{\rm Reg}}(f,\mathbf{s},\chi)$ in the region $-1/4 \lt {\rm Re}(s_1), {\rm Re}(s_2) \lt 1/4$ . Moreover, by [Reference YangYan23, Lemma 6.7], the integral $J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\mathbf{s},\chi)$ converges absolutely and thus defines a holomorphic function in this region.

Therefore, Proposition 3.1 follows.

3.5 The spectral side

Let ${f}_v$ and $f_v^{\dagger}$ be defined as in §§ 2.2.1–2.2.6. Let

\begin{align*}f^{\dagger}(g):=\bigotimes_{v}f_v^{\dagger},\end{align*}

where for $v\nmid\mathfrak{n}$ , $f_v^{\dagger}$ is defined by (2.9), (2.11), (2.15), (2.16), and (2.18); and for $v\mid\mathfrak{n}$ , $f_v^{\dagger}$ is defined by

\begin{align*}f_{v}^{\dagger}(g):=\int_{F_v^{\times}}\mathbf{1}_{K_v}(zg)\omega_v(z) \,d^{\times}z,\quad g\in G(F_v).\end{align*}

For an automorphic representation $\sigma\in\mathcal{A}([G],\omega)$ and $\varphi\in \sigma,$ we define $\varphi^{\ddagger}(x)$ by

\begin{align*}\int_{\overline{G}(\mathbb{A})}{f}^{\dagger}(g)\prod_{v\mid\mathfrak{Q}}\bigg[\frac{1}{\tau(\chi_v)}\sum_{\alpha\in\mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\chi_v(\alpha)\sigma_v\bigg(\!\!\begin{pmatrix} 1&\;\;\alpha \varpi_v^{-n_v}\\ &1\end{pmatrix}\!\!\bigg)\bigg]\sigma(g)\varphi(x)\, dg.\end{align*}

To simplify the notation, we shall still write $\Psi(s,\phi^{\ddagger},\chi)$ and $\Psi(s,E(\cdot,\phi,\lambda)^{\ddagger},\chi)$ for their meromorphic continuations, respectively.

Lemma 3.2. Let $\mathbf{s}=(s_1,s_2)$ with $-1/4\lt{\rm Re}(s_1),\ {\rm Re}(s_2)\lt1/4$ . Then

(3.11) \begin{equation}J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)=J_{0}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)+J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi),\end{equation}

where

\begin{align*}J_{0}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi):=\sum_{\sigma\in\mathcal{A}_0([G],\omega)}\sum_{\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}}\lambda_{\sigma}(\mathfrak{n})\Psi(s_1,\phi^{\ddagger},\chi)\overline{\Psi(\overline{s_2},\phi^{\ddagger},\chi)},\end{align*}

and $J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{s},\chi)$ is defined by

\begin{align*}\frac{1}{4\pi}\sum_{\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}}\int_{i\mathbb{R}}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\lambda_{\sigma_{\lambda,\eta}}(\mathfrak{n})\Psi(s_1,E(\cdot,\phi,\lambda)^{\ddagger},\chi)\overline{\Psi(\overline{s_2},E(\cdot,\phi,\lambda)^{\ddagger},\chi)}\,d\lambda.\end{align*}

In particular, we have

\begin{align*}J_{\textrm{Spec}}^{{\rm Reg},\heartsuit}(f,\textbf{0},\chi)=J_{0}^{{\rm Reg},\heartsuit}(f,\textbf{0},\chi)+J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{0},\chi),\end{align*}

where $\mathbf{0}=(0,0)$ , and

\begin{align*}&J_{0}^{{\rm Reg},\heartsuit}(f,\textbf{0},\chi):=\sum_{\sigma\in\mathcal{A}_0([G],\omega)}\lambda_{\sigma}(\mathfrak{n})\sum_{\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}}|\Psi(0,\phi^{\ddagger},\chi)|^2,\\&J_{{\rm Eis}}^{{\rm Reg},\heartsuit}(f,\textbf{0},\chi):=\frac{1}{4\pi}\sum_{\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}}\int_{i\mathbb{R}}\lambda_{\sigma_{\lambda,\eta}}(\mathfrak{n})\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}|\Psi(0,E(\cdot,\phi,\lambda)^{\ddagger},\chi)|^2 \, d\lambda.\end{align*}

• – Suppose that $\sigma\in\mathcal{A}_0([G],\omega)$ . Then

  (3.12) \begin{equation}\sum_{\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}}\Psi(s_1,\sigma(f)\phi)\Psi(s_2,\overline{\phi},\overline{\chi})=\sum_{\phi\in\mathfrak{B}_{\sigma}^{K[\ast]}}\lambda_{\sigma}(\mathfrak{n})\Psi(s_1,\phi^{\ddagger},\chi)\overline{\Psi(\overline{s_2},\phi^{\ddagger},\chi)}.\end{equation}

• – Suppose that $\lambda\in i\mathbb{R}$ . Then

  (3.13) \begin{align}&\sum_{\phi\in\mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\Psi(s_1,\sigma_{0,\eta}(f)E(\cdot,\phi,\lambda),\chi)\Psi(s_2,\overline{E(\cdot,\phi,\lambda)},\overline{\chi})\nonumber\\&\quad =\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}\lambda_{\sigma_{\lambda,\eta}}(\mathfrak{n})\Psi(s_1,E(\cdot,\phi,\lambda)^{\ddagger},\chi)\overline{\Psi(\overline{s_2},E(\cdot,\phi,\lambda)^{\ddagger},\chi)}.\end{align}

Therefore, (3.11) follows from (3.6), (3.12), and (3.13).

3.6 Lower bounds of period integrals

Recall that $\pi$ is defined in § 2.1. In this section, we establish a lower bound for the period integral $\Psi(0,\varphi^{\ddagger},\chi)$ , where

\begin{align*}\varphi=\begin{cases}\phi\in \mathfrak{B}_{\pi}^{K[\ast]} \ & \text{if $\pi$ is cuspidal,}\\E(\cdot,\phi,\lambda) \ & \text{if $\pi=\sigma_{\lambda,\eta}$, $\lambda\in i\mathbb{R}$, $\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}$, and $\phi \in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}$.}\end{cases}\end{align*}

Let $W_{\varphi}(g)=\prod_{v}W_{\varphi,v}(g_v)$ be the decomposition of the Whittaker function of $\varphi$ defined by (3.2).

By definition (3.3), we have

(3.14) \begin{equation}\Psi(0,\varphi^{\ddagger},\chi) = L(1/2,\pi\times\chi) \prod_{v\mid\infty} \Psi_v(0,\varphi^{\ddagger},\chi)\prod_{v\lt\infty} \frac{\Psi_v(0,\varphi^{\ddagger},\chi)}{L_v(1/2,\pi\times\chi)},\end{equation}

where $L_v(1/2,\pi\times\chi)$ denotes the local L-factor, and $\Psi_v(0,\varphi^{\ddagger},\chi)$ is the local period integral defined by (3.4).

Since $\Psi_v(0,\varphi^{\ddagger},\chi) / L_v(1/2,\pi\times\chi) \equiv W_{\phi,v}(I_2)$ for all but finitely many places, the product over $v\lt\infty$ in (3.14) reduces to a finite product. Moreover, by the bounds towards the Ramanujan conjecture in [Reference Kim and SarnakKS03], the local integral $\Psi_v(0,\varphi^{\ddagger},\chi)$ converges absolutely for all places $v\in \Sigma_F$ .

• – Suppose that $\pi$ is cuspidal. Then

  (3.15) \begin{equation}\sum_{\phi\in\mathfrak{B}_{\pi}^{K[\ast]}}|\Psi(0,\phi^{\ddagger},\chi)|^2\gg_{F,\varepsilon} C(\pi)^{-\varepsilon}|L(1/2,\pi\times\chi)|^2\prod_{v\mid\infty}T_v^{-\frac{1}{2}-\varepsilon}.\end{equation}

• – Suppose that $\pi=\sigma_{\lambda,\eta}$ for some $\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}$ , and $\lambda \in i\mathbb{R}$ . Then

  (3.16) \begin{equation}\sum_{\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}}|\Psi(0,E(\cdot,\phi,\lambda)^{\ddagger},\chi)|^2\gg_{F,\varepsilon} \frac{C(\sigma_{\lambda,\eta})^{-\varepsilon}|L(1/2,\sigma_{\lambda,\eta}\times\chi)|^2}{|L(1+2\lambda,\eta^2\omega^{-1})|^2}\prod_{v\mid\infty}T_v^{-\frac{1}{2}-\varepsilon}.\end{equation}

Proposition 3.3 will be proved in § 3.6.3, by collecting lemmas in §§ 3.6.1 and 3.6.2.

3.6.1 Lower bounds of $\Psi_v(s,\phi^{\ddagger},\chi)$ at $v\lt\infty$

Lemma 3.4. Suppose that $ \pi \in \mathcal{A}_0([G],\omega) $ and $ \phi \in \mathfrak{B}_{\pi}^{K[\ast]} $ . Assume that $ W_{\phi,v} $ is a local newform at all finite places $ v \lt \infty $ . Then, for $ v \in \Sigma_{F,\operatorname{fin}} - \{v_0\} $ , we have

(3.17) \begin{equation}\Psi_v(s,\phi^{\ddagger},\chi) = W_{\phi,v}(I_2) L_v(s+1/2,\pi_v\times\chi_v), \quad {\rm Re}(s) \geqslant 0.\end{equation}

Proof. When $v\nmid\mathfrak{Q},$ $\chi_v$ is unramified. By the explicit formula for the local new Kirillov vector (see, e.g., [Reference SchmidtSch02, § 2.4]), we have (3.17).

We now suppose that $v\mid\mathfrak{Q},$ i.e., $n_v\geqslant 1.$ Then

(3.18) \begin{align}&\Psi_v(s,\phi^{\ddagger},\chi)\nonumber\\&\quad =\frac{1}{\tau(\chi_v)}\sum_{\alpha\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\chi_v(\alpha)\int_{F_v^{\times}}W_{\phi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\begin{pmatrix} 1&\;\;\alpha \varpi_v^{-n_v}\\ &1\end{pmatrix}\!\!\bigg)\chi_v(x_v)|x_v|_v^s \, d^{\times}x_v.\end{align}

Note that

(3.19) \begin{equation}W_{\phi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\begin{pmatrix} 1&\;\;\alpha \varpi_v^{-n_v}\\ &1\end{pmatrix}\!\!\bigg)=\psi_v(\alpha x_v\varpi_v^{-n_v})W_{\phi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg).\end{equation}

Substituting (3.19) into (3.18) we then derive that

\begin{align*}\Psi_v(s,\phi^{\ddagger},\chi)=\frac{1}{\tau(\chi_v)}\sum_{\alpha\in\mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\int_{F_v^{\times}}W_{\phi,v}\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\chi_v(\alpha x_v)\psi_v(\alpha x_v\varpi_v^{-n_v})|x_v|_v^s \, d^{\times}x_v.\end{align*}

Note that $W_{\phi,v}\big(\!\big(\begin{smallmatrix} x_v\\ &\;\;1\end{smallmatrix}\big)\!\big)=0$ unless $e_v(x_v)\geqslant 0.$ Moreover,

\begin{align*}\sum_{\alpha\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\chi_v(\alpha x_v)\psi_v(\alpha x_v\varpi_v^{-n_v})=\begin{cases} \tau(\chi_v) \ &\text{if $e_v(x_v)=0,$}\\ 0 \ &\text{if $e_v(x_v)\geqslant 1$.}\end{cases}\end{align*}

Therefore, we have $\Psi_v(s,\phi^{\ddagger},\chi)=W_{\phi,v}(I_2)L_v(s+1/2,\pi_v\times\chi_v)$ .

Lemma 3.5. Let $\lambda \in i\mathbb{R}$ and $\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}$ . Let $\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}$ . Assume that $W_{\phi,v}$ is a local newform at all $v\lt\infty$ . Then, for $v\in\Sigma_{F,\operatorname{fin}}-\{v_0\}$ and ${\rm Re}(s)\geqslant 0,$ we have

\begin{align*}\Psi_v(s,E(\cdot,\phi,\lambda)^{\ddagger},\chi)=W_{\phi,v}(I_2)L_v(s+1/2+\lambda,\eta_v\chi_v)L_v(s+1/2-\lambda,\eta_v^{-1}\chi_v\omega_v).\end{align*}

Proof. The proof is similar to that of Lemma 3.4, so we omit it.

We now examine the local integral at $ v = v_0 $ , where $ v_0 $ is the fixed auxiliary unramified place defined in § 2.1.4. Recall that the test function at $ v_0 $ is chosen to eliminate contributions from the residues on the spectral side when shifting the contour from $ {\rm Re}(s) \gt 1 $ to $ {\rm Re}(s) \leqslant 0 $ . Our goal is to establish that the local integral at $ v = v_0 $ is nonvanishing.

Lemma 3.6. Suppose that $ \pi \in \mathcal{A}_0([G],\omega) $ and $ \phi \in \mathfrak{B}_{\pi}^{K[\ast]} $ . Assume that $ W_{\phi,v} $ is a local newform at all finite places $ v \lt \infty $ . Then, for $v=v_0$ , we have

(3.20) \begin{equation}|\Psi_v(0,\phi^{\ddagger},\chi)|\geqslant 10^{-1} |W_{\phi,v}(I_2)L(1/2,\pi_v\times\chi_v)|.\end{equation}

Proof. Let ${\rm Re}(s) \geqslant 0$ . By the definition of $f_v^{\dagger}$ in (2.13) and (2.15), we obtain

(3.21) \begin{equation}\Psi_v(s,\phi^{\ddagger},\chi) = \Psi_{v,1} + \Psi_{v,2},\end{equation}

where $\Psi_{v,1}$ is given by

\begin{align*}\sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\int_{F_v^{\times}}\int_{K_v} W_{\phi,v} \bigg( \!\!\begin{pmatrix}x_v&\;\; x_v\beta\varpi_v^{-1}\\&1\end{pmatrix} k_v\! \bigg) \, dk_v \chi_v(x_v) |x_v|_v^s \, d^{\times}x_v,\end{align*}

and $\Psi_{v,2}$ is given by

\begin{align*}\int_{F_v^{\times}}\int_{K_v} W_{\phi,v} \bigg(\!\!\begin{pmatrix}x_v\\& \;\;1\end{pmatrix} \begin{pmatrix}\varpi_v^{-1}\\&\;\;\varpi_v\end{pmatrix} k_v\! \bigg) \, dk_v \chi_v(x_v) |x_v|_v^s \, d^{\times}x_v.\end{align*}

By the definition of $K[\ast]$ in (3.1) and the assumption that $m_v = 0$ at $v = v_0$ , $W_{\phi,v}$ is spherical. Therefore, we obtain

\begin{align*}\Psi_{v,1} &= \sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\sum_{r\in \mathbb{Z}}\int_{\mathcal{O}_v^{\times}} W_{\phi,v} \bigg(\!\!\begin{pmatrix}\alpha\varpi_v^r&\; \alpha\beta\varpi_v^{r-1}\\&1\end{pmatrix}\!\!\bigg) \frac{\chi_v(\varpi_v)^r}{q_v^{rs}} \, d^{\times}\alpha,\\\Psi_{v,2} &= \int_{F_v^{\times}} W_{\phi,v} \bigg(\!\!\begin{pmatrix}x_v\varpi_v^{-1}\\&\;\;\varpi_v\end{pmatrix}\!\!\bigg) \chi_v(x_v) |x_v|_v^s \, d^{\times}x_v.\end{align*}

After the change of variable $x_v \mapsto \varpi_v^2 x_v$ , we obtain

(3.22) \begin{equation}\Psi_{v,2} = \omega_v(\varpi_v)\chi_v(\varpi_v)^2 q_v^{-2s} W_{\phi,v}(I_2) L(s+1/2,\pi_v\times\chi_v).\end{equation}

Moreover, decomposing the sum according to $r \geqslant 1$ and $r \leqslant 0$ , we get

\begin{align*}\Psi_{v,1} &= \sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\sum_{r\leqslant 0} \int_{\mathcal{O}_v^{\times}} W_{\phi,v} \bigg( \!\!\begin{pmatrix}\alpha\varpi_v^r&\;\; \alpha\beta\varpi_v^{r-1}\\&1\end{pmatrix}\!\!\bigg) \frac{\chi_v(\varpi_v)^r}{q_v^{rs}} \,d^{\times}\alpha\\&\quad+ \sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\varpi_v^{-1})\sum_{r\geqslant 1} \int_{\mathcal{O}_v^{\times}} W_{\phi,v} \bigg(\!\!\begin{pmatrix}\alpha\varpi_v^r&\;\; \alpha\beta\varpi_v^{r-1}\\&1\end{pmatrix}\!\!\bigg) \frac{\chi_v(\varpi_v)^r}{q_v^{rs}} \,d^{\times}\alpha.\end{align*}

Since $W_{\phi,v}(\mathrm{diag}(\alpha\varpi_v^r),1) \equiv 0$ for $r \leqslant -1$ , we obtain

\begin{align*}\Psi_{v,1} &= W_{\phi,v}(I_2) \int_{\mathcal{O}_v^{\times}} G_{\beta,1+\alpha} \, d^{\times}\alpha+ G_{\beta,1} \sum_{r\geqslant 1} W_{\phi,v} \bigg(\!\!\begin{pmatrix}\varpi_v^r\\& \;\;1\end{pmatrix}\!\!\bigg) \chi_v(\varpi_v)^r q_v^{-rs},\end{align*}

where

\begin{align*}G_{\beta,\gamma} := \sum_{\beta\in \mathcal{O}_v^{\times}/(1+\mathfrak{p}_v)}\psi_v(\beta\gamma\varpi_v^{-1}).\end{align*}

By a straightforward calculation, we deduce that

(3.23) \begin{equation}\Psi_{v,1} = \frac{W_{\phi,v}(I_2)}{q_v-1} + W_{\phi,v}(I_2) (1 - L(s+1/2,\sigma_v\times\chi_v)).\end{equation}

Combining (3.22) with (3.23), and applying the triangle inequality, we derive

(3.24) \begin{equation}\frac{|\Psi_v(0,\phi^{\ddagger},\chi)|}{|W_{\phi,v}(I_2)|} \geqslant |L(1/2,\pi_v\times\chi_v)| - \frac{1}{q_v-1} - |1 - L(1/2,\pi_v\times\chi_v)|.\end{equation}

By the bound towards the Ramanujan conjecture (see [Reference Kim and SarnakKS03]), we obtain

(3.25) \begin{equation}|1 - L(1/2,\pi_v\times\chi_v)| \leqslant 20 q_v^{-7/64}.\end{equation}

Therefore, (3.20) follows from (3.24) and (3.25).

Lemma 3.7. Let $\lambda \in i\mathbb{R}$ and $\eta\in \widehat{F^{\times}\backslash\mathbb{A}_F^{(1)}}$ . Let $\phi\in \mathfrak{B}_{\sigma_{0,\eta}}^{K[\ast]}$ . Assume that $W_{\phi,v}$ is a local newform at all $v\lt\infty$ . Then, for $v=v_0$ , we have

\begin{align*}|\Psi_v(0,E(\cdot,\phi,\lambda)^{\ddagger},\chi)|\geqslant 10^{-1}|W_{\phi,v}(I_2)L_v(1/2+\lambda,\eta_v\chi_v)L_v(1/2-\lambda,\eta_v^{-1}\chi_v\omega_v)|.\end{align*}

Proof. The proof is similar to that of Lemma 3.6, so we omit it.

3.6.2 Lower bounds of $\Psi_v(s,\phi^{\ddagger},\chi)$ at $v\mid\infty$

Proposition 3.8. Let $\varepsilon\gt0$ be a small constant, and let $T_v$ be defined as in § 2.1.2. There exists a unit vector $W_v^*$ in the Kirillov model of $\pi_v,$ such that

(3.26) \begin{equation}\bigg|\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v^*\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v\bigg|\gg T_v^{-\frac{1}{4}-\varepsilon}.\end{equation}

Proof. We consider the following scenarios according to the types of $\pi_v\otimes\chi_v$ .

• – Suppose that $\pi_v\otimes\chi_v$ is of type I. By the Kirillov theorem (see, e.g., [Reference NelsonNel21, Lemma 11.1 (vi)]), there exists a unit vector $W_v$ in the Whittaker model of $\pi_v,$ such that

  (3.27) \begin{equation}\bigg|\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v\bigg|\gg C_v(\pi \otimes\chi)^{-\frac{1}{8}-\varepsilon}\gg T_v^{-\frac{1}{4}-\varepsilon}.\end{equation}

• – Suppose that $\pi_v\otimes\chi_v$ is of type II. Define the vector $ W_v $ in the Kirillov model of $\pi_v$ by

  (3.28) \begin{equation}W_v\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg) = h_{1,v}(x_v) \chi_v^{-1}(x_v),\quad x_v\in F_v^{\times}.\end{equation}

Since $ \varepsilon_0 \gt 0 $ is sufficiently small and $ h_{1,v}(x_v) = 0 $ unless $ |x_v - 1|_v \leqslant \varepsilon_0 $ , it follows that $ W_v $ is a smooth function on $ F_v^{\times} $ with compact support, satisfying the bound

(3.29) \begin{equation}\langle W_v\otimes \chi_v,W_v\otimes \chi_v\rangle =\int_{F_v^{\times}}|h_{1,v}(x_v)|^2\,d^{\times}x_v\ll \varepsilon_0^{-1}\int_{F_v^{\times}}h_{1,v}(x_v)\,d^{\times}x_v=\varepsilon_0^{-1}.\end{equation}

By the definition (2.10) and (2.11), we obtain

\begin{align*}&\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\, d^{\times}x_v\\&\quad=T_v^{\frac{3}{4}}\int_{F_v}\int_{F_v}\int_{F_v^{\times}}\int_{F_v^{\times}}\int_{F_v^{\times}}h_{1,v}(z)h_{1,v}(a)h_{2,v}'(b)h_{3,v}(T_v^{1+\varepsilon}c)\\&\qquad\cdot(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\begin{pmatrix}a& \;\;b\\& 1\end{pmatrix}\begin{pmatrix}1& \\c & \;\;1\end{pmatrix}\!\!\bigg)d^{\times}x_v|a|_v^{-1}\, d^{\times}a\, d^{\times}z\, db\, dc.\end{align*}

Utilizing the definition of $h_{j,v}$ , $1\leqslant j\leqslant 3$ , and the property

\begin{align*}(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix}x_va&\;\; x_vb\\& 1\end{pmatrix}\begin{pmatrix}1& \\c &\;\; 1\end{pmatrix}\!\!\bigg)=\psi_v(x_vb)(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_va\\ &\;\;1\end{pmatrix}\begin{pmatrix}1& \\c & \;\;1\end{pmatrix}\!\!\bigg),\end{align*}

we derive that

\begin{align*}&\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)d^{\times}x_v\\&\quad =T_v^{\frac{3}{4}}\int_{F_v}\int_{F_v^{\times}}h_{1,v}(a)h_{3,v}(T_v^{1+\varepsilon}c)\\&\qquad \cdot\int_{F_v^{\times}}\widehat{h_{2,v}'}(x_v)(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_va\\ &\;\;1\end{pmatrix}\begin{pmatrix}1& \\c & \;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v|a|_v^{-1}\,d^{\times}a\,dc,\end{align*}

where $\widehat{h_{2,v}'}(x_v):=\int_{F_v}h_{2,v}'(b)\psi_v(x_vb_v)\,db_v$ is the Fourier transform. Define

\begin{align*}\mathcal{W}(x_va,c):=(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_va\\ &\;\;1\end{pmatrix}\begin{pmatrix}1& \\c & \;\;1\end{pmatrix}\!\!\bigg)-(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_va\\ &\;\;1\end{pmatrix}\!\!\bigg).\end{align*}

By the Cauchy–Schwarz inequality, we have

\begin{align*}\bigg|\int_{F_v^{\times}}\widehat{h_{2,v}'}(x_v)\mathcal{W}(x_va,c)\,d^{\times}x_v\bigg|\leqslant\bigg[\int_{F_v^{\times}}|\widehat{h_{2,v}'}(x_v)|^2\,d^{\times}x_v\bigg]^{\frac{1}{2}}\bigg[\int_{F_v^{\times}}|\mathcal{W}(x_va,c)|^2\,d^{\times}x_v\bigg]^{\frac{1}{2}}.\end{align*}

Note that $|\widehat{h_{2,v}'}(x_v)| = |2\pi x_v \widehat{h_{2,v}}(x_v)|$ . Hence,

(3.30) \begin{equation}\int_{F_v^{\times}} |\widehat{h_{2,v}'}(x_v)|^2 \,d^{\times}x_v\ll \int_{F_v^{\times}} |\widehat{h_{2,v}}(x_v)|^2 |x_v|^2 |x_v|_v^{-1} \,dx_v \ll_{h_{2,v}} 1.\end{equation}

Note that the implied constants are absolute since $ h_{2,v} $ is fixed.

Recall that $T_v=\varepsilon_0^{-100}C_v(\pi \otimes \chi)$ . Also, $h_{3,v}(T_v^{1+\varepsilon}c)\equiv 0$ unless $|c|\leqslant T_v^{-1-\varepsilon}$ . For $|c|\leqslant T_v^{-1-\varepsilon}$ , we have by [Reference Michel and VenkateshMV10, (3.19)] that

(3.31) \begin{equation}\bigg[\int_{F_v^{\times}}|\mathcal{W}(x_va,c)|^2\,d^{\times}x_v\bigg]^{\frac{1}{2}}\ll \varepsilon_0^{50}C_v(\pi \otimes \chi)^{-{\varepsilon}/{[F_v:\mathbb{R}]}}\mathcal{S}_{2}^{F_v^{\times}}(h_{1,v}),\end{equation}

where

\begin{align*}\mathcal{S}_{2}^{F_v^{\times}}(h_{1,v}):=\int_{F_v^{\times}}(|y|+|y|^{-1})^2\bigg|\sum_i\mathcal{D}_ih_{1,v}(y)\bigg|^2\,d^{\times}y,\end{align*}

with $\mathcal{D}_i$ ranging over a basis of $F_v^{\times}$ -invariant differential operators of degree $\leqslant 2$ . By the construction of $h_{1,v}$ , we have $\mathcal{S}_{2}^{F_v^{\times}}(h_{1,v})\ll \varepsilon_0^{-30}$ . Hence, it follows from (3.31) that

(3.32) \begin{equation}\bigg[\int_{F_v^{\times}}|\mathcal{W}(x_va,c)|^2\,d^{\times}x_v\bigg]^{\frac{1}{2}}\ll \varepsilon_0^{20}C_v(\pi \otimes \chi)^{-{\varepsilon}/{[F_v:\mathbb{R}]}}.\end{equation}

As a consequence of (3.30) and (3.32), along with $\int_{F_v} h_{3,v}(c) \,dc = 1$ and the definition (3.28), we derive

(3.33) \begin{align}&\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v\nonumber\\&\quad =T_v^{-\frac{1}{4}-\varepsilon}\int_{F_v^{\times}}h_{1,v}(a)\int_{F_v^{\times}}\widehat{h_{2,v}'}(x_v)h_{1,v}(x_va)\,d^{\times}x_v|a|_v^{-1}\,d^{\times}a\nonumber\\&\qquad+O(\varepsilon_0^{20}C_v(\pi\otimes \chi)^{-{\varepsilon}/{2}}T_v^{-\frac{1}{4}-\varepsilon}).\end{align}

By the definition of $h_{1,v}$ , $h_{1,v}(a)h_{1,v}(x_va)\equiv 0$ unless $|a-1|_v\leqslant \varepsilon_0$ and $|x_va-1|_v\leqslant \varepsilon_0$ . Therefore, by the mean value theorem, we have

(3.34) \begin{equation}h_{1,v}(a)h_{1,v}(x_va)|\widehat{h_{2,v}'}(x_v)-\widehat{h_{2,v}'}(1)|\ll h_{1,v}(a)h_{1,v}(x_va)\varepsilon_0.\end{equation}

Substituting (3.34) into (3.33) we deduce that

(3.35) \begin{align}&\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v\nonumber\\&\quad=T_v^{-\frac{1}{4}-\varepsilon}\widehat{h_{2,v}'}(1)\int_{F_v^{\times}}h_{1,v}(a)\int_{F_v^{\times}}h_{1,v}(x_va)\,d^{\times}x_v|a|_v^{-1}\,d^{\times}a\nonumber\\&\qquad +O(\mathcal{E})+O(\varepsilon_0^{20}C_v(\pi\otimes \chi)^{-{\varepsilon}/{2}}T_v^{-\frac{1}{4}-\varepsilon}),\end{align}

where

\begin{align*}\mathcal{E}:=\varepsilon_0T_v^{-\frac{1}{4}-\varepsilon}\int_{F_v^{\times}}\int_{F_v^{\times}}h_{1,v}(a)h_{1,v}(x_va)\,d^{\times}x_v|a|_v^{-1}\,d^{\times}a.\end{align*}

By the change of variable $x_v\mapsto x_va^{-1}$ and the support of $h_{1,v}$ ,

(3.36) \begin{equation}\mathcal{E}\ll \varepsilon_0T_v^{-\frac{1}{4}-\varepsilon}\bigg|\int_{F_v^{\times}}h_{1,v}(a)\,d^{\times}a\bigg|^2\ll T_v^{-\frac{1}{4}-\varepsilon}\varepsilon_0.\end{equation}

Combining (3.35) with (3.36), in conjunction with the change of variable $x_v\mapsto x_va^{-1}$ , and $\int_{F_v^{\times}}h_{1,v}(x_v)\,d^{\times}x_v=1$ , we deduce

\begin{align*}\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v=\frac{\widehat{h_{2,v}'}(1)}{T_v^{\frac{1}{4}+\varepsilon}}\int_{F_v^{\times}}\frac{h_{1,v}(a)}{|a|_v}\,d^{\times}a+O(T_v^{-\frac{1}{4}-\varepsilon}\varepsilon_0).\end{align*}

Since $ h_{1,v}(a) = 0 $ unless $ |a - 1|_v \leqslant \varepsilon_0 $ , then

\begin{align*}\int_{F_v^{\times}}h_{1,v}(a)|a|_v^{-1}\,d^{\times}a=\int_{F_v^{\times}}h_{1,v}(a)(1+O(\varepsilon_0))\,d^{\times}a=1+O(\varepsilon_0).\end{align*}

Therefore, it follows from the above estimates that

(3.37) \begin{equation}\int_{F_v^{\times}}\pi_v(f_{v}^{\dagger})(W_v\otimes \chi_v)\bigg(\!\!\begin{pmatrix} x_v\\ &\;\;1\end{pmatrix}\!\!\bigg)\,d^{\times}x_v=T_v^{-\frac{1}{4}-\varepsilon}\widehat{h_{2,v}'}(1)+O(T_v^{-\frac{1}{4}-\varepsilon}\varepsilon_0).\end{equation}

By the construction of $h_{2,v}$ , we have $|\widehat{h_{2,v}}(1)|=1$ . Hence, $\widehat{h_{2,v}'}(1)\gg 1$ .

Let $ W^* := W_v / \langle W_v, W_v \rangle $ , where $ W_v $ is given in (3.27) or (3.28), as appropriate. As a consequence of the above discussion, (3.26) follows from (3.27) and (3.37).

4.1 Local calculation at $v\in\Sigma_{F,\operatorname{fin}}-\{v_0,v_1\}$ and $v\nmid\mathfrak{n}$

Recall that $m_v=\max\{r_{\pi_v},n_v+r_{\omega_v}\}$ , where the parameters are defined in § 2.1.3.

Lemma 4.3. Let $v\in\Sigma_{F,\operatorname{fin}}-\{v_0,v_1\}$ and $v\nmid\mathfrak{n}$ . Then

(4.4) \begin{equation}\frac{\operatorname{Vol}(\overline{K_{0,v}[m_v]})J_{{\rm Small},v}(f,\textbf{s},\chi)}{N_{F_v}(\mathfrak{D}_{F_v})^{-1/2+s}\zeta_v(1+s)}=\begin{cases}q_v^{m_vs}(1-q_v^{-1-s}) & \text{if $n_v\geqslant 1,$}\\q_v^{m_vs} & \text{if $n_v=0$.}\end{cases}\end{equation}

In particular, $J_{{\rm Small},v}(f,\textbf{s},\chi)\zeta_v(1+s)^{-1}\equiv 1$ for all but finitely many $v\in \Sigma_F$ .

Proof. For simplicity, we define

(4.5) \begin{equation}\mathcal{I}_v(x_v):=\int_{F_v^{\times}}\int_{F_v}f_v\bigg(\!\!\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\!\!\bigg)\psi_v(x_vb_v)\overline{\chi}_v(y_v)\,db_v\,d^{\times}y_v.\end{equation}

By definition, we have $f_v\big(\!\big(\begin{smallmatrix} y_v&\;\;b_v\\ &1\end{smallmatrix}\big)\!\big)\neq 0$ if and only if there exists $z\in F_v^{\times}$ such that

$$z\begin{pmatrix} 1&\;\;\alpha \varpi_v^{-n_v}\\ &1\end{pmatrix}\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\begin{pmatrix} 1&\;\;\beta \varpi_v^{-n_v}\\ &1\end{pmatrix}\in K_{0,v}[m_v],$$

which implies that $z, y_v\in \mathcal{O}_v^{\times}.$ Making the change of variable $\beta\mapsto y_v^{-1}\beta,$

(4.6) \begin{align}\mathcal{I}_v(x_v)&=\frac{\operatorname{Vol}(\mathcal{O}_v^{\times})\cdot|\tau(\chi_v)|^{-2}}{\operatorname{Vol}(\overline{K_{0,v}[m_v]})}\nonumber\\&\quad\cdot\sum_{\alpha,\beta\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\int_{F_v} \mathbf{1}_{K_{0,v}[m_v]}\bigg(\!\!\begin{pmatrix} 1&\;\;b_v+(\alpha +\beta) \varpi_v^{-n_v}\\ &1\end{pmatrix}\!\!\bigg)\psi_v(x_vb_v)\chi_v(\alpha)\overline{\chi}_v(\beta)\,db_v.\end{align}

After the further change of variables $b_v\mapsto b_v-(\alpha+\beta)\varpi_v^{-n_v},$ (4.6) reduces to

(4.7) \begin{align}\mathcal{I}_v(x_v)&=\frac{\operatorname{Vol}(\mathcal{O}_v^{\times})|\tau(\chi_v)|^{-2}}{\operatorname{Vol}(\overline{K_{0,v}[m_v]})}\int_{\mathcal{O}_v}\psi_v(x_vb_v)\,db_v\nonumber\\&\quad\cdot\sum_{\alpha,\beta\in \mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\psi_v(-x_v(\alpha \varpi_v^{-n_v}+\beta \varpi_v^{-n_v}))\chi_v(\alpha)\overline{\chi}_v(\beta).\end{align}

Substituting the formula

\begin{align*}\int_{\mathcal{O}_v}\psi_v(x_vb_v)\,db_v=N_{F_v}(\mathfrak{D}_{F_v})^{-\frac{1}{2}}\mathbf{1}_{\mathfrak{D}_{F_v}^{-1}}(x_v)\end{align*}

into (4.7), we then obtain that

\begin{align*}\mathcal{I}_v(x_v)=\frac{\operatorname{Vol}(\overline{K_{0,v}[m_v]})^{-1}}{|\tau(\chi_v)|^2N_{F_v}(\mathfrak{D}_{F_v})}\mathbf{1}_{\mathfrak{D}_{F_v}^{-1}}(x_v)\bigg|\sum_{\alpha\in\mathcal{O}_v^{\times}/(1+\varpi_v^{n_v}\mathcal{O}_v^{\times})}\psi_v(x_v\alpha\varpi_v^{-n_v})\chi_v(\alpha)\bigg|^2.\end{align*}

As a consequence, we derive

\begin{align*}\mathcal{I}_v(x_v)=\begin{cases}N_{F_v}(\mathfrak{D}_{F_v})^{-1}\operatorname{Vol}(\overline{K_{0,v}[m_v]})^{-1}\mathbf{1}_{\mathfrak{D}_{F_v}^{-1}-\varpi_v\mathfrak{D}_{F_v}^{-1}}(x_v) \ & \text{if $v\mid\mathfrak{Q},$ i.e., $n_v\geqslant 1,$}\\N_{F_v}(\mathfrak{D}_{F_v})^{-1}\operatorname{Vol}(\overline{K_{0,v}[m_v]})^{-1}\mathbf{1}_{\mathfrak{D}_{F_v}^{-1}}(x_v) \ & \text{if $v\nmid\mathfrak{Q}$, i.e., $n_v=0$.}\end{cases}\end{align*}

Substituting this expression for $\mathcal{I}_v(x_v)$ into the definition

\begin{align*}J_{{\rm Small},v}(f,\mathbf{s},\chi) = \int_{F_v^{\times}} \mathcal{I}_v(x_v) |x_v|_v^{1+s} \,d^{\times}x_v,\end{align*}

we obtain the formula (4.4).

4.2 Local calculation at $v\mid \mathfrak{n}$ or $v\in \{v_0,v_1\}$

Let $\mathcal{I}_v(x_v)$ be defined as in (4.5). By definition, we have

(4.8) \begin{equation}J_{{\rm Small},v}(f,\mathbf{s},\chi) = \int_{F_v^{\times}} \mathcal{I}_v(x_v) |x_v|_v^{1+s} \,d^{\times}x_v.\end{equation}

Lemma 4.4. Let $v\mid\mathfrak{n}$ . Then

(4.9) \begin{equation}\frac{J_{{\rm Small},v}(f,\mathbf{s},\chi)}{\zeta_v(1+s)}=q_v^{-({e_v(\mathfrak{n})})/{2}}\overline{\chi}_v(\varpi_v)^{e_v(\mathfrak{n})}\sum_{0\leqslant r_3\leqslant e_v(\mathfrak{n})}\chi_v^2(\varpi_v^{r_3})\omega_v(\varpi_v^{r_3})q_v^{-r_3s}.\end{equation}

Proof. Denote $e_v(y) = r_2$ . Since $f_v$ is bi- $K_v$ -invariant, we have

(4.10) \begin{equation}\mathcal{I}_v(x_v) = \sum_{r_2 \in \mathbb{Z}} \overline{\chi}_v(\varpi_v^{r_2})\int_{F_v}f_v\bigg(\!\!\begin{pmatrix} \varpi_v^{r_2} &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg)\psi_v(x_v b_v) \,db_v.\end{equation}

By the definition of $f_v$ in (2.17), we have

(4.11) \begin{align}f_v\bigg(\!\!\begin{pmatrix} \varpi_v^{r_2} &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg)= q_v^{-({e_v(\mathfrak{n})})/{2}} \int_{F_v^{\times}} \omega_v(z)\sum_{\substack{i+j=e_v(\mathfrak{n}) \\ i\geqslant j\geqslant 0}}\mathbf{1}_{K_v \operatorname{diag}(\varpi_v^i, \varpi_v^j) K_v} \bigg(z \begin{pmatrix} \varpi_v^{r_2} &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg) \,d^{\times}z.\end{align}

Write $e_v(z) = r_3$ . By the Cartan decomposition, the sum

\[\sum_{\substack{i+j=e_v(\mathfrak{n}) \\ i\geqslant j\geqslant 0}} \mathbf{1}_{K_v \operatorname{diag}(\varpi_v^i, \varpi_v^j) K_v}\bigg(z \begin{pmatrix} \varpi_v^{r_2} &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg) \]

boils down to the indicator function of the constraints

\begin{align*}\begin{cases}2r_3 + r_2 = e_v(\mathfrak{n}), \\r_3 \geqslant 0, \quad r_3 + r_2 \geqslant 0, \quad r_3 + e_v(b_v) \geqslant 0.\end{cases}\end{align*}

As a consequence, we obtain from (4.11) that

(4.12) \begin{equation}f_v\bigg(\!\!\begin{pmatrix} \varpi_v^{r_2} &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg)= q_v^{-({e_v(\mathfrak{n})})/{2}}\sum_{0 \leqslant r_3 \leqslant e_v(\mathfrak{n})}\omega_v(\varpi_v)^{r_3}\mathbf{1}_{\substack{r_2 = e_v(\mathfrak{n}) - 2r_3 \\ e_v(b_v) \geqslant -r_3}}.\end{equation}

Substituting (4.12) into (4.10) gives

\begin{align*}\mathcal{I}_v(x_v) = q_v^{-({e_v(\mathfrak{n})})/{2}} \overline{\chi}_v(\varpi_v)^{e_v(\mathfrak{n})}\sum_{0 \leqslant r_3 \leqslant e_v(\mathfrak{n})} \chi_v^2(\varpi_v^{r_3}) \omega_v(\varpi_v^{r_3})\int_{\mathfrak{p}_v^{-r_3}} \psi_v(x_v b_v)\,db_v,\end{align*}

which simplifies to

(4.13) \begin{equation}\mathcal{I}_v(x_v) = q_v^{-({e_v(\mathfrak{n})})/{2}} \overline{\chi}_v(\varpi_v)^{e_v(\mathfrak{n})}\sum_{0 \leqslant r_3 \leqslant e_v(\mathfrak{n})}\chi_v^2(\varpi_v^{r_3}) \omega_v(\varpi_v^{r_3}) q_v^{r_3}\cdot\mathbf{1}_{e_v(x_v) \geqslant r_3}.\end{equation}

Therefore, (4.9) follows from substituting (4.13) into the definition (4.8).

Lemma 4.5. Let $v_0, v_1$ be the fixed places constructed in § 2.1.4, and let $v \in \{v_0, v_1\}$ . Suppose that $0 \lt \varepsilon \lt 10^{-1}$ and $|s| = \varepsilon$ . Then,

(4.14) \begin{equation}\bigg|\frac{J_{{\rm Small},v}(f,\mathbf{s},\chi)}{\zeta_v(1+s)}\bigg| \ll 1.\end{equation}

Proof. By the constructions in §§ 2.2.3 and 2.2.4, the local test function $f_v$ has compact support modulo the center. Hence, the function

\begin{align*}x_v \mapsto \mathcal{I}_v(x_v)=\int_{F_v^{\times}} \int_{F_v}f_v\bigg(\!\!\begin{pmatrix} y_v &\;\; b_v \\ & 1 \end{pmatrix}\!\!\bigg)\psi_v(x_v b_v) \overline{\chi}_v(y_v) \,db_v \,d^{\times}y_v,\end{align*}

as the Fourier transform of a Bruhat function, has compact support. Consequently, $J_{{\rm Small},v}(f,\mathbf{s},\chi)$ converges absolutely, implying (4.14).

4.3 Local estimates at Archimedean places

Let $v\mid \infty.$ Recall that

\begin{align*}\mathcal{I}_v(x_v)=\int_{F_v^{\times}}\int_{F_v}f_v\bigg(\!\!\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\!\!\bigg)\psi_v(x_vb_v)\overline{\chi}_v(y_v)\,db_v\,d^{\times}y_v,\end{align*}

and $J_{{\rm Small},v}(f,\mathbf{s},\chi)$ is defined by (4.8).

Lemma 4.6. Let $v\mid \infty.$ Suppose that $0 \lt \varepsilon \lt 10^{-1}$ and $|s| = \varepsilon$ . Then

(4.15) \begin{equation}J_{{\rm Small},v}(f,\mathbf{s},\chi)\ll T_v^{\frac{1}{2}+\varepsilon}.\end{equation}

– Suppose that $\pi_v\otimes\chi_v$ is of type I. Then $f_v$ is constructed in § 2.2.1. Note that the function

(4.16) \begin{equation}x_v\mapsto \int_{F_v}f_v\bigg(\!\!\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\!\!\bigg)\psi_v(x_vb_v)\,db_v\end{equation}

is the Fourier transform. By the construction of $f_v$ (see [Reference NelsonNel23, § 14] or [Reference Nelson and VenkateshNV21, § 2.5]), it follows that

(4.17) \begin{equation}f_v\bigg(\!\!\begin{pmatrix} y_v&\;\;b_v\\ &1\end{pmatrix}\!\!\bigg)\ll T_v^{-\infty}\quad \text{if}\|b_v|_v\gg T_v^{-\frac{1}{2}+\varepsilon}.\end{equation}

Moreover, the function in (4.16) vanishes uniformly for $x_v$ and $y_v$ unless $x_vT_v^{-1}-\gamma_v\ll T_v^{-\frac{1}{2}+\varepsilon}$ for some $\gamma_v$ , which is determined by the chosen element $\tau_v$ in the coadjoint orbit of $\pi_v \otimes \chi_v$ (see § 2.2.1). Consequently,

(4.18) \begin{equation}\mathcal{I}_v(x_v)\equiv 0\quad \text{unless } x_{v} T_v^{-1}-\gamma_v\ll T_v^{-\frac{1}{2}+\varepsilon}.\end{equation}

Therefore, it follows from (4.17) and (4.18) that

(4.19) \begin{equation}\mathcal{I}_v(x_v)\ll \mathbf{1}_{x_vT_v^{-1}-\gamma_v\ll T_v^{-\frac{1}{2}+\varepsilon}}\cdot T_{v}^{1+\varepsilon}\cdot T_v^{-\frac{1}{2}+\varepsilon}\cdot T_v^{-\frac{1}{2}+\varepsilon}+O(T_v^{-\infty}),\end{equation}

where the factor $T_{v}^{1+\varepsilon}$ comes from the sup-norm estimate of $f_v$ (see [Reference NelsonNel21, (14.13)]), the first factor $T_v^{-\frac{1}{2}+\varepsilon}$ comes from the range of $y_v$ according to the support of $f_v$ (see (2.6)), and the second $T_v^{-\frac{1}{2}+\varepsilon}$ comes from the essential range of $b_v,$ i.e., $|b_v|_v\ll T_v^{-\frac{1}{2}+\varepsilon}.$

As a consequence of (4.19), we obtain

\begin{align*}J_{{\rm Small},v}(f,\mathbf{s},\chi) = \int_{F_v^{\times}} \mathcal{I}_v(x_v) |x_v|_v^{1+s} \,d^{\times}x_v\ll T_v^{3\varepsilon} \int_{x_v T_v^{-1} - \gamma_v \ll T_v^{-1/2+\varepsilon}} |x_v|_v^{{\rm Re}(s)} \,dx_v,\end{align*}

which leads to (4.15) after a slight adjustment of $\varepsilon$ .

– Suppose that $\pi_v \otimes \chi_v$ is of type II. Then, $f_v$ is constructed in § 2.2.2. In this case, (4.15) follows from a straightforward calculation using the constructions in (2.10)–(2.12).

Therefore, the estimate (4.15) follows from the above discussions.

4.4 Proof of Proposition 4.1

Let $V:= \prod_{v \lt \infty} \operatorname{Vol}(\overline{K_{0,v}[m_v]})$ . Suppose that $0 \lt \varepsilon \lt 10^{-1}$ and $|s| = \varepsilon$ .

By combining Lemmas 4.3, 4.4, 4.5, and 4.6 with the expression (4.2), we obtain

\begin{align*}| J^{{\rm Reg}}_{{\rm Geo},{\rm Small}}(f,\mathbf{s},\chi) | \ll |\zeta_F(1+s)| V^{-1} N_F(\mathfrak{n})^{-\frac{1}{2}+\varepsilon} \prod_{v \mid \infty} T_v^{\frac{1}{2}+\varepsilon}.\end{align*}

Therefore, Proposition 4.1 follows from the bound $|\zeta_F(1+s)| \ll \varepsilon^{-1}$ and the following volume calculation:

(4.20) \begin{equation}V^{-1} = [M, M'Q] \prod_{v \mid [M, M'Q]} (1 + q_v^{-1}) \ll [M, M'Q]^{1+\varepsilon}.\end{equation}

6.1 Local estimates of $\mathcal{E}_v(t)$ at $v\in \Sigma_{F,\operatorname{fin}}$ : $v\nmid\mathfrak{Q}\mathfrak{n}$

Recall that $v_0$ and $v_1$ are the fixed auxiliary finite places defined in § 2.1.4. Let $\mathfrak{n}$ be the integral ideal defined in § 2.2.5, and let $\mathfrak{Q}$ be the conductor of $\chi$ , as defined in § 2.1.3.

• – Suppose that $v\not\in \{v_0,v_1\}$ . Then

  (6.3) \begin{equation}\mathcal{E}_v(t)\ll_{F_v} \frac{(1-e_v(1-t))(1+e_v(t)-2e_v(1-t))}{\operatorname{Vol}(\overline{K_{0,v}[m_v]})}\mathbf{1}_{\substack{e_v(t-1)\leqslant 0\\ e_v(t)-e_v(1-t)\geqslant m_v}}.\end{equation}

• – Moreover, $\mathcal{E}_v(t)=1$ if $e_v(t)=e_v(1-t)=0,$ and $v\nmid \mathfrak{D}_F.$ In particular, $\mathcal{E}_v(t)=1$ for all but finitely many $v\in \Sigma_{F,\operatorname{fin}}$ .

• – Suppose that $v=v_0$ . Then

  (6.4) \begin{equation}\mathcal{E}_v(t)\ll (1-e_v(1-t))(1+e_v(t)-2e_v(1-t))\mathbf{1}_{\substack{e_v(t-1)\leqslant 4\\ e_v(t)-e_v(1-t)\geqslant -4}},\end{equation}
  where implied constant depends only on $v_0$ and is therefore absolute.

• – Suppose that $v=v_1$ . Then

  (6.5) \begin{equation}\mathcal{E}_v(t)\ll (1+e_v(t))\mathbf{1}_{\substack{e_v(t)\geqslant 1}},\end{equation}
  where the implied constant depends only on $v_1$ and is therefore absolute.

– Suppose that $v\not\in \{v_0,v_1\}$ . By the construction of $f_v$ , $f_v\big(\!\big(\begin{smallmatrix} y_v&\;\;x_v^{-1}t\\ x_vy_v&\;\;1\end{smallmatrix}\big)\!\big)\equiv 0$ unless there exists some $z_v\in F_v^{\times}$ such that