# A nonabelian circle method

Nuno Arala  Mathematics Institute, Zeeman Building, University of Warwick,
Coventry CV4 7AL, United Kingdom<EMAIL_ADDRESS>,  Jayce R.
Getz  Department of Mathematics
Duke University
Durham, NC 27708<EMAIL_ADDRESS>,  Jiaqi Hou  Department of Mathematics
University of Wisconsin-Madison
Madison, WI 53706<EMAIL_ADDRESS>,  Chun-Hsien Hsu  Department of
Mathematics
Duke University
Durham, NC 27708<EMAIL_ADDRESS>,  Huajie Li  Department of
Mathematics
Johns Hopkins University
Baltimore, MD 21218<EMAIL_ADDRESS>and  Victor Y. Wang  Department of
Mathematics
IST Austria<EMAIL_ADDRESS>
###### Abstract.

We count integral quaternion zeros of
$\gamma_{1}^{2}\pm\dots\pm\gamma_{n}^{2}$, giving an asymptotic when $n\geq
9$, and a likely near-optimal bound when $n=8$. To do so, we introduce a new,
nonabelian delta symbol method, which is of independent interest. Our
asymptotic at height $X$ takes the form $cX^{4n-8}+O(X^{3n+\varepsilon})$ for
suitable $c\in\mathbb{C}$ and any $\varepsilon>0.$ We construct special
subvarieties implying that, in general, $3n+\varepsilon$ can be at best
improved to $3n-2.$

###### Contents

  1. 1 Introduction
  2. 2 Conventions
  3. 3 The $\delta$-symbol method
  4. 4 Poisson summation
  5. 5 Spectral bounds for split cases
  6. 6 Spectral bounds for nonsplit cases
  7. 7 General Hessian estimates
  8. 8 General exponential sums
  9. 9 Applying the geometry of numbers

##  1\. Introduction

Let $R$ be a ring and let

$\displaystyle\delta_{a}:=\begin{cases}1&\textrm{ if }a=0,\\\ 0&\textrm{ if
}a\neq 0,\end{cases}$

be the characteristic function of $\\{0\\}\subset R.$ A $\delta$-symbol method
amounts to an expression for $\delta$ that is “analytically tractable.” One
uses it to estimate how often $a=0$ as $a$ varies over a family of
arithmetically interesting sets.

When $R=\mathbb{Z}$ a $\delta$-symbol method was introduced by Duke,
Friedlander, and Iwaniec and refined by Heath-Brown [DFI93, HB96]. It yields a
form of the circle method that has been used to establish a quantitative form
of the Hasse principle for quadratic equations in as few as $4$ variables, or
$3$ variables if the equation is homogeneous. In [BV14] the $\delta$-symbol
method was generalized to the case $R=\mathcal{O}_{F},$ where
$\mathcal{O}_{F}$ is the ring of integers in a number field.

An adelic version of the $\delta$-symbol method was introduced in [Get18]. In
this case the ring $R$ is a number field $F.$ A similar method is possible for
global function fields $R$, though for $R=\mathbb{F}_{q}[t]$ a simpler
substitute is available, as illustrated by [BV15].

In all previous settings, the $\delta$-symbol was used to study arithmetic
questions related to commutative rings (typically $\mathbb{Z}$). In this paper
we move into the nonabelian setting. More specifically, we consider the case
that $R=D$ where $D$ is a central division algebra over a number field $F.$
Prospects for further generalizations are discussed in § 3.1.

Thus let $D$ be a central division algebra over a number field $F.$ Let

(1.0.1) $\displaystyle\mathbb{R}_{>0}\longrightarrow F_{\infty}^{\times}$

be the diagonal embedding and let $A_{\mathbb{G}_{m}}$ be its image. We often
identify $F_{\infty}^{\times}$ with the center of $D_{F_{\infty}}^{\times}$
and hence $A_{\mathbb{G}_{m}}$ with a central subgroup of
$D_{F_{\infty}}^{\times}.$

Let $F\langle Y_{1},\dots,Y_{n}\rangle$ be the associative algebra of
noncommutative polynomials in $Y_{1},\dots,Y_{n}$ with coefficients in $F$ and
let $P\in F\langle Y_{1},\dots,Y_{n}\rangle$. Let
$f\in\mathcal{S}(D_{\mathbb{A}_{F}}^{n}).$ Our motivation for the expression
for the $\delta$-symbol obtained in this work is to study the asymptotic
behavior of expressions of the form

(1.0.2) $\displaystyle\Sigma(X):=\sum_{\begin{subarray}{c}\gamma\in D^{n}\\\
P(\gamma)=0\end{subarray}}f\left(\frac{\gamma}{X}\right)$

for $X\in A_{\mathbb{G}_{m}}.$ To our knowledge, equations in division
algebras have so far mainly been studied qualitatively (see e.g. [Niv46,
Pol18, BHSW21] and references within). For instance, [Niv46] constructs a
solution to $\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}=t$ for each $t$ in
the ring of _Hurwitz quaternions_

$\mathbb{H}=\left\\{\frac{a+bi+cj+dk}{2}:a\equiv b\equiv c\equiv
d\bmod{2}\right\\}\quad\textnormal{(detailed in
\cite[cite]{[\@@bibref{}{voight2021quaternion}{}{}, Chapter~{}11]})},$

with the caveat that the constructed solutions
$\gamma_{1},\gamma_{2},\gamma_{3}\in\mathbb{H}$ generally have coordinates
that are larger than the classical Waring problem for squares would allow.

We focus on quantitative questions. For $\gamma=w+xi+yj+zk\in\mathbb{H},$ we
let

$\left\lVert\gamma\right\rVert=\max(\lvert w\rvert,\lvert x\rvert,\lvert
y\rvert,\lvert
z\rvert),\quad\mathrm{trd}(\gamma)=2w,\quad\mathrm{nrd}(\gamma)=w^{2}+x^{2}+y^{2}+z^{2}.$

The simplest problem we study below is to estimate the number of solutions to

(1.0.3) $\gamma_{1}^{2}\pm\dots\pm\gamma_{n}^{2}=0$

in Hurwitz quaternions $\gamma_{1},\dots,\gamma_{n}$ with
$\left\lVert\gamma_{1}\right\rVert,\dots,\left\lVert\gamma_{n}\right\rVert\leq
X$ as $X\to\infty$. Let us briefly explain some new difficulties of this
problem compared to the classical abelian case, i.e. when $D=F$.

One difficulty is that the set of solutions of (1.0.3) seems to have a less
useful automorphism group than in the abelian case. Even the diagonal action
of $D^{\times}$ on quadratic forms in $n$ variables over $D$ given by

(1.0.4) $\displaystyle g.P(Y_{1},\dots,Y_{n}):=P(Y_{1}g,\dots,Y_{n}g)$

for $g\in D^{\times}$ is nontrivial (i.e. $g.P\neq Pg^{2}$). Moreover, we know
of no transitive group action on the set of solutions of (1.0.3) when
$D^{\times}$ is not abelian.

Another difficulty stems from an excess of points on (1.0.3) when $n$ is
small. In the abelian case, it is well known that _Hypothesis K_ [VW02,
(17.2)] holds for squares in $F$. This allows for clever use of Hölder’s
inequality, as is illustrated in [Bro21, Theorem 4.1, (4.5), and Lemma 4.5] in
the setting of cubic norm forms. However, taking $n=4$ in the following
example shows that _Hypothesis K ∗_ (coined by [Hoo97], [Bro09, §1.3.2]) fails
for squares in $\mathbb{H}$, and therefore the stronger _Hypothesis K_ also
fails for squares in $\mathbb{H}$.

######  Example.

By Cayley–Hamilton, $x^{2}=\mathrm{trd}(x)x-\mathrm{nrd}(x)$ for
$x\in\mathbb{H}$. Therefore, if $n\geq 2$, then any equation of the form
$\gamma_{1}^{2}\pm\dots\pm\gamma_{n}^{2}=0$, with not all signs positive, has
$\asymp X^{3n-2}$ solutions of height $\leq X$ with
$\mathrm{trd}(\gamma_{1})=\dots=\mathrm{trd}(\gamma_{n})=0$.111Indeed, these
linear equations cut out a single quadric in $3n$ variables. This justifies
the assertion on special subvarieties at the end of the abstract. If $n\leq 5$
(resp. $n=6$), this exceeds (resp. matches)

$X^{\\#\textrm{variables}-\textrm{degree}.\\#\textrm{equations}}=X^{4n-8},$

the size of the usual heuristic main term in the the circle method.

Thus, at least for $n\leq 5,$ we cannot expect an asymptotic point count for
(1.0.3) of the shape $cX^{4n-8}+O_{\epsilon}(X^{2n+\epsilon})$ for all
$\epsilon>0$, with an error term matching the square-root of the trivial bound
$X^{4n}$. Briefly, square-root cancellation fails. This is in stark contrast
to the abelian case, where square-root cancellation holds in a strong form by
[HB96, Get18, Tra20].

In the setting of the Hamiltonians our main theorem is the following:

######  Theorem 1.1.

Fix $n\geq 8$ and $\upsilon_{1},\dots,\upsilon_{n}\in\\{\pm 1\\}$. For
$f_{\infty}\in C_{c}^{\infty}(D_{\infty})$ there is a constant
$c(f_{\infty}\mathbf{1}_{\widehat{\mathbb{H}}^{n}})$ such that

(1.0.5) $\sum_{\begin{subarray}{c}\gamma\in\mathbb{H}^{n}\\\
\upsilon_{1}\gamma_{1}^{2}+\dots+\upsilon_{n}\gamma_{n}^{2}=0\end{subarray}}f_{\infty}(\gamma/X)=c(f_{\infty}\mathbf{1}_{\widehat{\mathbb{H}}^{n}})X^{4n-8}+O_{f_{\infty},\epsilon}(X^{3n+\epsilon}).$

Here $\widehat{\mathbb{H}}:=\prod_{p}\mathbb{H}_{\mathbb{Z}_{p}}$, where $p$
ranges over all finite primes. We discuss the constant
$c(f_{\infty}\mathbf{1}_{\widehat{\mathbb{H}}^{n}})$ in Theorem 1.3 below. The
exponent $3n+\epsilon$ in (1.0.5) (and in (1.0.6) below) might improve to
$3n-2+\epsilon$ for $n\geq 6$ if we further optimized our methods. But given
the length of the paper, we have chosen to optimize our error term only
asymptotically in $n$.

In the discussion above we restricted to $\mathbb{H}$ merely for concreteness.
We actually work much more generally with quaternion algebras $D$ over
$\mathbb{Q}$ that are nonsplit at $2$ and $\infty.$

######  Theorem 1.2.

Let $D$ be a quaternion $\mathbb{Q}$-algebra that is nonsplit at a set of
places $S\supseteq\\{2,\infty\\}$. Fix a maximal order
$\mathcal{O}_{D}\subseteq D$. Fix $n\geq 8$ and
$\upsilon_{1},\dots,\upsilon_{n}\in\\{\pm 1\\}$. Let
$P(\gamma):=\upsilon_{1}\gamma_{1}^{2}+\dots+\upsilon_{n}\gamma_{n}^{2}$. For
$f_{\infty}\in C_{c}^{\infty}(D_{\infty})$ we have

(1.0.6) $\sum_{\begin{subarray}{c}\gamma\in\mathcal{O}_{D}^{n}\\\
P(\gamma)=0\end{subarray}}f_{\infty}(\gamma/X)=c(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{n}})X^{4n-8}+O_{f_{\infty},\epsilon}(X^{3n+\epsilon}).$

It should be possible, just with some more technical effort, to treat
arbitrary test functions $f\in
C_{c}^{\infty}(D_{\mathbb{A}_{\mathbb{Q}}}^{n}).$ From a classical perspective
this amounts to adding congruence conditions. Moreover one should be able to
replace the assumption $S\supseteq\\{2,\infty\\}$ with the simpler assumption
$S\neq\emptyset$ (which, by the Hasse–Minkowski theorem, is equivalent to the
statement that $D$ is nonsplit). With yet more technical effort one should
even be able to work over an arbitrary number field. In fact the only place
where we use the fact that $F=\mathbb{Q}$ is in the geometry of numbers
estimates in §9; this is why the majority of the paper is written in the
setting of a general number field.

Let
$\psi:=\otimes_{v}\psi_{v}:\mathbb{Q}\backslash\mathbb{A}_{\mathbb{Q}}\to\mathbb{C}^{\times}$
be an additive character that is unramified at all finite places.

######  Theorem 1.3.

Under the assumptions of Theorem 1.2, one has

$\displaystyle
c(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{n}})=c(f_{\infty})\prod_{p}c(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}}),$

where

(1.0.7)
$\displaystyle\begin{split}c(f_{\infty})&=\int_{D_{\infty}}\int_{D_{\infty}^{n}}f_{\infty}(Y)\psi_{\infty}\left(\langle
P(Y),Z\rangle\right)dYdZ,\\\
c(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}})&=\int_{D_{p}}\int_{\mathcal{O}_{D_{p}}^{n}}\psi_{p}\left(\langle
P(Y),Z\rangle\right)dYdZ\in\mathbb{R}_{>0}.\end{split}$

Here the measure on $D_{v}$ is normalized so the Fourier inversion holds.
Moreover, $f_{\infty}$ can be chosen so that $c(f_{\infty})\neq 0$.

The series (1.0.7) is reminiscent of singular series appearing in the abelian
circle method. It also equals

$\displaystyle\int_{U_{p}}^{\mathrm{reg}}\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}}(u)du$

defined in the sense of [GHL21, § 7], where $U_{p}<D^{n}_{p}$ is the closed
subset cut out by $P(Y)=0$ and $du$ is a properly normalized measure on
$U_{p}$.

In the rest of this introduction we will refer to the supplementary document
[AHH+24]. None of the results in this paper require the results of [AHH+24],
but the latter document may be helpful for researchers interested in
generalizations.

### Further context

We now embed our result into the literature. Taking
$\upsilon_{1}=\dots=\upsilon_{n}=1$ for simplicity, and writing
$\gamma=w+xi+yj+zk$ with suitable integrality conditions, the equation
$P(\gamma):=\gamma_{1}^{2}+\dots+\gamma_{n}^{2}=0$ treated in Theorem 1.1
becomes

(1.0.8) $\sum_{1\leq i\leq
n}(w_{i}^{2}-x_{i}^{2}-y_{i}^{2}-z_{i}^{2})=\sum_{1\leq i\leq
n}w_{i}x_{i}=\sum_{1\leq i\leq n}w_{i}y_{i}=\sum_{1\leq i\leq n}w_{i}z_{i}=0.$

A natural approach to this specific system of equations would be to fix $w$
and exploit the linearity of the last three equations in $x,y,z$, which when
combined with uniform results on quadrics, might lead to a version of Theorem
1.1 with a worse error term as $n\to\infty$. Since our main goal is to
introduce a more general method for nonabelian equations, with Theorem 1.1
being the simplest possible example illustrating the method, we do not discuss
this approach further.

There is much work on systems of $2$ or $3$ quadrics; see e.g. [Mun15, HBP17,
PSW16, Vis23, Ara23], and references within. The best results on systems of
$\geq 4$ quadrics that are applicable to our situation are due to Myerson
[RM18], which for $\geq 4$ quadrics always supersedes Birch’s earlier work
[Bir62]. Let
$Q_{1},Q_{2},Q_{3},Q_{4}\in\mathbb{Q}[w_{i},x_{i},y_{i},z_{i}]_{1\leq i\leq
n}$ be the four quadratic forms that are set equal to $0$ in (1.0.8). Let

$\varrho_{\mathbb{R}}:=\min_{W\in\mathbb{R}^{4}-\\{0\\}}\mathrm{rank}(W_{1}Q_{1}+W_{2}Q_{2}+W_{3}Q_{3}+W_{4}Q_{4})=\min_{W\in
D_{\infty}-\\{0\\}}\mathrm{rank}(\mathrm{trd}(WP(\gamma))).$

The form $Q_{2}=\sum_{1\leq i\leq n}w_{i}x_{i}$ has rank $2n$, so
$\varrho_{\mathbb{R}}\leq 2n$. In Lemma 7.1(2), we will see that
$\varrho_{\mathbb{R}}\geq 2n$. So $\varrho_{\mathbb{R}}=2n$. The quantity
$\varrho_{\mathbb{R}}$ is closely related to singular loci discussed in
[AHH+24, Appendix A]. Myerson’s work [RM18, Theorem 1.2, (1.5)] is applicable
when

(1.0.9) $\varrho_{\mathbb{R}}>8\cdot 4,$

i.e. $n\geq 17$. In comparison, Theorem 1.1 provides an asymptotic for $n\geq
9$.

As a last contextual comment, we discuss generalizations of Theorems 1.2 and
1.3 to arbitrary quadratic polynomials. Our present work should immediately
generalize to polynomials $P(\gamma)=\gamma^{t}M\gamma$, for _symmetric_
matrices $M\in\mathrm{GL}_{n}(F)$. These polynomials are precisely the
polynomials that can be diagonalized by an $F$-linear change of variables
$\gamma\mapsto A\gamma$ with $A\in\mathrm{GL}_{n}(F)$. Not all
(noncommutative) quadratic polynomials on $D^{n}$ are of this form.

There is some chance that our methods could be adapted to handle general
$M\in\mathrm{GL}_{n}(F)$, or even $M\in\mathrm{GL}_{n}(D)$. We have focused on
diagonal $P$ for simplicity.

### Proof strategy

As previously indicated, we develop and apply a nonabelian delta method to
give a convenient analytic expression isolating the
$\gamma^{\prime}\in\mathcal{O}_{D}^{n}$ satisfying $P(\gamma^{\prime})=0.$ The
delta method is initiated in § 3. In § 4, we then apply Poisson summation in
$\gamma^{\prime}.$ After Poisson summation, we obtain a dual sum over
$\gamma\in D^{n}$ in which there are two flavors of error terms we must treat.

The first flavor of error term is a new feature of our nonabelian setting.
After Poisson summation, we require a spectral expansion of a sum over
$\delta\in D^{\times}$ to further refine the $\gamma=0$ term. This leads
naturally to a spectral expansion in terms of automorphic representations of
$D^{\times}_{\mathbb{A}_{\mathbb{Q}}}.$ The relevant local spectral estimates
are established in §§ 5 and 6.

The second flavor of error term comes, as in the abelian case, from
$\gamma\neq 0.$ However the behavior of these terms is markedly different. The
reduced norm scales quadratically, i.e. with degree $d:=\sqrt{\dim{D}}=2.$
Thus Poisson summation leads to longer dual sums over $\gamma$ than in the
abelian case $d=1$. Usual heuristics suggest that the dual sum (i.e. the sum
after applying Poisson summation) is of the same complexity as the original.
However, we discover new, nonabelian _vanishing phenomena_ in § 8, which allow
for progress over $\gamma\neq 0$. The situation is still complicated by the
fact that our complete exponential sums fail square-root cancellation,
asymptotically in $n$ [AHH+24, Appendix B]. Nonetheless, by identifying
various geometric sources of vanishing, cancellation, and sparsity, we
eventually obtain an error term that has an exponent $\frac{3}{4}$ times the
exponent of the trivial bound $X^{4n}$.222This “slope” $\frac{3}{4}$ also
appears in work on cubics, such as [Dav05, HB83, Vau86, Hoo86, Hoo97, HB98].
The tools involved are diverse, featuring Cartan decomposition, matrix
identities, iterative Gauss sum calculations, the geometry of quadric
fibrations, and lattice-point methods (geometry of numbers).

Our work on $\gamma\neq 0$ appears in §§ 7–9. General Hessian-based analysis
is done in § 7, using a rank estimate proven via the Cayley–Hamilton theorem.
Deeper local integral estimates are proven in § 8. A final global application
of the geometry of numbers appears in § 9, based on some fortuitous bounds on
the successive minima of a relevant family of lattices.

We leave Kloosterman-type averaging as an interesting open question. By this,
we mean the use of nontrivial averaging over the Fourier inversion variable in
our local integrals. This would be analogous to Kloosterman’s idea of
averaging over numerators in the abelian case, as is done in [HB96, Get18],
explicitly or implicitly. We expect the prime-case formulas in [AHH+24,
Appendix B] to be useful for this purpose. Alternatively, one could improve on
Theorems 1.1 and 1.2 by sharpening our geometry of numbers estimates in § 9 on
average. We also suspect that a secondary term of order $X^{3n-2}$ may exist,
and may be detectable by methods of [HB96, HB98, Get18, Tra20, Wan23].

We also leave the case $d=\sqrt{\dim{D}}\geq 3$ open; it is unclear what
bounds to expect. Extrapolating from $d\in\\{1,2\\}$, a tentative guess is
that after accounting for nonabelian vanishing phenomena, we might morally
have sums of length $\ll(X^{d^{2}(d-1)})^{(d-1)/(d^{2}-d)}$ and local
cancellations of quality $\gg(X^{d})^{d-1}(X^{d})^{1/2}$, all raised to the
$n$th power, for a total saving of roughly $X^{dn/2}$ over the trivial bound
$X^{d^{2}n}$. If so, then we might hope for an asymptotic point count with
main term $\asymp X^{(n-2)d^{2}}$ when $n>4d$, even without Kloosterman-type
averaging. Yet by the methods of [AHH+24, Appendix A], we have
$\sigma_{\mathbb{R}}\geq((d-1)^{2}+1)n$ for $d\geq 2$, at least if
$D_{\infty}$ is split (which is always the case if $2\nmid d$). If
$\sigma_{\mathbb{R}}=(d^{2}-2d+c)n$, where $c\geq 2$, then [RM18] requires
$n>\frac{8d^{2}}{2d-c}\geq 4d+4+\frac{4}{d-1}$, and [Bir62] requires strictly
more.

Finally, we mention that our nonabelian delta method might have a natural
interpretation in terms of “nonabelian Dirichlet arcs” like $q\theta-a\ll
1/Q$, for various elements $a,q\in D$ and parameters $Q$. It is also possible
that “nonabelian exponential sums” on these arcs could be bounded without
using Poisson summation as we do. It may also be worth comparing our setting
to the _orbital circle method_ and _orbital exponential sums_ studied by
Bourgain, Kontorovich, et al. [Kon14]. However, a key difference is that the
equations discussed in [Kon14] are valued in $F$, whereas our equations are
valued in $D$.

These are all interesting questions, which we leave open.

### Acknowledgements

This project is the outcome of the 2023-2024 Duke Research Scholars program,
organized by J. R. G. and funded by the RTG DMS-2231514.

N. A. thanks Simon Rydin Myerson for helpful conversations, and was funded
through the Engineering and Physical Sciences Research Council Doctoral
Training Partnership at the University of Warwick. Part of this work was
performed while N. A. was in residence at the Mittag-Leffler Institute in
2024, the hospitality and financial support of which are gratefully
acknowledged.

J. R. G. is partially supported by DMS–2400550. He thanks Oscar Marmon and
Lillian Pierce for inviting him to the Hausdorff School on the circle method
in 2021. This is where he first announced Lemma 3.1, which ultimately led to
the present collaboration. He also acknowledges John Voight’s help with
notational conventions.

J. H. is partially supported by DMS–1902173.

V. Y. W. thanks Tim Browning, Jakob Glas, and Damaris Schindler for
encouragement and for interesting discussions on closely related topics, and
was supported in part by the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie Grant Agreement No.
101034413, and briefly by the Stanford Math Department. Thanks are also due to
Tim Browning for several helpful comments.

##  2\. Conventions

###  2.1. Local fields

Let $F$ be a global field and let $v$ be a place of $F.$ We let $|\cdot|_{v}$
be the number theorist’s norm on $F_{v}$. If $F_{v}\cong\mathbb{R}$ then
$|\cdot|_{v}$ is the absolute value, and if $F\cong\mathbb{C}$ then
$|z|_{v}=z\overline{z}$. If $F_{v}$ is non-Archimedean, let
$\mathcal{O}_{F_{v}}$ be its ring of integers and $\varpi_{v}$ be a
uniformizer. Then $|\varpi^{-1}_{v}|_{v}$ is the cardinality $q_{v}$ of the
residue field $\mathcal{O}_{F_{v}}/\varpi_{v}\mathcal{O}_{F_{v}}$. The
valuation associated to the place $v$ is denoted by the same symbol:
$v:F_{v}\to\mathbb{Z}\cup\\{\infty\\}.$ Thus $v(\varpi_{v})=1$.

The usual norm $z\mapsto(z\overline{z})^{1/2}$ on $\mathbb{C}$ is denoted by
$|\cdot|.$ Fix a place $v$ of $F$ and drop it from notation, writing
$F:=F_{v},$ $|\cdot|:=|\cdot|_{v},$ etc. This creates the possibility of
confusion when $F\cong\mathbb{C}.$ To alleviate this, we do not identify $F$
and $\mathbb{C}$ and we take the standard convention that $|x|=x\overline{x}$
if $x\in F$ and $|y|=(y\overline{y})^{1/2}$ if $y\in\mathbb{C}.$ In other
words, the meaning of $|\cdot|$ changes depending on whether we use the symbol
$F$ or $\mathbb{C}$ for the domain.

If $F$ is a finite extension of $\mathbb{Q}_{p}$, the _standard additive
character_ $\psi_{F}:F\to\mathbb{C}^{\times}$ is defined by the formula
$x\mapsto e^{2\pi
i(\operatorname{Tr}_{F/\mathbb{Q}_{p}}(x)\bmod{\mathbb{Z}_{p}})}$, where
$\operatorname{Tr}_{F/\mathbb{Q}_{p}}(x)\bmod{\mathbb{Z}_{p}}$ denotes the
image of $\operatorname{Tr}_{F/\mathbb{Q}_{p}}(x)$ in
$\mathbb{Q}_{p}/\mathbb{Z}_{p}\cong\mathbb{Z}[1/p]/\mathbb{Z}\subset\mathbb{R}/\mathbb{Z}$.
Here the first isomorphism is induced by the inclusion of $\mathbb{Z}[1/p]$
into $\mathbb{Q}_{p}.$ If $F$ is Archimedean, then the standard additive
character is defined by $x\mapsto e^{-2\pi
i\operatorname{Tr}_{F/\mathbb{R}}(x)}$. In general, a nontrivial additive
character on $F$ is given by $x\mapsto\psi_{F}(\alpha x)$ for some $\alpha\in
F^{\times}$.

Let $F$ be a finite extension of $\mathbb{Q}_{p}$. For a nontrivial additive
character $\psi$ on $F$, its conductor $c(\psi)$ is the integer such that
$\psi$ is trivial on $\varpi^{c(\psi)}\mathcal{O}_{F}$ but not on
$\varpi^{c(\psi)-1}\mathcal{O}_{F}$. We say $\psi$ is unramified if
$c(\psi)=0$.

###  2.2. Central simple algebras

For a central simple algebra $D$ over a field $F,$ let $d:=\sqrt{\dim_{F}D}$.
We let $\mathrm{trd}:D\to F$ and $\mathrm{nrd}:D\to F$ be the reduced trace
and the reduced norm. When $d=2$, we let $\dagger$ denote the unique standard
involution on $D$ [Voi21, Definition 3.2.4 and Corollary 3.4.4]. Then
$\mathrm{trd}(A)=A+A^{\dagger}$ and
$\mathrm{nrd}(A)=A^{\dagger}A=AA^{\dagger}$. When $F$ is a number field, we
use the same notation for the adelic and local versions of these maps. At all
split places $v$, we have $D_{v}:=D\otimes_{F}F_{v}\cong M_{d}(F_{v})$ by
definition. By the Skolem-Noether theorem, for any such isomorphism
$\mathrm{trd}$ is sent to $\mathrm{tr}$ and $\mathrm{nrd}$ is sent to $\det.$
Moreover, if $d=2$, then $\dagger$ is sent to the map sending an element of
$M_{2}(F_{v})$ to its adjugate, because the adjugate is the unique standard
involution on $M_{2}(F_{v})$ [Voi21, Example 3.2.8 and Corollary 3.4.4]. Any
maximal order of $M_{d}(F_{v})$ over $\mathcal{O}_{F_{v}}$ is isomorphic to
$M_{d}(\mathcal{O}_{F_{v}})$ by an inner automorphism of $M_{d}(F_{v})$
[Voi21, Chapter 10].

Let $F$ be a local field. For any integer $m\geq 0$, the box norm on $F^{m}$
is given by

(2.2.1) $\displaystyle\left\lVert(v_{1},\ldots,v_{m})\right\rVert:=\max_{1\leq
i\leq m}|v_{i}|.$

Often we will make an identification of $D$ with $F^{d^{2}}$ as $F$-vector
spaces and let $\left\lVert\cdot\right\rVert$ be the induced box norm on $D$.
We sometimes call $\left\lVert x\right\rVert$ the _magnitude_ of $x$, to avoid
linguistic confusion with the reduced norm $\mathrm{nrd}(x)$ for $x\in D$.

######  Remark.

If $F\cong\mathbb{C}$, then $\lvert\cdot\rvert$ and
$\left\lVert\cdot\right\rVert$ are not norms in the traditional sense (they do
not satisfy the triangle inequality).

###  2.3. Measures and Fourier transforms

Let $F$ be a local field. If $F$ is non-Archimedean, the Schwarz space
$\mathcal{S}(D)=C_{c}^{\infty}(D)$ is the space of compactly supported smooth
functions. If $F$ is Archimedean $\mathcal{S}(D)$ is the usual Schwartz space
on $D$ (viewed as an $F$-vector space). Fix a nontrivial additive character
$\psi:F\to\mathbb{C}^{\times}$. For a central simple $F$-algebra $D$, let

$\displaystyle\mathcal{F}_{D}(f)(y):=\int_{D}f(x)\psi(\mathrm{trd}(xy))\,dx$

be the Fourier transform on the Schwartz space $\mathcal{S}(D).$ Here the
measure $dx$ on $D$ is always normalized to be self-dual with respect to
$\psi.$ In other words, it is normalized so that the Fourier inversion holds:

(2.3.1) $\displaystyle
f(y)=\int_{D}\mathcal{F}_{D}(f)(x)\psi(-\mathrm{trd}(xy))\,dx.$

We will often drop the subscript and write $\mathcal{F}=\mathcal{F}_{D}$
whenever the context is clear. Any Haar measure on $D^{\times}$ is a
$\mathbb{R}_{>0}$ multiple of $|\mathrm{nrd}|^{-d}dx.$ We will specify the
constant when necessary.

Suppose $F$ is non-Archimedean. Let $\mathcal{O}_{D}$ be a maximal order of
$D$ over $\mathcal{O}_{F}$ and

$\displaystyle\mathcal{O}_{D}^{\\#}:=\big{\\{}y\in
D:\mathrm{trd}(xy)\in\mathcal{O}_{F}\,\forall x\in\mathcal{O}_{D}\big{\\}}$

be its dual. Note that $\mathcal{O}_{D}^{\\#\\#}=\mathcal{O}_{D}$ and

$\displaystyle\mathcal{F}(\mathbf{1}_{\mathcal{O}_{D}})=dx(\mathcal{O}_{D})\mathbf{1}_{\varpi^{c(\psi)}\mathcal{O}_{D}^{\\#}}.$

If $D$ is split and $\psi$ is unramified, then our normalizations ensure that
the function $\mathbf{1}_{\mathcal{O}_{D}}$ is self-dual, i.e.,
$\mathcal{F}(\mathbf{1}_{\mathcal{O}_{D}})=\mathbf{1}_{\mathcal{O}_{D}}$ and
$\mathrm{vol}(\mathcal{O}_{D})=1$.

###  2.4. Gauss sum

Generalized Gauss sums will be repeatedly used in our local estimates
throughout the paper. Assume $F$ is non-Archimedean and define for
$(a,t,\xi)\in F\times F^{\times}\times F$, the _generalized Gauss sum_

$\displaystyle\mathcal{G}(a,t,\xi):=\int_{\mathcal{O}_{F}}\psi\left(\frac{ay^{2}+\xi
y}{t}\right)\,dy.$

We emphasize that our Gauss sums are normalized so that the trivial bound is
$\lvert\mathcal{G}\rvert\leq\mathrm{vol}(\mathcal{O}_{F})$; in classical
settings, Gauss sums are often normalized differently.

######  Lemma 2.1.

Suppose $F$ is a finite extension of $\mathbb{Q}_{p}$ and $\psi$ is
unramified. In general there is a constant $c_{F}>0$, with $c_{F}=1$ if $p\neq
2$, such that

(2.4.1) $\displaystyle
c_{F}\,|\mathcal{G}(a,t,\xi)|_{\infty}\leq\min(1,|t/a|)^{1/2}=q^{\min(v(a)-v(t),0)/2},$

which is an equality if $v(\xi)\geq\min(v(a),v(t))$ and $p\neq 2$. If
$|t/a|\geq 1$, i.e. $v(a)\geq v(t)$, then

$\displaystyle\mathcal{G}(a,t,\xi)=\mathcal{G}(t,t,\xi)=\mathcal{G}(0,t,\xi)=\mathbf{1}_{\mathcal{O}_{F}}(\xi/t).$

Suppose $v(a)\leq v(t)$, i.e. $|t/a|\leq 1$. Then $\mathcal{G}(a,t,\xi)=0$
unless

(2.4.2) $\xi/a\in\mathcal{O}_{F},$

i.e. $|\xi/a|\leq 1$. In addition, if $\xi/a\in 2\mathcal{O}_{F}$, then

(2.4.3)
$\displaystyle\mathcal{G}(a,t,\xi)=\psi\left(-\frac{\xi^{2}}{4at}\right)\mathcal{G}(a,t,0).$

###### Proof.

Everything is clear if $v(a)\geq v(t)$, so we may assume $v(a)\leq v(t)$.
Dividing $(a,t,\xi)$ by $a\neq 0$, we may assume $a=1$. All but the bound for
$p=2$ follow from [Get18, Lemma 6.1] and its proof. For $p=2$, it follows from
the Van der Corput Lemma [Clu11, Proposition 3.3] ∎

###  2.5. Adelic measure and Fourier transform

Let $D$ be a central simple algebra over a number field $F$. Fix a maximal
order $\mathcal{O}_{D}\subset D$ over $\mathcal{O}_{F}$. At every finite place
$v$, the localization $\mathcal{O}_{D_{v}}\subset D_{F_{v}}$ is a maximal
order over $\mathcal{O}_{F_{v}}$ by [Voi21, Lemma 10.4.3].

The Schwartz space of $D_{\mathbb{A}_{F}}$ is defined as

$\displaystyle\mathcal{S}(D_{\mathbb{A}_{F}}):=\mathcal{S}(D_{F_{\infty}})\otimes\mathcal{S}(D_{\mathbb{A}_{F}^{\infty}}).$

Here

$\displaystyle\mathcal{S}(D_{\mathbb{A}_{F}^{\infty}}):=\sideset{}{{}^{\prime}}{\bigotimes}_{v\nmid\infty}\mathcal{S}(D_{F_{v}})=\sideset{}{{}^{\prime}}{\bigotimes}_{v\nmid\infty}C^{\infty}_{c}(D_{F_{v}})$

where the restricted tensor product is taken with respect to the basic
functions $\mathbf{1}_{\mathcal{O}_{D_{v}}}$, and

$\displaystyle\mathcal{S}(D_{F_{\infty}}):=\widehat{\bigotimes}_{v|\infty}\mathcal{S}(D_{F_{v}})$

where $\mathcal{S}(D_{F_{\infty}})$ and $\mathcal{S}(D_{F_{v}})$ are the usual
Schwartz spaces on the $\mathbb{R}$-vector spaces $D_{F_{\infty}}$ (resp.
$D_{F_{v}}$) with the natural Fréchet topology, and $\widehat{\bigotimes}$ is
the completion of the (projective) tensor product. Concretely, this means that
any element of $\mathcal{S}(D_{\mathbb{A}_{F}})$ is a finite sum of functions
of the form $f_{\infty}\otimes\left(\otimes_{v\nmid\infty}f_{v}\right)$ with
$f_{\infty}\in\mathcal{S}(D_{F_{\infty}})$ and
$f_{v}\in\mathcal{S}(D_{F_{v}})$ where
$f_{v}=\mathbf{1}_{\mathcal{O}_{D_{v}}}$ for all but finitely many $v.$

Let $\psi:F\backslash\mathbb{A}_{F}\to\mathbb{C}^{\times}$ be a nontrivial
additive character. Then $\psi=\otimes_{v}\psi_{v}$ and we have an adelic
Fourier transform
$\mathcal{F}_{D_{\mathbb{A}_{F}}}:=\otimes_{v}\mathcal{F}_{D_{v}}$.
Explicitly, it is given by

$\displaystyle\mathcal{F}_{D}:\mathcal{S}(D_{\mathbb{A}_{F}})$
$\displaystyle\tilde{\longrightarrow}\mathcal{S}(D_{\mathbb{A}_{F}})$
$\displaystyle f$
$\displaystyle\longmapsto\left(y\mapsto\int_{D_{\mathbb{A}_{F}}}f(x)\psi\left(\mathrm{trd}(yx)\right)dx\right)$

where $dx=\prod_{v}dx_{v}$.

On $\mathcal{S}(D_{\mathbb{A}_{F}}^{m})$ we let $\mathcal{F}_{i}$ be the
Fourier transform in the $i$th entry. For instance if
$\Phi\in\mathcal{S}(D_{\mathbb{A}_{F}}^{2})$, then

$\displaystyle\mathcal{F}_{2}(\Phi)(x,y)=\int_{D_{\mathbb{A}_{F}}}\Phi(x,z)\psi\left(\mathrm{trd}(yz)\right)dz.$

is the Fourier transform in the second variable. We use the same notation over
local fields.

###  2.6. Analytic number theory conventions

We use the phrase “dominated by” as a synonym for “is bounded by a constant
times.”

##  3\. The $\delta$-symbol method

Let $D$ be a central division algebra over a number field $F$. For $\alpha\in
D$ let

$\displaystyle\delta_{\alpha}:=\mathbf{1}_{\alpha=0}:=\begin{cases}1&\textrm{
if }\alpha=0,\\\ 0&\textrm{ otherwise.}\end{cases}$

The following is a nonabelian expansion of the $\delta$-symbol:

######  Lemma 3.1.

Let $\Phi\in\mathcal{S}(D^{2}_{\mathbb{A}_{F}})$. Assume $\Phi(t,0)=0$ for all
$t\in D_{\mathbb{A}_{F}}$ and $\mathcal{F}_{2}(\Phi)(0,0)\neq 0$. For all
$Q\in A_{\mathbb{G}_{m}}$,

(3.0.1) $b_{\Phi,Q}\,\delta_{\alpha}=\frac{1}{|Q|^{\dim_{F}D}}\sum_{\delta\in
D^{\times}}\left(\Phi\left(\frac{\alpha\delta^{-1}}{Q},\frac{\delta}{Q}\right)-\Phi\left(\frac{\delta}{Q},\frac{\delta^{-1}\alpha}{Q}\right)\right),$

where

$b_{\Phi,Q}:=\sum_{\delta\in
D}\mathcal{F}_{2}(\Phi)(0,Q\delta)=\mathcal{F}_{2}(\Phi)(0,0)+O_{N,\Phi}(|Q|^{-N})$

for all $N\geq 0$. Moreover, for $|Q|$ sufficiently large

(3.0.2) $\delta_{\alpha}=\frac{c_{\Phi,Q}}{|Q|^{\dim_{F}D}}\sum_{\delta\in
D^{\times}}\left(\Phi\left(\frac{\alpha\delta^{-1}}{Q},\frac{\delta}{Q}\right)-\Phi\left(\frac{\delta}{Q},\frac{\delta^{-1}\alpha}{Q}\right)\right),$

where
$c_{\Phi,Q}:=b_{\Phi,Q}^{-1}=\frac{1}{\mathcal{F}_{2}(\Phi)(0,0)}+O_{N,\Phi}(|Q|^{-N})$.

###### Proof.

If $\alpha=0$ then by our vanishing assumption $\Phi(t,0)=0$, we have

$\displaystyle\sum_{\delta\in
D^{\times}}\left(\Phi\left(\frac{\alpha\delta^{-1}}{Q},\frac{\delta}{Q}\right)-\Phi\left(\frac{\delta}{Q},\frac{\delta^{-1}\alpha}{Q}\right)\right)=\sum_{\delta\in
D^{\times}}\Phi\left(0,\frac{\delta}{Q}\right)=\sum_{\delta\in
D}\Phi\left(0,\frac{\delta}{Q}\right).$

By Poisson summation this is

$\displaystyle|Q|^{\dim_{F}D}\sum_{\delta\in
D}\mathcal{F}_{2}(\Phi)(0,Q\delta)=|Q|^{\dim_{F}D}\mathcal{F}_{2}(\Phi)(0,0)+O_{N,\Phi}(|Q|^{-N}).$

This establishes the equality (3.0.1) in the lemma when $\alpha=0$.

If $\alpha\neq 0,$ by changing variables $\delta\mapsto\delta^{-1}\alpha$ we
see that

$\displaystyle\sum_{\delta\in
D^{\times}}\Phi\left(\frac{\alpha\delta^{-1}}{Q},\frac{\delta}{Q}\right)=\sum_{\delta\in
D^{\times}}\Phi\left(\frac{\delta}{Q},\frac{\delta^{-1}\alpha}{Q}\right)$

and hence the sum on the right in (3.0.1) vanishes. Of course, (3.0.2) is
immediate from (3.0.1). ∎

######  Remark.

Lemma 3.1 can be shown to hold even if $D$ is an _alternative_ non-associative
division algebra, e.g. an octonion algebra. This is because in such an algebra
unique two-sided inverses of nonzero elements exist, and if $x$ and $y$ are
both nonzero one has $(xy)^{-1}=y^{-1}x^{-1}$ and $y=x^{-1}(xy).$

Since $D$ is a division algebra, for $\alpha\in D,$ $\alpha=0$ holds if and
only if $\mathrm{nrd}(\alpha)=0$. If we replace $D$ by the split central
simple algebra $M_{n}(F),$ then the idea behind Lemma 3.1 naturally produces
an identity where the condition $\alpha=0$ is replaced by the weaker condition
$\det(\alpha)=0.$ This observation leads to the generalization we explore in
the next subsection.

###  3.1. A remark on generalizations

Let $G$ be a linear algebraic group over $F$ acting on a quasi-affine scheme
$X$ over $F:$

(3.1.1) $\displaystyle X\times G\longrightarrow X.$

Assume that $X$ admits a unique open $G$-orbit $O,$ that $b_{0}\in O(F),$ and
that the stabilizer of $b_{0}$ in $G$ is trivial. Thus the action map induces
a bijection $G(F)\tilde{\to}O(F).$ Finally assume that there are automorphisms
(of schemes, not group schemes) $\iota:X\to X$ and $\iota:G\to G$ of order $2$
such that $\iota(gh)=\iota(h)\iota(g)$ and $\iota(b_{0}.g)=b_{0}.\iota(g).$

Let $\Phi:X(\mathbb{A}_{F})\to\mathbb{C}$ be a function and let $x\in X(F).$
Consider

(3.1.2) $\displaystyle f_{\Phi}(x):=\sum_{g\in
G(F)}\Phi\left(x.g^{-1},b_{0}.\iota(g)\right)-\sum_{g\in
G(F)}\Phi\left(b_{0}.g,\iota(x).\iota(g^{-1})\right).$

Here we assume that $\Phi$ is sufficiently nice that both sums converge
absolutely. For example, if $X$ is smooth we can take $\Phi\in
C_{c}^{\infty}(X(\mathbb{A}_{F}))=C_{c}^{\infty}(X(F_{\infty}))\otimes
C^{\infty}_{c}(X(\mathbb{A}_{F}^{\infty})).$

If $x\in O(F)$ we can choose an $h\in G(F)$ such that $b_{0}.h=x.$ We then
change variables $g\mapsto g^{-1}h$ in the first sum in the definition of
$f_{\Phi}(x)$ to see that $f_{\Phi}(x)=0$ for $x\in O(F).$ In other words,
$f_{\Phi}(x)$ vanishes on the open orbit $O(F).$ We hasten to point out that
$f_{\Phi}$ is not necessarily constant away from $O(F)$, unless $X(F)-O(F)$
consists of a single point.

Expressions of the type (3.1.2) could be useful for counting points of schemes
that do not lie in an open orbit under a group action. The $\delta$-symbol in
Lemma 3.1 is essentially the special case where $X$ and $G$ are the scheme and
group with points in an $F$-algebra $R$ given by $X(R):=D\otimes R,$
$G(R):=(D\otimes R)^{\times}.$ In this case $X(F)-O(F)=\\{0\\},$ we can take
$\iota$ to be induced by the standard involution $\dagger$ on $D,$ and
$b_{0}=I_{D}.$ In order to simplify the presentation we did not incorporate
the involution into Lemma 3.1.

Another natural case is when $G=\mathrm{GL}_{n}$ and $X=M_{n\times n}$
($n\times n$ matrices) equipped with the usual right action. In this case
$X(F)-O(F)=\\{Z\in M_{n\times n}(F):\det(Z)=0\\}$ and we can take $\iota$ to
be the transpose. In unpublished work Myerson and Vishe considered this
setting when $n=2.$

One could also work more generally with reductive monoids, and make use of the
Poisson summation formula for these objects conjectured to exist by Braverman
and Kazhdan [BK00, Ngô14]. The conjectural Poisson summation formula has been
established in certain cases, see [BK99, BK02, CG21, GH20, GHL21, GL19a,
GL19b].

##  4\. Poisson summation

Let $f\in C_{c}^{\infty}(D_{\mathbb{A}_{F}}^{n})$ and $X\in
A_{\mathbb{G}_{m}}.$ In this section we study the analytic behavior as
$|X|\to\infty$ of

$\displaystyle\Sigma(X):=\sum_{\begin{subarray}{c}\gamma^{\prime}\in D^{n}\\\
P(\gamma^{\prime})=0\end{subarray}}f\left(\frac{\gamma^{\prime}}{X}\right).$

Assume henceforth that $P$ is homogeneous of degree $2$ and that $D$ is a
quaternion division algebra over $F.$ This is the setting relevant for Theorem
1.2 and making this assumption now simplifies the notation required below.

Choose a pure tensor $\Phi=\otimes_{v}\Phi_{v}\in
C^{\infty}_{c}(D_{\mathbb{A}_{F}}^{2})$ such that the assumption of Lemma 3.1
is satisfied, i.e. $\Phi(t,0)=0$ for all $t\in D_{\mathbb{A}_{F}}$ and
$\int_{D_{\mathbb{A}_{F}}}\Phi(0,t)dt\neq 0.$ Applying the $\delta$-symbol
expansion of Lemma 3.1 (with $Q=X\gg_{\Phi}1$), we have

(4.0.1) $\displaystyle\Sigma(X)=\frac{c_{\Phi,X}}{|X|^{4}}\sum_{\gamma\in
D^{n}}f\left(\frac{\gamma^{\prime}}{X}\right)\sum_{\delta\in
D^{\times}}\Phi^{\operatorname{diff}}\left(\frac{\gamma^{\prime}}{X},\frac{\delta}{X}\right),$

where for convenience

(4.0.2)
$\displaystyle\Phi^{\operatorname{diff}}(\gamma^{\prime},\delta):=\Phi\left(P(\gamma^{\prime})\delta^{-1},\delta\right)-\Phi\left(\delta,\delta^{-1}P(\gamma^{\prime})\right)$

Here we have used the fact that
$P(\gamma^{\prime}/X)=P(\gamma^{\prime})/X^{2}$.

Writing $\gamma^{\prime}=(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime})$, we
define pairings

(4.0.3) $\displaystyle\begin{split}\cdot:D_{\mathbb{A}_{F}}^{n}\times
D_{\mathbb{A}_{F}}^{n}&\longrightarrow D_{\mathbb{A}_{F}}\\\
(\gamma,\gamma^{\prime})&\longmapsto\sum_{i=1}^{n}\gamma_{i}\gamma_{i}^{\prime},\\\
\langle\,,\,\rangle:D_{\mathbb{A}_{F}}^{n}\times
D_{\mathbb{A}_{F}}^{n}&\longrightarrow\mathbb{A}_{F}\\\
(\gamma,\gamma^{\prime})&\longmapsto\mathrm{trd}(\gamma\cdot\gamma^{\prime})=\sum_{i=1}^{n}\mathrm{trd}\,\gamma_{i}\gamma_{i}^{\prime}.\end{split}$

Applying Poisson summation in $\gamma^{\prime}\in D^{n}$ to (4.0.1), we have

$\displaystyle\Sigma(X)=\frac{c_{\Phi,X}}{|X|^{4}}\sum_{\gamma\in
D^{n}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(\frac{t}{X}\right)\sum_{\delta\in
D^{\times}}\Phi^{\operatorname{diff}}\left(\frac{t}{X},\frac{\delta}{X}\right)\psi\left(\langle\gamma,t\rangle\right)\,dt.$

By the change of variables $\gamma\mapsto\frac{\gamma}{\mathrm{nrd}(\delta)}$,
we arrive at

$\displaystyle\Sigma(X)=\frac{c_{\Phi,X}}{|X|^{4}}\sum_{\gamma\in
D^{n}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(\frac{t}{X}\right)\sum_{\delta\in
D^{\times}}\Phi^{\operatorname{diff}}\left(\frac{t}{X},\frac{\delta}{X}\right)\psi\left(\frac{\langle\gamma,t\rangle}{\mathrm{nrd}(\delta)}\right)\,dt.$

Changing variables $t\mapsto Xt$ and using the fact that
$\mathrm{nrd}(X)=X^{2}$, we obtain

(4.0.4) $\displaystyle\Sigma(X)=c_{\Phi,X}|X|^{4(n-1)}\sum_{\gamma\in
D^{n}}\int_{D_{\mathbb{A}_{F}}^{n}}f(t)\sum_{\delta\in
D^{\times}}\Phi^{\operatorname{diff}}\left(t,\frac{\delta}{X}\right)\psi\left(\frac{\langle\gamma/X,t\rangle}{\mathrm{nrd}(\delta/X)}\right)\,dt.$

For $(\delta,\gamma)\in D_{\mathbb{A}_{F}}^{\times}\times
D_{\mathbb{A}_{F}}^{n},$ define

(4.0.5)
$\displaystyle\begin{split}I_{0}(\delta,\gamma):=I_{0}(f,\Phi,\delta,\gamma)&:=\int_{D_{\mathbb{A}_{F}}^{n}}f(t)\Phi(\delta,\delta^{-1}P(t))\psi\left(\frac{\langle\gamma,t\rangle}{\mathrm{nrd}(\delta)}\right)\,dt,\\\
I_{1}(\delta,\gamma):=I_{1}(f,\Phi,\delta,\gamma)&:=\int_{D^{n}_{\mathbb{A}_{F}}}f(t)\Phi(P(t)\delta^{-1},\delta)\psi\left(\frac{\langle\gamma,t\rangle}{\mathrm{nrd}(\delta)}\right)\,dt.\end{split}$

Then (4.0.4) can be written as

(4.0.6) $\begin{split}\Sigma(X)&=c_{\Phi,X}X^{4(n-1)}\sum_{\gamma\in
D^{n}}\sum_{\delta\in
D^{\times}}\left(I_{1}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)-I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\right)\end{split}$

Defining (4.0.3) and (4.0.5) analogously over local fields, one has for pure
tensors $f=\otimes_{v}f$

$\displaystyle
I_{i}(f,\Phi,\delta,\gamma)=\prod_{v}I_{i,v}(f_{v},\Phi_{v},\delta,\gamma).$

To avoid potential confusion, we point out that since $X\in
A_{\mathbb{G}_{m}}\cong\mathbb{R}_{>0},$ if $f$ and $\Phi$ are pure tensors
then
$I_{i}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)=I_{i}\left(\delta,\gamma\right)$
for all $v\nmid\infty.$ In other words only the Archimedean factors
$I_{i,\infty}$ (i.e. those with $v\mid\infty$) in (4.0.6) depend on the
parameter $X.$

Moreover, by the change of variables $t\mapsto t^{\dagger}$ on
$D_{\mathbb{A}_{F}}^{n}$, and the cyclicity of $\mathrm{trd}$, we have

(4.0.7)
$I_{0}(f,\Phi,\delta,\gamma)=I_{1}(f\circ\dagger,\Phi^{\operatorname{sw}}\circ\dagger,\delta^{\dagger},\gamma^{\dagger}),$

where $f\circ\dagger$ is the map $t\mapsto f(t^{\dagger})$, and
$\Phi^{\operatorname{sw}}\circ\dagger$ is the map
$(x,y)\mapsto\Phi(y^{\dagger},x^{\dagger})$. So we may concentrate on either
$I_{0}$ or $I_{1}$ for the most part.

Write $\Sigma(X)=\Sigma_{0}(X)+E_{1}(X),$ where

(4.0.8) $\Sigma_{0}(X):=c_{\Phi,X}X^{4(n-1)}\sum_{\delta\in
D^{\times}}\left(I_{1}\left(\frac{\delta}{X},0\right)-I_{0}\left(\frac{\delta}{X},0\right)\right),$
(4.0.9) $E_{1}(X):=c_{\Phi,X}X^{4(n-1)}\sum_{\gamma\in
D^{n}-\\{0\\}}\sum_{\delta\in
D^{\times}}\left(I_{1}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)-I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\right).$

We expect that $E_{1}(X)\ll_{f,\Phi,\epsilon}X^{3n+\epsilon}$ for all
$\epsilon>0.$ In Theorem 9.1 we prove this when $F=\mathbb{Q},$ $D$ is
nonsplit at $\infty$ and $2,$ and $f=\otimes_{v}f_{v}$,
$\Phi:=\otimes_{v}\Phi_{v}$ with $f_{v}:=\mathbf{1}_{\mathcal{O}_{D_{v}}^{n}}$
and $\Phi_{v}=\mathbf{1}_{\mathcal{O}_{D_{v}}^{2}}$ for all $v$ where $D_{v}$
is split. This suffices for the proof of Theorem 1.2.

###  4.1. Contribution of $\Sigma_{0}$

For the remainder of this section we focus on $\Sigma_{0}(X).$ For this
contribution we do not assume that $F=\mathbb{Q}$ because we do not require
this assumption. For functions $\phi:D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times}\to\mathbb{C}^{\times}$ and $s\in\mathbb{C}$ let
$\phi_{s}(x):=\phi(x)|\mathrm{nrd}(x)|^{s}.$

In order to state a lemma let us discuss measures on $A_{\mathbb{G}_{m}}$ and
$D_{\mathbb{A}_{F}}^{\times}.$ Let

$(D_{\mathbb{A}_{F}}^{\times})^{1}:=\ker(|\mathrm{nrd}|:D_{\mathbb{A}_{F}}^{\times}\longrightarrow\mathbb{R}_{>0}).$

Endow $\mathbb{R}_{>0}$ with the usual measure $\frac{dt}{t}.$ Choose a Haar
measure on $D_{\mathbb{A}_{F}}^{\times}.$ We assume that the map
$|\mathrm{nrd}|:A_{\mathbb{G}_{m}}\tilde{\to}\mathbb{R}_{>0}$ is measure
preserving and that the natural isomorphisms
$D_{\mathbb{A}_{F}}^{\times}=A_{\mathbb{G}_{m}}\times(D_{\mathbb{A}_{F}}^{\times})^{1}$
and $(D_{\mathbb{A}_{F}}^{\times})^{1}\tilde{\to}A_{\mathbb{G}_{m}}\backslash
D_{\mathbb{A}_{F}}^{\times}$ are measure preserving. We then obtain a measure
on $A_{\mathbb{G}_{m}}D^{\times}\backslash D^{\times}_{\mathbb{A}_{F}}$ in the
usual manner since $D^{\times}<A_{\mathbb{G}_{m}}\backslash
D^{\times}_{\mathbb{A}_{F}}$ is discrete. We define orthonormal bases on
subspaces of $L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times})$ using this measure.

######  Lemma 4.1.

Let $h\in C_{c}^{\infty}(D_{\mathbb{A}_{F}}).$ Then

(4.1.1) $\displaystyle\sum_{\delta\in D^{\times}}h(\delta g)=\frac{1}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}\sum_{\pi}\sum_{\phi\in
B_{\pi}}\left(\int_{D_{\mathbb{A}_{F}}^{\times}}h(x)\overline{\phi}_{s}(x)\,d^{\times}x\right)\phi_{-s}(g)\,ds,$

for all $g\in D_{\mathbb{A}_{F}}^{\times}$ and $\sigma\gg 1$, where the sum
over $\pi$ is taken over isomorphic classes of automorphic representations of
$A_{\mathbb{G}_{m}}D^{\times}\backslash D^{\times}_{\mathbb{A}_{F}}$ and
$B_{\pi}$ is an orthonormal basis of $\pi$.

###### Proof.

By Mellin inversion we have

$\displaystyle\sum_{\delta\in D^{\times}}h(\delta g)=\frac{1}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}\left(\sum_{\delta\in
D^{\times}}\int_{A_{\mathbb{G}_{m}}}h(a\delta g)|a|^{s}d^{\times}a\right)ds$

for $\sigma\gg 1.$ On the other hand, for $\mathrm{Re}(s)>2$ the function
$g\mapsto\int_{A_{\mathbb{G}_{m}}}h(ag)|a|^{s}da$ lies in
$L^{1}((D_{\mathbb{A}_{F}}^{\times})^{1})$ [GJ72, Lemma 12.5]. Moreover since
$D$ is a division algebra, $A_{\mathbb{G}_{m}}D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times}$ is compact, and hence
$L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times})$
decomposes discretely as a representation of $D_{\mathbb{A}_{F}}^{\times}.$
Hence

$\displaystyle\sum_{\delta\in D^{\times}}\int_{A_{\mathbb{G}_{m}}}h(a\delta
g)|a|^{s}d^{\times}a$ $\displaystyle=\sum_{\pi}\sum_{\phi\in
B_{\pi}}\int_{(D_{\mathbb{A}_{F}}^{\times})^{1}}\int_{A_{\mathbb{G}_{m}}}h(ax)|a|^{s}da\overline{\phi}(x)d^{\times}x\phi_{-s}(g)$
$\displaystyle=\sum_{\pi}\sum_{\phi\in
B_{\pi}}\int_{D_{\mathbb{A}_{F}}^{\times}}h(x)\overline{\phi}_{s}(x)d^{\times}x\phi_{-s}(g).$

∎

Now apply (4.1.1) with $g=I_{D}$ to (4.0.8). We obtain

$\displaystyle\begin{split}\Sigma_{0}(X)&=\frac{c_{\Phi,X}|X|^{4(n-1)}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}\sum_{\pi}\sum_{\phi\in
B_{\pi}}\int_{D_{\mathbb{A}_{F}}^{\times}\times
D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\\\
&\times\left(\Phi\left(P(t)Xx^{-1},x\right)-\Phi\left(x,x^{-1}XP(t)\right)\right)\,dt\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}xds.\end{split}$

By the change of variables $x\mapsto Xx$, we obtain

$\displaystyle\begin{split}\Sigma_{0}(X)&=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}|X|^{2s}\sum_{\pi}\sum_{\phi\in B_{\pi}}\\\
&\times\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\left(\Phi\left(P(t)x^{-1},x\right)-\Phi\left(x,x^{-1}P(t)\right)\right)\,dt\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}xds\\\
&=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}|X|^{2s}\sum_{\pi}\sum_{\phi\in
B_{\pi}}\left(I^{1}(f,\Phi,\overline{\phi}_{s})-I^{0}(f,\Phi,\overline{\phi}_{s})\right)\,ds,\end{split}$

where

$\displaystyle\begin{split}I^{0}(f,\Phi,\overline{\phi}_{s})&:=\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\Phi\left(x,x^{-1}P(t)\right)\,dt\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}x\\\
&=\int_{D_{\mathbb{A}_{F}}^{\times}}I_{0}(f,\Phi,x,0)\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}x,\end{split}$
$\displaystyle\begin{split}I^{1}(f,\Phi,\overline{\phi}_{s})&:=\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\Phi\left(P(t)x^{-1},x\right)\,dt\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}x\\\
&=\int_{D_{\mathbb{A}_{F}}^{\times}}I_{1}(f,\Phi,x,0)\overline{\phi}_{s}(x)\phi(I_{D})\,d^{\times}x.\end{split}$

for cusp forms $\phi:A_{\mathbb{G}_{m}}D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times}\to\mathbb{C}.$

Let $K_{\infty}<D_{\infty}^{\times}$ be a maximal compact subgroup and let
$K:=K_{\infty}\widehat{\mathcal{O}}_{D}^{\times}=\prod_{v}K_{v}$. We assume
henceforth that $\Phi$ is bi-$K$-invariant in both entries. We also assume
that

(4.1.2) $\displaystyle
f(kt_{1}k^{-1},\dots,kt_{n}k^{-1})=f(t_{1},\dots,t_{n})$

for $(t_{1},\dots,t_{n})\in D^{n}_{\mathbb{A}_{F}}$ and $k\in K_{v}$ for $v$
such that $D_{v}$ is split. Then only $\pi$ with $\pi^{K}\neq 0$ contribute a
nonzero summand to $\Sigma_{0}(X)$ (this follows from the same argument
proving Lemma 6.1). For $\pi^{K}\neq 0,$ fix a vector $\phi_{\pi}\in\pi^{K}$
of $L^{2}$-norm $1.$ The vector $\phi_{\pi}$ is unique up to a constant of
norm $1.$ It follows that

(4.1.3) $\displaystyle\Sigma_{0}(X)$
$\displaystyle=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}\sum_{\pi:\pi^{K}\neq
0}|X|^{2s}\left(I^{1}(f,\Phi,\overline{\phi}_{\pi
s})-I^{0}(f,\Phi,\overline{\phi}_{\pi s})\right)ds.$

######  Lemma 4.2.

Assume that $\pi^{K}\neq 0.$ One has

$\displaystyle I^{1}(f,\Phi,\overline{\phi_{\pi s}})$
$\displaystyle=|\phi_{\pi}(I_{D})|^{2}\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f(t)\Phi\left(P(t)x^{-1},x\right)\overline{m_{\pi}(x)}|\mathrm{nrd}(x)|^{s}\,dtd^{\times}x$
$\displaystyle=|\phi_{\pi}(I_{D})|^{2}\prod_{v}\int_{D_{v}^{\times}}\int_{D_{v}^{n}}f_{v}(t_{v})\Phi_{v}\left(P(t_{v})x_{v}^{-1},x_{v}\right)\overline{m_{\pi_{v}}(x_{v})}|\mathrm{nrd}(x_{v})|_{v}^{s}\,dt_{v}d^{\times}x_{v}.$

Here $m_{\pi}(x)$ is the matrix coefficient
$\langle\pi(x)\phi_{\pi},\phi_{\pi}\rangle_{\pi}$, and $m_{\pi_{v}}$ is the
zonal spherical function of $\pi_{v}$. A similar formula holds for $I^{0}$.

In the lemma we realize $\pi$ as a subrepresentation of
$L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times}),$
and $\langle\,,\,\rangle_{\pi}$ is the pairing on the space of $\pi$ given by
restriction of the pairing on $L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times}).$

###### Proof.

By $K$-invariance,

$\displaystyle I^{1}(f,\Phi,\overline{\phi}_{\pi s})$
$\displaystyle=\frac{1}{\mathrm{vol}(K)}\phi_{\pi}(I_{D})\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\Phi\left(P(t)x^{-1},x\right)\left(\int_{K}\overline{\phi}_{\pi}(kx)dk\right)|\mathrm{nrd}(x)|^{s}\,dt\,d^{\times}x.$

We have

$\displaystyle\int_{K}\phi_{\pi}(kx)dk=\mathrm{vol}(K)\phi_{\pi}(I_{D})\langle\pi(x)\phi_{\pi},\phi_{\pi}\rangle_{\pi}$

since spherical functions are unique up to a constant multiple. Hence,

$\displaystyle I^{1}(f,\Phi,\overline{\phi_{\pi s}})$
$\displaystyle=|\phi_{\pi}(I_{D})|^{2}\int_{D_{\mathbb{A}_{F}}^{\times}}\int_{D_{\mathbb{A}_{F}}^{n}}f\left(t\right)\Phi\left(P(t)x^{-1},x\right)\overline{\langle\pi(x)\phi_{\pi},\phi_{\pi}\rangle_{\pi}}|\mathrm{nrd}(x)|^{s}\,dt\,d^{\times}x.$

The last identity in the lemma follows from the fact that the global matrix
coefficient attached to a cusp form is factorizable if the cusp form is
factorizable. ∎

Using notation from (5.0.1) below, we have shown

(4.1.4) $\displaystyle I^{i}(f,\Phi,\overline{\phi}_{\pi
s})=|\phi_{\pi}(I_{D})|^{2}\prod_{v}I^{i}(f_{v},\Phi_{v},\overline{m}_{\pi_{v}s}).$

Thus we can use the work in §5 and §6 to understand these functions.

At this point it is convenient to recall some basic automorphic representation
theory. Unitary automorphic representations of $D_{\mathbb{A}_{F}}^{\times}$
are either $1$-dimensional or infinite dimensional. The one-dimensional
representations are precisely of the form $\chi\circ\mathrm{nrd}$ for a
character
$\chi:F^{\times}\backslash\mathbb{A}_{F}^{\times}\to\mathbb{C}^{\times}.$ If
$\pi$ is the one-dimensional representation $\chi\circ\mathrm{nrd},$ then its
standard $L$-function is

$L(s,\pi)=L(s+\tfrac{1}{2},\chi)L(s-\tfrac{1}{2},\chi).$

The infinite dimensional representations correspond under the Jacquet-
Langlands correspondence to cuspidal automorphic representations of
$\mathrm{GL}_{2}(\mathbb{A}_{F})$ (although not all representations of
$\mathrm{GL}_{2}(\mathbb{A}_{F})$ are in the image of the Jacquet-Langlands
correspondence). The (standard) $L$-function $L(s,\pi)$ of an infinite-
dimensional automorphic representation of $D^{\times}_{\mathbb{A}_{F}}$ is by
definition the standard $L$-function of its Jacquet-Langlands transfer.

There is only one setting in which we have to understand the local Jacquet-
Langlands transfer of an irreducible admissible representation $\pi_{v}$ of
$D_{v}^{\times}$ when $D_{v}$ is nonsplit. This is the case where
$\pi_{v}\cong|\mathrm{nrd}|^{it}.$ In this case the Jacquet-Langlands transfer
is $\mathrm{St}|\det|^{it},$ where $\mathrm{St}$ is the Steinberg (also known
as special) representation [JL70, Above Proposition 15.5]. We point out that
$L(s,\mathrm{St}|\cdot|^{it})=\zeta_{v}(s+it+\tfrac{1}{2})$ [Gel75, Theorem
6.15].

We will also require the completed zeta function
$\zeta_{D}(s):=\prod_{v}\zeta_{D_{v}}(s)$ of $D.$ Let
$\Gamma_{\mathbb{R}}(s):=\pi^{-s/2}\Gamma(s/2).$ We then define

(4.1.5)
$\displaystyle\begin{split}\zeta_{D_{v}}(s):=\begin{cases}\Gamma_{\mathbb{R}}(s+\tfrac{1}{2})\Gamma_{\mathbb{R}}(s+\tfrac{3}{2})=2(2\pi)^{-(s+1/2)}\Gamma(s+\tfrac{1}{2})&\textrm{
if }F_{v}=\mathbb{R},D_{v}\textrm{ nonsplit, }\\\
\zeta_{v}\left(s+\frac{1}{2}\right)&\textrm{ if }v\nmid\infty,D_{v}\textrm{
nonsplit, }\\\
\zeta_{v}\left(s+\frac{1}{2}\right)\zeta_{v}\left(s-\frac{1}{2}\right)&\textrm{
if }D_{v}\textrm{ split. }\end{cases}\end{split}$

Up to harmless normalizations this is the same as the definition in [Voi21,
§29.5-29.6]. We note that $\zeta_{D}(s)\neq L(s,1\circ\mathrm{nrd}),$ although
the corresponding local identity is true for all places $v$ where $D_{v}$ is
split. The function $\zeta_{D}(s)$ is meromorphic. Because we assumed $D$ is
nonsplit it is holomorphic apart from simple poles at
$s=-\frac{1}{2},\frac{3}{2}.$

Let $Z_{D_{\infty}^{\times}}$ be the center of $D_{\infty}^{\times}.$ We point
out that if $F=\mathbb{Q},$ then $Z_{D_{\infty}^{\times}}\backslash
D_{\infty}^{\times}$ is compact if and only if $A_{\mathbb{G}_{m}}\backslash
D_{\infty}^{\times}$ is compact.

######  Theorem 4.3.

Suppose $n\geq 5$ and $\pi^{K}\neq 0$. Assume
$Z_{D_{\infty}^{\times}}\backslash D_{\infty}^{\times}$ is compact, and that
$(f,\Phi)=(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{n}},\Phi_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{2}})$
where $\Phi_{\infty}$ is bi-invariant under $K_{\infty}$ in both entries. The
quotient

$\displaystyle\frac{I^{i}(f,\Phi,\overline{\phi}_{\pi
s})}{L\left(s+\frac{3}{2},\pi\right)}$

is holomorphic on $\sigma=\mathrm{Re}(s)>-\frac{n-1}{2},$ and
$I^{i}(f,\Phi,\overline{\phi}_{\pi s})$ is rapidly decreasing on vertical
strips (away from its poles). If $\pi$ is the trivial representation, then the
same is true if we replace $L(s+\tfrac{3}{2},\pi)$ by
$\zeta_{D}(s+\tfrac{3}{2}).$

###### Proof.

By (4.1.4), the assertion follows from Propositions 5.8, 6.4, and 6.8, our
comments on the Jacquet Langlands correspondence above, and standard bounds on
$L$-functions [IK04, Lemma 5.2]. ∎

Let $1:D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times}\to\mathbb{C}$ be the
function that is identically $1.$ It spans the trivial subrepresentation of
$L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times}).$
We assume $1$ on $A_{\mathbb{G}_{m}}D^{\times}\backslash
D^{\times}_{\mathbb{A}_{F}}$ has $L^{2}$-norm $1,$ and thus the measure
$d^{\times}x$ in (4.1.1) is chosen so that
$\mathrm{vol}(A_{\mathbb{G}_{m}}D^{\times}\backslash
D^{\times}_{\mathbb{A}_{F}})=1$.

######  Lemma 4.4.

Suppose $n\geq 5$. Assume $A_{\mathbb{G}_{m}}\backslash D_{\infty}^{\times}$
is compact, $F=\mathbb{Q}$ and $(f,\Phi)$ is as above. For any $\epsilon>0$
one has

$\displaystyle\Sigma_{0}(X)=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}|X|^{2s}(I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s}))\,ds+O_{\epsilon,f,\Phi}(|X|^{3n-3+\epsilon}).$

###### Proof.

Consider

(4.1.6) $\displaystyle\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}\sum_{\pi:\pi\neq\mathrm{triv}}|X|^{2s}(I^{1}(f,\Phi,\overline{\phi}_{\pi
s})-I^{0}(f,\Phi,\overline{\phi}_{\pi s}))\,ds$

where the sum is over nontrivial automorphic representations of
$A_{\mathbb{G}_{m}}\backslash D^{\times}_{\mathbb{A}_{F}}$ such that $\pi^{K}$
is nonzero. The lemma is equivalent to the assertion that (4.1.6) is
$O_{\epsilon,f,\Phi}(|X|^{3n-3+\epsilon}).$

Since $A_{\mathbb{G}_{m}}\backslash D_{\infty}^{\times}$ is compact, the
quotient $A_{\mathbb{G}_{m}}D^{\times}\backslash
D_{\mathbb{A}_{F}}^{\times}/K$ is finite. Thus
$L^{2}(A_{\mathbb{G}_{m}}D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times}/K)$
is finite dimensional and we deduce that the sum over $\pi$ in (4.1.6) is
finite. Hence it suffices to prove that the contribution of any given $\pi$ is
$O_{\epsilon,f,\Phi,\pi}(|X|^{3n-3+\epsilon})$.

Since the complete $L$-function $L(s,\pi)$ (i.e. with the Archimedean factor)
is entire, by Theorem 4.3 we can perform a contour shift

$\displaystyle\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}|X|^{2s}(I^{1}(f,\Phi,\overline{\phi}_{\pi
s})-I^{0}(f,\Phi,\overline{\phi}_{\pi s}))\,ds$
$\displaystyle=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=-\tfrac{n-1}{2}+\epsilon}|X|^{2s}(I^{1}(f,\Phi,\overline{\phi}_{\pi
s})-I^{0}(f,\Phi,\overline{\phi}_{\pi s}))\,ds$
$\displaystyle=O_{\epsilon,f,\Phi,\pi}(|X|^{3n-3+2\epsilon}).$

∎

######  Remark.

With more local work at the Archimedean places one could remove the assumption
that $A_{\mathbb{G}_{m}}\backslash D_{\infty}^{\times}$ is compact and
$F=\mathbb{Q}.$

######  Lemma 4.5.

For $(f,\Phi)$ as in Theorem 4.3, one has

$\displaystyle\mathrm{Res}_{s=0}\left(I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s})\right)=0.$

###### Proof.

From (4.0.7), we deduce that

$\displaystyle
I^{0}(f_{v},\Phi_{v},\overline{1}_{s})=I^{1}(f_{v}\circ\dagger,\Phi^{\operatorname{sw}}_{v}\circ\dagger,\overline{1}_{s}).$

But for all finite $v,$ we have
$\mathbf{1}_{\mathcal{O}_{D_{v}}^{n}}=\mathbf{1}_{\mathcal{O}_{D_{v}}^{n}}\circ\dagger$
and
$\mathbf{1}_{\mathcal{O}_{D_{v}}^{2}}=\mathbf{1}_{\mathcal{O}_{D_{v}}^{2}}^{\mathrm{sw}}\circ\dagger$.
Therefore, by Theorem 4.3 and (4.1.5), it suffices to verify that

$\displaystyle
I^{1}(f_{\infty},\Phi_{\infty},\overline{1}_{0})-I^{0}(f_{\infty},\Phi_{\infty},\overline{1}_{0})=0.$

By Fourier inversion we have

$\displaystyle I^{1}(f_{\infty},\Phi_{\infty},\overline{1}_{0})$
$\displaystyle=\int_{D_{\infty}^{\times}}\int_{D_{\infty}^{n}}f_{\infty}(Y)\left(\int_{D_{\infty}}\psi\left(\langle
P(Y),Z\rangle\right)\mathcal{F}_{1}(\Phi_{\infty})(-xZ,x)|\mathrm{nrd}(x)|^{2}\,dZ\right)dYdx$

By the bounds in Lemma 6.7 all of the integrals here converge absolutely. Thus
by the Fubini-Tonelli theorem we can rearrange the integrals to see that the
above is

$\displaystyle\int_{D_{\infty}^{\times}\times
D_{\infty}}\left(\int_{D_{\infty}^{n}}f_{\infty}(Y)\psi\left(\langle
P(Y),Z\rangle\right)\,dY\right)\mathcal{F}_{1}(\Phi_{\infty})(-xZ,x)|\mathrm{nrd}(x)|^{2}\,dZdx.$

Here and for the rest of the proof we use capital letters to denote Haar
measures on the additive group $D_{\infty}$ and lowercase letters to denote
Haar measures on the multiplicative group $D^{\times}_{\infty}.$ Thus
$|\mathrm{nrd}(x)|^{2}dx=dX.$ The above is

$\displaystyle\int_{D_{\infty}\times
D_{\infty}}\left(\int_{D_{\infty}^{n}}f_{\infty}(Y)\psi\left(\langle
P(Y),Z\rangle\right)\,dY\right)\left(\int_{D_{\infty}}\Phi_{\infty}(H,X)\psi(\langle
H,-XZ\rangle)dHdX\right)dZ$ $\displaystyle=\int_{D_{\infty}\times
D_{\infty}}\left(\int_{D_{\infty}^{n}}f_{\infty}(Y)\psi\left(\langle
P(Y),Z\rangle\right)\,dY\right)\left(\int_{D_{\infty}}\Phi_{\infty}(H,X)\psi(\langle-
ZH,X\rangle)dXdH\right)dZ$ $\displaystyle=\int_{D_{\infty}^{\times}\times
D_{\infty}}\left(\int_{D_{\infty}^{n}}f_{\infty}(Y)\psi\left(\langle
P(Y),Z\rangle\right)\,dY\right)\mathcal{F}_{2}(\Phi_{\infty})(h,-Zh)|\mathrm{nrd}(h)|^{2}dZdh$
$\displaystyle=I^{0}(f_{\infty},\Phi_{\infty},\overline{1}_{0}).$

∎

###### Proof of Theorem 1.2.

We have $\Sigma(X)=\Sigma_{0}(X)+E_{1}(X)$ as explained above (4.0.8) and
(4.0.9). As mentioned earlier, $E_{1}(X)\ll_{\epsilon,f,\Phi}X^{3n+\epsilon}$
for all $\epsilon>0$ by Theorem 9.1. Thus by Lemma 4.4 we have

$\displaystyle\Sigma(X)=\frac{c_{\Phi,X}|X|^{4n-4}}{2\pi
i}\int_{\mathrm{Re}(s)=\sigma}|X|^{2s}(I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s}))\,ds+O_{\epsilon,f,\Phi}(|X|^{3n+\epsilon}).$

We perform a countour shift using Theorem 4.3 and Lemma 4.5 to obtain

(4.1.7) $\displaystyle\Sigma(X)$
$\displaystyle=c_{\Phi,X}\bigg{(}\mathrm{Res}_{s=-2}I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s})\bigg{)}|X|^{4n-8}+O_{\epsilon,f,\Phi}(|X|^{3n+\epsilon}).$

∎

Thus we have reduced Theorem 1.2 to a study of the Eulerian integrals
$I^{i}(f,\Phi,\phi_{s})$ and the error term $E_{1}(X).$ The error term
$E_{1}(X)$ will be treated in §§ 7–9. We treat the integrals
$I^{i}(f,\Phi,\phi_{s})$ first, starting with the non-Archimedean case in the
following section. Before this we pause to explain the proof of Theorem 1.3,
modulo some local results contained below. In contrast to the rest of this
section, for the proof of Theorem 1.3 we will make use of our assumption that
$F=\mathbb{Q},$ although with more effort this assumption could be removed.

###### Proof of Theorem 1.3.

By Lemma 3.1 and (4.1.7) the constant
$c(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{\infty}})$ is

$\displaystyle\frac{1}{\mathcal{F}_{2}(\Phi)(0,0)}\bigg{(}\mathrm{Res}_{s=-2}I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s})\bigg{)}.$

Using Theorem 4.3 this is

$\displaystyle\frac{\mathrm{Res}_{s=-2}\zeta_{D}(s+3/2)}{\mathcal{F}_{2}(\Phi_{\infty})(0,0)\mathrm{vol}(\widehat{\mathcal{O}}_{D})}\frac{I^{1}(f,\Phi,\overline{1}_{s})-I^{0}(f,\Phi,\overline{1}_{s})}{\zeta_{D}(s+3/2)}\Bigg{|}_{s=-2}$
$\displaystyle=\frac{\mathrm{Res}_{s=-1/2}\zeta_{D}(s)}{\mathcal{F}_{2}(\Phi_{\infty})(0,0)}\frac{I^{1}(f_{\infty},\Phi_{\infty},\overline{1}_{s})-I^{0}(f_{\infty},\Phi_{\infty},\overline{1}_{s})}{\zeta_{D_{\infty}}(s+3/2)}\Bigg{|}_{s=-2}\mathrm{vol}(\widehat{\mathcal{O}}_{D}^{\times})\prod_{p}c(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}}).$

Here we have used the fact that
$I^{1}(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}},\mathbf{1}_{\mathcal{O}_{D_{p}}^{2}},\overline{1}_{s})=I^{0}(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}},\mathbf{1}_{\mathcal{O}_{D_{p}}^{2}},\overline{1}_{s})$
for all $p,$ (5.2.3), (5.2.4), and (6.1.4).

Since $\Sigma(X)$ is independent of $\Phi_{\infty},$ Theorem 1.2 implies that
the factor at infinity above is also independent of $\Phi_{\infty}.$ We claim,
moreover, that it remains unchanged if we replace $\Phi_{\infty}$ by any
function $\Phi_{\infty}^{\prime}\in\mathcal{S}(D_{\infty}^{2})$ that is
bi-$K_{\infty}$-invariant in both variables. To see this, choose a sequence
$\\{\Phi_{i}\\}\subset C^{\infty}_{c}(D_{\infty}^{2})$ consisting of functions
that are bi-$K_{\infty}$-invariant in both variables such that
$\Phi_{i}\to\Phi_{\infty}^{\prime}$ in $\mathcal{S}(D_{\infty}^{2}).$ Then

$\displaystyle\frac{\mathrm{Res}_{s=-1/2}\zeta_{D}(s)}{\mathcal{F}_{2}(\Phi_{i})(0,0)}\frac{I^{1}(f_{\infty},\Phi_{i},\overline{1}_{s})-I^{0}(f_{\infty},\Phi_{i},\overline{1}_{s})}{\zeta_{D_{\infty}}(s+3/2)}\Bigg{|}_{s=-2}$

is independent of $\Phi_{i}.$ On the other hand, it converges to

(4.1.8)
$\displaystyle\frac{\mathrm{Res}_{s=-1/2}\zeta_{D}(s)}{\mathcal{F}_{2}(\Phi_{\infty}^{\prime})(0,0)}\frac{I^{1}(f_{\infty},\Phi_{\infty}^{\prime},\overline{1}_{s})-I^{0}(f_{\infty},\Phi_{\infty}^{\prime},\overline{1}_{s})}{\zeta_{D_{\infty}}(s+3/2)}\Bigg{|}_{s=-2}.$

This justifies the claim.

Using the notation in the proof of Lemma 6.10, take
$\Phi_{\infty}^{\prime}:=\Phi_{2,2}-\Phi_{2,1}$. Then using (6.2.4) and
(6.2.5) we see that (4.1.8) equals

$\displaystyle-\mathrm{Res}_{s=-\tfrac{1}{2}}\zeta_{D}(s)\frac{\mathrm{vol}((D^{\times}_{\infty})^{1})}{2}\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)\,dZ.$

By [Voi21, Theorem 29.10.23, Lemma 29.5.18, Lemma 29.8.24] we have

$\displaystyle\mathrm{Res}_{s=-\tfrac{1}{2}}\zeta_{D}(s)=-\frac{\mathrm{Res}_{s=1}\zeta_{\mathbb{Q}}(s)}{2\pi^{2}\widehat{\tau}^{\times}(\widehat{\mathcal{O}}_{D}^{\times})}=-\frac{1}{2\pi^{2}\widehat{\tau}^{\times}(\widehat{\mathcal{O}}_{D}^{\times})}.$

The definition of
$\widehat{\tau}^{\times}(\widehat{\mathcal{O}}_{D}^{\times})$ will be recalled
in a moment. We conclude that

(4.1.9) $\displaystyle
c(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{\infty}})=\frac{\mathrm{vol}(K)}{4\pi^{2}\widehat{\tau}^{\times}(\widehat{\mathcal{O}}_{D}^{\times})}c(f_{\infty})\prod_{p}c(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}}).$

Let us be a little more explicit about the measures in this expression. Above
Lemma 4.4 we assumed that the volume of
$A_{\mathbb{G}_{m}}D^{\times}\backslash D_{\mathbb{A}_{F}}^{\times}$ is $1.$
There is a unique Haar measure on $A_{\mathbb{G}_{m}}\backslash
D_{\mathbb{A}_{F}}^{\times}$ that induces this measure. By [Voi21, Theorem
29.11.3], this measure is the Tamagawa measure. It decomposes as a product of
a Haar measure on $A_{\mathbb{G}_{m}}\backslash D_{F_{\infty}}^{\times}$ and a
Haar measure $\widehat{\tau}^{\times}$ on
$D^{\times}_{\mathbb{A}_{F}^{\infty}}.$ Since $A_{\mathbb{G}_{m}}\backslash
D^{\times}_{\infty}$ is compact, the restriction of the Tamagawa measure on
$A_{\mathbb{G}_{m}}\backslash D^{\times}_{\mathbb{A}_{F}}$ to $K$ induces a
Haar measure on $K.$ This measure is used to compute the volume in (4.1.9).

By [Voi21, (29.8.15), Lemma 29.5.9], we have
$\mathrm{vol}(K)=4\pi^{2}\widehat{\tau}^{\times}(\widehat{\mathcal{O}}_{D}^{\times})$.
Thus
$c(f_{\infty}\mathbf{1}_{\widehat{\mathcal{O}}_{D}^{n}})=c(f_{\infty})\prod_{p}c(\mathbf{1}_{\mathcal{O}_{D_{p}}^{n}}).$
The nonvanishing statement follows from Corollary 5.9 and Lemmas 6.5 and 6.9.
∎

##  5\. Spectral bounds for split cases

Let $v$ be a place of $F$ which we drop from notation, writing
$F:=F_{v},D:=D_{v},$ etc. We do not yet make any assumption on $F$ or $D$. Let
$\pi$ be an irreducible admissible unitary representation of $D^{\times}.$ We
realize $\pi$ in the category of smooth representations. Let $m$ be a matrix
coefficient of $\pi$ and let

$\displaystyle m_{s}(g):=m(g)|\mathrm{nrd}(g)|^{s}.$

For $(f,\Phi)\in\mathcal{S}(D^{n})\times\mathcal{S}(D^{2})$ and
$s\in\mathbb{C}$, define

(5.0.1) $\displaystyle
I(f,\Phi,m_{s}):=\int_{D^{\times}}\int_{D^{n}}f(Y)\Phi\left(P(Y)g^{-1},g\right)m_{s}(g)\,dYdg.$

Since $\pi$ is unitary, the function
$g\mapsto\int_{D^{n}}f(Y)\Phi\left(P(Y)g^{-1},g\right)m(g)\,dY$ is bounded by
the restriction of a Schwartz function in $\mathcal{S}(D)$ to $D^{\times}.$
Hence the integral above converges absolutely for $\mathrm{Re}(s)>1$ [GJ72,
Proposition 1.1]. By Fourier inversion, for $\mathrm{Re}(s)>1$ the integral
(5.0.1) is

$\displaystyle\int_{D^{n}}f(Y)\int_{D^{\times}}\left(\int_{D}\mathcal{F}_{1}(\Phi)(Z,g)\psi(-\langle
P(Y)g^{-1},Z\rangle)m_{s}(g)\,dZ\right)\,dgdY$
$\displaystyle=\int_{D^{n}}f(Y)\left(\int_{D}\psi(\langle
P(Y),Z\rangle)\int_{D^{\times}}\mathcal{F}_{1}(\Phi)(-gZ,g)m_{s+2}(g)\,dgdZ\right)\,dY$

For a given $Z\in D$, define the zeta integral

$\displaystyle\mathcal{Z}_{Z}(\Phi,s,m):=\int_{D^{\times}}\mathcal{F}_{1}(\Phi)(-gZ,g)m_{s+2}(g)\,dg,$

and define

$\displaystyle
H_{Z}(\Phi,s,m):=\frac{\mathcal{Z}_{Z}(\Phi,s,m)}{L(s+3/2,\pi)}.$

When $\Phi$ is fixed, we often write
$\mathcal{Z}_{Z}(s,m):=\mathcal{Z}_{Z}(\Phi,s,m)$ and
$H_{Z}(s,m):=H_{Z}(\Phi,s,m)$.

######  Lemma 5.1.

If $F$ is non-Archimedean, $\mathcal{Z}_{Z}(\Phi,s,m)$ lies in
$\mathbb{C}(q^{-s})$ and $H_{Z}(\Phi,s,m)$ lies in $\mathbb{C}[q^{-s},q^{s}].$

###### Proof.

This follows from [GJ72, Theorem 3.3]. ∎

Assume now $F$ is non-Archimedean and $D$ is split, so we identify $D$ with
$M_{2}(F)$. Assume $\psi$ is unramified and fix
$\Phi=\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}}$. We normalize the measure on
$\mathrm{GL}_{2}(F)$ so that $dg(\mathrm{GL}_{2}(\mathcal{O}_{F}))=1$. In the
rest of the section, we make the following assumption:

$\displaystyle f\textrm{ is invariant under conjugation by
}\mathrm{GL}_{2}(\mathcal{O}_{F})\textrm{ (acting diagonally)}.$

Then $I(f,\Phi,m_{s})$ is nonzero only if $\pi$ is unramified and $m$ is left
and right $\mathrm{GL}_{2}(\mathcal{O}_{F})$-invariant. We henceforth assume
this. Moreover, we shall normalize $m$ so that $m(e)=1$. This is compatible
with Lemma 4.2.

We have
$\mathcal{F}(\mathbf{1}_{M_{2}(\mathcal{O}_{F})})=\mathbf{1}_{M_{2}(\mathcal{O}_{F})},$
so $H_{Z}(s,m)$ is invariant under $Z\mapsto Z+W$ for $W\in
M_{2}(\mathcal{O}_{F}).$ We make an additional assumption:

$\displaystyle\textrm{ If }f(Y)\neq 0,\textrm{ then }P(Y)\in
M_{2}(\mathcal{O}_{F}).$

Under this assumption, we can write

(5.0.2)
$\displaystyle\frac{I(f,\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},m_{s})}{L(s+3/2,\pi)}$
$\displaystyle=\int_{M_{2}(F)^{n}}f(Y)\left(\sum_{Z\in
M_{2}(F)/M_{2}(\mathcal{O}_{F})}\psi(\langle
P(Y),Z\rangle)H_{Z}(s,m)\right)\,dY.$

Recall the box norm defined in (2.2.1), which is the matrix norm in this case.

######  Lemma 5.2.

Let $Z\in M_{2}(F)$. There exists $W\in M_{2}(\mathcal{O}_{F})$ so that
$Z+W\neq 0,$ $|\det(Z+W)|\geq\left\lVert Z+W\right\rVert$ (or equivalently
$(Z+W)^{-1}\in M_{2}(\mathcal{O}_{F})$) and $|\mathrm{tr}(Z+W)|\geq|2|.$

###### Proof.

We can and do assume that $Z\in\mathrm{GL}_{2}(F)$. By the Cartan
decomposition, there exist $k_{1},k_{2}\in\mathrm{GL}_{2}(\mathcal{O}_{F})$
and integers $n_{1}\geq n_{2}$ such that
$Z=k_{1}\left(\begin{smallmatrix}\varpi^{n_{1}}&\\\
&\varpi^{n_{2}}\end{smallmatrix}\right)k_{2}$. Let
$W=k_{1}\left(\begin{smallmatrix}\lambda_{1}&\\\
&\lambda_{2}\end{smallmatrix}\right)k_{2}$, where

$\displaystyle\lambda_{i}=\begin{cases}1-\varpi^{n_{i}}&\text{ if }n_{i}>0,\\\
0&\text{ otherwise,}\end{cases}\quad\text{ for }i=1,2.$

It is clear that $W\in M_{2}(\mathcal{O}_{F})$ and $|\det(Z+W)|\geq\left\lVert
Z+W\right\rVert$.

We may therefore assume $Z=(z_{ij})$ satisfies $|\det(Z)|\geq\left\lVert
Z\right\rVert,$ so $|\det(Z)|\geq 1$. Suppose $|\mathrm{tr}(Z)|<|2|\leq 1$. If
$|z_{11}|<\left\lVert Z\right\rVert$, choose
$W=\left(\begin{smallmatrix}0&0\\\ 0&1\end{smallmatrix}\right),$ so

$\displaystyle|\mathrm{tr}(Z+W)|=|\mathrm{tr}(Z)+1|=1.$

Notice that $\det(Z+W)=\det(Z)+z_{11}$ and we have $|\det(Z)|\geq\left\lVert
Z\right\rVert>|z_{11}|$ and $|\det(Z)|\geq 1$, so

$\displaystyle|\det(Z+W)|=|\det(Z)|\geq\left\lVert
Z\right\rVert\geq\left\lVert Z+W\right\rVert.$

Assume $|z_{11}|=\left\lVert Z\right\rVert.$ In this case we claim that we can
take $W=\left(\begin{smallmatrix}0&0\\\ 0&\alpha\end{smallmatrix}\right)$ for
an appropriately chosen $\alpha.$ In more detail, we choose $\alpha=2$ if
$2|q.$ If $q$ is odd and $p$ is the prime dividing $q,$ we choose
$\alpha\in\\{1,\dots,p-1\\}$ so that $|\det(Z+W)|=|\det(Z)|.$ This is possible
because $\det(Z+W)=\det(Z)+\alpha z_{11}$ and $|\det(Z)|\geq|z_{11}|$. Since

$\displaystyle\left\lVert
Z+W\right\rVert=\max(|z_{11}|,|z_{22}+\alpha|)\leq\max(\left\lVert
Z\right\rVert,1)\leq|\det(Z)|,$

and $|\mathrm{tr}(Z+W)|=|\mathrm{tr}(Z)+\alpha|=|\alpha|\geq|2|$, the
assertion is justified. ∎

By the above lemma, for each coset in $M_{2}(F)/M_{2}(\mathcal{O}_{F})$, we
can and do choose a representative $Z$ such that $Z^{-1}\in
M_{2}(\mathcal{O}_{F})$. Then

$\displaystyle\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(gZ)=1\Rightarrow\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(g)=1,$

and

$\displaystyle\mathcal{Z}_{Z}(s,m)$
$\displaystyle=\int_{\mathrm{GL}_{2}(F)}\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(gZ)m_{s+2}(g)\,dg$
(5.0.3)
$\displaystyle=|\det(Z)|^{-(s+2)}\int_{\mathrm{GL}_{2}(F)}\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(g)m(gZ^{-1})|\det(g)|^{s+2}\,dg.$

######  Lemma 5.3.

We have

$\displaystyle H_{Z}(s,m)=m(Z^{-1})|\det(Z)|^{-(s+2)}.$

###### Proof.

According to (5), we need to compute

$\displaystyle\mathcal{Z}_{Z}(s,m)$
$\displaystyle=|\det(Z)|^{-(s+2)}\int_{\mathrm{GL}_{2}(F)}\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(g)m(gZ^{-1})|\det
g|^{s+2}\,dg$
$\displaystyle=|\det(Z)|^{-(s+2)}\int_{\mathrm{GL}_{2}(F)}\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(g)|\det
g|^{s+2}\int_{\mathrm{GL}_{2}(\mathcal{O}_{F})}m(gkZ^{-1})\,dkdg.$

Observe that the function

$\displaystyle
g\longmapsto\int_{\mathrm{GL}_{2}(\mathcal{O}_{F})}m(gkZ^{-1})\,dk$

is bi-$\mathrm{GL}_{2}(\mathcal{O}_{F})$-invariant and hence, by the
uniqueness of the spherical function, equals $m(Z^{-1})m(g)$. The assertion
then follows from the identity (see e.g. [GJ72, Lemma 6.10])

$\displaystyle
L\left(s+3/2,\pi\right)=\int_{\mathrm{GL}_{2}(F)}\mathbf{1}_{M_{2}(\mathcal{O}_{F})}(g)m(g)|\det
g|^{s+2}\,dg.$

∎

######  Lemma 5.4.

One has

$\displaystyle\frac{I(f,\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},m_{s})}{L(s+3/2,\pi)}$
$\displaystyle=\sum_{Z}m(Z^{-1})|\det(Z)|^{-(s+2)}\int_{D^{n}}f(Y)\psi(\langle
P(Y),Z\rangle)\,dY$

for $\mathrm{Re}(s)>2$. Here the sum in $Z$ is over a set of representatives
of cosets in $M_{2}(F)/M_{2}(\mathcal{O}_{F})$ such that $Z^{-1}\in
M_{2}(\mathcal{O}_{F}).$

###### Proof.

By Lemma 5.3 we can rewrite (5.0.2) as

$\displaystyle\frac{I(f,\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},m_{s})}{L(s+3/2,\pi)}$
$\displaystyle=\int_{M_{2}(F)^{n}}f(Y)\left(\sum_{Z}m(Z^{-1})|\det(Z)|^{-(s+2)}\psi(\langle
P(Y),Z\rangle)\right)\,dY.$

Since $f$ is Schwartz and $|m(Z^{-1})|\leq 1,$ by the Fubini-Tonelli theorem
to prove the assertion it suffices to show $\sum_{Z}|\det(Z)|^{-(\sigma+2)}$
converges for $\sigma=\mathrm{Re}(s)>2$.

We define for each $a\in\mathbb{Z}_{>0}$ the set

$\displaystyle E_{a}:=\\{Z\in M_{2}(F)/M_{2}(\mathcal{O}_{F}):\left\lVert
Z\right\rVert=q^{a}\\}.$

Note that $|E_{a}|\leq q^{4a}$. Since $|\det(Z)|\geq\left\lVert
Z\right\rVert$, for $\sigma>2$

$\displaystyle\sum_{Z}|\det(Z)|^{-(\sigma+2)}\leq
1+\sum_{a=1}^{\infty}|E_{a}|q^{-a(\sigma+2)}\leq
1+\sum_{a=1}^{\infty}q^{-a(\sigma-2)}<\infty$

∎

###  5.1. Bounding nonabelian Gauss sums

In view of Lemma 5.4, we need to understand the analytic behavior of the
integral

$\displaystyle\int_{M_{2}(F)^{n}}f(Y)\psi(\langle P(Y),Z\rangle)\,dY,$

which is a nonabelian analogue of a Gauss sum. For this we will use the
classical one-dimensional estimates of Lemma 2.1. We continue to assume $\psi$
is unramified.

######  Lemma 5.5.

Suppose $2\nmid q$. Let $Z\in M_{2}(F)$ with $|\mathrm{tr}(Z)|\geq 1$. Then

$\displaystyle\bigg{|}\int_{M_{2}(\mathcal{O}_{F})}\psi(\mathrm{tr}(Y^{2}Z))\,dY\bigg{|}$
$\displaystyle=|\mathrm{tr}(Z)|^{-1/2}\max(|\mathrm{tr}(Z)\det(Z)|,\left\lVert
Z\right\rVert^{2})^{-1/2}$
$\displaystyle\leq|\mathrm{tr}(Z)|^{-1/2}\left\lVert Z\right\rVert^{-1}.$

###### Proof.

Write $Y=(y_{ij})$ and $Z=(z_{ij})$. Below we identify $M_{2}(F)$ with
$F^{4}$, viewed as a space of column vectors. Then $\mathrm{tr}(Y^{2}Z)$ is a
quadratic form in $Y$ whose associated matrix is given by

$\displaystyle J=\left(\begin{matrix}2z_{11}&z_{21}&z_{12}&0\\\
z_{21}&0&z_{11}+z_{22}&z_{21}\\\ z_{12}&z_{11}+z_{22}&0&z_{12}\\\
0&z_{21}&z_{12}&2z_{22}\end{matrix}\right).$

That is $\mathrm{tr}(Y^{2}Z)=\frac{1}{2}Y^{t}JY$. Consider the matrix

$\displaystyle R=\left(\begin{matrix}0&0&1&-z_{22}\\\ 1&1&0&z_{12}\\\
1&-1&0&z_{21}\\\ 0&0&-1&-z_{11}\end{matrix}\right).$

Let $r:=\mathrm{tr}(Z),r_{+}:=z_{12}+z_{21},r_{-}:=z_{12}-z_{21}.$ One has

$\displaystyle R^{t}JR$ $\displaystyle=2r\left(\begin{matrix}1&&&\\\ &-1&&\\\
&&1&\\\ &&&\det(Z)\end{matrix}\right),$ $\displaystyle\det(R)$
$\displaystyle=2r,$ $\displaystyle R^{-1}$
$\displaystyle=\frac{-1}{2r}\left(\begin{matrix}-r_{+}&-r&-r&-r_{+}\\\
-r_{-}&-r&r&-r_{-}\\\ -2z_{11}&0&0&2z_{22}\\\ 2&0&0&2\end{matrix}\right).$

Since $2\nmid q$,

$\displaystyle R^{-1}\mathcal{O}_{F}^{4}$
$\displaystyle=\mathcal{O}_{F}e_{1}+\mathcal{O}_{F}e_{2}+\mathcal{O}_{F}e_{3}+r^{-1}\mathcal{O}_{F}\left(\begin{smallmatrix}r_{+}\\\
r_{-}\\\ 2z_{11}\\\ -2\end{smallmatrix}\right),$

Changing variables $Y\mapsto RY$, we have

$\displaystyle\int_{M_{2}(\mathcal{O}_{F})}\psi(\mathrm{tr}(Y^{2}Z))\,dY$
$\displaystyle=|r|\int_{r^{-1}\mathcal{O}_{F}}\int_{\mathcal{O}_{F}^{3}}\psi(r(w+r_{+}z)^{2}-r(x+r_{-}z)^{2}+r(y+2z_{11}z)^{2}+r\det(Z)4z^{2})\,dwdxdydz$
$\displaystyle=|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(r_{+}^{2}-r_{-}^{2}+(2z_{11})^{2}+4\det(Z))z^{2})$
$\displaystyle\times\int_{\mathcal{O}_{F}^{3}}\psi(r(w^{2}+2r_{+}wz)-r(x^{2}+2r_{-}xz)+r(y^{2}+4z_{11}yz))\,dwdxdydz$

Since $|r|\geq 1$, by Lemma 2.1 we have

$\displaystyle\left|\int_{M_{2}(\mathcal{O}_{F})}\psi(\mathrm{tr}(Y^{2}Z))\,dY\right|$
$\displaystyle=|r|\left|\int_{r^{-1}\mathcal{O}_{F}}\mathbf{1}_{\mathcal{O}}(r_{+}z)\mathbf{1}_{\mathcal{O}}(r_{-}z)\mathbf{1}_{\mathcal{O}}(2z_{11}z)\psi(4r\det(Z)z^{2})\,dz\right|\left|\int_{\mathcal{O}_{F}}\psi(rw^{2}-rx^{2}+ry^{2})\,dwdxdy\right|$
$\displaystyle=|r|^{-\frac{1}{2}}\left|\int_{\varpi^{m}\mathcal{O}_{F}}\psi(4r\det(Z)z^{2})\,dz\right|,$

where $m\geq 0$ is the integer such that

$\displaystyle q^{-m}=\max(|r|,|r_{+}|,|r_{-}|,|2z_{11}|)^{-1}=\left\lVert
Z\right\rVert^{-1}.$

Changing variables, we have again by Lemma 2.1

$\displaystyle\left|\int_{\varpi^{m}\mathcal{O}_{F}}\psi(4r\det(Z)z^{2})\,dz\right|$
$\displaystyle=q^{-m}\left|\int_{\mathcal{O}_{F}}\psi(4\varpi^{2m}r\det(Z)z^{2})\,dz\right|$
$\displaystyle=\max(|\mathrm{tr}(Z)\det(Z)|,\left\lVert
Z\right\rVert^{2})^{-1/2}$

and the assertion follows. ∎

######  Lemma 5.6.

Suppose $2|q$. Let $Z\in M_{2}(F)$ with $|\mathrm{tr}(Z)|\geq|2|$. Then

$\displaystyle\bigg{|}\int_{M_{2}(\mathcal{O}_{F})}\psi(\mathrm{tr}(Y^{2}Z))\,dY\bigg{|}\ll|\mathrm{tr}(Z)|^{-1/2}\left\lVert
Z\right\rVert^{-1}.$

###### Proof.

We retain the notation in the previous lemma. In this case

$\displaystyle
R^{-1}\mathcal{O}_{F}^{4}:=\mathcal{O}_{F}e_{1}+2^{-1}\mathcal{O}_{F}\left(\begin{smallmatrix}1\\\
1\\\ 0\\\ 0\\\
\end{smallmatrix}\right)+\mathcal{O}_{F}e_{3}+(2r)^{-1}\mathcal{O}_{F}\left(\begin{smallmatrix}r_{+}\\\
r_{-}\\\ 2z_{11}\\\ -2\end{smallmatrix}\right).$

Thus

$\displaystyle\int_{M_{2}(\mathcal{O}_{F})}\psi(\mathrm{tr}(Y^{2}Z))dY$
$\displaystyle=|r|\int_{r^{-1}\mathcal{O}_{F}}\int_{\mathcal{O}_{F}^{3}}\psi(r(w+2^{-1}(x+r_{+}z))^{2}-4^{-1}r(x+r_{-}z)^{2}+r(y+z_{11}z)^{2}+r(\det(Z)z^{2}))\,dwdxdydz$
$\displaystyle|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(4^{-1}r_{+}^{2}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})$
$\displaystyle\times\int_{\mathcal{O}_{F}^{3}}\psi(r(w^{2}+w(x+r_{+}z)+2^{-1}x(r_{+}-r_{-})z)+r(y^{2}+2yz_{11}z))\,dwdxdydz.$

If $|r|<1$, we can and do assume $\left\lVert Z\right\rVert\geq 1$. Then the
integral equals

$\displaystyle|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(4^{-1}r_{+}^{2}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})\mathbf{1}_{\mathcal{O}_{F}}(2rz_{11}z)$
$\displaystyle\times\int_{\mathcal{O}_{F}}\mathbf{1}_{\mathcal{O}_{F}}(r(x+r_{+}z))\psi(2^{-1}xr(r_{+}-r_{-})z))\,dxdz$
$\displaystyle=|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(4^{-1}r_{+}^{2}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})\mathbf{1}_{\mathcal{O}_{F}}(2rz_{11}z)\mathbf{1}_{\mathcal{O}_{F}}(rr_{+}z)\mathbf{1}_{\mathcal{O}_{F}}(2^{-1}r(r_{+}-r_{-})z)\,dz,$

which is dominated by $\left\lVert Z\right\rVert^{-1}$. Now suppose $|r|\geq
1.$ By Lemma 2.1 the integral equals

$\displaystyle|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(4^{-1}r_{+}^{2}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})\mathbf{1}_{\mathcal{O}_{F}}(2yz_{11})$
$\displaystyle\times\int_{\mathcal{O}_{F}^{3}}\psi(r(w^{2}+w(x+r_{+}z)+2^{-1}x(r_{+}-r_{-})z)+r(y^{2}+2yz_{11}z))\mathbf{1}_{\mathcal{O}_{F}}(x+r_{+}z)\,dwdxdydz$
$\displaystyle=|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(-4^{-1}r_{+}^{2}+2^{-1}r_{+}r_{-}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})\mathbf{1}_{\mathcal{O}_{F}}(2yz_{11})\mathbf{1}_{\mathcal{O}_{F}}(r_{+}z)$
$\displaystyle\times\int_{\mathcal{O}_{F}^{3}}\psi(r(w^{2}+wx+2^{-1}x(r_{+}-r_{-})z)+r(y^{2}+2yz_{11}z))\,dwdxdydz$

Observe that since $|r|\geq 1$

$\displaystyle\int_{\mathcal{O}_{F}^{2}}\psi(r(w^{2}+wx+2^{-1}x(r_{+}-r_{-})z)\,dwdx$
$\displaystyle=\int_{\mathcal{O}_{F}}\psi(rw^{2})\mathbf{1}_{\mathcal{O}_{F}}(r(w+2^{-1}(r_{+}-r_{-})z))\,dw$
$\displaystyle=\mathbf{1}_{\mathcal{O}_{F}}(2^{-1}(r_{+}-r_{-})z)\int_{\mathcal{O}_{F}}\psi(rw^{2})\mathbf{1}_{\mathcal{O}_{F}}(r(w+2^{-1}(r_{+}-r_{-})z))\,dw.$

Thus the original integral equals

$\displaystyle|r|\int_{r^{-1}\mathcal{O}_{F}}\psi(r(-4^{-1}r_{+}^{2}+2^{-1}r_{+}r_{-}-4^{-1}r_{-}^{2}+z_{11}^{2}+\det(Z))z^{2})\mathbf{1}_{\mathcal{O}_{F}}(2yz_{11})\mathbf{1}_{\mathcal{O}_{F}}(r_{+}z)\mathbf{1}_{\mathcal{O}_{F}}(2^{-1}(r_{+}-r_{-})z)$
$\displaystyle\times\int_{\mathcal{O}_{F}^{2}}\psi(rw^{2})\mathbf{1}_{\mathcal{O}_{F}}(r(w+2^{-1}(r_{+}-r_{-})z))\psi(r(y^{2}+2yz_{11}z))\,dwdydz$

The assertion follows by applying Lemma 2.1 to the integral over $y$ and then
applying the trivial bound. ∎

###  5.2. Analytic continuation

######  Lemma 5.7.

Suppose $n\geq 4$. For $\sigma\in\mathbb{R}$, consider the series

$\displaystyle\sum_{Z}|\det(Z)|^{-(\sigma+2)}|\mathrm{tr}(Z)|^{-n/2}\left\lVert
Z\right\rVert^{-n},$

where the sum in $Z$ is taken over a set of representatives of cosets in
$(M_{2}(F)-M_{2}(\mathcal{O}_{F}))/M_{2}(\mathcal{O}_{F})$ such that
$\det(Z)\geq\left\lVert Z\right\rVert$ and $|\mathrm{tr}(Z)|\geq|2|.$ The
series converges when $\sigma>-\frac{n+1}{2}$ and is bounded above by

$\displaystyle
c_{F}q^{-(\sigma+\frac{n+1}{2})}\zeta(n-3)\zeta(n/2-1)\zeta\left(\sigma+\frac{n+1}{2}\right),$

for some positive constant $c_{F},$ which is $1$ if $2\nmid q$.

###### Proof.

We prove only for the case we need, i.e. $2\nmid q$, and leave the case $2|q$
to the reader. We rewrite the series as

(5.2.1)
$\displaystyle\sum_{m=1}^{\infty}q^{-m(\sigma+2)}\sum_{Z:|\det(Z)|=q^{m}}|\mathrm{tr}(Z)|^{-\frac{n}{2}}\left\lVert
Z\right\rVert^{-n}.$

For a fixed $m\geq 1$, we first study the inner sum. We define for integers
$0\leq r\leq a$ the set

$\displaystyle E_{a,r}:=\\{Z:\left\lVert
Z\right\rVert=q^{a},|\mathrm{tr}(Z)|=q^{r}\\}.$

Note that $|E_{a,r}|\leq q^{3a+r}$. Since $\left\lVert
Z\right\rVert\leq|\det(Z)|\leq\left\lVert Z\right\rVert^{2},$ the inner sum in
(5.2.1) is bounded by

$\displaystyle\sum_{\begin{subarray}{c}0\leq a\leq m\leq 2a\\\ 0\leq r\leq
a\end{subarray}}q^{3a+r}q^{-\frac{rn}{2}}q^{-na}$
$\displaystyle\leq\zeta(n/2-1)\sum_{0\leq\lceil\frac{m}{2}\rceil\leq a\leq
m}q^{(3-n)a}$ $\displaystyle\leq q^{-m(n-3)/2}\zeta(n-3)\zeta(n/2-1).$

Thus (5.2.1) is convergent when $\sigma>-\frac{n+1}{2}$ and is bounded above
by

$\displaystyle\zeta(n-3)\zeta(n/2-1)\sum_{m=1}^{\infty}q^{-m(\sigma+\frac{n+1}{2})}=q^{-(\sigma+\frac{n+1}{2})}\zeta(n-3)\zeta(n/2-1)\zeta\left(\sigma+\frac{n+1}{2}\right).$

∎

######  Proposition 5.8.

Suppose $n\geq 4$ and coefficients of $P$ lie in $\mathcal{O}_{F}^{\times}.$
Then the function

$\displaystyle\frac{I(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}},\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},m_{s})}{L\left(s+\frac{3}{2},\pi\right)}$

extends to a holomorphic function in $s$ on
$\sigma=\mathrm{Re}(s)>-\frac{n+1}{2}$. Moreover,

$\displaystyle\left|\frac{I(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}},\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},m_{s})}{L\left(s+\frac{3}{2},\pi\right)}-1\right|\leq
c_{F}q^{-(\sigma+\frac{n+1}{2})}\zeta(n-3)\zeta(n/2-1)\zeta\left(\sigma+\frac{n+1}{2}\right),$

where $c_{F}>0$ is a constant that is $1$ if $2\nmid q$.

###### Proof.

By Lemma 5.4 it suffices to bound

(5.2.2)
$\displaystyle\sum_{Z}|\det(Z)|^{-(s+2)}\int_{M_{2}(\mathcal{O}_{F})^{n}}\psi(\langle
P(Y),Z\rangle)\,dY.$

Here, using Lemma 5.2, the sum in $Z$ can be taken over a set of
representatives of cosets in
$(M_{2}(F)-M_{2}(\mathcal{O}_{F}))/M_{2}(\mathcal{O}_{F})$ such that
$|\det(Z)|\geq\left\lVert Z\right\rVert$ and $|\mathrm{tr}(Z)|\geq|2|.$ By
Lemmas 5.5 and 5.6, the sum (5.2.2) is bounded by

$\displaystyle
C\sum_{Z}|\det(Z)|^{-(\sigma+2)}|\mathrm{tr}(Z)|^{-\frac{n}{2}}\left\lVert
Z\right\rVert^{-n}.$

for some constant $C>0$ that is $1$ if $2\nmid q$. The assertion then follows
from Lemma 5.7. ∎

For the remainder of this section we continue to assume that the coefficients
of $P$ lie in $\mathcal{O}_{F}^{\times}.$ Recall for the trivial
representation,
$L(s+3/2,1\circ\mathrm{nrd})=\zeta_{D}(s+3/2)=\zeta(s+2)\zeta(s+1).$ Motivated
by global considerations, we define the local density to be

(5.2.3) $\displaystyle
c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right):=\left.\frac{I(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}},\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{2}},1_{s})}{\zeta(s+2)\zeta(s+1)}\right|_{s=-2},$

which is well defined for $n\geq 4$ by Proposition 5.8. This definition is
consistent with the definition in Theorem 1.3, as we now explain. By the proof
of Proposition 5.8,

$\displaystyle
c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right)=\sum_{Z}\int_{M_{2}(\mathcal{O}_{F})^{n}}\psi(\langle
P(Y),Z\rangle)\,dY$

is an absolutely convergent series. As the integral on the right is invariant
under $Z\mapsto Z+W$ for $W\in M_{2}(\mathcal{O}_{F}),$ we have

(5.2.4) $\displaystyle
c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right)=\int_{M_{2}(F)}\int_{M_{2}(\mathcal{O}_{F})^{n}}\psi(\langle
P(Y),Z\rangle)\,dY.$

######  Corollary 5.9.

Suppose $2\nmid q$ and $n\geq 5$. Then
$c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right)\neq 0$.

###### Proof.

By Proposition 5.8, it suffices to check that

$\displaystyle
1>q^{-\frac{n-3}{2}}\zeta(n-3)\zeta(n/2-1)\zeta\left(\frac{n-3}{2}\right)$

when $q=3$ and $n=5,$ which is straightforward. ∎

We can alternatively write

$\displaystyle c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right)$
$\displaystyle=\int_{M_{2}(F)}\left(\int_{M_{2}(\mathcal{O}_{F})^{n}}\psi(\langle
P(Y),Z\rangle)dY\right)\,dZ$
$\displaystyle=\lim_{m\to\infty}\int_{M_{2}(F)}\mathbf{1}_{\varpi^{-m}M_{2}(\mathcal{O}_{F})}(Z)\left(\int_{M_{2}(\mathcal{O}_{F})^{n}}\psi(\langle
P(Y),Z\rangle)dY\right)\,dZ$
$\displaystyle=\lim_{m\to\infty}q^{4m}\int_{M_{2}(\mathcal{O}_{F})^{n}}\mathbf{1}_{\varpi^{m}M_{2}(\mathcal{O}_{F})}(P(Y))\,dY$
$\displaystyle=\lim_{m\to\infty}q^{4m(1-n)}\sum_{Y\in(M_{2}(\mathcal{O}_{F})/\varpi^{m}M_{2}(\mathcal{O}_{F}))^{n}}\mathbf{1}_{\varpi^{m}M_{2}(\mathcal{O}_{F})}(P(Y))>0$

when $2\nmid q$ and $n\geq 5$.

######  Corollary 5.10.

For $2\nmid q$ and $n\geq 5,$

$\displaystyle\\#\\{Y\in(M_{2}(\mathcal{O}_{F})/\varpi^{m}M_{2}(\mathcal{O}_{F}))^{n}:P(Y)\equiv
0\ (\mathop{\rm mod}\varpi^{m})\\}\sim
c\left(\mathbf{1}_{M_{2}(\mathcal{O}_{F})^{n}}\right)q^{4m(n-1)}.$

∎

##  6\. Spectral bounds for nonsplit cases

Let $F$ be any local field of characteristic zero and assume $D$ is nonsplit
(hence a division algebra). Then we have an exact sequence

$\displaystyle 1\longrightarrow(D^{\times})^{1}\longrightarrow
D^{\times}\longrightarrow\mathrm{nrd}(D^{\times})\longrightarrow 1,$

where $(D^{\times})^{1}$ is the subgroup of norm $1,$ which is also the
maximal compact subgroup of $D^{\times}$.

######  Lemma 6.1.

Suppose $\Phi$ is (bi-)$(D^{\times})^{1}$-invariant in both entries. Then
$I(f,\Phi,m_{s})=0$ unless $\pi$ is unramified.

###### Proof.

For $\mathrm{Re}(s)>0,$ we have for $k_{1},k_{2}\in(D^{\times})^{1}$

$\displaystyle I(f,\Phi,m_{s})$
$\displaystyle=\int_{D^{\times}}\int_{D^{n}}f(Y)\Phi\left(P(Y)k_{2}^{-1}g^{-1}k_{1}^{-1},k_{1}gk_{2}\right)m_{s}(k_{1}gk_{2})\,dYdg$
$\displaystyle=\int_{D^{\times}}\int_{D^{n}}f(Y)\Phi\left(P(Y)g^{-1}gk_{2}^{-1}g^{-1},g\right)m_{s}(k_{1}gk_{2})\,dYdg.$

Since $(D^{\times})^{1}$ is a normal subgroup of $D^{\times},$ by our
assumptions on $\Phi$ the integral equals

$\displaystyle\int_{D^{\times}}\int_{D^{n}}f(Y)\Phi\left(P(Y)g^{-1},g\right)m_{s}(k_{1}gk_{2})\,dYdg.$

∎

As in the split case we assume $\Phi$ is (bi-)$(D^{\times})^{1}$-invariant in
both entries, so $I(f,\Phi,m_{s})$ is zero unless $\pi$ is unramified by Lemma
6.1. The only unitary unramified representations $\pi$ of $D^{\times}$ are
characters $|\mathrm{nrd}|^{it}$ where $t\in\mathbb{R}.$ Thus for our purposes
in this section it is no loss of generality to assume that $\pi$ is the
trivial representation.

In view of this, we henceforth assume $\pi$ is the trivial representation and
the matrix coefficient $m$ is identically $1.$ For $\mathrm{Re}(s)>0$, we have
by Fourier inversion

$\displaystyle I(f,\Phi,1_{s})=\int_{D}\left(\int_{D^{n}}f(Y)\psi(\langle
P(Y),Z\rangle)\,dY\right)\left(\int_{D^{\times}}\mathcal{F}_{1}(\Phi)(-gZ,g)|\mathrm{nrd}(g)|^{s+2}\,dg\right)\,dZ.$

We will follow the strategy employed in the split case to prove the analytic
continuation of $I(f,\Phi,1_{s}).$ Recall the zeta integral

$\displaystyle\mathcal{Z}_{Z}(\Phi,s):=\mathcal{Z}_{Z}(\Phi,s,1)=\int_{D^{\times}}\mathcal{F}_{1}(\Phi)(-gZ,g)|\mathrm{nrd}(g)|^{s+2}\,dg,$

and

$\displaystyle
H_{Z}(\Phi,s):=H_{Z}(\Phi,s,1)=\frac{\mathcal{Z}_{Z}(\Phi,s,1)}{\zeta_{D}(s+3/2)}.$

Thus

(6.0.1)
$\displaystyle\frac{I(f,\Phi,1_{s})}{\zeta_{D}(s+3/2)}=\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)H_{Z}(\Phi,s)\,dZ.$

###  6.1. The non-Archimedean case

Suppose $F$ is non-Archimedean and $\psi$ is unramified. We begin by recalling
standard facts on quaternion division algebras to fix notations. We refer to
[Voi21, §6.1.5, §13.3.10] for details.

Let $\varpi_{D}\in\mathcal{O}_{D}$ be a uniformizer in $D$, chosen so that
$\varpi_{D}^{2}=\varpi$. Then $\varpi_{D}\mathcal{O}_{D}$ is a two-sided ideal
in $\mathcal{O}_{D}$ and $\mathcal{O}_{D}/\varpi_{D}\mathcal{O}_{D}$ is a
quadratic extension over $\mathbb{F}_{q}$. Let $K$ be the unique quadratic
unramified extension of $F$. We may identify $D$ with $K\oplus K\varpi_{D}$
equipped with the multiplication rule

$\displaystyle\varpi_{D}\alpha=\overline{\alpha}\varpi_{D}$

for $\alpha\in K$, where $\overline{\,\cdot\,}$ is the Galois conjugation for
$K/F$. Under this identification, the maximal order $\mathcal{O}_{D}$ can be
identified with $\mathcal{O}_{K}\oplus\mathcal{O}_{K}\varpi_{D}$, where
$\mathcal{O}_{K}$ is the ring of integers of $K$.

Let $u\in\mathcal{O}_{F}^{\times}-(\mathcal{O}_{F}^{\times})^{2}$ so that
$K=F[x]/(x^{2}-u)$. We may write $K=F\oplus F\sqrt{u}$, and

$\displaystyle D=K\oplus K\varpi_{D}=F\oplus F\sqrt{u}\oplus F\varpi_{D}\oplus
F\sqrt{u}\varpi_{D},$

with the multiplication rule $\varpi_{D}\sqrt{u}=-\sqrt{u}\varpi_{D}$. We will
often identify $D$ with $F^{4}$ as a vector space via the $F$-linear
isomorphism

$\displaystyle F^{4}$ $\displaystyle\tilde{\longrightarrow}D$
$\displaystyle(z_{1},z_{2},z_{3},z_{4})$ $\displaystyle\longmapsto
z_{1}+z_{2}\sqrt{u}+z_{3}\varpi_{D}+z_{4}\sqrt{u}\varpi_{D}.$

and define a vector norm on $D$ by

$\displaystyle\|Z\|:=\max(|z_{1}|,|z_{2}|,|z_{3}|,|z_{4}|).$

Note that $\mathrm{trd}(Z)=2z_{1},\mathrm{nrd}(Z)=z_{1}^{2}-z_{2}^{2}u-\varpi
z_{3}^{2}+\varpi uz_{4}^{2}$. Therefore, if $Z\neq 0$ then

(6.1.1) $\displaystyle 1\ll\frac{|\mathrm{nrd}(Z)|^{\frac{1}{2}}}{\left\lVert
Z\right\rVert}\leq 1.$

Indeed, the right inequality is obvious. To prove the left inequality we may
assume that $Z\in\mathcal{O}_{D}$ and some $z_{i}\in\mathcal{O}_{F}^{\times};$
hence $\left\lVert Z\right\rVert=1.$ The set of all $Z\in\mathcal{O}_{D}$ such
that $z_{i}\in\mathcal{O}_{F}^{\times}$ is a compact subset of $D.$ Since
$\mathrm{nrd}$ is anisotropic, $|\mathrm{nrd}|$ does not vanish on this
compact set and hence is bounded below.

######  Lemma 6.2.

Let $\ell,e\in\mathbb{Z}$. Let $Z\in D$ with $|\mathrm{trd}(Z)|\geq q^{e}$.
Then

$\displaystyle\left|\int_{\varpi_{D}^{\ell}\mathcal{O}_{D}}\psi(\mathrm{trd}(Y^{2}Z))\,dY\right|\ll_{e,\ell}|\mathrm{trd}(Z)|^{-\frac{1}{2}}\left\lVert
Z\right\rVert^{-1}.$

###### Proof.

By changing variables $Y\mapsto\varpi^{-2\lfloor\frac{\ell}{2}\rfloor}Y$ and
replacing $Z$ and $e$, we can and do assume $\ell\in\\{0,1\\}.$ Write
$Y=(y_{i})$ and $Z=(z_{i})$. Consider first $\ell=0$. We have

$\displaystyle\mathrm{trd}(Y^{2}Z)$
$\displaystyle=2z_{1}y_{1}^{2}+2uz_{1}y_{2}^{2}+2\varpi
z_{1}y_{3}^{2}-2u\varpi z_{1}y_{4}^{2}+4uz_{2}y_{1}y_{2}+4\varpi
z_{3}y_{1}y_{3}-4u\varpi z_{4}y_{1}y_{4}.$

Thus, up to a positive constant
$\int_{\mathcal{O}_{D}}\psi(\mathrm{trd}(Y^{2}Z))\,dY$ is

$\displaystyle\int_{\mathcal{O}_{F}^{4}}\psi(2(z_{1}y_{1}^{2}+uz_{1}y_{2}^{2}+\varpi
z_{1}y_{3}^{2}-u\varpi z_{1}y_{4}^{2}+2uz_{2}y_{1}y_{2}+2\varpi
z_{3}y_{1}y_{3}-2u\varpi z_{4}y_{1}y_{4}))\,dy_{1}dy_{2}dy_{3}dy_{4}$
$\displaystyle=\int_{\mathcal{O}_{F}}\psi(2z_{1}y_{1}^{2})\left(\int_{\mathcal{O}_{F}}\psi(2uz_{1}(y_{2}^{2}+2z_{1}^{-1}z_{2}y_{1}y_{2}))\,dy_{2}\right)$
$\displaystyle\times\left(\int_{\mathcal{O}_{F}}\psi(2\varpi
z_{1}(y_{3}^{2}+2z_{1}^{-1}z_{3}y_{1}y_{3}))\,dy_{3}\right)\left(\int_{\mathcal{O}_{F}}\psi(-2u\varpi
z_{1}(y_{4}^{2}+2z_{1}^{-1}z_{4}y_{1}y_{4}))\,dy_{4}\right)\,dy_{1}.$

By Lemma 2.1, for $2\leq i\leq 4$ the integral over $y_{i}$ vanishes unless
$|z_{1}^{-1}z_{i}y_{1}|\ll\min(1,|z_{1}|)^{-1}.$ When $z_{i}$ is nonzero this
bound is equivalent to $|y_{1}|\ll|z_{i}|^{-1}\max(|z_{1}|,1)$. By applying
(2.4.1) to the integrals over $y_{i}$ for $2\leq i\leq 4$ and the trivial
bound to the integral over $y_{1}$, one has

$\displaystyle\left|\int_{\mathcal{O}_{D}}\psi(\mathrm{trd}(Y^{2}Z))\,dY\right|\ll|z_{1}|^{-\frac{3}{2}}\min(1,\max(|z_{2}|,|z_{3}|,|z_{4}|)^{-1}\max(|z_{1}|,1))$

Here by convention if $z_{2}=z_{3}=z_{4}=0$ then
$\min(1,\max(|z_{2}|,|z_{3}|,|z_{4}|)^{-1}\max(|z_{1}|,1))=1.$ One easily
checks this is dominated by $|\mathrm{trd}(Z)|^{-\frac{1}{2}}\left\lVert
Z\right\rVert^{-1},$ and thus the assertion follows.

For $\ell=1$, by changing variables $Y\mapsto\varpi_{D}Y$, it suffices to
bound the integral

$\displaystyle\int_{\mathcal{O}_{D}}\psi(\mathrm{trd}((\varpi_{D}Y)^{2}Z))\,dY.$

By a direct calculation, we have

$\displaystyle\mathrm{trd}((\varpi_{D}Y)^{2}Z)$
$\displaystyle=2\varpi\left(z_{1}y_{1}^{2}-uz_{1}y_{2}^{2}+\varpi
z_{1}y_{3}^{2}+u\varpi z_{1}y_{4}^{2}+2\varpi z_{3}y_{1}y_{3}+2u\varpi
z_{4}y_{2}y_{3}-2u\varpi z_{2}y_{3}y_{4}\right)$

The assertion can be proved similarly as in the case $\ell=0$ and we leave it
to the reader. ∎

At this point it is helpful to recall that

(6.1.2) $\displaystyle\int_{\mathcal{O}_{D}\cap
D^{\times}}|\mathrm{nrd}(g)|^{s}d^{\times}g=d^{\times}g(\mathcal{O}_{D}^{\times})\zeta(s).$

We refer to [Voi21, Lemma 29.7.17] for the proof.

######  Lemma 6.3.

We have

$\displaystyle|H_{Z}(\Phi,s)|\ll\max(|\mathrm{nrd}(Z)|,1)^{-(\mathrm{Re}(s)+2)}J(q^{-\mathrm{Re}(s)}).$

for some polynomial $J\in\mathbb{R}[s]$ with positive coefficients that is
independent of $Z$.

###### Proof.

Assume first $|\mathrm{nrd}(Z)|\leq 1$. Then there exists a finite set
$A\subset\mathcal{S}(D)$ independent of $Z$ such that the function
$g\mapsto\mathcal{F}_{1}(\Phi)(-gZ,g)$ can be written as
$\sum_{\widetilde{\Phi}\in A_{Z}}\widetilde{\Phi}$ for some subset
$A_{Z}\subset A$. Thus there exists a finite set
$A^{\prime}\subset\mathbb{C}[q^{-s}]$ such that $H_{Z}(\Phi,s)=\sum_{J\in
A^{\prime}_{Z}}J(s)$ for some subset $A^{\prime}_{Z}\subset A^{\prime}$. The
assertion then follows easily. For $|\mathrm{nrd}(Z)|\geq 1$, a similar
argument applies after changing variables $g\mapsto gZ^{-1}$. ∎

######  Proposition 6.4.

Suppose $n\geq 3$. Then the function

$\displaystyle\frac{I(\mathbf{1}_{\mathcal{O}_{D}^{n}},\Phi,1_{s})}{\zeta_{D}\left(s+3/2\right)}$

extends to a holomorphic function in $s$ on
$\sigma=\mathrm{Re}(s)>-\frac{n+1}{2}$.

###### Proof.

By Lemma 6.3, it suffices to study the convergence of

$\displaystyle\int_{D}\max(|\mathrm{nrd}(Z)|,1)^{-(\sigma+2)}\left|\int_{\mathcal{O}_{D}^{n}}\psi(\langle
P(Y),Z\rangle)\,dY\right|\,dZ$
$\displaystyle=\mathrm{vol}(\mathcal{O}_{D})\left(1+\sum_{Z\in(D-\mathcal{O}_{D})/\mathcal{O}_{D}}|\mathrm{nrd}(Z)|^{-(\sigma+2)}\left|\int_{\mathcal{O}_{D}^{n}}\psi(\langle
P(Y),Z\rangle)\,dY\right|\right)$

For each coset in $(D-\mathcal{O}_{D})/\mathcal{O}_{D},$ we choose a
representative $Z$ with $|z_{11}|\geq 1$ so $|\mathrm{trd}(Z)|\geq|2|$. By
Lemma 6.2 and (6.1.1), it suffices to study the convergence of the series

$\displaystyle\sum_{Z}|\mathrm{trd}(Z)|^{-\frac{n}{2}}\left\lVert
Z\right\rVert^{-(2\sigma+4+n)}.$

It equals

$\displaystyle\sum_{a=1}^{\infty}q^{-a(2\sigma+4+n)}\sum_{Z:\left\lVert
Z\right\rVert=q^{a}}|\mathrm{trd}(Z)|^{-\frac{n}{2}}$
$\displaystyle\ll\sum_{a=1}^{\infty}q^{-a(2\sigma+4+n)}\sum_{r=0}^{a}q^{3a+r}q^{-r\frac{n}{2}}$
$\displaystyle\leq\zeta(n/2-1)\sum_{a=1}^{\infty}q^{-a(2\sigma+1+n)}$

which converges when $\sigma>-\frac{n+1}{2}$. ∎

Recall that $\zeta_{D}(s+3/2)=\zeta(s+2)$. Define the local density to be

(6.1.3) $\displaystyle
c\left(\mathbf{1}_{\mathcal{O}_{D}^{n}}\right):=\frac{1}{dg(\mathcal{O}_{D}^{\times})\mathrm{vol}(\mathcal{O}_{D})}\frac{I(\mathbf{1}_{\mathcal{O}_{D}^{n}},\mathbf{1}_{\mathcal{O}_{D}^{2}},1_{s})}{\zeta(s+2)}\Bigg{|}_{s=-2}.$

Since $\mathcal{O}_{D}^{\\#}=\varpi^{-1}_{D}\mathcal{O}_{D},$ we have

$\displaystyle\mathcal{Z}_{Z}(\mathbf{1}_{\mathcal{O}_{D}^{2}},s)$
$\displaystyle=\mathrm{vol}(\mathcal{O}_{D})\int_{D^{\times}\cap\mathcal{O}_{D}}\mathbf{1}_{\mathcal{O}_{D}}(\varpi_{D}gZ)|\mathrm{nrd}(g)|^{s+2}\,dg$
$\displaystyle=dg(\mathcal{O}_{D}^{\times})\mathrm{vol}(\mathcal{O}_{D})\max(|\mathrm{nrd}(\varpi_{D}Z)|,1)^{-(s+2)}\zeta(s+2).$

Therefore,

(6.1.4) $\displaystyle
c\left(\mathbf{1}_{\mathcal{O}_{D}^{n}}\right)=\int_{D}\left(\int_{\mathcal{O}_{D}^{n}}\psi(\langle
P(Y),Z\rangle)\,dY\right)\,dZ.$

Thus the definition (6.1.3) is consistent with the definition in Theorem 1.3.

######  Lemma 6.5.

Suppose $n\geq 5$ and coefficients of $P$ lie in $\mathcal{O}_{F}^{\times}$.
Then $c\left(\mathbf{1}_{\mathcal{O}_{D}^{n}}\right)>0$.

###### Proof.

We have that

$\displaystyle\int_{D}\left(\int_{\mathcal{O}_{D}^{n}}\psi(\langle
P(Y),Z\rangle)\,dY\right)\,dZ$
$\displaystyle=\lim_{m\to\infty}\int_{D}\mathbf{1}_{\varpi^{-m}\mathcal{O}_{D}}(Z)\left(\int_{\mathcal{O}_{D}^{n}}\psi(\langle
P(Y),Z\rangle)\,dY\right)\,dZ$
$\displaystyle=\mathrm{vol}(\mathcal{O}_{D})\lim_{m\to\infty}q^{4m}\int_{\mathcal{O}_{D}^{n}}\mathbf{1}_{\varpi_{D}^{2m-1}\mathcal{O}_{D}}(P(Y))\,dY$

is nonnegative. To prove the assertion, it suffices to show

(6.1.5)
$\displaystyle\int_{\mathcal{O}_{D}^{n}}\mathbf{1}_{\varpi_{D}^{2m-1}\mathcal{O}_{D}}(P(Y))\,dY\gg
q^{-4m}.$

Since $n\geq 5,$ by [Lam05, Chapter VI Theorem 2.12] there exists nonzero
$Y_{0}=(y_{1},\ldots,y_{n})\in\mathcal{O}_{F}^{n}$ such that $P(Y_{0})=0$. We
can and do assume $y_{1},y_{2}\in\mathcal{O}_{F}^{\times}$. By the discussion
above Theorem 4.5 in [Con12], the map $\varphi:D^{\times}\to D^{\times}$ given
by $\varphi(Y):=Y^{2}$ is open. Thus there exists $N\in\mathbb{Z}_{\geq 1}$
such that $1+\varpi^{2N}\mathcal{O}_{D}\leq\varphi(\mathcal{O}_{D}^{\times}).$
Let $U:=\varphi^{-1}(1+\varpi^{2N}\mathcal{O}_{D}).$ The integral on the left
in (6.1.5) is bounded below by

(6.1.6)
$\displaystyle\int_{y_{1}U}\left(\int_{y_{2}U\times\prod_{i=3}^{n}(y_{i}+\varpi^{2N}\mathcal{O}_{D})}\mathbf{1}_{\varpi_{D}^{2m-1}\mathcal{O}_{D}}(P(Y))\,dY_{n}\ldots
dY_{2}\right)\,dY_{1}.$

Since

$\displaystyle\\{\upsilon_{2}Y_{2}^{2}+\ldots+\upsilon_{n}Y_{n}^{2}:Y_{2}\in
y_{2}U,Y_{i}\in y_{i}+\varpi^{2N}\mathcal{O}_{D},i\geq
3\\}=-\upsilon_{1}y_{1}^{2}+\varpi^{2N}\mathcal{O}_{D},$

we have (6.1.6) is $\gg_{q,n}q^{-4m}$ for $m>2N+1$.

∎

###  6.2. The Archimedean case

In this subsection $F\cong\mathbb{R}$ and $D$ is the Hamiltonian quaternions.
Note that
$D^{\times}\cong\mathbb{R}_{>0}\times\mathrm{SU}(2)\cong\mathbb{R}_{>0}\times
S^{3}$. Let $\varepsilon>0.$ For $f$ in a Schwartz space we use the symbol
$\ll_{f,\varepsilon}$ to indicate that the implicit constant depends on
$\varepsilon$ and for each $\varepsilon$ it is continuous as a function of $f$
with respect to the usual Fréchet topology on the Schwartz space.

######  Lemma 6.6.

For $f\in\mathcal{S}(D)$ and $Z\in D^{\times}$ with $\mathrm{nrd}(Z)\geq 1$,
we have

$\displaystyle\left|\int_{D}f(Y)\psi(\mathrm{trd}(Y^{2}Z))\,dY\right|\ll_{f,\varepsilon}\max(|\mathrm{trd}(Z)|,1)^{-1}\mathrm{nrd}(Z)^{\varepsilon-\frac{1}{2}}.$

###### Proof.

Write $Y=y_{1}+y_{2}i+y_{3}j+y_{4}k$ and $Z=z_{1}+z_{2}i+z_{3}j+z_{4}k$. We
use these coordinates to identify $D$ with $\mathbb{R}^{4}.$ Then
$\mathrm{trd}(Z)=2z_{1}$ and
$\mathrm{nrd}(Z)=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}.$ We have

$\displaystyle
2^{-1}\mathrm{trd}(Y^{2}Z)=(y_{1}^{2}-y_{2}^{2}-y_{3}^{2}-y_{4}^{2})z_{1}-2y_{1}y_{2}z_{2}-2y_{1}y_{3}z_{3}-2y_{1}y_{4}z_{4}.$

After replacing $\psi(x)$ by $\psi(2x)$ the integral in the lemma is

(6.2.1)
$\displaystyle\int_{\mathbb{R}^{4}}f(y_{1},y_{2},y_{3},y_{4})\psi((y_{1}^{2}-y_{2}^{2}-y_{3}^{2}-y_{4}^{2})z_{1}-2y_{1}y_{2}z_{2}-2y_{1}y_{3}z_{3}-2y_{1}y_{4}z_{4})\,dy_{1}dy_{2}dy_{3}dy_{4}.$

Assume first that $|z_{1}|\leq\tfrac{1}{2}.$ Choose $2\leq i\leq 4$ such that
$\left\lVert Z\right\rVert=|z_{i}|.$ We then repeatedly apply integration by
parts to the expression

$\displaystyle\int_{\mathbb{R}^{4}}\left(f(y_{1},y_{2},y_{3},y_{4})\psi((y_{1}^{2}-y_{2}^{2}-y_{3}^{2}-y_{4}^{2})z_{1})\right)\psi(-2y_{1}y_{2}z_{2}-2y_{1}y_{3}z_{3}-2y_{1}y_{4}z_{4})\,dy_{1}dy_{2}dy_{3}dy_{4}$

where we differentiate the expression in parenthesis with respect to $y_{i}.$
For any $N\in\mathbb{Z}_{\geq 1}$ the contribution of
$|y_{1}|\geq|z_{i}|^{\varepsilon-1}$ is $O_{f,N}(|z_{i}|^{-N\epsilon}).$
Bounding the integral over $|y_{1}|<|z_{i}|^{\varepsilon-1}$ trivially we
obtain the lemma in this case when $|z_{1}|\leq\frac{1}{2}.$

Now assume that $|z_{1}|\geq\tfrac{1}{2}.$ In this case we apply the method of
stationary phase. The quadratic form
$(y_{1}^{2}-y_{2}^{2}-y_{3}^{2}-y_{4}^{2})z_{1}-2y_{1}y_{2}z_{2}-2y_{1}y_{3}z_{3}-2y_{1}y_{4}z_{4}$
in $(y_{1},y_{2},y_{3},y_{4})$ is nondegenerate. Thus the gradient vanishes
only at $(y_{1},y_{2},y_{3},y_{4})=0.$ The Hessian is

(6.2.2)
$\displaystyle\det\left(\begin{smallmatrix}2z_{1}&-2z_{2}&-2z_{3}&-2z_{2}\\\
-2z_{2}&-2z_{1}&0&0\\\ -2z_{3}&0&-2z_{1}&0\\\
-2z_{4}&0&0&-2z_{1}\end{smallmatrix}\right)=-8z_{1}^{2}(\mathrm{nrd}(Z))$

Thus the standard stationary phase bound [Hör03, Lemma 7.7.3] suffices in this
case. ∎

######  Lemma 6.7.

The function $H_{Z}(\Phi,s)$ is entire. Moreover, given any real numbers
$\sigma_{1}<\sigma_{2},$ $0<\epsilon,$ and polynomial $J\in\mathbb{C}[s],$ one
has

$\displaystyle|J(s)\mathcal{Z}_{Z}(\Phi,s)|\ll_{\sigma_{1},\sigma_{2},J,\epsilon,\Phi}\max(\mathrm{nrd}(Z),1)^{-(\mathrm{Re}(s)+2)}$

for $\sigma_{1}<\mathrm{Re}(s)<\sigma_{2}$ and $\epsilon<\mathrm{Im}(s).$

###### Proof.

Note that $\mathcal{F}_{1}(\Phi)$ is invariant under $S^{3}$ in both entries
as $\Phi$ is. Using spherical coordinates, $\mathcal{Z}_{Z}(\Phi,s)$ is up to
a positive constant

(6.2.3)
$\displaystyle\int_{0}^{\infty}\mathcal{F}_{1}(\Phi)(ar,r)r^{2s+3}\,dr.$

where $a:=\mathrm{nrd}(Z)^{\frac{1}{2}}$. By Tate’s thesis,
$\Gamma(s+2)^{-1}\mathcal{Z}_{Z}(\Phi,s)$ is entire and the first assertion
follows. The second assertion is a consequence of [Igu78, Chapter 1 Theorem
3.1] if $a\leq 1.$ For $a>1,$ the same argument applies after a change of
variables $r\mapsto a^{-1}r$. ∎

######  Proposition 6.8.

Suppose $n\geq 3$. For any $f\in\mathcal{S}(D^{n}),$ the function

$\displaystyle\frac{I(f,\Phi,1_{s})}{\zeta_{D}(s+3/2)}$

extends to a holomorphic function on $\sigma=\mathrm{Re}(s)>-\frac{n+1}{2}$.
Moreover, $I(f,\Phi,1_{s})$ is rapidly decreasing on vertical strips (away
from its poles).

###### Proof.

By (6.0.1) and Lemmas 6.6 and 6.7, it suffices to prove that

$\displaystyle\int_{\mathrm{nrd}(Z)\leq
1}dZ+\int_{\mathrm{nrd}(Z)>1}\max(|\mathrm{trd}(Z)|,1)^{-n}\mathrm{nrd}(Z)^{-(\sigma+2+\frac{n}{2}-\epsilon)}\,dZ.$

converges for $\sigma>-\frac{n+1}{2}+\epsilon$. Using spherical coordinates,
the integral above is equal to a positive constant times

$\displaystyle\int_{1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\max(2r|\cos(\theta_{1})|,1)^{-n}r^{-(n+1+2\sigma-2\epsilon)}\sin(\theta_{1})^{2}\sin(\theta_{2})\,d\theta_{3}d\theta_{2}d\theta_{1}dr$
$\displaystyle\ll\int_{1}^{\infty}\int_{\cos^{-1}(r^{-1})}^{\pi/2}r^{-(n+1+2\sigma-2\epsilon)}\sin(\theta)^{2}\,d\theta
dr$
$\displaystyle+\int_{1}^{\infty}\int_{0}^{\cos^{-1}(r^{-1})}r^{-(2n+1+2\sigma-2\epsilon)}(\sec(\theta)^{n}-\sec(\theta)^{n-2})\,d\theta
dr.$

We have

$\displaystyle\int_{\cos^{-1}(r^{-1})}^{\pi/2}\sin^{2}(\theta)\,d\theta=\frac{\theta-\sin(\theta)\cos(\theta)}{2}\bigg{|}_{\cos^{-1}(r^{-1})}^{\pi/2}\ll
r^{-1}$

and by the reduction formula for antiderivaties of $\sec^{n}$

$\displaystyle\int_{0}^{\cos^{-1}(r^{-1})}\left(\sec(\theta)^{n}-\sec(\theta)^{n-2}\right)\,d\theta\ll
r^{n-1}.$

Therefore, we are to examine the convergence of

$\displaystyle\int_{1}^{\infty}r^{-(n+2+2\sigma-2\epsilon)}\,dr.$

It converges when $n+2+2\sigma-2\epsilon>1$, i.e.,
$\sigma>-\frac{n+1}{2}+\epsilon$. ∎

######  Lemma 6.9.

Suppose $n\geq 4$. The linear operator on $\mathcal{S}(D^{n})$ given by

$\displaystyle f\mapsto
c(f):=\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)\,dZ$

is nonzero.

###### Proof.

Since we are working over $F\cong\mathbb{R},$ upon changing variables if
necessary we may assume $P(Y)=Y_{1}^{2}+\ldots+Y_{n}^{2}.$ Consider the closed
$F$-subscheme $U$ of $\mathbb{A}^{4n}$ cut out by the system of equations
(1.0.8). The Jacobian of (1.0.8) has full rank if $w_{i}\neq 0$ for some $i.$
Therefore, the $F$-points of the smooth locus $U^{\mathrm{sm}}(F)$ is an open
dense subset of $U(F)$. If $f\in C^{\infty}_{c}(D^{n})$ is a function whose
support does not intersect the subset

$\\{(w,x,y,z)\in(F^{4})^{n}:w=0\\}$

then $f|_{U(F)}$ is compactly supported on $U^{\mathrm{sm}}(F)$ and

$\displaystyle\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)\,dZ=\int_{U^{\mathrm{sm}}(F)}f(u)\,du$

for an appropriate measure $du$ on $U^{\mathrm{sm}}(F)$ (see [GHL21, §7]).
Therefore, the operator is easily seen to be nonzero. ∎

######  Lemma 6.10.

Suppose $n\geq 4$. Let $c_{1},c_{0}\in\mathbb{C}^{\times}$ be given. There
exist $(f,\Phi)\in C^{\infty}_{c}(D^{n})\times C_{c}^{\infty}(D^{2})$ such
that $\Phi$ is bi-invariant under $S^{3}$ in both entries, satisfies
$\mathcal{F}_{2}(\Phi)(0,0)\neq 0$ and $\Phi(t,0)=0$ for all $t\in D,$ and

$\displaystyle\frac{c_{1}I^{1}(f,\Phi,1_{s})-c_{0}I^{0}(f,\Phi,1_{s})}{\zeta_{D}(s+3/2)}\Bigg{|}_{s=-2}\neq
0.$

###### Proof.

It suffices to show there exists $f\in C^{\infty}_{c}(D^{n})$ such that the
$\mathbb{C}$-span of

$\displaystyle\zeta_{D}(s+3/2)^{-1}(I^{1}(f,\Phi,1_{s}),I^{0}(f,\Phi,1_{s}))\Bigg{|}_{s=-2}$

with $\Phi\in C^{\infty}_{c}(D^{2})$ satisfying the stated conditions is
$\mathbb{C}^{2}$. Furthermore, since both coordinates are continuous in
$\Phi,$ it suffices to show that $\mathbb{C}$-span as $\Phi$ ranges over all
functions in $\mathcal{S}(D^{2})$ satisfying the stated conditions is
$\mathbb{C}^{2}.$

For $a\in\mathbb{R}_{>0},$ consider $\Psi_{a}\in\mathcal{S}(D)$ given by
$\Psi_{a}(X)=e^{-a\mathrm{nrd}(X)}$. One has

$\displaystyle\int_{D^{\times}}\Psi_{a}(g)\mathrm{nrd}(g)^{s}dg=\mathrm{vol}((D^{\times})^{1})\int_{0}^{\infty}e^{-ar^{2}}|r|^{2s}d^{\times}(r^{2})=\frac{\mathrm{vol}((D^{\times})^{1})}{2}a^{-s}(2\pi)^{s}\zeta_{D}(s-1/2)$

For $a_{1},a_{2}\in\mathbb{R}_{>0}$, let
$\Phi_{a_{1},a_{2}}:=\Psi_{a_{1}}\otimes\Psi_{a_{2}}$. Then one has

$\displaystyle\mathcal{Z}_{Z}(\Phi_{a_{1},a_{2}},s)=\mathcal{F}_{1}(\Phi_{a_{1},a_{2}})(0,0)\frac{\mathrm{vol}((D^{\times})^{1})}{2}Q(\mathrm{nrd}(Z))^{-s-2}\zeta_{D}(s+3/2)$

for some linear function $Q$. Thus

(6.2.4)
$\displaystyle\frac{I^{1}(f,\Phi_{a_{1},a_{2}},1_{s})}{\zeta_{D}(s+3/2)}\Bigg{|}_{s=-2}=\mathcal{F}_{1}(\Phi_{a_{1},a_{2}})(0,0)\frac{\mathrm{vol}((D^{\times})^{1})}{2}\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)\,dZ.$

Similarly,

(6.2.5)
$\displaystyle\frac{I^{0}(f,\Phi_{a_{1},a_{2}},1_{s})}{\zeta_{D}(s+3/2)}\Bigg{|}_{s=-2}=\mathcal{F}_{2}(\Phi_{a_{1},a_{2}})(0,0)\frac{\mathrm{vol}((D^{\times})^{1})}{2}\int_{D}\left(\int_{D^{n}}f(Y)\psi\left(\left\langle
P(Y),Z\right\rangle\right)\,dY\right)\,dZ.$

By Lemma 6.9, we can choose $f$ so that the integral in (6.2.4) and (6.2.5)
does not vanish. Then

$\displaystyle\zeta_{D}(s+3/2)^{-1}(I^{1}(f,\Phi_{2,2}-\Phi_{2,1},1_{s}),I^{0}(f,\Phi_{2,2}-\Phi_{2,1},1_{s}))\Bigg{|}_{s=-2}$

and

$\displaystyle\zeta_{D}(s+3/2)^{-1}(I^{1}(f,\Phi_{2,2}-\Phi_{1,1},1_{s}),I^{0}(f,\Phi_{2,2}-\Phi_{1,1},1_{s}))\Bigg{|}_{s=-2}$

are two linear independent vectors in $\mathbb{C}^{2}$. This completes the
proof. ∎

##  7\. General Hessian estimates

In this section, we assume $P(Y)=\sum_{1\leq i\leq n}\upsilon_{i}Y_{i}^{2}$,
where $\upsilon_{i}\in F^{\times}$ are fixed. We fix a place $v$ of $F$ and
omit it from notation, writing $F:=F_{v},$ $D:=D_{F_{v}}$, etc. For any
integer $N\geq 0$, we fix vector norms $\left\lVert\cdot\right\rVert$ on
$F^{N}$ and $M_{N}(F)\cong F^{N^{2}}$, by the formula (2.2.1).

We need the following quadric fibration result. We emphasize that $D-\\{0\\}$
may be strictly larger than $D^{\times}$, since $D$ has nontrivial zero
divisors if $D$ is split.

######  Lemma 7.1.

Fix a choice of coordinates on $D$, i.e. an $F$-linear isomorphism $D\cong
F^{4}$. Let

$\displaystyle m:M_{4n}(F)$ $\displaystyle\longrightarrow F^{N}$
$\displaystyle A$ $\displaystyle\longmapsto(m_{1}(A),\dots,m_{N}(A))$

be the map sending $A\in M_{4n}(F)$ to its $N:=\binom{4n}{2n}^{2}$ size
$2n\times 2n$ minors. Let $W\in D$.

  1. (1)

If $W\neq 0$, then $\mathrm{trd}(WY^{2})$ is a quadratic form over $F$ of
Hessian rank $\geq 2$.

  2. (2)

If $W\neq 0$, then $\mathrm{trd}(WP(Y))$ is a quadratic form over $F$ of
Hessian rank $\geq 2n$.

  3. (3)

Let $H_{W}\in M_{4n}(F)$ be the Hessian matrix of $\mathrm{trd}(WP(Y))$. The
map

$D\to F^{N},\quad W\mapsto m(H_{W})$

sends any compact subset of $D-\\{0\\}$ to a compact subset of
$F^{N}-\\{0\\}$.

  4. (4)

Let $H,A_{1},A_{2}\in M_{4n}(F)$. Suppose $\left\lVert H\right\rVert\ll 1$ and
$\left\lVert m(H)\right\rVert\gg 1$. If $\left\lVert A_{i}\right\rVert\ll 1$
and $\lvert\det(A_{i})\rvert\asymp 1$, then $\left\lVert
A_{1}HA_{2}\right\rVert\ll 1$ and $\left\lVert m(A_{1}HA_{2})\right\rVert\gg
1$.

###### Proof.

We first remark that if $Q$ is a nonzero quadratic form on $F^{r}$ and $L$ is
a nonzero linear form on $F^{r}$, then

$\mathrm{rank}(Q|_{L=0})-\mathrm{rank}(Q)\in\\{0,-1,-2\\}.$

Indeed, after linearly changing variables so that $L$ is the projection to the
first coordinate, the Hessian matrix of $Q|_{L=0}$ is a sub-matrix of the
Hessian matrix of $Q$, with the first row and column deleted.

(1): If $\mathrm{trd}(Y)=0$, then $Y^{2}=-\mathrm{nrd}(Y)$ by Cayley–Hamilton,
so $\mathrm{trd}(WY^{2})=-\mathrm{trd}(W)\mathrm{nrd}(Y)$. If
$\mathrm{trd}(W)\neq 0$, then

$\mathrm{rank}(\mathrm{trd}(WY^{2})|_{\mathrm{trd}(Y)=0})=\mathrm{rank}(\mathrm{nrd}(Y)|_{\mathrm{trd}(Y)=0}).$

After passing to an algebraic closure we see that this rank is
$\mathrm{rank}(ad-bc|_{a+d=0})=3,$ so
$\mathrm{rank}(\mathrm{trd}(WY^{2}))\in\\{3,4\\}$. Now suppose
$\mathrm{trd}(W)=0$. Then by Cayley–Hamilton,

$\mathrm{trd}(WY^{2})=\mathrm{trd}(Y)\mathrm{trd}(WY)-\mathrm{nrd}(Y)\mathrm{trd}(W)=\mathrm{trd}(Y)\mathrm{trd}(WY).$

But the linear forms $\mathrm{trd}(Y)$ and $\mathrm{trd}(WY)$ are linearly
independent, because $\mathrm{trd}(W)=0$ and $W\neq 0$.333If we had
$\lambda\,\mathrm{trd}(Y)=\mu\,\mathrm{trd}(WY)$, then taking $Y=1$ would
force $\lambda=0$. But $\mathrm{trd}(WY)$ is a nonzero linear form, since
$W\neq 0$; so we would then have $\mu=0$ as well. Therefore,
$\mathrm{rank}(\mathrm{trd}(WY^{2}))=2$ in the case $\mathrm{trd}(W)=0$.

(2): Immediate from (1), since $P(Y)=\sum_{1\leq i\leq
n}\upsilon_{i}Y_{i}^{2}$.

(3): Immediate from (2), the determinantal definition of rank, and
compactness.

(4): Immediate from the determinantal definition of rank, and compactness. ∎

In addition, we need a _uniform_ diagonalization result for quadratic forms
over local fields.

######  Lemma 7.2.

Let $c:=1$ unless $F$ has residue characteristic $2$, in which case let
$c:=\lvert 2\rvert_{F}^{-1}$. Let $H=H^{t}\in M_{N}(F)$ be a symmetric matrix.
Then there exists a matrix $A\in M_{N}(F)$ with entries of absolute value less
than or equal to $c^{N}$ and with $\lvert\det{A}\rvert=1$, such that $A^{t}HA$
is diagonal.

###### Proof.

We may assume $H\neq 0$. If $F=\mathbb{R}$ (resp. $F=\mathbb{C}$), the
spectral theorem (resp. Autonne–Takagi factorization) for symmetric matrices
over $F$ lets us take $A\in\mathrm{GL}_{N}(F)\cap U(N)$ unitary. Now assume
$F$ non-Archimedean. By compactness, there exists a point
$x=x_{0}\in\mathcal{O}_{F}^{N}$ maximizing the quantity $\lvert
x^{t}Hx\rvert$. Because $H$ is nonzero $x_{0}\neq 0$. Since dividing $x_{0}$
by $\varpi_{F}$ cannot decrease $\lvert x_{0}^{t}Hx_{0}\rvert$, we may further
assume $x_{0}$ primitive. So by an $\mathcal{O}_{F}$-linear change of
variables, we may assume $x_{0}=(0,\dots,0,1)$. By completing the square in
the last variable of the quadratic form $x^{t}Hx$, there exists a matrix
$A_{0}\in 2^{-1}M_{N}(\mathcal{O}_{F})$ with $\det(A_{0})=1$ such that
$A_{0}^{t}HA_{0}=\left(\begin{smallmatrix}H_{0}&0\\\
0&x_{0}^{t}Hx_{0}\end{smallmatrix}\right)$. By induction on $N$, there exists
a matrix $A_{1}\in 2^{1-N}M_{N-1}(\mathcal{O}_{F})$ with $\det(A_{1})=1$ such
that $A_{1}^{t}H_{0}A_{1}$ is diagonal. To complete the proof, take
$A=A_{0}\left(\begin{smallmatrix}A_{1}&0\\\ 0&1\end{smallmatrix}\right)$. ∎

We also need stationary phase in the clean form suggested by [BV15, Lemma
2.7].

######  Lemma 7.3.

Assume $F\cong\mathbb{R}$. Fix an integer $N\geq 0$. Let $w,\Psi\in
C^{\infty}(F^{N})$. Suppose $w$ is supported on
$\left\lVert\bm{x}\right\rVert\leq R\ll 1$, and that its $k$th order
derivatives are $\ll_{k}1$ for $\bm{x}\in F^{N}$, for $k\geq 0$. Suppose the
$k$th order derivatives of $\Psi$ are $\ll_{k}1$ for
$\left\lVert\bm{x}\right\rVert\leq R+1$, for $k\geq 0$. If $T\geq 1$, then

$\int_{F^{N}}w(\bm{x})\psi(T\Psi(\bm{x}))\,d\bm{x}\ll_{A,\epsilon}\mathrm{vol}(\Omega_{T,\epsilon})+T^{-A}$

for all $A\geq 0$ and $\epsilon\in(0,\frac{1}{2})$, where
$\Omega_{T,\epsilon}:=\\{\bm{x}\in\operatorname{Supp}(w):\left\lVert\nabla{\Psi}(\bm{x})\right\rVert\leq
T^{-1/2+\epsilon}\\}$.

###### Proof.

Fix a smooth partition of unity $1=\nu_{0}+\nu_{1}+\dots+\nu_{N}$ of $F^{N}$
such that $\nu_{0}$ is supported on $\left\lVert\bm{x}\right\rVert\leq 1$, and
$\nu_{i}$ for $1\leq i\leq N$ is supported on $\lvert
x_{i}\rvert\geq\frac{1}{2}$. Trivially

$\int_{F^{N}}\nu_{0}(T^{1/2-\epsilon}\nabla{\Psi}(\bm{x}))w(\bm{x})\psi(T\Psi(\bm{x}))\,d\bm{x}\ll\mathrm{vol}(\Omega_{T,\epsilon}).$

Also, if $1\leq i\leq N$ then by non-stationary phase in $\bm{x}$ [Hör03,
Theorem 7.7.1]

$\int_{F^{N}}\nu_{i}(T^{1/2-\epsilon}\nabla{\Psi}(\bm{x}))w(\bm{x})\psi(T\Psi(\bm{x}))\,d\bm{x}\ll_{A,\epsilon}T^{-A}.$

∎

Recall for
$(f,\Phi,\delta,\gamma)\in\mathcal{S}(D^{n})\times\mathcal{S}(D^{2})\times
D^{\times}\times D^{n}$ we define

(7.0.1)
$\begin{split}I_{0}(\delta,\gamma):=I_{0}(f,\Phi,\delta,\gamma)&:=\int_{D^{n}}f(Y)\Phi(\delta,\delta^{-1}P(Y))\psi(\mathrm{trd}(\gamma\cdot
Y)/\mathrm{nrd}(\delta))\,dY.\end{split}$

We now have the following integral result at infinite places $v$. A similar
result holds when $F\cong\mathbb{C}$, but we focus on $F\cong\mathbb{R}$ for
notational simplicity.

######  Proposition 7.4.

Assume $F\cong\mathbb{R}$. Fix $f\in C^{\infty}_{c}(D^{n})$ and $\Phi\in
C^{\infty}_{c}(D^{2})$. Let $\delta\in D^{\times}$ and $\gamma\in D^{n}$. Then
$I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)=0$ unless
$\left\lVert\delta/X\right\rVert\ll 1$. Moreover, for all $A,\epsilon>0$,

$I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\ll_{A,\epsilon}(1+\left\lVert\gamma\right\rVert/\left\lVert\delta\right\rVert)^{-A}(1+\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert)^{\epsilon-n}.$

###### Proof.

The first claim is clear since $\Phi$ is assumed to have compact support. It
remains to prove the second claim, assuming that
$\left\lVert\delta/X\right\rVert\ll 1$, so $1\ll\left\lVert
X\delta^{-1}\right\rVert.$ Fix $A,\epsilon>0$ and $B\gg_{A,\epsilon}1$. By
non-stationary phase in $Y$ (i.e. repeated integration by parts, integrating
the $\psi$ factor and differentiating the complementary factor), we have

$I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\ll_{B}\int_{D^{n}}\mathbf{1}_{\left\lVert
Y\right\rVert\ll 1}\frac{\left\lVert X\delta^{-1}\right\rVert^{B}}{\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert^{B}}\,dY,$

because differentiating $X\delta^{-1}P(Y)$ introduces a factor of
$\ll\left\lVert X\delta^{-1}\right\rVert.$ But
$\delta^{-1}=\delta^{\dagger}/\mathrm{nrd}(\delta)$ and
$\left\lVert\delta^{\dagger}\right\rVert\asymp\left\lVert\delta\right\rVert$,
so this implies

$I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\ll_{B}\frac{\left\lVert\delta\right\rVert^{B}}{\left\lVert\gamma\right\rVert^{B}}.$

This estimate, combined with the trivial estimate $I_{0}\ll 1$, already
suffices, except in the case where we have, say,

(7.0.2) $\left\lVert\gamma\right\rVert\leq(1+\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert)^{\omega}\left\lVert\delta\right\rVert,$

where $\omega:=\min(\frac{\epsilon}{2n+A},\frac{1}{100})$ is a parameter whose
purpose will become clear later.

To go further, we will use Lemma 7.1. First, by Fourier inversion, i.e.
(2.3.1), we have

$\displaystyle
I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)=\int_{D}\mathcal{F}_{2}(\Phi)(\delta/X,Z)\int_{D^{n}}f(Y)\overline{\psi}\left(\frac{\mathrm{trd}(ZX\delta^{\dagger}P(Y)-X\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY\,dZ.$

By continuity, we may restrict to $Z\neq 0$. After identifying $Y\in D^{n}$
with $\bm{x}\in F^{4n}$, let

$\bm{u}:=X\gamma/\mathrm{nrd}(\delta)\in F^{4n},\quad\lambda:=X\left\lVert
Z\delta^{\dagger}\right\rVert/\mathrm{nrd}(\delta)\in F,\quad
G(\bm{x}):=\mathrm{trd}(Z\delta^{\dagger}P(Y))/\left\lVert
Z\delta^{\dagger}\right\rVert.$

We now apply stationary phase, in the form of Lemma 7.3, to the inner integral
over $Y$, with $T=\max(1,\lvert\lambda\rvert,\left\lVert\bm{u}\right\rVert)$
and $\Psi(\bm{x})=\frac{\lambda G(\bm{x})-\bm{u}\cdot\bm{x}}{T}$. We may
assume $\left\lVert\bm{u}\right\rVert\gg 1,$ or there is nothing to prove.
Lemma 7.2, together with parts (3) and (4) of Lemma 7.1, implies that up to a
bounded change of variables over $F$, the quadratic form $G(\bm{x})$ is of the
form $\sum_{1\leq i\leq 4n}a_{i}x_{i}^{2}$, where $\lvert
a_{1}\rvert,\dots,\lvert a_{2n}\rvert\asymp 1$ and $a_{2n+1},\dots,a_{4n}\ll
1$. The measure of the “near-stationary set”

$\\{\bm{x}\in\operatorname{Supp}(f):\left\lVert\lambda\nabla{G}(\bm{x})-\bm{u}\right\rVert\leq
T^{1/2+\omega}\\}$

is $0$ if $\lambda\ll\left\lVert\bm{u}\right\rVert$ (with a small enough
implied constant), and is $\ll\prod_{1\leq i\leq
4n}\min(1,\frac{T^{1/2+\omega}}{\lvert\lambda
a_{i}\rvert})\ll(T^{-1/2+\omega})^{2n}$ for all other $\lambda$. Since
$T\geq\left\lVert\bm{u}\right\rVert$, we conclude that

$I_{0}\left(\frac{\delta}{X},\frac{\gamma}{X}\right)\ll_{B,\omega}\int_{D}(1+\left\lVert
Z\right\rVert)^{-B}\left(\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert^{-n+2n\omega}+\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert^{-B}\right)\,dZ.$

But
$(1+\left\lVert\gamma\right\rVert/\left\lVert\delta\right\rVert)^{A}\ll_{A}(1+\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert)^{A\omega}$ by (7.0.2). So the last
display is

$\ll_{A,\epsilon}(1+\left\lVert\gamma\right\rVert/\left\lVert\delta\right\rVert)^{-A}(1+\left\lVert
X\gamma/\mathrm{nrd}(\delta)\right\rVert)^{\epsilon-n},$

because $2n\omega+A\omega\leq\epsilon$ by definition of $\omega$. This
completes the proof of the proposition. ∎

Now assume $F$ is non-Archimedean for the rest of the section. For $x\in D$,
let

$v(x):=\max\\{e\in\mathbb{Z}\cup\\{\infty\\}:x\in\varpi^{e}\mathcal{O}_{D}\\}.$

Let

$v(\gamma):=\min(v(\gamma_{1}),\dots,v(\gamma_{n})),\quad\left\lVert\gamma\right\rVert:=q^{-v(\gamma)}.$

We have the following analogue of Proposition 7.4.

######  Proposition 7.5.

Let $(f,\Phi)\in C_{c}^{\infty}(D^{n})\times C_{c}^{\infty}(D^{2}).$

  1. (1)

If $I_{0}(\delta,\gamma)\neq 0$, then $v(\delta)\geq-O_{\Phi}(1)$ and
$v(\gamma)\geq v(\delta)-O_{f,\Phi}(1)$.

  2. (2)

$I_{0}(\delta,\gamma)\ll_{f,\Phi}q^{\min(v(\gamma)-v(\mathrm{nrd}(\delta)),0)\,n}$.

###### Proof.

By Fourier inversion, i.e. (2.3.1), we have

$I_{0}(\delta,\gamma)=\int_{D}\mathcal{F}_{2}(\Phi)(\delta,Z)\int_{D^{n}}f(Y)\overline{\psi}\left(\frac{\mathrm{trd}(Z\delta^{\dagger}P(Y)-\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY\,dZ.$

Here $\mathcal{F}_{2}(\Phi)(\delta,Z)\,f(t)=0$ unless $(\delta,Z)\in
K_{\Phi}^{2}$ and $t\in K_{f}^{n}$, where $K_{\Phi}\subset D$ and
$K_{f}\subset D^{n}$ are suitable compact sets depending on $\Phi$ and $f$,
respectively.

(1): First, $\delta\in K_{\Phi}$ implies $v(\delta)\geq-O_{\Phi}(1)$. Also,
for any $W\in\mathcal{O}_{D}$, the shift $Y\mapsto
Y+\varpi^{v(\mathrm{nrd}(\delta))-v(\delta)+k}W$ leaves $f(Y)$ and the residue
$Z\delta^{\dagger}P(Y)\bmod{\mathrm{nrd}(\delta)}$ invariant provided $k$ in
$\mathbb{Z}$ is sufficiently large in a sense depending on $f$, $K_{\Phi}$,
and $P$. Therefore, on averaging over $W\in\mathcal{O}_{D}$ using the
definition of
$\mathcal{F}_{D}(\mathbf{1}_{\mathcal{O}_{D}})(\varpi^{v(\mathrm{nrd}(\delta))-v(\delta)+k}\gamma/\mathrm{nrd}(\delta))$,
we get $I_{0}=0$ unless
$\varpi^{-v(\delta)+k}\gamma\in\operatorname{Supp}(\mathcal{F}_{D}(\mathbf{1}_{\mathcal{O}_{D}}))$.
It follows that $v(\gamma)\geq v(\delta)-k-1$.

(2): By linearity in $f$, we may assume $f$ is the indicator function of a
small box. Fix an identification $D\cong F^{4}$. For each $W\in D$, let
$H_{W}\in M_{4n}(F)$ be the Hessian matrix of the $4n$-variable quadratic form
$WP(t)$ over $F$. Given $(\delta,Z)\in K_{\Phi}^{2}$, let
$W:=Z\delta^{\dagger}/\varpi^{v(Z\delta^{\dagger})}\in\mathcal{O}_{D}$
(assuming $Z\neq 0$). This definition ensures that $v(W)=0$ and

$I(Z):=\int_{D^{n}}f(Y)\overline{\psi}\left(\frac{\mathrm{trd}(Z\delta^{\dagger}P(Y)-\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY=\int_{D^{n}}f(Y)\overline{\psi}\left(\frac{\mathrm{trd}(\varpi^{v(Z\delta^{\dagger})}WP(Y)-\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY.$

By Lemma 7.2, there exists a matrix $A\in 2^{-4n}M_{4n}(\mathcal{O}_{F})$,
with $\lvert\det{A}\rvert=1$, such that $A^{t}H_{W}A$ is diagonal. By Lemma
7.1(3)–(4), the diagonal matrix $A^{t}H_{W}A$ has at least $2n$ entries of
absolute value $\gg 1$, since $\left\lVert W\right\rVert=1$. Thus by (2.4.1)
we have

$I(Z)\ll_{q,f}(q^{\min(v(Z\delta^{\dagger})-v(\mathrm{nrd}(\delta)),0)/2})^{2n}.$

On the other hand, arguing as in (1), we have $I(Z)=0$ unless
$v(\gamma)\geq\min(v(Z\delta^{\dagger}),v(\mathrm{nrd}(\delta)))-O_{f,\Phi}(1)$.
Thus $I(Z)\ll_{f,\Phi}(q^{\min(v(\gamma)-v(\mathrm{nrd}(\delta)),0)})^{n}$.
This extends to $Z=0$ by continuity. ∎

When $D$ is nonsplit, we will ultimately not need much more than Proposition
7.5. In fact, for analytic purposes, nonsplit $D$ turn out to be simpler than
split $D$, because fewer moduli appear. The key observation is this:

######  Lemma 7.6.

If $D$ is nonsplit, then $2v(\delta)+1\geq v(\mathrm{nrd}(\delta))\geq
2v(\delta).$

###### Proof.

By scaling $\delta$ by an appropriate power of $\varpi$, we may assume
$v(\delta)=0$, i.e. $\delta\in\mathcal{O}_{D}-\varpi\mathcal{O}_{D}$. If $D$
is nonsplit, then $\mathcal{O}_{D}=\\{x\in D:v(\mathrm{nrd}(x))\geq 0\\}$ by
[Voi21, § 13.3, Proposition 13.3.4]. So $\varpi\mathcal{O}_{D}=\\{x\in
D:v(\mathrm{nrd}(x))\geq 2\\}$. Since
$\delta\in\mathcal{O}_{D}-\varpi\mathcal{O}_{D}$, it follows that $0\leq
v(\mathrm{nrd}(\delta))\leq 1$. ∎

Split $D$ are much more intricate. They are the focus of the next section.

##  8\. General exponential sums

Adopt the setting of Proposition 7.5 from the last section, with $F$ non-
Archimedean, $\gamma\in D^{n}$, and $\delta\in D^{\times}$. Our goal in this
section is to obtain sharper integral estimates when $D$ is split. We assume
$P(Y)=\sum_{1\leq i\leq n}\upsilon_{i}Y_{i}^{2}$, where
$\upsilon_{i}\in\mathcal{O}_{F}^{\times}.$ With more notation, we could likely
handle general $\upsilon_{i}\in F^{\times}$, but this would obscure the main
ideas. Our main result in this setting is the estimate of Theorem 8.2 below.

Throughout this section, we assume $D$ is split and we will often identify
$D\cong M_{2}(F)$ over $F$ and $\mathcal{O}_{D}\cong M_{2}(\mathcal{O}_{F})$
over $\mathcal{O}_{F}$. We also assume $f=\mathbf{1}_{\mathcal{O}_{D}^{n}}$,
$\Phi=\mathbf{1}_{\mathcal{O}_{D}^{2}},$ and that $\psi$ is unramified. Then
by (7.0.1), we have

(8.0.1)
$\begin{split}I_{0}(\delta,\gamma)&=\mathbf{1}_{\mathcal{O}_{D}}(\delta)\int_{\mathcal{O}_{D}^{n}}\mathbf{1}_{\mathcal{O}_{D}}(\delta^{-1}P(Y))\psi(\mathrm{trd}(\gamma\cdot
Y)/\mathrm{nrd}(\delta))\,dY.\end{split}$

In view of (8.0.1), we will henceforth assume $\delta\in\mathcal{O}_{D}\cap
D^{\times}$. By Fourier inversion, self-duality of
$\mathbf{1}_{\mathcal{O}_{D}}$, and the identity
$\delta^{-1}=\delta^{\dagger}/\mathrm{nrd}(\delta)$, we have

(8.0.2)
$I_{0}(\delta,\gamma)=\int_{\mathcal{O}_{D}}I_{0}(Z,\delta,\gamma)\,dZ,$

where

$I_{0}(Z,\delta,\gamma):=\int_{\mathcal{O}_{D}^{n}}\psi\left(\frac{\mathrm{trd}(Z\delta^{\dagger}P(Y)+\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY.$

Via the invertible affine map $Y_{i}\mapsto Y_{i}+\delta W_{i}$ on
$\mathcal{O}_{D}$ (where $W_{i}\in\mathcal{O}_{D}$),

$I_{0}(Z,\delta,\gamma)=\int_{\mathcal{O}_{D}^{n}}\int_{\mathcal{O}_{D}^{n}}\psi\left(\frac{\mathrm{trd}(Z\delta^{\dagger}P(Y+\delta
W)+\gamma\cdot(Y+\delta W))}{\mathrm{nrd}(\delta)}\right)\,dY\,dW.$

But $\delta^{\dagger}P(Y+\delta W)\equiv\delta^{\dagger}P(Y)+\sum_{1\leq i\leq
n}\upsilon_{i}\delta^{\dagger}Y_{i}\delta W_{i}\bmod{\mathrm{nrd}(\delta)}$,
since $\delta^{\dagger}\delta\equiv 0\bmod{\mathrm{nrd}(\delta)}$. So on
averaging over $W$, we conclude that

(8.0.3)
$I_{0}(Z,\delta,\gamma)=\int_{\mathcal{O}_{D}^{n}}\mathbf{1}_{\mathrm{nrd}(\delta)\mathcal{O}_{D}^{n}}(\upsilon_{i}Z\delta^{\dagger}Y_{i}\delta+\gamma_{i}\delta)\,\psi\left(\frac{\mathrm{trd}(Z\delta^{\dagger}P(Y)+\gamma\cdot
Y)}{\mathrm{nrd}(\delta)}\right)\,dY.$

The condition on
$(\upsilon_{i}Z\delta^{\dagger}Y_{i}\delta+\gamma_{i}\delta)\in\mathcal{O}_{D}^{n}$
leads to new, nonabelian _vanishing phenomena_ , which we alluded to in the
introduction of the paper, for the local integrals $I_{0}$.

This condition leads us to study the image of the set
$\eta^{\dagger}\mathcal{O}_{D}\eta:=\\{\eta^{\dagger}A\eta:A\in\mathcal{O}_{D}\\}$
in $\mathcal{O}_{D}/\mathrm{nrd}(\eta)\mathcal{O}_{D},$ denoted by
$\eta^{\dagger}\frac{\mathcal{O}_{D}}{\mathrm{nrd}(\eta)\mathcal{O}_{D}}\eta$.

######  Lemma 8.1.

Suppose $\eta\in\mathcal{O}_{D}-\varpi\mathcal{O}_{D}$. We have an isomorphism
of $\mathcal{O}_{F}$-modules

$\eta^{\dagger}\frac{\mathcal{O}_{D}}{\mathrm{nrd}(\eta)\mathcal{O}_{D}}\eta\cong\mathcal{O}_{F}/\mathrm{nrd}(\eta)\mathcal{O}_{F}.$

In particular,
$\eta^{\dagger}\frac{\mathcal{O}_{D}}{\mathrm{nrd}(\eta)\mathcal{O}_{D}}\eta$
is a cyclic $\mathcal{O}_{F}$-module, generated by some
$N\in\eta^{\dagger}\mathcal{O}_{D}\eta$.

###### Proof.

Let $m=\mathrm{nrd}(\eta)$. If $\alpha,\beta\in\mathcal{O}_{D}^{\times}$, then
$(\alpha\eta\beta)^{\dagger}\frac{\mathcal{O}_{D}}{m\mathcal{O}_{D}}\alpha\eta\beta=\beta^{\dagger}\eta^{\dagger}\frac{\mathcal{O}_{D}}{m\mathcal{O}_{D}}\eta\beta\cong\eta^{\dagger}\frac{\mathcal{O}_{D}}{m\mathcal{O}_{D}}\eta$.
Thus, by the Cartan decomposition we may assume
$\eta=\left(\begin{smallmatrix}m&0\\\ 0&1\end{smallmatrix}\right)$, since
$\eta\in\mathcal{O}_{D}-\varpi\mathcal{O}_{D}$. If
$A=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$, then
$\eta^{\dagger}A\eta=\left(\begin{smallmatrix}1&0\\\
0&m\end{smallmatrix}\right)\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}m&0\\\
0&1\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}0&b\\\
0&0\end{smallmatrix}\right)\bmod{m}$. ∎

Write $\delta=\varpi^{v(\delta)}\eta$, so
$\eta\in\mathcal{O}_{D}-\varpi\mathcal{O}_{D}$ and
$\mathrm{nrd}(\delta)=\varpi^{2v(\delta)}\mathrm{nrd}(\eta)$. The restriction
$\upsilon_{i}Z\delta^{\dagger}Y_{i}\delta+\gamma_{i}\delta\equiv
0\bmod{\mathrm{nrd}(\delta)}$ in (8.0.3) implies first that

(8.0.4)
$\upsilon_{i}Z\delta^{\dagger}Y_{i}+\gamma_{i}\in\mathcal{O}_{D}\delta^{\dagger},\textnormal{
whence }\gamma_{i}=\varpi^{v(\delta)}\gamma^{\prime}_{i}$

for some $\gamma^{\prime}_{i}\in\mathcal{O}_{D}$, and second that

(8.0.5) $\upsilon_{i}Z\eta^{\dagger}Y_{i}\eta+\gamma^{\prime}_{i}\eta\equiv
0\bmod{\mathrm{nrd}(\eta)}.$

Letting $N=\eta^{\dagger}N_{0}\eta\in\eta^{\dagger}\mathcal{O}_{D}\eta$
generate
$\eta^{\dagger}\frac{\mathcal{O}_{D}}{\mathrm{nrd}(\eta)\mathcal{O}_{D}}\eta$
as in Lemma 8.1, it follows that there exist
$\lambda_{1},\dots,\lambda_{n}\in\mathcal{O}_{F}/\mathrm{nrd}(\eta)\mathcal{O}_{F}$
such that