2.1 Extension of Zagier’s regularized integrals

In this subsection we recall and summarize our extension of the theory of regularized integrals in [Reference WuWu18, §§ 2 and 3] without proofs. This extension fits well in the context of the Rankin–Selberg trace formula. It could not be well understood in the framework of the subconvexity problem. Hence we encourage the interested reader to read [Reference WuWu18, §§ 2 and 3] for a better understanding.

We begin by recalling the following space of functions on the automorphic quotient of $\text{GL}_{2}$ over a general number field $\mathbf{F}$ with the ring of adeles  $\mathbb{A}$ .

Definition 2.1 [Reference WuWu18, Definition 2.14].

Let $\unicode[STIX]{x1D714}$ be a unitary character of $\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }$ . Let $\unicode[STIX]{x1D711}$ be a smooth function on $\text{GL}_{2}(\mathbf{F})\backslash \text{GL}_{2}(\mathbb{A})$ with central character  $\unicode[STIX]{x1D714}$ . We call $\unicode[STIX]{x1D711}$ finitely regularizable if there exist unitary characters $\unicode[STIX]{x1D712}_{i}:\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }\rightarrow \mathbb{C}^{(1)}$ , $\unicode[STIX]{x1D6FC}_{i}\in \mathbb{C}$ , $n_{i}\in \mathbb{N}$ , and smooth functions $f_{i}\in \text{Ind}_{\mathbf{B}(\mathbb{A})\cap \mathbf{K}}^{\mathbf{K}}(\unicode[STIX]{x1D712}_{i},\unicode[STIX]{x1D714}\unicode[STIX]{x1D712}_{i}^{-1})$ for $1\leqslant i\leqslant l$ , such that:

1. (i) for any $M\gg 1$ ,

   $$\begin{eqnarray}\unicode[STIX]{x1D711}(n(x)a(y)k)=\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(n(x)a(y)k)+O(|y|_{\mathbb{ A}}^{-M})\quad \text{as}~|y|_{\mathbb{ A}}\rightarrow \infty ;\end{eqnarray}$$

2. (ii) we can differentiate the above equality with respect to the universal enveloping algebra of the Lie algebra of $\text{GL}_{2}(\mathbb{A}_{\infty })$ .

Here we have written/defined the essential constant term

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(n(x)a(y)k)=\unicode[STIX]{x1D711}_{\boldsymbol{ N}}^{\ast }(a(y)k)=\mathop{\sum }_{i=1}^{l}\unicode[STIX]{x1D712}_{i}(y)|y|_{\mathbb{A}}^{1/2+\unicode[STIX]{x1D6FC}_{i}}\log ^{n_{i}}|y|_{\mathbb{A}}\cdot f_{i}(k).\end{eqnarray}$$

In this case, we call ${\mathcal{E}}\text{x}(\unicode[STIX]{x1D711})=\{\unicode[STIX]{x1D712}_{i}|\cdot |^{1/2+\unicode[STIX]{x1D6FC}_{i}}:1\leqslant i\leqslant l\}$ the exponent set of  $\unicode[STIX]{x1D711}$ , and define

$$\begin{eqnarray}{\mathcal{E}}\text{x}^{+}(\unicode[STIX]{x1D711})=\{\unicode[STIX]{x1D712}_{i}|\cdot |^{1/2+\unicode[STIX]{x1D6FC}_{i}}\in {\mathcal{E}}\text{x}(\unicode[STIX]{x1D711}):\Re \unicode[STIX]{x1D6FC}_{i}\geqslant 0\},\quad {\mathcal{E}}\text{x}^{-}(\unicode[STIX]{x1D711})=\{\unicode[STIX]{x1D712}_{i}|\cdot |^{1/2+\unicode[STIX]{x1D6FC}_{i}}\in {\mathcal{E}}\text{x}(\unicode[STIX]{x1D711}):\Re \unicode[STIX]{x1D6FC}_{i}\leqslant 0\}.\end{eqnarray}$$

The space of finitely regularizable functions with central character $\unicode[STIX]{x1D714}$ is denoted by ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ .

Obviously ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ is stable under the right regular translation of $\text{GL}_{2}(\mathbb{A})$ and contains the Schwartz space with central character  $\unicode[STIX]{x1D714}$ , hence the space of smooth cusp forms. It also contains any finite product of Eisenstein series [Reference WuWu18, Remark 2.19]. In the case $\unicode[STIX]{x1D714}=1$ and for any $\unicode[STIX]{x1D711}\in {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},1)$ , the integral

$$\begin{eqnarray}R(s,\unicode[STIX]{x1D711}):=\int _{\mathbb{A}^{\times }\times \mathbf{K}}(\unicode[STIX]{x1D711}_{\mathbf{N}}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y)\unicode[STIX]{x1D705})|y|_{\mathbb{ A}}^{s-1/2}\,d^{\times }y\,d\unicode[STIX]{x1D705}\end{eqnarray}$$

is convergent for any $\Re s\gg 1$ and admits meromorphic continuation. We use it to define the regularized integral as

$$\begin{eqnarray}\displaystyle & {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},1)\rightarrow \mathbb{C}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D711}\mapsto \int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}(g)\,dg:=\frac{1}{\text{Vol}([\text{PGL}_{2}])}\biggl(\text{Res}_{s=1/2}R(s,\unicode[STIX]{x1D711})+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FC}_{i}=-1 \\ n_{i}=0 \\ \unicode[STIX]{x1D712}_{1}(\mathbb{A}^{(1)})=1}}\int _{\mathbf{K}}f_{i}(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\biggr). & \displaystyle \nonumber\end{eqnarray}$$

If $f\in \text{Ind}_{\mathbf{B}(\mathbb{A})}^{\text{GL}_{2}(\mathbb{A})}(\unicode[STIX]{x1D712}_{1},\unicode[STIX]{x1D712}_{2})$ such that $\unicode[STIX]{x1D712}_{1}\unicode[STIX]{x1D712}_{2}^{-1}=|\cdot |_{\mathbb{A}}^{i\unicode[STIX]{x1D707}}$ for some $\unicode[STIX]{x1D707}\in \mathbb{R}$ , we introduce the regularizing Eisenstein series as [Reference WuWu18, Definition 2.16]

(2.1) $$\begin{eqnarray}\text{E}^{\text{reg}}(s,f)(g)=\text{E}(s,f)(g)-\frac{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(1-2s-i\unicode[STIX]{x1D707}_{j})}{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(1+2s+i\unicode[STIX]{x1D707}_{j})}\int _{\mathbf{K}}f(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\cdot \unicode[STIX]{x1D712}_{1}^{-1}(\det g)|\det g|_{\mathbb{ A}}^{i\unicode[STIX]{x1D707}_{j}/2}.\end{eqnarray}$$

For any $\unicode[STIX]{x1D711}\in {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ with auxiliary data given in Definition 2.1 we define [Reference WuWu18, (2.3)]

(2.2) $$\begin{eqnarray}{\mathcal{E}}(\unicode[STIX]{x1D711})=\mathop{\sum }_{\substack{ \Re \unicode[STIX]{x1D6FC}_{j}>0 \\ \unicode[STIX]{x1D6FC}_{j}\neq 1/2+i\unicode[STIX]{x1D707}_{j}}}\frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}\text{E}(\unicode[STIX]{x1D6FC}_{j},f_{j})+\mathop{\sum }_{\substack{ \Re \unicode[STIX]{x1D6FC}_{j}>0 \\ \unicode[STIX]{x1D6FC}_{j}=1/2+i\unicode[STIX]{x1D707}_{j}}}\frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}\text{E}^{\text{reg}}(\unicode[STIX]{x1D6FC}_{j},f_{j}),\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{j}\in \mathbb{R}$ is defined only if $\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}_{j}^{2}(y)=|y|_{\mathbb{A}}^{-2i\unicode[STIX]{x1D707}_{j}}$ . This defines a linear map

$$\begin{eqnarray}{\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})\rightarrow {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714}),\quad \unicode[STIX]{x1D711}\mapsto {\mathcal{E}}(\unicode[STIX]{x1D711}),\end{eqnarray}$$

such that $\unicode[STIX]{x1D711}-{\mathcal{E}}(\unicode[STIX]{x1D711})\in \text{L}^{1}(\text{GL}_{2},\unicode[STIX]{x1D714})$ , which is $\text{GL}_{2}(\mathbb{A})$ -intertwining when ${\mathcal{E}}\text{x}(\unicode[STIX]{x1D711})$ does not contain $|\cdot |_{\mathbb{A}}$ . We denote the image by ${\mathcal{E}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ . Moreover, if ${\mathcal{E}}\text{x}^{+}(\unicode[STIX]{x1D711})\cap {\mathcal{E}}\text{x}^{-}(\unicode[STIX]{x1D711})=\emptyset$ then ${\mathcal{E}}(\unicode[STIX]{x1D711})$ is the unique element in ${\mathcal{E}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ such that $\unicode[STIX]{x1D711}-{\mathcal{E}}(\unicode[STIX]{x1D711})\in \text{L}^{2}(\text{GL}_{2},\unicode[STIX]{x1D714})$ [Reference WuWu18, Proposition 2.25], and we call it the $\text{L}^{2}$ -residue of $\unicode[STIX]{x1D711}$ [Reference WuWu18, Definition 2.26]. In the case $\unicode[STIX]{x1D714}=1$ , ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},1)$ is in the range of applicability of the regularized integral and [Reference WuWu18, Proposition 2.27]

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}(g)\,dg=\int _{[\text{PGL}_{2}]}(\unicode[STIX]{x1D711}(g)-{\mathcal{E}}(\unicode[STIX]{x1D711})(g))\,dg.\end{eqnarray}$$

In particular, the above equation proves the $\text{GL}_{2}(\mathbb{A})$ -invariance of the regularized integral as a functional on ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},1)$ , when ${\mathcal{E}}\text{x}(\unicode[STIX]{x1D711})$ does not contain $|\cdot |_{\mathbb{A}}$ . In this case the above equality was originally due to Zagier [Reference ZagierZag82]. In [Reference WuWu18, Theorem 2.12 and Definition 2.13] we carefully generalized this theory into the adelic setting and proved the above equality without constraint on ${\mathcal{E}}\text{x}(\unicode[STIX]{x1D711})$ .

In view of the inclusion [Reference WuWu18, Remark 2.19]

$$\begin{eqnarray}{\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714}_{1})\cdot {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714}_{2})\subset {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}_{2}),\end{eqnarray}$$

we can consider the following bilinear form. Let $\unicode[STIX]{x1D70B}_{j}$ , $j=1,2$ , be two principal series representations with central character $\unicode[STIX]{x1D714}_{j}$ satisfying $\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}_{2}=1$ . Let $V_{j}$ be the vector space of $\unicode[STIX]{x1D70B}_{j}$ realized in the induced model from $\mathbf{B}(\mathbb{A})$ with subspace of smooth vectors $V_{j}^{\infty }$ . We then get a $\text{GL}_{2}(\mathbb{A})$ -invariant bilinear form

$$\begin{eqnarray}V_{1}^{\infty }\times V_{2}^{\infty }\rightarrow \mathbb{C},\quad (f_{1},f_{2})\mapsto \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}(f_{1})(g)\text{E}(f_{2})(g)\,dg,\end{eqnarray}$$

where $\text{E}(f_{j})$ should be suitably regularized if $\unicode[STIX]{x1D70B}_{j}$ is at a position which creates a pole/zero for the relevant Eisenstein series. In [Reference WuWu18, Theorem 3.5] we succeeded in identifying this bilinear form in the induced model. In order to present the result, we need to introduce some extra notation. If we identify for any $s\in \mathbb{C}$ the space of functions $\unicode[STIX]{x1D70B}_{s}$ with  $H$ , where

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{s}:=\text{Ind}_{\mathbf{B}(\mathbb{A})}^{\text{GL}_{2}(\mathbb{A})}(|\cdot |_{\mathbb{ A}}^{s},|\cdot |_{\mathbb{ A}}^{-s}),\quad H:=\text{Ind}_{\boldsymbol{ B}(\mathbb{A})\cap \mathbf{K}}^{\mathbf{K}}1,\end{eqnarray}$$

then we can regard the intertwining operator ${\mathcal{M}}_{s}:\unicode[STIX]{x1D70B}_{s}\rightarrow \unicode[STIX]{x1D70B}_{-s}$ as a map from $H$ to itself. Using the flat section map $H\rightarrow \unicode[STIX]{x1D70B}_{s}$ , $f\mapsto f_{s}$ , we mean

$$\begin{eqnarray}({\mathcal{M}}_{s}f_{s})(a(y)\unicode[STIX]{x1D705})=:|y|_{\mathbb{A}}^{1/2-s}({\mathcal{M}}_{s}f)(\unicode[STIX]{x1D705}),\quad \text{i.e.,}~{\mathcal{M}}_{s}f_{s}=({\mathcal{M}}_{s}f)_{-s}.\end{eqnarray}$$

Let $e_{0}\in H$ be the constant function taking value 1. Define

$$\begin{eqnarray}\text{P}_{\mathbf{K}}:H\rightarrow \mathbb{C},\quad f\mapsto \int _{\mathbf{K}}f(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705},\end{eqnarray}$$

where $d\unicode[STIX]{x1D705}$ is the probability Haar measure on $\mathbf{K}$ . We obtain a map from $H$ to itself,

$$\begin{eqnarray}\widetilde{{\mathcal{M}}}_{s}:={\mathcal{M}}_{s}\circ (I-\text{P}_{\mathbf{K}}e_{0}),\end{eqnarray}$$

where $I$ is the identity map. Since ${\mathcal{M}}_{s}$ is ‘diagonalizable’, we obtain the Taylor expansion as operators

$$\begin{eqnarray}{\mathcal{M}}_{s}f=\mathop{\sum }_{n=0}^{\infty }\frac{s^{n}}{n!}{\mathcal{M}}_{0}^{(n)}f,\quad \biggl(\text{respectively}~\widetilde{{\mathcal{M}}}_{1/2+s}f=\mathop{\sum }_{n=0}^{\infty }\frac{s^{n}}{n!}\widetilde{{\mathcal{M}}}_{1/2}^{(n)}f\biggr).\end{eqnarray}$$

Theorem 2.2 [Reference WuWu18, Theorem 3.5].

The regularized integral of the product of two unitary Eisenstein series is computed as follows.

1. (i) Suppose $\unicode[STIX]{x1D709}_{1}\neq \unicode[STIX]{x1D709}_{2}$ . If $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2}),\unicode[STIX]{x1D70B}_{2}=\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709}_{1}^{-1},\unicode[STIX]{x1D709}_{2}^{-1})$ (respectively, $\unicode[STIX]{x1D70B}_{2}=\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709}_{2}^{-1},\unicode[STIX]{x1D709}_{1}^{-1})$ ), then

   $$\begin{eqnarray}\displaystyle & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}(0,f_{1})\text{E}(0,f_{2})=\frac{2\unicode[STIX]{x1D706}_{\mathbf{F}}^{(0)}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\text{P}_{\mathbf{K}}(f_{1}f_{2})-\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(1)}f_{1}\cdot {\mathcal{M}}_{0}f_{2}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \biggl(\text{respectively}~\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(0)}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}(\text{P}_{\mathbf{K}}(f_{1}{\mathcal{M}}_{0}f_{2})+\text{P}_{\mathbf{K}}(f_{2}{\mathcal{M}}_{0}f_{1}))-\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(1)}f_{1}\cdot f_{2})\biggr). & \displaystyle \nonumber\end{eqnarray}$$

2. (ii) If $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}),\unicode[STIX]{x1D70B}_{2}=\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709}^{-1},\unicode[STIX]{x1D709}^{-1})$ , then

   $$\begin{eqnarray}\displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{(1)}(0,f_{1})\text{E}^{(1)}(0,f_{2}) & = & \displaystyle \frac{4\unicode[STIX]{x1D706}_{\mathbf{F}}^{(2)}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{-1}(0)}\text{P}_{\mathbf{K}}(f_{1}f_{2})+\frac{4\unicode[STIX]{x1D706}_{\mathbf{F}}^{(2)}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{-1}(0)}\text{P}_{\mathbf{K}}(f_{1}\cdot {\mathcal{M}}_{0}^{(1)}f_{2})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(0)}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{-1}(0)}\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(1)}f_{1}\cdot {\mathcal{M}}_{0}^{(1)}f_{2})-\frac{1}{3}\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(3)}f_{1}\cdot f_{2})\nonumber\\ \displaystyle & & \displaystyle -\,\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(2)}f_{1}\cdot {\mathcal{M}}_{0}^{(1)}f_{2}).\nonumber\end{eqnarray}$$

Here we have written [Reference WuWu18, (2.2)]

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\mathbf{F}}(s):=\frac{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(-2s)}{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(2+2s)}=\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}{s}+\mathop{\sum }_{n=0}^{\infty }\frac{s^{n}}{n!}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n)}(0).\end{eqnarray}$$

2.2 Regularized triple product formula

Let us first complete the analysis for products of two Eisenstein series. We recall a lemma.

Lemma 2.3 [Reference WuWu18, Lemma 3.4].

Let $f,f_{1},f_{2}\in \text{Res}_{\mathbf{K}}^{\text{GL}_{2}(\mathbb{A})}\unicode[STIX]{x1D70B}(1,1)$ . For $0\neq s\in \mathbb{C}$ small, we have, for any $n,n_{1},n_{2}\in \mathbb{N}$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2}+s,f\biggr)=-\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n)}(s)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\text{P}_{\mathbf{K}}(f); & \displaystyle \nonumber\\ \displaystyle & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg},(n_{1})}\biggl(\frac{1}{2}+s,f_{1}\biggr)\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2},f_{2}\biggr)=0. & \displaystyle \nonumber\end{eqnarray}$$

We also recall the technique of deformation [Reference WuWu18, (3.1)], inspired by the work of Michel and Venkatesh. In general, if $\unicode[STIX]{x1D711}\in {\mathcal{A}}^{\text{fr}}(\text{PGL}_{2})$ , ${\mathcal{E}}\in {\mathcal{E}}(\text{PGL}_{2})$ are given, so that $\unicode[STIX]{x1D711}-{\mathcal{E}}\in \text{L}^{1}([\text{PGL}_{2}])$ , and if we can find continuous families $\unicode[STIX]{x1D711}_{s}\in {\mathcal{A}}^{\text{fr}}(\text{PGL}_{2})$ , ${\mathcal{E}}_{s}\in {\mathcal{E}}(\text{PGL}_{2})$ which coincide with $\unicode[STIX]{x1D711}$ , ${\mathcal{E}}$ at $s=0$ , then we have

(2.3) $$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}=\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}-{\mathcal{E}}=\lim _{s\rightarrow 0}\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}_{s}-{\mathcal{E}}_{s}=\lim _{s\rightarrow 0}\biggl(\int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}_{s}-\int _{[\text{PGL}_{2}]}^{\text{reg}}{\mathcal{E}}_{s}\biggr).\end{eqnarray}$$

We turn to the study of regularized integrals of the form

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}\biggl(\frac{1}{2},f_{1}\biggr)\text{E}\biggl(\frac{1}{2},f_{2}\biggr).\end{eqnarray}$$

Denote $e=e_{1}$ . We can write

$$\begin{eqnarray}\displaystyle \text{E}_{\mathbf{N}}^{\text{reg}}(s,f) & = & \displaystyle f_{s}+(\widetilde{{\mathcal{M}}}_{s}f)_{-s}+\unicode[STIX]{x1D706}_{\mathbf{F}}\biggl(s-\frac{1}{2}\biggr)\text{P}_{\mathbf{K}}(f)(e_{-s}-e_{-1/2});\nonumber\\ \displaystyle \text{E}_{\mathbf{N}}^{\text{reg},(n)}\biggl(\frac{1}{2},f\biggr) & = & \displaystyle f_{1/2}^{(n)}+\mathop{\sum }_{k=0}^{n}\binom{n}{k}(-1)^{k}(\widetilde{{\mathcal{M}}}_{1/2}^{(n-k)}f)_{-1/2}^{(k)}\nonumber\\ \displaystyle & & \displaystyle +\,\text{P}_{\mathbf{K}}(f)\cdot \bigg\{\frac{(-1)^{n+1}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}{n+1}e_{-1/2}^{(n+1)}+\mathop{\sum }_{k=1}^{n}\binom{n}{k}(-1)^{k}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n-k)}(0)e_{-1/2}^{k}\bigg\},\nonumber\end{eqnarray}$$

from which one can easily deduce $\text{E}_{\mathbf{N}}^{\text{reg}}(1/2+s,f_{1})\text{E}_{\mathbf{N}}^{\text{reg},(n_{2})}(1/2,f_{2})$ . We tentatively define

$$\begin{eqnarray}\displaystyle {\mathcal{E}}^{\text{reg}}(s) & := & \displaystyle \text{E}^{(n_{2})}\biggl(\frac{3}{2}+s,f_{1}f_{2}\biggr)+\mathop{\sum }_{k=0}^{n_{2}}\binom{n_{2}}{k}(-1)^{k}\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2}+s,f_{1}\widetilde{{\mathcal{M}}}_{1/2}^{(n_{2}-k)}f_{2}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\text{P}_{\mathbf{K}}(f_{2})\cdot \bigg\{\frac{(-1)^{n_{2}+1}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}{n_{2}+1}\text{E}^{\text{reg},(n_{2}+1)}\biggl(\frac{1}{2}+s,f_{1}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{k=1}^{n_{2}}\binom{n_{2}}{k}(-1)^{k}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{2}-k)}(0)\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2}+s,f_{1}\biggr)\bigg\}\nonumber\\ \displaystyle & & \displaystyle +\,\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2}-s,f_{2}\widetilde{{\mathcal{M}}}_{1/2+s}f_{1}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D706}_{\mathbf{F}}(s)\text{P}_{\mathbf{K}}(f_{1})\cdot \bigg\{\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2}-s,f_{2}\biggr)-\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2},f_{2}\biggr)\bigg\}.\nonumber\end{eqnarray}$$

Applying Lemma 2.3 with $n_{1}=0$ together with (2.3), we get

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg}}\biggl(\frac{1}{2},f_{1}\biggr)\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2},f_{2}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\lim _{s\rightarrow 0}\int _{[\text{PGL}_{2}]}\text{E}^{\text{reg}}\biggl(\frac{1}{2}+s,f_{1}\biggr)\text{E}^{\text{reg},(n_{2})}\biggl(\frac{1}{2},f_{2}\biggr)-{\mathcal{E}}^{\text{reg}}(s)\nonumber\\ \displaystyle & & \displaystyle \quad =\lim _{s\rightarrow 0}\mathop{\sum }_{k=0}^{n_{2}}\binom{n_{2}}{k}\frac{(-1)^{k}}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(k)}(s)\text{P}_{\mathbf{K}}(f_{1}\widetilde{{\mathcal{M}}}_{1/2}^{(n_{2}-k)}f_{2})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{2})}(-s)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\text{P}_{\mathbf{K}}(f_{2}\widetilde{{\mathcal{M}}}_{1/2+s}f_{1})+\text{P}_{\mathbf{K}}(f_{1})\text{P}_{\mathbf{K}}(f_{2})\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\bigg\{\frac{(-1)^{n_{2}+1}\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{2}+1)}(s)}{n_{2}+1}+\mathop{\sum }_{k=1}^{n_{2}}\binom{n_{2}}{k}(-1)^{k}\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{2}-k)}(0)\unicode[STIX]{x1D706}_{\boldsymbol{ F}}^{(k)}(s)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}+\unicode[STIX]{x1D706}_{\mathbf{F}}(s)\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{2})}(-s)\bigg\}.\nonumber\end{eqnarray}$$

Taking Laurent expansions, we verify that the function in $s$ in the range of the above limit is regular at $s=0$ , contrary to appearances. The symmetry

$$\begin{eqnarray}\text{P}_{\mathbf{K}}(f_{1}\widetilde{{\mathcal{M}}}_{1/2}^{(k)}f_{2})=\text{P}_{\mathbf{K}}(f_{2}\widetilde{{\mathcal{M}}}_{1/2}^{(k)}f_{1}),\quad \forall k\in \mathbb{N},\end{eqnarray}$$

must be used. Moreover, it can be differentiated $n_{1}$ times to deduce (iii) in the following theorem.

Next, we give a complement of the main theorem of regularized integrals [Reference WuWu18, Theorem 2.12]. Let $\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D714}$ be Hecke characters with $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}\unicode[STIX]{x1D714}=1$ . Let $f\in \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ and $\unicode[STIX]{x1D711}\in \text{C}^{\infty }(\text{GL}_{2},\unicode[STIX]{x1D714})$ , that is, a smooth function on $\text{GL}_{2}(\mathbf{F})\backslash \text{GL}_{2}(\mathbb{A})$ with central character $\unicode[STIX]{x1D714}$ . Suppose $\unicode[STIX]{x1D711}$ is finitely regularizable, as defined in Definition 2.1.

Proposition 2.5. For $\Re s\gg 1$ sufficiently large,

$$\begin{eqnarray}R(s,\unicode[STIX]{x1D711};f):=\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\int _{\mathbf{K}}(\unicode[STIX]{x1D711}_{\mathbf{N}}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y)\unicode[STIX]{x1D705})f(\unicode[STIX]{x1D705})\unicode[STIX]{x1D709}_{1}(y)|y|_{\mathbb{A}}^{s-1/2}\,d\unicode[STIX]{x1D705}\,d^{\times }y\end{eqnarray}$$

is absolutely convergent. It has a meromorphic continuation to $s\in \mathbb{C}$ . If, in addition,

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:=\max _{j}\{\Re \unicode[STIX]{x1D6FC}_{j}\}<0,\end{eqnarray}$$

then we have, with the right-hand side absolutely converging,

$$\begin{eqnarray}R(s,\unicode[STIX]{x1D711};f)=\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\cdot \text{E}(s,f),\quad \unicode[STIX]{x1D6E9}<\Re s<-\unicode[STIX]{x1D6E9}.\end{eqnarray}$$

In the above region, the possible poles of $R(s,\unicode[STIX]{x1D711};f)$ are:

• ∙ $1/2+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}^{-1})$ if $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}^{-1}$ is trivial on $\mathbb{A}^{(1)}$ ;

• ∙ $(\unicode[STIX]{x1D70C}-1)/2$ where $\unicode[STIX]{x1D70C}$ runs over the non-trivial zeros of $L(s,\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}^{-1})$ .

In particular, $R(s,\unicode[STIX]{x1D711};f)$ is holomorphic for $0\leqslant \Re s<\min (-\unicode[STIX]{x1D6E9},1/2)$ .

Proof. The proof is quite similar to that of [Reference WuWu18, Theorem 2.12(3)], except that ${\mathcal{M}}f_{s}$ is no longer explicitly computable. In fact, we have, for $T>1$ , $\Re s\gg 1$ , using the standard Rankin–Selberg unfolding,

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\cdot \unicode[STIX]{x1D6EC}^{T}\text{E}(s,f)\nonumber\\ \displaystyle & & \displaystyle \quad =R(s,\unicode[STIX]{x1D711};f)-\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\biggl(\int _{\mathbf{K}}(\unicode[STIX]{x1D711}_{\mathbf{N}}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y)\unicode[STIX]{x1D705})f(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\biggr)\unicode[STIX]{x1D709}_{1}(y)|y|_{\mathbb{A}}^{s-1/2}1_{|y|_{\mathbb{A}}>T}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\biggl(\int _{\mathbf{K}}(\unicode[STIX]{x1D711}_{\mathbf{N}}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y)\unicode[STIX]{x1D705}){\mathcal{M}}f_{s}(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\biggr)\unicode[STIX]{x1D709}_{2}(y)|y|_{\mathbb{A}}^{-s-1/2}1_{|y|_{\mathbb{A}}>T}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\text{Vol}(\mathbf{F}^{\times }\backslash \mathbb{A}^{(1)})\biggl(\mathop{\sum }_{j=1}^{l}\int _{\mathbf{K}}f_{j}(\unicode[STIX]{x1D705})f(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\cdot 1_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D709}_{1}(\mathbb{A}^{(1)})=1}\cdot \frac{1}{n_{j}!}\frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}\biggl(\frac{T^{s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}_{j}}}{s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}_{j}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\mathop{\sum }_{j=1}^{l}\int _{\mathbf{K}}f_{j}(\unicode[STIX]{x1D705}){\mathcal{M}}f_{s}(\unicode[STIX]{x1D705})\,d\unicode[STIX]{x1D705}\cdot 1_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D709}_{2}(\mathbb{A}^{(1)})=1}\cdot \frac{(-1)^{n_{j}}}{n_{j}!}\frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}\biggl(\frac{T^{-s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}_{j}^{\prime }}}{-s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}_{j}^{\prime }}\biggr)\biggr),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{j}$ (respectively, $\unicode[STIX]{x1D707}_{j}^{\prime }$ ) is such that

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D709}_{1}(y)=|y|_{\mathbb{A}}^{i\unicode[STIX]{x1D707}_{j}},\quad (\text{respectively,}~\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D709}_{2}(y)=|y|_{\mathbb{A}}^{i\unicode[STIX]{x1D707}_{j}^{\prime }}).\end{eqnarray}$$

We conclude by first shifting $s$ to the desired region, then letting $T\rightarrow \infty$ . The possible poles are encoded in the possible poles of ${\mathcal{M}}f_{s}$ , which are included in those of $L(1+2s,\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}^{-1})^{-1}$ in the above region (cf., for example, [Reference WuWu17b, Corollaries 3.7, 3.10 and Lemma 3.18]).◻

Proposition 2.6. Retain the notation of the previous proposition, with $\unicode[STIX]{x1D6E9}\leqslant -1/2$ . Recall

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(n(x)a(y)k)=\unicode[STIX]{x1D711}_{\boldsymbol{ N}}^{\ast }(a(y)k)=\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D712}_{j}(y)|y|_{\mathbb{A}}^{1/2+\unicode[STIX]{x1D6FC}_{j}}\log ^{n_{i}}|y|_{\mathbb{A}}f_{j}(k).\end{eqnarray}$$

1. (i) If $\unicode[STIX]{x1D709}_{1}\neq \unicode[STIX]{x1D709}_{2}$ , then

   $$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x2202}^{n}R}{\unicode[STIX]{x2202}s^{n}}\biggr)^{\text{hol}}\biggl(\frac{1}{2},\unicode[STIX]{x1D711};f\biggr)=\int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}\cdot \text{E}^{(n)}\biggl(\frac{1}{2},f\biggr)-\mathop{\sum }_{j}^{\prime }\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n+n_{j})}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\text{P}_{\mathbf{K}}(f_{j}f),\end{eqnarray}$$
   where the summation is over $j$ such that $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D712}_{j}(\mathbb{A}^{(1)})=1$ , $\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D712}_{j})=-1/2$ .

2. (ii) If $\unicode[STIX]{x1D709}_{1}=\unicode[STIX]{x1D709}_{2}=\unicode[STIX]{x1D709}$ , then

   $$\begin{eqnarray}\displaystyle \biggl(\frac{\unicode[STIX]{x2202}^{n}R}{\unicode[STIX]{x2202}s^{n}}\biggr)^{\text{hol}}\biggl(\frac{1}{2},\unicode[STIX]{x1D711};f\biggr) & = & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}\cdot \text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},f\biggr)-\mathop{\sum }_{j}^{\prime }\frac{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n+n_{j})}(0)}{\unicode[STIX]{x1D706}_{\mathbf{F}}^{(-1)}(0)}\text{P}_{\mathbf{K}}(f_{j}f)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n)}(0)\cdot \text{P}_{\mathbf{K}}(f\cdot (\unicode[STIX]{x1D709}^{-1}\circ \det ))\cdot \int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\cdot (\unicode[STIX]{x1D709}\circ \det ),\nonumber\end{eqnarray}$$
   where the summation over $j$ is as in the previous case.

Proof. The case (i) being simpler, we only give details for (ii). By twisting, we may assume $\unicode[STIX]{x1D709}=1$ . Let $s$ be small with $\Re s<0$ . The $\text{L}^{2}$ -residue of $\unicode[STIX]{x1D711}\cdot \text{E}(1/2+s,f)$ is given by

$$\begin{eqnarray}{\mathcal{E}}(s):=\mathop{\sum }_{j}\text{E}^{(n_{j})}(s+1+\unicode[STIX]{x1D6FC}_{j},f_{j}f),\end{eqnarray}$$

where the summation is over $j$ such that $\Re \unicode[STIX]{x1D6FC}_{j}>-1$ . Define

$$\begin{eqnarray}{\mathcal{E}}^{\text{reg}}(s):=\mathop{\sum }_{j}^{\prime }\text{E}^{\text{reg},(n_{j})}(s+1+\unicode[STIX]{x1D6FC}_{j},f_{j}f)+\mathop{\sum }_{j}^{\ast }\text{E}^{(n_{j})}(s+1+\unicode[STIX]{x1D6FC}_{j},f_{j}f),\end{eqnarray}$$

where $\sum _{j}^{\prime }$ is the summation as in the statement and $\sum _{j}^{\ast }$ is the rest. By the previous proposition, we have

$$\begin{eqnarray}\displaystyle R\biggl(\frac{1}{2}+s,\unicode[STIX]{x1D711};f\biggr) & = & \displaystyle \!\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\cdot \text{E}\biggl(\frac{1}{2}+s,f\biggr)=\!\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\cdot \text{E}^{\text{reg}}\biggl(\frac{1}{2}+s,f\biggr)+\unicode[STIX]{x1D706}_{\mathbf{F}}(s)\text{P}_{\mathbf{K}}(f)\cdot \!\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle \!\int _{[\text{PGL}_{2}]}\biggl(\unicode[STIX]{x1D711}\cdot \text{E}^{\text{reg}}\biggl(\frac{1}{2}+s,f\biggr)-{\mathcal{E}}^{\text{reg}}(s)\biggr)-\int _{[\text{PGL}_{2}]}({\mathcal{E}}(s)-{\mathcal{E}}^{\text{reg}}(s))\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D706}_{\mathbf{F}}(s)\text{P}_{\mathbf{K}}(f)\cdot \!\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}.\nonumber\end{eqnarray}$$

Since ${\mathcal{E}}^{\text{reg},(n)}(s)$ is the $\text{L}^{2}$ -residue of $\unicode[STIX]{x1D711}\cdot \text{E}^{\text{reg},(n)}(1/2+s,f)$ , we can compare the finite parts of both sides and conclude by

$$\begin{eqnarray}\hspace{132.0pt}{\mathcal{E}}(s)-{\mathcal{E}}^{\text{reg}}(s)=\mathop{\sum }_{j}^{\prime }\unicode[STIX]{x1D706}_{\mathbf{F}}^{(n_{j})}(s)\text{P}_{\boldsymbol{ K}}(f_{j}f).\hspace{128.39996pt}\square\end{eqnarray}$$

Finally, we state and prove a special case of regularized triple product formulas. The method used in the proof is applicable in any general case but only the one treated in the following theorem is used in the current paper.

Theorem 2.7. Let $f_{j}\in \unicode[STIX]{x1D70B}(1,1),j=1,2,3$ . Then, for any $n\in \mathbb{N}$ ,

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\ast }(0,f_{1})\cdot \text{E}^{\ast }(0,f_{2})\cdot \text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},f_{3}\biggr)\end{eqnarray}$$

is the sum of

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x2202}^{n}R}{\unicode[STIX]{x2202}s^{n}}\biggr)^{\text{hol}}\biggl(\frac{1}{2},\text{E}^{\ast }(0,f_{1})\cdot \text{E}^{\ast }(0,f_{2});f_{3}\biggr)\end{eqnarray}$$

and a weighted sum with coefficients depending only on $\unicode[STIX]{x1D706}_{\mathbf{F}}(s)$ of

$$\begin{eqnarray}\displaystyle & \text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(l)}f_{1}\cdot f_{2})\text{P}_{\mathbf{K}}(f_{3}),\quad 0\leqslant l\leqslant 3; & \displaystyle \nonumber\\ \displaystyle & \text{P}_{\mathbf{K}}(f_{1}\cdot f_{2}\cdot \widetilde{{\mathcal{M}}}_{1/2}^{(l)}f_{3}),\quad 0\leqslant l\leqslant \max (2,n),~l=n+3; & \displaystyle \nonumber\\ \displaystyle & \text{P}_{\mathbf{K}}((f_{1}{\mathcal{M}}_{0}f_{2}+f_{2}{\mathcal{M}}_{0}f_{1})\cdot \widetilde{{\mathcal{M}}}_{1/2}^{(l)}f_{3}),\quad 0\leqslant l\leqslant \max (1,n),~l=n+2; & \displaystyle \nonumber\\ \displaystyle & \text{P}_{\mathbf{K}}({\mathcal{M}}_{0}f_{1}\cdot {\mathcal{M}}_{0}f_{2}\cdot \widetilde{{\mathcal{M}}}_{1/2}^{(l)}f_{3}),\quad 0\leqslant l\leqslant n,~l=n+1. & \displaystyle \nonumber\end{eqnarray}$$

Proof. We shall only point out how the computation is effectuated, since the precise formulas are quite long, of no interest for the purpose of the current paper, and would only obscure matters.

$$\begin{eqnarray}\displaystyle {\mathcal{E}}(f_{1},f_{2}) & := & \displaystyle (\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast })^{2}\cdot \big\{\text{E}^{\text{reg},(2)}({\textstyle \frac{1}{2}},f_{1}f_{2})+{\textstyle \frac{1}{2}}\text{E}^{\text{reg},(1)}({\textstyle \frac{1}{2}},f_{1}\cdot {\mathcal{M}}_{0}^{(1)}f_{2}+{\mathcal{M}}_{0}^{(1)}f_{1}\cdot f_{2})\nonumber\\ \displaystyle & & \displaystyle +\,{\textstyle \frac{1}{4}}\text{E}^{\text{reg}}({\textstyle \frac{1}{2}},{\mathcal{M}}_{0}^{(1)}f_{1}\cdot {\mathcal{M}}_{0}^{(1)}f_{2})\big\}\nonumber\end{eqnarray}$$

is the $\text{L}^{2}$ -residue of $\text{E}^{\ast }(0,f_{1})\cdot \text{E}^{\ast }(0,f_{2})$ . Let $\unicode[STIX]{x1D711}:=\text{E}^{\ast }(0,f_{1})\cdot \text{E}^{\ast }(0,f_{2})-{\mathcal{E}}(f_{1},f_{2})$ . Then we need to compute

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\unicode[STIX]{x1D711}\cdot \text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},f_{3}\biggr)+\int _{[\text{PGL}_{2}]}^{\text{reg}}{\mathcal{E}}(f_{1},f_{2})\cdot \text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},f_{3}\biggr).\end{eqnarray}$$

The first term is computed by Proposition 2.6(ii), involving

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}\unicode[STIX]{x1D711}=\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\ast }(0,f_{1})\cdot \text{E}^{\ast }(0,f_{2})=\frac{(\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast })^{2}}{4}\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{(1)}(0,f_{1})\cdot \text{E}^{(1)}(0,f_{2}),\end{eqnarray}$$

which is treated in Theorem 2.2(ii). The second term is treated in Theorem 2.4(iii). ◻

2.3 Extension of global zeta integral

Fixing a central Hecke character $\unicode[STIX]{x1D714}$ over a number field  $\mathbf{F}$ , we extend the global part of the Hecke–Jacquet–Langlands theory to ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ (Definition 2.1), as well as an analogue of the ‘approximate functional equation’. Note that Eisenstein series are in ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ .

Definition 2.8. Let $\unicode[STIX]{x1D711}\in {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ for some Hecke character $\unicode[STIX]{x1D714}$ . For a Hecke character $\unicode[STIX]{x1D712}$ and $s\in \mathbb{C}$ , $\Re s\gg 1$ , we define the zeta functional by

$$\begin{eqnarray}\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})=\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{ A}}^{s-1/2}\,d^{\times }y,\end{eqnarray}$$

where we recall the essential constant term

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(n(x)a(y)k)=\unicode[STIX]{x1D711}_{\boldsymbol{ N}}^{\ast }(a(y)k)=\mathop{\sum }_{i=1}^{l}\unicode[STIX]{x1D712}_{i}(y)|y|_{\mathbb{A}}^{1/2+\unicode[STIX]{x1D6FC}_{i}}\log ^{n_{i}}|y|_{\mathbb{A}}f_{i}(k).\end{eqnarray}$$

Proposition 2.9. $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})$ has a meromorphic continuation to $s\in \mathbb{C}$ with functional equation

$$\begin{eqnarray}\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})=\unicode[STIX]{x1D701}(1-s,\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711}).\end{eqnarray}$$

It has possible poles at $s=-\unicode[STIX]{x1D6FC}_{j}-i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712})$ with $\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}(\mathbb{A}^{(1)})=1$ (respectively, $s=1+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1})$ with $\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}(\mathbb{A}^{(1)})=1$ ), with pure order $n_{j}+1$ . Here $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712})\in \mathbb{R}$ is defined for $\unicode[STIX]{x1D712}(\mathbb{A}^{(1)})=1$ as $\unicode[STIX]{x1D712}(t)=|t|_{\mathbb{A}}^{i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712})}$ .

Proof. By the invariance of $\unicode[STIX]{x1D711}$ at left by $w$ , we can rewrite the zeta integral as

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}) & = & \displaystyle \int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\geqslant 1}}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\leqslant 1}}(w.\unicode[STIX]{x1D711}-w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y^{-1}))\unicode[STIX]{x1D714}\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle -\,\int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\leqslant 1}}\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\leqslant 1}}w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(a(y^{-1}))\unicode[STIX]{x1D714}\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y.\nonumber\end{eqnarray}$$

We can calculate the integral concerning $\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }$ and get

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\geqslant 1}}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int _{\substack{ y\in \mathbf{F}^{\times }\backslash \mathbb{A}^{\times } \\ |y|_{\mathbb{A}}\geqslant 1}}(w.\unicode[STIX]{x1D711}-w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}(y)|y|_{\mathbb{A}}^{1/2-s}\,d^{\times }y+\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\biggl(\mathop{\sum }_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}|_{\mathbb{ A}^{(1)}}=1}\frac{(-1)^{n_{j}+1}f_{j}(1)}{(s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}))^{n_{j}+1}}+\mathop{\sum }_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}|_{\mathbb{ A}^{(1)}}=1}\frac{(-1)^{n_{j}+1}f_{j}(w)}{(1-s+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}))^{n_{j}+1}}\biggr),\nonumber\end{eqnarray}$$

from which we easily deduce all the assertions. ◻

We turn to the special case $\unicode[STIX]{x1D711}(g)=\text{E}^{\ast }(s_{0},\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1};f)(g)$ the usual completed Eisenstein series (respectively, $\text{E}^{\text{reg}}(s_{0},\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1};f)(g)$ defined in equation (2.1)), for which the local computation is the same as for a cusp form. Note that in this case $\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }=\unicode[STIX]{x1D711}_{\mathbf{N}}$ , hence $\unicode[STIX]{x1D711}(a(y))-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(a(y))=\sum _{\unicode[STIX]{x1D6FC}\in \mathbf{F}^{\times }}W_{\unicode[STIX]{x1D711}}(a(\unicode[STIX]{x1D6FC}y))$ .

Proposition 2.10. Let $\unicode[STIX]{x1D711}(g)=\text{E}^{\ast }(s_{0},\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1};f)(g)$ (respectively, $\text{E}^{\text{reg}}(s_{0},\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1};f)(g)$ ) where $\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}$ are Hecke characters and $f\in \unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1}}$ . The zeta functional has a decomposition as an Euler product in which only a finite number of terms are not equal to  $1$ :

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}) & = & \displaystyle \unicode[STIX]{x1D6EC}(s+s_{0},\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})\unicode[STIX]{x1D6EC}(s-s_{0},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1}\unicode[STIX]{x1D712})\cdot \mathop{\prod }_{v}\frac{L_{v}(1+2s_{0},\unicode[STIX]{x1D714}_{v}^{-1}\unicode[STIX]{x1D709}_{v}^{2})}{L_{v}(s+s_{0},\unicode[STIX]{x1D709}_{v}\unicode[STIX]{x1D712}_{v})L_{v}(s-s_{0},\unicode[STIX]{x1D714}_{v}\unicode[STIX]{x1D709}_{v}^{-1}\unicode[STIX]{x1D712}_{v})}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\int _{\mathbf{F}_{v}^{\times }}W_{f_{v}}^{(s_{0})}(a(y_{v}))|y_{v}|_{v}^{s-1/2}\,d^{\times }y_{v},\quad \biggl(\text{respectively}\nonumber\\ \displaystyle \unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}) & = & \displaystyle \frac{\unicode[STIX]{x1D6EC}(s+s_{0},\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})\unicode[STIX]{x1D6EC}(s-s_{0},\unicode[STIX]{x1D714}\unicode[STIX]{x1D709}^{-1}\unicode[STIX]{x1D712})}{\unicode[STIX]{x1D6EC}(1+2s_{0},\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D709}^{2})}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\mathop{\prod }_{v}\frac{L_{v}(1+2s_{0},\unicode[STIX]{x1D714}_{v}^{-1}\unicode[STIX]{x1D709}_{v}^{2})}{L_{v}(s+s_{0},\unicode[STIX]{x1D709}_{v}\unicode[STIX]{x1D712}_{v})L_{v}(s-s_{0},\unicode[STIX]{x1D714}_{v}\unicode[STIX]{x1D709}_{v}^{-1}\unicode[STIX]{x1D712}_{v})}\int _{\mathbf{F}_{v}^{\times }}W_{f_{v}}^{(s_{0})}(a(y_{v}))|y_{v}|_{v}^{s-1/2}\,d^{\times }y_{v}\biggr).\nonumber\end{eqnarray}$$

The method of truncation used in the proof of Proposition 2.9 for the analytic continuation of the global zeta functional is not the only possible one. Another version of truncation on the integral is closely related to the classical approximate functional equation. Let $h_{0}$ be a smooth function supported in the inteval $[0,2)$ , being equal to 1 on $[0,1]$ . For any $A>0$ , we denote by $h_{0,A}$ the function $t\mapsto h_{0}(t/A)$ . We then have, for $\Re s\gg 1$ ,

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}\biggr) & = & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s}(1-h_{0,A}(|y|_{\mathbb{A}}))\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(w.\unicode[STIX]{x1D711}-w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}(y)|y|_{\mathbb{A}}^{-s}h_{0,A}(|y|_{\mathbb{A}}^{-1})\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle -\,\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(a(y))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s}h_{0,A}(|y|_{\mathbb{A}})\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }(a(y))\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}(y)|y|_{\mathbb{A}}^{-s}h_{0,A}(|y|_{\mathbb{A}}^{-1})\,d^{\times }y.\end{eqnarray}$$

For the last two lines, it is not hard to compute their analytic continuation using the form of $\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }$ and the analytic continuation of the Mellin transform of $h_{0}$ as (since $h_{0}^{\prime }$ is of compact support contained in $(1,2)$ )

$$\begin{eqnarray}\displaystyle \mathfrak{M}(h_{0})(s) & = & \displaystyle \int _{0}^{\infty }h_{0}(t)t^{s}\frac{dt}{t}=\frac{1}{s}-\frac{\mathfrak{M}(h_{0}^{\prime })(s+1)+1}{s}\nonumber\\ \displaystyle & = & \displaystyle (-1)^{N}\mathop{\prod }_{j=0}^{N-1}(s+j)^{-1}\mathfrak{M}(h_{0}^{(N)})(s+N),\quad \forall N\in \mathbb{N},s\in \mathbb{C}.\nonumber\end{eqnarray}$$

Remark 2.11. We also have, first for $\Re s\ll -1$ then for $s\in \mathbb{C}$ ,

$$\begin{eqnarray}\mathfrak{M}(1-h_{0})(s)=\int _{0}^{\infty }(1-h_{0})(t)t^{s}\frac{dt}{t}=\frac{\mathfrak{M}(h_{0}^{\prime })(s+1)}{s}=-\mathfrak{M}(h_{0})(s).\end{eqnarray}$$

Then, writing $s_{j}=1/2+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712})$ , the last two lines of (2.4) are defined for $s\in \mathbb{C}$ as

$$\begin{eqnarray}\displaystyle & & \displaystyle -\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(1)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}\frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}(A^{s+s_{j}}\mathfrak{M}(h_{0})(s+s_{j}))\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(1)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}\frac{(-1)^{n_{j}+1}}{(s+s_{j})^{n_{j}+1}}-\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(1)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\mathop{\sum }_{k=0}^{n_{j}}\binom{n_{j}}{k}(\log A)^{n_{j}-k+1}\int _{0}^{\infty }h_{0}^{\prime }(t)t^{s+s_{j}}\log ^{k}t\,dt\cdot \int _{0}^{1}A^{\unicode[STIX]{x1D6FF}(s+s_{j})}\unicode[STIX]{x1D6FF}^{n_{j}-k}\,d\unicode[STIX]{x1D6FF}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(1)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}\int _{0}^{\infty }h_{0}^{\prime }(t)\biggl(\int _{0}^{1}\unicode[STIX]{x1D6FF}^{n_{j}}t^{\unicode[STIX]{x1D6FF}(s+s_{j})}\,d\unicode[STIX]{x1D6FF}\biggr)\log ^{n_{j}+1}t\,dt,\nonumber\end{eqnarray}$$

and, writing $s_{j}^{\prime }=1/2+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1})$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(w)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}(-1)^{n_{j}}\cdot \frac{\unicode[STIX]{x2202}^{n_{j}}}{\unicode[STIX]{x2202}s^{n_{j}}}(A^{s-s_{j}^{\prime }}\mathfrak{M}(h_{0})(s-s_{j}^{\prime }))\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(w)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}\frac{(-1)^{n_{j}+1}}{(s_{j}^{\prime }-s)^{n_{j}+1}}+\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(w)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}(-1)^{n_{j}}\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\mathop{\sum }_{k=0}^{n_{j}}\binom{n_{j}}{k}(\log A)^{n_{j}-k+1}\int _{0}^{\infty }h_{0}^{\prime }(t)t^{s-s_{j}^{\prime }}\log ^{k}t\,dt\cdot \int _{0}^{1}A^{\unicode[STIX]{x1D6FF}(s-s_{j}^{\prime })}\unicode[STIX]{x1D6FF}^{n_{j}-k}\,d\unicode[STIX]{x1D6FF}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\mathop{\sum }_{j=1}^{l}f_{j}(w)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}(-1)^{n_{j}}\int _{0}^{\infty }h_{0}^{\prime }(t)\biggl(\int _{0}^{1}\unicode[STIX]{x1D6FF}^{n_{j}}t^{\unicode[STIX]{x1D6FF}(s-s_{j}^{\prime })}\,d\unicode[STIX]{x1D6FF}\biggr)\log ^{n_{j}+1}t\,dt.\nonumber\end{eqnarray}$$

We separate the terms in the sum $\sum _{j=1}^{l}$ according as $s_{j}=0$ and $s_{j}\neq 0$ (respectively, $s_{j}^{\prime }=0$ and $s_{j}^{\prime }\neq 0$ ). For $s_{j}=0$ (respectively, $s_{j}^{\prime }=0$ ), the finite part at $s=0$ is bounded, with implied constants depending only on $\mathbf{F},n_{j},h_{0}$ , by

$$\begin{eqnarray}\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}1_{s_{j}=0}O(|f_{j}(1)\log ^{n_{j}+1}A|)\quad \biggl(\text{respectively}~\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}1_{s_{j}^{\prime }=0}O(|f_{j}(w)\log ^{n_{j}+1}A|)\biggr).\end{eqnarray}$$

For $s_{j}\neq 0$ (respectively, $s_{j}^{\prime }\neq 0$ ), they are of size at $s=0$ , with implied constants depending only on $\mathbf{F},\unicode[STIX]{x1D6FC}_{j},n_{j},h_{0}$ and an arbitrary $N\in \mathbb{N}$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}1_{s_{j}\neq 0}O\biggl(\!\frac{A^{\Re s_{j}}|f_{j}(1)\log ^{n_{j}}A|}{|s_{j}|^{N}}\!\biggr), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \biggl(\text{respectively}~\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}1_{s_{j}^{\prime }\neq 0}O\biggl(\!\frac{A^{-\Re s_{j}^{\prime }}|f_{j}(w)\log ^{n_{j}}A|}{|s_{j}^{\prime }|^{N}}\!\biggr)\biggr). & \displaystyle \nonumber\end{eqnarray}$$

The first (respectively, second) line of (2.4) is supported in $|y|_{\mathbb{A}}\in [A,\infty )$ (respectively, $[(2A)^{-1},\infty )$ ), hence is well defined for all $s\in \mathbb{C}$ by the rapid decay of $\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast }$ . For the second line at $s=0$ , we can apply Mellin inversion to see

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(w.\unicode[STIX]{x1D711}-w.\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}(y)h_{0,A}(|y|_{\mathbb{A}}^{-1})\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D701}_{\mathbf{F}}^{\ast }(1)\int _{\Re s_{1}=c_{1}\gg 1}A^{s_{1}}\unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s_{1},\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711}\biggr)\mathfrak{M}(h_{0})(s_{1})\frac{ds_{1}}{2\unicode[STIX]{x1D70B}i},\nonumber\end{eqnarray}$$

which is bounded, with implied constant depending only on $\mathbf{F},h_{0}$ and an arbitrary $N\in \mathbb{N}$ , as

$$\begin{eqnarray}A^{c_{1}}O\biggl(\int _{\Re s_{1}=c_{1}\gg 1}\frac{|\unicode[STIX]{x1D701}(1/2+s_{1},\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711})|}{\mathop{\prod }_{m=0}^{N-1}|s_{1}+m|}\frac{|ds_{1}|}{2\unicode[STIX]{x1D70B}}\biggr),\end{eqnarray}$$

where $c_{1}$ can be chosen as any real number such that the integral defining $\unicode[STIX]{x1D701}(1/2+s_{1},\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711})$ is absolutely convergent for $\Re s_{1}\geqslant c_{1}$ . Similarly, we have, for any $B>0$ , that

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\unicode[STIX]{x1D712}(y)(1-h_{0,B})(|y|_{\mathbb{A}})\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \quad =-\int _{\Re s_{2}=c_{2}\gg 1}B^{-s_{2}}\unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s_{2},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}\biggr)\mathfrak{M}(h_{0})(-s_{2})\frac{ds_{2}}{2\unicode[STIX]{x1D70B}i}\nonumber\end{eqnarray}$$

is bounded, with implied constant depending only on $\mathbf{F},h_{0}$ and an arbitrary $N\in \mathbb{N}$ as

$$\begin{eqnarray}B^{-c_{2}}O\biggl(\int _{\Re s_{2}=c_{2}\gg 1}\frac{|\unicode[STIX]{x1D701}(1/2+s_{2},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})|}{\mathop{\prod }_{m=0}^{N-1}|s_{2}+m|}\frac{|ds_{2}|}{2\unicode[STIX]{x1D70B}}\biggr),\end{eqnarray}$$

where $c_{2}$ can be chosen as any real number such that the integral defining $\unicode[STIX]{x1D701}(1/2+s_{2},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})$ is absolutely convergent for $\Re s_{2}\geqslant c_{2}$ .

Definition 2.12. For any function $h:\mathbb{R}_{+}\rightarrow \mathbb{C}$ and any Hecke character $\unicode[STIX]{x1D712}$ , we define the $h$ -truncated (zeta) integral on ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ as

$$\begin{eqnarray}\unicode[STIX]{x1D701}(h,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})=\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\unicode[STIX]{x1D712}(y)(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{\mathbf{N}}^{\ast })(a(y))\,d^{\times }y.\end{eqnarray}$$

In summary, we have obtained the following proposition.

Proposition 2.13. Take $h_{0}$ as indicated at the beginning, some positive constants $0<A<B$ , and define $h(t)=h_{0,B}(t)-h_{0,A}(t)$ , $t>0$ . For any $\unicode[STIX]{x1D711}\in {\mathcal{A}}^{\text{fr}}(\text{GL}_{2},\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D712}_{i},\unicode[STIX]{x1D6FC}_{i},n_{i}$ given in Definition 2.8, we write

$$\begin{eqnarray}s_{j}=1/2+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712})\quad \text{(respectively,}~s_{j}^{\prime }=1/2+\unicode[STIX]{x1D6FC}_{j}+i\unicode[STIX]{x1D707}(\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1})\text{)}\end{eqnarray}$$

if $\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}$ (respectively, $\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}$ ) is trivial on $\mathbb{A}^{(1)}$ , and $\unicode[STIX]{x1D707}$ defined in Proposition 2.9. Then the difference

$$\begin{eqnarray}\unicode[STIX]{x1D701}^{\text{hol}}({\textstyle \frac{1}{2}},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})-\unicode[STIX]{x1D701}(h,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})\end{eqnarray}$$

is bounded, with implied constants depending only on $\mathbf{F},\unicode[STIX]{x1D6FC}_{j},n_{j},h_{0}$ and an arbitrary $N\in \mathbb{N}$ , as the sum of:

1. (i) degenerate polar part,

   $$\begin{eqnarray}\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}1_{s_{j}=0}O(|f_{j}(1)\log ^{n_{j}+1}A|)+\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}1_{s_{j}^{\prime }=0}O(|f_{j}(w)\log ^{n_{j}+1}A|);\end{eqnarray}$$

2. (ii) normal polar part,

   $$\begin{eqnarray}\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D712}}1_{s_{j}\neq 0}O\biggl(\frac{A^{\Re s_{j}}|f_{j}(1)\log ^{n_{j}}A|}{|s_{j}|^{N}}\biggr)+\mathop{\sum }_{j=1}^{l}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D712}_{j}\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1}}1_{s_{j}^{\prime }\neq 0}O\biggl(\frac{A^{-\Re s_{j}^{\prime }}|f_{j}(w)\log ^{n_{j}}A|}{|s_{j}^{\prime }|^{N}}\biggr);\end{eqnarray}$$

3. (iii) lower part,

   $$\begin{eqnarray}A^{c_{1}}O\biggl(\int _{\Re s=c_{1}\gg 1}\frac{|\unicode[STIX]{x1D701}(1/2+s,\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711})|}{\mathop{\prod }_{m=0}^{N-1}|s+m|}\frac{|ds|}{2\unicode[STIX]{x1D70B}}\biggr);\end{eqnarray}$$

4. (iv) upper part,

   $$\begin{eqnarray}B^{-c_{2}}O\biggl(\int _{\Re s=c_{2}\gg 1}\frac{|\unicode[STIX]{x1D701}(1/2+s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})|}{\mathop{\prod }_{m=0}^{N-1}|s+m|}\frac{|ds|}{2\unicode[STIX]{x1D70B}}\biggr).\end{eqnarray}$$

In (iii) (respectively, (iv)), $c_{1}>0$ (respectively, $c_{2}>0$ ) is any real number such that the integral defining $\unicode[STIX]{x1D701}(1/2+s,\unicode[STIX]{x1D714}^{-1}\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711})$ (respectively, $\unicode[STIX]{x1D701}(1/2+s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})$ ) is absolutely convergent for $\Re s\geqslant c_{1}$ (respectively, $\Re s\geqslant c_{2}$ ).

2.4 Classical vectors in spherical series

Let $\mathbf{F}$ be a non-archimedean local field with uniformizer $\unicode[STIX]{x1D71B}$ , absolute valuation $|\cdot |$ , valuation ring  $\mathfrak{o}$ , ideal $\mathfrak{p}$ and cardinality of the residue class field  $q$ . Denote by $\unicode[STIX]{x1D70B}_{s}=\text{Ind}_{\mathbf{B}(\mathbf{F})}^{\text{GL}_{2}(\mathbf{F})}(|\cdot |^{s},|\cdot |^{-s})$ the principal series representation of $\text{PGL}_{2}(\mathbf{F})$ , where $s\in \mathbb{C}$ . For $s\in i\mathbb{R}$ , the underlying Hilbert spaces for the unitary representation $\unicode[STIX]{x1D70B}_{s}$ can be identified with each other. We denote this common Hilbert space by  $H$ , that is,

$$\begin{eqnarray}\text{Res}_{\mathbf{K}}^{\text{GL}_{2}(\mathbf{F})}\unicode[STIX]{x1D70B}_{s}=\text{Ind}_{\mathbf{B}(\mathbf{F})\cap \mathbf{K}}^{\mathbf{K}}(1,1)=:H,\quad \text{where}~\mathbf{K}=\text{GL}_{2}(\mathfrak{o}).\end{eqnarray}$$

We regard $\unicode[STIX]{x1D70B}_{s},s\in \mathbb{C}$ as a family of representations of $\text{GL}_{2}(\mathbf{F})$ on  $H$ . By the branching law, there is a canonical decomposition as $\mathbf{K}$ -representations,

$$\begin{eqnarray}H=\bigoplus _{n\geqslant 0}H_{n},\end{eqnarray}$$

where $H_{n}$ is an irreducible $\mathbf{K}$ -subspace of $H$ generated by a unitary vector $e_{n}$ , such that $\{e_{0},\ldots ,e_{m}\}$ form an orthonormal basis of the $\mathbf{K}_{0}[\mathfrak{p}^{m}]$ -invariant subspace of $H$ for any $m\in \mathbb{N}$ . These vectors $e_{n}$ , called ‘classical vectors’Footnote ² in [Reference WuWu14, Definition 5.4], are defined up to a factor of modulus 1. We determine/choose them as follows. First of all, we impose

$$\begin{eqnarray}e_{0}(\unicode[STIX]{x1D705})=1,\quad \forall \unicode[STIX]{x1D705}\in \mathbf{K}.\end{eqnarray}$$

Lemma 2.15. Let $e_{n}^{\prime }\in \unicode[STIX]{x1D70B}_{s}$ be defined by

$$\begin{eqnarray}\displaystyle & \displaystyle e_{1}^{\prime }=\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-1})).e_{0}-\frac{q^{s}+q^{-s}}{q^{1/2}+q^{-1/2}}e_{0}, & \displaystyle \nonumber\\ \displaystyle & e_{n}^{\prime }=\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}-q^{-1/2}(q^{s}+q^{-s})\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n+1})).e_{0}+q^{-1}\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n+2})).e_{0},\quad \forall n\geqslant 2. & \displaystyle \nonumber\end{eqnarray}$$

Then if $s\in i\mathbb{R}$ , $\{e_{0},e_{1}^{\prime },\ldots ,e_{m}^{\prime }\}$ is an orthogonal basis of the $\mathbf{K}_{0}[\mathfrak{p}^{m}]$ -invariant subspace of $H$ for any $m\in \mathbb{N}$ .

Lemma 2.16. If we define

$$\begin{eqnarray}e_{1}:=\frac{q^{1/2}+q^{-1/2}}{q^{s+1/2}-q^{-s-1/2}}e_{1}^{\prime },\quad e_{n}:=\sqrt{\frac{q+1}{q-1}}\frac{q^{-ns}}{1-q^{-1-2s}}e_{n}^{\prime },\quad \forall n\geqslant 2,\end{eqnarray}$$

then $e_{n}$ is independent of $s$ and $\{e_{0},\ldots ,e_{m}\}$ form an orthonormal basis of the $\mathbf{K}_{0}[\mathfrak{p}^{m}]$ -invariant subspace of $H$ for any $m\in \mathbb{N}$ . Moreover, the dimension $d_{n}$ of $H_{n}$ is given by

$$\begin{eqnarray}d_{0}=1,\quad d_{1}=q,\quad d_{n}=q^{n}-q^{n-2},\quad n\geqslant 2.\end{eqnarray}$$

Proof. If we write $\mathfrak{o}_{n}=\unicode[STIX]{x1D71B}^{n}\mathfrak{o}-\unicode[STIX]{x1D71B}^{n+1}\mathfrak{o}$ , $n\geqslant 1$ , then

$$\begin{eqnarray}\displaystyle & \displaystyle D_{0}=\mathbf{B}(\mathfrak{o})w\mathbf{N}(\mathfrak{o})=\left\{\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \mathbf{K}:c\in \mathfrak{o}^{\times }\right\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle D_{n}=\mathbf{B}(\mathfrak{o})\mathbf{N}_{-}(\mathfrak{o}_{n})=\left\{\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \mathbf{K}:c\in \mathfrak{o}_{n}\right\},\quad 1\leqslant n\leqslant m, & \displaystyle \nonumber\\ \displaystyle & D_{m}^{\prime }=\mathbf{K}_{0}[m]=\mathop{\bigcup }_{n=m}^{\infty }D_{n} & \displaystyle \nonumber\end{eqnarray}$$

are the double cosets of $\mathbf{K}$ with respect to $\mathbf{B}(\mathfrak{o})$ and $\mathbf{K}_{0}[m]$ . From the computation

$$\begin{eqnarray}\displaystyle & \displaystyle \left(\begin{array}{@{}cc@{}}a\unicode[STIX]{x1D71B}^{-n} & b\\ c\unicode[STIX]{x1D71B}^{-n} & d\end{array}\right)\left(\begin{array}{@{}cc@{}}c^{-1}d\unicode[STIX]{x1D71B}^{n} & 1\\ -1 & 0\end{array}\right)=\left(\begin{array}{@{}cc@{}}c^{-1}(ad-bc) & \ast \\ 0 & c\unicode[STIX]{x1D71B}^{-n}\end{array}\right), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \left(\begin{array}{@{}cc@{}}a\unicode[STIX]{x1D71B}^{-n} & b\\ c\unicode[STIX]{x1D71B}^{-n} & d\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & 0\\ -d^{-1}c\unicode[STIX]{x1D71B}^{-n} & 1\end{array}\right)=\left(\begin{array}{@{}cc@{}}d^{-1}\unicode[STIX]{x1D71B}^{-n}(ad-bc) & \ast \\ 0 & d\end{array}\right), & \displaystyle \nonumber\end{eqnarray}$$

one easily deduces that

$$\begin{eqnarray}\displaystyle & \displaystyle e_{1}^{\prime }|_{D_{0}}=\frac{q^{s+1/2}-q^{-s-1/2}}{q^{1/2}+q^{-1/2}}\cdot (-q^{-1/2}),\quad e_{1}^{\prime }|_{D_{k}}=\frac{q^{s+1/2}-q^{-s-1/2}}{q^{1/2}+q^{-1/2}}\cdot q^{1/2},\quad k\geqslant 1; & \displaystyle \nonumber\\ \displaystyle & \displaystyle e_{n}^{\prime }|_{D_{k}}=0,\quad 0\leqslant k\leqslant n-2,\quad e_{n}^{\prime }|_{D_{n-1}}=(1-q^{-1-2s})q^{n(s+1/2)}(-q^{-1}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle e_{n}^{\prime }|_{D_{k}}=(1-q^{-1-2s})q^{n(s+1/2)}(1-q^{-1}),\quad k\geqslant n. & \displaystyle \nonumber\end{eqnarray}$$

The assertion follows since the mass $w_{n}$ of $D_{n}$ , assuming the mass of $\mathbf{K}$ is 1, is given by

$$\begin{eqnarray}w_{0}=\frac{q}{q+1},\quad w_{n}=\frac{q^{-(n-1)}}{q+1}(1-q^{-1}),\quad n\geqslant 1.\end{eqnarray}$$

For the ‘moreover’ part, it suffices to notice

(2.5) $$\begin{eqnarray}e_{n}(\unicode[STIX]{x1D705})=d_{n}^{1/2}\langle \unicode[STIX]{x1D705}.e_{n},e_{n}\rangle\end{eqnarray}$$

and to evaluate the above equation at $\unicode[STIX]{x1D705}=1$ .◻

Corollary 2.17. We record some special values of $e_{n}$ :

$$\begin{eqnarray}e_{n}(1)=\left\{\begin{array}{@{}ll@{}}1 & n=0,\\ q^{1/2} & n=1,\\ (q^{n}-q^{n-2})^{1/2} & n\geqslant 2,\end{array}\right.\quad e_{n}(w)=\left\{\begin{array}{@{}ll@{}}1 & n=0,\\ -q^{-1/2} & n=1,\\ 0 & n\geqslant 2.\end{array}\right.\end{eqnarray}$$

Proof. (i) is merely a restatement of Lemmas 2.15 and 2.16. For (ii), we first use Lemma 2.15 to deduce a relation of two formal power series

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=2}^{\infty }e_{n}^{\prime }X^{n} & = & \displaystyle \biggl(\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}X^{n}\biggr)(1-q^{-1/2}(q^{s}+q^{-s})X+q^{-1}X^{2})\nonumber\\ \displaystyle & & \displaystyle -\,e_{0}-\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-1})).e_{0}X+q^{-1/2}(q^{s}+q^{-s})e_{0}X\nonumber\\ \displaystyle & = & \displaystyle \biggl(\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}X^{n}\biggr)(1-q^{-1/2}(q^{s}+q^{-s})X+q^{-1}X^{2})\nonumber\\ \displaystyle & & \displaystyle -\,e_{0}-\biggl(e_{1}^{\prime }-q^{-1}\frac{q^{s}+q^{-s}}{q^{1/2}+q^{-1/2}}e_{0}\biggr)X.\nonumber\end{eqnarray}$$

Reverting this relation, we obtain

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}X^{n} & = & \displaystyle \biggl(\mathop{\sum }_{n=0}^{\infty }q^{-n/2}\frac{q^{(n+1)s}-q^{-(n+1)s}}{q^{s}-q^{-s}}X^{n}\biggr)\nonumber\\ \displaystyle & & \displaystyle \cdot \,\biggl(e_{0}+\biggl(e_{1}^{\prime }-q^{-1}\frac{q^{s}+q^{-s}}{q^{1/2}+q^{-1/2}}e_{0}\biggr)X+\mathop{\sum }_{n=2}^{\infty }e_{n}^{\prime }X^{n}\biggr)\nonumber\end{eqnarray}$$

and conclude by inserting Lemma 2.16. (iii) follows from (ii) by noting

$$\begin{eqnarray}\hspace{44.39996pt}\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{n})).e_{0}=\unicode[STIX]{x1D70B}_{s}(wa(\unicode[STIX]{x1D71B}^{-n})w^{-1})\unicode[STIX]{x1D70B}_{s}\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D71B}^{n} & \\ & \unicode[STIX]{x1D71B}^{n}\end{array}\right)\right).e_{0}=w.\unicode[STIX]{x1D70B}_{s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}.\hspace{39.60004pt}\square\end{eqnarray}$$

Corollary 2.19. If ${\mathcal{R}}(s):\unicode[STIX]{x1D70B}_{s}\rightarrow \unicode[STIX]{x1D70B}_{-s}$ is the normalized intertwining operator sending $e_{0}$ to  $e_{0}$ , then ${\mathcal{R}}(s)$ acts on $H_{n}$ by multiplication by

$$\begin{eqnarray}\unicode[STIX]{x1D707}(n;s)=\unicode[STIX]{x1D707}_{\mathfrak{p}}(n,s)=q^{-2ns}\frac{1-q^{-(1-2s)}}{1-q^{-(1+2s)}}.\end{eqnarray}$$

Proof. This is a special case of the computation in [Reference WuWu17b, § 3.4.3]. Here is another proof. ${\mathcal{R}}(s)e_{n}$ is equal to

$$\begin{eqnarray}\displaystyle & & \displaystyle \sqrt{\frac{q+1}{q-1}}\frac{q^{-ns}}{1-q^{-1-2s}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}-q^{-1/2}(q^{s}+q^{-s})\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n+1})).e_{0}+q^{-1}\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n+2})).e_{0})\nonumber\\ \displaystyle & & \displaystyle \quad =q^{-2ns}\frac{1-q^{-(1-2s)}}{1-q^{-(1+2s)}}\sqrt{\frac{q+1}{q-1}}\frac{q^{ns}}{1-q^{-1+2s}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n})).e_{0}-q^{-1/2}(q^{s}+q^{-s})\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n+1})).e_{0}+q^{-1}\unicode[STIX]{x1D70B}_{-s}(a(\unicode[STIX]{x1D71B}^{-n+2})).e_{0}),\nonumber\end{eqnarray}$$

the last line being equal to $e_{n}$ since it is independent of  $s$ .◻

Definition 2.21. For $n,l\in \mathbb{Z}$ , we write $l\preccurlyeq n$ to mean either $0\leqslant l\leqslant n$ or $n\leqslant l\leqslant 0$ . We extend the definition of $e_{n}$ , $\unicode[STIX]{x1D707}(n;s)$ and $c(n,l;s)$ for $n,l\in \mathbb{N}$ to $n,l\in \mathbb{Z},l\preccurlyeq n$ by requiring

$$\begin{eqnarray}e_{-n}:=w.e_{n},\quad \unicode[STIX]{x1D707}(-n;s)=\unicode[STIX]{x1D707}(n;s),\quad c(-n,-l;s)=c(n,l;s).\end{eqnarray}$$

3.1 Non-archimedean places for exceptional part

We work on a non-archimedean place $\mathfrak{p}$ and omit the subscript $\mathfrak{p}$ for simplicity of notation. Recall $e_{n}$ defined in Lemma 2.16 and Definition 2.21, but change $s$ to  $s_{0}$ . To emphasize the dependence on  $s_{0}$ , we write $e_{n,s_{0}}\in \unicode[STIX]{x1D70B}_{s_{0}}$ for the flat section associated with  $e_{n}$ , and $W_{n}(s_{0},\cdot )$ the associated Kirillov function in the Kirillov model ${\mathcal{K}}(\unicode[STIX]{x1D70B}_{s_{0}},\unicode[STIX]{x1D713})$ of $\unicode[STIX]{x1D70B}_{s_{0}}$ , with respect to an unramified additive character $\unicode[STIX]{x1D713}$ of  $\mathbf{F}$ . Recall the local zeta functional

$$\begin{eqnarray}\unicode[STIX]{x1D701}(s,W):=\int _{\mathbf{F}^{\times }}W(y)|y|^{s-1/2}\,d^{\times }y,\quad W\in {\mathcal{K}}(\unicode[STIX]{x1D70B}_{s_{0}},\unicode[STIX]{x1D713}).\end{eqnarray}$$

Lemma 3.1. The ratios of zeta functions

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{\mathfrak{p},n}(s,s_{0})=\unicode[STIX]{x1D701}_{n}(s,s_{0}):=\frac{\unicode[STIX]{x1D701}(1/2+s,W_{n}(s_{0},\cdot ))}{\unicode[STIX]{x1D701}(1/2+s,W_{0}(s_{0},\cdot ))},\quad n\in \mathbb{Z},\end{eqnarray}$$

are determined by:

• ∙ $\unicode[STIX]{x1D701}_{0}(s,s_{0})=1$ and

  $$\begin{eqnarray}\unicode[STIX]{x1D701}_{1}(s,s_{0})=\frac{q^{1/2}+q^{-1/2}}{q^{s_{0}+1/2}-q^{-(s_{0}+1/2)}}q^{-s}-\frac{q^{s_{0}}+q^{-s_{0}}}{q^{s_{0}+1/2}-q^{-(s_{0}+1/2)}};\end{eqnarray}$$

• ∙ if $n\geqslant 2$ , then

  $$\begin{eqnarray}\unicode[STIX]{x1D701}_{n}(s,s_{0})=\sqrt{\frac{q+1}{q-1}}\frac{q^{-ns_{0}}}{1-q^{-1-2s_{0}}}(q^{-ns}-q^{-1/2}(q^{s_{0}}+q^{-s_{0}})q^{-(n-1)s}+q^{-1-(n-2)s});\end{eqnarray}$$

• ∙ if $n<0$ , then $\unicode[STIX]{x1D701}_{n}(s,s_{0})=\unicode[STIX]{x1D701}_{-n}(-s,s_{0})$ .

Proof. From the relation of zeta functions

$$\begin{eqnarray}\unicode[STIX]{x1D701}(s,a(\unicode[STIX]{x1D71B}^{n}).W)=|\unicode[STIX]{x1D71B}|^{-ns}\unicode[STIX]{x1D701}(s,W)=q^{ns}\unicode[STIX]{x1D701}(s,W),\end{eqnarray}$$

the desired formulas are simple consequences of those in Lemma 2.18(i) and (iii). ◻

Corollary 3.2. Assume $\Re s=\unicode[STIX]{x1D716}>0$ small, $|n|$ is bounded by a constant and $0\leqslant k\leqslant 2$ . Then we have

$$\begin{eqnarray}\biggl|\frac{\unicode[STIX]{x2202}^{k}}{\unicode[STIX]{x2202}s_{0}^{k}}\unicode[STIX]{x1D701}_{n}(s,1/2)\biggr|\ll _{\unicode[STIX]{x1D716}}q^{n\unicode[STIX]{x1D716}-|n|/2}(\log q)^{k}.\end{eqnarray}$$

3.2 Non-archimedean places for $\text{L}^{4}$ -norms

Using the notation of the previous subsection, we define the Rankin–Selberg local zeta ratios for $n_{1},n_{2},n\in \mathbb{Z}$ and $s_{1},s_{2},s\in \mathbb{C}$ ,

(3.1)

Proof. (i) is a consequence of the $w$ -invariance of the Rankin–Selberg local zeta functional. For (ii), we may assume $n\geqslant 0$ by (i). It suffices to notice that

$$\begin{eqnarray}\int _{\mathbf{K}}e_{n}(\unicode[STIX]{x1D705})\overline{W_{n_{1}}(s_{1},g\unicode[STIX]{x1D705})}\,d\unicode[STIX]{x1D705}\end{eqnarray}$$

is non-vanishing only if $|n_{1}|=|n|$ , since by (2.5), $\int _{\mathbf{K}}\overline{e_{n}(\unicode[STIX]{x1D705})}\unicode[STIX]{x1D705}\,d\unicode[STIX]{x1D705}$ is $d_{n}^{-1/2}$ times the orthogonal projection onto the $e_{n}$ -vector of  $H_{n}$ . In particular, we deduce, for $n\geqslant 0$ ,

$$\begin{eqnarray}\int _{\mathbf{K}}e_{n}(\unicode[STIX]{x1D705})\overline{W_{n}(s_{1},g\unicode[STIX]{x1D705})}\,d\unicode[STIX]{x1D705}=d_{n}^{-1/2}\overline{W_{n}(s_{1},g)}.\end{eqnarray}$$

Hence for (iii), we are reduced to computing

$$\begin{eqnarray}\int _{\mathbf{F}^{\times }}\overline{W_{n}(0,a(y))}W_{0}(0,a(y))|y|^{s-1/2}\,d^{\times }y.\end{eqnarray}$$

By Lemma 2.18(i), we are again reduced to computing

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbf{F}^{\times }}\overline{W_{0}(0,a(y\unicode[STIX]{x1D71B}^{-n}))}W_{0}(0,a(y))|y|^{s-1/2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{m=0}^{\infty }q^{-m/2}(m+1)\cdot q^{-(n+m)/2}(n+m+1)\cdot q^{-(n+m)(s-1/2)}\nonumber\\ \displaystyle & & \displaystyle \quad =q^{-ns}\cdot \frac{2+(n-1)(1-q^{-(s+1/2)})}{(1-q^{-(s+1/2)})^{3}},\nonumber\end{eqnarray}$$

and conclude from it. ◻

Corollary 3.4. For any integer $k\geqslant 0$ , we have

4.1 Reduction to global period bound

The fixed $\text{GL}_{2}$ automorphic representation $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}(1,1)$ is realized as the completed Eisenstein series $\text{E}^{\ast }(0,\cdot )$ . We imitate the cuspidal case by choosing

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}=\text{E}^{\ast }(0,f_{0})\quad \unicode[STIX]{x1D711}=n(T).\unicode[STIX]{x1D711}_{0},\end{eqnarray}$$

where $f_{0}$ is the spherical function taking value 1 on $\mathbf{K}$ in the induced model of $\unicode[STIX]{x1D70B}(1,1)$ . Writing the normalized Whittaker functions as

$$\begin{eqnarray}W_{0,v}^{\ast }:=\unicode[STIX]{x1D701}_{v}(1)W_{0,v}=\unicode[STIX]{x1D701}_{v}(1)W_{f_{0},v},\end{eqnarray}$$

we get by Proposition 2.10 an expression for the relevant $L$ -function.

$$\begin{eqnarray}L\biggl(\frac{1}{2},\unicode[STIX]{x1D712}\biggr)^{2}=\biggl[\mathop{\prod }_{v\mid \infty }\int _{\mathbf{F}_{v}^{\times }}n(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\,d^{\times }y_{v}\mathop{\prod }_{v<\infty }\frac{\int _{\mathbf{F}_{v}^{\times }}n(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\,d^{\times }y_{v}}{L_{v}(1/2,\unicode[STIX]{x1D712}_{v})^{2}}\biggr]^{-1}\unicode[STIX]{x1D701}\biggl(\frac{1}{2},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}\biggr),\end{eqnarray}$$

where the global zeta integral is defined in Definition 2.8 and reduces in our case to

$$\begin{eqnarray}\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})=\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}(\unicode[STIX]{x1D711}(a(y))-\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y)))\unicode[STIX]{x1D712}(y)|y|_{\mathbb{A}}^{s-1/2}\,d^{\times }y,\end{eqnarray}$$

whose value at $s=1/2$ must be interpreted via analytic continuation, unlike the cuspidal case.

Proposition 4.1. We can choose $T_{v}$ with $|T_{v}|_{v}\in [\mathbf{C}(\unicode[STIX]{x1D712}_{v}),2\mathbf{C}(\unicode[STIX]{x1D712}_{v})]$ such that

$$\begin{eqnarray}\mathop{\prod }_{v\mid \infty }\int _{\mathbf{F}_{v}^{\times }}n(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\,d^{\times }y_{v}\mathop{\prod }_{v<\infty }\frac{\int _{\mathbf{F}_{v}^{\times }}n(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\,d^{\times }y_{v}}{L_{v}(1/2,\unicode[STIX]{x1D712}_{v})^{2}}\gg _{\mathbf{F}}Q^{-1/2},\end{eqnarray}$$

where $Q=\mathbf{C}(\unicode[STIX]{x1D712})=\unicode[STIX]{x1D6F1}_{v}\mathbf{C}(\unicode[STIX]{x1D712}_{v})$ is the analytic conductor of $\unicode[STIX]{x1D712}$ (we keep this notation in what follows).

4.2 Reduction to bound of truncated integral

We are reduced to bounding $\unicode[STIX]{x1D701}(1/2,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})$ . This can be defined only via analytic continuation. However, we can still approximate it with a truncated integral, just as the classical approximate functional equation does. The outcome is that we essentially only need to estimate a compact domain integral, which is equivalent to a finite sum in the classical setting. Recall Definition 2.12. We shall apply Proposition 2.13 with $A=Q^{-\unicode[STIX]{x1D705}-1}$ , $B=Q^{\unicode[STIX]{x1D705}-1}$ for some $\unicode[STIX]{x1D705}\in (0,1)$ to be chosen later, with $c_{1}=c_{2}=1/2+\unicode[STIX]{x1D716}$ and with $h_{0}$ and $h$ specified there.

Lemma 4.2. Assume $Q$ is bounded away from $0$ . Then we have, for any small $\unicode[STIX]{x1D716}>0$ ,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}\biggl(\frac{1}{2},\unicode[STIX]{x1D712},\unicode[STIX]{x1D711}\biggr) & = & \displaystyle \unicode[STIX]{x1D701}(\unicode[STIX]{x1D70E}\ast h,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})+O_{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(Q^{-\unicode[STIX]{x1D705}/2+\unicode[STIX]{x1D716}})\nonumber\\ \displaystyle & = & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\unicode[STIX]{x1D70E}\ast h(|y|_{\mathbb{A}})\unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y+O_{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(Q^{-\unicode[STIX]{x1D705}/2+\unicode[STIX]{x1D716}}+Q^{(\unicode[STIX]{x1D705}-1)/2+\unicode[STIX]{x1D716}}),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}$ is the average of Dirac measures given by

$$\begin{eqnarray}\unicode[STIX]{x1D70E}=\frac{1}{M_{E}^{2}}\mathop{\sum }_{v,v^{\prime }\in I_{E}}\unicode[STIX]{x1D6FF}_{|\unicode[STIX]{x1D71B}_{v}|_{v}|\unicode[STIX]{x1D71B}_{v^{\prime }}|_{v^{\prime }}^{-1}},\end{eqnarray}$$

with a parameter $E>0$ to be chosen later, and

$$\begin{eqnarray}I_{E}=\{v<\infty :q_{v}\in [E,2E],T_{v}=0\},\quad M_{E}=|I_{E}|\gg \frac{E}{\log E}.\end{eqnarray}$$

Proof. We only need to consider the case for $h$ since $\unicode[STIX]{x1D70E}\ast h$ gives bounded translations. Compared to [Reference WuWu14, Lemma 3.2], the new situation is as follows.

• ∙ The normal polar part is non-vanishing.

• ∙ At $v\mid \infty$ , $W_{0,v}^{\ast }$ is no longer of compact support in $\mathbf{F}_{v}^{\times }$ , hence [Reference WuWu14, Corollary 4.3] used in [Reference WuWu14, § 6.1] for local archimedean bounds on the vertical line $\Re s=-1/2-\unicode[STIX]{x1D716}$ needs to be reconsidered.

• ∙ There is a new passage from the first line to the second line, for which the estimation of an integral of the constant term $\unicode[STIX]{x1D711}_{\mathbf{N}}$ needs to be done.

We proceed to bound each part appearing in Proposition 2.13.

(0) The degenerate polar part is vanishing.

(1) The normal polar part and the integral of $\unicode[STIX]{x1D711}_{\mathbf{N}}$ are non-vanishing only if $\unicode[STIX]{x1D712}=|\cdot |^{i\unicode[STIX]{x1D707}}$ for some $\unicode[STIX]{x1D707}\in \mathbb{R}$ , in which case $\mathbf{C}(\unicode[STIX]{x1D712}_{v})=1$ for all $v<\infty$ and $\mathbf{C}(\unicode[STIX]{x1D712}_{v})\asymp |\unicode[STIX]{x1D707}|^{[\mathbf{F}_{v}:\mathbb{R}]}$ for all $v\mid \infty$ . Hence $Q\asymp |\unicode[STIX]{x1D707}|^{[\mathbf{F}:\mathbb{Q}]}$ . Note that by [Reference WuWu18, Proposition 5.33], there are $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}\in \mathbb{C}$ depending only on $\mathbf{F}$ such that

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{0,\mathbf{N}}(zn(x)a(y)\unicode[STIX]{x1D705})=\unicode[STIX]{x1D707}_{1}|y|_{\mathbb{A}}^{1/2}+\unicode[STIX]{x1D707}_{2}|y|_{\mathbb{A}}^{1/2}\log |y|_{\mathbb{ A}},\quad \forall z\in \mathbf{Z}(\mathbb{A}),x\in \mathbb{A},y\in \mathbb{A}^{\times },\unicode[STIX]{x1D705}\in \mathbf{K}.\end{eqnarray}$$

If we write

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\mathbf{N}}(zn(x)a(y)\unicode[STIX]{x1D705})=|y|_{\mathbb{A}}^{1/2}f_{1}(\unicode[STIX]{x1D705})+|y|_{\mathbb{A}}^{1/2}\log |y|_{\mathbb{ A}}f_{2}(\unicode[STIX]{x1D705}),\end{eqnarray}$$

then we can easily calculate

$$\begin{eqnarray}\displaystyle & \displaystyle f_{1}(1)=\unicode[STIX]{x1D707}_{1},\quad f_{2}(1)=\unicode[STIX]{x1D707}_{2},\quad f_{1}(w)=\unicode[STIX]{x1D707}_{1}\mathop{\prod }_{v\mid \infty }(1+|T_{v}|^{2})^{-[\mathbf{F}_{v}:\mathbb{R}]/2}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle f_{2}(w)=\unicode[STIX]{x1D707}_{2}\mathop{\prod }_{v\mid \infty }(1+|T_{v}|^{2})^{-[\mathbf{F}_{v}:\mathbb{R}]/2}\log \biggl(\mathop{\prod }_{v\mid \infty }(1+|T_{v}|^{2})^{-[\mathbf{F}_{v}:\mathbb{R}]/2}\biggr). & \displaystyle \nonumber\end{eqnarray}$$

We thus find that the normal polar part, which is of the form

$$\begin{eqnarray}O\biggl(\frac{A^{1/2}|f_{1}(1)|}{|1/2+i\unicode[STIX]{x1D707}|^{N}}+\frac{A^{1/2}|f_{2}(1)\log A|}{|1/2+i\unicode[STIX]{x1D707}|^{N}}\biggr)+O\biggl(\frac{A^{-1/2}|f_{1}(w)|}{|1/2-i\unicode[STIX]{x1D707}|^{N}}+\frac{A^{-1/2}|f_{2}(w)\log A|}{|1/2-i\unicode[STIX]{x1D707}|^{N}}\biggr),\end{eqnarray}$$

can be bounded as $O(Q^{-N})$ for any $N\in \mathbb{N}$ , due to the arbitrarily large denominators. For the integral of $\unicode[STIX]{x1D711}_{\mathbf{N}}$ , since $h(t)$ has support contained in $[Q^{-\unicode[STIX]{x1D705}-1},Q^{\unicode[STIX]{x1D705}-1}]$ with $|h(t)|\leqslant 1$ , we find that (note that $n(T).\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y))=\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y))$ )

(4.2) $$\begin{eqnarray}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y=O_{\boldsymbol{ F},h_{0},\unicode[STIX]{x1D716}}(Q^{(\unicode[STIX]{x1D705}-1)/2+\unicode[STIX]{x1D716}}).\end{eqnarray}$$

(2) We turn to the lower part. Recall the choice $c_{1}=1/2+\unicode[STIX]{x1D716}$ . The relevant zeta function has a decomposition as a finite product

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s,\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711}\biggr) & = & \displaystyle L\biggl(\frac{1}{2}+s,\unicode[STIX]{x1D712}^{-1}\biggr)^{2}\cdot \mathop{\prod }_{v\mid \infty }\int _{\mathbf{F}_{v}^{\times }}wn(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\unicode[STIX]{x1D712}_{v}^{-1}(y_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\mathop{\prod }_{v<\infty }\frac{\int _{\mathbf{F}_{v}^{\times }}wn(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\unicode[STIX]{x1D712}_{v}^{-1}(y_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}}{L_{v}(1/2+s,\unicode[STIX]{x1D712}_{v}^{-1})^{2}}\nonumber\end{eqnarray}$$

At an archimedean place $v$ , say $\mathbf{F}_{v}=\mathbb{R}$ , [Reference WuWu17a, Lemma 3.12(2)] gives the relevant local integral

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbb{R}^{\times }}wn(T_{v}).W_{0,v}^{\ast }(a(y_{v}))\unicode[STIX]{x1D712}_{v}^{-1}(y_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D712}_{v}(1+T_{v}^{2})^{-1}(1+T_{v}^{2})^{s}\int _{\mathbb{R}^{\times }}W_{0,v}^{\ast }(a(y))\unicode[STIX]{x1D713}_{v}(yT_{v})\unicode[STIX]{x1D712}_{v}^{-1}(y_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}.\nonumber\end{eqnarray}$$

a bound $O(|T_{v}|_{v}^{1/2+\unicode[STIX]{x1D716}})$ . For $\mathbf{F}_{v}=\mathbb{C}$ the argument is similar, using [Reference WuWu17a, Lemma 3.13(2)]. At $v<\infty$ , [Reference WuWu14, Corollary 4.8] is still applicable. We thus deduce that, using the convex bound of $L(1/2+s,\unicode[STIX]{x1D712}^{-1})$ ,

$$\begin{eqnarray}|\unicode[STIX]{x1D701}({\textstyle \frac{1}{2}}+s,\unicode[STIX]{x1D712}^{-1},w.\unicode[STIX]{x1D711})|\ll _{\boldsymbol{ F},\unicode[STIX]{x1D716}}|{\textstyle \frac{1}{2}}+s|^{\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/2+\unicode[STIX]{x1D716}}.\end{eqnarray}$$

The desired bound is thus $O(Q^{-(\unicode[STIX]{x1D705}+1)/2}Q^{1/2+\unicode[STIX]{x1D716}})=O(Q^{-\unicode[STIX]{x1D705}/2+\unicode[STIX]{x1D716}})$ .

(3) The treatment of the upper part is similar and simpler. It gives the desired bound $O(Q^{-\unicode[STIX]{x1D705}/2+\unicode[STIX]{x1D716}})$ .◻

4.3 Interlude: failure of truncation on Eisenstein series

We have approximated $\unicode[STIX]{x1D701}(1/2,\unicode[STIX]{x1D712},\unicode[STIX]{x1D711})$ by some smoothly truncated integral

$$\begin{eqnarray}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}^{h}\unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y\quad \biggl(\text{where}~\int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}^{h}:=\int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}h(|\cdot |)\biggr)\end{eqnarray}$$

as in the cuspidal case. We then would like to apply the Cauchy–Schwarz inequality

$$\begin{eqnarray}\biggl|\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}^{h}\unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y\biggr|^{2}\leqslant \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}^{h}\,d^{\times }y\cdot \int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}^{h}n(T).|\unicode[STIX]{x1D711}_{0}(a(y))|^{2}\,d^{\times }y,\end{eqnarray}$$

and apply Fourier inversion to $|\unicode[STIX]{x1D711}_{0}|^{2}$ , interchange the order of summation and estimate each component as in the cuspidal case. This is not possible because $\unicode[STIX]{x1D711}_{0}$ is no longer of rapid decay hence $|\unicode[STIX]{x1D711}_{0}|^{2}$ is not square-integrable any more. A first idea, already employed in [Reference Michel and VenkateshMV10, § 5.1.7], is to (smoothly) truncate the Eisenstein series $\unicode[STIX]{x1D711}_{0}$ up to some height  $X$ , denoted by $\unicode[STIX]{x1D6EC}^{X}\unicode[STIX]{x1D711}_{0}$ . For example, if we naively choose $X$ no less than the height of the truncation on the integral (namely, $Q^{\unicode[STIX]{x1D705}+1}$ in the notation of [Reference WuWu14]), we find

$$\begin{eqnarray}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}^{h}\unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y=\int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}^{h}n(T).\unicode[STIX]{x1D6EC}^{X}\unicode[STIX]{x1D711}_{0}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y.\end{eqnarray}$$

We could continue the argument by replacing $\unicode[STIX]{x1D711}_{0}$ with $\unicode[STIX]{x1D6EC}^{X}\unicode[STIX]{x1D711}_{0}$ . But then some $\text{L}^{2}$ -Sobolev norm of $|\unicode[STIX]{x1D6EC}^{X}\unicode[STIX]{x1D711}_{0}|^{2}$ would come in as a multiplicative factor of the final estimation of the above integral. This causes no problem in the cuspidal case since the relevant norm is bounded by some $\text{L}^{4}$ -Sobolev norms of  $\unicode[STIX]{x1D711}_{0}$ , which depend only on  $\unicode[STIX]{x1D70B}$ . This is no longer the case for $\unicode[STIX]{x1D6EC}^{X}\unicode[STIX]{x1D711}_{0}$ since its norms all depend on $X$ , hence some positive power of $Q=\mathbf{C}(\unicode[STIX]{x1D712})$ . This means that in the final optimization just before [Reference WuWu14, Remark 3.11], we would have to replace $EQ^{-1/4+\unicode[STIX]{x1D703}/2}$ by something like $XEQ^{-1/4+\unicode[STIX]{x1D703}/2}$ , which completely destroys the Burgess-like quality. Indeed, in the thesis version of [Reference WuWu14] we have pursued this idea and were only able to obtain a saving $(1-2\unicode[STIX]{x1D703})/12$ instead of the Burgess-like saving $(1-2\unicode[STIX]{x1D703})/8$ .

A better way, which is also the main innovation of this paper, is to generalize the spectral decomposition/Fourier inversion into a space of functions suitably larger than the square-integrable ones. As we have seen in § 2.2, ${\mathcal{A}}^{\text{fr}}(\text{GL}_{2},1)$ is a good candidate: it contains $|\unicode[STIX]{x1D711}_{0}|^{2}$ and differs from smooth vectors in the $\text{L}^{2}$ -space only by (non-unitary) Eisenstein series. Specifically, we shall decompose

$$\begin{eqnarray}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}^{h}n(T).|\unicode[STIX]{x1D711}_{0}(a(y))|^{2}\,d^{\times }y=\int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}^{h}n(T).(|\unicode[STIX]{x1D711}_{0}|^{2}-{\mathcal{E}})(a(y))\,d^{\times }y+\int _{\boldsymbol{ F}^{\times }\backslash \mathbb{A}^{\times }}^{h}n(T).{\mathcal{E}}(a(y))\,d^{\times }y,\end{eqnarray}$$

where we have written ${\mathcal{E}}={\mathcal{E}}(|\unicode[STIX]{x1D711}_{0}|^{2})$ for simplicity. Without amplification, the norms of $|\unicode[STIX]{x1D711}_{0}|^{2}-{\mathcal{E}}$ depend only on $\unicode[STIX]{x1D711}_{0}$ , hence  $\unicode[STIX]{x1D70B}$ ; with amplification the relevant norms have contributions as small as $(\log E)^{3}$ (see Theorem 5.4) where $E$ denotes the length of the amplifiers, which is negligible. Hence we can treat the term related to $|\unicode[STIX]{x1D711}_{0}|^{2}-{\mathcal{E}}$ in the same way as in the cuspidal case without harming the quality of the bound. Since ${\mathcal{E}}$ is determined explicitly by  $\unicode[STIX]{x1D711}_{0}$ , the term related to it is explicitly estimable. We will treat the estimation and see that its contribution does not harm the quality of the final bound either. Namely, the generalized spectral decomposition fits as well with the estimation of the integrals as the ordinary one in the cuspidal case. Note that the simpler $\unicode[STIX]{x1D711}_{0}$ is, the simpler ${\mathcal{E}}$ is.

4.4 Regroupment of generalized Fourier inversion

We make the strategy described in the above remark more precise. Introducing

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{\prime }=\frac{1}{M_{E}^{2}}\mathop{\sum }_{\mathfrak{p},\mathfrak{p}^{\prime }\in I_{E}}\unicode[STIX]{x1D712}\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}^{\prime }}}\biggr)\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D71B}_{\mathfrak{p}}\unicode[STIX]{x1D71B}_{\mathfrak{p}^{\prime }}^{-1}},\end{eqnarray}$$

we can apply the Cauchy–Schwarz inequality

$$\begin{eqnarray}\displaystyle \biggl|\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}\unicode[STIX]{x1D70E}\ast h(|y|_{\mathbb{A}})\unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y\biggr|^{2} & = & \displaystyle \biggl|\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{\prime }\ast \unicode[STIX]{x1D711}(a(y))\unicode[STIX]{x1D712}(y)\,d^{\times }y\biggr|^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\,d^{\times }y\cdot \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})|\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{\prime }\ast \unicode[STIX]{x1D711}(a(y))|^{2}\,d^{\times }y.\nonumber\end{eqnarray}$$

The first integral in the last line is of size $O_{\mathbf{F}}(\log Q)$ , hence negligible. Opening the square, we get

$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{\prime }\ast \unicode[STIX]{x1D711}(a(y))|^{2} & = & \displaystyle \frac{1}{M_{E}^{4}}\mathop{\sum }_{\mathfrak{p}_{1},\mathfrak{p}_{1}^{\prime },\mathfrak{p}_{2},\mathfrak{p}_{2}^{\prime }\in I_{E}}\unicode[STIX]{x1D712}\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr)\unicode[STIX]{x1D712}^{-1}\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr)\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr).\unicode[STIX]{x1D711}\cdot \overline{a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr).\unicode[STIX]{x1D711}}\biggr)(a(y))\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr)n(T).\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{\unicode[STIX]{x1D711}_{0}}\biggr)(a(y)),\nonumber\end{eqnarray}$$

where we have abbreviated

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}:=\unicode[STIX]{x1D712}\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr)\unicode[STIX]{x1D712}^{-1}\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr)\in \mathbb{C}^{(1)}.\end{eqnarray}$$

Decomposing the non-square-integrable function as

(4.3) $$\begin{eqnarray}a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{\unicode[STIX]{x1D711}_{0}}=a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}\biggr).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})+{\mathcal{E}}_{0}(\vec{\mathfrak{p}}),\end{eqnarray}$$

where the $\text{L}^{2}$ -residual part (2.2) or [Reference WuWu18, Definition 2.26] is given an abbreviated notation

(4.4) $$\begin{eqnarray}{\mathcal{E}}_{0}(\vec{\mathfrak{p}}):={\mathcal{E}}\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{\unicode[STIX]{x1D711}_{0}}\biggr),\end{eqnarray}$$

applying to $\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})=\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\mathbf{N}}+\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{cusp}}+\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{Eis}}$ the Fourier inversion decomposition in the sense of [Reference WuWu14, Theorem 2.18] and regrouping the two constant terms, we can rewrite the second integral as

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})|\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{\prime }\ast \unicode[STIX]{x1D711}(a(y))|^{2}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}}\biggr)_{\mathbf{N}}(a(y))\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{cusp}})+\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{Eis}})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h_{|\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}/\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}|_{\mathbb{A}}},1,n(T).{\mathcal{E}}_{0}(\vec{\mathfrak{p}})),\nonumber\end{eqnarray}$$

where we have used the $h$ -truncated zeta integral in Definition 2.12. Note that we can drop $\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}/\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}$ in the second integrand, since its adelic norm is contained in $[1/2,2]$ .

4.5 Bounds for each part

Lemma 4.3. We have, for any $\unicode[STIX]{x1D716}>0$ ,

$$\begin{eqnarray}\biggl|\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}}\biggr)_{\mathbf{N}}(a(y))\,d^{\times }y\biggr|\ll _{\boldsymbol{ F},\unicode[STIX]{x1D716}}E^{-2+\unicode[STIX]{x1D716}}Q^{\unicode[STIX]{x1D716}}+Q^{\unicode[STIX]{x1D705}-1+\unicode[STIX]{x1D716}}.\end{eqnarray}$$

Proof. We write and decompose

$$\begin{eqnarray}\displaystyle \displaystyle S_{\mathbf{N}}(\vec{\mathfrak{p}};h) & := & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})\biggl(a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}\cdot \overline{a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{0}}\biggr)_{\mathbf{N}}(a(y))\,d^{\times }y,\nonumber\\ \displaystyle \displaystyle S_{\mathbf{N}}(\vec{\mathfrak{p}};h) & = & \displaystyle S_{\mathbf{N}}^{{\mathcal{W}}}(\vec{\mathfrak{p}};h)+S_{\mathbf{N}}^{\ast }(\vec{\mathfrak{p}};h),\nonumber\\ \displaystyle \displaystyle S_{\mathbf{N}}^{\ast }(\vec{\mathfrak{p}};h) & = & \displaystyle \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y))\cdot \overline{a\biggl(\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr).\unicode[STIX]{x1D711}_{\mathbf{N}}(a(y))}\,d^{\times }y.\nonumber\end{eqnarray}$$

The treatment of

$$\begin{eqnarray}\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}S_{\mathbf{N}}^{{\mathcal{W}}}(\vec{\mathfrak{p}};h)\end{eqnarray}$$

is the same as [Reference WuWu14, Lemma 3.4], which gives a term $\ll _{\mathbf{F},\unicode[STIX]{x1D716}}E^{-2+\unicode[STIX]{x1D716}}Q^{\unicode[STIX]{x1D716}}$ . Using (4.1), we find

$$\begin{eqnarray}\displaystyle S_{\mathbf{N}}^{\ast }(\vec{\mathfrak{p}};h) & = & \displaystyle \biggl(|\unicode[STIX]{x1D707}_{1}|^{2}+\unicode[STIX]{x1D707}_{1}\overline{\unicode[STIX]{x1D707}_{2}}\log \biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr|_{\mathbb{A}}+\unicode[STIX]{x1D707}_{2}\overline{\unicode[STIX]{x1D707}_{1}}\log \biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr|_{\mathbb{A}}\biggr)\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})|y|_{\mathbb{A}}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D707}_{1}\overline{\unicode[STIX]{x1D707}_{2}}+\unicode[STIX]{x1D707}_{2}\overline{\unicode[STIX]{x1D707}_{1}})\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\cdot \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})|y|_{\mathbb{A}}\log |y|_{\mathbb{A}}\,d^{\times }y\nonumber\\ \displaystyle & & \displaystyle +\,|\unicode[STIX]{x1D707}_{2}|^{2}\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{1}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\biggl|\frac{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}}}{\unicode[STIX]{x1D71B}_{\mathfrak{p}_{2}^{\prime }}}\biggr|_{\mathbb{A}}^{1/2}\cdot \int _{\mathbf{F}^{\times }\backslash \mathbb{A}^{\times }}h(|y|_{\mathbb{A}})|y|_{\mathbb{A}}\log ^{2}|y|_{\mathbb{A}}\,d^{\times }y,\nonumber\end{eqnarray}$$

from which we easily see, by the same consideration of (4.2),

$$\begin{eqnarray}\hspace{84.0pt}|S_{\mathbf{N}}^{\ast }(\vec{\mathfrak{p}};h)|\ll _{\boldsymbol{ F},\unicode[STIX]{x1D716}}Q^{\unicode[STIX]{x1D705}-1+\unicode[STIX]{x1D716}},\quad \biggl|\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}S_{\mathbf{N}}^{\ast }(\vec{\mathfrak{p}};h)\biggr|\ll _{\boldsymbol{ F},\unicode[STIX]{x1D716}}Q^{\unicode[STIX]{x1D705}-1+\unicode[STIX]{x1D716}}.\hspace{60.0pt}\square\end{eqnarray}$$

Lemma 4.5. For any $\unicode[STIX]{x1D716}>0$ , we have

$$\begin{eqnarray}\biggl|\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\mathfrak{p}}\unicode[STIX]{x1D701}(h,1,n(T).{\mathcal{E}}_{0}(\vec{\mathfrak{p}}))\biggr|\ll _{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(EQ)^{\unicode[STIX]{x1D716}}(Q^{\unicode[STIX]{x1D705}-1}E^{2}+E^{-2}).\end{eqnarray}$$

The estimation of $(1/M_{E}^{4})\sum _{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{cusp}})$ and $(1/M_{E}^{4})\sum _{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{Eis}})$ is essentially the same as the cuspidal case given in [Reference WuWu14, §§ 6.3 and 6.4]. In fact, the only difference appears in [Reference WuWu14, (6.16)], where we could bound $\Vert \unicode[STIX]{x1D6E5}_{\infty }^{A^{\prime }}a(\unicode[STIX]{x1D71B}_{v_{1}}/\unicode[STIX]{x1D71B}_{v_{1}^{\prime }}).\unicode[STIX]{x1D711}_{0}\cdot \overline{a(\unicode[STIX]{x1D71B}_{v_{2}}/\unicode[STIX]{x1D71B}_{v_{2}^{\prime }}).\unicode[STIX]{x1D711}_{0}}\Vert$ easily by $\text{L}^{4}$ -norm of  $\unicode[STIX]{x1D711}_{0}$ . For the current case, we need to bound $\Vert \unicode[STIX]{x1D6E5}_{\infty }^{A^{\prime }}\unicode[STIX]{x1D711}_{0}(v_{1},v_{1}^{\prime },v_{2},v_{2}^{\prime })\Vert$ defined in (4.3). Decomposing the relevant function into $\mathbf{K}_{\infty }$ -isotypic parts, we can apply Theorem 5.4. Thus unlike the cuspidal case, we get an extra $(\log E)^{3}$ in our estimation, which is harmless. Hence [Reference WuWu14, Lemmas 3.6 and 3.7] remain valid in the current case, giving the following results.

Lemma 4.6. For any $\unicode[STIX]{x1D716}>0$ , we have

$$\begin{eqnarray}\biggl|\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{cusp}})\biggr|\ll _{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(EQ)^{\unicode[STIX]{x1D716}}E^{2}Q^{1/2-\unicode[STIX]{x1D703}}.\end{eqnarray}$$

Lemma 4.7. For any $\unicode[STIX]{x1D716}>0$ , we have

$$\begin{eqnarray}\biggl|\frac{1}{M_{E}^{4}}\mathop{\sum }_{\vec{\mathfrak{p}}\in I_{E}^{4}}\unicode[STIX]{x1D712}_{\vec{\mathfrak{p}}}\unicode[STIX]{x1D701}(h,1,n(T).\unicode[STIX]{x1D711}_{0}(\vec{\mathfrak{p}})_{\text{Eis}})\biggr|\ll _{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(EQ)^{\unicode[STIX]{x1D716}}EQ^{(\unicode[STIX]{x1D705}-1)/2}.\end{eqnarray}$$

We are finally led to establishing (1.1) by

$$\begin{eqnarray}\min _{\unicode[STIX]{x1D705},E}\max (E^{-1},EQ^{-1/4+\unicode[STIX]{x1D703}/2},Q^{-\unicode[STIX]{x1D705}/2},Q^{(\unicode[STIX]{x1D705}-1)/2},E^{1/2}Q^{(\unicode[STIX]{x1D705}-1)/4},EQ^{(\unicode[STIX]{x1D705}-1)/2})=Q^{-(1-2\unicode[STIX]{x1D703})/8},\end{eqnarray}$$

with an optimal choice given by

$$\begin{eqnarray}E=Q^{(1-2\unicode[STIX]{x1D703})/8},\quad \unicode[STIX]{x1D705}=\frac{1}{4}+\frac{\unicode[STIX]{x1D703}}{6}.\end{eqnarray}$$

5.1 Estimation for exceptional part

Recall ${\mathcal{E}}_{0}(\vec{\mathfrak{p}})$ defined in (4.4).

Definition 5.1. For $\vec{\mathfrak{p}}$ , we define $\vec{n}(\vec{\mathfrak{p}})=(n_{v})_{v}$ for $v$ running over the set of places of $\mathbf{F}$ such that:

• ∙ $n_{v}=0$ for $v\mid \infty$ and $v\notin \{\mathfrak{p}_{1},\mathfrak{p}_{1}^{\prime },\mathfrak{p}_{2},\mathfrak{p}_{2}^{\prime }\}$ ;

• ∙ $n_{\mathfrak{p}}=1$ if $\mathfrak{p}\in \{\mathfrak{p}_{1}^{\prime },\mathfrak{p}_{2}\}$ , $n_{\mathfrak{p}}=-1$ if $\mathfrak{p}\in \{\mathfrak{p}_{1},\mathfrak{p}_{2}^{\prime }\}$ .

For $\vec{n}=(n_{v})_{v},\vec{l}=(l_{v})_{v}$ with components in $\mathbb{Z}$ , we write $\vec{l}\preccurlyeq \vec{n}$ to mean $l_{v}\preccurlyeq n_{v}$ at each $v$ , defined in Definition 2.21. We define, for $\vec{l},\preccurlyeq \vec{n}$

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}(\vec{l})=\mathop{\sum }_{v}l_{v},\quad \Vert \vec{l}\Vert =\mathop{\sum }_{v}|l_{v}|,\quad e_{\vec{n}}=\bigotimes _{v}^{\prime }e_{n_{v}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D707}(\vec{n};s)=\mathop{\prod }_{v}\unicode[STIX]{x1D707}_{v}(n_{v};s),\quad c(\vec{n},\vec{l};s):=\mathop{\prod }_{\mathfrak{p}<\infty }c_{\mathfrak{p}}(n_{\mathfrak{p}},l_{\mathfrak{p}};s), & \displaystyle \nonumber\end{eqnarray}$$

where the local components are defined in Lemma 2.18(ii), Corollary 2.19, Remark 2.20 and Definition 2.21. We write the Laurent expansion at $s=1$ of the complete zeta function $\unicode[STIX]{x1D6EC}_{\mathbf{F}}(s)$ as

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{F}}(s)=\frac{1}{s-1}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }+\unicode[STIX]{x1D6FE}_{\boldsymbol{ F}}+O((s-1)).\end{eqnarray}$$

We recall $\text{E}^{\text{reg}}(s,f)$ in (2.1) or [Reference WuWu18, Definition 2.16] as well as the abbreviation

$$\begin{eqnarray}\text{E}^{\text{reg},(n)}(s,f):=\frac{\unicode[STIX]{x2202}^{n}}{\unicode[STIX]{x2202}s^{n}}\text{E}^{\text{reg}}(s,f).\end{eqnarray}$$

We compute ${\mathcal{E}}_{0}(\vec{\mathfrak{p}})$ explicitly as

$$\begin{eqnarray}\displaystyle {\mathcal{E}}_{0}(\vec{\mathfrak{p}}) & = & \displaystyle \mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}(\vec{\mathfrak{p}})}c(\vec{n}(\vec{\mathfrak{p}}),\vec{l};0)\bigg\{|\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }|^{2}\text{E}^{\text{reg},(2)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)+\biggl(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-\frac{1}{2}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\biggr)\overline{2\unicode[STIX]{x1D6FE}_{\boldsymbol{ F}}}\text{E}^{\text{reg}}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\biggl(2\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\overline{\unicode[STIX]{x1D6FE}_{\mathbf{F}}}+\overline{\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }}\biggl(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-\frac{1}{2}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\biggr)\biggr)\text{E}^{\text{reg},(1)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\bigg\},\nonumber\end{eqnarray}$$

where the derivative $\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)$ is taken with respect to $s$ in $\unicode[STIX]{x1D707}(\vec{l};s)$ . Consequently, we obtain

(5.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(h,1,n(T).{\mathcal{E}}_{0}(\vec{\mathfrak{p}})) & = & \displaystyle \mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}(\vec{\mathfrak{p}})}c(\vec{n}(\vec{\mathfrak{p}}),\vec{l};0)\bigg\{|\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }|^{2}\unicode[STIX]{x1D701}\biggl(h,1,n(T).\text{E}^{\text{reg},(2)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\biggl(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-\frac{1}{2}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\biggr)\overline{2\unicode[STIX]{x1D6FE}_{\boldsymbol{ F}}}\unicode[STIX]{x1D701}\biggl(h,1,n(T).\text{E}^{\text{reg}}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\biggl(2\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\overline{\unicode[STIX]{x1D6FE}_{\mathbf{F}}}+\overline{\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }}\biggl(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-\frac{1}{2}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\biggr)\biggr)\unicode[STIX]{x1D701}\biggl(h,1,n(T).\text{E}^{\text{reg},(1)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\bigg\}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Thus we are reduced to bounding, for $n=0,1,2$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D701}(h,1,n(T).\text{E}^{\text{reg},(n)}({\textstyle \frac{1}{2}},e_{\vec{l}})).\end{eqnarray}$$

Lemma 5.2. For $n=0,1,2$ and any $\unicode[STIX]{x1D716}>0$ sufficiently small, we have

$$\begin{eqnarray}|\unicode[STIX]{x1D701}(h,1,n(T).\text{E}^{\text{reg},(n)}({\textstyle \frac{1}{2}},e_{\vec{l}}))|\ll _{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(EQ)^{\unicode[STIX]{x1D716}}(Q^{\unicode[STIX]{x1D705}-1}E^{-(1/2)\unicode[STIX]{x1D70E}(\vec{l})}+E^{-(1/2)\Vert \vec{l}\Vert }).\end{eqnarray}$$

Proof. At the left-hand side, the Mellin transform of the integrand is related to $L$ -functions. We make use of this fact by applying the Mellin inversion formula to get

$$\begin{eqnarray}\unicode[STIX]{x1D701}\biggl(h,1,n(T).\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)=\int _{\Re s\gg 1}\mathfrak{M}(h)(-s)\unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s,1,n(T).\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\frac{ds}{2\unicode[STIX]{x1D70B}i},\end{eqnarray}$$

and then shift the vertical line of integration to the left. There are poles of the integrand determined by Proposition 2.9. We calculate the constant terms in order to analyze the poles. We have

$$\begin{eqnarray}\displaystyle & \displaystyle \text{E}_{\mathbf{N}}^{\text{reg}}\biggl(\frac{1}{2}+s,e_{\vec{l}}\biggr)(a(y)k)=|y|_{\mathbb{A}}^{1+s}e_{\vec{l}}(k)+\frac{2s\unicode[STIX]{x1D6EC}_{\mathbf{F}}(-2s)}{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(2+2s)}\frac{\unicode[STIX]{x1D707}(\vec{l};1/2+s)}{2s}|y|_{\mathbb{A}}^{-s}e_{\vec{l}}(k),\quad \vec{l}\neq 0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{E}_{\mathbf{N}}^{\text{reg}}\biggl(\frac{1}{2}+s\biggr)(a(y)k,e_{\vec{0}})=|y|_{\mathbb{A}}^{1+s}e_{\vec{0}}(1)+\frac{2s\unicode[STIX]{x1D6EC}_{\mathbf{F}}(-2s)}{\unicode[STIX]{x1D6EC}_{\mathbf{F}}(2+2s)}\frac{|y|_{\mathbb{A}}^{-s}-1}{2s}e_{\vec{0}}(1). & \displaystyle \nonumber\end{eqnarray}$$

Hence we get, for $n=0,1,2$ , $\vec{l}\neq \vec{0}$ and with constants $c_{k}$ depending only on  $\mathbf{F}$ ,

$$\begin{eqnarray}\displaystyle n(T).\text{E}_{\mathbf{N}}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)(a(y)) & = & \displaystyle |y|_{\mathbb{A}}\log ^{n}|y|_{\mathbb{ A}}e_{\vec{l}}(1)+\mathop{\sum }_{k=\Vert \vec{l}\Vert -1}^{n}c_{k}E^{-\Vert \vec{l}\Vert }(\log E)^{k-\Vert \vec{l}\Vert +1}\log ^{n-k}|y|_{\mathbb{A}}e_{\vec{l}}(1),\nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle n(T).\text{E}_{\mathbf{N}}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)(a(y)w) & = & \displaystyle (\text{Ht}(wn(T))|y|_{\mathbb{A}})\log ^{n}(\text{Ht}(wn(T))|y|_{\mathbb{ A}})e_{\vec{l}}(w)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{k=\Vert \vec{l}\Vert -1}^{n}c_{k}E^{-\Vert \vec{l}\Vert }(\log E)^{k-\Vert \vec{l}\Vert +1}\log ^{n-k}(|y|_{\mathbb{A}}\text{Ht}(wn(T)))e_{\vec{l}}(w);\nonumber\end{eqnarray}$$

while for $\vec{l}=\vec{0}$ ,

$$\begin{eqnarray}\displaystyle n(T).\text{E}_{\mathbf{N}}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{0}}\biggr)(a(y)) & = & \displaystyle |y|_{\mathbb{A}}\log ^{n}|y|_{\mathbb{ A}}e_{\vec{0}}(1)+\mathop{\sum }_{k=0}^{n}c_{k}\log ^{n-k+1}|y|_{\mathbb{A}}e_{\vec{0}}(1),\nonumber\\ \displaystyle n(T).\text{E}_{\mathbf{N}}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{0}}\biggr)(a(y)w) & = & \displaystyle (\text{Ht}(wn(T))|y|_{\mathbb{A}})(\log \text{Ht}(wn(T))|y|_{\mathbb{A}})^{n}e_{\vec{0}}(1)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{k=0}^{n}c_{k}(\log |y|_{\mathbb{A}}\text{Ht}(wn(T)))^{n-k+1}e_{\vec{0}}(1).\nonumber\end{eqnarray}$$

By Proposition 2.9, $\unicode[STIX]{x1D701}(1/2+s,1,n(T).\text{E}^{\text{reg},(n)}(1/2,e_{\vec{l}}))$ has:

• ∙ a pole at $s=1$ with residue equal to

  $$\begin{eqnarray}\text{Ht}(wn(T))\log ^{n}\text{Ht}(wn(T))e_{\vec{l}}(w),\end{eqnarray}$$
  which is bounded, using Corollary 2.17, as
  $$\begin{eqnarray}Q^{-2}\log ^{n}Q\cdot E^{-(1/2)\unicode[STIX]{x1D70E}(\vec{l})};\end{eqnarray}$$

• ∙ a pole at $s=0$ ;

• ∙ a pole at $s=-1$ .

We can thus write, for $0<\unicode[STIX]{x1D716}<1$ , using [Reference WuWu14, (6.1) and (6.2)] and Proposition 2.9,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}\biggl(h,1,n(T).\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr) & = & \displaystyle \int _{\Re s=\unicode[STIX]{x1D716}}\unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s,1,n(T).\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\mathfrak{M}(h)(-s)\frac{ds}{2\unicode[STIX]{x1D70B}i}\nonumber\\ \displaystyle & & \displaystyle +\,O_{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(Q^{\unicode[STIX]{x1D705}-1+\unicode[STIX]{x1D716}}E^{-(1/2)\unicode[STIX]{x1D70E}(\vec{l})}).\nonumber\end{eqnarray}$$

To bound the integral on the vertical line $\Re s=\unicode[STIX]{x1D716}$ , we have, by Proposition 2.10,

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D701}\biggl(\frac{1}{2}+s,1,n(T).\text{E}^{\text{reg},(n)}\biggl(\frac{1}{2},e_{\vec{l}}\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}^{n}}{\unicode[STIX]{x2202}s_{0}^{n}}\,\biggr|_{s_{0}=1/2}\,\bigg\{\frac{\unicode[STIX]{x1D701}_{\mathbf{F}}(1/2+s+s_{0})\unicode[STIX]{x1D701}_{\mathbf{F}}(1/2+s-s_{0})}{\unicode[STIX]{x1D701}_{\mathbf{F}}(1+2s_{0})}\cdot \mathop{\prod }_{v\mid \infty }\int _{\mathbf{F}_{v}^{\times }}W_{0,v}(s_{0},a(y_{v}))\unicode[STIX]{x1D713}_{v}(y_{v}T_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\mathop{\prod }_{\mathfrak{p}<\infty ,T_{\mathfrak{p}}\neq 0}\frac{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1+2s_{0})}{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s+s_{0})\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s-s_{0})}\int _{\mathbf{F}_{\mathfrak{p}}^{\times }}W_{0,\mathfrak{p}}(s_{0},a(y_{\mathfrak{p}}))\unicode[STIX]{x1D713}_{\mathfrak{p}}(y_{\mathfrak{p}}T_{\mathfrak{p}})|y_{\mathfrak{p}}|_{\mathfrak{p}}^{s}\,d^{\times }y_{\mathfrak{p}}\nonumber\\ \displaystyle & & \displaystyle \qquad \cdot \,\mathop{\prod }_{\mathfrak{p}\in \{\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{1}^{\prime },\mathfrak{p}_{2}^{\prime }\}}\mathbf{C}(\unicode[STIX]{x1D713}_{\mathfrak{p}})^{s-s_{0}}\unicode[STIX]{x1D701}_{\mathfrak{p},l_{\mathfrak{p}}}(s,s_{0})\bigg\},\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{\mathfrak{p},l_{\mathfrak{p}}}(s,s_{0})$ is defined in Lemma 3.1. From Corollary 3.2 we deduce

$$\begin{eqnarray}\biggl|\frac{\unicode[STIX]{x2202}^{k}}{\unicode[STIX]{x2202}s_{0}^{k}}\biggr|_{s_{0}=1/2}\,\mathop{\prod }_{\mathfrak{p}\in \{\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{1}^{\prime },\mathfrak{p}_{2}^{\prime }\}}\mathbf{C}(\unicode[STIX]{x1D713}_{\mathfrak{p}})^{s-s_{0}}\unicode[STIX]{x1D701}_{\mathfrak{p},l_{\mathfrak{p}}}(s,s_{0})\biggr|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}E^{-(1/2)\Vert \vec{l}\Vert +\unicode[STIX]{x1D716}},\quad k\leqslant n\leqslant 2.\end{eqnarray}$$

At $\mathfrak{p}<\infty$ , $T_{\mathfrak{p}}\neq 0$ , assuming $T_{\mathfrak{p}}=\unicode[STIX]{x1D71B}_{\mathfrak{p}}^{-k_{\mathfrak{p}}}$ , we can calculate explicitly (using, for example, [Reference WuWu14, Lemma 4.7])

$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1+2s_{0})}{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s+s_{0})\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s-s_{0})}\int _{\mathbf{F}_{\mathfrak{p}}^{\times }}W_{0,\mathfrak{p}}(s_{0},a(y_{\mathfrak{p}}))\unicode[STIX]{x1D713}_{\mathfrak{p}}(y_{\mathfrak{p}}T_{\mathfrak{p}})|y_{\mathfrak{p}}|_{\mathfrak{p}}^{s}\,d^{\times }y_{\mathfrak{p}}\nonumber\\ \displaystyle & & \displaystyle \quad =q_{\mathfrak{p}}^{-k_{\mathfrak{p}}(1/2+s-s_{0})}\frac{1-q_{\mathfrak{p}}^{-1/2-s-s_{0}}}{1-q_{\mathfrak{p}}^{-2s_{0}}}-q_{\mathfrak{p}}^{-k_{\mathfrak{p}}(1/2+s+s_{0})-2s_{0}}\frac{1-q_{\mathfrak{p}}^{-1/2-s+s_{0}}}{1-q_{\mathfrak{p}}^{-2s_{0}}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,q_{\mathfrak{p}}^{-(k_{\mathfrak{p}}-1)(1/2+s-s_{0})}\frac{(1-q_{\mathfrak{p}}^{-1/2-s-s_{0}})(1-q_{\mathfrak{ p}}^{-1/2-s+s_{0}})}{q_{\mathfrak{p}}-1}\frac{1-q_{\mathfrak{p}}^{-2k_{\mathfrak{p}}s_{0}}}{1-q_{\mathfrak{p}}^{-2s_{0}}}.\nonumber\end{eqnarray}$$

Thus we obtain the following bound

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\frac{\unicode[STIX]{x2202}^{k}}{\unicode[STIX]{x2202}s_{0}^{k}}\biggr|_{s_{0}=1/2}\,\frac{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1+2s_{0})}{\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s+s_{0})\unicode[STIX]{x1D701}_{\mathfrak{p}}(1/2+s-s_{0})}\int _{\mathbf{F}_{\mathfrak{p}}^{\times }}W_{0,\mathfrak{p}}(s_{0},a(y_{\mathfrak{p}}))\unicode[STIX]{x1D713}_{\mathfrak{p}}(y_{\mathfrak{p}}T_{\mathfrak{p}})|y_{\mathfrak{p}}|_{\mathfrak{p}}^{s}\,d^{\times }y_{\mathfrak{p}}\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \ll q_{\mathfrak{p}}^{-k_{\mathfrak{p}}\unicode[STIX]{x1D716}}(k_{\mathfrak{p}}\log q_{\mathfrak{p}})^{k}\ll _{k}\unicode[STIX]{x1D716}^{-k}.\nonumber\end{eqnarray}$$

At $v\mid \infty$ , we can trivially bound

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\frac{\unicode[STIX]{x2202}^{k}}{\unicode[STIX]{x2202}s_{0}^{k}}\biggr|_{s_{0}=1/2}\,\int _{\mathbf{F}_{v}^{\times }}W_{0,v}(s_{0},a(y_{v}))\unicode[STIX]{x1D713}_{v}(y_{v}T_{v})|y_{v}|_{v}^{s}\,d^{\times }y_{v}\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \int _{\mathbf{F}_{v}^{\times }}\biggl|\frac{\unicode[STIX]{x2202}^{k}}{\unicode[STIX]{x2202}s_{0}^{k}}\biggr|_{s_{0}=1/2}\,W_{0,v}(s_{0},a(y_{v}))\biggr||y_{v}|_{v}^{\unicode[STIX]{x1D716}}\,d^{\times }{y_{v}\ll }_{\unicode[STIX]{x1D716}}1\nonumber\end{eqnarray}$$

using classical asymptotic estimation for Whittaker functions (or [Reference Jacquet, Hida, Ramakrishnan and ShahidiJac04, Proposition 4.1]). Together with convex bounds for $\unicode[STIX]{x1D701}_{\mathbf{F}}$ , we see, for $n=0,1,2$ ,

$$\begin{eqnarray}|\unicode[STIX]{x1D701}({\textstyle \frac{1}{2}}+s,1,n(T).\text{E}^{\text{reg},(n)}({\textstyle \frac{1}{2}},e_{\vec{l}}))|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}(1+|s|)^{[\mathbf{F}:\mathbb{Q}]/2+\unicode[STIX]{x1D716}}E^{-(1/2)\Vert \vec{l}\Vert +\unicode[STIX]{x1D716}}.\end{eqnarray}$$

We get the desired bound using [Reference WuWu14, (6.1) and (6.2)] again. ◻

Corollary 5.3. For a typical pattern, we have, for any $\unicode[STIX]{x1D716}>0$ ,

$$\begin{eqnarray}|\unicode[STIX]{x1D701}(h,1,n(T).{\mathcal{E}}_{0}(\vec{\mathfrak{p}}))|\ll _{\mathbf{F},h_{0},\unicode[STIX]{x1D716}}(EQ)^{\unicode[STIX]{x1D716}}(Q^{\unicode[STIX]{x1D705}-1}E^{2}+E^{-2}).\end{eqnarray}$$

Proof. This follows from (5.2) and the following bounds resulting from Lemma 2.18(ii), Corollary 2.19 and Remark 2.20:

$$\begin{eqnarray}\hspace{81.60004pt}\unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\ll _{\unicode[STIX]{x1D716}}E^{\unicode[STIX]{x1D716}},\quad c(\vec{n}(\vec{\mathfrak{p}}),\vec{l};0)\ll E^{-(1/2)(4-\Vert \vec{l}\Vert )}=E^{-2+(1/2)\Vert \vec{l}\Vert }.\hspace{69.60004pt}\square\end{eqnarray}$$

5.2 Estimation for regularized $\text{L}^{4}$ -norms

Recall $E>0$ defined in Lemma 4.2. Choose a uniformizer $\unicode[STIX]{x1D71B}_{\mathfrak{p}}$ at every finite place  $\mathfrak{p}$ . Let $\vec{n}=(n_{\mathfrak{p}})_{\mathfrak{p}},n_{\mathfrak{p}}\in \mathbb{N}$ such that:

• ∙ $n_{\mathfrak{p}}=0$ unless $E\leqslant q_{\mathfrak{p}}<2E$ ;

• ∙ both the number of $\mathfrak{p}$ such that $n_{\mathfrak{p}}\neq 0$ and $|n_{\mathfrak{p}}|$ are bounded by some absolute constant.

Associated to $\vec{n}$ we define $t=t(\vec{n})\in \mathbb{A}^{\times }$ such that $t_{v}=1$ , $v\mid \infty$ and $t_{\mathfrak{p}}=\unicode[STIX]{x1D71B}_{\mathfrak{p}}^{-n_{\mathfrak{p}}}$ . Take $\unicode[STIX]{x1D711}_{j}=\text{E}^{\ast }(0,f_{j})$ , for $j=1,2$ , where:

• ∙ $f_{j,v}\in \unicode[STIX]{x1D70B}_{v}(1,1)$ is a unitary vector, spherical at each $v=\mathfrak{p}<\infty$ ;

• ∙ at each $v\mid \infty$ , $f_{j,v}$ lies in some $\mathbf{K}_{v}$ -isotypic $H_{n_{j,v}}$ , whose definition is recalled in Remark 2.20;

• ∙ write $\vec{n}_{j}=(n_{j,v})_{v\mid \infty }$ or $(n_{j,v})_{v}$ with $n_{j,\mathfrak{p}}=0$ for $\mathfrak{p}<\infty$ ;

• ∙ $|n_{j,v}|,v\mid \infty$ are bounded by some absolute constant.

Recall [Reference WuWu18, Definition 2.26] and write, for $j=1,2$ ,

$$\begin{eqnarray}{\mathcal{E}}_{t}={\mathcal{E}}(a(t)\unicode[STIX]{x1D711}_{1}\cdot \overline{\unicode[STIX]{x1D711}_{2}}),\quad \unicode[STIX]{x1D711}_{t}=a(t)\unicode[STIX]{x1D711}_{1}\cdot \overline{\unicode[STIX]{x1D711}_{2}}-{\mathcal{E}}_{t},\quad {\mathcal{E}}_{j}={\mathcal{E}}(|\unicode[STIX]{x1D711}_{j}|^{2}).\end{eqnarray}$$

We are interested in the $\text{L}^{2}$ -norm of $\unicode[STIX]{x1D711}_{t}$ in terms of  $E$ .

Theorem 5.4. With the above notation and conditions, we have

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D711}_{t}\Vert \ll (\log E)^{3}.\end{eqnarray}$$

Proof. Note that

(5.3) $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D711}_{t}\Vert ^{2} & = & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}(a(t)|\unicode[STIX]{x1D711}_{1}|^{2}-a(t){\mathcal{E}}_{1})(g)(|\unicode[STIX]{x1D711}_{2}|^{2}-{\mathcal{E}}_{2})(g)\,dg+\int _{[\text{PGL}_{2}]}^{\text{reg}}|\unicode[STIX]{x1D711}_{2}|^{2}(g)a(t){\mathcal{E}}_{1}(g)\,dg\nonumber\\ \displaystyle & & \displaystyle -\,\int _{[\text{PGL}_{2}]}^{\text{reg}}a(t){\mathcal{E}}_{1}(g){\mathcal{E}}_{2}(g)\,dg+\int _{[\text{PGL}_{2}]}^{\text{reg}}|{\mathcal{E}}_{t}(g)|^{2}\,dg\nonumber\\ \displaystyle & & \displaystyle -\,2\Re \int _{[\text{PGL}_{2}]}^{\text{reg}}a(t)\unicode[STIX]{x1D711}_{1}(g)\overline{\unicode[STIX]{x1D711}_{2}(g)}\overline{{\mathcal{E}}_{t}(g)}\,dg+\int _{[\text{PGL}_{2}]}^{\text{reg}}a(t)|\unicode[STIX]{x1D711}_{1}|^{2}(g){\mathcal{E}}_{2}(g)\,dg.\end{eqnarray}$$

We bound the right-hand side term by term, which will occupy the rest of this subsection. Note that only the fourth and sixth terms have growing contribution as $(\log E)^{3}$ and $(\log E)^{6}$ , respectively.◻

For the first term in (5.3), we use the Cauchy–Schwarz inequality to get

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}(a(t)|\unicode[STIX]{x1D711}_{1}|^{2}-a(t){\mathcal{E}}_{1})(g)(|\unicode[STIX]{x1D711}_{2}|^{2}-{\mathcal{E}}_{2})(g)\,dg\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{[\text{PGL}_{2}]}a(t)(|\unicode[STIX]{x1D711}_{1}|^{2}-{\mathcal{E}}_{1})(g)(|\unicode[STIX]{x1D711}_{2}|^{2}-{\mathcal{E}}_{2})(g)\,dg\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \Vert a(t)(|\unicode[STIX]{x1D711}_{1}|^{2}-{\mathcal{E}}_{1})\Vert \cdot \Vert |\unicode[STIX]{x1D711}_{2}|^{2}-{\mathcal{E}}_{2}\Vert =\Vert |\unicode[STIX]{x1D711}_{1}|^{2}-{\mathcal{E}}_{1}\Vert \cdot \Vert |\unicode[STIX]{x1D711}_{2}|^{2}-{\mathcal{E}}_{2}\Vert ,\nonumber\end{eqnarray}$$

which is independent of $t$ , hence $E$ .

For the second and third terms in (5.3), first notice that we can write

$$\begin{eqnarray}{\mathcal{E}}_{1}=\mathop{\sum }_{k=0}^{2}{\mathcal{E}}_{1,k}^{(k)}\biggl(\frac{1}{2}\biggr),\end{eqnarray}$$

where ${\mathcal{E}}_{1,k}(s)$ is a regularizing Eisenstein series (2.1) and the superscript $(k)$ means taking derivative $k$ times with respect to  $s$ . Moreover, ${\mathcal{E}}_{1,k}(s)$ are spherical at finite places. We notice further that although the regularized integral is not $\text{GL}_{2}(\mathbb{A})$ -invariant, it is still $\mathbf{K}$ -invariant [Reference WuWu18, Proposition 2.27(2)]. Hence if we write the Hecke operator associated with $\vec{n}$ as

$$\begin{eqnarray}T(\vec{n}):=\int _{\mathbf{K}_{\text{fin}}\times \mathbf{K}_{\text{fin}}}\text{R}(\unicode[STIX]{x1D705}_{1}a(t)\unicode[STIX]{x1D705}_{2})\,d\unicode[STIX]{x1D705}_{1}\,d\unicode[STIX]{x1D705}_{2}=\mathop{\prod }_{\mathfrak{p}<\infty }T(\mathfrak{p}^{|n_{\mathfrak{p}}|}),\end{eqnarray}$$

where $\text{R}(\cdot )$ denotes the $\text{GL}_{2}(\mathbb{A})$ -translation, we have

$$\begin{eqnarray}\displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}|\unicode[STIX]{x1D711}_{2}|^{2}(g)a(t){\mathcal{E}}_{1}(g)\,dg & = & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}|\unicode[STIX]{x1D711}_{2}|^{2}(g)(T(\vec{n}){\mathcal{E}}_{1}(g))\,dg,\nonumber\\ \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}a(t){\mathcal{E}}_{1}(g){\mathcal{E}}_{2}(g)\,dg & = & \displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}(T(\vec{n}){\mathcal{E}}_{1}(g)){\mathcal{E}}_{2}(g)\,dg.\nonumber\end{eqnarray}$$

${\mathcal{E}}_{1,k}(s)$ is a generalized eigenvector of $T(\vec{n})$ with eigenvalue [Reference BumpBum98, Theorem 4.6.6]

(5.4) $$\begin{eqnarray}\unicode[STIX]{x1D706}(\vec{n};s)=\mathop{\prod }_{\mathfrak{p}}\unicode[STIX]{x1D706}_{\mathfrak{p}}(|n_{\mathfrak{p}}|;s),\quad \unicode[STIX]{x1D706}_{\mathfrak{p}}(n;s)=\frac{q_{\mathfrak{p}}^{-n/2}}{1+q_{\mathfrak{p}}^{-1}}\biggl(\frac{q_{\mathfrak{p}}^{(n+1)s}-q_{\mathfrak{p}}^{-(n+1)s}}{q_{\mathfrak{p}}^{s}-q_{\mathfrak{p}}^{-s}}-q_{\mathfrak{p}}^{-1}\frac{q_{\mathfrak{p}}^{(n-1)s}-q_{\mathfrak{ p}}^{-(n-1)s}}{q_{\mathfrak{p}}^{s}-q_{\mathfrak{p}}^{-s}}\biggr),\end{eqnarray}$$

in the sense that (cf. [Reference WuWu18, Remark 2.17])

$$\begin{eqnarray}T(\vec{n}){\mathcal{E}}_{1,k}(1/2+s)=\unicode[STIX]{x1D706}(\vec{n};1/2+s){\mathcal{E}}_{1,k}(1/2+s)+c_{k}(\unicode[STIX]{x1D706}(\vec{n};1/2+s)-1)\unicode[STIX]{x1D706}_{\mathbf{F}}(s),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{\mathbf{F}}(s)$ is a ratio of zeta functions of $\mathbf{F}$ (see, for example, Theorem 2.2) and $c_{k}\in \mathbb{C}$ is some constant depending on $\unicode[STIX]{x1D711}_{1}$ . Note that $\unicode[STIX]{x1D706}(\vec{n};1/2)=1$ . Consequently, we get

$$\begin{eqnarray}\displaystyle T(\vec{n}){\mathcal{E}}_{1,k}^{(k)}(1/2) & = & \displaystyle \mathop{\sum }_{l=0}^{k}\binom{k}{l}\unicode[STIX]{x1D706}^{(l)}(\vec{n};1/2){\mathcal{E}}_{1,k}^{(k-l)}(1/2)\nonumber\\ \displaystyle & & \displaystyle +\,c_{k}\mathop{\sum }_{l=0}^{k-1}\binom{k}{l+1}\unicode[STIX]{x1D706}^{(l+1)}(\vec{n};1/2)\unicode[STIX]{x1D706}_{\boldsymbol{ F}}^{(k-l)}(0)+\frac{\unicode[STIX]{x1D706}^{(k+1)}(\vec{n};1/2)\unicode[STIX]{x1D706}_{\boldsymbol{ F}}^{(-1)}(0)}{k+1}.\nonumber\end{eqnarray}$$

Inserting the obvious bound $\unicode[STIX]{x1D706}^{(l)}(\vec{n};1/2)\ll (\log E)^{l}$ , we see that the second and third term in (5.3) are bounded as $\ll (\log E)^{3}$ .

For the fourth term in (5.3), we first make ${\mathcal{E}}_{t}$ explicit.

We can thus write

(5.5) $$\begin{eqnarray}a(t).\unicode[STIX]{x1D711}_{1}=\mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}}c(\vec{n},\vec{l};0)\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})\quad \Rightarrow \quad {\mathcal{E}}_{t}=\mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}}c(\vec{n},\vec{l};0){\mathcal{E}}(\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})\overline{\unicode[STIX]{x1D711}_{2}}).\end{eqnarray}$$

Recall the tensor product formulas for representations of $\mathbf{K}_{v},v\mid \infty$ :

• ∙ $\mathbf{F}_{v}=\mathbb{R}$ : $H_{n}\otimes H_{m}\simeq H_{n+m}$ ;

• ∙ $\mathbf{F}_{v}=\mathbb{C}$ : $H_{n}\otimes H_{m}\simeq H_{n+m}\oplus H_{n+m-2}\oplus \cdots \oplus H_{|n-m|}$ .

Together with the explicit determination of constant terms

$$\begin{eqnarray}\displaystyle & \displaystyle \text{E}_{\mathbf{N}}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})(a(y)\unicode[STIX]{x1D705})=\big\{\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }(1)|y|_{\mathbb{A}}^{1/2}\log |y|_{\mathbb{A}}+(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }(1)\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{1}+\vec{l};0))|y|_{\mathbb{A}}^{1/2}\big\}e_{\vec{n}_{1}+\vec{l}}(\unicode[STIX]{x1D705}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D711}_{2,\mathbf{N}}(a(y)\unicode[STIX]{x1D705})=\big\{\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }(1)|y|_{\mathbb{A}}^{1/2}\log |y|_{\mathbb{A}}+(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }(1)\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{2};0))|y|_{\mathbb{A}}^{1/2}\big\}f_{2}(\unicode[STIX]{x1D705}), & \displaystyle \nonumber\end{eqnarray}$$

we deduce the existence of $c_{\vec{m}}$ such that

(5.6) $$\begin{eqnarray}{\mathcal{E}}(\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})\overline{\unicode[STIX]{x1D711}_{2}})=\mathop{\sum }_{\vec{m}\preccurlyeq \vec{n}_{1}+\vec{n}_{2}}c_{\vec{m}}(c_{2}\text{E}^{\text{reg},(2)}({\textstyle \frac{1}{2}},e_{\vec{m}+\vec{l}})+c_{1}(\vec{l})\text{E}^{\text{reg},(1)}({\textstyle \frac{1}{2}},e_{\vec{m}+\vec{l}})+c_{0}(\vec{l})\text{E}^{\text{reg}}({\textstyle \frac{1}{2}},e_{\vec{m}+\vec{l}})),\end{eqnarray}$$

where we have written

$$\begin{eqnarray}\displaystyle & \displaystyle c_{2}=c_{2}(\vec{l})=|\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }|^{2},\quad c_{0}(\vec{l})=(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{1}+\vec{l};0))\overline{(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{2};0))}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle c_{1}(\vec{l})=(2\unicode[STIX]{x1D6FE}_{\mathbf{F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{1}+\vec{l};0))\overline{\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }}+\unicode[STIX]{x1D6EC}_{\boldsymbol{ F}}^{\ast }\overline{(2\unicode[STIX]{x1D6FE}_{\boldsymbol{ F}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6EC}_{\mathbf{F}}^{\ast }\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{2};0))}. & \displaystyle \nonumber\end{eqnarray}$$

Lemma 5.7. For $k_{1},k_{2}\geqslant 0$ , the regularized integral

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg},(k_{1})}\biggl(\frac{1}{2},e_{\vec{m}_{1}+\vec{l}_{1}}\biggr)(g)\overline{\text{E}^{\text{reg},(k_{2})}\biggl(\frac{1}{2},e_{\vec{m}_{2}+\vec{l}_{2}}\biggr)(g)}\,dg\end{eqnarray}$$

is non-vanishing only if $\vec{m}_{1}=\vec{m}_{2},\vec{l}_{1}=\vec{l}_{2}$ . We also have

$$\begin{eqnarray}\biggl|\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg},(k_{1})}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)\overline{\text{E}^{\text{reg},(k_{2})}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)}\,dg\biggr|\ll E^{-\Vert \vec{l}\Vert }(\log E)^{k_{1}+k_{2}+2}.\end{eqnarray}$$

Proof. This is a direct consequence of Theorem 2.4, together with

$$\begin{eqnarray}\hspace{99.60004pt}\widetilde{{\mathcal{M}}}_{1/2}^{(k)}e_{\vec{m}+\vec{l}}=\frac{d^{k}}{ds^{k}}\biggr|_{s=1/2}(\unicode[STIX]{x1D706}_{\mathbf{F}}(s-1/2)\unicode[STIX]{x1D707}(\vec{m}+\vec{l};s))e_{\vec{m}+\vec{l}}.\hspace{88.79993pt}\square\end{eqnarray}$$

Applying the lemma and the estimations

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\prime }(\vec{n}_{1}+\vec{l};0)=\unicode[STIX]{x1D707}^{\prime }(\vec{n_{1}};0)+\unicode[STIX]{x1D707}^{\prime }(\vec{l};0),\quad \unicode[STIX]{x1D707}^{\prime }(\vec{l};0)\ll \log E,\quad |c(\vec{n},\vec{l};0)|\ll E^{-(\Vert \vec{n}\Vert -\Vert \vec{l}\Vert )/2},\end{eqnarray}$$

we finally get

$$\begin{eqnarray}\displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}|{\mathcal{E}}_{t}(g)|^{2}\,dg & = & \displaystyle \mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}}\mathop{\sum }_{\vec{m}\preccurlyeq \vec{n}_{1}+\vec{n}_{2}}|c(\vec{n},\vec{l};0)c_{\vec{m}}|^{2}\mathop{\sum }_{k_{1},k_{2}=0}^{2}c_{k_{1}}(\vec{l})\overline{c_{k_{2}}(\vec{l})}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\text{reg},(k_{1})}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)\overline{\text{E}^{\text{reg},(k_{2})}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)}\,dg\nonumber\\ \displaystyle & \ll & \displaystyle E^{-\Vert \vec{n}\Vert }(\log E)^{6}.\nonumber\end{eqnarray}$$

For the fifth term in (5.3), note that we have computed/decomposed $a(t).\unicode[STIX]{x1D711}_{1}$ and ${\mathcal{E}}_{t}$ in (5.5) and (5.6).

Lemma 5.8. For any $k\geqslant 0$ , the regularized integral

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}_{1}})(g)\overline{\unicode[STIX]{x1D711}_{2}(g)}\overline{\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}_{2}}\biggr)(g)}\,dg\end{eqnarray}$$

is non-vanishing only if $\vec{l}_{1}=\vec{l}_{2}$ . We also have

$$\begin{eqnarray}\biggl|\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})(g)\overline{\unicode[STIX]{x1D711}_{2}(g)}\overline{\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)}\,dg\biggr|\ll _{k}E^{-\Vert \vec{l}\Vert }(\log E)^{k+4}.\end{eqnarray}$$

Proof. We apply Theorem 2.7. The ‘degenerate part’, that is, the weighted sum involving $\text{P}_{\mathbf{K}}$ , is easily seen to be non-vanishing only if $\vec{l}_{1}=\vec{l}_{2}$ , and in the case $\vec{l}_{1}=\vec{l}_{2}=\vec{l}$ it is bounded by $E^{-\Vert \vec{l}\Vert }(\log E)^{k+4}$ since $\unicode[STIX]{x1D707}^{(n)}(\vec{l};1/2)\ll E^{-\Vert l\Vert }(\log E)^{n}$ . For the ‘main part’, we note that

$$\begin{eqnarray}R({\textstyle \frac{1}{2}}+\bar{s},\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}_{1}})(g)\overline{\unicode[STIX]{x1D711}_{2}(g)};\overline{e_{\vec{m}+\vec{l}_{2}}})\end{eqnarray}$$

is the product of $\overline{\unicode[STIX]{x1D701}_{\mathbf{F}}(s+1)^{4}\unicode[STIX]{x1D701}_{\mathbf{F}}(2s+2)^{-1}}$ , some local components at $v\mid \infty$ irrelevant for estimation, and

defined in (3.1). Lemma 3.3(ii) implies the non-vanishing assertion. We need to estimate

$$\begin{eqnarray}\biggl(\frac{\unicode[STIX]{x2202}^{k}R}{\unicode[STIX]{x2202}s^{k}}\biggr)^{\text{hol}}\biggl(\frac{1}{2},\ldots \biggr)=\frac{k!}{(k+4)!}\cdot \frac{\unicode[STIX]{x2202}^{k+4}}{\unicode[STIX]{x2202}s^{k+4}}\biggr|_{s=0}\biggl(s^{4}R\biggl(\frac{1}{2}+s,\cdots \,\biggr)\biggr).\end{eqnarray}$$

Corollary 3.4 concludes the bound. ◻

We deduce that

$$\begin{eqnarray}\displaystyle \int _{[\text{PGL}_{2}]}^{\text{reg}}a(t)\unicode[STIX]{x1D711}_{1}(g)\overline{\unicode[STIX]{x1D711}_{2}(g)}\overline{{\mathcal{E}}_{t}(g)}\,dg & = & \displaystyle \mathop{\sum }_{\vec{m}\preccurlyeq \vec{n}_{1}\cdot \vec{n}_{2}}\mathop{\sum }_{\vec{l}\preccurlyeq \vec{n}}c_{\vec{m}}|c(\vec{n},\vec{l};0)|^{2}\cdot \mathop{\sum }_{k=0}^{2}c_{k}(\vec{l})\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{[\text{PGL}_{2}]}^{\text{reg}}\text{E}^{\ast }(0,e_{\vec{n}_{1}+\vec{l}})(g)\overline{\unicode[STIX]{x1D711}_{2}(g)}\overline{\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2},e_{\vec{m}+\vec{l}}\biggr)(g)}\,dg\nonumber\\ \displaystyle & \ll & \displaystyle E^{-\Vert \vec{n}\Vert }(\log E)^{6}.\nonumber\end{eqnarray}$$

For the last term in (5.3), first notice that we can write

$$\begin{eqnarray}{\mathcal{E}}_{2}=\mathop{\sum }_{k=0}^{2}\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2},h_{k}\biggr),\end{eqnarray}$$

where $h_{k}$ are spherical at $\mathfrak{p}$ for which $n_{\mathfrak{p}}\neq 0$ . We are reduced to bounding

$$\begin{eqnarray}\int _{[\text{PGL}_{2}]}^{\text{reg}}|a(t).\text{E}^{\ast }(0,f_{1})(g)|^{2}\text{E}^{\text{reg},(k)}\biggl(\frac{1}{2},h_{k}\biggr)(g)\,dg,\end{eqnarray}$$

to which Theorem 2.7 applies directly.

The degenerate part contributes $(\log E)^{3}$ . In fact, only

$$\begin{eqnarray}\text{P}_{\mathbf{K}}({\mathcal{M}}_{0}^{(l)}a(t)f_{1}\cdot \overline{a(t)f_{1}})\text{P}_{\mathbf{K}}(h_{k}),\quad 0\leqslant l\leqslant 3,\end{eqnarray}$$

has non-constant contribution. For the main part, using the $\text{GL}_{2}(\mathbf{F}_{\mathfrak{p}})$ -invariance of the local Rankin–Selberg zeta functions, we easily deduce

$$\begin{eqnarray}R\biggl(\frac{1}{2}+s,|a(t).\text{E}^{\ast }(0,f_{1})|^{2};h_{k}\biggr)=\frac{\unicode[STIX]{x1D701}_{\mathbf{F}}(s+1)^{4}}{\unicode[STIX]{x1D701}_{\mathbf{F}}(2s+2)}\cdot \unicode[STIX]{x1D706}\biggl(\vec{n};\frac{1}{2}+s\biggr)\cdot \frac{\unicode[STIX]{x1D701}_{\mathbf{F}}(2s+2)R(1/2+s,|\text{E}^{\ast }(0,f_{1})|^{2};h_{k})}{\unicode[STIX]{x1D701}_{\mathbf{F}}(s+1)^{4}},\end{eqnarray}$$

where only the term $\unicode[STIX]{x1D706}(\vec{n};1/2+s)$ defined in (5.4) depends on  $E$ . We bound

$$\begin{eqnarray}\frac{k!}{(k+4)!}\frac{\unicode[STIX]{x2202}^{k+4}}{\unicode[STIX]{x2202}s^{k+4}}\biggr|_{s=0}s^{4}R\biggl(\frac{1}{2}+s,|a(t).\text{E}^{\ast }(0,f_{1})|^{2};h_{k}\biggr)\ll (\log E)^{k+4}\end{eqnarray}$$

from $\unicode[STIX]{x1D706}^{(l)}(\vec{n};1/2)\ll (\log E)^{l}$ , and conclude that the last term in (5.3) is bounded as $\ll (\log E)^{6}$ .