On a Generalization of Motohashi’s Formula: Non-archimedean Weight Functions

Han Wu School of Mathematical Sciences, University of Scinece and Technology of China, 230026 Hefei, P. R. China wuhan1121@yahoo.com

Abstract.

This is a continuation of the adelic version of Kwan’s formula. At non-archimedean places we give a bound of the weight function on the mixed moment side, when the weight function on the $\displaystyle{\rm PGL}_{3}\times{\rm PGL}_{2}$ side is nearly the characteristic function of a short family. Our method works for any tempered representation $\displaystyle\Pi$ of $\displaystyle{\rm PGL}_{3}$ , and reveals the structural reason for the appearance of Katz’s hypergeometric sums in a previous joint work with P.Xi.

1. Introduction

1.1. Main Results

This paper is the first follow-up of our previous [19]. Let $\displaystyle\mathbf{F}$ be a number field with adele ring $\displaystyle\mathbb{A}$ . Let $\displaystyle\psi$ be the additive character of $\displaystyle\mathbf{F}\backslash\mathbb{A}$ à la Tate. Let $\displaystyle S_{\mathbf{F}}$ be the set of places of $\displaystyle\mathbf{F}$ , and $\displaystyle S_{\infty}$ be the subset of archimedean places. Fix a cuspidal automorphic representation $\displaystyle\Pi$ of $\displaystyle{\rm PGL}_{3}(\mathbb{A})$ . We established an adelic version of Kwan’s spectral reciprocity identity (see [19, Theorem 1.1 & (5.19)-(5.21)]), which we recall in the special case of trivial central characters as follows.

Theorem 1.1.

Let $\displaystyle S_{\infty}\subset S\subset S_{\mathbf{F}}$ be any finite subset. At every place $\displaystyle v\in S$ there is a pair of weight functions $\displaystyle h_{v}$ and $\displaystyle\widetilde{h}_{v}$ with the auxiliary normalized ones at finite places $\displaystyle\mathfrak{p}<\infty$

(1.1)

H_{\mathfrak{p}}(\pi_{\mathfrak{p}}):=h_{\mathfrak{p}}(\pi_{\mathfrak{p}})\frac{L(1,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}})}{L(1/2,\Pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}})},\quad\widetilde{H}_{\mathfrak{p}}(\chi_{\mathfrak{p}}):=\frac{\widetilde{h}_{\mathfrak{p}}(\chi_{\mathfrak{p}})}{L(1/2,\Pi_{\mathfrak{p}}\times\chi_{\mathfrak{p}}^{-1})L(1/2,\chi_{\mathfrak{p}})},

so that the following equation holds (where $\displaystyle\pi$ runs through cuspidal automorphic forms of $\displaystyle{\rm PGL}_{2}(\mathbb{A})$ )

\sum_{\pi}\frac{L(1/2,\Pi\times\pi)}{2\Lambda_{\mathbf{F}}(2)L(1,\pi,\mathrm{Ad})}\cdot\prod_{v\mid\infty}h_{v}(\pi_{v})\cdot\prod_{\mathfrak{p}\in S}H_{\mathfrak{p}}(\pi_{\mathfrak{p}})+\\ \sum_{\chi\in\widehat{\mathbb{R}_{+}\mathbf{F}^{\times}\backslash\mathbb{A}^{\times}}}\int_{-\infty}^{\infty}\frac{\left\lvert L(1/2+i\tau,\Pi\times\chi)\right\rvert^{2}}{2\Lambda_{\mathbf{F}}(2)\left\lvert L(1+2i\tau,\chi^{2})\right\rvert^{2}}\cdot\prod_{v\mid\infty}h_{v}(\pi(\chi_{v},i\tau))\cdot\prod_{\mathfrak{p}\in S}H_{\mathfrak{p}}(\pi(\chi_{\mathfrak{p}},i\tau))\frac{\mathrm{d}\tau}{2\pi}\\ =\frac{1}{\zeta_{\mathbf{F}}^{*}}\sum_{\chi\in\widehat{\mathbb{R}_{+}\mathbf{F}^{\times}\backslash\mathbb{A}^{\times}}}\int_{-\infty}^{\infty}L(1/2-i\tau,\Pi\times\chi^{-1})L(1/2+i\tau,\chi)\cdot\prod_{v\mid\infty}\widetilde{h}_{v}(\chi_{v}\lvert\cdot\rvert_{v}^{i\tau})\cdot\prod_{\mathfrak{p}\in S}\widetilde{H}_{\mathfrak{p}}(\chi_{\mathfrak{p}}\lvert\cdot\rvert_{\mathfrak{p}}^{i\tau})\frac{\mathrm{d}\tau}{2\pi}+\\ \frac{1}{\zeta_{\mathbf{F}}^{*}}\sum_{\pm}{\rm Res}_{s_{1}=\pm\frac{1}{2}}L(1/2-s_{1},\Pi)\zeta_{\mathbf{F}}(1/2+s_{1})\cdot\prod_{v\mid\infty}\widetilde{h}_{v}(\lvert\cdot\rvert_{v}^{s_{1}})\cdot\prod_{\mathfrak{p}\in S}\widetilde{H}_{\mathfrak{p}}(\lvert\cdot\rvert_{\mathfrak{p}}^{s_{1}}),

where we have used the abbreviation $\displaystyle\pi(\chi_{v},s):=\pi(\chi_{v}\lvert\cdot\rvert_{v}^{s},\chi_{v}^{-1}\lvert\cdot\rvert_{v}^{-s})$ .

We only mention that the weight functions $\displaystyle h_{v}$ and $\displaystyle\widetilde{h}_{v}$ are integrals of a smooth Whittaker function $\displaystyle W_{v}\in\mathcal{W}(\Pi_{v}^{\infty},\psi_{v})$ . More details from [19] will be recalled when we need them.

In this paper we focus on the weight functions at a non-archimedean place $\displaystyle\mathfrak{p}<\infty$ . We omit the subscript $\displaystyle\mathfrak{p}$ for simplicity. Let $\displaystyle\mathbf{F}$ be a non-archimedean local field of characteristic $\displaystyle 0$ with valuation ring $\displaystyle\mathcal{O}_{\mathbf{F}}$ , and write $\displaystyle q={\rm Nr}(\mathfrak{p})$ from now on. In the case $\displaystyle\chi=\lvert\cdot\rvert_{\mathbf{F}}^{s}$ , the function $\displaystyle\widetilde{H}(\lvert\cdot\rvert_{\mathbf{F}}^{s})$ is a polynomial in $\displaystyle\mathbb{C}[q^{s},q^{-s}]$ , hence is entire. We introduce its Taylor expansion at any point $\displaystyle s_{0}$ as

(1.2)

\widetilde{H}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=\sideset{}{{}_{k\geqslant 0}}{\sum}\widetilde{H}(k;s_{0})\left(s-s_{0}\right)^{k}.

We summarize our main results (namely Proposition 5.13, 5.16, 6.10 & 6.12) as follows.

Theorem 1.2.

Suppose the residual characteristic of $\displaystyle\mathbf{F}$ is not $\displaystyle 2$ . Let $\displaystyle\Pi\in\widehat{{\rm PGL}_{3}(\mathbf{F})}$ be generic and tempered. Let $\displaystyle\pi_{0}\in\widehat{{\rm PGL}_{2}(\mathbf{F})}$ with $\displaystyle\mathfrak{c}(\pi_{0})>1$ . We can choose $\displaystyle W\in\mathcal{W}(\Pi^{\infty},\psi)$ so that:

(1)

The weight function satisfies $\displaystyle h(\pi)\geqslant 0$ for any $\displaystyle\pi\in\widehat{{\rm PGL}_{2}(\mathbf{F})}$ and $\displaystyle h(\pi_{0})>0$ ;

(2)

For any unitary $\displaystyle\chi\in\widehat{\mathbf{F}^{\times}}$ and $\displaystyle\epsilon>0$ the dual weight function satisfies

\displaystyle h(\pi_{0})^{-1}\widetilde{h}(\chi)\ll_{\epsilon}\mathbf{C}(\Pi)^{2+\epsilon}\cdot\left\{\mathbbm{1}_{\leqslant\max(\lfloor\frac{\mathfrak{c}(\pi_{0})}{2}\rfloor,6\mathfrak{c}(\Pi))}(\mathfrak{c}(\chi))+q^{\frac{1}{2}}\mathbbm{1}_{(-1)^{\frac{q-1}{2}}=\varepsilon_{0}}\mathbbm{1}_{2\nmid\mathfrak{c}(\chi)=\frac{\mathfrak{c}(\pi_{0})}{2}\geqslant 3}\mathbbm{1}_{\boldsymbol{\mathcal{E}}(\pi_{0})}(\chi^{2})\right\},

where $\displaystyle\varepsilon_{0}=-1$ if $\displaystyle\pi_{0}$ is dihedral supercuspidal, and $\displaystyle\varepsilon_{0}=1$ otherwise; and the exceptional set of characters $\displaystyle\boldsymbol{\mathcal{E}}(\pi_{0})$ of $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ has size $\displaystyle O(q^{-1}\mathbf{C}(\pi_{0})^{\frac{1}{2}})$ and is given below in Lemma 5.7 (2);

(3)

For any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$ and $\displaystyle\epsilon>0$ the normalized unramified dual weight function satisfies the bounds

\displaystyle h(\pi_{0})^{-1}\widetilde{H}(k;\pm 1/2)\ll_{\epsilon}\mathbf{C}(\Pi)^{4+\epsilon}q^{\lceil\frac{\mathfrak{c}(\pi_{0})}{2}\rceil+\epsilon}.

Remark 1.3.

Our primary goal is to reveal the structural reason for the bounds of the dual weight functions. Our main discovery is the quadratic elementary functions given in (1.6), which are “building blocks” of the Bessel functions of the relevant representations. See §1.3 for more details. Further extension of our method to the non-dihedral supercuspidal and Steinberg $\displaystyle\pi_{0}$ requires only plugging in the integral representation of the Bessel functions of such $\displaystyle\pi_{0}$ , analogous to [19, Theorem 1.6].

Remark 1.4.

Specializing to $\displaystyle\mathbf{F}=\mathbb{Q}_{p}$ and $\displaystyle\Pi=\mathbbm{1}\boxplus\mathbbm{1}\boxplus\mathbbm{1}$ , Theorem 1.2 corresponds to the main local non-archimedean computation of the recent work of Hu–Petrow–Young [8] in the case $\displaystyle p\neq 2$ .

1.2. Notation and Convention

For a locally compact group $\displaystyle G$ , let $\displaystyle\widehat{G}$ be the topological dual of unitary irreducible representations. For $\displaystyle\pi\in\widehat{G}$ , we write $\displaystyle V_{\pi}$ for the underlying Hilbert space, and write $\displaystyle V_{\pi}^{\infty}\subset V_{\pi}$ for the subspace of smooth vectors if $\displaystyle G$ carries extra structure to make sense of the notion.

Throughout the paper $\displaystyle\mathbf{F}$ is a local field of characteristic $\displaystyle 0$ , with residual characteristic $\displaystyle\neq 2$ . Let $\displaystyle\lvert\cdot\rvert_{\mathbf{F}}$ (resp. $\displaystyle v_{\mathbf{F}}$ ) be the valuation (resp. normalized additive valuation) of $\displaystyle\mathbf{F}$ . Fix $\displaystyle\psi$ an additive character of conductor exponent $\displaystyle 0$ and normalize the measures accordingly. The valuation ring of $\displaystyle\mathbf{F}$ is $\displaystyle\mathcal{O}_{\mathbf{F}}$ , while the valuation ideal is $\displaystyle\mathcal{P}_{\mathbf{F}}$ . We choose a uniformizer $\displaystyle\varpi_{\mathbf{F}}\in\mathcal{P}_{\mathbf{F}}-\mathcal{P}_{\mathbf{F}}^{2}$ . Different choices of $\displaystyle\varpi_{\mathbf{F}}$ give different ramified quadratic extensions of $\displaystyle\mathbf{F}$ . Write $\displaystyle\mathbf{G}_{d}:={\rm GL}_{d}$ for simplicity, introduce some compact open subgroups of $\displaystyle\mathbf{G}_{2}(\mathbf{F})$ as

\displaystyle\mathbf{K}:={\rm GL}_{2}(\mathcal{O}_{\mathbf{F}});\quad\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n}]:=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathbf{K}\ \middle|\ c\in\mathcal{P}_{\mathbf{F}}^{n}\right\},\ \forall n\in\mathbb{Z}_{\geqslant 0};

and some algebraic subgroups of $\displaystyle\mathbf{G}_{2}(\mathbf{F})$ as

\displaystyle\mathbf{Z}=\mathbf{Z}_{2}(\mathbf{F}):=\left\{z\mathbbm{1}_{2}\ \middle|\ z\in\mathbf{F}^{\times}\right\},\quad\mathbf{N}_{2}(\mathbf{F}):=\left\{n(x):=\begin{pmatrix}1&x\\ &1\end{pmatrix}\ \middle|\ x\in\mathbf{F}\right\},

\displaystyle\mathbf{A}_{2}(\mathbf{F}):=\left\{\begin{pmatrix}t_{1}&\\ &t_{2}\end{pmatrix}\ \middle|\ t_{1},t_{2}\in\mathbf{F}^{\times}\right\},\quad\mathbf{B}_{2}(\mathbf{F}):=\mathbf{A}_{2}(\mathbf{F})\mathbf{N}_{2}(\mathbf{F}).

For integers $\displaystyle n,m\geqslant 1$ we write $\displaystyle\mathcal{S}(n\times m,\mathbf{F})$ for the space of Schwartz–Bruhat functions on $\displaystyle{\rm M}(n\times m,\mathbf{F})$ the $\displaystyle n\times m$ matrices with entries in $\displaystyle\mathbf{F}$ . The (inverse) $\displaystyle\psi$ -Fourier transform is denoted and defined by

\displaystyle\widehat{\Psi}(X)=\mathfrak{F}_{\psi}(\Psi)(-X)=\int_{{\rm M}(n\times m,\mathbf{F})}\Psi(Y)\psi\left({\rm Tr}(XY^{T})\right)\mathrm{d}Y.

If no confusion occurs, we may omit $\displaystyle\psi$ from the notation. If this is the case, then the inverse Fourier transform is denoted by $\displaystyle\overline{\mathfrak{F}}=\mathfrak{F}_{\overline{\psi}}=\mathfrak{F}_{\psi^{-1}}$ . We introduce some elementary operators on the space of functions on $\displaystyle\mathbf{F}^{\times}$ :

•

For functions $\displaystyle\phi$ on $\displaystyle\mathbf{F}^{\times}$ , its extension by $\displaystyle 0$ to $\displaystyle\mathbf{F}$ is denoted by $\displaystyle\mathrm{e}(\phi)$ , and its inverse is $\displaystyle\mathrm{Inv}(\phi)(t):=\phi(t^{-1})$ ; for functions $\displaystyle\phi$ on $\displaystyle\mathbf{F}$ , its restriction to $\displaystyle\mathbf{F}^{\times}$ is denoted by $\displaystyle\mathrm{r}(\phi)$ , and the operator $\displaystyle\mathrm{i}$ is

$\displaystyle\mathrm{i}=\mathrm{e}\circ\mathrm{Inv}\circ\mathrm{r}.$
•

For $\displaystyle s\in\mathbb{C}$ , $\displaystyle\mu\in\widehat{\mathbf{F}^{\times}}$ and functions $\displaystyle\phi$ on $\displaystyle\mathbf{F}$ , we introduce the operator $\displaystyle\mathfrak{m}_{s}(\mu)$ by

$\displaystyle\mathfrak{m}_{s}(\mu)(\phi)(t)=\phi(t)\mu(t)\lvert t\rvert_{\mathbf{F}}^{s}.$
•

For $\displaystyle\delta\in\mathbf{F}^{\times}$ we introduce the operator $\displaystyle\mathfrak{t}(\delta)$ by

$\displaystyle\mathfrak{t}(\delta)(\phi)(y)=\phi(y\delta).$
•

Let $\displaystyle I_{n}=I_{n,\mathbf{F}}:{\rm C}(\mathbf{F}^{\times})\to{\rm C}({\rm GL}_{n}(\mathbf{F}))$ be given by $\displaystyle I_{n}(h)(g):=h(\det g)$ .

These notation apply to finite (field) extensions of $\displaystyle\mathbf{F}$ .

We introduce the standard involution of inverse-transpose on $\displaystyle{\rm GL}_{n}(R)$ as $\displaystyle g^{\iota}:={}^{t}g^{-1}$ .

If $\displaystyle(U,\mathrm{d}u)$ is a measured space with finite total mass, we introduce the normalized integral as

(1.3)

\oint_{U}f(u)\mathrm{d}u:=\frac{1}{{\rm Vol}(U,\mathrm{d}u)}\int_{U}f(u)\mathrm{d}u.

For $\displaystyle n\in\mathbb{Z}_{\geqslant 1}$ we will frequently perform the following process of regularization to an integral

(1.4)

\int_{\mathbf{F}}f(y)\mathrm{d}y=\int_{\mathbf{F}}\left(\oint_{\mathcal{O}_{\mathbf{F}}}f(y(1+\varpi_{\mathbf{F}}^{n}x))\mathrm{d}x\right)\mathrm{d}y,

which we shall refer to as the level $\displaystyle n$ regularization (with respect) to $\displaystyle\mathrm{d}y$ .

1.3. Outline of Proof

From the local main result [19, Theorem 1.4] the weight $\displaystyle h(\pi)$ and the dual weight $\displaystyle\widetilde{h}(\chi)$ functions are related to each other by a (hidden) relative orbital integral $\displaystyle H(y)$ via

(1.5)

h(\pi)=\int_{\mathbf{F}^{\times}}H(y)\cdot\mathrm{j}_{\widetilde{\pi},\psi^{-1}}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\frac{\mathrm{d}^{\times}y}{\lvert y\rvert_{\mathbf{F}}},\quad\widetilde{h}(\chi)=\int_{\mathbf{F}^{\times}}\psi(-y)\chi^{-1}(y)\lvert y\rvert_{\mathbf{F}}^{-\frac{1}{2}}\cdot\widetilde{\mathcal{V}}_{\Pi}(H)(y)\mathrm{d}^{\times}y

where $\displaystyle\mathrm{j}_{\widetilde{\pi},\psi^{-1}}$ is the Bessel function of the contragredient representation $\displaystyle\widetilde{\pi}$ in the sense of [2, §3.5], and $\displaystyle\widetilde{\mathcal{V}}_{\Pi}$ is the extended Voronoi transform characterized by the equations

\displaystyle\widetilde{\mathcal{VH}}_{\Pi}:=\widetilde{\mathcal{V}}_{\Pi}\circ\mathfrak{m}_{1},\quad\mathfrak{m}_{-2}(\beta)\circ I_{3}\circ\widetilde{\mathcal{VH}}_{\Pi}=\overline{\mathfrak{F}}\circ\mathfrak{m}_{-1}(\beta^{\iota})\circ I_{3}

for all smooth matrix coefficients $\displaystyle\beta$ of $\displaystyle\Pi$ .

We need to choose $\displaystyle H(y)$ so that $\displaystyle h(\pi)\geqslant 0$ selects a short family containing $\displaystyle\pi_{0}$ . In other words the weight function $\displaystyle h(\pi)$ should be an approximation of the characteristic functions of $\displaystyle\{\pi_{0}\}$ . From the first formula of (1.5) and the orthogonality of Bessel functions, one should expect that any such $\displaystyle H(y)$ is an approximation of $\displaystyle\mathrm{j}_{\pi_{0},\psi}\begin{pmatrix}&-y\\ 1&\end{pmatrix}$ . If $\displaystyle\pi_{0}$ is supercuspidal, the asymptotic analysis of both functions (in §2.1 and Lemma 3.10) show that the two functions can be equal. In general, we construct $\displaystyle H(y)$ from some test function $\displaystyle\phi_{0}^{\iota}*\phi_{0}$ on $\displaystyle{\rm PGL}_{2}(\mathbf{F})$ of positive type in order to ensure $\displaystyle h(\pi)\geqslant 0$ ; we also require $\displaystyle\phi_{0}$ to be left covariant with respect to some character of an open compact subgroup, which determines the (minimal $\displaystyle\mathbf{K}$ -)type of $\displaystyle\pi_{0}$ in the sense of [12]. We use $\displaystyle\phi_{0}$ to deduce some integral representation (3.5) of $\displaystyle H(y)$ which approximates the one of $\displaystyle\mathrm{j}_{\pi_{0},\psi}\begin{pmatrix}&-y\\ 1&\end{pmatrix}$ obtained in [19, Theorem 1.6]. The integral representation of $\displaystyle H(y)$ in all cases is summarized in Corollary 3.12, and is the departure point of our refined analysis of the dual weight function $\displaystyle\widetilde{h}(\chi)$ described below.

Our key observation on the dual weight function is the following decomposition into three parts. The first part is the contribution from $\displaystyle H_{\infty}(y)$ , essentially the restriction of $\displaystyle H(y)$ to $\displaystyle v_{\mathbf{F}}(y)\leqslant-\mathfrak{c}(\pi_{0})$ (see (4.1)). In this region $\displaystyle H_{\infty}(y)$ has a stable behavior regardless $\displaystyle\pi_{0}$ , i.e., the germ of any Bessel relative orbital integral at infinity. To treat the corresponding $\displaystyle\widetilde{h}_{\infty}(\chi)$ we crucially rely on the extension of the Voronoi–Hankel transform developed in our previous paper [19, Theorem 1.3]. The remaining part $\displaystyle\widetilde{h}_{c}(\chi)$ corresponding to $\displaystyle H_{c}:=H-H_{\infty}$ can be written as $\displaystyle\widetilde{h}_{c}(\chi)=\widetilde{h}_{c}^{+}(\chi)+\widetilde{h}_{c}^{-}(\chi)$ , where

\displaystyle\widetilde{h}_{c}^{+}(\chi)=\int_{\mathcal{O}_{\mathbf{F}}}\chi^{-1}(y)\lvert y\rvert_{\mathbf{F}}^{-\frac{1}{2}}\cdot\widetilde{\mathcal{V}}_{\Pi}(H)(y)\mathrm{d}^{\times}y,\quad\widetilde{h}_{c}^{-}(\chi)=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\psi(-y)\chi^{-1}(y)\lvert y\rvert_{\mathbf{F}}^{-\frac{1}{2}}\cdot\widetilde{\mathcal{V}}_{\Pi}(H)(y)\mathrm{d}^{\times}y.

The bound of $\displaystyle\widetilde{h}_{c}^{+}$ is offered by the local functional equations via Lemma 4.2. The method via the local functional equations can also offer a crude bound in Lemma 4.8 of $\displaystyle\widetilde{h}_{c}^{-}$ when $\displaystyle\mathfrak{c}(\pi_{0})\ll\mathfrak{c}(\Pi)$ .

To refine the bound of $\displaystyle\widetilde{h}_{c}^{-}$ in the case $\displaystyle\mathfrak{c}(\pi_{0})\gg\mathfrak{c}(\Pi)$ we observe that $\displaystyle H_{c}(y)$ is a linear combination of translations of the following quadratic elementary functions (see Definition 5.1 & 6.1, (5.1)-(5.4) & (6.1)-(6.3) for more details)

(1.6)

F_{n},G_{n}(y^{2})=\mathbbm{1}_{v(y)=-n}\cdot\sideset{}{{}_{\pm}}{\sum}\eta_{\mathbf{L}/\mathbf{F}}(\pm y)\psi(\pm y),

where $\displaystyle\mathbf{L}/\mathbf{F}$ is the quadratic algebra extension associated with the type of $\displaystyle\pi_{0}$ , and $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}$ is the corresponding quadratic character of $\displaystyle\mathbf{F}^{\times}$ . We essentially change the order of integrations by first computing the dual weight of (the translates of) the quadratic elementary functions. In particular, the translation pattern of the above quadratic elementary functions is responsible for the appearance of Katz’s hypergeometric sums in our previous joint work with Xi [21].

Remark 1.5.

Xi [22] has yet another transformation of the algebraic exponential sums in Petrow–Young’s work [14]. This is mysterious and seems to lie beyond the framework of $\displaystyle{\rm GL}_{2}$ or $\displaystyle{\rm GL}_{3}$ .

Remark 1.6.

In the case of principal series $\displaystyle\pi_{0}$ , our $\displaystyle\phi_{0}$ coincides with Nelson’s test function in [13]. But we do not view it within the theory of micro-localized vectors, nor does our method of bounding the dual weight function rely on anything in that theory. Our choice of test function follows the idea of a previous work [1] in the real case by analogy in terms of “minimal $\displaystyle\mathbf{K}$ -type” [12].

1.4. Acknowledgement

We thank Zhi Qi and Ping Xi for discussions related to the topics of the paper.

2. Local Weight Transforms Revisited

We have expressed the local weight transforms in terms of the extended Voronoi transforms in [19, Theorem 1.4]. In that version, a test function $\displaystyle H(y)$ is some integral of a Kirillov function of a generic unitary irreducible $\displaystyle\Pi$ (namely the restriction of a $\displaystyle W\in\mathcal{W}(\Pi^{\infty},\psi)$ to the left-upper embedding of $\displaystyle\mathbf{G}_{2}(\mathbf{F})$ ). As explained in [19, (6.16)], the space of test functions $\displaystyle H(y)$ contains the Bessel orbital integrals of $\displaystyle{\rm C}_{c}^{\infty}(\mathbf{G}_{2}(\mathbf{F}))$ . We shall restrict to the latter subspace of test functions and get more refined information on the local weight transforms in the case of trivial central characters.

2.1. Relative Orbital Integrals

For any $\displaystyle m\in\mathbb{Z}_{\geqslant 0}$ we introduce a function $\displaystyle E_{m}\in{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})$ supported in the subset of square elements of $\displaystyle\mathbf{F}^{\times}$ given by

(2.1)

E_{m}(y^{2}):=\mathbbm{1}_{v_{\mathbf{F}}(y)=-m}\cdot\lvert y\rvert_{\mathbf{F}}\sum_{\pm}\int_{\pm 1+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{m}{2}\rfloor}}\psi\left(y(u+u^{-1})\right)\mathrm{d}u.

Recall that for any $\displaystyle f\in{\rm C}_{c}^{\infty}({\rm GL}_{2}(\mathbf{F}))$ , the Bessel orbital integral [19, (5.17)] is given by

(2.2)

h(y)=\int_{\mathbf{F}^{\times}}\int_{\mathbf{F}^{2}}f\left(\begin{pmatrix}1&x_{1}\\ &1\end{pmatrix}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&x_{2}\\ &1\end{pmatrix}\begin{pmatrix}z&\\ &z\end{pmatrix}\right)\psi(-x_{1}-x_{2})\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}^{\times}z.

Proposition 2.1.

As $\displaystyle f$ traverses $\displaystyle{\rm C}_{c}^{\infty}(\mathbf{G}_{2}(\mathbf{F}))$ , the Bessel orbital integrals $\displaystyle h$ traverses

\displaystyle{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})\bigoplus\mathbb{C}E_{\geqslant n}

where $\displaystyle n\in\mathbb{Z}_{\geqslant 0}$ can be chosen arbitrarily and we have written

(2.3)

E_{\geqslant n}(y^{2}):=\sideset{}{{}_{m=n}^{\infty}}{\sum}E_{m}(y^{2})=\mathbbm{1}_{v_{\mathbf{F}}(y)\leqslant-n}\cdot\lvert y\rvert_{\mathbf{F}}\sideset{}{{}_{\pm}}{\sum}\int_{\pm 1+\mathcal{P}_{\mathbf{F}}^{\lfloor-\frac{v_{\mathbf{F}}(y)}{2}\rfloor}}\psi\left(y(u+u^{-1})\right)\mathrm{d}u.

Proof.

If $\displaystyle f$ traverses $\displaystyle{\rm C}_{c}^{\infty}(\mathbf{B}_{2}(\mathbf{F})w\mathbf{N}_{2}(\mathbf{F}))$ , then clearly $\displaystyle h$ traverses $\displaystyle{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})$ . Consider a function $\displaystyle f\in{\rm C}_{c}^{\infty}(\mathbf{B}_{2}(\mathbf{F})\mathbf{N}_{2}(\mathbf{F})^{t})$ and let $\displaystyle\phi\in{\rm C}_{c}^{\infty}(\mathbf{F}^{\times}\times\mathbf{F})$ be defined by

\displaystyle\phi(y,x):=\int_{\mathbf{F}^{\times}}\int_{\mathbf{F}^{2}}f\left(\begin{pmatrix}1&x_{1}\\ &1\end{pmatrix}\begin{pmatrix}y&\\ &1\end{pmatrix}\begin{pmatrix}1&\\ x&1\end{pmatrix}\begin{pmatrix}z&\\ &z\end{pmatrix}\right)\psi(-x_{1})\mathrm{d}x_{1}\mathrm{d}^{\times}z.

From the equation of matrices

\displaystyle\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&x_{2}\\ &1\end{pmatrix}=\begin{pmatrix}1&-y/x_{2}\\ &1\end{pmatrix}\begin{pmatrix}y/x_{2}&\\ &x_{2}\end{pmatrix}\begin{pmatrix}1&\\ 1/x_{2}&1\end{pmatrix}

we easily deduce that the relative orbital integral (2.2) is given by

\displaystyle h(y)=\int_{\mathbf{F}}\phi\left(\frac{y}{x_{2}^{2}},\frac{1}{x_{2}}\right)\psi\left(-\frac{y}{x_{2}}-x_{2}\right)\mathrm{d}x_{2}=\int_{\mathbf{F}}\phi(yu^{2},u)\psi(-yu-u^{-1})\lvert u\rvert_{\mathbf{F}}^{-2}\mathrm{d}u.

Let $\displaystyle\tau\in\{1,\varepsilon\}$ , we obtain by an obvious change of variables

\displaystyle h(\tau y^{2})=\lvert y\rvert_{\mathbf{F}}\int_{\mathbf{F}}\phi(\tau u^{2},y^{-1}u)\psi(-y(\tau u+u^{-1}))\lvert u\rvert_{\mathbf{F}}^{-2}\mathrm{d}u.

Choose $\displaystyle i_{0}\in\mathbb{Z}_{\geqslant 1}$ and $\displaystyle k_{0},m_{0}\in\mathbb{Z}$ such that:

•

$\displaystyle\phi(y(1+\delta_{1}),x(1+\delta_{2}))=\phi(y,x),\quad\forall\ \delta_{1},\delta_{2}\in\mathfrak{p}^{i_{0}}$ ;
•

for any $\displaystyle y\in\mathfrak{p}^{2+2k_{0}}$ and any $\displaystyle x\in\mathbf{F}$ , we have $\displaystyle\phi(y,x)=0$ ;
•

for any $\displaystyle x\in\mathfrak{p}^{m_{0}}$ and any $\displaystyle y\in\mathbf{F}^{\times}$ , we have $\displaystyle\phi(y,x)=\phi(y,0)$ .

Let $\displaystyle v_{\mathbf{F}}(y)=-m$ for some $\displaystyle m\geqslant\max(m_{0},2i_{0}+k_{0},2i_{0})$ , and take any $\displaystyle n\in\mathbb{Z}_{\geqslant 1}$ satisfying

\displaystyle 2n\geqslant m+k_{0},\quad n\geqslant i_{0},\quad m-n\geqslant i_{0}.

Performing the level $\displaystyle n$ regularization to $\displaystyle\mathrm{d}y$ we get

	$\displaystyle\displaystyle h(\tau y^{2})$	$\displaystyle\displaystyle=\lvert y\rvert_{\mathbf{F}}\int_{\mathbf{F}-\mathcal{P}_{\mathbf{F}}^{1+k_{0}}}\phi(\tau u^{2},y^{-1}u)\left[\oint_{\mathcal{O}_{\mathbf{F}}}\psi\left(-y\left(\tau u(1+\varpi_{\mathbf{F}}^{n}x)+u^{-1}(1+\varpi_{\mathbf{F}}^{n}x)^{-1}\right)\right)\mathrm{d}x\right]\lvert u\rvert_{\mathbf{F}}^{-2}\mathrm{d}u$
		$\displaystyle\displaystyle=\lvert y\rvert_{\mathbf{F}}\int_{\mathbf{F}-\mathcal{P}_{\mathbf{F}}^{1+k_{0}}}\phi(\tau u^{2},y^{-1}u)\psi(-y(\tau u+u^{-1}))\left[\oint_{\mathcal{O}_{\mathbf{F}}}\psi\left(-y\varpi_{\mathbf{F}}^{n}(\tau u-u^{-1})x\right)\mathrm{d}x\right]\lvert u\rvert_{\mathbf{F}}^{-2}\mathrm{d}u.$

The non-vanishing of the inner integral implies

\displaystyle v_{\mathbf{F}}(\tau u-u^{-1})\geqslant m-n\geqslant i_{0},

which can be satisfied only if $\displaystyle\tau$ is a square modulo $\displaystyle\mathcal{P}_{\mathbf{F}}$ , i.e., $\displaystyle\tau=1$ . Moreover, we have $\displaystyle v_{\mathbf{F}}(u-u^{-1})\geqslant i_{0}\Leftrightarrow u\in\pm 1+\mathcal{P}_{\mathbf{F}}^{i_{0}}$ . We therefore get $\displaystyle h(\varepsilon y^{2})=0$ and

	$\displaystyle\displaystyle h(y^{2})$	$\displaystyle\displaystyle=\mathbbm{1}_{k_{0}\geqslant 0}\phi(1,0)\cdot\lvert y\rvert_{\mathbf{F}}\sum_{\pm}\int_{\pm 1+\mathcal{P}_{\mathbf{F}}^{i_{0}}}\psi(-y(u+u^{-1}))\mathrm{d}u$
		$\displaystyle\displaystyle=\mathbbm{1}_{k_{0}\geqslant 0}\phi(1,0)\cdot\lvert y\rvert_{\mathbf{F}}\sum_{\pm}\int_{\pm 1+\mathcal{P}_{\mathbf{F}}^{i_{0}}}\psi(-y(u+u^{-1}))\left[\oint_{\mathcal{O}_{\mathbf{F}}}\psi\left(-y\varpi_{\mathbf{F}}^{n}(u-u^{-1})x\right)\mathrm{d}x\right]\mathrm{d}u,$

where we have performed the level $\displaystyle n=\lceil m/2\rceil\geqslant i_{0}$ regularization to $\displaystyle\mathrm{d}u$ . Again the inner integral is non-vanishing only if

\displaystyle v_{\mathbf{F}}(u-u^{-1})\geqslant m-n=\lfloor m/2\rfloor\quad\Leftrightarrow\quad u\in\pm 1+\mathcal{P}_{\mathbf{F}}^{\lfloor m/2\rfloor}.

We have obtained

\displaystyle h(y^{2})=\mathbbm{1}_{k_{0}\geqslant 0}\phi(1,0)\cdot\lvert y\rvert_{\mathbf{F}}\sum_{\pm}\int_{\pm 1+\mathcal{P}_{\mathbf{F}}^{\lfloor-\frac{v_{\mathbf{F}}(y)}{2}\rfloor}}\psi(-y(u+u^{-1}))\mathrm{d}u,

hence $\displaystyle h$ lies in the desired space of functions. Applying a smooth partition of unity to the open covering

\displaystyle\mathbf{G}_{2}(\mathbf{F})=\mathbf{B}_{2}(\mathbf{F})w\mathbf{N}_{2}(\mathbf{F})\bigcup\mathbf{B}_{2}(\mathbf{F})\mathbf{N}_{2}(\mathbf{F})^{t}

we conclude the proof. ∎

Definition 2.2.

We call $\displaystyle E_{\geqslant n}$ the elementary Bessel orbital integrals, abbreviated as EBOIs.

2.2. Voronoi–Hankel Transforms of EBOIs

We notice that the Mellin transform of $\displaystyle E_{m}$ is simple.

Lemma 2.3.

Let $\displaystyle m\geqslant 2$ and $\displaystyle\chi$ be a quasi-character. We have

\displaystyle\int_{\mathbf{F}^{\times}}E_{m}(y)\chi(y)\mathrm{d}^{\times}y=\mathbbm{1}_{\mathfrak{c}(\chi)=m}\cdot\zeta_{\mathbf{F}}(1)\gamma(1/2,\chi^{-1},\psi)^{2}.

Proof.

Applying the change of variables $\displaystyle y\mapsto y^{2}$ and the level $\displaystyle\lceil m/2\rceil$ regularization to $\displaystyle\mathrm{d}u$ we get

\int_{\mathbf{F}^{\times}}E_{m}(y)\chi(y)\mathrm{d}^{\times}y=\frac{1}{2}\int_{\varpi_{\mathbf{F}}^{-m}\mathcal{O}_{\mathbf{F}}^{\times}}E_{m}(y^{2})\chi^{2}(y)\mathrm{d}^{\times}y\\ =q^{m}\int_{1+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{m}{2}\rfloor}}\left(\int_{\varpi_{\mathbf{F}}^{-m}\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(y(u+u^{-1})\right)\chi^{2}(y)\mathrm{d}^{\times}y\right)\mathrm{d}u\\ =q^{m}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\left(\int_{\varpi_{\mathbf{F}}^{-m}\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(y(u+u^{-1}\right)\chi^{2}(y)\mathrm{d}^{\times}y\right)\mathrm{d}u\\ =\zeta_{\mathbf{F}}(1)q^{m}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\frac{y^{2}u+u^{-1}}{\varpi_{\mathbf{F}}^{m}}\right)\chi^{2}\left(\frac{y}{\varpi_{\mathbf{F}}^{m}}\right)\mathrm{d}y\mathrm{d}u.

While the level $\displaystyle\lceil m/2\rceil$ regularization to $\displaystyle\mathrm{d}u$ also implies

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\frac{\varepsilon y^{2}u+u^{-1}}{\varpi_{\mathbf{F}}^{m}}\right)\chi\left(\frac{\varepsilon y^{2}}{\varpi_{\mathbf{F}}^{2m}}\right)\mathrm{d}y\mathrm{d}u=0,

we can sum the above two equations to get

\int_{\mathbf{F}^{\times}}E_{m}(y)\chi(y)\mathrm{d}^{\times}y=\zeta_{\mathbf{F}}(1)q^{m}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\frac{yu+u^{-1}}{\varpi_{\mathbf{F}}^{m}}\right)\chi\left(\frac{y}{\varpi_{\mathbf{F}}^{2m}}\right)\mathrm{d}y\mathrm{d}u\\ =\zeta_{\mathbf{F}}(1)q^{m}\left(\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\frac{t}{\varpi_{\mathbf{F}}^{m}}\right)\chi\left(\frac{t}{\varpi_{\mathbf{F}}^{m}}\right)\mathrm{d}t\right)^{2}.

The last integral was studied in [20, Proposition 4.6], which is non-vanishing if and only if $\displaystyle\mathfrak{c}(\chi)=m$ . Its relation with the local gamma factor

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\frac{t}{\varpi_{\mathbf{F}}^{m}}\right)\chi\left(\frac{t}{\varpi_{\mathbf{F}}^{m}}\right)\mathrm{d}t=\gamma(1,\chi^{-1},\psi)

is the content of [5, Exercise 23.5]. ∎

Proposition 2.4.

There is $\displaystyle a=a(\Pi)\in\mathbb{Z}_{\geqslant 2}$ , called the stability barrier of $\displaystyle\Pi$ , such that for any $\displaystyle m\geqslant a$

\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(E_{m})(t)=\psi(t)\cdot\mathbbm{1}_{\varpi^{-m}\mathcal{O}_{\mathbf{F}}^{\times}}(t),\quad\widetilde{\mathcal{VH}}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(E_{\geqslant m})(t)=\psi(t)\cdot\mathbbm{1}_{\mathcal{P}_{\mathbf{F}}^{m}}(t^{-1}).

Moreover, we have $\displaystyle a\leqslant\max(2\mathfrak{c}(\Pi),1)$ . More precisely, we have:

(1)

If $\displaystyle\Pi$ is equal to or is included in $\displaystyle\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ we can take $\displaystyle a=\max(2\mathfrak{c}(\mu_{1}),2\mathfrak{c}(\mu_{2}),2\mathfrak{c}(\mu_{3}),1)$ ;
(2)

If $\displaystyle\Pi=\pi\boxplus\mu$ for a supercuspidal $\displaystyle\pi$ of $\displaystyle\mathbf{G}_{2}(\mathbf{F})$ we can take $\displaystyle a=\max(\mathfrak{c}(\pi),2\mathfrak{c}(\mu))$ ;
(3)

If $\displaystyle\Pi$ is supercuspidal we have $\displaystyle a\leqslant 2\mathfrak{c}(\Pi)$ .

Proof.

By Lemma 2.3 and the local functional equation, the integral

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(E_{m})(t)\chi^{-1}(t)\lvert t\rvert_{\mathbf{F}}^{-s}\mathrm{d}^{\times}t=\gamma(s,\Pi\times\chi,\psi)\int_{\mathbf{F}^{\times}}E_{m}(y)\chi(y)\lvert y\rvert_{\mathbf{F}}^{s-1}\mathrm{d}^{\times}y

is vanishing for any $\displaystyle\chi$ with $\displaystyle\mathfrak{c}(\chi)\neq m$ . By the stability of the local gamma factors (see [11, Proposition (2.2)], [5, Theorem 23.8] and [5, Exercise 23.5]), there is $\displaystyle a\in\mathbb{Z}_{\geqslant 2}$ depending only on $\displaystyle\Pi$ so that the factors

\displaystyle\gamma(s,\Pi\times\chi,\psi)=\gamma(s,\chi,\psi)^{3}

depend only on $\displaystyle\omega_{\Pi}=\mathbbm{1}$ if $\displaystyle\mathfrak{c}(\chi)\geqslant a$ . Hence for $\displaystyle\mathfrak{c}(\chi)=m\geqslant a$ we have by Lemma 2.3

\displaystyle\gamma(s,\Pi\times\chi,\psi)\int_{\mathbf{F}^{\times}}E_{m}(y)\chi(y)\lvert y\rvert_{\mathbf{F}}^{s-1}\mathrm{d}^{\times}y=\gamma(s,\chi,\psi)^{3}\cdot\zeta_{\mathbf{F}}(1)\gamma(3/2-s,\chi^{-1},\psi)^{2}=\zeta_{\mathbf{F}}(1)\gamma(s+1,\chi,\psi).

One verifies easily the desired formula for $\displaystyle\mathcal{VH}_{\Pi,\psi}(E_{m})$ by comparing their Mellin transforms. The desired formula for $\displaystyle E_{\geqslant m}$ follows easily by taking limits in the sense of tempered distributions on $\displaystyle{\rm M}_{3}(\mathbf{F})$ via $\displaystyle I_{3}:{\rm C}(\mathbf{F}^{\times})\to{\rm C}({\rm GL}_{3}(\mathbf{F}))$ by [19, Theorem 1.3]. In the “moreover” part, (1) and (2) follow from the effective versions of the stability theorems for $\displaystyle\mathbf{G}_{1}$ and $\displaystyle\mathbf{G}_{2}$ [5, Theorem 23.8 & 25.7] together with the multiplicativity of the local gamma factors. For (3) we can take $\displaystyle a=\max(2\mathfrak{c}(\Pi),6)$ by examining the proof in [11, §2]. Now that a supercuspidal $\displaystyle\Pi$ of $\displaystyle\mathbf{G}_{3}$ has $\displaystyle\mathfrak{c}(\Pi)\geqslant 3$ , we conclude the last assertion. ∎

Remark 2.5.

It would be interesting to know a sharp bound for $\displaystyle a$ , say in general for stability theorems for $\displaystyle\mathbf{G}_{n}\times\mathbf{G}_{t}$ . It could follow from the work of Bushnell–Henniart–Kutzko [6].

3. Local Weight Functions

3.1. Choice of Test Functions

The target representation $\displaystyle\pi_{0}\in\widehat{{\rm PGL}_{2}(\mathbf{F})}$ are:

(1)

(Split) $\displaystyle\pi(\chi_{0},\chi_{0}^{-1})$ with a ramified and $\displaystyle\vartheta_{3}$ -tempered quasi-character $\displaystyle\chi_{0}$ of $\displaystyle\mathbf{F}^{\times}$ ;
(2)

(Special) the quadratic twists $\displaystyle\mathrm{St}_{\eta}$ of the Steinberg representation $\displaystyle\mathrm{St}$ ;
(3)

(Dihedral) $\displaystyle\pi_{\beta}$ with a unitary regular character $\displaystyle\beta$ of $\displaystyle\mathbf{E}^{\times}$ , $\displaystyle\mathbf{E}/\mathbf{F}$ being a quadratic field extension.

Note that $\displaystyle\mathrm{St}_{\eta}$ is a sub-representation of $\displaystyle\pi(\eta\lvert\cdot\rvert_{\mathbf{F}}^{1/2},\eta\lvert\cdot\rvert_{\mathbf{F}}^{-1/2})$ .

Let $\displaystyle\mathbf{L}$ be a separable quadratic algebra extension of $\displaystyle\mathbf{F}$ . The non-trivial element of the group $\displaystyle\mathrm{Aut}_{\mathbf{F}}(\mathbf{L})$ is denoted by $\displaystyle\mathbf{L}\to\mathbf{L},x\mapsto\bar{x}$ . Define

\displaystyle{\rm Tr}={\rm Tr}_{\mathbf{L}/\mathbf{F}}:\mathbf{L}\to\mathbf{F},\ x\mapsto x+\bar{x};\quad{\rm Nr}={\rm Nr}_{\mathbf{L}/\mathbf{F}}:\mathbf{L}^{\times}\to\mathbf{F}^{\times},\ x\mapsto x\bar{x};\quad\lvert x\rvert_{\mathbf{L}}:=\lvert{\rm Nr}(x)\rvert.

We associate to each target $\displaystyle\pi_{0}$ a parameter $\displaystyle(\mathbf{L},\beta)$ by:

(1)

$\displaystyle\pi(\chi_{0},\chi_{0}^{-1})$ : Let $\displaystyle\mathbf{L}\simeq\mathbf{F}\oplus\mathbf{F}$ and $\displaystyle\beta:\mathbf{L}^{\times}\simeq\mathbf{F}^{\times}\times\mathbf{F}^{\times}\to\mathrm{S}^{1},(t_{1},t_{2})\mapsto\chi_{0}(t_{1}t_{2}^{-1})$ ;
(2)

$\displaystyle\mathrm{St}_{\eta}$ : Let $\displaystyle\mathbf{L}\simeq\mathbf{F}\oplus\mathbf{F}$ and $\displaystyle\beta:\mathbf{L}^{\times}\simeq\mathbf{F}^{\times}\times\mathbf{F}^{\times}\to\mathbb{C}^{\times},(t_{1},t_{2})\mapsto\eta(t_{1}t_{2}^{-1})\lvert t_{1}t_{2}^{-1}\rvert_{\mathbf{F}}^{1/2}$ ;
(3)

$\displaystyle\pi_{\beta}$ : Let $\displaystyle\mathbf{L}=\mathbf{E}$ and $\displaystyle\beta$ be the obvious one.

We equip $\displaystyle\mathbf{L}$ with the self-dual Haar measure $\displaystyle\mathrm{d}z$ with respect to $\displaystyle\psi_{\mathbf{L}}:=\psi\circ{\rm Tr}$ . Write

\displaystyle\mathcal{O}_{\mathbf{L}}:=\left\{x\in\mathbf{L}\ \middle|\ {\rm Tr}(x),{\rm Nr}(x)\in\mathcal{O}_{\mathbf{F}}\right\}.

Definition 3.1.

Define $\displaystyle\mathcal{P}_{\mathbf{L}}:=\varpi_{\mathbf{L}}\mathcal{O}_{\mathbf{L}}$ and $\displaystyle\varpi_{\mathbf{L}}:=\varpi_{\mathbf{F}}$ if $\displaystyle\mathbf{L}$ is split, otherwise $\displaystyle\varpi_{\mathbf{L}}$ is a uniformizer of $\displaystyle\mathbf{L}$ . For any (quasi-)character $\displaystyle\beta$ of $\displaystyle\mathbf{L}^{\times}$ we define its $\displaystyle\mathbf{F}$ -norm- $\displaystyle 1$ conductor as

\displaystyle\mathfrak{c}_{1}(\beta):=\min\left\{n\ \middle|\ \beta\left(\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n})\right)=\{1\}\right\}.

If $\displaystyle\mathfrak{c}_{1}(\beta)>0$ we say that $\displaystyle\beta$ is a regular character.

Remark 3.2.

Directly from the definition we have for any (quasi-)character $\displaystyle\beta$ of $\displaystyle\mathbf{L}^{\times}$

•

$\displaystyle\mathfrak{c}_{1}(\beta)\leqslant\mathfrak{c}(\beta)$ ;
•

$\displaystyle\mathfrak{c}_{1}(\beta)=\mathfrak{c}_{1}(\beta\cdot(\chi\circ{\rm Nr}))$ for any (quasi-)character $\displaystyle\chi$ of $\displaystyle\mathbf{F}^{\times}$ .

Lemma 3.3.

If $\displaystyle\beta$ is a regular character of $\displaystyle\mathbf{L}^{\times}$ so that the restriction $\displaystyle\beta\mid_{\mathbf{F}^{\times}}=\eta_{\mathbf{L}/\mathbf{F}}$ coincides with the quadratic character associated with the quadratic extension $\displaystyle\mathbf{L}/\mathbf{F}$ , then we have $\displaystyle\mathfrak{c}_{1}(\beta)=\mathfrak{c}(\beta)$ .

Proof.

Write $\displaystyle n_{0}:=\mathfrak{c}(\beta)(\geqslant\mathfrak{c}_{1}(\beta)\geqslant 1)$ . We shall prove that $\displaystyle\mathfrak{c}_{1}(\beta)<n_{0}$ is impossible.

(1) If $\displaystyle\mathbf{L}\simeq\mathbf{F}\oplus\mathbf{F}$ is split then $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}=\mathbbm{1}$ . The assertion follows from $\displaystyle\mathfrak{c}(\chi_{0})=\mathfrak{c}(\chi_{0}^{2})$ , because taking square is a group automorphism of $\displaystyle 1+\mathcal{P}_{\mathbf{F}}$ and the regularity of $\displaystyle\beta$ is equivalent with $\displaystyle\mathfrak{c}(\chi_{0}^{2})>0$ .

(2) Assume $\displaystyle\mathbf{L}/\mathbf{F}$ is non-split. Then $\displaystyle\beta$ coincides with $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}$ on $\displaystyle\mathbf{F}^{\times}$ . If $\displaystyle n_{0}=1$ then the assertion follows from the condition $\displaystyle\beta\mid_{\mathbf{E}^{1}}\neq\mathbbm{1}$ (equivalent to $\displaystyle\beta$ being regular). Assume $\displaystyle n_{0}\geqslant 2$ from now on. We choose the uniformizers $\displaystyle\varpi_{\mathbf{F}}$ and $\displaystyle\varpi_{\mathbf{L}}$ of $\displaystyle\mathcal{O}_{\mathbf{F}}$ and $\displaystyle\mathcal{O}_{\mathbf{L}}$ respectively so that

(3.1)

\begin{cases}\varpi_{\mathbf{L}}=\varpi_{\mathbf{F}}&\text{if }e=e(\mathbf{L}/\mathbf{F})=1\\ \overline{\varpi_{\mathbf{L}}}=-\varpi_{\mathbf{L}},\varpi_{\mathbf{F}}={\rm Nr}(\varpi_{\mathbf{L}})=-\varpi_{\mathbf{L}}^{2}&\text{if }e=e(\mathbf{E}/\mathbf{F})=2\end{cases}.

Claim: We have $\displaystyle 2\mid n_{0}$ in the ramified case.

Proof of Claim:.

In fact, if $\displaystyle\beta$ is trivial on $\displaystyle 1+\mathcal{P}_{\mathbf{L}}^{2n+1}$ with $\displaystyle n\geqslant 1$ , then for any element $\displaystyle\alpha\in 1+\mathcal{P}_{\mathbf{L}}^{2n}$ we can find $\displaystyle u_{0}\in\mathcal{O}_{\mathbf{F}}$ and $\displaystyle u_{1}\in\mathcal{O}_{\mathbf{L}}$ such that $\displaystyle\alpha=1+\varpi_{\mathbf{L}}^{2n}u_{0}+\varpi_{\mathbf{L}}^{2n+1}u_{1}$ , since the residue class fields of $\displaystyle\mathbf{L}$ and $\displaystyle\mathbf{F}$ are isomorphic. Then $\displaystyle\beta(\alpha)=\eta_{\mathbf{L}/\mathbf{F}}(1+(-\varpi_{\mathbf{F}})^{n}u_{0})=1$ . Hence $\displaystyle\beta$ is also trivial on $\displaystyle 1+\mathcal{P}_{\mathbf{L}}^{2n}$ . ∎

With these reductions, we find $\displaystyle\alpha\in 1+\mathcal{P}_{\mathbf{L}}^{n_{0}-1}$ such that $\displaystyle\beta(\alpha)\neq 1$ . But $\displaystyle{\rm Nr}(\alpha)\in\mathbf{F}\cap(1+\mathcal{P}_{\mathbf{L}}^{n_{0}-1})=1+\mathcal{P}_{\mathbf{F}}^{\lceil(n_{0}-1)/e\rceil}$ is a square. Hence $\displaystyle{\rm Nr}(\alpha)=k^{2}$ for some $\displaystyle k\in 1+\mathcal{P}_{\mathbf{F}}^{\lceil(n_{0}-1)/e\rceil}\subset 1+\mathcal{P}_{\mathbf{L}}^{n_{0}-1}$ , and $\displaystyle\beta(k^{-1}\alpha)=\beta(\alpha)\neq 1$ with $\displaystyle k^{-1}\alpha\in\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n_{0}-1})$ , proving the assertion. ∎

Corollary 3.4.

Any dihedral supercuspidal representations $\displaystyle\pi_{\beta}$ with trivial central character (the case (3) in the beginning of this subsection) is twisted minimal.

Proof.

For any (quasi-)character $\displaystyle\chi$ of $\displaystyle\mathbf{F}^{\times}$ we have $\displaystyle\mathfrak{c}(\beta)=\mathfrak{c}_{1}(\beta)=\mathfrak{c}_{1}\left(\beta\cdot(\chi\circ{\rm Nr})\right)\leqslant\mathfrak{c}\left(\beta\cdot(\chi\circ{\rm Nr})\right)$ . For any regular (quasi-)character $\displaystyle\beta$ of $\displaystyle\mathbf{E}^{\times}$ we have by [10, Theorem 4.7] with $\displaystyle e=e(\mathbf{E}/\mathbf{F})$

(3.2)

\mathfrak{c}(\pi_{\beta})=\mathfrak{c}(\beta)f(\mathbf{E}/\mathbf{F})+\mathfrak{c}(\psi_{\mathbf{E}})=\tfrac{2n_{0}}{e}+e-1,

where $\displaystyle f(\mathbf{E}/\mathbf{F})$ is the residual field index and $\displaystyle\psi_{\mathbf{E}}=\psi_{\mathbf{F}}\circ{\rm Tr}_{\mathbf{E}/\mathbf{F}}$ . The desired assertion follows readily. ∎

Proposition 3.5.

Let $\displaystyle\mathbf{L}^{1}$ be the kernel of the norm map $\displaystyle{\rm Nr}$ .

(1)

For any $\displaystyle x\in\mathbf{L}^{\times}$ , the following map

$\displaystyle\iota_{x}:\mathbf{F}^{\times}\times\mathbf{L}^{1}\to\mathbf{L}^{\times},\quad(r,\alpha)\mapsto xr\alpha$

is a $\displaystyle 2$ -to- $\displaystyle 1$ covering map onto an open subset of $\displaystyle\mathbf{L}^{\times}$ .

(2)

(Polar decomposition) The Haar measure $\displaystyle\mathrm{d}\alpha$ on $\displaystyle\mathbf{L}^{1}$ determined by

(3.3)

\lvert z\rvert_{\mathbf{L}}^{-1}\mathrm{d}z=\lvert r\rvert_{\mathbf{F}}^{-1}\mathrm{d}r\mathrm{d}\alpha,\quad z=\iota_{x}(r,\alpha)=xr\alpha,

where $\displaystyle\mathrm{d}r$ is the self-dual Haar measure of $\displaystyle\mathbf{F}$ with respect to $\displaystyle\psi$ , is independent of $\displaystyle x$ , and satisfies

\displaystyle{\rm Vol}(\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}},\mathrm{d}\alpha)=2^{e-1}\frac{{\rm Vol}(\mathcal{O}_{\mathbf{L}}^{\times},\mathrm{d}z)}{{\rm Vol}(\mathcal{O}_{\mathbf{F}}^{\times},\mathrm{d}r)},

where the number $\displaystyle e=e(\mathbf{L}/\mathbf{F})$ is the generalized ramification index of $\displaystyle\mathbf{L}/\mathbf{F}$ given by

(3.4)

e=\begin{cases}1&\text{if }\mathbf{L}/\mathbf{F}\text{ is not ramified}\\ 2&\text{if }\mathbf{L}/\mathbf{F}\text{ is ramified}\end{cases}.

(3)

For any $\displaystyle n\in\mathbb{Z}_{\geqslant 1}$ we have $\displaystyle{\rm Vol}(\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n}),\mathrm{d}\alpha)=q^{-\lfloor\frac{n}{e}\rfloor-\frac{e-1}{2}}$ .

Proof.

The proof of (1) is easy and omitted. The relation $\displaystyle\Im(\iota_{x})=x\Im(\iota_{1})$ implies the independence of $\displaystyle x$ in (3.3), since the measure $\displaystyle\lvert z\rvert_{\mathbf{L}}^{-1}\mathrm{d}z$ is invariant by multiplication by elements in $\displaystyle\mathbf{L}^{\times}$ . To show that (3.3) implies the stated formula for $\displaystyle{\rm Vol}(\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}},\mathrm{d}\alpha)$ , it suffices to show

\displaystyle{\rm Vol}(\iota_{1}(\mathcal{O}_{\mathbf{F}}^{\times}\times(\mathcal{O}_{\mathbf{L}}\cap\mathbf{L}^{1})))=2^{e-2}{\rm Vol}(\mathcal{O}_{\mathbf{L}}^{\times}).

Now that $\displaystyle\iota_{1}(\mathcal{O}_{\mathbf{F}}^{\times}\times(\mathcal{O}_{\mathbf{L}}\cap\mathbf{L}^{1}))=\mathcal{O}_{\mathbf{L}}^{\times}\cap{\rm Nr}^{-1}((\mathcal{O}_{\mathbf{F}}^{\times})^{2})$ , we get

\displaystyle[\mathcal{O}_{\mathbf{L}}^{\times}:\iota_{1}(\mathcal{O}_{\mathbf{F}}^{\times}\times(\mathcal{O}_{\mathbf{L}}\cap\mathbf{L}^{1}))]=[{\rm Nr}(\mathcal{O}_{\mathbf{L}}^{\times}):(\mathcal{O}_{\mathbf{F}}^{\times})^{2}]=\frac{[\mathcal{O}_{\mathbf{F}}^{\times}:(\mathcal{O}_{\mathbf{F}}^{\times})^{2}]}{[\mathcal{O}_{\mathbf{F}}^{\times}:{\rm Nr}(\mathcal{O}_{\mathbf{L}}^{\times})]}=\frac{2}{2^{e-1}}

and conclude the proof of (2). To prove (3) we note that the map

\displaystyle\sigma:\mathcal{O}_{\mathbf{L}}^{\times}\to\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}},\quad x\mapsto x/\bar{x}

is surjective onto $\displaystyle\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n})$ , whose pre-image is $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}(1+\mathcal{P}_{\mathbf{L}}^{n})$ . It is also easy to see that the image of $\displaystyle\sigma$ is a subgroup of $\displaystyle\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}$ with index $\displaystyle 2^{e-1}$ , by a refinement of Hilbert’s 90. Therefore we get

\displaystyle\frac{\Im(\sigma)}{\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n})}\simeq\frac{\mathcal{O}_{\mathbf{L}}^{\times}}{\mathcal{O}_{\mathbf{F}}^{\times}(1+\mathcal{P}_{\mathbf{L}}^{n})}\simeq\frac{(\mathcal{O}_{\mathbf{L}}/\mathcal{P}_{\mathbf{L}}^{n})^{\times}}{(\mathcal{O}_{\mathbf{F}}/\mathcal{P}_{\mathbf{F}}^{\lceil n/e\rceil})^{\times}}

and deduce from it the measure relation

\displaystyle\frac{{\rm Vol}(\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}},\mathrm{d}\alpha)}{{\rm Vol}(\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n}),\mathrm{d}\alpha)}=2^{e-1}\frac{q^{2n/e}}{q^{\lceil n/e\rceil}}\cdot\frac{{\rm Vol}(\mathcal{O}_{\mathbf{L}}^{\times},\mathrm{d}z)}{{\rm Vol}(\mathcal{O}_{\mathbf{F}}^{\times},\mathrm{d}r)}\cdot\frac{{\rm Vol}(\mathcal{O}_{\mathbf{F}},\mathrm{d}r)}{{\rm Vol}(\mathcal{O}_{\mathbf{L}},\mathrm{d}z)}.

We conclude the proof of (3) by comparing with (2) and $\displaystyle{\rm Vol}(\mathcal{O}_{\mathbf{L}},\mathrm{d}z)=q^{-\frac{e-1}{2}}$ . ∎

Let $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}$ be the quadratic character of $\displaystyle\mathbf{F}^{\times}$ associated with $\displaystyle\mathbf{L}/\mathbf{F}$ , which is trivial on $\displaystyle{\rm Nr}(\mathbf{L}^{\times})$ . We have $\displaystyle\beta\mid_{\mathbf{F}^{\times}}=\eta_{\mathbf{L}/\mathbf{F}}$ since $\displaystyle\pi_{0}$ has trivial central character. Write $\displaystyle\lambda(\mathbf{L}/\mathbf{F},\psi)$ for the Weil index, which is equal to $\displaystyle 1$ if $\displaystyle\mathbf{L}/\mathbf{F}$ is not ramified (see [7, Corollary 3.2]). Let $\displaystyle\tau$ run through a system of representatives of $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}/(\mathcal{O}_{\mathbf{F}}^{\times})^{2}$ (split), resp. $\displaystyle{\rm Nr}(\mathbf{L}^{\times})/(\mathbf{F}^{\times})^{2}$ (non-split), and choose any $\displaystyle x_{\tau}\in\mathbf{L}^{\times}$ s.t. $\displaystyle{\rm Nr}(x_{\tau})=\tau$ ; if $\displaystyle\mathbf{L}/\mathbf{F}$ is split we also require $\displaystyle{\rm Tr}(x_{\tau})=1+\tau$ . Define a function $\displaystyle H$ on $\displaystyle\mathbf{F}^{\times}$ with support contained in $\displaystyle{\rm Nr}(\mathbf{L}^{\times})$ by

(3.5)

H(\tau y^{2})=\lambda(\mathbf{L}/\mathbf{F},\psi)\lvert\tau\rvert_{\mathbf{F}}^{\frac{1}{2}}\cdot\lvert y\rvert_{\mathbf{F}}\cdot\eta_{\mathbf{L}/\mathbf{F}}(y)\cdot\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(x_{\tau}\alpha)\psi({\rm Tr}(x_{\tau}\alpha)y)\mathrm{d}\alpha.

It is clear that $\displaystyle H$ is independent of any choice made in the definition. For definiteness, we fix a $\displaystyle\varepsilon\in\mathcal{O}_{\mathbf{F}}^{\times}-(\mathcal{O}_{\mathbf{F}}^{\times})^{2}$ and choose a system $\displaystyle\{\tau\}$ of representatives of $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}/(\mathcal{O}_{\mathbf{F}}^{\times})^{2}$ , resp. $\displaystyle{\rm Nr}(\mathbf{L}^{\times})/(\mathbf{F}^{\times})^{2}$ as

(1)

$\displaystyle\{1,\varepsilon\}$ if $\displaystyle\mathbf{L}/\mathbf{F}$ is split or unramified (in the latter case $\displaystyle\mathbf{L}=\mathbf{F}[\sqrt{\varepsilon}]$ ),
(2)

$\displaystyle\{1,-\varpi_{\mathbf{F}}\}$ if $\displaystyle\mathbf{L}/\mathbf{F}$ is ramified (with $\displaystyle\mathbf{L}=\mathbf{F}[\sqrt{\varpi_{\mathbf{F}}}]$ and we fix a uniformizer $\displaystyle\varpi_{\mathbf{L}}=\sqrt{\varpi_{\mathbf{F}}}$ of $\displaystyle\mathbf{L}$ ).

The (partial) Mellin transform of $\displaystyle H$ will be fundamental for our analysis. We record it as follows.

Lemma 3.6.

Let $\displaystyle\chi\in\widehat{\mathbf{F}^{\times}}$ and $\displaystyle n\in\mathbb{Z}_{\geqslant 1}$ . We have

\varepsilon_{n}(\chi,H):=\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}H(y)\chi(y)\mathrm{d}^{\times}y=\zeta_{\mathbf{F}}(1)\cdot\\ \begin{cases}\mathbbm{1}_{2\mid n,\mathfrak{c}(\chi_{0}\chi)=\mathfrak{c}(\chi_{0}^{-1}\chi)=n/2}\cdot\varepsilon(1/2,\chi_{0}\chi^{-1},\psi)\varepsilon(1/2,\chi_{0}^{-1}\chi^{-1},\psi)&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }\mathfrak{c}(\chi_{0}\chi),\mathfrak{c}(\chi_{0}^{-1}\chi)\geqslant 1\\ -\mathbbm{1}_{n=2,\mathfrak{c}(\chi_{0})=1,\mathfrak{c}(\chi_{0}^{2})\neq 0}\cdot q^{-1}\chi_{0}\chi(\varpi_{\mathbf{F}})\varepsilon(1/2,\chi_{0}^{-1}\chi^{-1},\psi)&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }\mathfrak{c}(\chi_{0}\chi)=0\neq\mathfrak{c}(\chi_{0}^{-1}\chi)\\ -\mathbbm{1}_{n=2,\mathfrak{c}(\chi_{0})=1,\mathfrak{c}(\chi_{0}^{2})\neq 0}\cdot q^{-1}\chi_{0}^{-1}\chi(\varpi_{\mathbf{F}})\varepsilon(1/2,\chi_{0}\chi^{-1},\psi)&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }\mathfrak{c}(\chi_{0}^{-1}\chi)=0\neq\mathfrak{c}(\chi_{0}\chi)\\ -\mathbbm{1}_{n=2,\mathfrak{c}(\chi_{0})=1,\mathfrak{c}(\chi_{0}^{2})=0}\cdot q^{-2}\chi^{2}(\varpi_{\mathbf{F}})&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }\mathfrak{c}(\chi_{0}\chi)=0=\mathfrak{c}(\chi_{0}^{-1}\chi)\\ \mathbbm{1}_{2\mid en,\mathfrak{c}(\beta\cdot(\chi\circ{\rm Nr}))=en/2+1-e}\cdot\varepsilon(1/2,\pi_{\beta^{-1}}\otimes\chi^{-1},\psi)&\text{if }\mathbf{L}/\mathbf{F}\text{ non-split}\end{cases}.

In particular, we have $\displaystyle\left\lvert\varepsilon_{n}(\chi,H)\right\rvert\leqslant\zeta_{\mathbf{F}}(1)$ .

Proof.

We divide the domain of integration into cosets of square elements.

(A) In the split case, we have

	$\displaystyle\displaystyle\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}H(y)\chi(y)\mathrm{d}^{\times}y$	$\displaystyle\displaystyle=\mathbbm{1}_{2\mid n}\cdot\frac{1}{2}\sum_{\tau\in\{1,\varepsilon\}}\int_{\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}}\chi(\tau y^{2})\lvert y\rvert\left(\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}(\tau\alpha^{2})\psi\left((\tau\alpha+\alpha^{-1})y\right)\mathrm{d}\alpha\right)\mathrm{d}^{\times}y$
		$\displaystyle\displaystyle=\mathbbm{1}_{2\mid n}\cdot\zeta_{\mathbf{F}}(1)^{-1}\int_{\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}\times\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}}\chi(t_{1}t_{2})\lvert t_{1}t_{2}\rvert^{\frac{1}{2}}\cdot\chi_{0}(t_{1}t_{2}^{-1})\psi(t_{1}+t_{2})\mathrm{d}^{\times}t_{1}\mathrm{d}^{\times}t_{2}$
		$\displaystyle\displaystyle=\mathbbm{1}_{2\mid n}\cdot\zeta_{\mathbf{F}}(1)q^{\frac{n}{2}}\left(\int_{\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(t)\cdot\chi\chi_{0}(t)\frac{\mathrm{d}t}{\lvert t\rvert}\right)\left(\int_{\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(t)\cdot\chi\chi_{0}^{-1}(t)\frac{\mathrm{d}t}{\lvert t\rvert}\right),$

where we have taken into account that for each $\displaystyle\tau$ the map

\displaystyle\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}\times\mathcal{O}_{\mathbf{F}}^{\times}\to\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times}\times\varpi_{\mathbf{F}}^{-\frac{n}{2}}\mathcal{O}_{\mathbf{F}}^{\times},\quad(y,\alpha)\mapsto(\tau y\alpha,y\alpha^{-1})

is $\displaystyle 2$ -to- $\displaystyle 1$ . The stated formulae then follow from the relation between the Gauss integrals and the local epsilon-factors given in [5, Exercise 23.5], and a direct computation in the unramified case.

(B) Similarly in the non-split case we have

	$\displaystyle\displaystyle\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}H(y)\chi(y)\mathrm{d}^{\times}y$	$\displaystyle\displaystyle=\mathbbm{1}_{2\mid en}\cdot\frac{\zeta_{\mathbf{F}}(1)}{{\rm Vol}(\mathbf{L}^{1})}\int_{\varpi_{\mathbf{L}}^{-\frac{en}{2}}\mathcal{O}_{\mathbf{L}}^{\times}}H({\rm Nr}(z))\chi({\rm Nr}(z))\frac{\mathrm{d}z}{\lvert z\rvert_{\mathbf{L}}}$
		$\displaystyle\displaystyle=\mathbbm{1}_{2\mid en}\cdot\zeta_{\mathbf{F}}(1)q^{\frac{n}{2}}\lambda(\mathbf{L}/\mathbf{F},\psi)\int_{\varpi_{\mathbf{L}}^{-\frac{en}{2}}\mathcal{O}_{\mathbf{L}}^{\times}}\beta(z)\chi({\rm Nr}(z))\cdot\psi_{\mathbf{L}}(z)\frac{\mathrm{d}z}{\lvert z\rvert_{\mathbf{L}}}$
		$\displaystyle\displaystyle=\mathbbm{1}_{2\mid en}\cdot\zeta_{\mathbf{F}}(1)q^{\frac{n}{2}}\lambda(\mathbf{L}/\mathbf{F},\psi)\cdot\mathbbm{1}_{\mathfrak{c}\left(\beta\cdot(\chi\circ{\rm Nr})\right)+e-1=\frac{en}{2}}\varepsilon(1,\beta^{-1}\cdot(\chi^{-1}\circ{\rm Nr}),\psi_{\mathbf{L}}),$

where we have applied [5, Exercise 23.5] in the last line. Now that [10, Theorem 4.7] implies

\displaystyle\lambda(\mathbf{L}/\mathbf{F},\psi)\varepsilon(s,\beta\cdot(\chi\circ{\rm Nr}),\psi_{\mathbf{L}})=\varepsilon(s,\pi_{\beta}\otimes\chi,\psi)=q^{n\cdot(\frac{1}{2}-s)}\varepsilon(\tfrac{1}{2},\pi_{\beta}\otimes\chi,\psi),

we conclude the desired formula. Note that in this case $\displaystyle\beta\cdot(\chi\circ{\rm Nr})$ is never unramified. ∎

We give a first asymptotic analysis of $\displaystyle H$ as follows.

Lemma 3.7.

Let $\displaystyle\alpha\in\mathbf{L}^{1}$ and $\displaystyle n_{1}\in\mathbb{Z}_{\geqslant 1}$ . Then $\displaystyle\alpha-\overline{\alpha}\in\mathcal{P}_{\mathbf{L}}^{n_{1}}$ if and only if $\displaystyle\alpha\in\pm 1+\mathcal{P}_{\mathbf{L}}^{n_{1}}$ .

Proof.

If $\displaystyle\alpha\in\pm 1+\mathcal{P}_{\mathbf{L}}^{n_{1}}$ , then it clear that $\displaystyle\alpha-\overline{\alpha}\in\mathcal{P}_{\mathbf{L}}^{n_{1}}$ . The converse is justified as follows.

(a) If $\displaystyle\mathbf{L}/\mathbf{F}$ is split, we write $\displaystyle\alpha=(t,t^{-1})$ with $\displaystyle t\in\mathbf{F}^{\times}$ . Then $\displaystyle t-t^{-1}\in\mathcal{P}_{\mathbf{F}}^{n_{1}}$ . We must have $\displaystyle t\in\mathcal{O}_{\mathbf{F}}^{\times}$ , and $\displaystyle t\equiv t^{-1}\pmod{\mathcal{P}_{\mathbf{F}}}$ . Hence $\displaystyle t\in\pm 1+\mathcal{P}_{\mathbf{F}}$ . Writing $\displaystyle t=\pm 1+\varpi_{\mathbf{F}}^{k}u$ for some $\displaystyle k\in\mathbb{Z}_{\geqslant 1}$ and $\displaystyle u\in\mathcal{O}_{\mathbf{F}}^{\times}$ we infer that $\displaystyle t-t^{-1}\in 2\varpi_{\mathbf{F}}^{k}u+\mathcal{P}_{\mathbf{F}}^{k+1}\subset\mathcal{P}_{\mathbf{F}}^{k}-\mathcal{P}_{\mathbf{F}}^{k+1}$ . Hence $\displaystyle k\geqslant n_{1}$ , and $\displaystyle t\in\pm 1+\mathcal{P}_{\mathbf{F}}^{n-1}$ .

(b) If $\displaystyle\mathbf{L}/\mathbf{F}$ is a field, then $\displaystyle\overline{\alpha}=\alpha^{-1}$ . We repeat the above argument with (subscript) $\displaystyle\mathbf{F}$ replaced by $\displaystyle\mathbf{L}$ . ∎

Lemma 3.8.

There is $\displaystyle c_{\beta}\in\mathcal{O}_{\mathbf{F}}^{\times}$ , called the additive parameter and is uniquely determined up to multiplication by elements in $\displaystyle 1+\mathcal{P}_{\mathbf{F}}^{\lceil\frac{n_{0}}{2e}\rceil}$ , so that for all $\displaystyle u\in\mathcal{P}_{\mathbf{L}}^{\lfloor\frac{n_{0}}{2}\rfloor}$

\displaystyle\beta(1+u)=\begin{cases}\psi\left(\varpi_{\mathbf{F}}^{-n_{0}}c_{\beta}(u-\bar{u})\right)&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }2\mid n_{0},\\ \psi\left(\varpi_{\mathbf{F}}^{-n_{0}}c_{\beta}\left((u-\bar{u})-2^{-1}(u^{2}-\bar{u}^{2})\right)\right)&\text{if }\mathbf{L}/\mathbf{F}\text{ split and }2\nmid n_{0};\\ \psi\left(\varpi_{\mathbf{F}}^{-n_{0}}\sqrt{\varepsilon}c_{\beta}(u-\bar{u})\right)&\text{if }\mathbf{L}/\mathbf{F}\text{ unramified and }2\mid n_{0},\\ \psi\left(\varpi_{\mathbf{F}}^{-n_{0}}\sqrt{\varepsilon}c_{\beta}\left((u-\bar{u})-2^{-1}(u^{2}-\bar{u}^{2})\right)\right)&\text{if }\mathbf{L}/\mathbf{F}\text{ unramified and }2\nmid n_{0};\\ \psi\left(\varpi_{\mathbf{L}}^{-n_{0}-1}c_{\beta}(u-\bar{u})\right)&\text{if }\mathbf{L}/\mathbf{F}\text{ ramified}.\end{cases}

Proof.

We only treat the cases $\displaystyle 2\nmid n_{0}:=2m+1$ if $\displaystyle\mathbf{L}/\mathbf{F}$ is not ramified, the even ones being simpler. The following map

\displaystyle\log_{\mathbf{F}}:(1+\mathcal{P}_{\mathbf{F}}^{m})/(1+\mathcal{P}_{\mathbf{F}}^{2m+1})\to\mathcal{P}_{\mathbf{F}}^{m}/\mathcal{P}_{\mathbf{F}}^{2m+1},\quad 1+x\mapsto x-2^{-1}x^{2}

is a group isomorphism. If $\displaystyle\mathbf{L}/\mathbf{F}$ is split there is $\displaystyle c_{\beta}^{\prime}\in\mathcal{O}_{\mathbf{F}}^{\times}/(1+\mathcal{P}_{\mathbf{F}}^{m+1})$ such that

(3.6)

\beta(1+\varpi_{\mathbf{F}}^{m}u)=\psi\left(c_{\beta}^{\prime}\left(\varpi_{\mathbf{F}}^{-(m+1)}u-2^{-1}\varpi_{\mathbf{F}}^{-1}u^{2}\right)\right),\quad\forall u\in\mathcal{O}_{\mathbf{L}}\left(\simeq\mathcal{O}_{\mathbf{F}}\times\mathcal{O}_{\mathbf{F}}\right).

One checks easily that $\displaystyle c_{\beta}:=c_{\beta}^{\prime}$ is the required one. If $\displaystyle\mathbf{L}/\mathbf{F}$ is unramified we get an analogue of (3.6)

(3.7)

\beta(1+\varpi_{\mathbf{F}}^{m}u)=\psi\left({\rm Tr}\left(c_{\beta}^{\prime}\left(\varpi_{\mathbf{F}}^{-(m+1)}u-2^{-1}\varpi_{\mathbf{F}}^{-1}u^{2}\right)\right)\right),\quad\forall u\in\mathcal{O}_{\mathbf{L}}.

Now that $\displaystyle\beta$ is trivial on $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ and $\displaystyle\log_{\mathbf{F}}$ is surjective, implying $\displaystyle\psi\circ{\rm Tr}(c_{\beta}^{\prime}\mathcal{P}_{\mathbf{F}}^{-m-1})=1$ , i.e., $\displaystyle c_{\beta}^{\prime}+\overline{c_{\beta}^{\prime}}\in\mathcal{P}_{\mathbf{F}}^{m+1}$ . Therefore we may write $\displaystyle c_{\beta}^{\prime}\in c_{\beta}\sqrt{\varepsilon}+\mathcal{P}_{\mathbf{F}}^{m+1}$ for some $\displaystyle c_{\beta}\in\mathcal{O}_{\mathbf{F}}^{\times}/(1+\mathcal{P}_{\mathbf{F}}^{m+1})$ . If $\displaystyle\mathbf{L}/\mathbf{F}$ is ramified, then $\displaystyle 2\mid n_{0}$ by the claim proved in Lemma 3.3 and $\displaystyle\psi_{\mathbf{L}}:=\psi\circ{\rm Tr}_{\mathbf{L}/\mathbf{F}}$ has conductor exponent $\displaystyle-1$ . We similarly get

(3.8)

\beta(1+u)=\psi\left(\varpi_{\mathbf{F}}^{-\frac{n_{0}}{2}}\varpi_{\mathbf{L}}^{-1}(c_{\beta}^{\prime}u-\overline{c_{\beta}^{\prime}}\bar{u})\right),\quad\forall u\in\mathcal{P}_{\mathbf{L}}^{\frac{n_{0}}{2}}

for some $\displaystyle c_{\beta}^{\prime}\in\mathcal{O}_{\mathbf{L}}^{\times}/(1+\mathcal{P}_{\mathbf{L}}^{\frac{n_{0}}{2}})$ . Note that $\displaystyle\beta$ is trivial on $\displaystyle 1+\mathcal{P}_{\mathbf{F}}$ , implying $\displaystyle\psi\left(\varpi_{\mathbf{F}}^{-\frac{n_{0}}{2}}\varpi_{\mathbf{L}}^{-1}(c_{\beta}^{\prime}-\overline{c_{\beta}^{\prime}})\mathcal{P}_{\mathbf{F}}^{\lceil\frac{n_{0}}{4}\rceil}\right)=1$ , i.e., $\displaystyle c_{\beta}^{\prime}-\overline{c_{\beta}^{\prime}}\in\mathcal{P}_{\mathbf{L}}^{2\lfloor\frac{n_{0}}{4}\rfloor+1}$ . Therefore we may write $\displaystyle c_{\beta}^{\prime}\in c_{\beta}+\mathcal{P}_{\mathbf{L}}^{2\lfloor\frac{n_{0}}{4}\rfloor+1}$ , hence may take $\displaystyle c_{\beta}^{\prime}=c_{\beta}$ in (3.8), for some $\displaystyle c_{\beta}\in\mathcal{O}_{\mathbf{F}}^{\times}/(1+\mathcal{P}_{\mathbf{F}}^{\lceil\frac{n_{0}}{4}\rceil})$ . ∎

Remark 3.9.

In the split case the character $\displaystyle\beta$ is constructed from a character $\displaystyle\chi_{0}$ of $\displaystyle\mathbf{F}^{\times}$ . We also call $\displaystyle c_{\beta}=c_{\chi_{0}}$ the additive parameter of $\displaystyle\chi_{0}$ .

Lemma 3.10.

Let $\displaystyle e$ be the generalized ramification index given in (3.4), and write $\displaystyle n_{0}:=\mathfrak{c}(\beta)$ .

(1)

In the domain $\displaystyle v_{\mathbf{F}}(y)\leqslant-4n_{0}/e-2(e-1)$ we have $\displaystyle H(y)=E_{\geqslant 2n_{0}/e+e-1}(y)$ .
(2)

Assume $\displaystyle\beta$ is regular. For any $\displaystyle\tau$ we have $\displaystyle H(\tau y^{2})=0$ if $\displaystyle v_{\mathbf{F}}(y)\geqslant(2-n_{0})e^{-1}-1$ .
(3)

If $\displaystyle\tau\neq 1$ , then $\displaystyle H(\tau y^{2})=0$ unless $\displaystyle v_{\mathbf{F}}(y)=1-e-e^{-1}n_{0}$ .

(4)

Suppose $\displaystyle e=1,n_{0}\geqslant 2$ . If $\displaystyle\tau=1$ and $\displaystyle v_{\mathbf{F}}(y)=-m$ with $\displaystyle n_{0}\leqslant m\leqslant 2n_{0}-1$ , then we may put the following condition in the domain of integration in (3.5):

\displaystyle{\rm Tr}(\alpha)\in\begin{cases}\pm 2(1+\mathcal{P}_{\mathbf{F}}^{2(n_{0}-1)})&\text{if }m=2n_{0}-1\\ \pm 2(1+\varpi_{\mathbf{F}}^{2(m-n_{0})}\mathcal{O}_{\mathbf{F}}^{\times})&\text{if }n_{0}<m<2n_{0}-1\\ \mathcal{O}_{\mathbf{F}}-\sideset{}{{}_{\pm}}{\bigcup}\pm 2(1+\mathcal{P}_{\mathbf{F}})&\text{if }m=n_{0}\end{cases}.

(5)

Suppose $\displaystyle e=2$ . We may put the following condition in the domain of integration in (3.5):

\displaystyle\begin{cases}{\rm Tr}(\alpha)\in\pm 2(1+\mathcal{P}_{\mathbf{F}}^{n_{0}-1})&\text{if }\tau=1,\text{ and }-v_{\mathbf{F}}(y):=m=n_{0}\\ {\rm Tr}(\alpha)\in\pm 2(1+\varpi_{\mathbf{F}}^{2m-n_{0}+1}\mathcal{O}_{\mathbf{F}}^{\times})&\text{if }\tau=1,\text{ and }\tfrac{n_{0}}{2}+1\leqslant-v_{\mathbf{F}}(y):=m\leqslant n_{0}-1\end{cases}.

Proof.

(1) We only consider non-split $\displaystyle\mathbf{L}$ , leaving the simpler split case as an exercise. Let $\displaystyle n=-v_{\mathbf{F}}(y)\geqslant 4n_{0}/e+2(e-1)$ . For $\displaystyle\chi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}$ , by Lemma 3.6 the integral $\displaystyle\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}H(y)\chi(y)\mathrm{d}^{\times}y\neq 0$ is non-zero only if

\displaystyle\tfrac{en}{2}=\mathfrak{c}(\beta(\chi\circ{\rm Nr}))+\mathfrak{c}(\psi_{\mathbf{L}})\quad\Leftrightarrow\quad\max(\mathfrak{c}(\beta),\mathfrak{c}(\chi\circ{\rm Nr}))\geqslant\mathfrak{c}(\beta(\chi\circ{\rm Nr}))=\begin{cases}n-1&\text{if }e=2\\ n/2&\text{if }e=1\end{cases}.

By [16, Proposition \@slowromancapv@.2.3 & Corollay \@slowromancapv@.3.3] we have

(3.9)

\mathfrak{c}(\chi\circ{\rm Nr})=\begin{cases}\max(2\mathfrak{c}(\chi)-1,0)&\text{if }e=2\\ \mathfrak{c}(\chi)&\text{if }e=1\end{cases}.

In view of the assumption on $\displaystyle n$ (implying $\displaystyle\mathfrak{c}(\beta(\chi\circ{\rm Nr}))>n_{0}$ ), the non-vanishing condition is equivalent to $\displaystyle 2\mid n\ \&\ \mathfrak{c}(\chi)=n/2$ . Now that by [10, Theorem 4.7] $\displaystyle\ell(\chi)=\mathfrak{c}(\chi)-1=n/2-1>2\ell(\pi_{\beta})=\mathfrak{c}(\pi_{\beta})-2=2\mathfrak{c}(\beta)/e+e-3=2n_{0}/e+e-3$ , we can apply the stability theorem [5, Theorem 25.7] and obtain

(3.10)

\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}H(y)\chi(y)\mathrm{d}^{\times}y=\mathbbm{1}_{2\mid n,\mathfrak{c}(\chi)=\frac{n}{2}}\cdot\zeta_{\mathbf{F}}(1)\varepsilon(1/2,\chi^{-1},\psi)^{2}.

We conclude by comparing with Lemma 2.3 and applying the Mellin inversion on $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ .

(2) We average over the change of variables $\displaystyle\alpha\mapsto\alpha\delta$ for $\displaystyle\delta\in\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n_{0}-1})=:U$ . Note that

\displaystyle yx_{\tau}\alpha(\delta-1)\in\mathcal{P}_{\mathbf{L}}^{1-e},\quad{\rm Tr}(\mathcal{P}_{\mathbf{L}}^{1-e})\subset\mathcal{P}_{\mathbf{L}}^{1-e}\cap\mathbf{F}=\mathcal{O}_{\mathbf{F}}\quad\Rightarrow\quad\psi({\rm Tr}(x_{\tau}\alpha\delta)y)=\psi({\rm Tr}(x_{\tau}\alpha)y)

for $\displaystyle y$ satisfying the stated condition. By Lemma 3.3, the character $\displaystyle\beta$ is non-trivial on $\displaystyle U$ , hence

\displaystyle H(\tau y^{2})=\lambda(\mathbf{L}/\mathbf{F},\psi)\lvert\tau\rvert_{\mathbf{F}}^{\frac{1}{2}}\cdot\lvert y\rvert_{\mathbf{F}}\cdot\eta_{\mathbf{L}/\mathbf{F}}(y)\cdot\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(x_{\tau}\alpha)\psi({\rm Tr}(x_{\tau}\alpha)y)\cdot\left(\oint_{U}\beta(\delta)\mathrm{d}\delta\right)\mathrm{d}\alpha=0.

(3) The idea is to average over similar change of variables $\displaystyle\alpha\mapsto\alpha\delta$ for $\displaystyle\delta\in\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n})=:U_{n}$ with

(3.11)

n\geqslant\max(n_{0},(1-e-ev_{\mathbf{F}}(y))/2)

and perform a further change of variables $\displaystyle\delta=\delta(u)$

\displaystyle\delta=\frac{1+\varpi_{\mathbf{L}}^{n}u}{1+\overline{\varpi_{\mathbf{L}}^{n}u}},\ u\in\mathcal{O}_{\mathbf{L}}\quad\Rightarrow\quad\delta-1\in\varpi_{\mathbf{L}}^{n}u-\overline{\varpi_{\mathbf{L}}^{n}u}+y^{-1}\mathcal{P}_{\mathbf{L}}^{1-e}.

Then we have $\displaystyle\beta(\delta)=1$ , $\displaystyle\psi({\rm Tr}(x_{\tau}\alpha(\delta-1))y)=\psi_{\mathbf{L}}(yx_{\tau}\alpha(\varpi_{\mathbf{L}}^{n}u-\overline{\varpi_{\mathbf{L}}^{n}u}))$ and get

(3.12)

H(\tau y^{2})=\lambda(\mathbf{L}/\mathbf{F},\psi)\lvert\tau\rvert_{\mathbf{F}}^{\frac{1}{2}}\cdot\lvert y\rvert_{\mathbf{F}}\cdot\eta_{\mathbf{L}/\mathbf{F}}(y)\cdot\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(x_{\tau}\alpha)\psi({\rm Tr}(x_{\tau}\alpha)y)\cdot\\ \left[\oint_{\mathcal{O}_{\mathbf{L}}}\psi\left(y(x_{\tau}\alpha-\overline{x_{\tau}\alpha})(\varpi_{\mathbf{L}}^{n}u-\overline{\varpi_{\mathbf{L}}^{n}u})\right)\mathrm{d}u\right]\mathrm{d}\alpha.

For any $\displaystyle m\in\mathbb{Z}$ we introduce $\displaystyle\mathcal{P}_{\mathbf{L}}^{m,-}:=\left\{x\in\mathcal{P}_{\mathbf{L}}^{m}\ \middle|\ \bar{x}=-x\right\}$ . We see that

\displaystyle\mathcal{O}_{\mathbf{L}}\to\mathcal{P}_{\mathbf{L}}^{n,-},\quad u\mapsto\varpi_{\mathbf{L}}^{n}u-\overline{\varpi_{\mathbf{L}}^{n}u}

is a surjective group homomorphism. Therefore the inner integral in (3.12) is non-vanishing only if

(3.13)

y(x_{\tau}\alpha-\overline{x_{\tau}\alpha})\in\mathcal{P}_{\mathbf{L}}^{1-e-n,-}\quad\Leftrightarrow\quad v_{\mathbf{L}}(x_{\tau}\alpha-\overline{x_{\tau}\alpha})\geqslant 1-e-n-ev_{\mathbf{F}}(y).

We choose $\displaystyle n=1-2e-ev_{\mathbf{F}}(y)$ , which is consistent with the condition (3.11) if

(3.14)

v_{\mathbf{F}}(y)\leqslant\lfloor\min(e^{-1}-2-e^{-1}n_{0},e^{-1}-3)\rfloor=-e-e^{-1}n_{0},

so that (3.13) becomes $\displaystyle v_{\mathbf{L}}(x_{\tau}\alpha-\overline{x_{\tau}\alpha})\geqslant e$ or $\displaystyle v_{\mathbf{F}}((x_{\tau}\alpha-\overline{x_{\tau}\alpha})^{2})\geqslant 2$ . In view of the equality

(3.15)

\tau=x_{\tau}\alpha\cdot\overline{x_{\tau}\alpha}=\tfrac{1}{4}\left\{(x_{\tau}\alpha+\overline{x_{\tau}\alpha})^{2}-(x_{\tau}\alpha-\overline{x_{\tau}\alpha})^{2}\right\}

and $\displaystyle v_{\mathbf{F}}(\tau)\in\{0,1\}$ , we deduce that $\displaystyle v_{\mathbf{F}}(\tau)=v_{\mathbf{F}}((x_{\tau}\alpha+\overline{x_{\tau}\alpha})^{2})\in 2\mathbb{Z}$ . Hence $\displaystyle v_{\mathbf{F}}(\tau)=0$ and $\displaystyle\tau$ is a square in $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ . This is possible only if $\displaystyle\tau=1$ . In other words, if $\displaystyle\tau\neq 1$ and $\displaystyle v_{\mathbf{F}}(y)$ satisfies (3.14) then $\displaystyle H(\tau y^{2})=0$ . We conclude because the only integer not satisfying (3.14) nor the inequality in (1) is $\displaystyle v_{\mathbf{F}}(y)=1-e-e^{-1}n_{0}$ .

(4) Suppose $\displaystyle\mathbf{L}/\mathbf{F}$ is unramified. Let $\displaystyle n:=\lceil\tfrac{m}{2}\rceil$ . Recall $\displaystyle c_{\beta}$ defined in Lemma 3.8 (1). We average over the change of variables $\displaystyle\alpha\mapsto\alpha(1+u)(1+\bar{u})^{-1}$ for $\displaystyle u\in\mathcal{P}_{\mathbf{L}}^{n}$ and get

\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\mathrm{d}\alpha=\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\left\{\oint_{\mathcal{P}_{\mathbf{L}}^{n}}\psi\left(\tfrac{2c_{\beta}\sqrt{\varepsilon}(u-\bar{u})}{\varpi_{\mathbf{F}}^{n_{0}}}+y{\rm Tr}\left(\alpha\cdot\tfrac{u-\bar{u}}{1+\bar{u}}\right)\right)\mathrm{d}u\right\}\mathrm{d}\alpha\\ =\int_{\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\left\{\oint_{\mathcal{O}_{\mathbf{L}}}\psi\left((2\sqrt{\varepsilon}c_{\beta}\varpi_{\mathbf{F}}^{n-n_{0}}+y\varpi_{\mathbf{F}}^{n}(\alpha-\bar{\alpha}))(u-\bar{u})\right)\mathrm{d}u\right\}\mathrm{d}\alpha.

The non-vanishing of the inner integral implies

\left(2\sqrt{\varepsilon}c_{\beta}\varpi_{\mathbf{F}}^{n-n_{0}}+y\varpi_{\mathbf{F}}^{n}(\alpha-\bar{\alpha})\right)\sqrt{\varepsilon}\in\mathcal{O}_{\mathbf{F}}\ \Rightarrow\\ (\alpha-\bar{\alpha})\sqrt{\varepsilon}\in\begin{cases}\varpi_{\mathbf{F}}^{m-n_{0}}\mathcal{O}_{\mathbf{F}}^{\times}&\text{if }m<2n_{0}-1\\ \mathcal{P}_{\mathbf{F}}^{n_{0}-1}&\text{if }m=2n_{0}-1\end{cases}\Rightarrow\ (\alpha+\bar{\alpha})^{2}\in\begin{cases}4+\mathcal{P}_{\mathbf{F}}^{2(n_{0}-1)}&\text{if }m=2n_{0}-1\\ 4+\varpi_{\mathbf{F}}^{2(m-n_{0})}\mathcal{O}_{\mathbf{F}}^{\times}&\text{if }n_{0}<m<2n_{0}-1\\ \mathcal{O}_{\mathbf{F}}-4(1+\mathcal{P}_{\mathbf{F}})&\text{if }m=n_{0}\end{cases}

from which we conclude. Replacing $\displaystyle\sqrt{\varepsilon}$ by $\displaystyle 1$ we obtain the proof in the split case.

(5) In the case $\displaystyle\tau=1$ and $\displaystyle\tfrac{n_{0}}{2}+1\leqslant m\leqslant n_{0}-1$ , we average over the change of variables $\displaystyle\alpha\mapsto\alpha(1+u)(1+\bar{u})^{-1}$ for $\displaystyle u\in\mathcal{P}_{\mathbf{L}}^{m}$ , noting that $\displaystyle\alpha\tfrac{u-\bar{u}}{1+\bar{u}}\in\alpha(u-\bar{u})+y^{-1}\mathcal{O}_{\mathbf{L}}$ , and get

\int_{\mathbf{L}^{1}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\mathrm{d}\alpha=\int_{\mathbf{L}^{1}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\left\{\oint_{\mathcal{P}_{\mathbf{L}}^{m}}\psi\left(\tfrac{2c_{\beta}(u-\bar{u})}{\varpi_{\mathbf{L}}^{n_{0}+1}}+y{\rm Tr}\left(\alpha\cdot\tfrac{u-\bar{u}}{1+\bar{u}}\right)\right)\mathrm{d}u\right\}\mathrm{d}\alpha\\ =\int_{\mathbf{L}^{1}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\left\{\oint_{\mathcal{P}_{\mathbf{L}}^{m}}\psi\left(\left(\tfrac{2c_{\beta}}{\varpi_{\mathbf{L}}^{n_{0}+1}}+(\alpha-\bar{\alpha})y\right)(u-\bar{u})\right)\mathrm{d}u\right\}\mathrm{d}\alpha\\ =\int_{\mathbf{L}^{1}}\beta(\alpha)\psi(y{\rm Tr}(\alpha))\left\{\oint_{\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{m}{2}\rfloor}}\psi\left(\left(\tfrac{2c_{\beta}}{\varpi_{\mathbf{F}}^{n_{0}/2}}+\varpi_{\mathbf{L}}(\alpha-\bar{\alpha})y\right)u\right)\mathrm{d}u\right\}\mathrm{d}\alpha.

The non-vanishing of the inner integral implies

\displaystyle 2c_{\beta}\varpi_{\mathbf{F}}^{-\frac{n_{0}}{2}}+\varpi_{\mathbf{L}}(\alpha-\bar{\alpha})y\in\mathcal{P}_{\mathbf{F}}^{-\lfloor\frac{m}{2}\rfloor}\ \Rightarrow\ \begin{cases}(\alpha-\bar{\alpha})\varpi_{\mathbf{L}}^{-1}\in\varpi_{\mathbf{F}}^{m-\frac{n_{0}}{2}-1}\mathcal{O}_{\mathbf{F}}^{\times}&\text{if }m<n_{0}\\ (\alpha-\bar{\alpha})\varpi_{\mathbf{L}}^{-1}\in\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-1}\mathcal{O}_{\mathbf{F}}&\text{if }m=n_{0}\end{cases}.

We conclude by noting that for any $\displaystyle m\in\mathbb{Z}_{\geqslant 0}$ we have

\displaystyle(\alpha-\bar{\alpha})\varpi_{\mathbf{L}}^{-1}\in\mathcal{P}_{\mathbf{F}}^{m}\ \Leftrightarrow\ {\rm Tr}(\alpha)\in\pm 2(1+\mathcal{P}_{\mathbf{F}}^{2m+1}),

and that $\displaystyle{\rm Tr}(\alpha)^{2}-4=(\alpha-\bar{\alpha})^{2}$ never has even valuation. ∎

Remark 3.11.

If $\displaystyle\beta$ is not regular, which can happen only if $\displaystyle\mathbf{L}/\mathbf{F}$ is split with $\displaystyle\mathfrak{c}(\chi_{0}^{2})=0$ , then (2) fails. In fact, it is easy to see that $\displaystyle H(\tau y^{2})$ is proportional to $\displaystyle\chi_{0}(\tau)\lvert y\rvert$ as $\displaystyle\lvert y\rvert\to 0$ . In this case we modify the definition of $\displaystyle H$ by truncating it so that (2) still holds.

Corollary 3.12.

The test function $\displaystyle H$ defined by (3.5) and Remark 3.11 is a Bessel orbital integral.

Proof.

This is a direct consequence of Lemma 3.10 (1) & (2) and Proposition 2.1. ∎

3.2. Weight Functions

To every target representation $\displaystyle\pi_{0}$ we have associated a parameter $\displaystyle(\mathbf{L},\beta)$ and a test function $\displaystyle H$ in (3.5).

Lemma 3.13.

(1) In the split case $\displaystyle\pi_{0}=\pi(\chi_{0},\chi_{0}^{-1})$ consider the function $\displaystyle\phi_{0}\in{\rm C}_{c}^{\infty}({\rm PGL}_{2}(\mathbf{F}))$ given by

\displaystyle\phi_{0}\begin{pmatrix}x_{1}&x_{2}\\ x_{3}&x_{4}\end{pmatrix}=\chi_{0}\left(\frac{x_{4}}{x_{1}}\right)\mathbbm{1}_{\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]}\begin{pmatrix}x_{1}&x_{2}\\ x_{3}&x_{4}\end{pmatrix}.

The partial integral $\displaystyle I_{0}$ defined by

\displaystyle I_{0}(g):=\int_{\mathbf{F}}\phi_{0}\left(g\begin{pmatrix}1&x\\ &1\end{pmatrix}\right)\psi(-x)\mathrm{d}x

has support contained in $\displaystyle\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]\mathbf{N}(\mathbf{F})$ , and satisfies

\displaystyle I_{0}(\kappa n(x))=\phi_{0}(\kappa)\psi(x),\quad\forall\kappa\in\mathbf{Z}\mathbf{K_{0}},x\in\mathbf{F}.

(2) We can identify $\displaystyle H$ as a Bessel orbital integral

\displaystyle H(y)=\int_{\mathbf{F}^{2}}\phi_{0}\left(\begin{pmatrix}1&x_{1}\\ &1\end{pmatrix}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&x_{2}\\ &1\end{pmatrix}\right)\psi(-x_{1}-x_{2})\mathrm{d}x_{1}\mathrm{d}x_{2}.

(3) The weight function $\displaystyle h(\pi)$ is non-negative. We have $\displaystyle h(\pi)\neq 0$ only if $\displaystyle\mathfrak{c}(\pi\otimes\chi_{0}^{-1})\leqslant n_{0}$ , in which case we have a lower bound $\displaystyle h(\pi)\gg q^{-n_{0}}$ .

Proof.

(1) Note that $\displaystyle\phi_{0}$ is a character upon restriction to $\displaystyle\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]$ . Hence $\displaystyle I_{0}(g)$ satisfies

\displaystyle I_{0}(\kappa gn(x))=\phi_{0}(\kappa)\psi(x)I_{0}(g),\quad\forall\kappa\in\mathbf{Z}\mathbf{K_{0}},x\in\mathbf{F}.

From the Cartan decomposition $\displaystyle\sideset{}{{}_{n\in\mathbb{Z}_{\geqslant 0}}}{\bigsqcup}\mathbf{Z}\mathbf{K}\begin{pmatrix}\varpi_{\mathbf{F}}^{n}&\\ &1\end{pmatrix}\mathbf{N}(\mathbf{F})$ we deduce $\displaystyle\mathrm{supp}(I_{0})\subset\mathbf{Z}\mathbf{K}\mathbf{N}(\mathbf{F})$ . We have

\displaystyle\mathbf{Z}\mathbf{K}\mathbf{N}(\mathbf{F})=\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]\begin{pmatrix}&1\\ 1&\end{pmatrix}\mathbf{N}(\mathbf{F})\sqcup\sideset{}{{}_{u\in\mathcal{P}_{\mathbf{F}}/\mathcal{P}_{\mathbf{F}}^{n_{0}}}}{\bigcup}\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]\begin{pmatrix}1&\\ u&1\end{pmatrix}\mathbf{N}(\mathbf{F})

by the Bruhat decomposition over the residue field of $\displaystyle\mathbf{F}$ . We easily verify for $\displaystyle u\notin\mathcal{P}_{\mathbf{F}}^{n_{0}}$

\displaystyle\begin{pmatrix}&1\\ 1&\end{pmatrix}\mathbf{N}(\mathbf{F})\cap\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]=\emptyset,\quad\begin{pmatrix}1&\\ u&1\end{pmatrix}\mathbf{N}(\mathbf{F})\cap\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]=\emptyset.

Hence we deduce $\displaystyle\mathrm{supp}(I_{0})\subset\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]\mathbf{N}(\mathbf{F})$ and conclude by $\displaystyle I_{0}(\mathbbm{1})={\rm Vol}(\mathcal{O}_{\mathbf{F}})=1$ .

(2) Let $\displaystyle\phi^{\iota}(g):=\overline{\phi(g^{-1})}$ for all $\displaystyle\phi\in{\rm C}_{c}^{\infty}({\rm PGL}_{2}(\mathbf{F}))$ . We have $\displaystyle\phi_{0}^{\iota}=\phi_{0}$ and $\displaystyle\phi_{0}^{\iota}*\phi_{0}={\rm Vol}(\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}])\phi_{0}$ since $\displaystyle\phi_{0}$ is a unitary character upon restriction to $\displaystyle\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}]$ . Here the convolution is taken over the group $\displaystyle G:={\rm PGL}_{2}(\mathbf{F})$ . We can rewrite the relevant Bessel orbital integral as

{\rm Vol}(\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}])^{-1}\int_{\mathbf{F}^{2}}\int_{G}\overline{\phi_{0}(g)}\phi_{0}\left(g\begin{pmatrix}1&x_{1}\\ &1\end{pmatrix}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&x_{2}\\ &1\end{pmatrix}\right)\psi(-x_{1}-x_{2})\mathrm{d}g\mathrm{d}x_{1}\mathrm{d}x_{2}\\ ={\rm Vol}(\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}])^{-1}\int_{G}\\ \left\{\int_{\mathbf{F}}\overline{\phi_{0}\left(g\begin{pmatrix}1&-x_{1}\\ &1\end{pmatrix}\right)}\psi(-x_{1})\mathrm{d}x_{1}\right\}\cdot\left\{\int_{\mathbf{F}}\phi_{0}\left(g\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&x_{2}\\ &1\end{pmatrix}\right)\psi(-x_{2})\mathrm{d}x_{2}\right\}\mathrm{d}g\\ ={\rm Vol}(\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}])^{-1}\int_{G}\overline{I_{0}(g)}I_{0}\left(g\begin{pmatrix}&-y\\ 1&\end{pmatrix}\right)\mathrm{d}g=\int_{\mathbf{F}}\psi(-u)I_{0}\left(\begin{pmatrix}1&u\\ &1\end{pmatrix}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\right)\mathrm{d}u.

Considering the support of $\displaystyle I_{0}$ , the above integral is non-vanishing only if for some $\displaystyle u^{\prime}\in\mathbf{F}$

\displaystyle\begin{pmatrix}1&u\\ &1\end{pmatrix}\begin{pmatrix}&-y\\ 1&\end{pmatrix}\begin{pmatrix}1&u^{\prime}\\ &1\end{pmatrix}=\begin{pmatrix}u&uu^{\prime}-y\\ 1&u^{\prime}\end{pmatrix}\in\mathbf{Z}\mathbf{K}_{0}[\mathcal{P}_{\mathbf{F}}^{n_{0}}],

implying $\displaystyle y\in\varpi_{\mathbf{F}}^{-2n}\mathcal{O}_{\mathbf{F}}^{\times}$ for some $\displaystyle n\geqslant n_{0}$ , and $\displaystyle u=\varpi^{-n}u_{1}$ , $\displaystyle u^{\prime}=\varpi^{-n}u_{2}$ for some $\displaystyle u_{1},u_{2}\in\mathcal{O}_{\mathbf{F}}^{\times}$ satisfying $\displaystyle u_{1}u_{2}-\varpi^{2n}y\in\mathcal{P}_{\mathbf{F}}^{n}$ . Writing $\displaystyle y=\varpi_{\mathbf{F}}^{-2n}y_{0}$ for some $\displaystyle y_{0}\in\mathcal{O}_{\mathbf{F}}^{\times}$ we may take $\displaystyle u_{2}=u_{1}^{-1}y_{0}$ . We recognize the Bessel orbital integral as $\displaystyle H(y)$ by the following equation, which is easy to verify

\displaystyle q^{n}\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}(u_{1}^{-2}y_{0})\psi\left(-\tfrac{u_{1}+u_{1}^{-1}y_{0}}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}u_{1}=H(y).

(3) Up to notation this is due to Nelson [13, Theorem 3.1]. Our former work [1, Lemma 4.1] contains more details in a similar situation. ∎

Lemma 3.14.

Let $\displaystyle\pi$ be a supercuspidal representation of $\displaystyle{\rm GL}_{2}(\mathbf{F})$ . Let $\displaystyle C(g)$ be a matrix coefficient for smooth vectors in $\displaystyle\pi$ . Take a non-trivial additive character $\displaystyle\psi$ of $\displaystyle\mathbf{F}$ . Then the (relative) orbital integral

\displaystyle I(g):=\int_{\mathbf{F}^{2}}C(n(u_{1})gn(u_{2}))\psi(-u_{1}-u_{2})du_{1}du_{2}

is equal to the $\displaystyle\psi$ -Bessel function of $\displaystyle\pi$ up to a constant.

Proof.

Assume $\displaystyle C(g)=\langle\pi(g)v_{1},v_{2}\rangle$ for smooth vectors $\displaystyle v_{1}$ and $\displaystyle v_{2}$ . One checks that the functional

\displaystyle V_{\pi}^{\infty}\to\mathbb{C},\quad v\mapsto\int_{\mathbf{F}}\langle\pi(n(u_{1})).v,v_{2}\rangle\psi(-u_{1})du_{1}

is a $\displaystyle\psi$ -Whittaker functional. Hence the following function

\displaystyle W_{v_{1}}(g):=\int_{\mathbf{F}}C(n(u_{1})g)\psi(-u_{1})du_{1}

is the/a Whittaker function of $\displaystyle v_{1}$ . We then apply [17, Lemma 4.1] to get

\displaystyle I(g)=\mathrm{j}_{\pi,\psi}(g)\cdot W_{v_{1}}(\mathbbm{1}),\quad\forall g\in\mathbf{B}(\mathbf{F})w_{2}\mathbf{B}(\mathbf{F}).

The integral converges absolutely because $\displaystyle C$ is smooth of compact support modulo the center. ∎

Lemma 3.15.

In the dihedral case $\displaystyle\pi_{0}=\pi_{\beta}$ the weight function $\displaystyle h(\pi)\neq 0$ is non-vanishing only if $\displaystyle\pi\simeq\pi_{\beta}$ , in which case it is positive and satisfies a lower bound $\displaystyle h(\pi_{\beta})\gg q^{-e^{-1}n_{0}+1-e}$ .

Proof.

Up to some constant approximately equal to $\displaystyle 1$ we recognize $\displaystyle H(y)$ as $\displaystyle\mathrm{j}_{\pi_{\beta},\psi}\begin{pmatrix}&-y\\ 1&\end{pmatrix}$ by [3, Theorem 1.1]. By Lemma 3.14 there exists some $\displaystyle\phi_{0}\in{\rm C}_{c}^{\infty}({\rm GL}_{2}(\mathbf{F}))$ , whose integral along the center $\displaystyle\phi_{1}\in{\rm C}_{c}^{\infty}({\rm PGL}_{2}(\mathbf{F}))$ is a matrix coefficient of $\displaystyle\pi_{\beta}$ , so that $\displaystyle H$ is the Bessel orbital integral of $\displaystyle\phi_{0}$ as given in Lemma 3.13 (1). Now that $\displaystyle\pi(\phi_{1})\neq 0$ only if $\displaystyle\pi\simeq\pi_{\beta}$ by Schur’s lemma, it remains to show that $\displaystyle h(\pi_{\beta})$ is positive with the stated lower bound. By (1.5) the value $\displaystyle h(\pi_{\beta})$ is the square of the $\displaystyle{\rm L}^{2}$ -norm of $\displaystyle H(y)$ up to a constant essentially equal to $\displaystyle 1$ . Taking Lemma 3.10 (2) into account we have

	$\displaystyle\displaystyle\int_{\mathbf{F}^{\times}}\left\lvert H(y)\right\rvert^{2}\tfrac{\mathrm{d}^{\times}y}{\lvert y\rvert_{\mathbf{F}}}$	$\displaystyle\displaystyle=\frac{1}{2}\sum_{\tau}\int_{\mathbf{F}^{\times}}\left\lvert H(\tau y^{2})\right\rvert^{2}\tfrac{\mathrm{d}^{\times}y}{\lvert\tau y^{2}\rvert_{\mathbf{F}}}$
		$\displaystyle\displaystyle=\frac{1}{2}\sum_{\tau}\int_{\mathbf{F}^{\times}}\left(\int_{\mathbf{L}^{1}\times\mathbf{L}^{1}}\beta(x_{\tau}\alpha_{1})\overline{\beta(x_{\tau}\alpha_{2})}\psi_{\mathbf{L}}(x_{\tau}(\alpha_{1}-\alpha_{2})y)\mathrm{d}\alpha_{1}\mathrm{d}\alpha_{2}\right)\mathrm{d}^{\times}y$
		$\displaystyle\displaystyle=\frac{1}{2}\sum_{n\geqslant(n_{0}-2)e^{-1}+2}\int_{\mathbf{L}^{1}}\beta(\alpha)\left(\sum_{\tau}\int_{\mathbf{L}^{1}}\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}\psi_{\mathbf{L}}(x_{\tau}\alpha_{2}y(\alpha-1))\mathrm{d}^{\times}y\mathrm{d}\alpha_{2}\right)\mathrm{d}\alpha,$

where we applied the change of variables $\displaystyle\alpha_{1}\mapsto\alpha\alpha_{2}$ in the last line. By measure relation (3.3) we get

\displaystyle\frac{1}{2}\sum_{\tau}\int_{\mathbf{L}^{1}}\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}\psi_{\mathbf{L}}(x_{\tau}\alpha_{2}y(\alpha-1))\mathrm{d}^{\times}y\mathrm{d}\alpha_{2}=\zeta_{\mathbf{F}}(1)\int_{-en\leqslant v_{\mathbf{L}}(z)<e-en}\psi_{\mathbf{L}}(z(\alpha-1))\frac{\mathrm{d}z}{\lvert z\rvert_{\mathbf{L}}}.

Therefore we continue the calculation as

	$\displaystyle\displaystyle\int_{\mathbf{F}^{\times}}\left\lvert H(y)\right\rvert^{2}\tfrac{\mathrm{d}^{\times}y}{\lvert y\rvert_{\mathbf{F}}}$	$\displaystyle\displaystyle=\zeta_{\mathbf{F}}(1)\sum_{n\geqslant n_{0}-1+e}\int_{\mathbf{L}^{1}}\beta(\alpha)\left(\int_{\varpi_{\mathbf{L}}^{-n}\mathcal{O}_{\mathbf{L}}^{\times}}\psi_{\mathbf{L}}(z(\alpha-1))\tfrac{\mathrm{d}z}{\lvert z\rvert_{\mathbf{L}}}\right)\mathrm{d}\alpha$
		$\displaystyle\displaystyle=\tfrac{\zeta_{\mathbf{F}}(1)}{\zeta_{\mathbf{L}}(1)}q_{\mathbf{L}}^{-\frac{e-1}{2}}\sum_{n\geqslant n_{0}-1+e}\int_{\mathbf{L}^{1}}\beta(\alpha)\left(\mathbbm{1}_{\mathcal{P}_{\mathbf{L}}^{n+1-e}}(\alpha-1)-q_{\mathbf{L}}^{-1}\mathbbm{1}_{\mathcal{P}_{\mathbf{L}}^{n-e}}(\alpha-1)\right)\mathrm{d}\alpha$
		$\displaystyle\displaystyle=\tfrac{\zeta_{\mathbf{F}}(1)}{\zeta_{\mathbf{L}}(1)^{2}}q_{\mathbf{L}}^{-\frac{e-1}{2}}\sum_{n\geqslant n_{0}}{\rm Vol}(\mathbf{L}^{1}\cap(1+\mathcal{P}_{\mathbf{L}}^{n})),$

and conclude the lower bound by Proposition 3.5 (3) and the fact $\displaystyle e\mid n_{0}$ (see Claim in Lemma 3.3). ∎

Remark 3.16.

By (3.2), the bounds in Lemma 3.13 (1) & 3.15 can be rewritten as $\displaystyle h(\pi_{0})\gg q^{-\lceil\frac{\mathfrak{c}(\pi_{0})}{2}\rceil}$ .

4. Dual Weight Functions: Common Reductions

4.1. Infinite Part

Recall the constant $\displaystyle a(\Pi)$ for the stability range defined in Proposition 2.4. Take the “infinite part” of the test function as $\displaystyle H_{\infty}=E_{\geqslant n_{1}}$ for some

(4.1)

n_{1}\geqslant\max(2n_{0}/e+e-1,a(\Pi))(\geqslant 2).

Recall the formula for the dual weight function

(4.2)

\widetilde{h}(\chi)=\int_{\mathbf{F}^{\times}}\widetilde{\mathcal{VH}}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t.

Lemma 4.1.

The partial dual weight function

\displaystyle\widetilde{h}_{\infty}(\chi):=\int_{\mathbf{F}^{\times}}\widetilde{\mathcal{VH}}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{\infty})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t

is non-vanishing only if $\displaystyle\chi$ is unramified, in which case we have

\displaystyle\widetilde{h}_{\infty}(\chi)=\frac{\chi(\varpi_{\mathbf{F}})^{n_{1}}q^{-\frac{n_{1}}{2}}}{1-\chi(\varpi_{\mathbf{F}})q^{-\frac{1}{2}}}.

Proof.

By Proposition 2.4 we have

\displaystyle\widetilde{\mathcal{VH}}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(E_{\geqslant n_{1}})(t)=\psi(t)\cdot\mathbbm{1}_{\mathcal{P}_{\mathbf{F}}^{n_{1}}}(t^{-1}).

The desired formula follows readily from the above one. ∎

Taking into account Lemma 3.10 (2) we introduce the “finite part” of the test function as

(4.3)

H_{c}:=H-H_{\infty}=\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}H_{n},\qquad H_{n}:=H\cdot\mathbbm{1}_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}.

Since each function $\displaystyle H_{n}\in{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})$ , we may replace the extended Voronoi–Hankel transform with its original version in the dual weight formula and turn to the study of

(4.4)

\widetilde{h}_{n}(\chi):=\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{n})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t.

We divide the domain of integration into two parts: $\displaystyle\mathcal{O}_{\mathbf{F}}-\{0\}$ and $\displaystyle\mathbf{F}-\mathcal{O}_{\mathbf{F}}$ , giving the decomposition

(4.5)

\widetilde{h}_{n}(\chi)=\widetilde{h}_{n}^{+}(\chi)+\widetilde{h}_{n}^{-}(\chi).

Accordingly the dual weight functions (see (1.1)) have the following decomposition

(4.6)

\widetilde{h}(\chi)=\widetilde{h}_{\infty}(\chi)+\widetilde{h}_{c}(\chi),\quad\widetilde{h}_{c}(\chi)=\widetilde{h}_{c}^{+}(\chi)+\widetilde{h}_{c}^{-}(\chi),\quad\widetilde{h}_{c}^{\pm}(\chi)=\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}\widetilde{h}_{n}^{\pm}(\chi),

(4.7)

\widetilde{H}(\chi)=\widetilde{H}_{\infty}(\chi)+\widetilde{H}_{c}(\chi),\quad\widetilde{H}_{c}(\chi)=\widetilde{H}_{c}^{+}(\chi)+\widetilde{H}_{c}^{-}(\chi),\quad\widetilde{H}_{c}^{\pm}(\chi)=\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}\widetilde{H}_{n}^{\pm}(\chi).

We call those (partial) dual weight functions with “ $\displaystyle+$ ” (resp. “ $\displaystyle-$ ”) the postive (resp. negative) part.

4.2. Positive Part

We first study $\displaystyle\widetilde{h}_{n}^{+}(\chi)$ . Since $\displaystyle\psi$ is trivial on $\displaystyle\mathcal{O}_{\mathbf{F}}$ , we have $\displaystyle\widetilde{h}_{n}^{+}(\chi)=\widetilde{h}_{n}(1/2,\chi)$ for

(4.8)

\widetilde{h}_{n}(s,\chi):=\int_{\mathcal{O}_{\mathbf{F}}-\{0\}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{n})(t)\cdot\chi^{-1}(t)\lvert t\rvert^{-s}\mathrm{d}^{\times}t,

which is simply the partial sum of non-negative powers of $\displaystyle X=q^{s}$ in the Laurent series expansion of

(4.9)

f_{n}(q^{s};\chi,H)=\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{n})(t)\cdot\chi^{-1}(t)\lvert t\rvert^{-s}\mathrm{d}^{\times}t.

By the local functional equation we have (see Lemma 3.6)

(4.10)

f_{n}(q^{s};\chi,H)=\varepsilon_{n}(\chi,H)q^{n(s-1)}\cdot\varepsilon\left(s,\Pi\otimes\chi,\psi\right)\frac{L(1-s,\widetilde{\Pi}\otimes\chi^{-1})}{L(s,\Pi\otimes\chi)}.

To relate $\displaystyle f_{n}(q^{s};\chi,H)$ with $\displaystyle\widetilde{h}_{n}(s,\chi)$ we need the following crucial lemma.

Lemma 4.2.

Let $\displaystyle f(X)=\sideset{}{{}_{n>-\infty}}{\sum}a_{n}X^{n}$ be a Laurent series converging in $\displaystyle 0<\lvert X\rvert<\rho$ for some $\displaystyle\rho>1$ . Let $\displaystyle f_{+}(X)=\sideset{}{{}_{n>-\infty}}{\sum}a_{n}X^{n}$ . Let $\displaystyle D=X\tfrac{\mathrm{d}}{\mathrm{d}X}$ . Assume $\displaystyle f(X)$ (hence $\displaystyle f_{+}(X)$ ) has a meromorphic continuation to $\displaystyle X\in\mathbb{C}$ .

(0)

For any $\displaystyle X$ and any $\displaystyle\epsilon<\min(1,\rho\lvert X\rvert^{-1})$ we have the relation

$\displaystyle f_{+}(X)=f(X)-\int_{\lvert z\rvert=\epsilon}\frac{f(Xz)}{1-z}\frac{\mathrm{d}z}{2\pi i}.$

(1)

For $\displaystyle 0<\lvert X\rvert<\rho$ and any $\displaystyle 1<r<\rho\lvert X\rvert^{-1}$ we have the relation

\displaystyle f_{+}(X)=\int_{\lvert z\rvert=r}\frac{f(Xz)}{z-1}\frac{\mathrm{d}z}{2\pi i}=f(X)+\int_{\lvert z\rvert=1}\frac{f(Xz)-f(X)}{z-1}\frac{\mathrm{d}z}{2\pi i}.

Consequently, for $\displaystyle 0<\lvert X\rvert<\rho$ and $\displaystyle n\in\mathbb{Z}_{\geqslant 0}$ we have the bound

\displaystyle\left\lvert(D^{n}f_{+})(X)\right\rvert\ll\left\lvert(D^{n}f)(X)\right\rvert+\sideset{}{{}_{\lvert z\rvert=1}}{\sup}\left\lvert(D^{n+1}f)(Xz)\right\rvert.

(2)

Suppose $\displaystyle f(X)=Q(X)P(X)^{-1}$ for some $\displaystyle Q\in\mathbb{C}[X,X^{-1}],P\in\mathbb{C}[X]$ . Let

\displaystyle P(X)=\sideset{}{{}_{j=1}^{r}}{\prod}(1-b_{j}X)^{m_{j}}

with $\displaystyle m_{j}\in\mathbb{Z}_{\geqslant 1}$ and distinct $\displaystyle b_{j}\neq 0$ . Introduce

\displaystyle P_{j}(X):=\sideset{}{{}_{i\neq j}}{\prod}(1-b_{i}X)^{m_{i}},\quad C_{j,k}:=\frac{(-1)^{k}}{k!\cdot b_{j}^{k}}\left(\frac{Q}{P_{j}}\right)^{(k)}(b_{j}^{-1}).

Suppose the highest power $\displaystyle X^{m}$ in $\displaystyle Q$ satisfies $\displaystyle m<\deg P$ . Then we have

\displaystyle P(X)f_{+}(X)=\sideset{}{{}_{j=1}^{r}}{\sum}\sideset{}{{}_{k=0}^{m_{j}-1}}{\sum}C_{j,k}\cdot(1-b_{j}X)^{k}P_{j}(X).

Proof.

(0) The stated formula follows from the residue theorem via

\displaystyle\int_{\lvert z\rvert=\epsilon}\frac{f(Xz)}{1-z}\frac{\mathrm{d}z}{2\pi i}=\int_{\lvert z\rvert=\epsilon}\left(\sum_{n>-\infty}a_{n}X^{n}z^{n}\right)\left(\sum_{k=0}^{\infty}z^{k}\right)\frac{\mathrm{d}z}{2\pi i}=\sum_{n<0}a_{n}X^{n},

since only the terms for $\displaystyle n+k=-1$ give non-zero contribution.

(1) Clearly the stated formula follows from the one in (0) by the residue theorem. To see the bound, we introduce $\displaystyle g(z)=f(Xz)$ and rewrite the integral as

\displaystyle\int_{\lvert z\rvert=1}\frac{f(Xz)-f(X)}{z-1}\frac{\mathrm{d}z}{2\pi i}=\int_{-\pi}^{\pi}\frac{g(e^{i\theta})-g(1)}{e^{i\theta}-1}\frac{\mathrm{d}\theta}{2\pi}=\int_{-\pi}^{\pi}\frac{i}{e^{i\theta}-1}\left(\int_{0}^{\theta}g^{\prime}(e^{it})e^{it}\mathrm{d}t\right)\frac{\mathrm{d}\theta}{2\pi}.

Note that $\displaystyle zg^{\prime}(z)=Df(Xz)$ . Therefore we obtain the formula

\displaystyle f_{+}(X)=f(X)+\int_{-\pi}^{\pi}\frac{i}{e^{i\theta}-1}\left(\int_{0}^{\theta}(Df)(Xe^{it})\mathrm{d}t\right)\frac{\mathrm{d}\theta}{2\pi}.

Applying $\displaystyle D^{n}$ on both sides we get a formula and conclude the stated bound as

(D^{n}f_{+})(X)=(D^{n}f)(X)+\int_{-\pi}^{\pi}\frac{i}{e^{i\theta}-1}\left(\int_{0}^{\theta}(D^{n+1}f)(Xe^{it})\mathrm{d}t\right)\frac{\mathrm{d}\theta}{2\pi}\\ \ll\left\lvert(D^{n}f)(X)\right\rvert+\sideset{}{{}_{\lvert z\rvert=1}}{\sup}\left\lvert(D^{n+1}f)(Xz)\right\rvert\cdot\int_{-\pi}^{\pi}\left\lvert\frac{\theta}{e^{i\theta}-1}\right\rvert\frac{\mathrm{d}\theta}{2\pi}.

(2) We may assume $\displaystyle 0<\lvert X\rvert<\rho$ and depart from the first equation in (1), then move the contour to $\displaystyle\lvert z\rvert=r$ for $\displaystyle r\gg 1$ , picking up the residues. The contour integral tends to $\displaystyle 0$ as $\displaystyle r\to+\infty$ due to the assumption $\displaystyle m<\deg P$ . Multiplying the resulted formula by $\displaystyle P(X)$ , we get the stated formula. ∎

Definition 4.3.

For any generic irreducible representation $\displaystyle\Pi$ of $\displaystyle{\rm GL}_{r}(\mathbf{F})$ , let $\displaystyle d(\Pi)$ be the degree of the $\displaystyle L$ -function $\displaystyle L(s,\Pi)$ , namely the degree of the polynomial $\displaystyle L(s,\Pi)^{-1}$ in $\displaystyle X:=q^{-s}$ . Write

\displaystyle\mathfrak{\rho}(\Pi):=\mathfrak{c}(\Pi)+d(\Pi).

Introduce the set of exponents of $\displaystyle\Pi$ by

\displaystyle\mathrm{E}(\Pi):=\left\{\xi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\ \middle|\ d(\Pi\otimes\xi)>0\right\}.

Lemma 4.4.

Let $\displaystyle\Pi$ be a generic irreducible representation of $\displaystyle{\rm GL}_{3}(\mathbf{F})$ . We have $\displaystyle\mathfrak{\rho}(\Pi)\geqslant 3$ , $\displaystyle\left\lvert\mathrm{E}(\Pi)\right\rvert\leqslant 3$ . For any $\displaystyle\xi\in\mathrm{E}(\Pi)$ we have the bounds

\displaystyle\mathfrak{c}(\xi)\leqslant\mathfrak{c}(\Pi),\quad\mathfrak{c}(\Pi\otimes\xi)\leqslant 2\mathfrak{c}(\Pi),\quad\mathfrak{\rho}(\Pi\otimes\xi)\leqslant 2\mathfrak{c}(\Pi)+3.

Proof.

The upper bound for $\displaystyle\mathfrak{\rho}(\Pi\otimes\xi)$ obviously follows from the one for $\displaystyle\mathfrak{c}(\Pi\otimes\xi)$ since $\displaystyle d(\Pi\otimes\xi)\leqslant 3$ in any case. For the other bounds we distinguish cases for $\displaystyle\Pi$ .

(1) If $\displaystyle\mathfrak{c}(\Pi)=0$ , then $\displaystyle\Pi$ is spherical, $\displaystyle\mathrm{E}(\Pi)=\{\mathbbm{1}\}$ and $\displaystyle d(\Pi)=3$ . The stated bounds clearly hold.

(2) If $\displaystyle\mathfrak{c}(\Pi)>0$ and if $\displaystyle\Pi=\chi_{1}\boxplus\chi_{2}\boxplus\chi_{3}$ is induced from the Borel subgroup, then we have

\displaystyle\mathfrak{\rho}(\Pi)=\sideset{}{{}_{i=1}^{3}}{\sum}\mathfrak{\rho}(\chi_{i})=\sideset{}{{}_{i=1}^{3}}{\sum}\max(\mathfrak{c}(\chi_{i}),1)\geqslant 3.

We also have $\displaystyle\mathrm{E}(\Pi)\subset\{\xi_{1}^{-1},\xi_{2}^{-1},\xi_{3}^{-1}\}$ with $\displaystyle\chi_{j}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}=\xi_{j}$ , from which we deduce the bound for $\displaystyle\left\lvert\mathrm{E}(\Pi)\right\rvert$ . Obviously we have $\displaystyle\mathfrak{c}(\xi_{j})\leqslant\mathfrak{c}(\Pi)$ . Taking $\displaystyle\xi=\xi_{1}^{-1}$ for example, we have the second stated bound as

\displaystyle\mathfrak{c}(\Pi\otimes\xi)=\mathfrak{c}(\chi_{2}\xi_{1}^{-1})+\mathfrak{c}(\chi_{3}\xi_{1}^{-1})\leqslant 2\max\{\mathfrak{c}(\chi_{j}):1\leqslant j\leqslant 3\}\leqslant 2\mathfrak{c}(\Pi).

(3) If $\displaystyle\mathfrak{c}(\Pi)>0$ and if $\displaystyle\Pi=\pi\boxplus\chi$ for some supercuspidal $\displaystyle\pi$ of $\displaystyle{\rm GL}_{2}(\mathbf{F})$ , then the central character of $\displaystyle\pi$ is $\displaystyle\chi^{-1}$ . We have $\displaystyle\mathfrak{\rho}(\Pi)=\mathfrak{c}(\pi)+\max(\mathfrak{c}(\chi),1)\geqslant 2+1=3$ . We also have $\displaystyle\mathrm{E}(\Pi)=\{\chi^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}\}$ . For $\displaystyle\xi=\chi^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ we have $\displaystyle\mathfrak{c}(\xi)=\mathfrak{c}(\chi)\leqslant\mathfrak{c}(\Pi)$ and get the second stated bound via $\displaystyle\pi\otimes\chi^{-1}\simeq\widetilde{\pi}$ as

\displaystyle\mathfrak{c}(\Pi\otimes\xi)=\mathfrak{c}(\pi\otimes\chi^{-1})=\mathfrak{c}(\widetilde{\pi})=\mathfrak{c}(\pi)\leqslant\mathfrak{c}(\Pi).

(4) If $\displaystyle\mathfrak{c}(\Pi)>0$ and if $\displaystyle\Pi=\mathrm{St}_{\eta}\boxplus\chi$ , then $\displaystyle\chi\eta^{2}=\mathbbm{1}$ . By [15, §8 Proposition & §10 Proposition] we have

\displaystyle\mathfrak{\rho}(\Pi)=\begin{cases}2\mathfrak{c}(\eta)&\text{if }\mathfrak{c}(\eta)\geqslant 1\\ 1&\text{if }\mathfrak{c}(\eta)=0\end{cases}+d(\eta)+\max(\mathfrak{c}(\chi),1)\geqslant 3.

We also have $\displaystyle\mathrm{E}(\Pi)=\{\chi^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}},\eta^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}\}$ . For $\displaystyle\xi=\chi^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ we argue the same as in the proof of (3). For $\displaystyle\xi=\eta^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ we have

\displaystyle\mathfrak{c}(\xi)=\mathfrak{c}(\eta)<1+\mathfrak{c}(\eta)=\mathfrak{c}(\Pi\otimes\xi)\leqslant\max(2\mathfrak{c}(\eta),1)+\mathfrak{c}(\chi)=\mathfrak{c}(\Pi).

(5) If $\displaystyle\mathfrak{c}(\Pi)>0$ and if $\displaystyle\Pi=\mathrm{St}_{\eta}$ is a twist of the Steinberg representation of $\displaystyle{\rm PGL}_{3}(\mathbf{F})$ , then by [15, §8 Proposition & §10 Proposition] we have

\displaystyle\mathfrak{\rho}(\Pi)=\begin{cases}3\mathfrak{c}(\eta)&\text{if }\mathfrak{c}(\eta)\geqslant 1\\ 2&\text{if }\mathfrak{c}(\eta)=0\end{cases}+d(\eta)\geqslant 3.

We also have $\displaystyle\mathrm{E}(\Pi)=\{\eta^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}\}$ . For $\displaystyle\xi=\eta^{-1}\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ we have

\displaystyle\mathfrak{c}(\xi)=\mathfrak{c}(\eta)\leqslant\max(3\mathfrak{c}(\eta),2)=\mathfrak{c}(\Pi),\quad\mathfrak{c}(\Pi\otimes\xi)=2\leqslant\mathfrak{c}(\Pi).

(6) If $\displaystyle\Pi$ is supercuspidal, then $\displaystyle\mathrm{E}(\Pi)=\emptyset$ . We have $\displaystyle\mathfrak{\rho}(\Pi)=\mathfrak{c}(\Pi)\geqslant 3$ . ∎

The following subset of $\displaystyle\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}$ will turn out to be important for the bound of $\displaystyle\widetilde{h}_{n}^{+}(\chi)$ :

(4.11)

\mathcal{A}_{n}=\mathcal{A}_{n}(\beta,\Pi):=\left\{\xi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\ \middle|\ \mathfrak{\rho}(\Pi\otimes\xi)\leqslant n,\ \varepsilon_{n}(\xi,H)\neq 0\right\}.

Lemma 4.5.

For any $\displaystyle\xi\in\mathcal{A}_{n}$ we have $\displaystyle\mathfrak{c}(\xi)\leqslant\max\left(\tfrac{n_{0}}{e},2\mathfrak{c}(\Pi)\right)$ .

Proof.

Suppose $\displaystyle\xi\in\mathcal{A}_{n}$ and $\displaystyle\mathfrak{c}(\xi)>\max\left(\tfrac{n_{0}}{e},2\mathfrak{c}(\Pi)\right)$ . In particular, we have $\displaystyle\mathfrak{c}(\xi)\geqslant a(\Pi)$ , the stability barrier of $\displaystyle\Pi$ , by Proposition 2.4. Therefore we get $\displaystyle\mathfrak{c}(\Pi\otimes\xi)=3\mathfrak{c}(\xi)$ , $\displaystyle d(\Pi\otimes\xi)=0$ and deduce

(4.12)

3\mathfrak{c}(\xi)=\mathfrak{\rho}(\Pi\otimes\xi)\leqslant n.

On the other hand, we claim that the condition $\displaystyle\mathfrak{c}(\xi)>\tfrac{n_{0}}{e}$ and $\displaystyle\varepsilon_{n}(\xi,H)\neq 0$ imply

(4.13)

\mathfrak{c}(\xi)=\tfrac{n}{2},

and conclude the proof by comparing (4.12) and (4.13) taking into account $\displaystyle n\geqslant\tfrac{2}{e}(n_{0}-2)+3>0$ . In fact, the proof of (4.13) is case-by-case check with Lemma 3.6: (1) If $\displaystyle\mathbf{L}/\mathbf{F}$ is split, then $\displaystyle e=1$ and $\displaystyle\mathfrak{c}(\xi)>n_{0}=\mathfrak{c}(\chi_{0}^{\pm 1})$ . Only the first case in Lemma 3.6 is possible, yielding (4.13). (2) If $\displaystyle\mathbf{L}/\mathbf{F}$ is unramified, then $\displaystyle e=1$ and by (3.9) we have $\displaystyle\mathfrak{c}(\xi\circ{\rm Nr})=\mathfrak{c}(\xi)>n_{0}=\mathfrak{c}(\beta)$ . The last case in Lemma 3.6 yields (4.13). (3) If $\displaystyle\mathbf{L}/\mathbf{F}$ is ramified, then $\displaystyle e=2$ and by (3.9) we have $\displaystyle\mathfrak{c}(\xi\circ{\rm Nr})\geqslant 2\mathfrak{c}(\xi)-1\geqslant 2(n_{0}/2+1)-1=n_{0}+1>\mathfrak{c}(\beta)$ . The last case in Lemma 3.6 yields (4.13). ∎

Lemma 4.6.

Write $\displaystyle\xi=\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ . For any $\displaystyle\epsilon>0$ sufficiently small we have the bound

\left\lvert\widetilde{h}_{c}^{+}(\chi)\right\rvert\leqslant\sideset{}{{}_{\frac{2}{e}(n_{0}-2)+3\leqslant n<\mathfrak{\rho}(\Pi\otimes\xi)}}{\sum}\left\lvert\widetilde{h}_{n}^{+}(\chi)\right\rvert+\sideset{}{{}_{n\geqslant\mathfrak{\rho}(\Pi\otimes\chi)}}{\sum}\left\lvert\widetilde{h}_{n}^{+}(\chi)\right\rvert\\ \ll_{\epsilon}\mathbf{C}(\Pi)^{\epsilon}q^{-\frac{n_{0}}{e}+\frac{1-e}{2}}\cdot\mathbbm{1}_{\leqslant\max\left(\frac{n_{0}}{e},2\mathfrak{c}(\Pi)\right)}(\mathfrak{c}(\xi)).

Proof.

Write the relevant $\displaystyle L$ -functions of $\displaystyle\Pi\otimes\chi$ as

\displaystyle L(s,\Pi\otimes\chi)=\sideset{}{{}_{j=1}^{d}}{\prod}(1-a_{j}q^{-s})^{-1}

where $\displaystyle d=d(\Pi\otimes\chi)\in\{0,1,2,3\}$ is the degree, and $\displaystyle a_{j}=a_{j}(\Pi\otimes\chi)$ are the generalized Satake parameters of $\displaystyle\Pi\otimes\chi$ satisfying $\displaystyle\lvert a_{j}\rvert\in\{1,q^{-\frac{1}{2}},q^{-1}\}$ by temperedness. We can rewrite (4.10) as

(4.14)

f_{n}(X;\chi,H)=\varepsilon_{n}(\chi,H)\varepsilon\left(\tfrac{1}{2},\Pi\otimes\chi,\psi\right)q^{\frac{\mathfrak{c}(\Pi\otimes\chi)}{2}-n}X^{n-\mathfrak{\rho}(\Pi\otimes\chi)}\cdot\sideset{}{{}_{j=1}^{d}}{\prod}\tfrac{X-a_{j}}{1-\overline{a_{j}}q^{-1}X}.

If $\displaystyle n\geqslant\mathfrak{\rho}:=\mathfrak{\rho}(\Pi\otimes\chi)$ , then $\displaystyle f_{n}(X;\chi,H)\neq 0$ implies $\displaystyle\varepsilon_{n}(\chi,H)\neq 0$ , hence $\displaystyle\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}\in\mathcal{A}_{n}$ by definition. The Laurent expansion of $\displaystyle f_{n}(X;\chi,H)$ contains only non-negative powers of $\displaystyle X$ . Therefore the right hand side is equal to $\displaystyle\widetilde{h}_{n}(s,\chi)$ by definition (4.8). Consequently we get the formula and deduce the bound

\displaystyle\widetilde{h}_{n}(s,\chi)=\varepsilon_{n}(\chi,H)\varepsilon\left(\tfrac{1}{2},\Pi\otimes\chi,\psi\right)q^{\frac{\mathfrak{c}(\Pi\otimes\chi)}{2}-n}q^{s(n-\mathfrak{c}(\Pi\otimes\chi))}\cdot\sideset{}{{}_{j=1}^{d}}{\prod}\tfrac{1-a_{j}q^{-s}}{1-\overline{a_{j}}q^{s-1}},

(4.15)

\left\lvert\widetilde{h}_{n}^{+}(\chi)\right\rvert=\left\lvert\widetilde{h}_{n}(1/2,\chi)\right\rvert\ll_{\vartheta_{3}}\mathbbm{1}_{\mathcal{A}_{n}}(\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}})\cdot q^{-\frac{n}{2}}.

If $\displaystyle n<\mathfrak{\rho}$ , then the Laurent expansion of $\displaystyle f_{n}(X;\chi,H)$ contains non-negative powers of $\displaystyle X$ only if $\displaystyle d=d(\Pi\otimes\chi)>0$ , i.e. $\displaystyle\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}\in\mathrm{E}(\Pi)$ where $\displaystyle\mathrm{E}(\Pi)$ is the set of exponents of $\displaystyle\Pi$ introduced in Definition 4.3. By Lemma 4.4 and the summation range of $\displaystyle n$ we have

(4.16)

\tfrac{2}{e}(n_{0}-2)+3\leqslant n\leqslant 2\mathfrak{c}(\Pi)+2.

Note that the Laurent expansion of $\displaystyle f_{n}(X;\chi,H)$ is absolutely convergent for $\displaystyle\lvert X\rvert<q$ . Note also that $\displaystyle Xf_{n}^{\prime}(X;\chi,H)$ and $\displaystyle f_{n}(X;\chi,H)$ have the same type of bound for $\displaystyle\lvert X\rvert=q^{\frac{1}{2}}$ . By Lemma 4.2 (1) and (4.8) we deduce

(4.17)

\left\lvert\widetilde{h}_{n}^{+}(\chi)\right\rvert=\left\lvert\widetilde{h}_{n}(1/2,\chi)\right\rvert\leqslant\left\lvert f_{n}(q^{\frac{1}{2}};\chi,H)\right\rvert+\sideset{}{{}_{\lvert X\rvert=q^{\frac{1}{2}}}}{\sup}\left\lvert Xf_{n}^{\prime}(X;\chi,H)\right\rvert\\ \ll\mathbbm{1}_{\frac{2}{e}(n_{0}-2)+3\leqslant n\leqslant\mathfrak{\rho}(\Pi\otimes\chi)-1}\mathbbm{1}_{\mathrm{E}(\Pi)}(\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}})\cdot q^{-\frac{n}{2}}\cdot\left(1+\mathfrak{\rho}(\Pi\otimes\chi)-n\right)\\ \ll\mathbbm{1}_{\frac{2}{e}(n_{0}-2)+3\leqslant n\leqslant 2\mathfrak{c}(\Pi)+2}\mathbbm{1}_{\mathrm{E}(\Pi)}(\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}})\cdot q^{-\frac{n}{2}}\cdot(2\mathfrak{c}(\Pi)+3).

by Lemma 3.6 & 4.4. We conclude by summing over $\displaystyle n$ (for $\displaystyle 2\mid en$ ) the bounds (4.15) and (4.17), taking into account the bound for $\displaystyle\mathbbm{1}_{\mathcal{A}_{n}}$ given by Lemma 4.5. ∎

4.3. Negative Part

We turn to the study of a “trivial” bound of $\displaystyle\widetilde{h}_{n}^{-}(\chi)$ . We introduce

(4.18)

\mathcal{B}_{n}=\mathcal{B}_{n}(\beta,\Pi):=\left\{\xi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\ \middle|\ \mathfrak{\rho}(\Pi\otimes\xi)>n,\ \varepsilon_{n}(\xi,H)\neq 0\right\}.

Lemma 4.7.

Recall $\displaystyle e=e(\mathbf{L}/\mathbf{F})$ . We have $\displaystyle\mathcal{B}_{n}\subset\left\{\xi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\ \middle|\ \mathfrak{c}(\xi)\leqslant n/2\right\}$ and the bound $\displaystyle\left\lvert\mathcal{B}_{n}\right\rvert\ll q^{\frac{n}{2}}$ .

Proof.

For $\displaystyle\xi\in\mathcal{B}_{n}$ , the condition $\displaystyle\varepsilon_{n}(\xi,H)\neq 0$ implies $\displaystyle\mathfrak{c}(\xi)\leqslant\tfrac{n}{2}$ by checking Lemma 3.6 case-by-case, taking into account (3.9). For example in the case $\displaystyle\mathbf{L}/\mathbf{F}$ is ramified, we have $\displaystyle n\geqslant n_{0}+1$ and

\displaystyle 2\mathfrak{c}(\xi)-1\leqslant\mathfrak{c}(\xi\circ{\rm Nr})\leqslant\max\left(\mathfrak{c}(\beta),\mathfrak{c}(\beta\cdot(\xi\circ{\rm Nr}))\right)=\max(n_{0},n-1)=n-1.

Therefore we get $\displaystyle\left\lvert\mathcal{B}_{n}\right\rvert\leqslant\left\lvert\mathcal{O}_{\mathbf{F}}^{\times}/(1+\mathcal{P}_{\mathbf{F}}^{n/2})\right\rvert\ll q^{\frac{n}{2}}$ . ∎

Lemma 4.8.

We have the bound

\displaystyle\widetilde{h}_{n}^{-}(\chi)\ll q^{-\frac{1}{2}}\cdot\mathbbm{1}_{\mathfrak{c}(\chi)\leqslant\mathfrak{c}(\Pi)+\frac{n}{2}}+\mathbbm{1}_{n\leqslant 2\mathfrak{c}(\Pi)+2}\cdot q^{-\frac{n}{2}}\sideset{}{{}_{m=1}^{2\mathfrak{c}(\Pi)+3-n}}{\sum}q^{-\frac{m}{2}}\mathbbm{1}_{\mathfrak{c}(\chi)\leqslant\max(\mathfrak{c}(\Pi),m)}.

Consequently if $\displaystyle\mathfrak{c}(\Pi)>0$ and $\displaystyle n_{0}\leqslant A\mathfrak{c}(\Pi)$ for some constant $\displaystyle A\geqslant 1$ then we have

\displaystyle\widetilde{h}_{c}^{-}(\chi)\ll\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}\left\lvert\widetilde{h}_{n}^{-}(\chi)\right\rvert\ll\mathfrak{c}(\Pi)\mathbbm{1}_{\mathfrak{c}(\chi)\leqslant 3A\mathfrak{c}(\Pi)}.

Proof.

Write $\displaystyle\chi_{0}=\chi\mid_{\mathcal{O}_{\mathbf{F}}^{\times}}$ . By definition and the Plancherel formula on $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ we can write

\displaystyle\widetilde{h}_{n}^{-}(\chi)={\rm Vol}(\mathcal{O}_{\mathbf{F}}^{\times},\mathrm{d}^{\times}t)^{-1}q^{-n}\sideset{}{{}_{m=1}^{\infty}}{\sum}\chi(\varpi_{\mathbf{F}})^{m}q^{-\frac{m}{2}}\cdot C_{m}(n,\chi_{0}^{-1}),

\displaystyle C_{m}(n,\chi_{0}^{-1}):=\sideset{}{{}_{\xi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}}}{\sum}a_{m}(\xi;n)\cdot b_{m}(\xi\chi_{0}^{-1}),

where we have put

\displaystyle a_{m}(\xi;n):=\int_{\varpi_{\mathbf{F}}^{-m}\mathcal{O}_{\mathbf{F}}^{\times}}\mathcal{VH}_{\Pi,\psi}(H_{n})(t)\xi^{-1}(t)\mathrm{d}^{\times}t,\quad b_{m}(\xi):=\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi(-\varpi_{\mathbf{F}}^{-m}t)\xi(t)\mathrm{d}^{\times}t.

Since $\displaystyle m\geqslant 1$ the integral $\displaystyle b_{m}(\xi)$ is essentially a Gauss sum, which we can easily bound as

(4.19)

b_{m}(\xi)\ll q^{-\frac{m}{2}}\cdot\mathbbm{1}_{\mathfrak{c}(\xi)=m}+\mathbbm{1}_{m=1}\cdot q^{-1}\cdot\mathbbm{1}_{\mathfrak{c}(\xi)=0}\ll q^{-\frac{m}{2}}\cdot\mathbbm{1}_{\mathfrak{c}(\xi)\leqslant m}.

The defining formula for $\displaystyle a_{m}(\xi;n)$ makes sense for all $\displaystyle m\in\mathbb{Z}$ , and we have (writing $\displaystyle X:=q^{s}$ )

\sideset{}{{}_{m\in\mathbb{Z}}}{\sum}a_{m}(\xi;n)X^{-m}=\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(H_{n})(t)\xi^{-1}(t)\lvert t\rvert^{-s}\mathrm{d}^{\times}t=\gamma\left(s,\Pi\otimes\xi,\psi\right)\cdot\varepsilon_{n}(\xi,H)X^{n}\\ =\varepsilon_{n}(\xi,H)\varepsilon\left(\tfrac{1}{2},\Pi\otimes\xi,\psi\right)q^{\frac{\mathfrak{c}(\Pi\otimes\xi)}{2}}X^{n-\mathfrak{\rho}}\cdot\sideset{}{{}_{j=1}^{d}}{\prod}\tfrac{X-a_{j}}{1-\overline{a_{j}}q^{-1}X},

which is equivalent to (4.14). If $\displaystyle\xi\notin E(\Pi)$ then $\displaystyle d:=d(\Pi\otimes\xi)=0$ , and we get for $\displaystyle m\geqslant 1$

(4.20)

a_{m}(\xi;n)\ll\mathbbm{1}_{\mathcal{B}_{n}}(\xi)\cdot\mathbbm{1}_{m+n}(\mathfrak{c}(\Pi\otimes\xi))\cdot q^{\frac{m+n}{2}}.

If $\displaystyle\xi\in E(\Pi)$ then we apply the residue theorem (as $\displaystyle\lvert a_{j}\rvert\in\{1,q^{-\frac{1}{2}},q^{-1}\}$ ) to get

	$\displaystyle\displaystyle a_{m}(\xi;n)$	$\displaystyle\displaystyle=\varepsilon_{n}(\xi,H)\varepsilon\left(\tfrac{1}{2},\Pi\otimes\xi,\psi\right)q^{\frac{\mathfrak{c}(\Pi\otimes\xi)}{2}}\int_{\lvert X\rvert=q^{\frac{1}{2}}}X^{n-\mathfrak{c}(\Pi\otimes\xi)+m-1}\sideset{}{{}_{j=1}^{d}}{\prod}\tfrac{1-a_{j}X^{-1}}{1-\overline{a_{j}}q^{-1}X}\frac{\mathrm{d}X}{2\pi i}$
(4.21)			$\displaystyle\displaystyle\ll\mathbbm{1}_{\mathcal{B}_{n}}(\xi)\cdot\mathbbm{1}_{\geqslant m+n}(\mathfrak{\rho}(\Pi\otimes\xi))\cdot q^{\frac{n+m}{2}}.$

Combining the bounds (4.19)-(4.21), Lemma 4.4 & 4.7 and $\displaystyle\mathfrak{c}(\Pi\otimes\xi)\leqslant\mathfrak{c}(\Pi)+3\mathfrak{c}(\xi)$ (see [4]) we get

	$\displaystyle\displaystyle C_{m}(n,\chi_{0}^{-1})$	$\displaystyle\displaystyle\ll q^{\frac{n}{2}}\sideset{}{{}_{\begin{subarray}{c}\xi\in\mathcal{B}_{n}-E(\Pi)\\ \mathfrak{c}(\xi\chi_{0}^{-1})\leqslant m\end{subarray}}}{\sum}\mathbbm{1}_{\mathfrak{c}(\Pi\otimes\xi)=m+n}+q^{\frac{n}{2}}\sideset{}{{}_{\begin{subarray}{c}\xi\in\mathcal{B}_{n}\cap E(\Pi)\\ \mathfrak{c}(\xi\chi_{0}^{-1})\leqslant m\end{subarray}}}{\sum}\mathbbm{1}_{\geqslant m+n}(\mathfrak{\rho}(\Pi\otimes\xi))$
		$\displaystyle\displaystyle\ll q^{\frac{n}{2}}\sideset{}{{}_{\begin{subarray}{c}\mathfrak{c}(\xi)\leqslant n/2\\ \mathfrak{c}(\xi\chi_{0}^{-1})\leqslant m\end{subarray}}}{\sum}\mathbbm{1}_{m\leqslant\mathfrak{c}(\Pi)+\frac{n}{2}}+q^{\frac{n}{2}}\mathbbm{1}_{\leqslant 2\mathfrak{c}(\Pi)+3}(m+n)\mathbbm{1}_{\leqslant\max(\mathfrak{c}(\Pi),m)}(\mathfrak{c}(\chi_{0}))$
		$\displaystyle\displaystyle\leqslant q^{\frac{n}{2}+\min(\frac{n}{2},m)}\mathbbm{1}_{m\leqslant\mathfrak{c}(\Pi)+\frac{n}{2}}\mathbbm{1}_{\leqslant\mathfrak{c}(\Pi)+\frac{n}{2}}(\mathfrak{c}(\chi_{0}))+q^{\frac{n}{2}}\mathbbm{1}_{\leqslant 2\mathfrak{c}(\Pi)+3}(m+n)\mathbbm{1}_{\leqslant\max(\mathfrak{c}(\Pi),m)}(\mathfrak{c}(\chi_{0})),$

and conclude the first bound. To derive the second bound, one simply notice $\displaystyle n_{1}$ can be taken as $\displaystyle 2A\mathfrak{c}(\Pi)$ , since under the condition $\displaystyle\mathfrak{c}(\Pi)>0$ we have $\displaystyle a(\Pi)\leqslant\mathfrak{c}(\Pi)$ by the “moreover” part of Proposition 2.4. ∎

Remark 4.9.

The bound established in Lemma 4.8 is far from being optimal. For example in the case $\displaystyle e=1$ for $\displaystyle\xi\in\mathcal{B}_{n}$ the typical size of $\displaystyle\mathfrak{c}(\Pi\otimes\xi)$ should be $\displaystyle 3n/2$ , hence the term $\displaystyle q^{-\frac{\mathfrak{c}(\Pi\otimes\xi)}{2}}$ could be bounded as $\displaystyle q^{-\frac{3n}{4}}$ . But even with this improvement the individual bound of $\displaystyle\widetilde{h}_{n}^{-}(\chi)$ is too weak to apply in the case $\displaystyle n_{0}\gg\mathfrak{c}(\Pi)$ . It would be interesting to recover the cancellation in the sum of $\displaystyle a_{m}(\xi;n)b_{m}(\xi)$ over $\displaystyle\xi$ by refining the above method.

4.4. For Unramified Characters

We turn to the unramified case $\displaystyle\chi=\lvert\cdot\rvert_{\mathbf{F}}^{s}$ . According to the decomposition (4.7) of $\displaystyle\widetilde{H}$ , the notation $\displaystyle\widetilde{H}_{\infty}(k;s_{0})$ , $\displaystyle\widetilde{H}_{c}(k;s_{0})$ and $\displaystyle\widetilde{H}_{n}^{\pm}(k;s_{0})$ have obvious meaning and

\displaystyle\widetilde{H}(k;s_{0})=\widetilde{H}_{\infty}(k;s_{0})+\sideset{}{{}_{\pm}}{\sum}\widetilde{H}_{c}^{\pm}(k;s_{0}),\quad\widetilde{H}_{c}^{\pm}(k;s_{0})=\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}\widetilde{H}_{n}^{\pm}(k;s_{0}).

Lemma 4.10.

For any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$ we have the following bounds.

(1)

$\displaystyle\widetilde{H}_{\infty}(k;\tfrac{1}{2})\ll_{k,\epsilon}q^{-n_{1}(1-\epsilon)}$ , $\displaystyle\widetilde{H}_{\infty}(k;-\tfrac{1}{2})\ll_{k,\epsilon}q^{n_{1}\epsilon}$ .

(2)

$\displaystyle\widetilde{H}_{n}^{+}(k;\pm\tfrac{1}{2})=0$ unless $\displaystyle n=\tfrac{2n_{0}}{e}+e-1$ . We have

\displaystyle\widetilde{H}_{c}^{+}(k;\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\frac{1}{2}}\cdot q^{\left(\frac{2n_{0}}{e}+e-1\right)\epsilon},\quad\widetilde{H}_{c}^{+}(k;-\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\frac{1}{2}}\cdot q^{-\left(\frac{2n_{0}}{e}+e-1\right)(1-\epsilon)}.

Proof.

(1) By Lemma 4.1 we have $\displaystyle\widetilde{H}_{\infty}(s)=q^{-\left(\frac{1}{2}+s\right)n_{1}}L\left(\tfrac{1}{2}-s,\widetilde{\Pi}\right)^{-1}$ . The stated bounds follow readily.

(2) By Lemma 3.6 and (4.10) we have

f_{n}(q^{\frac{1}{2}};\lvert\cdot\rvert_{\mathbf{F}}^{s},H)=f_{n}(q^{s+\frac{1}{2}};\mathbbm{1},H)=\\ q^{n\left(s-\frac{1}{2}\right)}\varepsilon\left(s+\tfrac{1}{2},\Pi,\psi\right)\frac{L(\tfrac{1}{2}-s,\widetilde{\Pi})}{L(\tfrac{1}{2}+s,\Pi)}\cdot\begin{cases}\mathbbm{1}_{n=2n_{0}}\cdot\zeta_{\mathbf{F}}(1)\chi_{0}(-1)&\text{if }\mathbf{L}/\mathbf{F}\text{ split}\\ \mathbbm{1}_{n=\frac{2n_{0}}{e}+e-1}\cdot\zeta_{\mathbf{F}}(1)\varepsilon(1/2,\pi_{\beta},\psi)&\text{if }\mathbf{L}/\mathbf{F}\text{ non-split}\end{cases}.

We introduce $\displaystyle a_{j}$ ( $\displaystyle\lvert a_{j}\rvert\in\{1,q^{-\frac{1}{2}},q^{-1}\}$ ) as the parameters of $\displaystyle L(s,\Pi)$ so that $\displaystyle L(\tfrac{1}{2}-s,\widetilde{\Pi})=P(X)^{-1}$ with

\displaystyle P(X):=\sideset{}{{}_{j=1}^{d(\Pi)}}{\prod}\left(1-\overline{a_{j}}q^{-\frac{1}{2}}X\right),\quad f(X):=f_{n}(q^{\frac{1}{2}}X;\mathbbm{1},H)=X^{n-\mathfrak{\rho}(\Pi)}\sideset{}{{}_{j=1}^{d(\Pi)}}{\prod}\tfrac{X-a_{j}q^{-\frac{1}{2}}}{1-\overline{a_{j}}q^{-\frac{1}{2}}X}\cdot q^{-\frac{n}{2}}\delta

for some $\displaystyle\lvert\delta\rvert=1$ . We then rewrite, with $\displaystyle X=q^{s}$

\displaystyle\widetilde{H}_{n}^{+}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=\tfrac{f_{+}(q^{s})}{L(\tfrac{1}{2}-s,\widetilde{\Pi})\zeta_{\mathbf{F}}(\tfrac{1}{2}+s)}=(1-q^{-\frac{1}{2}}X^{-1})P(X)f_{+}(X).

The bounds at $\displaystyle s=-1/2$ then follows readily from Lemma 4.2 (1) via the bounds

\displaystyle\sideset{}{{}_{\lvert z\rvert=1}}{\sup}\left\lvert D^{k}f(q^{-\frac{1}{2}}z)\right\rvert\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\frac{1}{2}}q^{-n(1-\epsilon)},\quad D^{k}\left((1-q^{-\frac{1}{2}}X^{-1})P(X)\right)\mid_{X=q^{-\frac{1}{2}}}\ll_{k}1.

We now consider the bounds at $\displaystyle s=1/2$ . If $\displaystyle n=\tfrac{2n_{0}}{e}+e-1\geqslant\mathfrak{\rho}(\Pi)$ , then $\displaystyle f_{+}(X)=f(X)$ , implying

\displaystyle\widetilde{H}_{n}^{+}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=q^{-\frac{n}{2}}\delta\cdot X^{n-\mathfrak{c}(\Pi)}(1-q^{-\frac{1}{2}}X^{-1})\sideset{}{{}_{j=1}^{d(\Pi)}}{\prod}\left(1-a_{j}q^{-\frac{1}{2}}X^{-1}\right),

\displaystyle D^{k}\widetilde{H}_{n}^{+}(\lvert\cdot\rvert_{\mathbf{F}}^{s})\mid_{s=1/2}\ll_{k,\epsilon}\mathbf{C}(\Pi)^{-\frac{1}{2}}q^{n\epsilon}.

If $\displaystyle n=\tfrac{2n_{0}}{e}+e-1<\mathfrak{\rho}(\Pi)=:\mathfrak{\rho}$ , then $\displaystyle Q(X):=P(X)f(X)$ has highest term $\displaystyle X^{m}$ with $\displaystyle m<d(\Pi)=\deg P$ . We also note that $\displaystyle f_{+}(X)=0$ unless $\displaystyle d:=d(\Pi)\geqslant 1$ . We apply Lemma 4.2 (2) distinguishing cases:

(i) $\displaystyle d=1$ . Necessarily we have $\displaystyle c:=c(\Pi)\geqslant 2$ , $\displaystyle n-c\leqslant 0$ . We get and conclude the stated bounds by

\displaystyle P(X)f_{+}(X)=\overline{a_{1}}^{c-n}\delta\cdot q^{-\frac{c}{2}}(1-\lvert a_{1}\rvert^{2}q^{-1})\ll\mathbf{C}(\Pi)^{-\frac{1}{2}}.

(ii) $\displaystyle d=2$ . Necessarily $\displaystyle\Pi=\mathrm{St}_{\chi}\boxplus\chi^{-2}$ for some unramified unitary character $\displaystyle\chi$ . We have $\displaystyle c=1$ and $\displaystyle n-c\leqslant 1$ , and may assume $\displaystyle\lvert a_{1}\rvert=q^{-\frac{1}{2}}$ and $\displaystyle\lvert a_{2}\rvert=1$ . We get the formula

	$\displaystyle\displaystyle P(X)f_{+}(X)$	$\displaystyle\displaystyle=\overline{a_{1}}^{c-n}\delta\cdot q^{-\frac{c}{2}}\frac{(1-\lvert a_{1}\rvert^{2}q^{-1})(1-\overline{a_{1}}a_{2}q^{-1})}{1-\overline{a_{2}}\overline{a_{1}}^{-1}}(1-\overline{a_{2}}q^{-\frac{1}{2}}X)+$
		$\displaystyle\displaystyle\quad\overline{a_{2}}^{c-n}\delta\cdot q^{-\frac{c}{2}}\frac{(1-\lvert a_{2}\rvert^{2}q^{-1})(1-\overline{a_{2}}a_{1}q^{-1})}{1-\overline{a_{1}}\overline{a_{2}}^{-1}}(1-\overline{a_{1}}q^{-\frac{1}{2}}X)$

and conclude the stated bounds by

(4.22)

D^{k}\left(P(X)f_{+}(X)\right)\mid_{X=q^{\frac{1}{2}}}\ll_{k}1.

(iii) $\displaystyle d=3$ . Necessarily $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ for some unramified unitary characters $\displaystyle\mu_{i}$ . We have $\displaystyle c=0$ and $\displaystyle n\leqslant 2$ , and may assume $\displaystyle\lvert a_{j}\rvert=1$ with $\displaystyle a_{1}a_{2}a_{3}=1$ . It is easy to see $\displaystyle C_{j,k}\ll 1$ . Therefore (4.22) still holds and we conclude the stated bounds in the same way. ∎

Lemma 4.11.

For any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$ we have the following bounds

\displaystyle\widetilde{H}_{n}^{-}(k;\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\epsilon},\quad\widetilde{H}_{n}^{-}(k;-\tfrac{1}{2})\ll_{k,\epsilon}\left(\mathbf{C}(\Pi)^{2}q^{\frac{n}{2}}\right)^{1+\epsilon}.

Consequently if $\displaystyle\mathfrak{c}(\Pi)>0$ and $\displaystyle n_{0}\leqslant A\mathfrak{c}(\Pi)$ for some constant $\displaystyle A\geqslant 1$ then we have

\displaystyle\widetilde{H}_{c}^{-}(k;\delta\cdot\tfrac{1}{2})\leqslant\sideset{}{{}_{\begin{subarray}{c}n=\frac{2}{e}(n_{0}-2)+3\\ 2\mid en\end{subarray}}^{2n_{1}-1}}{\sum}\left\lvert\widetilde{H}_{n}^{-}(k;\delta\cdot\tfrac{1}{2})\right\rvert\ll_{\epsilon}\begin{cases}\mathbf{C}(\Pi)^{\epsilon}&\text{if }\delta=1\\ \mathbf{C}(\Pi)^{\left(2+2Ae^{-1}\right)(1+\epsilon)}&\text{if }\delta=-1\end{cases}.

Proof.

With similar argument as in the proof of Lemma 4.8 we get

\displaystyle\widetilde{h}_{n}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=q^{-n}\sideset{}{{}_{m=1}^{\infty}}{\sum}q^{m\left(\frac{1}{2}-s\right)}C_{m}(n),\quad C_{m}(n)\ll q^{\frac{n}{2}+\min(m,\frac{n}{2})}\mathbbm{1}_{m\leqslant\mathfrak{c}(\Pi)+\frac{n}{2}}+q^{\frac{n}{2}}\mathbbm{1}_{m+n\leqslant 2\mathfrak{c}(\Pi)+3}.

The stated bounds follow readily. ∎

5. Dual Weight Functions: Split and Unramified Cases

5.1. First Quadratic Elementary Functions

Definition 5.1.

For any $\displaystyle n\in\mathbb{Z}_{\geqslant 0}$ we define the first quadratic elementary functions $\displaystyle F_{n}\in{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})$ by

\displaystyle F_{n}(y^{2}):=\mathbbm{1}_{v(y)=-n}\cdot\sideset{}{{}_{\pm}}{\sum}\psi(\pm y),

and are supported in the subset of square elements of $\displaystyle\mathbf{F}^{\times}$ .

Definition 5.2.

Let $\displaystyle\eta_{0}$ be the character of $\displaystyle\mathbf{F}^{\times}$ associated with the (ramified) quadratic extension $\displaystyle\mathbf{F}[\sqrt{-\varpi_{\mathbf{F}}}]/\mathbf{F}$ . Explicitly it is given by the following formula for all $\displaystyle n\in\mathbb{Z}$

\displaystyle\eta_{0}(\varpi_{\mathbf{F}}^{n}u)=\begin{cases}1&\text{if }u\in(\mathcal{O}_{\mathbf{F}}^{\times})^{2}\\ -1&\text{if }u\in\mathcal{O}_{\mathbf{F}}^{\times}-(\mathcal{O}_{\mathbf{F}}^{\times})^{2}\end{cases}.

Denote by $\displaystyle\tau_{0}=\tau(\eta_{0},\psi;\varpi_{\mathbf{F}})$ the quadratic Gauss sum given by

\displaystyle\tau_{0}:=\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\psi\left(\tfrac{u}{\varpi_{\mathbf{F}}}\right)\eta_{0}(u)\mathrm{d}u=\int_{\mathcal{O}_{\mathbf{F}}}\psi\left(\tfrac{u^{2}}{\varpi_{\mathbf{F}}}\right)\mathrm{d}u.

Lemma 5.3.

Let $\displaystyle\chi$ be a quasi-character of $\displaystyle\mathbf{F}^{\times}$ . We have

\displaystyle\int_{\mathbf{F}^{\times}}F_{n}(y)\chi(y)\mathrm{d}^{\times}y=\begin{cases}\mathbbm{1}_{\mathfrak{c}(\chi^{2})=n}\cdot\zeta_{\mathbf{F}}(1)\gamma(1,\chi^{-2},\psi)&\text{if }n\geqslant 2\\ \mathbbm{1}_{\mathfrak{c}(\chi^{2})=1}\cdot\zeta_{\mathbf{F}}(1)\gamma(1,\chi^{-2},\psi)-\mathbbm{1}_{\mathfrak{c}(\chi^{2})=0}\cdot\zeta_{\mathbf{F}}(1)q^{-1}\chi(\varpi_{\mathbf{F}})^{-2}&\text{if }n=1\\ \mathbbm{1}_{\mathfrak{c}(\chi^{2})=0}&\text{if }n=0\end{cases}.

Note that we can replace the condition $\displaystyle\mathfrak{c}(\chi^{2})=n$ with $\displaystyle\mathfrak{c}(\chi)=n$ if $\displaystyle n\geqslant 1$ .

Proof.

The case for $\displaystyle n=0$ is simple and omitted. For $\displaystyle n\geqslant 1$ with the change of variables $\displaystyle y\to y^{2}$

\displaystyle\int_{\mathbf{F}^{\times}}F_{n}(y)\chi(y)\mathrm{d}^{\times}y=\int_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(y)\chi^{2}(y)\mathrm{d}^{\times}y,

the desired formula follows readily from [5, Exercise 23.5]. ∎

Corollary 5.4.

Let $\displaystyle n\geqslant a(\Pi)(\geqslant 2)$ , the stability barrier of $\displaystyle\Pi$ (see Proposition 2.4). We have

\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{n})(y)=\mathbbm{1}_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\cdot\begin{cases}q^{\frac{3n}{2}}\psi(4y)&\text{if }2\mid n\\ \tau_{0}q^{\lceil\frac{3n}{2}\rceil}\psi(4y)\eta_{0}(4y)&\text{if }2\nmid n\end{cases}.

Proof.

By the local functional equation, Lemma 5.3 and Proposition 2.4 we have for any $\displaystyle\chi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}$

\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{n})(y)\chi(y)^{-1}\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\zeta_{\mathbf{F}}(1)\mathbbm{1}_{\mathfrak{c}(\chi)=n}\cdot\gamma(s,\chi,\psi)^{3}\gamma(1-2s,\chi^{-2},\psi)\\ =\zeta_{\mathbf{F}}(1)\mathbbm{1}_{\mathfrak{c}(\chi)=n}\gamma(1,\chi,\psi)^{3}\gamma(1,\chi^{-2},\psi)\cdot q^{3n-ns}.

Therefore the support of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{n})$ is contained in $\displaystyle\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}$ . Applying [5, Exercise 23.5] we find

\displaystyle\zeta_{\mathbf{F}}(1)\mathbbm{1}_{\mathfrak{c}(\chi)=n}\gamma(1,\chi,\psi)^{3}\gamma(1,\chi^{-2},\psi)\cdot q^{3n}=\mathbbm{1}_{\mathfrak{c}(\chi)=n}\cdot q^{3n}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{4}}\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{4}}{\varpi_{\mathbf{F}}^{n}}\right)\chi^{-1}\left(\tfrac{t_{1}t_{2}t_{3}}{\varpi_{\mathbf{F}}^{n}t_{4}^{2}}\right)\mathrm{d}\vec{t},

where we have written $\displaystyle\mathrm{d}\vec{t}:=\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}$ for simplicity. The Fourier inversion on $\displaystyle\mathcal{O}_{\mathbf{F}}^{\times}$ yields for $\displaystyle y\in\mathcal{O}_{\mathbf{F}}^{\times}$

\mathcal{VH}_{\Pi,\psi}(F_{n})\left(\tfrac{y}{\varpi_{\mathbf{F}}^{n}}\right)=\zeta_{\mathbf{F}}(1)q^{3n}\sideset{}{{}_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\\ \mathfrak{c}(\chi)=n\end{subarray}}}{\sum}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{4}}\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{4}}{\varpi_{\mathbf{F}}^{n}}\right)\chi\left(\tfrac{t_{4}^{2}y}{t_{1}t_{2}t_{3}}\right)\mathrm{d}\vec{t}\\ =\zeta_{\mathbf{F}}(1)q^{3n}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{4}}\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{4}}{\varpi_{\mathbf{F}}^{n}}\right)\left(\left\lvert\left(\mathcal{O}_{\mathbf{F}}/\mathcal{P}_{\mathbf{F}}^{n}\right)^{\times}\right\rvert\mathbbm{1}_{1+\mathcal{P}_{\mathbf{F}}^{n}}-\left\lvert\left(\mathcal{O}_{\mathbf{F}}/\mathcal{P}_{\mathbf{F}}^{n-1}\right)^{\times}\right\rvert\mathbbm{1}_{1+\mathcal{P}_{\mathbf{F}}^{n-1}}\right)\left(\tfrac{t_{4}^{2}y}{t_{1}t_{2}t_{3}}\right)\mathrm{d}\vec{t}\\ =q^{3n}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\psi\left(\tfrac{t_{2}+t_{3}+t_{4}+t_{2}^{-1}t_{3}^{-1}t_{4}^{2}y}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}-\\ q^{4n-1}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}\times\mathcal{P}_{\mathbf{F}}^{n-1}}\psi\left(\tfrac{t_{2}+t_{3}+t_{4}+t_{2}^{-1}t_{3}^{-1}t_{4}^{2}y}{\varpi_{\mathbf{F}}^{n}}\right)\psi\left(\tfrac{t_{2}^{-1}t_{3}^{-1}t_{4}^{2}yu}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}\mathrm{d}u.

The last integral is vanishing because $\displaystyle\mathfrak{c}(\psi)=0$ . Re-numbering the variables we get

\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{n})\left(\tfrac{y}{\varpi_{\mathbf{F}}^{n}}\right)=q^{3n}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{1}^{-1}t_{2}^{-1}t_{3}^{2}y}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}.

Performing the level $\displaystyle\lceil\tfrac{n}{2}\rceil$ regularization to $\displaystyle\mathrm{d}t_{j}$ we see that the non-vanishing of the above integral implies

\displaystyle t_{1}-\tfrac{t_{3}^{2}y}{t_{1}t_{2}}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor},\quad t_{2}-\tfrac{t_{3}^{2}y}{t_{1}t_{2}}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor},\quad t_{3}+\tfrac{2t_{3}^{2}y}{t_{1}t_{2}}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}.

Writing $\displaystyle a=\tfrac{t_{3}^{2}y}{t_{1}t_{2}}\in\mathcal{O}_{\mathbf{F}}^{\times}$ , we see that the above conditions imply $\displaystyle a\equiv 4y\pmod{\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}$ , hence

\displaystyle t_{1}\in 4y\left(1+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}\right),\quad t_{2}\in 4y\left(1+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}\right),\quad t_{3}\in-8y\left(1+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}\right).

The change of variables $\displaystyle t_{1}=4y(1+u_{1})$ , $\displaystyle t_{2}=4y(1+u_{2})$ and $\displaystyle t_{3}=-8y(1+u_{3})$ with $\displaystyle u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}$ gives

\mathcal{VH}_{\Pi,\psi}(F_{n})\left(\tfrac{y}{\varpi_{\mathbf{F}}^{n}}\right)=q^{3n}\int_{u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\frac{4y(u_{1}+u_{2}-2u_{3})+\frac{4(1+u_{3})^{2}y}{(1+u_{1})(1+u_{2})}}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}u_{1}\mathrm{d}u_{2}\mathrm{d}u_{3}\\ =q^{3n}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\right)\int_{u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{1}u_{2}-2u_{1}u_{3}-2u_{2}u_{3}\right)\right)\mathrm{d}u_{1}\mathrm{d}u_{2}\mathrm{d}u_{3}\\ =q^{3n}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\right)\int_{u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\left(u_{1}^{2}+u_{2}^{2}+u_{1}u_{2}-u_{1}u_{3}\right)\right)\mathrm{d}u_{1}\mathrm{d}u_{2}\mathrm{d}u_{3}\\ =q^{3n-\lfloor\frac{n}{2}\rfloor}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\right)\int_{u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\left(u_{1}^{2}+u_{2}^{2}+u_{1}u_{2}\right)\right)\mathbbm{1}_{\mathcal{P}_{\mathbf{F}}^{\lceil\frac{n}{2}\rceil}}(u_{1})\mathrm{d}u_{1}\mathrm{d}u_{2}\\ =q^{2n}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\right)\int_{\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}u_{2}^{2}\right)\mathrm{d}u_{2},

where we made the change of variables $\displaystyle u_{2}\mapsto u_{2}+u_{3}$ in the third line. The desired formula follows. ∎

Remark 5.5.

If we compute the Mellin transform of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{n})(y)$ with the given formula in Corollary 5.4, we get in the case $\displaystyle 2\mid n\geqslant 2$ and for $\displaystyle\mathfrak{c}(\chi)=n$ an interesting equation

\displaystyle\chi(4)\gamma\left(\tfrac{1}{2},\chi^{-1},\psi\right)=\gamma\left(\tfrac{1}{2},\chi,\psi\right)^{3}\gamma\left(\tfrac{1}{2},\chi^{-2},\psi\right).

Corollary 5.6.

Suppose $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ with $\displaystyle\mathfrak{c}(\mu_{j})=0$ and $\displaystyle\mu_{1}\mu_{2}\mu_{3}=\mathbbm{1}$ . Let $\displaystyle E_{i}(y):=\mu_{i}^{-1}(y)\lvert y\rvert\mathbbm{1}_{\mathcal{O}_{\mathbf{F}}}(y)$ and $\displaystyle f_{i}:=(1-\mu_{i}(\varpi_{\mathbf{F}})\mathfrak{t}(\varpi_{\mathbf{F}})).E_{i}$ . For $\displaystyle f,g\in{\rm L}^{1}(\mathbf{F}^{\times})$ define $\displaystyle f*g(y):=\int_{\mathbf{F}^{\times}}f(yt^{-1})g(t)\mathrm{d}^{\times}t$ . We have

	$\displaystyle\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})(y)$	$\displaystyle\displaystyle=(f_{1}f_{2}f_{3})(y)+\tau_{0}q^{2}\cdot\eta_{0}(-y)\mathbbm{1}_{\varpi_{\mathbf{F}}^{-3}\mathcal{O}_{\mathbf{F}}^{\times}}(y),$
	$\displaystyle\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})(y)$	$\displaystyle\displaystyle=-q^{-1}\cdot(f_{1}f_{2}f_{3})(\varpi_{\mathbf{F}}^{-2}y)-\zeta_{\mathbf{F}}(1)\cdot\mathbbm{1}_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}(y)$
		$\displaystyle\displaystyle\quad+\tau_{0}q\mathbbm{1}_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\int_{(\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times})^{2}}\psi\left(t_{1}+t_{2}-\tfrac{t_{1}t_{2}}{4y}\right)\eta_{0}\left(\tfrac{4y}{t_{1}t_{2}}\right)\mathrm{d}t_{1}\mathrm{d}t_{2}.$

Proof.

By the local functional equation and Lemma 5.3 we have for any $\displaystyle\chi\in\widehat{\mathbf{F}^{\times}}$

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{0})(y)\chi(y)^{-1}\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\begin{cases}\sideset{}{{}_{i=1}^{3}}{\prod}\tfrac{L(1-s,\mu_{i}^{-1}\chi^{-1})}{L(s,\mu_{i}\chi)}&\text{if }\mathfrak{c}(\chi)=0\\ q^{3(\frac{1}{2}-s)}\gamma\left(\tfrac{1}{2},\chi,\psi\right)^{3}&\text{if }\mathfrak{c}(\chi\eta_{0})=0\\ 0&\text{otherwise}\end{cases}.

Therefore we can write $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})=\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}+\mathcal{VH}_{\Pi,\psi}(F_{0})_{1}$ with the properties

\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}(y\delta)=\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}(y),\quad\mathcal{VH}_{\Pi,\psi}(F_{0})_{1}(y\delta)=\mathcal{VH}_{\Pi,\psi}(F_{0})_{1}(y)\eta_{0}(\delta),\quad\forall y\in\mathbf{F}^{\times},\delta\in\mathcal{O}_{\mathbf{F}}^{\times};

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\sideset{}{{}_{i=1}^{3}}{\prod}\tfrac{L(1-s,\mu_{i}^{-1})}{L(s,\mu_{i})}=\sideset{}{{}_{i=1}^{3}}{\prod}\int_{\mathbf{F}^{\times}}f_{i}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y,

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}(y)\eta_{0}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=q^{3(\frac{1}{2}-s)}\gamma\left(\tfrac{1}{2},\eta_{0},\psi\right)^{3}=\eta_{0}(-1)\tau_{0}q^{2}\cdot\int_{\varpi_{\mathbf{F}}^{-3}\mathcal{O}_{\mathbf{F}}^{\times}}\lvert y\rvert^{-s}\mathrm{d}^{\times}y.

Hence we identify $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})_{0}=f_{1}*f_{2}*f_{3}$ and $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})_{1}(y)=\tau_{0}q^{2}\cdot\eta_{0}(-y)\mathbbm{1}_{\varpi_{\mathbf{F}}^{-3}\mathcal{O}_{\mathbf{F}}^{\times}}(y)$ . Similarly we can write $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})=\mathcal{VH}_{\Pi,\psi}(F_{1})_{0}+\mathcal{VH}_{\Pi,\psi}(F_{1})_{1}$ with the properties

\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})_{0}(y\delta)=\mathcal{VH}_{\Pi,\psi}(F_{1})_{0}(y),\quad\forall\delta\in\mathcal{O}_{\mathbf{F}}^{\times};

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{1})_{0}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=-q^{2s-1}\sideset{}{{}_{i=1}^{3}}{\prod}\int_{\mathbf{F}^{\times}}f_{i}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y,

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{1})_{1}(y)\chi^{-1}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\mathbbm{1}_{\mathfrak{c}(\chi)=1}\cdot q^{3-s}\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(y)\chi^{2}(y)\mathrm{d}^{\times}y\cdot\left(\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(y)\chi^{-1}(y)\mathrm{d}^{\times}y\right)^{3}.

We easily identify $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})_{0}(y)=-q^{-1}\cdot(f_{1}*f_{2}*f_{3})(\varpi_{\mathbf{F}}^{-2}y)$ . The argument in the proof of Corollary 5.4 shows that $\displaystyle\mathrm{supp}\left(\mathcal{VH}_{\Pi,\psi}(F_{1})_{1}\right)\subset\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}$ , and for $\displaystyle y\in\mathcal{O}_{\mathbf{F}}^{\times}$ that

\mathcal{VH}_{\Pi,\psi}(F_{1})_{1}\left(\tfrac{y}{\varpi_{\mathbf{F}}}\right)=q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\psi\left(\tfrac{t_{2}+t_{3}+t_{4}+t_{2}^{-1}t_{3}^{-1}t_{4}^{2}y}{\varpi_{\mathbf{F}}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}-\zeta_{\mathbf{F}}(1)q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{4}}\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{4}}{\varpi_{\mathbf{F}}}\right)\mathrm{d}\vec{t}\\ =\tau_{0}q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{2}}\psi\left(\tfrac{t_{2}+t_{3}-\frac{t_{2}t_{3}}{4y}}{\varpi_{\mathbf{F}}}\right)\eta_{0}\left(\tfrac{y}{t_{2}t_{3}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}-q^{2}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{2}}\psi\left(\tfrac{t_{2}+t_{3}}{\varpi_{\mathbf{F}}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}-\zeta_{\mathbf{F}}(1)q^{-1}\\ =\tau_{0}q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{2}}\psi\left(\tfrac{t_{2}+t_{3}-\frac{t_{2}t_{3}}{4y}}{\varpi_{\mathbf{F}}}\right)\eta_{0}\left(\tfrac{y}{t_{2}t_{3}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}-\zeta_{\mathbf{F}}(1).

Re-numbering the variables and inserting the formula of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})$ we get the formula of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})$ . ∎

5.2. Further Reductions

The functions $\displaystyle F_{n}$ are “building blocks” of our test functions $\displaystyle H_{c}$ in (4.3) when $\displaystyle\mathbf{L}/\mathbf{F}$ is not ramified. In fact writing $\displaystyle\varepsilon_{\mathbf{L}}:=\eta_{\mathbf{L}/\mathbf{F}}(\varpi_{\mathbf{F}})\in\{\pm 1\}$ and applying Lemma 3.10 we can rewrite the summands of $\displaystyle H_{c}$ as

(5.1)		$\displaystyle\displaystyle H_{2n+1}=0,$
(5.2)		$\displaystyle\displaystyle H_{2n}=\begin{cases}E_{n}&\text{if }2n_{0}\leqslant n\leqslant n_{1}-1\\ (\varepsilon_{\mathbf{L}}q)^{n}\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\mathcal{P}_{\mathbf{F}}^{2(n_{0}-1)})\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).F_{n}\right)\mathrm{d}\alpha&\text{if }n=2n_{0}-1\\ (\varepsilon_{\mathbf{L}}q)^{n}\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2(n-n_{0})}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).F_{n}\right)\mathrm{d}\alpha&\text{if }n_{0}+1\leqslant n<2n_{0}-1\end{cases}.$

The decomposition of $\displaystyle H_{2n_{0}}$ is subtler and really goes in the direction of expression in terms of the quadratic elementary functions. We shall write (the second numeric parameter $\displaystyle m$ in subscript always indicates the parameter of the relevant quadratic elementary function)

	$\displaystyle\displaystyle H_{2n_{0}}$	$\displaystyle\displaystyle=H_{2n_{0}}^{a}+H_{2n_{0}}^{b},$
(5.3)		$\displaystyle\displaystyle H_{2n_{0}}^{a}$	$\displaystyle\displaystyle=\tfrac{(\varepsilon_{\mathbf{L}}q)^{n_{0}}}{2}\mathbbm{1}_{\eta_{0}(-1)=\varepsilon_{\mathbf{L}}}\cdot\sideset{}{{}_{m=0}^{n_{0}-1}}{\sum}H_{2n_{0},m}^{a,1}+\tfrac{(\varepsilon_{\mathbf{L}}q)^{n_{0}}}{2}\mathbbm{1}_{\eta_{0}(-1)=-\varepsilon_{\mathbf{L}}}\cdot\sideset{}{{}_{m=0}^{n_{0}-1}}{\sum}H_{2n_{0},m}^{a,\varepsilon},$
(5.4)		$\displaystyle\displaystyle H_{2n_{0}}^{b}$	$\displaystyle\displaystyle=\tfrac{(\varepsilon_{\mathbf{L}}q)^{n_{0}}}{2}\cdot\left\{H_{2n_{0},n_{0}}^{b,\varepsilon}+H_{2n_{0},n_{0}}^{b,1}\right\},$

where the summands are given by (below we assume $\displaystyle m\geqslant 1$ and $\displaystyle\tau\in\{1,\varepsilon\}$ )

\displaystyle H_{2n_{0},0}^{a,\tau}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\tau}\alpha)\in\mathcal{P}_{\mathbf{F}}^{n_{0}}\end{subarray}}\beta(x_{\tau}\alpha)\mathrm{d}\alpha\cdot\left(\mathfrak{t}\left(\tau^{-1}\varpi_{\mathbf{F}}^{2n_{0}}\right).F_{0}\right),

\displaystyle H_{2n_{0},m}^{a,\tau}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\tau}\alpha)\in\varpi_{\mathbf{F}}^{n_{0}-m}\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\tau}\alpha)\cdot\left(\mathfrak{t}\left(\tau^{-1}{\rm Tr}(x_{\tau}\alpha)^{2}\right).F_{m}\right)\mathrm{d}\alpha;

\displaystyle H_{2n_{0},n_{0}}^{b,\varepsilon}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\varepsilon}\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\varepsilon}\alpha)\cdot\left(\mathfrak{t}\left(\varepsilon^{-1}{\rm Tr}(x_{\varepsilon}\alpha)^{2}\right).F_{n_{0}}\right)\mathrm{d}\alpha,

\displaystyle H_{2,1}^{b,1}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).F_{1}\right)\mathrm{d}\alpha,

\displaystyle H_{2n_{0},n_{0}}^{b,1}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)^{2}\in\mathcal{O}_{\mathbf{F}}^{\times}-4(1+\mathcal{P}_{\mathbf{F}})\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).F_{n_{0}}\right)\mathrm{d}\alpha,\ n_{0}\geqslant 2.

Lemma 5.7.

Let $\displaystyle\chi$ be a (unitary) character of $\displaystyle\mathbf{F}^{\times}$ with $\displaystyle\mathfrak{c}(\chi)=n$ . Recall the additive parameter $\displaystyle c_{\beta}$ (resp. $\displaystyle c_{\chi}$ ) in Lemma 3.8 (resp. Remark 3.9).

(1) Assume $\displaystyle 0\leqslant n<n_{0}$ . For $\displaystyle\mathbf{L}/\mathbf{F}$ split resp. unramified we have

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}\left(\tfrac{1+\varpi_{\mathbf{F}}^{n_{0}-n}t}{1-\varpi_{\mathbf{F}}^{n_{0}-n}t}\right)\chi(t)\mathrm{d}t\quad{\rm resp.}\quad\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon})\chi(t)\mathrm{d}t\ll q^{-\frac{n}{2}}.

(2) Assume $\displaystyle n=n_{0}\geqslant 2$ . For $\displaystyle\mathbf{L}/\mathbf{F}$ split resp. unramified we have for any $\displaystyle k\in\{0,1\}$

\int_{\mathcal{O}_{\mathbf{F}}^{\times}-\sideset{}{{}_{\pm}}{\bigcup}(\pm 1+\mathcal{P}_{\mathbf{F}})}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\chi(t)\eta_{0}^{k}(1-t^{2})\mathrm{d}t\quad{\rm resp.}\quad\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\sqrt{\varepsilon})\chi(t)\eta_{0}^{k}(1-t^{2}\varepsilon)\mathrm{d}t\\ \ll\begin{cases}q^{-\frac{n_{0}}{2}}&\text{if }\chi\notin\boldsymbol{\mathcal{E}}(\beta)\\ q^{-\frac{n_{0}}{2}+\frac{1}{2}}&\text{if }\chi\in\boldsymbol{\mathcal{E}}(\beta)\end{cases},

where $\displaystyle\boldsymbol{\mathcal{E}}(\beta)\neq\emptyset$ only if $\displaystyle 2\nmid n_{0}$ and $\displaystyle\eta_{0}(-1)=\varepsilon_{\mathbf{L}}$ , under which condition it is given by

\displaystyle\boldsymbol{\mathcal{E}}(\beta)=\begin{cases}\left\{\chi\ \middle|\ (c_{\chi}c_{\beta}^{-1})^{2}\equiv-1\pmod{\mathcal{P}_{\mathbf{F}}}\right\}&\text{if }\mathbf{L}/\mathbf{F}\text{ split}\\ \left\{\chi\ \middle|\ (c_{\chi}c_{\beta}^{-1})^{2}\equiv-\varepsilon\pmod{\mathcal{P}_{\mathbf{F}}}\right\}&\text{if }\mathbf{L}/\mathbf{F}\text{ unramified}\end{cases}.

Proof.

We omit the proof of the split case, which is similar and simpler. The case of $\displaystyle n=0<n_{0}$ is easy and omitted, since the integrand is constant.

(1) If $\displaystyle n=1$ , then $\displaystyle t\mapsto\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)$ is a non-trivial additive character of $\displaystyle\mathcal{O}_{\mathbf{F}}$ and the bound follows from the one for Gauss sums. Assume $\displaystyle n\geqslant 2$ . We perform a level $\displaystyle\lceil\tfrac{n}{2}\rceil$ regularization to $\displaystyle\mathrm{d}t$ and get

\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)\chi(t)\mathrm{d}t\\ =\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)\chi(t)\cdot\left(\oint_{\mathcal{O}_{\mathbf{F}}}\beta\left(1+\tfrac{\varpi_{\mathbf{F}}^{n_{0}-\lfloor\frac{n}{2}\rfloor}}{1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}}tu\sqrt{\varepsilon}\right)\chi(1+\varpi_{\mathbf{F}}^{\lceil\frac{n}{2}\rceil}u)\mathrm{d}u\right)dt.

The integrands of the inner integral, denoted by $\displaystyle I(t;n)$ , are additive characters of $\displaystyle\mathcal{O}_{\mathbf{F}}$ . We have

\displaystyle I(t;n)=\oint_{\mathcal{O}_{\mathbf{F}}}\psi\left(\tfrac{1}{\varpi_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\cdot\tfrac{2c_{\beta}tu\varepsilon}{1-\varpi_{\mathbf{F}}^{2(n_{0}-n)}t^{2}\varepsilon}\right)\psi\left(\tfrac{c_{\chi}u}{\varpi_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\right)\mathrm{d}u.

The non-vanishing of $\displaystyle I(t;n)$ implies the congruence condition

\displaystyle\tfrac{2c_{\beta}t\varepsilon}{1-\varpi_{\mathbf{F}}^{2(n_{0}-n)}t^{2}\varepsilon}+c_{\chi}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}\quad\Leftrightarrow\quad 2c_{\beta}t\varepsilon+c_{\chi}(1-\varpi_{\mathbf{F}}^{2(n_{0}-n)}t^{2}\varepsilon)\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor},

which has a unique solution $\displaystyle t\in t_{0}+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}$ with $\displaystyle t_{0}\in\mathcal{O}_{\mathbf{F}}^{\times}$ by Hensel’s lemma. Consequently we get

(5.5)

\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)\chi(t)\mathrm{d}t=\int_{t_{0}+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)\chi(t)\mathrm{d}t\ll q^{-\lfloor\frac{n}{2}\rfloor}.

If $\displaystyle 2\mid n$ then we are done (for both (1) and (2)). Otherwise let $\displaystyle n=2m+1$ . We may assume

\displaystyle 2c_{\beta}t_{0}\varepsilon+c_{\chi}(1-\varpi_{\mathbf{F}}^{2(n_{0}-n)}t_{0}^{2}\varepsilon)=0,

make the change of variables $\displaystyle t=t_{0}(1+\varpi_{\mathbf{F}}^{m}u)$ , and continue (5.5) to conclude by

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)\chi(t)\mathrm{d}t=q^{-m}\beta(1+\varpi_{\mathbf{F}}^{n_{0}-n}t_{0}\sqrt{\varepsilon})\chi(t_{0})\int_{\mathcal{O}_{\mathbf{F}}}\psi\left(-\tfrac{c_{\chi}u^{2}}{2\varpi_{\mathbf{F}}}\right)du\ll q^{-\frac{n}{2}}.

(2) Let $\displaystyle n=2m+1$ with $\displaystyle m\geqslant 1$ . With the change of variables $\displaystyle t=t_{0}(1+\varpi_{\mathbf{F}}^{m}u)$ for $\displaystyle t_{0}\in\mathcal{O}_{\mathbf{F}}^{\times}$ satisfying

(5.6)

2c_{\beta}t_{0}\varepsilon+c_{\chi}(1-t_{0}^{2}\varepsilon)=0,

which is solvable only if $\displaystyle\eta_{0}(-1)=\varepsilon_{\mathbf{L}}$ , we get the equation

\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\sqrt{\varepsilon})\chi(t)\eta_{0}^{k}(1-t^{2}\varepsilon)\mathrm{d}t=\sideset{}{{}_{t_{0}}}{\sum}\beta(1+t_{0}\sqrt{\varepsilon})\chi(t_{0})\eta_{0}^{k}(1-t_{0}^{2}\varepsilon)\cdot\int_{\mathcal{O}_{\mathbf{F}}}\psi\left(\tfrac{u^{2}}{\varpi}\left(\tfrac{2c_{\beta}t_{0}^{3}\varepsilon}{(1-t_{0}^{2}\varepsilon)^{2}}-\tfrac{c_{\chi}}{2}\right)\right)\mathrm{d}u\\ =\sideset{}{{}_{t_{0}}}{\sum}\beta(1+t_{0}\sqrt{\varepsilon})\chi(t_{0})\eta_{0}^{k}(1-t_{0}^{2}\varepsilon)\cdot\int_{\mathcal{O}_{\mathbf{F}}}\psi\left(\tfrac{c_{\chi}u^{2}}{2\varpi}\left(\tfrac{c_{\chi}}{c_{\beta}}t_{0}-1\right)\right)\mathrm{d}u.

The above integral is $\displaystyle\ll q^{-\frac{1}{2}}$ unless $\displaystyle t_{0}\equiv c_{\chi}^{-1}c_{\beta}\pmod{\mathcal{P}_{\mathbf{F}}}$ , in which case (5.6) implies $\displaystyle\chi\in\boldsymbol{\mathcal{E}}(\beta)$ . ∎

Lemma 5.8.

Suppose $\displaystyle n_{0}+1\leqslant n\leqslant 2n_{0}-1$ (hence $\displaystyle n_{0}\geqslant 2$ ) and $\displaystyle n\geqslant a(\Pi)$ . Then we have

\displaystyle\widetilde{h}_{2n}^{-}(\chi)\ll q^{-n_{0}}\mathbbm{1}_{2n_{0}-n}(\mathfrak{c}(\chi\eta_{0}^{n}))+q^{-n_{0}}\mathbbm{1}_{n=2n_{0}-1}\mathbbm{1}_{\mathfrak{c}(\chi\eta_{0})=0}.

Proof.

(1) First consider $\displaystyle n<2n_{0}-1$ . By Corollary 5.4 we have for any $\displaystyle\delta\in 1+\varpi_{\mathbf{F}}^{2(n-n_{0})}\mathcal{O}_{\mathbf{F}}^{\times}$

(5.7)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(4\delta\right).F_{n}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\\ =q^{-n}\zeta_{\mathbf{F}}(1)\cdot\begin{cases}\chi^{-1}\left(\tfrac{\delta}{1-\delta}\right)\cdot\gamma(1,\chi,\psi)\mathbbm{1}_{2n_{0}-n}(\mathfrak{c}(\chi))&\text{if }2\mid n\\ \chi^{-1}\eta_{0}\left(\tfrac{\delta}{1-\delta}\right)\cdot\tau_{0}q^{\frac{1}{2}}\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{2n_{0}-n}(\mathfrak{c}(\chi\eta_{0}))&\text{if }2\nmid n\end{cases},

where we used $\displaystyle\eta_{0}(\delta)=1$ . Inserting (5.7) with $\displaystyle\delta=4^{-1}{\rm Tr}(\alpha)^{2}$ into (5.2) we get

(5.8)

\widetilde{h}_{2n}^{-}(\chi)=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2(n-n_{0})}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\chi^{-1}\eta_{0}^{n}\left(\tfrac{{\rm Tr}(\alpha)^{2}}{4-{\rm Tr}(\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{2n_{0}-n}{2}}\mathbbm{1}_{2n_{0}-n}(\mathfrak{c}(\chi\eta_{0}^{n})).

In the case of split, resp. unramified $\displaystyle\mathbf{L}/\mathbf{F}$ we apply the change of variables $\displaystyle t=\tfrac{\alpha-\alpha^{-1}}{\alpha+\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{\alpha-\alpha^{-1}}{\alpha+\alpha^{-1}}\tfrac{1}{\sqrt{\varepsilon}}$ . The inner integral in (5.8) becomes

\displaystyle 2\chi\eta_{0}^{n}(-1)\int_{\varpi_{\mathbf{F}}^{n-n_{0}}\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\chi^{2}(t)\mathrm{d}t\quad{\rm resp.}\quad\chi\eta_{0}^{n}(-\varepsilon)\int_{\varpi_{\mathbf{F}}^{n-n_{0}}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\sqrt{\varepsilon})\chi^{2}(t)\mathrm{d}t.

We apply Lemma 5.7 (1) to bound the above integrals and conclude.

(2) Consider $\displaystyle n=2n_{0}-1$ . We have for any $\displaystyle\delta\in 1+\mathcal{P}_{\mathbf{F}}^{2(n_{0}-1)}$

(5.9)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(4\delta\right).F_{n}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=q^{-(2n_{0}-1)}\cdot\tau_{0}q^{\frac{1}{2}}\cdot\\ \begin{cases}\chi\eta(\varpi_{\mathbf{F}})^{2n_{0}-1}\cdot\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0}))&\text{if }\delta\in 1+\mathcal{P}_{\mathbf{F}}^{2n_{0}-1}\\ \chi^{-1}\eta_{0}\left(\tfrac{\delta}{1-\delta}\right)\cdot\zeta_{\mathbf{F}}(1)\left\{\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{1}(\mathfrak{c}(\chi\eta_{0}))-q^{-1}\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0}))\right\}&\text{if }\delta\in 1+\varpi_{\mathbf{F}}^{2(n_{0}-1)}\mathcal{O}_{\mathbf{F}}^{\times}\end{cases}.

Inserting (5.9) with $\displaystyle\delta=4^{-1}{\rm Tr}(\alpha)^{2}$ into (5.2) we get

(5.10)

\widetilde{h}_{2n}^{-}(\chi)=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2(n_{0}-1)}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\chi^{-1}\eta_{0}\left(\tfrac{{\rm Tr}(\alpha)^{2}}{4-{\rm Tr}(\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{1}{2}}\mathbbm{1}_{1}(\mathfrak{c}(\chi\eta_{0}))+\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\mathcal{P}_{\mathbf{F}}^{2n_{0}-1})\end{subarray}}\beta(\alpha)\mathrm{d}\alpha-\tfrac{\chi\eta_{0}(\varpi_{\mathbf{F}})^{-1}}{q-1}\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2(n_{0}-1)}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\mathrm{d}\alpha\right\rvert\cdot\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0})).

The first summand is bounded the same way as before. Note that $\displaystyle{\rm Tr}(\alpha)\in 2(1+\mathcal{P}_{\mathbf{F}}^{2n_{0}-1})$ is equivalent to $\displaystyle\alpha\in 1+\mathcal{P}_{\mathbf{L}}^{n_{0}}$ . With the same change of variables the inner integrals in the second summand become

\displaystyle 2q^{-n_{0}}-\tfrac{2\chi\eta_{0}(\varpi_{\mathbf{F}})^{-1}}{q-1}\int_{\varpi_{\mathbf{F}}^{n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\mathrm{d}t\quad{\rm resp.}\quad q^{-n_{0}}-\tfrac{\chi\eta_{0}(\varpi_{\mathbf{F}})^{-1}}{q-1}\int_{\varpi_{\mathbf{F}}^{n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\sqrt{\varepsilon})\mathrm{d}t.

They are of size $\displaystyle O(q^{-n_{0}})$ since the integrands are additive characters of conductor exponent $\displaystyle n_{0}$ . ∎

Lemma 5.9.

Suppose $\displaystyle(2\leqslant)a(\Pi)\leqslant m<n_{0}$ . Then we have for unitary $\displaystyle\chi$ and $\displaystyle\tau\in\{1,\varepsilon\}$

\displaystyle\widetilde{h}_{2n_{0},m}^{a,\tau}(\chi):=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(H_{2n_{0},m}^{a,\tau}\right)(t)\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll q^{-2n_{0}}\mathbbm{1}_{m>\frac{2n_{0}}{3}}\mathbbm{1}_{m}(\mathfrak{c}(\chi)).

Proof.

We first use Corollary 5.4 to obtain for any $\displaystyle\delta\in\varpi_{\mathbf{F}}^{2(n_{0}-m)}\mathcal{O}_{\mathbf{F}}^{\times}$ (note that $\displaystyle\eta_{0}(4-\delta)=1$ )

(5.11)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(\delta\right).F_{m}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\\ =q^{-n_{0}}\zeta_{\mathbf{F}}(1)\mathbbm{1}_{m>\frac{2n_{0}}{3}}\cdot\begin{cases}\chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\gamma(1,\chi,\psi)\mathbbm{1}_{m}(\mathfrak{c}(\chi))&\text{if }2\mid m\\ \chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\tau_{0}q^{\frac{1}{2}}\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{m}(\mathfrak{c}(\chi))&\text{if }2\nmid m\end{cases}.

Inserting (5.11) with $\displaystyle\delta=\tau^{-1}{\rm Tr}(x_{\tau}\alpha)^{2}$ into the integral representation of $\displaystyle H_{2n_{0},m}^{a,\tau}$ we get

(5.12)

\widetilde{h}_{2n_{0},m}^{a,\tau}(\chi)\ll\mathbbm{1}_{m>\frac{2n_{0}}{3}}\cdot\left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\tau}\alpha)\in\varpi_{\mathbf{F}}^{n_{0}-m}\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\tau}\alpha)\cdot\chi\left(\tfrac{4\tau-{\rm Tr}(x_{\tau}\alpha)^{2}}{{\rm Tr}(x_{\tau}\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-n_{0}-\frac{m}{2}}\mathbbm{1}_{m}(\mathfrak{c}(\chi)).

In the case of split, resp. unramified $\displaystyle\mathbf{L}/\mathbf{F}$ we apply the change of variables $\displaystyle t=\tfrac{\alpha+\alpha^{-1}}{\alpha-\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{\alpha+\alpha^{-1}}{\alpha-\alpha^{-1}}\tfrac{1}{\sqrt{\varepsilon}}$ for $\displaystyle\widetilde{h}_{2n_{0},m}^{a,1}(\chi)$ ; $\displaystyle t=\tfrac{\alpha+\varepsilon\alpha^{-1}}{\alpha-\varepsilon\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{x_{\varepsilon}\alpha+\overline{x_{\varepsilon}\alpha}}{x_{\varepsilon}\alpha-\overline{x_{\varepsilon}\alpha}}\tfrac{1}{\sqrt{\varepsilon}}$ for $\displaystyle\widetilde{h}_{2n_{0},m}^{a,\varepsilon}(\chi)$ . The inner integrals in (5.12) become

\displaystyle 2\chi_{0}\chi(-1)\int_{\varpi_{\mathbf{F}}^{n_{0}-m}\mathcal{O}_{\mathbf{F}}^{\times}}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\chi^{-2}(t)\mathrm{d}t\quad{\rm resp.}\quad 2\beta(\sqrt{\varepsilon})\chi^{-1}(-\varepsilon)\int_{\varpi_{\mathbf{F}}^{n_{0}-m}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\sqrt{\varepsilon})\chi^{-2}(t)\mathrm{d}t.

We apply Lemma 5.7 (1) to bound the above integrals and conclude the desired inequalities. ∎

Lemma 5.10.

Suppose $\displaystyle n_{0}\geqslant a(\Pi)(\geqslant 2)$ . Let $\displaystyle\tau\in\{1,\varepsilon\}$ . With $\displaystyle\boldsymbol{\mathcal{E}}(\beta)$ defined in Lemma 5.7 (2) we have

\widetilde{h}_{2n_{0},n_{0}}^{b,\tau}(\chi):=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(H_{2n_{0},n_{0}}^{b,\tau}\right)(t)\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\\ \ll q^{-2n_{0}}\mathbbm{1}_{n_{0}}(\mathfrak{c}(\chi))\left(\mathbbm{1}_{\boldsymbol{\mathcal{E}}(\beta)^{c}}(\chi^{2})+q^{\frac{1}{2}}\mathbbm{1}_{\boldsymbol{\mathcal{E}}(\beta)}(\chi^{2})\right).

Proof.

We first use Corollary 5.4 to obtain for any $\displaystyle\delta\in\mathcal{O}_{\mathbf{F}}^{\times}-4(1+\mathcal{P}_{\mathbf{F}})$

(5.13)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(\delta\right).F_{m}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\\ =q^{-n_{0}}\zeta_{\mathbf{F}}(1)\cdot\begin{cases}\chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\gamma(1,\chi,\psi)\mathbbm{1}_{n_{0}}(\mathfrak{c}(\chi))&\text{if }2\mid n_{0}\\ \eta_{0}(4-\delta)\chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\tau_{0}q^{\frac{1}{2}}\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{n_{0}}(\mathfrak{c}(\chi))&\text{if }2\nmid n_{0}\end{cases}.

Inserting (5.13) with $\displaystyle\delta=\tau^{-1}{\rm Tr}(x_{\tau}\alpha)^{2}$ into the integral representation of $\displaystyle H_{2n_{0},n_{0}}^{b,\tau}$ we get

(5.14)

\widetilde{h}_{2n_{0},n_{0}}^{b,1}(\chi)\ll\left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)^{2}\in\mathcal{O}_{\mathbf{F}}^{\times}-4(1+\mathcal{P}_{\mathbf{F}})\end{subarray}}\beta(\alpha)\cdot\chi\left(\tfrac{4-{\rm Tr}(\alpha)^{2}}{{\rm Tr}(\alpha)^{2}}\right)\eta_{0}^{n_{0}}\left(4-{\rm Tr}(\alpha)^{2}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{3n_{0}}{2}}\mathbbm{1}_{n_{0}}(\mathfrak{c}(\chi)).

(5.15)

\widetilde{h}_{2n_{0},n_{0}}^{b,\varepsilon}(\chi)\ll\left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\varepsilon}\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\varepsilon}\alpha)\cdot\chi\left(\tfrac{4\tau-{\rm Tr}(x_{\varepsilon}\alpha)^{2}}{{\rm Tr}(x_{\varepsilon}\alpha)^{2}}\right)\eta_{0}^{n_{0}}\left(4-\tau^{-1}{\rm Tr}(x_{\varepsilon}\alpha)^{2}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{3n_{0}}{2}}\mathbbm{1}_{n_{0}}(\mathfrak{c}(\chi)).

In the case of split, resp. unramified $\displaystyle\mathbf{L}/\mathbf{F}$ we apply the change of variables $\displaystyle t=\tfrac{\alpha+\alpha^{-1}}{\alpha-\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{\alpha+\alpha^{-1}}{\alpha-\alpha^{-1}}\tfrac{1}{\sqrt{\varepsilon}}$ for $\displaystyle\widetilde{h}_{2n_{0},n_{0}}^{b,1}(\chi)$ ; $\displaystyle t=\tfrac{\alpha+\varepsilon\alpha^{-1}}{\alpha-\varepsilon\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{x_{\varepsilon}\alpha+\overline{x_{\varepsilon}\alpha}}{x_{\varepsilon}\alpha-\overline{x_{\varepsilon}\alpha}}\tfrac{1}{\sqrt{\varepsilon}}$ for $\displaystyle\widetilde{h}_{2n_{0},n_{0}}^{b,\varepsilon}(\chi)$ . The inner integrals in (5.14) and (5.15), denoted by $\displaystyle I_{1}$ and $\displaystyle I_{\varepsilon}$ respectively, become

I_{1}=I_{\varepsilon}=2\chi_{0}\chi(-1)\int_{\mathcal{O}_{\mathbf{F}}^{\times}-\sideset{}{{}_{\pm}}{\bigcup}(\pm 1+\mathcal{P}_{\mathbf{F}})}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\chi^{-2}(t)\eta_{0}^{n_{0}}(1-t^{2})\mathrm{d}t\quad{\rm resp.}\\ \begin{cases}I_{1}=2\beta(\sqrt{\varepsilon})\chi^{-1}(-\varepsilon)\eta_{0}^{n_{0}}(-\varepsilon)\int_{\begin{subarray}{c}\mathcal{O}_{\mathbf{F}}^{\times}\\ 1-\varepsilon t^{2}\in-\varepsilon(\mathcal{O}_{\mathbf{F}}^{\times})^{2}\end{subarray}}\beta(1+t\sqrt{\varepsilon})\chi^{-2}(t)\mathrm{d}t\\ I_{\varepsilon}=2\beta(\sqrt{\varepsilon})\chi^{-1}(-\varepsilon)\eta_{0}^{n_{0}}(-1)\int_{\begin{subarray}{c}\mathcal{O}_{\mathbf{F}}^{\times}\\ 1-\varepsilon t^{2}\in-(\mathcal{O}_{\mathbf{F}}^{\times})^{2}\end{subarray}}\beta(1+t\sqrt{\varepsilon})\chi^{-2}(t)\mathrm{d}t\end{cases}.

We apply Lemma 5.7 (2) to bound the above integrals and conclude the desired inequalities. ∎

Lemma 5.11.

Let $\displaystyle n_{0}\geqslant\max(2\mathfrak{c}(\Pi),\mathfrak{c}(\Pi)+2)$ and $\displaystyle m\leqslant\mathfrak{c}(\Pi)$ . Then for any $\displaystyle\delta\in\varpi_{\mathbf{F}}^{2(n_{0}-m)}\mathcal{O}_{\mathbf{F}}^{\times}$ we have

\displaystyle\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta).F_{m}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=0.

Consequently, we get for $\displaystyle\tau\in\{1,\varepsilon\}$ and any unitary $\displaystyle\chi$ the vanishing of $\displaystyle\widetilde{h}_{2n_{0},m}^{a,\tau}(\chi)=0$ .

Proof.

For simplicity we only consider the case $\displaystyle m\geqslant 2$ . By the local functional equation and Lemma 5.3 we have (recall the parameters of $\displaystyle L(s,\Pi\otimes\xi)$ introduced in (4.14))

\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{m})(t)\xi^{-1}(t)\lvert t\rvert^{-s}\mathrm{d}^{\times}t=\mathbbm{1}_{\mathfrak{c}(\xi)=m}\cdot\zeta_{\mathbf{F}}(1)\gamma\left(1,\xi^{-2},\psi\right)\varepsilon\left(\tfrac{1}{2},\Pi\otimes\xi,\psi\right)q^{\frac{\mathfrak{c}(\Pi\otimes\xi)}{2}}\\ \cdot q^{(2m-\mathfrak{\rho}(\Pi\otimes\xi))s}\sideset{}{{}_{j=1}^{d(\Pi\otimes\xi)}}{\prod}\frac{q^{s}-a_{j}}{1-\overline{a_{j}}q^{-1+s}}.

As a Laurent series in $\displaystyle X:=q^{s}$ , the right hand side contains $\displaystyle X^{k}$ only for $\displaystyle k\geqslant 2m-\mathfrak{\rho}(\Pi\otimes\xi)$ . We have

\displaystyle\mathfrak{\rho}(\Pi\otimes\xi)\begin{cases}=\mathfrak{c}(\Pi\otimes\xi)\leqslant\mathfrak{c}(\Pi)+3\mathfrak{c}(\xi)\leqslant 4\mathfrak{c}(\Pi)&\text{if }\xi\notin E(\Pi)\\ \leqslant 2\mathfrak{c}(\Pi)+3&\text{if }\xi\in E(\Pi)\end{cases}

by [4] and Lemma 4.4 respectively. Therefore the support of $\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta).F_{m}\right)$ is contained in

\displaystyle\delta\mathcal{P}_{\mathbf{F}}^{2m-\max(4\mathfrak{c}(\Pi),2\mathfrak{c}(\Pi)+3)}=\mathcal{P}_{\mathbf{F}}^{2n_{0}-\max(4\mathfrak{c}(\Pi),2\mathfrak{c}(\Pi)+3)}\subset\mathcal{O}_{\mathbf{F}},

proving the desired vanishing. ∎

5.3. The Bounds of Dual Weight

Lemma 5.12.

Suppose $\displaystyle\mathfrak{c}(\Pi)=0$ and $\displaystyle n_{0}=1$ . Then we have $\displaystyle\widetilde{h}_{2}^{-}(\chi)\leqslant q^{-\frac{3}{2}}\mathbbm{1}_{\mathfrak{c}(\chi)=0}+q^{-1}\mathbbm{1}_{\mathfrak{c}(\chi)=1}$ .

Proof.

Necessarily we have $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ for unramified $\displaystyle\mu_{j}$ . Note that $\displaystyle H_{2}^{a}$ (resp. $\displaystyle H_{2}^{b}$ ) is related to $\displaystyle F_{0}$ (resp. $\displaystyle F_{1}$ ) by the formulae:

\displaystyle H_{2}^{a}=\varepsilon_{\mathbf{L}}\mathbbm{1}_{\eta_{0}(-1)=\varepsilon_{\mathbf{L}}}\beta(\sqrt{-1})\cdot\left(\mathfrak{t}(\varpi_{\mathbf{F}}^{2}).F_{0}\right)+\varepsilon_{\mathbf{L}}\mathbbm{1}_{\eta_{0}(-1)=-\varepsilon_{\mathbf{L}}}\beta(\sqrt{-\varepsilon})\cdot\left(\mathfrak{t}(\varepsilon^{-1}\varpi_{\mathbf{F}}^{2}).F_{0}\right),

\displaystyle H_{2}^{b}=\tfrac{\varepsilon_{\mathbf{L}}q}{2}\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}({\rm Tr}(\alpha)^{2}).F_{1}\right)\mathrm{d}\alpha+\tfrac{\varepsilon_{\mathbf{L}}q}{2}\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\varepsilon}\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\varepsilon}\alpha)\cdot\left(\mathfrak{t}(\varepsilon^{-1}{\rm Tr}(x_{\varepsilon}\alpha)^{2}).F_{1}\right)\mathrm{d}\alpha.

For any $\displaystyle\delta_{0}\in\varpi_{\mathbf{F}}^{2}\mathcal{O}_{\mathbf{F}}^{\times}$ and $\displaystyle\delta_{1}\in\mathcal{O}_{\mathbf{F}}^{\times}$ we have

\displaystyle\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta_{0}).F_{0}\right)(t)\psi(-t)\chi(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=q^{-\frac{5}{2}}\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{0})(\delta_{0}^{-1}t)\psi(-t)\chi(t)\mathrm{d}^{\times}t,

\displaystyle\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta_{1}).F_{1}\right)(t)\psi(-t)\chi(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=q^{-\frac{5}{2}}\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\mathcal{VH}_{\Pi,\psi}(F_{1})(\delta_{1}^{-1}t)\psi(-t)\chi(t)\mathrm{d}^{\times}t,

by inspecting the supports of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})$ and $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})$ given in Corollary 5.6. Define

\displaystyle K(y):=\tau_{0}q\mathbbm{1}_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\int_{(\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times})^{2}}\psi\left(t_{1}+t_{2}-\tfrac{t_{1}t_{2}}{4y}\right)\eta_{0}\left(\tfrac{4y}{t_{1}t_{2}}\right)\mathrm{d}t_{1}\mathrm{d}t_{2},

\displaystyle\widetilde{K}(\delta,\chi):=q^{-\frac{5}{2}}\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}K(\delta^{-1}t)\psi(-t)\chi(t)\mathrm{d}^{\times}t.

\displaystyle I_{1}(\chi):=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\alpha)\widetilde{K}({\rm Tr}(\alpha)^{2},\chi)\mathrm{d}\alpha,\quad I_{2}(\chi):=\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\varepsilon}\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\varepsilon}\alpha)\widetilde{K}(\varepsilon^{-1}{\rm Tr}(x_{\varepsilon}\alpha)^{2},\chi)\mathrm{d}\alpha.

The formulae of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{0})$ and $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{1})$ in Corollary 5.6 easily imply the bound

(5.16)

\widetilde{h}_{2}^{-}(\chi)\ll q^{-\frac{5}{2}}\mathbbm{1}_{\mathfrak{c}(\chi)\leqslant 1}+q\left\lvert I_{1}(\chi)+I_{2}(\chi)\right\rvert.

An easy change of variables gives

(5.17)

\widetilde{K}(\delta,\chi)=\tau_{0}q^{\frac{1}{2}}\zeta_{\mathbf{F}}(1)\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\psi\left(\tfrac{t_{1}+t_{2}-\frac{t_{1}t_{2}\delta}{4t}-t}{\varpi_{\mathbf{F}}}\right)\eta_{0}\left(\tfrac{4t}{\delta t_{1}t_{2}}\right)\chi\left(\tfrac{t}{\varpi_{\mathbf{F}}}\right)\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t\\ =\tau_{0}q^{\frac{3}{2}}\zeta_{\mathbf{F}}(1)\int_{\begin{subarray}{c}(\mathcal{O}_{\mathbf{F}}^{\times})^{4}\\ t_{1}t_{2}\equiv 4\delta^{-1}t_{3}t_{4}\pmod{\mathcal{P}_{\mathbf{F}}}\end{subarray}}\psi\left(\tfrac{t_{1}+t_{2}-t_{3}-t_{4}}{\varpi_{\mathbf{F}}}\right)\chi\left(\tfrac{t_{3}}{\varpi_{\mathbf{F}}}\right)\eta_{0}(t_{4})\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}.

Note that the function $\displaystyle t\mapsto K(\delta^{-1}t)\psi(-t)$ is invariant by $\displaystyle 1+\mathcal{P}_{\mathbf{F}}$ , hence $\displaystyle\widetilde{K}(\delta,\chi)$ is non-zero only if $\displaystyle\mathfrak{c}(\chi)\leqslant 1$ . If $\displaystyle\mathfrak{c}(\chi)=0$ then one simplifies (5.17) by performing the integrals over $\displaystyle t_{1},t_{3},t_{4},t_{2}$ in order

(5.18)

\widetilde{K}(\delta,\chi)=\chi(\varpi_{\mathbf{F}})^{-1}\zeta_{\mathbf{F}}(1)\left(q^{-\frac{5}{2}}+q^{-\frac{3}{2}}\cdot\mathbbm{1}_{\mathcal{O}_{\mathbf{F}}^{\times}}(\delta-4)\eta_{0}\left(\tfrac{\delta}{\delta-4}\right)\right).

Inserting (5.18) into (5.16) we see that the integrands $\displaystyle\widetilde{K}(\cdot)$ are constant, equal to $\displaystyle\chi(\varpi_{\mathbf{F}})^{-1}\zeta_{\mathbf{F}}(1)q^{-\frac{3}{2}}(q^{-1}+\varepsilon_{\mathbf{L}})$ , in both integrals. We obtain the bound $\displaystyle\widetilde{h}_{2}^{-}(\chi)\ll q^{-\frac{3}{2}}$ by

\displaystyle\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\alpha)\mathrm{d}\alpha=-q^{-1}\mathbbm{1}_{\eta_{0}(-1)=\varepsilon_{\mathbf{L}}}\beta(\sqrt{-1}),\quad\int_{\begin{subarray}{c}\mathbf{L}^{1}\cap\mathcal{O}_{\mathbf{L}}\\ {\rm Tr}(x_{\varepsilon}\alpha)\in\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(x_{\varepsilon}\alpha)\mathrm{d}\alpha=-q^{-1}\mathbbm{1}_{\eta_{0}(-1)=-\varepsilon_{\mathbf{L}}}\beta(\sqrt{-\varepsilon}).

If $\displaystyle\mathfrak{c}(\chi)=1$ , then regarding $\displaystyle\chi$ and $\displaystyle\eta_{0}$ as characters of $\displaystyle\mathbb{F}_{q}^{\times}$ we rewrite $\displaystyle\widetilde{K}$ as

\displaystyle\widetilde{K}(\delta,\chi)=-\tau_{0}q^{-1}\zeta_{\mathbf{F}}(1)\cdot H(4\delta^{-1},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta_{0})),

where $\displaystyle H(\cdot)$ is precisely the hypergeometric sum of Katz appeared in [21, §5.4]. In the case of split, resp. unramified $\displaystyle\mathbf{L}/\mathbf{F}$ we apply the change of variables $\displaystyle t=\tfrac{\alpha-\alpha^{-1}}{\alpha+\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{\alpha-\alpha^{-1}}{\alpha+\alpha^{-1}}\sqrt{\varepsilon}$ for $\displaystyle I_{1}(\chi)$ ; $\displaystyle t=\tfrac{\alpha-\varepsilon\alpha^{-1}}{\alpha+\varepsilon\alpha^{-1}}$ , resp. $\displaystyle t=\tfrac{x_{\varepsilon}\alpha-\overline{x_{\varepsilon}\alpha}}{x_{\varepsilon}\alpha+\overline{x_{\varepsilon}\alpha}}\sqrt{\varepsilon}$ for $\displaystyle I_{2}(\chi)$ . These integrals become

I_{1}(\chi)=I_{2}(\chi)=2\int_{\mathcal{O}_{\mathbf{F}}-\sideset{}{{}_{\pm}}{\bigcup}(\pm 1+\mathcal{P}_{\mathbf{F}})}\chi_{0}\left(\tfrac{1+t}{1-t}\right)\widetilde{K}\left(\tfrac{4}{1-t^{2}},\chi\right)\mathrm{d}t\quad{\rm resp.}\\ \begin{cases}I_{1}(\chi)=2\beta(\sqrt{\varepsilon})^{-1}\int_{\mathcal{O}_{\mathbf{F}}}\beta(t+\sqrt{\varepsilon})\widetilde{K}\left(\tfrac{4}{1-\varepsilon^{-1}t^{2}},\chi\right)\mathbbm{1}_{(\mathcal{O}_{\mathbf{F}}^{\times})^{2}}(1-\varepsilon^{-1}t^{2})\mathrm{d}t\\ I_{2}(\chi)=2\beta(\sqrt{\varepsilon})^{-1}\int_{\mathcal{O}_{\mathbf{F}}}\beta(t+\sqrt{\varepsilon})\widetilde{K}\left(\tfrac{4}{1-\varepsilon^{-1}t^{2}},\chi\right)\mathbbm{1}_{\varepsilon(\mathcal{O}_{\mathbf{F}}^{\times})^{2}}(1-\varepsilon^{-1}t^{2})\mathrm{d}t\end{cases}.

In the case of unramified $\displaystyle\mathbf{L}/\mathbf{F}$ we recognize and get the bound (with $\displaystyle H(\cdot)$ defined in [21, §5.3.2])

\displaystyle q\left\lvert I_{1}(\chi)+I_{2}(\chi)\right\rvert=2\zeta_{\mathbf{F}}(1)q^{-\frac{3}{2}}\left\lvert\sideset{}{{}_{\alpha\in\mathbb{F}_{q}}}{\sum}\beta(\alpha+\sqrt{\varepsilon})H(1-\alpha^{2}\varepsilon^{-1},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta_{0}))\right\rvert\ll q^{-1}

by [21, Lemma 5.10]. In the case of split $\displaystyle\mathbf{L}/\mathbf{F}$ we recognize and claim a bound of

(5.19)

q\left\lvert I_{1}(\chi)+I_{2}(\chi)\right\rvert=4\zeta_{\mathbf{F}}(1)q^{-\frac{3}{2}}\left\lvert\sideset{}{{}_{\alpha\in\mathbb{F}_{q}-\{\pm 1\}}}{\sum}\chi_{0}\left(\tfrac{\alpha+1}{\alpha-1}\right)H(1-\alpha^{2},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta_{0}))\right\rvert\ll q^{-1}.

The method of the proof of [21, Lemma 5.10] works through to bound the above sum. In fact the essence of that proof is to show:

(1)

the $\displaystyle\ell$ -adic sheaf associated with the function $\displaystyle\alpha\mapsto H(1-\alpha^{2}\varepsilon^{-1},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta_{0}))$ has rank $\displaystyle 2$ ,
(2)

while the sheaf associated with the function $\displaystyle\alpha\mapsto\beta(\alpha+\sqrt{\varepsilon})$ has rank $\displaystyle 1$ .

The analogues for the above split case are

(1’)

the $\displaystyle\ell$ -adic sheaf associated with the function $\displaystyle\alpha\mapsto H(1-\alpha^{2},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta_{0}))$ has rank $\displaystyle 2$ ,
(2’)

while the sheaf associated with the function $\displaystyle\alpha\mapsto\chi_{0}\left(\tfrac{\alpha+1}{\alpha-1}\right)$ has rank $\displaystyle 1$ .

Note that (1’) and (1) follow from the same argument that the sheaf associated with $\displaystyle t\mapsto H(t,q;\cdots)$ has geometric monodromy group $\displaystyle{\rm SL}_{2}$ , which does not admit finite index subgroup; while (2’) is trivially true because the Kummer sheaf has rank $\displaystyle 1$ . In fact, both (2) and (2’) are known to Weil [18, Appendix \@slowromancapv@]. We conclude the claimed bound (5.19). ∎

Proposition 5.13.

With the test function $\displaystyle H$ given by (3.5) the dual weight function is bounded as

\displaystyle\widetilde{h}(\chi)\ll_{\epsilon}\mathbf{C}(\Pi)^{2+\epsilon}q^{-n_{0}+\epsilon}\left(\mathbbm{1}_{\leqslant\max(n_{0},6\mathfrak{c}(\Pi))}(\mathfrak{c}(\chi))+q^{\frac{1}{2}}\mathbbm{1}_{\eta_{0}(-1)=\varepsilon_{\mathbf{L}}}\mathbbm{1}_{2\nmid n_{0}\geqslant 3}\mathbbm{1}_{\mathfrak{c}(\chi)=n_{0}}\mathbbm{1}_{\boldsymbol{\mathcal{E}}(\beta)}(\chi^{2})\right),

where we recall $\displaystyle\varepsilon_{\mathbf{L}}=\eta_{\mathbf{L}/\mathbf{F}}(\varpi_{\mathbf{F}})$ , and $\displaystyle\boldsymbol{\mathcal{E}}(\beta)$ is given in Lemma 5.7 (2).

Proof.

We distinguish two cases $\displaystyle\mathfrak{c}(\Pi)>0$ and $\displaystyle\mathfrak{c}(\Pi)=0$ .

(1) Suppose $\displaystyle\mathfrak{c}(\Pi)>0$ . We have $\displaystyle a(\Pi)\leqslant\mathfrak{c}(\Pi)$ by Proposition 2.4. If $\displaystyle n_{0}\leqslant 2\mathfrak{c}(\Pi)$ , then we may take $\displaystyle n_{1}=4\mathfrak{c}(\Pi)$ . Applying Lemma 4.1, 4.6 and 4.8 we deduce

\displaystyle\widetilde{h}(\chi)\ll\left\lvert\widetilde{h}_{\infty}(\chi)\right\rvert+\left\lvert\widetilde{h}_{c}^{+}(\chi)\right\rvert+\left\lvert\widetilde{h}_{c}^{-}(\chi)\right\rvert\ll\mathfrak{c}(\Pi)\mathbbm{1}_{\leqslant 6\mathfrak{c}(\Pi)}(\mathfrak{c}(\chi)),

the “worst” bound being offered by Lemma 4.8. If $\displaystyle n_{0}\geqslant 2\mathfrak{c}(\Pi)$ , then we may take $\displaystyle n_{1}=2n_{0}$ . Applying Lemma 4.1, 4.6 and Lemma 5.8-5.11 we deduce

\displaystyle\widetilde{h}(\chi)\ll q^{-n_{0}}\left(\mathbbm{1}_{\leqslant n_{0}}(\mathfrak{c}(\chi))+q^{\frac{1}{2}}\mathbbm{1}_{\eta_{0}(-1)=\varepsilon_{\mathbf{L}}}\mathbbm{1}_{2\nmid n_{0}\geqslant 3}\mathbbm{1}_{\mathfrak{c}(\chi)=n_{0}}\mathbbm{1}_{\boldsymbol{\mathcal{E}}(\beta)}(\chi)\right).

The stated bound is simply a common upper bound of the above two.

(2) Suppose $\displaystyle\mathfrak{c}(\Pi)=0$ . Necessarily we have $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ for unramified $\displaystyle\mu_{j}$ and $\displaystyle a(\Pi)=2$ . The argument of the second part in (1) works through also for $\displaystyle n_{0}\geqslant 2$ . It remains to consider the case $\displaystyle n_{0}=1$ . Note that $\displaystyle n_{1}=2$ , hence the contribution of $\displaystyle H_{\infty}$ is $\displaystyle\widetilde{h}_{\infty}(\chi)\ll q^{-1}\mathbbm{1}_{\mathfrak{c}(\chi)=0}$ by Lemma 4.1. We have

\displaystyle\widetilde{h}(\chi)=\widetilde{h}_{\infty}(\chi)+\widetilde{h}_{2}^{+}(\chi)+\widetilde{h}_{2}^{-}(\chi)

by (4.3) and the condition $\displaystyle 2\mid n$ following (5.1). By Lemma 4.6 we have $\displaystyle\widetilde{h}_{2}^{+}(\chi)\ll_{\epsilon}q^{-1+\epsilon}\mathbbm{1}_{\mathfrak{c}(\chi)=0}$ . By Lemma 5.12 we have $\displaystyle\widetilde{h}_{2}^{-}(\chi)\ll q^{-\frac{3}{2}}\mathbbm{1}_{\mathfrak{c}(\chi)=0}+q^{-1}\mathbbm{1}_{\mathfrak{c}(\chi)=1}$ . The stated bound follows. ∎

5.4. The Bounds of Unramified Dual Weight

Lemma 5.14.

Suppose $\displaystyle n_{0}+1\leqslant n\leqslant 2n_{0}-1$ and $\displaystyle n\geqslant a(\Pi)$ . Then the function $\displaystyle\widetilde{h}_{2n}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})\neq 0$ is non-vanishing only if $\displaystyle n=2n_{0}-1$ . Moreover for any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$ we have

\displaystyle\widetilde{H}_{2(2n_{0}-1)}^{-}(k;\tfrac{1}{2})\ll_{k,\epsilon}q^{-\frac{1}{2}+\left(2n_{0}-1\right)\epsilon},\quad\widetilde{H}_{2(2n_{0}-1)}^{-}(k;-\tfrac{1}{2})\ll_{k,\epsilon}q^{\frac{1}{2}-2n_{0}+\left(2n_{0}-1\right)\epsilon}.

Proof.

By Corollary 5.4 and (5.2) the support of $\displaystyle\mathcal{VH}_{\Pi,\psi}(F_{n})$ , hence also $\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})$ are contained in $\displaystyle\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}$ . Therefore we get by (5.8) the relation $\displaystyle\widetilde{h}_{2n}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=q^{ns}\widetilde{h}_{2n}^{-}(\mathbbm{1})$ . The stated results follow readily from Lemma 5.8. ∎

Lemma 5.15.

Suppose $\displaystyle\mathfrak{c}(\Pi)=0$ and $\displaystyle n_{0}=1$ . Then we have for any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$

\displaystyle\widetilde{H}_{2}^{-}(k;\tfrac{1}{2})\ll_{k,\epsilon}q^{-(1-\epsilon)},\quad\widetilde{H}_{2}^{-}(k;-\tfrac{1}{2})\ll_{k,\epsilon}q^{-2(1-\epsilon)}.

Proof.

Similarly as the previous lemma, we have $\displaystyle\widetilde{h}_{2}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=q^{s}\cdot\widetilde{h}_{2}^{-}(\mathbbm{1})$ by the inspection of the supports of $\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2})$ via Corollary 5.6 in the proof of Lemma 5.12, from which the stated bounds follow. ∎

Proposition 5.16.

With the test function $\displaystyle H$ given by (3.5) we have for any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$

\displaystyle\widetilde{H}(k;\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\frac{1}{2}}\cdot q^{2n_{0}\epsilon},\quad\widetilde{H}(k;-\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{4+\epsilon}\cdot q^{2n_{0}\epsilon}.

Proof.

The argument is quite similar to Proposition 5.13. We distinguish two cases $\displaystyle\mathfrak{c}(\Pi)>0$ and $\displaystyle\mathfrak{c}(\Pi)=0$ .

(1) Suppose $\displaystyle\mathfrak{c}(\Pi)>0$ . We take $\displaystyle n_{1}=4\mathfrak{c}(\Pi)$ if $\displaystyle n_{0}\leqslant 2\mathfrak{c}(\Pi)$ and apply Lemma 4.10 & 4.11. If $\displaystyle n_{0}\geqslant 2\mathfrak{c}(\Pi)$ we take $\displaystyle n_{1}=2n_{0}$ . The proofs of Lemma 5.8-5.11 imply $\displaystyle\widetilde{h}_{2n_{0}}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=0$ . Together with Lemma 4.10 & 5.14 we conclude the stated bounds in this case.

(2) Suppose $\displaystyle\mathfrak{c}(\Pi)=0$ . Necessarily we have $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ for unramified $\displaystyle\mu_{j}$ and $\displaystyle a(\Pi)=2$ . The argument of the second part in (1) works through also for $\displaystyle n_{0}\geqslant 2$ . It remains to consider the case $\displaystyle n_{0}=1$ . Note that $\displaystyle n_{1}=2$ . We conclude the stated bounds by Lemma 4.10 & 5.15. ∎

6. Dual Weight Functions: Ramified Cases

6.1. Second Quadratic Elementary Functions

Definition 6.1.

Let $\displaystyle\mathbf{L}=\mathbf{F}[\sqrt{\varpi_{\mathbf{F}}}]$ and recall the associated quadratic character $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}$ . For any $\displaystyle n\in\mathbb{Z}_{\geqslant 0}$ we define the second quadratic elementary functions $\displaystyle G_{n}\in{\rm C}_{c}^{\infty}(\mathbf{F}^{\times})$ by

\displaystyle G_{n}(y^{2}):=\mathbbm{1}_{v(y)=-n}\cdot\sideset{}{{}_{\pm}}{\sum}\eta_{\mathbf{L}/\mathbf{F}}(\pm y)\psi(\pm y),

and are supported in the subset of square elements of $\displaystyle\mathbf{F}^{\times}$ .

Lemma 6.2.

Let $\displaystyle\chi$ be a quasi-character of $\displaystyle\mathbf{F}^{\times}$ . We have

\int_{\mathbf{F}^{\times}}G_{n}(y)\chi(y)\mathrm{d}^{\times}y=\\ \begin{cases}\mathbbm{1}_{\mathfrak{c}(\chi^{2})=n}\cdot\zeta_{\mathbf{F}}(1)\gamma(1,\eta_{\mathbf{L}/\mathbf{F}}\chi^{-2},\psi)&\text{if }n\geqslant 2\\ \mathbbm{1}_{\mathfrak{c}(\eta_{\mathbf{L}/\mathbf{F}}\chi^{2})=1}\cdot\zeta_{\mathbf{F}}(1)\gamma(1,\eta_{\mathbf{L}/\mathbf{F}}\chi^{-2},\psi)-\mathbbm{1}_{\mathfrak{c}(\eta_{\mathbf{L}/\mathbf{F}}\chi^{2})=0}\cdot\zeta_{\mathbf{F}}(1)q^{-1}(\eta_{\mathbf{L}/\mathbf{F}}\chi^{-2})(\varpi_{\mathbf{F}})&\text{if }n=1\\ \mathbbm{1}_{\mathfrak{c}(\eta_{\mathbf{L}/\mathbf{F}}\chi^{2})=0}&\text{if }n=0\end{cases}.

Proof.

The proof is quite similar to Lemma 5.3 and is omitted. ∎

Corollary 6.3.

Let $\displaystyle n\geqslant a(\Pi)(\geqslant 2)$ , the stability barrier of $\displaystyle\Pi$ (see Proposition 2.4). We have

\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{n})(y)=\eta_{0}(-1)^{n}\eta_{0}(-2)\mathbbm{1}_{\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\cdot\begin{cases}q^{\frac{3n}{2}}\psi(4y)\eta_{0}(4y)&\text{if }2\mid n\\ \tau_{0}q^{\lceil\frac{3n}{2}\rceil}\psi(4y)&\text{if }2\nmid n\end{cases}.

Proof.

The proof is quite similar to Corollary 5.4. We only emphasize the difference. We get

\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{n})\left(\tfrac{y}{\varpi_{\mathbf{F}}^{n}}\right)=q^{3n}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\eta_{\mathbf{L}/\mathbf{F}}\left(\tfrac{t_{3}}{\varpi_{\mathbf{F}}^{n}}\right)\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{1}^{-1}t_{2}^{-1}t_{3}^{2}y}{\varpi_{\mathbf{F}}^{n}}\right)\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}.

Performing the level $\displaystyle\lceil\tfrac{n}{2}\rceil$ regularization to $\displaystyle\mathrm{d}t_{j}$ we see that the non-vanishing of the above integral implies $\displaystyle t_{1}=4y(1+u_{1})$ , $\displaystyle t_{2}=4y(1+u_{2})$ and $\displaystyle t_{3}=-8y(1+u_{3})$ with $\displaystyle u_{j}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}$ and obtain

\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{n})\left(\tfrac{y}{\varpi_{\mathbf{F}}^{n}}\right)=q^{2n}\eta_{\mathbf{L}/\mathbf{F}}\left(\tfrac{-8y}{\varpi_{\mathbf{F}}^{n}}\right)\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}\right)\int_{\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\psi\left(\tfrac{4y}{\varpi_{\mathbf{F}}^{n}}u_{2}^{2}\right)\mathrm{d}u_{2}.

We conclude by noting $\displaystyle\eta_{\mathbf{L}/\mathbf{F}}\left(\varpi_{\mathbf{F}}^{-n}y\right)=\eta_{0}(-1)^{n}\eta_{0}(y)$ for $\displaystyle y\in\mathcal{O}_{\mathbf{F}}^{\times}$ . ∎

Corollary 6.4.

(1)

We have $\displaystyle G_{0}=0$ unless $\displaystyle 4\mid q-1$ , in which case $\displaystyle\left\{\chi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}\ \middle|\ \eta_{0}=\chi^{2}\right\}=\{\eta_{1},\eta_{1}^{-1}\}$ . Extending $\displaystyle\eta_{1}$ to $\displaystyle\mathbf{F}^{\times}$ by $\displaystyle\eta_{1}(\varpi_{\mathbf{F}})=1$ we get

\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{0})(y)=\mathbbm{1}_{\varpi_{\mathbf{F}}^{-3}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\cdot q^{\frac{3}{2}}\cdot\left\{\gamma(\tfrac{1}{2},\eta_{1},\psi)^{3}\eta_{1}(y)+\gamma(\tfrac{1}{2},\eta_{1}^{-1},\psi)^{3}\eta_{1}^{-1}(y)\right\}.

(2)

We have the formula

\mathcal{VH}_{\Pi,\psi}(G_{1})(y)=\zeta_{\mathbf{F}}(1)\gamma(1,\eta_{\mathbf{L}/\mathbf{F}},\psi)\cdot\left(f_{1}*f_{2}*f_{3}\right)(\varpi_{\mathbf{F}}^{-2}y)+\\ \eta_{\mathbf{L}/\mathbf{F}}(-1)\mathbbm{1}_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}(y)\cdot\left\{\int_{(\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\eta_{0}(t_{3})\psi\left(t_{1}+t_{2}+t_{3}+\tfrac{t_{3}^{2}y}{t_{1}t_{2}}\right)\mathrm{d}\vec{t}+\zeta_{\mathbf{F}}(1)\tau_{0}\right\}.

Proof.

(1) From Lemma 6.2 and the local functional equation we deduce for any $\displaystyle\chi\in\widehat{\mathcal{O}_{\mathbf{F}}^{\times}}$

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(G_{0})(y)\chi^{-1}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\mathbbm{1}_{\chi=\eta_{1}^{\pm 1}}\cdot q^{3\left(\frac{1}{2}-s\right)}\gamma\left(\tfrac{1}{2},\chi,\psi\right)^{3}.

We readily deduce the desired formula for $\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{0})$ .

(2) We can write $\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{1})=\mathcal{VH}_{\Pi,\psi}(G_{1})_{0}+\mathcal{VH}_{\Pi,\psi}(G_{1})_{1}$ with the properties

\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{1})_{0}(y\delta)=\mathcal{VH}_{\Pi,\psi}(G_{1})_{0}(y),\quad\forall y\in\mathbf{F}^{\times},\delta\in\mathcal{O}_{\mathbf{F}}^{\times};

\displaystyle\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(G_{1})_{0}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\zeta_{\mathbf{F}}(1)\gamma(1,\eta_{\mathbf{L}/\mathbf{F}},\psi)\cdot q^{2s}\cdot\sideset{}{{}_{i=1}^{3}}{\prod}\int_{\mathbf{F}^{\times}}f_{i}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y,

\int_{\mathbf{F}^{\times}}\mathcal{VH}_{\Pi,\psi}(G_{1})_{1}(y)\chi^{-1}(y)\lvert y\rvert^{-s}\mathrm{d}^{\times}y=\\ \mathbbm{1}_{\mathfrak{c}(\chi)=1}\cdot q^{3-s}\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(y)\eta_{\mathbf{L}/\mathbf{F}}\chi^{2}(y)\mathrm{d}^{\times}y\cdot\left(\int_{\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}}\psi(y)\chi^{-1}(y)\mathrm{d}^{\times}y\right)^{3}.

We easily identify $\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{1})_{0}(y)=\zeta_{\mathbf{F}}(1)\gamma(1,\eta_{\mathbf{L}/\mathbf{F}},\psi)\cdot\left(f_{1}*f_{2}*f_{3}\right)(\varpi_{\mathbf{F}}^{-2}y)$ . We also deduce that $\displaystyle\mathrm{supp}(\mathcal{VH}_{\Pi,\psi}(G_{1})_{1})\subset\varpi_{\mathbf{F}}^{-1}\mathcal{O}_{\mathbf{F}}^{\times}$ , and for $\displaystyle y\in\mathcal{O}_{\mathbf{F}}^{\times}$

\mathcal{VH}_{\Pi,\psi}(G_{1})_{1}\left(\tfrac{y}{\varpi_{\mathbf{F}}}\right)=q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{3}}\eta_{\mathbf{L}/\mathbf{F}}\left(\tfrac{t_{4}}{\varpi_{\mathbf{F}}}\right)\psi\left(\tfrac{t_{2}+t_{3}+t_{4}+t_{2}^{-1}t_{3}^{-1}t_{4}^{2}y}{\varpi_{\mathbf{F}}}\right)\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}\\ -\zeta_{\mathbf{F}}(1)q^{3}\int_{(\mathcal{O}_{\mathbf{F}}^{\times})^{4}}\eta_{\mathbf{L}/\mathbf{F}}\left(\tfrac{t_{4}}{\varpi_{\mathbf{F}}}\right)\psi\left(\tfrac{t_{1}+t_{2}+t_{3}+t_{4}}{\varpi_{\mathbf{F}}}\right)\mathrm{d}\vec{t}.

The second summand is equal to $\displaystyle\zeta_{\mathbf{F}}(1)\eta_{\mathbf{L}/\mathbf{F}}(-1)\tau_{0}$ . We conclude by re-numbering the variables. ∎

6.2. Further Reductions

The functions $\displaystyle G_{n}$ are “building blocks” of our test functions $\displaystyle H_{c}$ in (4.3) when $\displaystyle\mathbf{L}/\mathbf{F}$ is ramified. In fact writing $\displaystyle\lambda_{\mathbf{L}}:=\lambda(\mathbf{L}/\mathbf{F},\psi)\eta_{\mathbf{L}/\mathbf{F}}(2)$ and applying Lemma 3.10 we can rewrite the summands of $\displaystyle H_{c}$ as

(6.1)		$\displaystyle\displaystyle H_{2n+1}=0\quad\text{if }n\neq n_{0}/2,$
(6.2)		$\displaystyle\displaystyle H_{2n}=\begin{cases}E_{n}&\text{if }n_{0}+1\leqslant n\leqslant n_{1}-1\\ \lambda_{\mathbf{L}}q^{n}\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\mathcal{P}_{\mathbf{F}}^{n_{0}-1})\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).G_{n}\right)\mathrm{d}\alpha&\text{if }n=n_{0}\\ \lambda_{\mathbf{L}}q^{n}\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2n-n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\left(\mathfrak{t}\left({\rm Tr}(\alpha)^{2}\right).G_{n}\right)\mathrm{d}\alpha&\text{if }\tfrac{n_{0}}{2}+1\leqslant n\leqslant n_{0}-1\end{cases}.$

The decomposition of $\displaystyle H_{n_{0}+1}$ is subtler and really goes in the direction of expression in terms of the quadratic elementary functions. We shall write (the second numeric parameter $\displaystyle m$ in subscript always indicates the parameter of the relevant quadratic elementary function)

(6.3)

H_{n_{0}+1}=\tfrac{\lambda(\mathbf{L}/\mathbf{F},\psi)}{2}q^{\frac{n_{0}+1}{2}}\cdot\sideset{}{{}_{m=0}^{\frac{n_{0}}{2}}}{\sum}H_{n_{0}+1,m},

where the summands are given by (below $\displaystyle m\geqslant 1$ )

\displaystyle H_{n_{0}+1,m}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\varpi_{\mathbf{L}}\alpha)\in\varpi_{\mathbf{F}}^{n_{0}/2+1-m}\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\varpi_{\mathbf{L}}\alpha)\cdot\eta_{\mathbf{L}/\mathbf{F}}\left({\rm Tr}(\varpi_{\mathbf{L}}\alpha)\right)\cdot\mathfrak{t}\left(-\varpi_{\mathbf{F}}^{-1}{\rm Tr}(\varpi_{\mathbf{L}}\alpha)^{2}\right).G_{m}\mathrm{d}\alpha,

\displaystyle H_{n_{0}+1,0}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\varpi_{\mathbf{L}}\alpha)\in\mathcal{P}_{\mathbf{F}}^{n_{0}/2+1}\end{subarray}}\beta(\varpi_{\mathbf{L}}\alpha)\mathrm{d}\alpha\cdot\mathfrak{t}\left((-\varpi_{\mathbf{F}})^{n_{0}+1}\right).G_{0}.

Lemma 6.5.

Let $\displaystyle\chi$ be a (unitary) character of $\displaystyle\mathbf{F}^{\times}$ with $\displaystyle\mathfrak{c}(\chi)=n\leqslant\tfrac{n_{0}}{2}$ . Recall the additive parameter $\displaystyle c_{\beta}$ (resp. $\displaystyle c_{\chi}$ ) in Lemma 3.8 (resp. Remark 3.9). We have

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-n}t\varpi_{\mathbf{L}}\right)\chi(t)\mathrm{d}t\ll q^{-\frac{n}{2}}.

Proof.

Since the proof is quite similar to the one of Lemma 5.7 (1), we skip some details. The case for $\displaystyle n=0$ is easy. If $\displaystyle n=1$ , then $\displaystyle t\mapsto\beta\left(1+\varpi_{\mathbf{F}}^{n_{0}-n}t\sqrt{\varepsilon}\right)$ is a non-trivial additive character of $\displaystyle\mathcal{O}_{\mathbf{F}}$ and the bound follows from the one for Gauss sums. Assume $\displaystyle n\geqslant 2$ . We perform a level $\displaystyle\lceil\tfrac{n}{2}\rceil$ regularization to $\displaystyle\mathrm{d}t$ and get the non-vanishing condition

\displaystyle\tfrac{2c_{\beta}t}{1-\varpi_{\mathbf{F}}^{n_{0}+1-2-n}t^{2}}+c_{\chi}\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}\quad\Leftrightarrow\quad 2c_{\beta}t+c_{\chi}(1-\varpi_{\mathbf{F}}^{n_{0}+1-2n}t^{2})\in\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor},

(6.4)

\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-n}t\varpi_{\mathbf{L}}\right)\chi(t)\mathrm{d}t=\int_{t_{0}+\mathcal{P}_{\mathbf{F}}^{\lfloor\frac{n}{2}\rfloor}}\beta\left(1+\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-n}t\varpi_{\mathbf{L}}\right)\chi(t)\mathrm{d}t\ll q^{-\lfloor\frac{n}{2}\rfloor}.

If $\displaystyle 2\mid n$ then we are done (for both (1) and (2)). Otherwise let $\displaystyle n=2m+1$ . We may assume

\displaystyle 2c_{\beta}t_{0}+c_{\chi}(1-\varpi_{\mathbf{F}}^{n_{0}+1-2n}t_{0}^{2}\varepsilon)=0,

make the change of variables $\displaystyle t=t_{0}(1+\varpi_{\mathbf{F}}^{m}u)$ , and continue (6.4) as

\displaystyle\int_{\mathcal{O}_{\mathbf{F}}^{\times}}\beta\left(1+\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-n}t\varpi_{\mathbf{L}}\right)\chi(t)\mathrm{d}t=q^{-m}\beta\left(1+\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-n}t_{0}\varpi_{\mathbf{L}}\right)\chi(t_{0})\int_{\mathcal{O}_{\mathbf{F}}}\psi\left(-\tfrac{c_{\chi}u^{2}}{2\varpi_{\mathbf{F}}}\right)du\ll q^{-\frac{n}{2}}

and conclude the proof. ∎

Lemma 6.6.

Suppose $\displaystyle\tfrac{n_{0}}{2}+1\leqslant n\leqslant n_{0}$ and $\displaystyle n\geqslant a(\Pi)$ . Then we have

\displaystyle\widetilde{h}_{2n}^{-}(\chi)\ll q^{-\frac{n_{0}+1}{2}}\mathbbm{1}_{n_{0}+1-n}(\mathfrak{c}(\chi\eta_{0}^{n+1}))+q^{-\frac{n_{0}+1}{2}}\mathbbm{1}_{n=n_{0}}\mathbbm{1}_{\mathfrak{c}(\chi\eta_{0})=0}.

Proof.

(1) First consider $\displaystyle n<n_{0}$ . By Corollary 6.3 we have for any $\displaystyle\delta\in 1+\varpi_{\mathbf{F}}^{2n-n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times}$

(6.5)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(4\delta\right).G_{n}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=\\ q^{-n}\eta_{0}(2(-1)^{n+1})\zeta_{\mathbf{F}}(1)\cdot\begin{cases}\chi^{-1}\eta_{0}\left(\tfrac{\delta}{1-\delta}\right)\cdot\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{n_{0}+1-n}(\mathfrak{c}(\chi\eta_{0}))&\text{if }2\mid n\\ \chi^{-1}\left(\tfrac{\delta}{1-\delta}\right)\cdot\tau_{0}q^{\frac{1}{2}}\cdot\gamma(1,\chi,\psi)\mathbbm{1}_{n_{0}+1-n}(\mathfrak{c}(\chi))&\text{if }2\nmid n\end{cases},

where we used $\displaystyle\eta_{0}(\delta)=1$ . Inserting (6.5) with $\displaystyle\delta=4^{-1}{\rm Tr}(\alpha)^{2}$ into (6.2) we get

(6.6)

\widetilde{h}_{2n}^{-}(\chi)=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{2n-n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\chi^{-1}\eta_{0}^{n+1}\left(\tfrac{{\rm Tr}(\alpha)^{2}}{4-{\rm Tr}(\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{n_{0}+1-n}{2}}\mathbbm{1}_{n_{0}+1-n}(\mathfrak{c}(\chi\eta_{0}^{n+1})).

Applying the change of variables $\displaystyle t=\tfrac{\alpha-\alpha^{-1}}{\alpha+\alpha^{-1}}\tfrac{1}{\varpi_{\mathbf{L}}}$ the inner integral in (6.6) becomes, taking into account the measure normalization in Proposition 3.5 (3)

\displaystyle q^{-\frac{1}{2}}\cdot\chi\eta_{0}^{n+1}(-1)\int_{\varpi_{\mathbf{F}}^{n-\frac{n_{0}}{2}-1}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\varpi_{\mathbf{L}})\chi^{2}(t)\mathrm{d}t.

We apply Lemma 6.5 (1) to bound the above integrals and conclude.

(2) Consider $\displaystyle n=n_{0}$ . We have for any $\displaystyle\delta\in 1+\mathcal{P}_{\mathbf{F}}^{n_{0}-1}$

(6.7)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(4\delta\right).G_{n}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=q^{-n_{0}}\eta_{0}(-2)\cdot\tau_{0}q^{\frac{1}{2}}\cdot\\ \begin{cases}\chi^{-1}\eta_{0}\left(\tfrac{\delta}{1-\delta}\right)\cdot\zeta_{\mathbf{F}}(1)\left\{\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{1}(\mathfrak{c}(\chi\eta_{0}))-q^{-1}\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0}))\right\}&\text{if }\delta\in 1+\varpi_{\mathbf{F}}^{n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times}\\ \chi\eta_{0}(\varpi_{\mathbf{F}})^{n_{0}}\cdot\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0}))&\text{if }\delta\in 1+\mathcal{P}_{\mathbf{F}}^{n_{0}}\end{cases}.

Inserting (6.7) with $\displaystyle\delta=4^{-1}{\rm Tr}(\alpha)^{2}$ into (6.2) we get

(6.8)

\widetilde{h}_{2n}^{-}(\chi)=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\cdot\chi^{-1}\eta_{0}\left(\tfrac{{\rm Tr}(\alpha)^{2}}{4-{\rm Tr}(\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert\cdot q^{-\frac{1}{2}}\mathbbm{1}_{1}(\mathfrak{c}(\chi\eta_{0}))+\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\mathcal{P}_{\mathbf{F}}^{n_{0}})\end{subarray}}\beta(\alpha)\mathrm{d}\alpha-\tfrac{\chi\eta_{0}(\varpi_{\mathbf{F}})^{-1}}{q-1}\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\alpha)\in 2(1+\varpi_{\mathbf{F}}^{n_{0}-1}\mathcal{O}_{\mathbf{F}}^{\times})\end{subarray}}\beta(\alpha)\mathrm{d}\alpha\right\rvert\cdot\mathbbm{1}_{0}(\mathfrak{c}(\chi\eta_{0})).

The first summand is bounded the same way as before. With the same change of variables the inner integrals of the second summand become

\displaystyle q^{-\frac{1}{2}}\cdot\left(q^{-\frac{n_{0}}{2}}-\tfrac{\chi\eta_{0}(\varpi_{\mathbf{F}})^{-1}}{q-1}\int_{\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-1}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\varpi_{\mathbf{L}})\mathrm{d}t\right).

It is of size $\displaystyle O(q^{-\frac{n_{0}+1}{2}})$ since the integrand is an additive character of conductor exponent $\displaystyle n_{0}/2$ . ∎

Lemma 6.7.

Suppose $\displaystyle(2\leqslant)a(\Pi)\leqslant m\leqslant\tfrac{n_{0}}{2}$ . Then we have for unitary $\displaystyle\chi$

\displaystyle\widetilde{h}_{n_{0}+1,m}(\chi):=\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(H_{n_{0}+1,m}\right)(t)\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\ll q^{-n_{0}-1}\mathbbm{1}_{m>\frac{n_{0}+1}{3}}\mathbbm{1}_{m}(\mathfrak{c}(\chi)).

Proof.

We first use Corollary 6.3 to obtain for any $\displaystyle\delta\in\varpi_{\mathbf{F}}^{n_{0}+1-2m}\mathcal{O}_{\mathbf{F}}^{\times}$ (note that $\displaystyle\eta_{0}(4-\delta)=1$ )

(6.9)

\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}\left(\delta\right).G_{m}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t\\ =q^{-\frac{n_{0}+1}{2}}\eta_{0}(2(-1)^{m+1})\zeta_{\mathbf{F}}(1)\mathbbm{1}_{m>\frac{n_{0}+1}{3}}\cdot\begin{cases}\chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\gamma(1,\chi\eta_{0},\psi)\mathbbm{1}_{m}(\mathfrak{c}(\chi))&\text{if }2\mid m\\ \chi\left(\tfrac{4-\delta}{\delta}\right)\cdot\tau_{0}q^{\frac{1}{2}}\gamma(1,\chi,\psi)\mathbbm{1}_{m}(\mathfrak{c}(\chi))&\text{if }2\nmid m\end{cases}.

Inserting (6.9) with $\displaystyle\delta=-\varpi_{\mathbf{F}}^{-1}{\rm Tr}(\varpi_{\mathbf{L}}\alpha)^{2}$ into the integral representation of $\displaystyle H_{n_{0}+1,m}$ we get

(6.10)

\widetilde{h}_{n_{0}+1,m}(\chi)\ll\mathbbm{1}_{m>\frac{n_{0}+1}{3}}\cdot q^{-\frac{n_{0}+1+m}{2}}\mathbbm{1}_{m}(\mathfrak{c}(\chi))\cdot\\ \left\lvert\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\varpi_{\mathbf{L}}\alpha)\in\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}+1-m}\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\varpi_{\mathbf{L}}\alpha)\eta_{\mathbf{L}/\mathbf{F}}\left({\rm Tr}(\varpi_{\mathbf{L}}\alpha)\right)\cdot\chi\left(\tfrac{4\varpi_{\mathbf{F}}+{\rm Tr}(\varpi_{\mathbf{L}}\alpha)^{2}}{-{\rm Tr}(\varpi_{\mathbf{L}}\alpha)^{2}}\right)\mathrm{d}\alpha\right\rvert.

Applying the change of variables $\displaystyle t=\tfrac{\varpi_{\mathbf{L}}\alpha+\overline{\varpi_{\mathbf{L}}\alpha}}{\varpi_{\mathbf{L}}\alpha-\overline{\varpi_{\mathbf{L}}\alpha}}\tfrac{1}{\varpi_{\mathbf{L}}}$ the inner integral in (6.10) becomes, taking into account the measure normalization in Proposition 3.5 (3)

\displaystyle q^{-\frac{1}{2}}\cdot 2\beta(\varpi_{\mathbf{L}})\eta_{\mathbf{L}/\mathbf{F}}(-2)\chi^{-1}(-\varpi_{\mathbf{F}})\int_{\varpi_{\mathbf{F}}^{\frac{n_{0}}{2}-m}\mathcal{O}_{\mathbf{F}}^{\times}}\beta(1+t\varpi_{\mathbf{L}})\eta_{\mathbf{L}/\mathbf{F}}\chi^{-2}(t)\mathrm{d}t.

We apply Lemma 6.5 (1) to bound the above integrals and conclude the desired inequalities. ∎

Lemma 6.8.

Let $\displaystyle n_{0}\geqslant\max(4\mathfrak{c}(\Pi),2\mathfrak{c}(\Pi)+2)$ and $\displaystyle m\leqslant\mathfrak{c}(\Pi)$ . Then for any $\displaystyle\delta\in\varpi_{\mathbf{F}}^{n_{0}+1-2m}\mathcal{O}_{\mathbf{F}}^{\times}$ we have

\displaystyle\int_{\mathbf{F}-\mathcal{O}_{\mathbf{F}}}\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta).G_{m}\right)(t)\cdot\psi(-t)\chi^{-1}(t)\lvert t\rvert^{-\frac{1}{2}}\mathrm{d}^{\times}t=0.

Consequently, we get for any unitary $\displaystyle\chi$ the vanishing of $\displaystyle\widetilde{h}_{n_{0}+1,m}(\chi)=0$ .

Proof.

The proof is quite similar to Lemma 5.11. One shows that the support of $\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}\left(\mathfrak{t}(\delta).G_{m}\right)$ is contained in $\displaystyle\delta\mathcal{P}_{\mathbf{F}}^{2m-\max(4\mathfrak{c}(\Pi),2\mathfrak{c}(\Pi)+3)}\subset\mathcal{O}_{\mathbf{F}}$ under the assumption. ∎

6.3. The Bounds of Dual Weight

Lemma 6.9.

Suppose $\displaystyle\mathfrak{c}(\Pi)=0$ and $\displaystyle n_{0}=2$ . We have $\displaystyle\widetilde{h}_{3}^{-}(\chi)=0$ for any unitary character $\displaystyle\chi$ .

Proof.

Necessarily we have $\displaystyle\Pi=\mu_{1}\boxplus\mu_{2}\boxplus\mu_{3}$ for unramified $\displaystyle\mu_{j}$ . Note that $\displaystyle H_{3,0}$ (resp. $\displaystyle H_{3,1}$ ) is related to $\displaystyle G_{0}$ (resp. $\displaystyle G_{1}$ ) by the formulae

\displaystyle H_{3,0}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\varpi_{\mathbf{L}}\alpha)\in\mathcal{P}_{\mathbf{F}}^{2}\end{subarray}}\beta(\varpi_{\mathbf{L}}\alpha)\mathrm{d}\alpha\cdot\mathfrak{t}\left((-\varpi_{\mathbf{F}})^{3}\right).G_{0},

\displaystyle H_{3,1}=\int_{\begin{subarray}{c}\mathbf{L}^{1}\\ {\rm Tr}(\varpi_{\mathbf{L}}\alpha)\in\varpi_{\mathbf{F}}\mathcal{O}_{\mathbf{F}}^{\times}\end{subarray}}\beta(\varpi_{\mathbf{L}}\alpha)\cdot\eta_{\mathbf{L}/\mathbf{F}}\left({\rm Tr}(\varpi_{\mathbf{L}}\alpha)\right)\cdot\mathfrak{t}\left(-\varpi_{\mathbf{F}}^{-1}{\rm Tr}(\varpi_{\mathbf{L}}\alpha)^{2}\right).G_{1}\mathrm{d}\alpha.

Inspecting the supports of $\displaystyle G_{0}$ and $\displaystyle G_{1}$ given in Corollary 6.4 we see $\displaystyle\mathrm{supp}(H_{3,m})\subset\mathcal{O}_{\mathbf{F}}$ for $\displaystyle m\in\{0,1\}$ . ∎

Proposition 6.10.

With the test function $\displaystyle H$ given by (3.5) the dual weight function is bounded as

\displaystyle\widetilde{h}(\chi)\ll_{\epsilon}\mathbf{C}(\Pi)^{2+\epsilon}q^{-\frac{n_{0}+1}{2}+\epsilon}\mathbbm{1}_{\leqslant\max(\frac{n_{0}}{2},6\mathfrak{c}(\Pi))}(\mathfrak{c}(\chi)).

Proof.

The argument is quite similar to and simpler than Proposition 5.13. In the case $\displaystyle\mathfrak{c}(\Pi)>0$ we distinguish $\displaystyle n_{0}\leqslant 4\mathfrak{c}(\Pi)$ , resp. $\displaystyle n_{0}\geqslant 4\mathfrak{c}(\Pi)$ . We apply Lemma 4.1, 4.6 and 4.8 (with $\displaystyle A=4$ ), resp. Lemma 4.1, 4.6 and Lemma 6.6-6.8. In the case $\displaystyle\mathfrak{c}(\Pi)=0$ , we apply Lemma 4.1, 4.6 and 6.9. We leave the details to the reader. ∎

6.4. The Bounds of Unramified Dual Weight

Lemma 6.11.

Suppose $\displaystyle\tfrac{n_{0}}{2}+1\leqslant n\leqslant n_{0}$ and $\displaystyle n\geqslant a(\Pi)$ . Then the function $\displaystyle\widetilde{h}_{2n}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})\neq 0$ is non-vanishing only if $\displaystyle n=n_{0}$ . Moreover for any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$ we have

\displaystyle\widetilde{H}_{2n_{0}}^{-}(k;\tfrac{1}{2})\ll_{k,\epsilon}q^{-\frac{1}{2}+n_{0}\epsilon},\quad\widetilde{H}_{2n_{0}}^{-}(k;-\tfrac{1}{2})\ll_{k,\epsilon}q^{-n_{0}-\frac{1}{2}+n_{0}\epsilon}.

Proof.

By Corollary 6.3 and (6.2) the support of $\displaystyle\mathcal{VH}_{\Pi,\psi}(G_{n})$ , hence also $\displaystyle\mathcal{VH}_{\Pi,\psi}\circ\mathfrak{m}_{-1}(H_{2n})$ are contained in $\displaystyle\varpi_{\mathbf{F}}^{-n}\mathcal{O}_{\mathbf{F}}^{\times}$ . Therefore we get by (6.6) the relation $\displaystyle\widetilde{h}_{2n}^{-}(\lvert\cdot\rvert_{\mathbf{F}}^{s})=q^{ns}\widetilde{h}_{2n}^{-}(\mathbbm{1})$ . The stated results follow readily from Lemma 6.6. ∎

Proposition 6.12.

With the test function $\displaystyle H$ given by (3.5) we have for any $\displaystyle k\in\mathbb{Z}_{\geqslant 0}$

\displaystyle\widetilde{H}(k;\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{\frac{1}{2}}\cdot q^{(n_{0}+1)\epsilon},\quad\widetilde{H}(k;-\tfrac{1}{2})\ll_{k,\epsilon}\mathbf{C}(\Pi)^{4+\epsilon}\cdot q^{(n_{0}+1)\epsilon}.

Proof.

The proof is quite similar to and simpler than the one of Proposition 5.16. We simply note that we can take $\displaystyle n_{1}=n_{0}+1$ , and leave the details to the reader. ∎

Appendix A Relation with Petrow–Young’s Exponential Sums

For simplicity of notation we shall write $\displaystyle\sideset{}{{}_{t}}{\sum}$ for $\displaystyle\sideset{}{{}_{t\in\mathbb{F}_{q}}}{\sum}$ , and use the convention $\displaystyle\rho(0)=0$ for any character $\displaystyle\rho$ of $\displaystyle\mathbb{F}_{q}^{\times}$ (even if $\displaystyle\rho=\mathbbm{1}$ is the trivial one). In (5.19) we have encountered the following algebraic exponential sum

(A.1)

S=S(\chi_{0},\chi):=\sideset{}{{}_{\alpha\in\mathbb{F}_{q}-\{\pm 1\}}}{\sum}\chi_{0}\left(\tfrac{\alpha+1}{\alpha-1}\right)H(1-\alpha^{2},q;(\mathbbm{1},\mathbbm{1}),(\chi^{-1},\eta)),

where $\displaystyle\chi_{0},\chi$ are non-trivial characters of $\displaystyle\mathbb{F}_{q}^{\times}$ and $\displaystyle\eta$ is the unique non-trivial quadratic character of $\displaystyle\mathbb{F}_{q}^{\times}$ . Note that the setting of the relevant dual weight specializes to the Petrow–Young’s [14] upon taking $\displaystyle\Pi=\mathbbm{1}\boxplus\mathbbm{1}\boxplus\mathbbm{1}$ . It is a natural question to relate $\displaystyle S$ with their algebraic exponential sum

(A.2)

T=T(\chi_{0},\chi):=\sideset{}{{}_{u,v}}{\sum}\chi\left(\tfrac{u(u+1)}{v(v+1)}\right)\chi_{0}(uv-1).

The purpose of this appendix is to give such an explicit relation. Our approach will take into account the recent discovery of Xi [22], which relates $\displaystyle T$ to a special value of a hypergeometric sum of Katz.

We need to rewrite a hyper-Kloosterman sum via the duplication formula of Gauss sums.

Definition A.1.

For a character $\displaystyle\rho$ of $\displaystyle\mathbb{F}_{q}^{\times}$ and a non-trivial character $\displaystyle\psi$ of $\displaystyle\mathbb{F}_{q}$ we have the Gauss sum

\displaystyle\tau(\rho)=\tau(\rho,\psi):=\sideset{}{{}_{t}}{\sum}\rho(t)\psi(t).

Lemma A.2.

We have the relation $\displaystyle\tau(\rho^{2})\tau(\eta)=\rho(4)\tau(\rho)\tau(\rho\eta)$ for any character $\displaystyle\rho$ of $\displaystyle\mathbb{F}_{q}^{\times}$ .

Proof.

If $\displaystyle\rho=\mathbbm{1}$ or $\displaystyle\eta$ , the stated relation trivially holds true. Assume $\displaystyle\rho\neq\mathbbm{1},\eta$ . We have the classical relation of Gauss and Jacobi sums [9, §8.3 Theorem 1]

\displaystyle\tfrac{\tau(\rho)^{2}}{\tau(\rho^{2})}=J(\rho,\rho)=\sideset{}{{}_{t}}{\sum}\rho(t(1-t)),\quad\tfrac{\tau(\rho)\tau(\eta)}{\tau(\rho\eta)}=J(\rho,\eta)=\sideset{}{{}_{t}}{\sum}\eta(t)\rho(1-t).

Since $\displaystyle q$ is odd, we can rewrite by completing the square

J(\rho,\rho)=\sideset{}{{}_{t}}{\sum}\rho(t(1-t))=\rho(4)^{-1}\sideset{}{{}_{t}}{\sum}\rho(1-t^{2})\\ =\rho(4)^{-1}\sideset{}{{}_{t}}{\sum}(1+\eta(t))\rho(1-t)=\rho(4)^{-1}\sideset{}{{}_{t}}{\sum}\eta(t)\rho(1-t)=\rho(4)^{-1}J(\rho,\eta).

The stated relation follows readily since $\displaystyle\tau(\rho)\neq 0$ . ∎

Corollary A.3.

For any $\displaystyle\delta\in\mathbb{F}_{q}^{\times}$ we have the equality

\displaystyle\sideset{}{{}_{u,t\neq 0}}{\sum}\eta\left(1-u\right)\psi\left(\tfrac{\delta}{t^{2}u}+2t\right)=\sideset{}{{}_{x_{1}x_{2}x_{3}=\delta}}{\sum}\psi(x_{1}+x_{2}+x_{3}).

Proof.

The right hand side is $\displaystyle\mathrm{Kl}_{3}(\delta)$ , a hyper-Kloosterman sum, whose Mellin transform satisfies

\displaystyle\sideset{}{{}_{\delta\in\mathbb{F}_{q}^{\times}}}{\sum}\mathrm{Kl}_{3}(\delta)\rho(\delta)=\tau(\rho)^{3},\quad\forall\rho\in\widehat{\mathbb{F}_{q}^{\times}}.

It suffices to identify the Mellin transform of the left hand side as $\displaystyle\tau(\rho)^{3}$ . We have

\sideset{}{{}_{\delta}}{\sum}\sideset{}{{}_{u,t\neq 0}}{\sum}\eta\left(1-u\right)\psi\left(\tfrac{\delta}{t^{2}u}+2t\right)\rho(\delta)=\tau(\rho)\sideset{}{{}_{u,t}}{\sum}\eta\left(1-u\right)\psi\left(2t\right)\rho(t^{2}u)\\ =\rho(4)^{-1}\tau(\rho)\tau(\rho^{2})\sideset{}{{}_{u}}{\sum}\eta\left(1-u\right)\rho(u)=\tfrac{\tau(\rho)^{2}\tau(\rho^{2})\tau(\eta)}{\rho(4)\tau(\rho\eta)}.

Lemma A.2 identifies the above right hand side precisely as $\displaystyle\tau(\rho)^{3}$ . ∎

For $\displaystyle S=S(\chi_{0},\chi)$ , we make the change of variables $\displaystyle u=1+\alpha$ and $\displaystyle v=1-\alpha$ , and detect the condition $\displaystyle u+v=2$ with the additive character $\displaystyle\psi$ to get

	$\displaystyle\displaystyle S$	$\displaystyle\displaystyle=-q^{-\frac{3}{2}}\chi_{0}(-1)\sideset{}{{}_{u+v=2}}{\sum}\chi_{0}(u)\overline{\chi_{0}}(v)\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i}\\ x_{1}x_{2}=uvy_{1}y_{2}\end{subarray}}}{\sum}\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2})$
		$\displaystyle\displaystyle=-q^{-\frac{5}{2}}\chi_{0}(-1)\sideset{}{{}_{t}}{\sum}\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i},u,v\\ x_{1}x_{2}=uvy_{1}y_{2}\end{subarray}}}{\sum}\chi_{0}(u)\overline{\chi_{0}}(v)\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2}+tu+tv-2t)$
		$\displaystyle\displaystyle=-q^{-\frac{5}{2}}(S_{1}+S_{2}).$

In the above the sum $\displaystyle S_{1}$ is defined and computed as

	$\displaystyle\displaystyle S_{1}$	$\displaystyle\displaystyle:=\chi_{0}(-1)\sideset{}{{}_{t\in\mathbb{F}_{q}^{\times}}}{\sum}\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i},u,v\\ x_{1}x_{2}=uvy_{1}y_{2}\end{subarray}}}{\sum}\chi_{0}(u)\overline{\chi_{0}}(v)\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2}-tu-tv+2t)$
		$\displaystyle\displaystyle=\chi_{0}(-1)\sideset{}{{}_{t\in\mathbb{F}_{q}^{\times}}}{\sum}\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i},u,v\\ t^{2}x_{1}x_{2}=uvy_{1}y_{2}\end{subarray}}}{\sum}\chi_{0}(u)\overline{\chi_{0}}(v)\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2}-u-v+2t)$
		$\displaystyle\displaystyle=\chi_{0}(-1)\sideset{}{{}_{t\in\mathbb{F}_{q}^{\times}}}{\sum}\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i},u,v\\ t^{2}x_{1}x_{2}=uvy_{1}\end{subarray}}}{\sum}\eta(y_{2})\psi(y_{2}(x_{2}-1))\cdot\chi_{0}(u)\overline{\chi_{0}}(v)\chi(y_{1})\psi(x_{1}+2t-u-v-y_{1})$
		$\displaystyle\displaystyle=\chi_{0}(-1)\tau(\eta)\sideset{}{{}_{t\in\mathbb{F}_{q}^{\times}}}{\sum}\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i}\\ t^{2}x_{1}x_{2}=y_{1}y_{2}y_{3}\end{subarray}}}{\sum}\eta(x_{2}-1)\chi(y_{1})\chi_{0}(y_{2})\overline{\chi_{0}}(y_{3})\psi(x_{1}+2t-y_{1}-y_{2}-y_{3})$
		$\displaystyle\displaystyle=\chi_{0}\eta(-1)\tau(\eta)\sideset{}{{}_{y_{i}}}{\sum}\chi(y_{1})\chi_{0}(y_{2})\overline{\chi_{0}}(y_{3})\psi(-y_{1}-y_{2}-y_{3})\sideset{}{{}_{x_{2},t\neq 0}}{\sum}\eta(1-x_{2})\psi\left(\tfrac{y_{1}y_{2}y_{3}}{t^{2}x_{2}}+2t\right).$

Re-naming the variable $\displaystyle u=x_{2}$ we identify the inner sum as $\displaystyle\mathrm{Kl}_{3}(y_{1}y_{2}y_{3})$ by Corollary A.3 so that

	$\displaystyle\displaystyle S_{1}$	$\displaystyle\displaystyle=\chi_{0}\eta(-1)\tau(\eta)\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i}\\ x_{1}x_{2}x_{3}=y_{1}y_{2}y_{3}\end{subarray}}}{\sum}\chi(y_{1})\chi_{0}(y_{2})\overline{\chi_{0}}(y_{3})\psi(x_{1}+x_{2}+x_{3}-y_{1}-y_{2}-y_{3})$
		$\displaystyle\displaystyle=-q^{\frac{5}{2}}\chi_{0}\eta(-1)\tau(\eta)H(1,q;(\mathbbm{1},\mathbbm{1},\mathbbm{1}),(\overline{\chi},\chi_{0},\overline{\chi_{0}}))=-q^{2}\cdot T(\chi_{0},\chi),$

where we have applied [22, Theorem 1.1] to get the last equality. The sum $\displaystyle S_{2}$ is defined as

\displaystyle S_{2}:=\chi_{0}(-1)\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i},u,v\\ x_{1}x_{2}=uvy_{1}y_{2}\end{subarray}}}{\sum}\chi_{0}(u)\overline{\chi_{0}}(v)\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2}).

Let $\displaystyle t=uv$ , so $\displaystyle u=tv^{-1}$ , and sum first over $\displaystyle v\in\mathbb{F}_{q}^{\times}$ . We see that $\displaystyle S_{2}\neq 0$ only if $\displaystyle\chi_{0}=\eta$ , and

	$\displaystyle\displaystyle S_{2}$	$\displaystyle\displaystyle=\delta_{\chi_{0}=\eta}\cdot\eta(-1)(q-1)\sideset{}{{}_{\begin{subarray}{c}t,x_{i},y_{i}\\ x_{1}x_{2}=ty_{1}y_{2}\end{subarray}}}{\sum}\chi(y_{1})\eta(ty_{2})\psi(x_{1}+x_{2}-y_{1}-y_{2})$
		$\displaystyle\displaystyle=\delta_{\chi_{0}=\eta}\cdot\eta(-1)(q-1)\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i}\\ x_{1}x_{2}=y_{1}y_{2}\end{subarray}}}{\sum}\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1})\sideset{}{{}_{t\neq 0}}{\sum}\psi(-t^{-1}y_{2})$
		$\displaystyle\displaystyle=-\delta_{\chi_{0}=\eta}\cdot\eta(-1)(q-1)\sideset{}{{}_{\begin{subarray}{c}x_{i},y_{i}\\ x_{1}x_{2}=y_{1}y_{2}\end{subarray}}}{\sum}\chi(y_{1})\eta(y_{2})\psi(x_{1}+x_{2}-y_{1})$
		$\displaystyle\displaystyle=-\delta_{\chi_{0}=\eta}\cdot\eta(-1)(q-1)\sideset{}{{}_{x_{1},x_{2},y_{1}}}{\sum}\chi(y_{1})\eta\left(\tfrac{x_{1}x_{2}}{y_{1}}\right)\psi(x_{1}+x_{2}-y_{1})$
		$\displaystyle\displaystyle=-\delta_{\chi_{0}=\eta}\cdot\eta(-1)(q-1)\tau(\eta)^{2}\overline{\tau(\eta\chi^{-1})}=-\delta_{\chi_{0}=\eta}\cdot q(q-1)\overline{\tau(\eta\chi^{-1})}.$

We summarize the above computation as the following result.

Proposition A.4.

The two algebraic exponential sums $\displaystyle S(\chi_{0},\chi)$ in (A.1) and $\displaystyle T(\chi_{0},\chi)$ in (A.2) satisfy

\displaystyle S(\chi_{0},\chi)=q^{-\frac{1}{2}}\cdot T(\chi_{0},\chi)+\delta_{\chi_{0}=\eta}\cdot q^{-\frac{1}{2}}(1-q^{-1})\overline{\tau(\eta\chi^{-1})}.

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