2.1.1 Generalities

Let ${\mathbf {f}}$ denote the finite places of ${\mathbb {Q}}$ (which we identify with the set of primes) and $\infty$ the archimedean place. We let ${\mathbb {A}}$ denote the ring of adeles over ${\mathbb {Q}}$ and ${\mathbb {A}}_{\mathbf {f}}$ the ring of finite adeles. Given an algebraic group $H$ defined over ${\mathbb {Q}}$, a place $v$ of ${\mathbb {Q}}$, a subset of places $U$ of ${\mathbb {Q}}$, and a positive integer $M$, we denote $H_v:= H({\mathbb {Q}}_v)$, $H_U:=\prod _{v \in U} H_v$, $H_M :=\prod _{p|M} H_p$. Given an element $g$ in $H({\mathbb {Q}})$ (respectively, in $H({\mathbb {A}})$), we will use $g_p$ to denote the image of $g$ in $H_p$ (respectively, the $p$-component of $g$); more generally, for any set of places $U$, we let $g_U$ denote the image of $g$ in $H_U$.

Given two integers $a$ and $b$, we use $a|b$ to denote that $a$ divides $b$, and we use $a|b^\infty$ to denote that $a|b^n$ for some positive integer $n$. For any real number $\alpha$, we let $\lfloor \alpha \rfloor$ denote the greatest integer less than or equal to $\alpha$ and we let $\lceil \alpha \rceil$ denote the smallest integer greater than or equal to $\alpha$. For any integer $A=\prod _{p\in {\mathbf {f}}} p^{a_p}$, we write

(4)\begin{equation} A_1 = \prod_{p\in {\mathbf{f}}} p^{\lceil {a_p}/2 \rceil}. \end{equation}

In other words, $A_1$ is the smallest integer such that $A$ divides $A_1^2$.

All representations of (topological) groups are assumed to be continuous and over the field of complex numbers.

2.1.2 Quaternions, orders, and groups

Throughout this paper, we fix an indefinite quaternion division algebra $D$ over ${\mathbb {Q}}$. We fix once and for all a maximal order ${\mathcal {O}}^\textrm {max}$ of $D$. All constants in the bounds in this paper will be allowed to depend on $D$ without explicit mention. We let $d$ denote the reduced discriminant of $D$, i.e. the product of all primes such that $D_p$ is a division algebra. We let $\textrm {nr}$ be the reduced norm on $D^\times$.

We denote $G=D^\times$ and $G' =PD^\times = D^\times /Z$, where $Z$ denotes the centre of $D^\times$. For each prime $p$, let $K_p=({\mathcal {O}}^\textrm {max}_p)^\times$ and let $K_p'$ denote the image of $K_p$ in $G'_p$. Given an order ${\mathcal {O}}$ of $D$, we define a compact open subgroup of $G({\mathbb {A}}_{\mathbf {f}})$ by

\[ K_{\mathcal{O}} =\prod_{p \in {\mathbf{f}}} {\mathcal{O}}_p^\times. \]

For each place $v$ that is not among the primes dividing $d$, fix once and for all an isomorphism $\iota _v: D_v \xrightarrow {\cong } M(2,{\mathbb {Q}}_v)$. We assume that these isomorphisms are chosen such that for each finite prime $p\nmid d$, we have $\iota _p({\mathcal {O}}_p) = M(2,{\mathbb {Z}}_p)$. By abuse of notation, we also use $\iota _v$ to denote the composition map $D({\mathbb {Q}}) \rightarrow D_v \rightarrow M(2,{\mathbb {Q}}_v)$.

For any lattice ${\mathcal {L}}\subseteq {\mathcal {O}}^\textrm {max}$ of $D$, we get a local lattice ${\mathcal {L}}_p$ of $D_p$ by localizing at each prime $p$. These collections of lattices satisfy

(5)\begin{equation} {\mathcal{L}} = \{g \in D: g_p \in {\mathcal{L}}_p \text{ for all primes }p\}. \end{equation}

Conversely, if we are given a collection of local lattices $\{{\mathcal {L}}_p\}_{p \in {\mathbf {f}}}$ such that ${\mathcal {L}}_p \subseteq {\mathcal {O}}^\textrm {max}_p$ for all $p$ and ${\mathcal {L}}_p = {\mathcal {O}}^\textrm {max}_p$ for all but finitely many $p$, then there exists a unique lattice ${\mathcal {L}}\subseteq {\mathcal {O}}^\textrm {max}$ of $D$ defined via (5) and whose localizations at primes $p$ are precisely the ${\mathcal {L}}_p$. We will refer to ${\mathcal {L}}$ as the global lattice corresponding to the collection of local lattices $\{{\mathcal {L}}_p\}_{p \in {\mathbf {f}}}$. More generally, given a finite subset $S \subseteq {\mathbf {f}}$ and a collection of local lattices $\{{\mathcal {L}}_p\}_{p \in S}$, we can construct the (unique) lattice whose localization at a prime $p$ equals ${\mathcal {L}}_p$ if $p\in S$ and equals ${\mathcal {O}}_p^\textrm {max}$ if $p \notin S$; we will refer to this lattice as the global lattice corresponding to $\{{\mathcal {L}}_p\}_{p \in S}$.

Let ${\mathcal {L}}$ be a lattice in $D$ such that ${\mathcal {L}}\subseteq {\mathcal {O}}^\textrm {max}$. We say that ${\mathcal {L}}$ is tidy in ${\mathcal {O}}^\textrm {max}$ if ${\mathcal {L}}$ contains 1 and $M_3^2$ divides $N=[{\mathcal {O}}^\textrm {max}: {\mathcal {L}}]$, where $(M_1, M_2, M_3)$ is the unique triple of positive integers such that $M_1|M_2|M_3$ and ${\mathcal {O}}^\textrm {max} /{\mathcal {L}} \simeq ({\mathbb {Z}} / M_1{\mathbb {Z}}) \times ({\mathbb {Z}} / M_2{\mathbb {Z}}) \times ({\mathbb {Z}} / M_3{\mathbb {Z}}).$ Note that since $N=M_1M_2M_3$, $M_3^2$ divides $N$ if and only if $N$ divides $(M_1M_2)^2$ if and only if $M_3$ divides $M_1M_2$. Let ${\mathcal {L}}_p$ be a lattice of $D_p$ such that ${\mathcal {L}}_p\subseteq {\mathcal {O}}_p^\textrm {max}$. We say that ${\mathcal {L}}_p$ is tidy in ${\mathcal {O}}_p^\textrm {max}$ if $1 \in {\mathcal {L}}_p$ and $m_3 \leqslant m_1+m_2$, where $(m_1, m_2, m_3)$ is the unique triple of non-negative integers such that $m_1\leqslant m_2 \leqslant m_3$ and ${\mathcal {O}}^\textrm {max}_p/{\mathcal {L}}_p \simeq ({\mathbb {Z}}_p / p^{m_1}{\mathbb {Z}}_p) \times ({\mathbb {Z}}_p / p^{m_2}{\mathbb {Z}}_p) \times ({\mathbb {Z}}_p / p^{m_3}{\mathbb {Z}}_p).$ It is clear that a global lattice ${\mathcal {L}}$ is tidy in ${\mathcal {O}}^\textrm {max}$ if and only if all the corresponding local lattices ${\mathcal {L}}_p$ are tidy in ${\mathcal {O}}^\textrm {max}_p$.

For each $g \in G({\mathbb {A}}_{\mathbf {f}})$ and a lattice ${\mathcal {L}}$ of $D$, we let ${}^g{\mathcal {L}}$ denote the lattice whose localization at each prime $p$ equals $g_p {\mathcal {L}}_p g_p^{-1}$. Note that if $g \in K_{{\mathcal {O}}^\textrm {max}}$ and ${\mathcal {L}}$ is tidy in ${\mathcal {O}}^\textrm {max}$, then ${}^g{\mathcal {L}}$ is also tidy in ${\mathcal {O}}^\textrm {max}$.

2.1.3 Haar measures

We fix the Haar measure on each group $G_p$ such that $\operatorname {vol}(K_p)=1$. We fix a Haar measure on ${\mathbb {Q}}_p^\times$ such that $\operatorname {vol}({\mathbb {Z}}_p^\times )=1$. This gives us resulting Haar measures on each group $G'_p$ such that $\operatorname {vol}(K'_p)=1$. Fix any Haar measure on $G_\infty$ and take the Haar measure on ${\mathbb {R}}^\times$ to be equal to ${d}x/|x|$, where $dx$ is the Lebesgue measure. This gives us a Haar measure on $G'_\infty$. Take the measures on $G({\mathbb {A}})$ and $G'({\mathbb {A}})$ to be given by the product measure.

For each continuous function $\phi$ on the space $G({\mathbb {A}})$, we let $R(g)$ denote the right-regular action, given by $(R(g)\phi )(h) = \phi (hg)$. If a continuous function $\phi$ on $G({\mathbb {A}})$ satisfies that $|\phi |$ is left $Z({\mathbb {A}})G({\mathbb {Q}})$ invariant, define

(6)\begin{equation} \|\phi\|_2 = \bigg(\int_{G'({\mathbb{Q}})\backslash G'({\mathbb{A}})} |\phi(g)|^2\,dg\bigg)^{1/2}. \end{equation}

Note above that $G'({\mathbb {Q}})\backslash G'({\mathbb {A}})$ is compact, so convergence of the integral is not an issue.

2.1.4 Asymptotic notation

We use the notation $A \ll _{x,..,y} B$ to signify that there exists a quantity $C$ depending only on $x,..,y$ (and possibly on any objects fixed throughout the paper) so that $|A| \leqslant C |B|$. We use $A \asymp _{x,..y} B$ to mean that $A \ll _{x,..y} B$ and $B \ll _{x,..y} A$. The symbol $\epsilon$ will denote a small positive quantity whose value may change from line to line; a statement such as $A \ll _{\epsilon , x,..} B$ should be read as ‘For all small $\epsilon >0$, there is a quantity $C$ that depends only on $\epsilon , x,..,$ and on any objects fixed throughout the paper such that $|A| \leqslant C |B|$.’ An assertion such as $A \ll _{x,..y} D^{O(1)} B$ means that there is a constant $C$ such that $|A| \ll _{x,..y} |D|^{C} |B|$.

4.3.1 Test functions

We define a test function $\kappa$ on $G({\mathbb {A}})$, which will be essentially the same as the one used in [Reference SahaSah20]. Let $S = \textbf {S}(\pi ) \cup \{p \in {\mathbf {f}}: p|d\}$. Let $\textbf {ur} = {\mathbf {f}} \setminus S$ be the set of primes not in $S$. We will choose $\kappa$ of the form $\kappa = \kappa _S \kappa _\textbf {ur} \kappa _\infty$. For convenience, we denote $G_S = \prod _{p \in S}G_p$, ${\mathbb {Q}}_S^\times = \prod _{p \in S} {\mathbb {Q}}_p^\times$, and ${\mathcal {O}}_S^\times = \prod _{p\in S}{\mathcal {O}}_p^\times$. By assumption, the action of ${\mathcal {O}}_S^\times$ on $\phi '$ generates an irreducible representation of dimension $\ll _{\epsilon } C_1(\pi )^{\delta + \epsilon }$.

We define the function $\kappa _S$ on $G_S$ as follows:

\[ \kappa_S (g_S)= \begin{cases} 0 & \text{if } g_S \notin {\mathbb{Q}}_S^\times{\mathcal{O}}_S^\times, \\ \omega_{\pi}^{-1}(z) \langle \phi', \pi(k) \phi' \rangle & \text{if } g_S= zk, \quad z \in {\mathbb{Q}}_S^\times,\quad \ k \in {\mathcal{O}}_S^\times. \end{cases} \]

Then, as in § 4.1 of [Reference SahaSah20], we have

(18)\begin{equation} R(\kappa_S)\phi':=\int_{{\mathbb{Q}}_S^\times \backslash G_S} \kappa_S(g)(\pi(g)\phi') \,dg = \lambda_S \phi', \quad \text{where}\ \lambda_S \gg_{\epsilon} \frac{1}{NC_1(\pi)^{\delta +\epsilon}}. \end{equation}

Next we move on to the primes in $\textbf {ur}$. We define $\kappa _\textbf {ur}$ exactly as in § 4.1 of [Reference SahaSah20]. The definition of $\kappa _\textbf {ur}$ depends on a parameter $\Lambda$ that we will fix later. As shown in [Reference SahaSah20],

(19)\begin{equation} R(\kappa_\textbf{ur} )\phi' = \lambda_\textbf{ur} \phi', \quad \lambda_\textbf{ur} \gg_\epsilon \Lambda^{2-\epsilon}. \end{equation}

Finally, we consider the infinite place. As we are not looking for a bound in the archimedean aspect, the choice of $\kappa _\infty$ is unimportant. However, for definiteness, let us fix the function $\kappa _\infty$ as follows. Let $f:{\mathbb {R}}_{\geqslant 0} \rightarrow [0,1]$ be a smooth non-increasing function such that $f(x)=1$ if $x \in [0, \frac 12]$ and $f(x)=0$ if $x \geqslant 1$. Let $g \in \textrm {GL}_2({\mathbb {R}})^+$ and define $u(g) = {|g(i) - i|^2}/{4 \,\mathrm {Im}(g(i))}$. Define

\[ \kappa_\infty(g) = f(u(g)) \langle \phi' , \pi(g) \phi'\rangle \]

for $g \in \textrm {GL}_2({\mathbb {R}})^+$ and define $\kappa _\infty$ to be equal to identically zero on $\textrm {GL}_2({\mathbb {R}})^-$. Then we have that

\[ \kappa_\infty(g) \neq 0 \Rightarrow \det(\iota_\infty(g_\infty))>0, \quad u(\iota_\infty(g_\infty)) \leqslant 1 \]

and furthermore the operator $R(\kappa _\infty )$ satisfies

(20)\begin{equation} R(\kappa_\infty)\phi' = \lambda_\infty \phi', \quad \lambda_\infty \gg_{T} 1. \end{equation}

We define the automorphic kernel $K_\kappa (g_1, g_2)$ for $g_1, g_2 \in G({\mathbb {A}})$ via

\[ K_\kappa(g_1, g_2) = \sum_{\gamma \in G'({\mathbb{Q}})} \kappa(g_1^{-1} \gamma g_2). \]

Now, as in § 4.2 of [Reference SahaSah20], we get

(21)\begin{equation} |\phi'(g)|^2 \ll_{T, \epsilon} NC_1(\pi)^{\delta + \epsilon}\Lambda^{-2 + \epsilon} K_\kappa(g,g).\end{equation}

On the other hand, we have by construction

(22)\begin{equation} K_\kappa(g,g) \leqslant \sum_{1 \leqslant \ell \leqslant 16\Lambda^4} \frac{y_\ell}{\ell^{1/2}}\sum_{\substack{\gamma \in G'({\mathbb{Q}})\\ \kappa_\ell(\gamma) \neq 0 \\ \kappa_\infty(g_\infty^{-1}\gamma_\infty g_\infty) \neq 0} } \big|\kappa_S(g_S^{-1}\gamma_S g_S) \big|, \end{equation}

where the $y_\ell$ satisfy

(23)\begin{equation} |y_\ell| \ll \begin{cases}\Lambda, & \ell=1,\\1, & \ell = \ell_1 \ell_2 \text{ or } \ell = \ell_1^2\ell_2^2 \text{ with } \ell_1, \ell_2 \in \mathcal{P}, \\ 0, & \text{otherwise}\end{cases}\end{equation}

with $\mathcal {P}= \{\ell : \ell \text { prime, } \ell \in \textbf {ur}, \ \Lambda \leqslant \ell \leqslant 2\Lambda \}$ and where $\kappa _\ell = \prod _{p \in \textbf {ur}}\kappa _{\ell , p}$ is a function on $\prod _{p \in \textbf {ur}} G({\mathbb {Q}}_p)$ that is defined in § 3.5 of [Reference SahaSah17] (see also § 4.1 of [Reference SahaSah20]); we recall that $\kappa _{\ell , p}$ is supported on ${\mathbb {Q}}_p^\times {\mathcal {O}}_p(\ell )$, where ${\mathcal {O}}_p(\ell ) = \{ \alpha \in {\mathcal {O}}_p: \textrm {nr}(\alpha ) \in \ell {\mathbb {Z}}_p^\times \}$.

Let us look at (22) more carefully. First of all, note that if $\kappa _\ell (\gamma ) \kappa _\infty (g_\infty ^{-1}\gamma _\infty g_\infty ) \neq 0$, then:

1. (a) $\gamma _p \in {\mathbb {Q}}_p^\times {\mathcal {O}}_p(\ell ) \quad \forall p \in \textbf {ur}$;

2. (b) $\det (\iota _\infty (\gamma _\infty ))>0$, $u(z, \iota _\infty (\gamma _\infty ) z) \leqslant 1$, where $z = g_\infty i$.

Looking at the primes $p |d$, we see that $\kappa _S(g_S^{-1}\gamma _S g_S) \neq 0$ implies that:

1. (c) $\gamma _p \in {\mathbb {Q}}_p^\times {\mathcal {O}}_p^\times \quad \forall p |d$.

(We remind the reader here that ${\mathcal {O}}_p = {\mathcal {O}}^\textrm {max}_p$ if $p \in \textbf {ur}$ or if $p|d$.)

Consider the primes $p \in \textbf {S}(\pi )$. If $\kappa _p(g_p^{-1}\gamma _p g_p) \neq 0$, then clearly $g_p^{-1}\gamma _p g_p \in {\mathbb {Q}}_p^\times {\mathcal {O}}_p^\times$ or, equivalently, $\gamma _p \in {\mathbb {Q}}_p^\times ({}^g{\mathcal {O}})_p^\times$. So far, we have not at all used condition $(2)$ of Definition 3.6. We now do so. For each prime $p \in \textbf {S}(\pi )$, define $r_p = a_1(\pi _p)+1.$ Define $R_p = \{1, \ldots , r_p\}$ and let $R$ be the set-theoretic product $\prod _{p \in \textbf {S}(\pi )}R_p$. For each $u = (u_p)_{p\in \textbf {S}(\pi )} \in R$, where each $u_p \in R_p$, associate another tuple $H_u = (\eta _{p,u_p})_{p\in \textbf {S}(\pi )}$ as follows: $\eta _{p, 1} = \eta _1$ and $\eta _{p,i} = \eta _1 + (i-1) (({\eta _2-\eta _1})/{a_1(\pi _p))}$ for all $1\leqslant i \leqslant r_p$.

Now consider a $\gamma \in G'({\mathbb {Q}})$ which satisfies (a)–(c) above and such that $\gamma _p \in {\mathbb {Q}}_p^\times ({}^g{\mathcal {O}})_p^\times$ for each $p \in \textbf {S}(\pi )$. It is clear that for any such $\gamma$, there exists a unique tuple $u \in R$ such that:

1. (d) $g_p^{-1}\gamma _p g_p \in {\mathbb {Q}}_p^\times ({\mathcal {L}}^{\eta _{p,u_p}}_p \cap {\mathcal {O}}_p^\times )$, $g_p^{-1}\gamma _p g_p \notin {\mathbb {Q}}_p^\times ({\mathcal {L}}^{\eta _{p,u_p+1}}_p \cap {\mathcal {O}}_p^\times ) \quad \forall p \in \textbf {S}(\pi )$.

Above, we adopt the convention that ${\mathcal {L}}_p^{\eta _{p,r_p+1}}$ is the empty set for each $p \in \textbf {S}(\pi )$, so that the second part of condition (d) is automatic for the primes where $u_p=r_p$.

It is clear from the above discussion that the contribution to the right-most sum in (22) only comes from those $\gamma$ for which the conditions (a)–(d) above are satisfied for some tuple $u \in R$. Furthermore, whenever the conditions (a)–(d) above are satisfied for a particular $u$, condition 2(c) of Definition 3.6 implies that

\[ \big|\kappa_S(g_S^{-1}\gamma_S g_S) \big| \ll C'(\pi)^{O(1)} \prod_{p\in \textbf{S}(\pi)}p^{(\eta_2 - \eta_1)(u_p - a_1(\pi_p))}. \]

For each tuple $u$, recall the definition of the lattice ${\mathcal {L}}^{H_u}$, which is precisely the global lattice corresponding to the collection of local lattices $\{{\mathcal {L}}_p^{\eta _{p,u_p}}\}_{p\in \textbf {S}(\pi )}$. Define

\[ {}^g{\mathcal{L}}^{H_u}(\ell;z, 1) =\{\alpha \in {}^g{\mathcal{L}}^{H_u}: \textrm{nr}(\alpha) = \ell, u(z, \iota_\infty(\alpha) z) \leqslant 1\}. \]

By Proposition 4.2 of [Reference SahaSah20], the number of $\gamma \in G'({\mathbb {Q}})$ satisfying (a)–(d) above is bounded by the size of $|{}^g{\mathcal {L}}^{H_u}(\ell ;z, 1)|$.

Therefore, we conclude that

(24)\begin{equation} K_\kappa(g,g) \ll C'(\pi)^{O(1)} \sum_{u\in R} \sum_{1 \leqslant \ell \leqslant 16\Lambda^4} \frac{y_\ell}{\ell^{1/2}}|{}^g{\mathcal{L}}^{H_u}(\ell;z, 1)| \prod_{p\in \textbf{S}(\pi)}p^{(\eta_2 - \eta_1)(u_p - a_1(\pi_p))}. \end{equation}

Now, using the facts that the lattice ${}^g{\mathcal {L}}^{H_u}$ is tidy in ${\mathcal {O}}^\textrm {max}$ and has index $N^{H_u}$ in ${\mathcal {O}}^\textrm {max}$, we use Proposition 2.1 and (15) to obtain for each $1 \leqslant L \leqslant C(\pi )^{O(1)}$,

(25)\begin{gather} \sum_{1\leqslant m \leqslant L } |{}^g{\mathcal{L}}^{H_u}(m;z, 1)| \ll_{\epsilon} C(\pi)^{\epsilon} \biggl( L +C'(\pi)^{O(1)}\frac{L^2}{N\prod_{p \in \textbf{S}(\pi)} p^{(\eta_2 - \eta_1)(u_p - 1)} } \biggr), \end{gather}

(26)\begin{gather} \sum_{1\leqslant m \leqslant L } |{}^g{\mathcal{L}}^{H_u}(m^2;z, 1)| \ll_{\epsilon}C(\pi)^{\epsilon} \biggl( L +C'(\pi)^{O(1)}\frac{L^3}{N\prod_{p \in \textbf{S}(\pi)} p^{(\eta_2 - \eta_1)(u_p - 1)} } \biggr). \end{gather}

Combining (23), (24), (25), and (26), we get

(27)\begin{align} K_\kappa(g,g) &\ll_{ \epsilon} C'(\pi)^{O(1)}C(\pi)^{\epsilon} \biggl(\Lambda + \frac{\Lambda^4}{NC_1(\pi)^{\eta_2-\eta_1}}\biggr) \sum_{u\in R } 1 \nonumber\\ &\ll_\epsilon C'(\pi)^{O(1)} C(\pi)^\epsilon \biggl(\Lambda + \frac{\Lambda^4}{NC_1(\pi)^{\eta_2-\eta_1}}\biggr) \end{align}

since $|R| \ll _\epsilon C(\pi )^\epsilon$.

From (21) and (27), we obtain the pivotal inequality

(28)\begin{equation} |\phi'(g)|^2 \ll_{T, \epsilon} C_1(\pi)^{\eta_1+\delta + \epsilon}C'(\pi)^{O(1)}\biggl( \frac{1}{\Lambda} + \frac{\Lambda^2}{C_1(\pi)^{\eta_2} } \biggr). \end{equation}

Now, putting $\Lambda = C_1(\pi )^{{\eta _2}/3}$, we immediately obtain (16), as required.

5.5.1 Principal series representation

Let $\pi =\pi (\mu _1,\mu _2)$ be a principal series representation. In this case $n$ is even and we take $\mu _2 = \mu _1^{-1} = \mu$, $a(\mu )=n_1=n_0=n/2$. Denote

(35)\begin{equation} C_0=\int_{u\in {\mathfrak o}^\times} \mu( u)\psi(-\varpi^{-n_0} u)\,du. \end{equation}

Using the usual interpretation as a Gauss sum (see, e.g., [Reference SahaSah16, (6)]), we see that $|C_0|\asymp {1}/{q^{n_0/2}}$.

By [Reference HuHu17, Lemma 2.12], we have the following lemma.

Lemma 5.5 When $n_0 < i\leqslant n$, $W^{(i)}(x)$ is supported on $x \in {\mathfrak o}^\times$ and for $x \in {\mathfrak o}^\times$ we have

(36)\begin{equation} W^{(i)}( x )=C_0^{-1}\int_{u\in {\mathfrak o}^\times} \mu(1+u\varpi^{i-n_0})\mu( x u)\psi(-\varpi^{-n_0} x u)\,du. \end{equation}

Note that the condition $a(\pi )\geqslant 3$ implies that $a(\mu )\geqslant 2$. Let ${\alpha }$ be the constant associated to $\mu$ by Lemma 5.1. Then $v({\alpha })=-n_0$.

By the results of [Reference HuHu18], $\Phi _\pi ^{(i)}(a,m)$ is supported on $v(a)=0$ and $v(m)=i-n>-n_0$. Then, by (34),

(37)\begin{align} \Phi_\pi^{(i)}(a,m)&=C_0^{-1}\int_{v(x)=0}\psi(mx)\int_{u\in {\mathfrak o}^\times} \mu(1+u\varpi^{i-n_0})\mu(a x u)\psi(-\varpi^{-n_0}a x u)\,du\,d^\times x \nonumber\\ &=C_0^{-1}\iint_{v(x)=v(u)=0} \psi(mxu)\mu(1+u^{-1}\varpi^{i-n_0})\mu(a x )\psi(-\varpi^{-n_0}a x )\,du\,d^\times x. \end{align}

The idea is to break the above integral into small intervals, on each of which we can apply Lemma 5.1 to analyse the integral and get easy vanishing for most of the small intervals. This is the exact analogue of the archimedean stationary phase analysis. In the integrand in (37), write $u=u_0(1+{{\Delta }} u)$ for $u_0\in {\mathfrak o}^\times / (1+\mathfrak p^{\lceil (n-i)/2\rceil })$ and ${{\Delta }} u\in \mathfrak p^{\lceil (n-i)/2\rceil }$, and $x=x_0(1+{{\Delta }} x)$ for $x_0\in {\mathfrak o}^\times /(1+\mathfrak p^{\lceil n_0/2\rceil })$ and ${{\Delta }} x\in \mathfrak p^{\lceil n_0/2\rceil }$. Using Lemma 5.1 and the invariance properties of $\psi$ and $\mu$, we get

(38)\begin{align} \Phi_\pi^{(i)}(a,m)&=C_0^{-1}\sum_{x_0,u_0} \psi(mx_0u_0)\mu(1+u_0^{-1}\varpi^{i-n_0})\mu(a x_0 )\psi(-\varpi^{-n_0}a x_0 ) \nonumber\\ & \quad \times \int_{\mathfrak p^{\lceil ({n-i})/2\rceil} } \int_{\mathfrak p^{\lceil ({n_0})/2\rceil} } \psi\Big(\vphantom{\frac{\varpi^{i-n_0}u_0^{-1}}{1+\varpi^{i-n_0}u_0^{-1}}}mu_0x_0{{\Delta}} x+mx_0u_0{{\Delta}} u\nonumber\\ &\quad -{\alpha}\frac{\varpi^{i-n_0}u_0^{-1}}{1+\varpi^{i-n_0}u_0^{-1}} {{\Delta}} u+{\alpha} {{\Delta}} x-\varpi^{-n_0}ax_0{{\Delta}} x\Big) d{{\Delta}} x \,d{{\Delta}} u. \end{align}

For the innermost integral involving ${{\Delta }} x$, ${{\Delta }} u$ to be non-zero, we must have that

(39)\begin{gather} mu_0x_0+{\alpha}-\varpi^{-n_0}ax_0 \equiv 0 \mod \varpi^{-\lceil {n_0}/{2}\rceil}, \end{gather}

(40)\begin{gather} mx_0u_0 -{\alpha}\frac{\varpi^{i-n_0}u_0^{-1}}{1+\varpi^{i-n_0}u_0^{-1}}\equiv 0\mod \varpi^{-\lceil({n-i})/{2}\rceil}. \end{gather}

From the first equation, we get that

(41)\begin{equation} x_0\equiv -\frac{{\alpha}}{mu_0-\varpi^{-n_0}a}\mod{\varpi^{\lfloor {n_0}/{2}\rfloor}}. \end{equation}

So, there is a unique $x_0 \mod \varpi ^{\lfloor {n_0}/{2}\rfloor }$ for each $u_0\mod \varpi ^{\lceil ({n-i})/{2}\rceil }$ satisfying the above. As a trivial consequence, there are at most $q$ solutions of $x_0\mod \varpi ^{\lceil {n_0}/{2}\rceil }$ for each $u_0\mod \varpi ^{\lceil ({n-i})/{2}\rceil }$.

Next, by computing (39)$\,\times\, mu_0$ – (40)$\,\times\, (mu_0-\varpi ^{-n_0}a)$, we get the following necessary condition for non-vanishing:

(42)\begin{equation} \alpha\bigg(mu_0+\frac{\varpi^{i-n_0}u_0^{-1}(mu_0-\varpi^{-n_0}a)}{1+\varpi^{i-n_0}u_0^{-1}}\bigg)\equiv 0\mod \varpi^{-\lceil ({n-i})/{2}\rceil -n_0}. \end{equation}

Here we have used that $-\lceil {n_0}/{2}\rceil +i-n\geqslant -\lceil ({n-i})/{2}\rceil -n_0$. This congruence is equivalent to

(43)\begin{equation} mu_0^2+2\varpi^{i-n_0}m u_0-\varpi^{i-n}a \equiv 0 \mod \varpi^{-\lceil({n-i})/{2}\rceil}, \end{equation}

as $v({\alpha })=-n_0$. Note that $v(mu_0^2)=v(\varpi ^{i-n}a)=i-n < v(2\varpi ^{i-n_0}m u_0)$. So, this quadratic equation is not degenerate when $p\neq 2$, and we can solve for at most two solutions of $u_0\mod \varpi ^{\lfloor ({n-i})/{2}\rfloor }$ and consequently at most $2q$ solutions of $u_0\mod \varpi ^{\lceil ({n-i})/{2}\rceil }$.

In summary, we have that there are $\leqslant 2q^2$ pairs $(x_0, u_0)$ contributing to (38) and so we get

(44)\begin{equation} |\Phi_\pi^{(i)}(a,m)|\ll |C_0^{-1}| 2q^2\text{Vol}({{\Delta}} x)\text{Vol}({{\Delta}} u)\asymp q^{({i-n})/{2}+ O(1)}, \end{equation}

as required.

5.5.2 Supercuspidal representations

When $2\nmid q$, $\pi$ is associated by compact induction theory to a character $\theta$ over a quadratic field extension ${E}/{F}$ with ramification index $e_{E}$. Their relations are given explicitly as follows (see [Reference Bushnell and HenniartBH06]).

1. (i) $a(\pi )=n=2n_0$ corresponds to $e_{E}=1$ and $a(\theta )=n_0$.

2. (ii) $n=2n_0+1$ corresponds to $e_{E}=2$ and $a(\theta )=2n_0$.

In the following we shall give uniform formulations and estimates for both of these cases, which one can verify case by case according to this classification. For simplicity, let ${E}={F}(\sqrt {D})$ with $v_{F}(D)=e_{E}-1$. We let ${\mathfrak o}_E$ denote the ring of integers of $E$, $\varpi _E$ denote a uniformizer of $E$, and $\mathfrak p_E = \varpi _E{\mathfrak o}_E$. Let $\psi _{E}=\psi \circ \textrm {tr}_{{E}/{F}}$. It is easy to check that $a(\psi _{E})=-e_{E}+1$ since $a(\psi )=0$. Let

(45)\begin{equation} C_0=\int_{v_{E}(u)=-a(\theta)-e_{E}+1}\theta^{-1}(u)\psi_{E}(u)\,d^\times u. \end{equation}

Again, by the usual interpretation as a Gauss sum, we get $|C_0|\asymp {1}/{q^{a(\pi )/2}}$. Checking case by case, one can also see that for $u$ in the domain of the integral,

(46)\begin{equation} v(N_{E/F}(u))=-n. \end{equation}

The following lemma is a reformulation of [Reference AssingAss19, Lemma 3.1].

Lemma 5.7 When $i>n_0$, $W^{(i)}(x)$ is supported on $v(x)=0$ and, on the support,

(47)\begin{equation} W^{(i)}(x)=C_0^{-1}\int_{v_{E}(u)=-a(\theta)-e_{E}+1}\theta^{-1}(u) \psi \biggl(-\frac{1}{x}\varpi^{i}N_{{E}/{F}}(u)\biggr)\psi_{E}(u)\,d^\times u. \end{equation}

Again by [Reference HuHu18] the matrix coefficient $\Phi _\pi ^{(i)}(a,m)$ is supported on $v(a)=0$, $v(m)=i-n$ when $n_0 < i < n-1$. On the support, by the above lemma and (34),

(48)\begin{align} \Phi_\pi^{(i)}(a,m)&=C_0^{-1}\int_{v(x)=0}\psi(mx)\int_{v_{E}(u)=-a(\theta)-e_{E}+1}\theta^{-1}(u) \psi \biggl(-\frac{1}{ax}\varpi^{i}N_{{E}/{F}}(u)\biggr)\psi_{E}(u)\,d^\times u\,d^\times x\nonumber\\ &=C_0^{-1}\int_{v(x)=0}\psi \biggl(m\frac{1}{x}\biggr)\int_{v_{E}(u)=-a(\theta)-e_{E}+1}\theta^{-1}(u) \psi \biggl(-\frac{x}{a}\varpi^{i}N_{{E}/{F}}(u)\biggr)\psi_{E}(u)\,d^\times u\,d^\times x. \end{align}

Since $a(\pi )\geqslant 3$, we have $a(\theta )\geqslant 2$. Let ${\alpha }\in {E}^\times$ be the constant associated to $\theta$ by Lemma 5.1; then $v_{E}({\alpha })=-a(\theta )+a(\psi _{E})=-a(\theta )-e_{E}+1$. As $\theta |_{{F}^\times }$ is essentially the central character $w_\pi$, which is trivial, we can assume that ${\alpha }$ is purely imaginary in ${E}^\times$. In the integrand in (48), write $x=x_0(1+{{\Delta }} x)$ with $x_0\in {\mathfrak o}^\times /(1+\mathfrak p^{\lceil (n-i)/2\rceil })$, ${{\Delta }} x\in \mathfrak p^{\lceil (n-i)/2\rceil }$, and

\[ u=u_0(1+{{\Delta}} u)=(a_0+\sqrt{D} b_0)(1+{{\Delta}} a+\sqrt{D}{{\Delta}} b) \]

for $u_0\in (\varpi _E^{-a(\theta )-e_{E}+1}{\mathfrak o}_E/\varpi _E^{-\lfloor a(\theta )/2\rfloor -e_{E}+1}{\mathfrak o}_E)^\times$, ${{\Delta }} u = {{\Delta }} a+\sqrt {D}{{\Delta }} b\in \varpi _{E}^{\lceil a(\theta )/2\rceil } {\mathfrak o}_{E}$.

Then

(49)\begin{align} \Phi_\pi^{(i)}(a,m)&=C_0^{-1}\sum_{x_0,u_0}\psi \bigg(\frac{m}{x_0}\bigg)\theta^{-1}(u_0)\psi \biggl(-\frac{x_0}{a}\varpi^i N_{{E}/{F}}(u_0)\biggr)\psi_{E}(u_0) \nonumber\\ &\quad \times \iint_{\substack{{{\Delta}} x \in \mathfrak p^{\lceil ({n-i})/2\rceil} \\ {{\Delta}} u \in \mathfrak p_{E}^{\lceil {a(\theta)}/2\rceil} }}F_{x_0, u_0}({{\Delta}} x, {{\Delta}} u) \, d{{\Delta}} x \,d{{\Delta}} u, \end{align}

where $F_{x_0, u_0}({{\Delta }} x, {{\Delta }} u)= \psi (-({m}/{x_0}){{\Delta }} x-2{\alpha } \sqrt {D}{{\Delta }} b-({x_0}/{a})\varpi ^i N_{{E}/F}(u_0)({{\Delta }} x - 2 {{\Delta }} a)+2a_0{{\Delta }} a+2Db_0{{\Delta }} b )$ for ${{\Delta }} u= {{\Delta }} a+\sqrt {D}{{\Delta }} b$ with ${{\Delta }} a$, ${{\Delta }} b$ in ${\mathfrak o}$. Here we have used that

\begin{gather*} \theta^{-1}(1+{{\Delta}} u)=\psi_E(-\alpha {{\Delta}} u)=\psi_E(-\alpha {{\Delta}} a-\alpha\sqrt{D}{{\Delta}} b)=\psi(-2\alpha\sqrt{D}{{\Delta}} b),\\ \psi_E(u_0{{\Delta}} u)=\psi(2a_0{{\Delta}} a+2Db_0{{\Delta}} b). \end{gather*}

Using the fact that $a(\psi )=0$, we observe that in order for the integral in (49) to be non-zero, we need the following conditions to hold: For all $x_1$, $a_1$, $b_1$ in ${\mathfrak o}$ such that $x_1 \in \mathfrak p^{\lceil ({n-i})/2\rceil }$, $a_1+\sqrt {D}b_1 \in \mathfrak p_{E}^{\lceil {a(\theta )}/2\rceil }$, we have

(50)\begin{equation} \biggl(\frac{m}{x_0}+\frac{x_0}{a}\varpi^i N_{{E}/F}(u_0)\biggr) x_1 \in {\mathfrak o}, \end{equation}

(51)\begin{equation} \biggl(a_0-\frac{x_0}{a}\varpi^i N_{{E}/F}(u_0)\biggr) a_1 \in {\mathfrak o}, \end{equation}

(52)\begin{equation}(Db_0-\alpha\sqrt{D})\, b_1 \in {\mathfrak o}. \end{equation}

Now, using a very similar analysis as in the principal series case, we shall see that the number of pairs $(x_0, u_0)$ satisfying (50), (51), and (52) is $\ll q^{O(1)}.$

Consider the number of $b_0$ satisfying (52) first. When $e_E=1$, or $e_{E}=2$ and $a(\theta )/2$ is odd, we can choose $a_1$, $b_1$ in ${\mathfrak o}$ such that $a_1+\sqrt {D}b_1\in \mathfrak p_{E}^{\lceil a(\theta )/2\rceil }$, $b_1 \in \varpi _{E}^{\lceil a(\theta )/2\rceil -e_E+1}{\mathfrak o}_{E}^\times \cap {\mathfrak o}$, which combined with (52) gives us

(53)\begin{equation} b_0\equiv \frac{\alpha}{\sqrt{D}} \mod \mathfrak p_{E}^{-\lceil a(\theta)/2\rceil-e_E+1} \end{equation}

while by the definition of $u_0$, $b_0\sqrt {D}$ is well defined up to $\mathfrak p_{E}^{-\lfloor a(\theta )/2\rfloor -e_E+1}$. Thus,

(54)\begin{equation} \sharp \{b_0 \text{satisfying (52)}\}\ll q^{O(1)}. \end{equation}

When $e_{E}=2$ and $a(\theta )/2$ is even, we can choose $b_1\in \varpi _{E}^{\lceil a(\theta )/2\rceil }{\mathfrak o}_E^\times \cap {\mathfrak o}$ and this time (52) gives us $b_0\equiv ({\alpha }/{\sqrt {D}}) \mod \mathfrak p_{E}^{-\lceil a(\theta )/2\rceil -2e_E+2}$. By the same argument as above, (54) still holds in this case.

Similarly, for each fixed $u_0$, there exist solutions for $x_0$ from (50) if and only if

\[ -\frac{am}{\varpi^iN_{E/F}(u_0)} \]

is a square modulo $\varpi ^{\lfloor (n-i)/2\rfloor }$. In that case we obtain

(55)\begin{equation} x_0\equiv \pm\sqrt{-\frac{am}{\varpi^iN_{E/F}(u_0)}} \mod \varpi^{\lfloor (n-i)/2\rfloor}. \end{equation}

Here we have used that $p\neq 2$. So, by the definition of $x_0$,

(56)\begin{equation} \sharp \{x_0 \text{satisfying (50) for fixed } u_0 \}\ll q^{O(1)}. \end{equation}

Finally, we come to counting $a_0$. When $e_{E}=1$, or $e_{E}=2$ and $a(\theta )/2$ is even, we can choose $a_1$, $b_1$ so that $a_1\in \varpi _{E}^{\lceil a(\theta )/2\rceil }{\mathfrak o}_{E}^\times \cap {\mathfrak o}$, $a_1+\sqrt {D} b_1\in \mathfrak p_{E}^{\lceil a(\theta )/2\rceil }$, so that from (51) we now deduce that

(57)\begin{equation} a_0-\frac{x_0}{a}\varpi^iN_{E/F}(u_0)\equiv 0 \mod \varpi_E^{-\lceil a(\theta)/2\rceil}. \end{equation}

Note that if $v_E(({x_0}/{a})\varpi ^iN_{E/F}(u_0))=e_E(i-n)\geqslant -\lceil a(\theta )/2\rceil$, we get a unique solution $a_0\equiv 0\mod \varpi _E^{-\lceil a(\theta )/2\rceil }$ and by the definition of $a_0$ and the previous results,

(58)\begin{equation} \sharp \{(a_0, b_0, u_0) \text{satisfying (50)--(52)}\}\ll q^{O(1)}. \end{equation}

Otherwise when $e_E(i-n) < -\lceil a(\theta )/2\rceil$, (57) is a non-trivial congruence relation and $v(a_0)=i-n$. As $p\neq 2$, we have for any solution $a_0$,

\[ a_0+\frac{x_0}{a}\varpi^iN_{E/F}(u_0)\equiv 0\mod \varpi^{i-n}. \]

Multiplying it with (57) and substituting (55), we get

(59)\begin{equation} a_0^2\equiv -\frac{m}{a }\varpi^iN_{E/F}(u_0)=-\frac{m}{a }\varpi^i(a_0^2-b_0^2D) \mod \varpi_E^{-\lceil a(\theta)/2\rceil} \varpi^{i-n}. \end{equation}

One can get at most two solutions of $a_0\mod \varpi _E^{-\lceil a(\theta )/2\rceil }$ for each fixed $b_0$. So, (58) is still true.

If $e_{E}=2$ and $a(\theta )/2$ is odd, we can instead choose $a_1\in \varpi _{E}^{\lceil a(\theta )/2\rceil +1}{\mathfrak o}^\times _{E} \cap {\mathfrak o}$ in the argument above (57). The rest of the discussions are similar and (58) still holds.

In conclusion, we get that

(60)\begin{equation} |\Phi_\pi^{(i)}(a,m)|\ll q^{O(1)} |C_0^{-1}|\text{Vol}({{\Delta}} x)\text{Vol}({{\Delta}} u)\asymp \frac{1}{q^{(n-i)/2 + O(1)}}, \end{equation}

as required.