# On the Cubic Shimura lift to $\operatorname{PGL}(3)$: The Fundamental Lemma

Solomon Friedberg  Department of Mathematics, Boston College, Chestnut Hill,
MA 02467<EMAIL_ADDRESS>and  Omer Offen  Department of Mathematics,
Brandeis University, Waltham MA 02453<EMAIL_ADDRESS>
###### Abstract.

The classical Shimura correspondence lifts automorphic representations on the
double cover of $\operatorname{SL}_{2}$ to automorphic representations on
$\operatorname{PGL}_{2}$. Here we take key steps towards establishing a
relative trace formula that would give a new global Shimura lift, from the
triple cover of $\operatorname{SL}_{3}$ to $\operatorname{PGL}_{3}$, and also
characterize the image of the lift. The characterization would be through the
nonvanishing of a certain global period involving a function in the space of
the automorphic minimal representation $\Theta_{\operatorname{SO}_{8}}$ for
split $\operatorname{SO}_{8}(\mathbb{A})$, consistent with a 2001 conjecture
of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global
distribution on $\operatorname{PGL}_{3}(\mathbb{A})$ involving this period and
show that it is a sum of factorizable orbital integrals. The same is true for
the Kuznetsov distribution attached to the triple cover of
$\operatorname{SL}_{3}(\mathbb{A})$. We then match the corresponding local
orbital integrals for the unit elements of the spherical Hecke algebras; that
is, we establish the Fundamental Lemma.

###### Key words and phrases:

Shimura correspondence, metaplectic group, cubic cover, relative trace
formula, minimal representation, period

###### 2020 Mathematics Subject Classification:

Primary 11F70; Secondary 11F27; 11F67; 11F72; 22E50; 22E55

This work was supported by the NSF, grant numbers DMS-1801497 and DMS-2100206,
by a Simons Fellowship in Mathematics, award number 816862 (Friedberg) and by
a Simons Collaboration Grant, award number 709678 (Offen).

###### Contents

  1. 1 Introduction
  2. 2 An unfolding principle
  3. 3 The geometrization of the distribution $I(f,\Theta)$
  4. 4 Determination of the relevant orbits and orbital integrals for $I$
  5. 5 Unfolding the distribution $J$
  6. 6 Notation and main result
  7. 7 Ingredients needed for the comparison
  8. 8 The integral $J(a,b)$
  9. 9 The integral $I(a,b)$
  10. 10 The comparison of $I(a,b)$ and $J(a,b)$: Proof of Theorem 6.1
  11. 11 The comparison of orbital integrals for the other relevant orbits

##  1\. Introduction

The classical Shimura correspondence [Shi73] lifts automorphic representations
on the double cover of $\operatorname{SL}_{2}$ to automorphic representations
on $\operatorname{PGL}_{2}$. Niwa [Niw75] (working classically) and
Waldspurger [Wal80] (working adelically) showed that the map may be obtained
as a theta lifting. Waldspurger also showed that the image of the lift is
characterized by the nonvanishing of a certain period (the integral of an
automorphic form over a cycle). Jacquet [Jac87] then used his relative trace
formula, which compares distributions on two different groups, one involving
the relevant period, to give another proof of these facts.

Let $n\geq 2$ and let $G$ be a split connected reductive algebraic group over
a global field $F$ with a full set of $n$-th roots of unity $\mu_{n}\subset
F$. Let $\mathbb{A}=\mathbb{A}_{F}$ be the ring of adeles of $F$. Then one may
define an $n$-fold cover $\widetilde{G}^{(n)}(\mathbb{A})$ of the adelic
points of $G$, and it is natural to ask if there is an analogue of the Shimura
map for $\widetilde{G}^{(n)}(\mathbb{A})$. The local Shimura correspondence
was investigated by Savin [Sav88], who proved (in generality) an isomorphism
of Iwahori Hecke algebras. Based on this work, for Cartan type A one expects a
global functorial lift from genuine cuspidal automorphic representations on
$\widetilde{\operatorname{SL}}_{r}^{(n)}(\mathbb{A})$ to automorphic
representations on $\operatorname{PGL}_{r}(\mathbb{A})$ if $n$ divides $r$,
and a lift to automorphic representations on
$\operatorname{SL}_{r}(\mathbb{A})$ if $(n,r)=1$. (See also Bump, Friedberg
and Ginzburg [BFG01], Section 1, for a discussion of this.) Such lifts have
been studied since the 1980s, but progress in establishing a global Shimura
correspondence has been obtained only in the cases $n=2$ or $r=2$. In this
paper we establish the Fundamental Lemma for a relative trace formula that
will give the global Shimura lift in the case $n=r=3$; moreover this project
will characterize the image of the lift by means of a period involving a
function in the space of the automorphic minimal representation for
$\operatorname{SO}_{8}(\mathbb{A})$, confirming a 2001 conjecture of Bump,
Friedberg and Ginzburg.

To put this work on context, recall that Ginzburg, Rallis and Soudry [GRS97a]
observed that $\operatorname{SL}_{2}$ and its 3-fold cover
$\widetilde{\operatorname{SL}}_{2}^{(3)}$ form a dual pair in the 3-fold cover
of the exceptional group $G_{2}$, and used this to establish a lifting of
genuine cuspidal automorphic representations on
$\widetilde{\operatorname{SL}}_{2}^{(3)}(\mathbb{A})$ to automorphic
representations on $\operatorname{SL}_{2}(\mathbb{A})$ and to determine its
image. Let $\text{Sym}^{3}:\operatorname{SL}_{2}\to\operatorname{Sp}_{4}$ be
the symmetric cube map, and $\Theta_{\operatorname{Sp}_{4}}$ be the theta
representation on the metaplectic double cover of
$\operatorname{Sp}_{4}(\mathbb{A})$. This cover splits on the image of
$\text{Sym}^{3}$. Then they showed that an irreducible cuspidal automorphic
representation $\tau$ of $\operatorname{SL}_{2}(\mathbb{A})$ is in the image
of the rank-one cubic Shimura map if and only if the period integral

(1)
$\int_{\operatorname{SL}_{2}(F)\backslash\operatorname{SL}_{2}(\mathbb{A})}\varphi(g)\theta(\text{Sym}^{3}(g))\,dg$

is nonzero for some $\varphi$ in the space of $\tau$ and some $\theta$ in the
space of $\Theta_{\operatorname{Sp}_{4}}$. Another proof of these results was
given by Mao and Rallis [MR99] via a relative trace formula. However, for
higher degree covers or higher rank special linear groups there is no similar
dual pair.

By comparing orbital integrals, Flicker [Fli80] succeeded in establishing a
Shimura-type correspondence for the $n$-fold cover of
$\operatorname{GL}_{2}(\mathbb{A})$ (there is more than one cover, and he
treated a specific one); in this generality there is no known general
characterization of the image by periods. Kazhdan and Patterson [KP86] and
Flicker and Kazhdan [FK86] studied these orbital integrals for higher rank
general linear groups, but were not able to obtain sufficient control to
establish a correspondence. As for double covers, Mao [Mao98] established the
fundamental lemma for a relative trace formula for the double cover of
$\operatorname{GL}_{3}$, and the case of the double cover of
$\operatorname{GL}_{n}$ has recently been treated by Do by first establishing
the equal characteristic case [Do15] and then using model theoretic methods to
move to characteristic zero [Do20].

Here we are concerned with the conjectured Shimura map from the cubic cover of
$\operatorname{SL}_{3}$ to $\operatorname{PGL}_{3}$. The question of
characterizing its image was considered by Bump, Friedberg and Ginzburg
[BFG01]. Let $\Theta_{\operatorname{SO}_{8}}$ be the automorphic minimal
representation on the split special orthogonal group
$\operatorname{SO}_{8}(\mathbb{A})$. This representation was constructed by
Ginzburg, Rallis and Soudry [GRS97b] as a multi-residue of a Borel Eisenstein
series on $\operatorname{SO}_{8}$. Let $\operatorname{Ad}$ denote the adjoint
representation
$\operatorname{Ad}:\operatorname{PGL}_{3}\to\operatorname{SO}_{8}$ (see
Subsection 3.1 below). Supposing that a Shimura lift exists in this case, they
then conjectured

######  Conjecture 1.1 (Bump, Friedberg, Ginzburg).

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\operatorname{PGL}_{3}(\mathbb{A})$. Then $\pi$ is in the image of the cubic
Shimura correspondence from
$\widetilde{\operatorname{SL}}_{3}^{(3)}(\mathbb{A})$ if and only if the
period

(2)
$\int_{\operatorname{PGL}_{3}(F)\backslash\operatorname{PGL}_{3}(\mathbb{A})}\varphi(g)\,\theta(\operatorname{Ad}(g))\,dg$

is nonzero for some $\varphi$ in the space of $\pi$ and some $\theta$ in
$\Theta_{\operatorname{SO}_{8}}$.

Those authors presented two pieces of evidence for this conjecture, the first
from finite fields, and the second by supposing that $\pi$ was not cuspidal
but rather an Eisenstein series induced from cuspidal data $\tau$ on
$\operatorname{GL}_{2}(\mathbb{A})$, formally unfolding the period (2) in this
case, and showing that the resulting integral is nonvanishing for some choice
of data if and only if the period (1) for $\tau$ is nonvanishing for some
choice of data.

The study of periods in the context of Langlands type correspondences is a
main theme of contemporary research; see for example Sakellaridis and
Venkatesh [SV17]. However, the extension to metaplectic covers is not well
developed. In particular, we do not know of an extension of Conjecture 1.1 to
higher rank special linear groups or higher degree covers.

Here we take key steps towards establishing the existence of the global
Shimura correspondence with $n=r=3$ and towards characterizing its image as in
Conjecture 1.1 above. Our approach, following Jacquet, Mao and Rallis, and
Mao, is to establish a comparison of relative trace formulas. For an algebraic
group $H$ defined over $F$ we denote its automorphic quotient space by
$[H]=H(F)\backslash H(\mathbb{A})$. For an affine variety $X$ defined over $F$
denote by $\mathcal{S}(X(\mathbb{A}))$ the space of Schwartz-Bruhat functions
on $X(\mathbb{A})$. The action of $f\in\mathcal{S}(H(\mathbb{A}))$ on
$L^{2}([H])$ obtained by averaging the right regular representation

$R(f)\phi(h)=\int_{H(\mathbb{A})}\phi(hy)f(y)\,dy,$

is realized by the kernel function

$\sum_{\gamma\in G(F)}f(h_{1}^{-1}\gamma h_{2}),\quad h_{1},h_{2}\in[H].$

The relative trace formula compares two distributions obtained as integrals of
these kernel functions for different groups, one integral involving the period
and a Whittaker character and the other involving solely Whittaker characters.

Let $G=\operatorname{PGL}(3)$ considered as an algebraic group defined over
$F$ and $N$ the standard maximal unipotent subgroup of $G$ realized as the
group of upper triangular $3\times 3$ unipotent matrices. Fix a non-trivial
character $\psi$ of $F\backslash\mathbb{A}$. By abuse of notation, we further
denote by $\psi$ the generic character $\psi(n)=\psi(n_{1,2}+n_{2,3})$ of
$[N]$. For $\Theta\in\Theta_{\operatorname{SO}_{8}}$ we consider the
distribution $I(\Theta)$ on $G(\mathbb{A})$ defined by

(3) $I(f,\Theta)=\int_{{[N]}}\int_{[G]}\Big{\\{}\sum_{\gamma\in
G(F)}f(g^{-1}\gamma n)\Big{\\}}\Theta(\operatorname{Ad}(g))\psi(n)\ dg\ dn,\ \
\ f\in\mathcal{S}(G(\mathbb{A})).$

Let $G^{\prime}(\mathbb{A})$ be the adelic $3$-fold metaplectic cover of
$\operatorname{SL}_{3}(\mathbb{A})$. It is a central extension of
$\operatorname{SL}_{3}(\mathbb{A})$ satisfying the exact sequence

$1\rightarrow\mu_{3}\rightarrow
G^{\prime}(\mathbb{A})\rightarrow\operatorname{SL}_{3}(\mathbb{A})\rightarrow
1.$

(Since $G^{\prime}(\mathbb{A})$ is not the adelic points of an algebraic group
this is an abuse of notation.) Denote by $\mathcal{S}(G^{\prime}(\mathbb{A}))$
the space of genuine (that is, $\mu_{3}$-equivariant) Schwartz-Bruhat
functions on $G^{\prime}(\mathbb{A})$. There is a splitting of
$\operatorname{SL}_{3}(F)$ in $G^{\prime}(\mathbb{A})$ and we denote by
$G^{\prime}(F)$ its image. The group $N(\mathbb{A})$ also splits in
$G^{\prime}(\mathbb{A})$ and we continue to denote by $N(\mathbb{A})$ the
image of this splitting. The Kuznetsov trace formula is realized by the
distribution on $G^{\prime}(\mathbb{A})$ defined by

(4) $J(f^{\prime})=\int_{[N]}\int_{[N]}\Big{\\{}\sum_{\gamma\in
G^{\prime}(F)}f^{\prime}(n_{1}\gamma n_{2})\Big{\\}}\psi(n_{1}n_{2})\ dn_{1}\
dn_{2},\ \ \ f^{\prime}\in\mathcal{S}(G^{\prime}(\mathbb{A})).$

Our goal is to study a comparison between the two distributions $I(f,\Theta)$
and $J(f^{\prime})$.

The first step is to _geometerize_ the distribution $I(\Theta)$, that is,
express it as a sum (over some geometric orbits) of distributions that are
factorizable. In most relative trace formulas this step is straightforward:
one writes the sums using the Bruhat decomposition or some variant and unfolds
to get a sum over double cosets of adelic orbital integrals, with only the
relevant double cosets contributing. For factorizable test functions, each
side is then factorizable.

The obstacle in our case is the automorphic minimal representation that
appears in the distribution (3). In contrast to the rank one case, this
representation is not directly obtained from the Weil representation, so there
is no easy way to express the functions in $\Theta_{\operatorname{SO}_{8}}$ as
sums that may then be unfolded. To address this, we make use of the dual pair
$(\operatorname{SL}_{2},\operatorname{SO}_{8})$ inside
$\operatorname{Sp}_{16}$, and realize the automorphic minimal representation
$\Theta_{\operatorname{SO}_{8}}$ as the theta lift of the trivial
representation from $\operatorname{SL}_{2}(\mathbb{A})$ to
$\operatorname{SO}_{8}(\mathbb{A})$. This realization is due to Ginzburg,
Rallis and Soudry [GRS97b, Theorem 6.9]; their proof relies on the work of
Kudla and Rallis [KR94] (or in a classical language over $\mathbb{Q}$, the
work of Deitmar and Krieg [DK91]). When applied to the groups at hand, these
works establish that the suitably regularized theta lift of an Eisenstein
series on $\operatorname{SL}_{2}$ is an Eisenstein series on
$\operatorname{SO}_{8}$; then taking a residue, one obtains the desired
realization of $\Theta_{\operatorname{SO}_{8}}$. Since we are aiming for a
factorizable integral, it is perhaps surprising that it is helpful to
introduce an additional integration over an automorphic quotient
$[\operatorname{SL}_{2}]$. Nonetheless, it turns out that this realization
allows us to do an unfolding and to express the distribution $I(f,\Theta)$ as
a sum of factorizable orbital integrals.

An analogous geometric expansion for $J(f^{\prime})$ follows directly from the
Bruhat decomposition on $G^{\prime}$ (and is well-known). This allows us to
establish a one-to-one correspondence for the relevant orbits of the two
distributions. To effect a global comparison, we must then achieve a
comparison of the corresponding local orbital integrals for each family of
relevant orbits up to a transfer factor whose product over all places is $1$.

The main local result of this paper is a matching of these orbital integrals
for the unit element of the spherical Hecke algebra, that is, the Fundamental
Lemma for this relative trace formula. The calculations are rather elaborate,
but as in prior work on relative trace formulas that give Shimura
correspondences, there is an algebraic fact that is key. In Jacquet’s work on
the Shimura correspondence, this was the computation of a certain Salié sum;
this fact was also key in Iwaniec’s work [Iwa87] on the same topic with an eye
towards analytic number theoretic applications. For the work of Mao and
Rallis, key use was made of an identity relating a cubic exponential sum to a
Kloosterman sum with a cubic character that is due to Duke and Iwaniec [DI93]
(as these authors remark, the analogous fact at a real archimedean place is
Nicholson’s formula for the Airy integral!). The proof of that result in turn
is based on the Davenport-Hasse relation. Their result applies when the
additive character is of conductor 1, that is, to exponential sums of the form

$\sum_{x\bmod p}\exp\left(2\pi i\frac{ax^{3}+bx}{p}\right)$

where $p$ in our computation is the cardinality of the residue field. These
sums are shown to be equal to certain Kloosterman sums modulo $p$ with a cubic
character. Remarkably, we observe that a similar relation is true for
exponential sums that involve additive characters of higher conductor
(Corollary 7.5). For higher conductor, the proof is based on the method of
stationary phase.

In our work the orbital integrals for the big cell frequently reduce to
integrals of pairs of Kloosterman integrals with cubic characters. These
integrals appear intractable. However, using this identity twice to make them
into integrals of pairs of cubic exponential integrals, we are able to effect
the desired comparison. It also sometimes happens that we encounter
Kloosterman integrals without a cubic character. In this case for conductor 1
the result of Duke-Iwaniec is not applicable, but another identity, which
expresses the characteristic function of the local condition
$(ab^{-1})^{3}\equiv-1\bmod\mathfrak{p}$ in terms of a sum of three cubic
exponential integrals, is used instead (Lemma 7.6).

We describe the contents of this paper. The first part of this paper gives the
geometric expansions of the distributions we have described above. Our first
task is to express the distribution $I(f,\Theta)$ as a sum of factorizable
orbital integrals. In Section 2, we formulate a general unfolding principle
and recall the definition of relevant orbits. Then, in Section 3, we use this
principle and the theta lift to $\operatorname{SO}_{8}$ to unfold the
distribution $I(f,\Theta)$ and to express it as a sum of factorizable orbital
integrals over relevant orbits (Proposition 3.2). In the following Section 4,
we work with flag varieties to determine these relevant orbits explicitly
(Theorem 4.2) and also write the orbital integrals for each relevant orbit in
a form amenable for computation. We also need the geometric expansion for the
Kuznetsov distribution $J(f^{\prime})$ and this is presented in Section 5.
This completes the reduction of the comparison of spectral sides of the global
distributions to a comparison of a collection of explicitly given geometric
local distributions.

The second part of this paper establishes the Fundamental Lemma. We begin with
the big cell comparison, which is stated in Theorem 6.1, and whose proof
occupies the following four sections. In Section 7, we develop the necessary
local ingredients, including the comparison of cubic exponential and
Kloosterman integrals, which is given (as a consequence of prior computations
that are closely related to the method of stationary phase) in Subsection 7.5.
The companion computation of cubic integrals when Duke-Iwaniec does not apply
is in Subsection 7.6. The orbital integral $J(a,b)$ attached to the big cell
is then evaluated in Section 8. This is the piece of the calculation that
requires computations in the metaplectic group. Specifically, if $\mathcal{O}$
denotes the local ring of integers then the embedding of
$\operatorname{SL}_{3}(\mathcal{O})$ in the local covering group is given by
means of a section that must be computed in order to work with the unit
element in the corresponding Hecke algebra. In doing so we make use of an
algorithm for computing the local two-cocycle (defined by Matsumoto [Mat69])
that is given in Bump and Hoffstein [BH87]. This leads to the complicated
orbital integrals involving pairs of Kloosterman integrals that we then
evaluate. The orbital integral $I(a,b)$ is evaluated in Section 9. The
comparison is then completed in the brief Section 10. The final section,
Section 11, matches the remaining orbital integrals with an explicit transfer
factor. For the two one-parameter families of relevant orbits, the comparison
once again requires replacing a Kloosterman integral by a cubic exponential
integral, but this time only once.

## Part I. Global Theory: The Geometric Expansions

##  2\. An unfolding principle

Let $F$ be a global field, $\mathbb{A}=\mathbb{A}_{F}$ the ring of adeles of
$F$, $X$ an affine $F$-variety and $G$ an affine algebraic $F$-group acting on
$X$ on the right. We denote the action by $(x,g)\mapsto x^{g}$, $x\in X$,
$g\in G$. Let $\delta_{G}$ be the modular quasicharacter of $G(\mathbb{A})$
and $[G]=G(F)\backslash G(\mathbb{A})$.

For $\xi\in X(F)$ we denote by $G_{\xi}$ the stabilizer of $\xi$ in $G$, an
affine algebraic group defined over $F$. We say that a function
$\mathfrak{c}:X(\mathbb{A})\times G(\mathbb{A})\rightarrow\mathbb{C}$ is an
_automorphic cocycle_ if it satisfies the following two conditions:

  * •

$\mathfrak{c}(\xi,\gamma)=1$, $\xi\in X(F)$, $\gamma\in G(F)$;

  * •

$\mathfrak{c}(\xi,gh)=\mathfrak{c}(\xi,g)\mathfrak{c}(\xi^{g},h)$, $\xi\in
X(F)$, $g,\,h\in G(\mathbb{A})$.

This implies that $g\mapsto\mathfrak{c}(\xi,g)$ restricts to an automorphic
character of $G_{\xi}(\mathbb{A})$ for every $\xi\in X(F)$.

Assume that $\mathfrak{c}$ is an automorphic cocycle such that for every
$\xi\in X(F)$

(5)
$\int_{[G_{\xi}]}\left|{\mathfrak{c}(\xi,h)}\right|\delta_{G}(h)\delta_{G_{\xi}}(h)^{-1}\
dh<\infty.$

Consequently,

(6) $\int_{[G_{\xi}]}\mathfrak{c}(\xi,h)\delta_{G}(h)\delta_{G_{\xi}}(h)^{-1}\
dh=\begin{cases}\operatorname{vol}([G_{\xi}])&(\mathfrak{c}(\xi,\cdot)\delta_{G})|_{G_{\xi}(\mathbb{A})}\equiv\delta_{G_{\xi}}\\\
0&\text{otherwise.}\end{cases}$

######  Remark 2.1.

Note that if $[G_{\xi}]$ is compact for every $\xi\in X(F)$ then the
assumption (5) clearly holds. If $G(\mathbb{A})$ and $G_{\xi}(\mathbb{A})$ are
unimodular and $\mathfrak{c}(\xi,\cdot)|_{G_{\xi}(\mathbb{A})}$ is a unitary
character then for (5) to hold it is enough to assume that $[G_{\xi}]$ is of
finite volume.

######  Remark 2.2.

Note that if $\mathfrak{c}:X(\mathbb{A})\times
G(\mathbb{A})\rightarrow\mathbb{C}$ is an automorphic cocycle and $\chi$ is an
automorphic character of $G(\mathbb{A})$ then the function
$\mathfrak{c}_{\chi}$ defined by
$\mathfrak{c}_{\chi}(x,g)=\chi(g)\mathfrak{c}(x,g)$ is also an automorphic
cocycle.

We say that $\xi\in X(F)$ is _relevant_ if
$\mathfrak{c}(\xi,\cdot)\delta_{G}\delta_{G_{\xi}}^{-1}$ is trivial on
$G_{\xi}(\mathbb{A})$. By our assumptions, for $\xi\in X(F)$, $\gamma\in G(F)$
and $h\in G_{\xi}(\mathbb{A})$ we have
$\mathfrak{c}(\xi^{\gamma},\gamma^{-1}h\gamma)=\mathfrak{c}(\xi,h)$. It is
also clear that $\delta_{G}(\gamma^{-1}h\gamma)=\delta_{G}(h)$ and
$\delta_{G_{\xi^{\gamma}}}(\gamma^{-1}h\gamma)=\delta_{G_{\xi}}(h)$. Therefore
$\xi$ is relevant if and only if $\xi^{\gamma}$ is relevant. We denote by
$(X/G)$ a set of representatives for the $G(F)$-orbits in $X(F)$ and by
$(X/G)_{\operatorname{rel}}$ the subset of relevant elements in $(X/G)$. We
note the following formal unfolding principle. At this generality we ignore
convergence issues, hence the term principle. For
$\phi\in\mathcal{S}(X(\mathbb{A}))$, formally, we have

(7) $\int_{[G]}\sum_{\xi\in X(F)}\phi(\xi^{g})\mathfrak{c}(\xi,g)\
dg=\sum_{\xi\in(X/G)_{\operatorname{rel}}}\operatorname{vol}([G_{\xi}])\int_{G_{\xi}(\mathbb{A})\backslash
G(\mathbb{A})}\phi(\xi^{g})\mathfrak{c}(\xi,g)\ dg.$

Indeed, let $(X/G)$ be a set of representatives for the $G(F)$-orbits in
$X(F)$. Then

$\sum_{\xi\in
X(F)}\phi(\xi^{g})\mathfrak{c}(\xi,g)=\sum_{\xi\in(X/G)}\sum_{\gamma\in
G_{\xi}(F)\backslash G(F)}\phi(\xi^{\gamma
g})\mathfrak{c}(\xi^{\gamma},g)=\sum_{\xi\in(X/G)}\sum_{\gamma\in
G_{\xi}(F)\backslash G(F)}\phi(\xi^{\gamma g})\mathfrak{c}(\xi,\gamma g).$

On the left hand side of (7) we exchange the order of integration over $[G]$
and summation over $(X/G)$ and then unfold $\gamma$ with $g$ to obtain

$\sum_{\xi\in(X/G)}\int_{G_{\xi}(F)\backslash
G(\mathbb{A})}\phi(\xi^{g})\mathfrak{c}(\xi,g)\ dg.$

Integrating in stages this equals

$\sum_{\xi\in(X/G)}\int_{G_{\xi}(\mathbb{A})\backslash
G(\mathbb{A})}\phi(\xi^{g})\int_{[G_{\xi}]}\mathfrak{c}(\xi,hg)\delta_{G}(h)\delta_{G_{\xi}}(h)^{-1}\
dh\ dg=\\\ \sum_{\xi\in(X/G)}\int_{G_{\xi}(\mathbb{A})\backslash
G(\mathbb{A})}\phi(\xi^{g})\mathfrak{c}(\xi,g)\
dg\int_{[G_{\xi}]}\mathfrak{c}(\xi,h)\delta_{G}(h)\delta_{G_{\xi}}(h)^{-1}\
dh.$

It follows from (6) that this equals the right hand side of (7).

##  3\. The geometrization of the distribution $I(f,\Theta)$

Throughout the rest of Part I we suppose that the global field $F$ contains a
primitive cube root of unity $\rho$ (that is, there exits $\rho\in F$ such
that $\rho^{2}+\rho+1=0$), and we set $\mu_{3}=\langle\rho\rangle$. Our goal
is to geometrize the distribution $I(f,\Theta)$ given by (3).

###  3.1. The Adjoint embedding

For $n\geq 2$ let $w_{n}$ denote the $n\times n$ antidiagonal matrix
$w_{n}=(\delta_{i,n+1-j})\in\operatorname{GL}_{n}$. We realize the Adjoint
representation of $\operatorname{PGL}(3)$ as the map
$\operatorname{PGL}(3)\to\operatorname{GL}({\mathfrak{sl}}(3))$ defined by
conjugation. Considering ${\mathfrak{sl}}(3)$ as a quadratic space with
respect to the bilinear form $<x,y>=\text{Tr}(xy)$, the image lies in the
special orthogonal group $\operatorname{SO}({\mathfrak{sl}}(3))$. With respect
to the basis

$\\{e_{1,3},e_{2,3},e_{1,2},e_{1,1}-e_{2,2},e_{2,2}-e_{3,3},e_{2,1},e_{3,2},e_{3,1}\\}$

of ${\mathfrak{sl}}(3)$ where the $e_{i,j}$ are the standard elementary
matrices, $\operatorname{SO}({\mathfrak{sl}}(3))$ is isomorphic to
$\operatorname{SO}(J)$ where

$J=\begin{pmatrix}&&&w_{3}\\\ &2&-1&\\\ &-1&2&\\\ w_{3}&&&\end{pmatrix}.$

Since $F$ contains the cube roots of unity, the group $\operatorname{SO}(J)$
is split over $F$ and may be conjugated to
$\operatorname{SO}_{8}:=\operatorname{SO}(w_{8})$; indeed if
$g_{0}=-\tfrac{1}{3}\left(\begin{smallmatrix}1-\rho&2+\rho\\\
-(1+2\rho)&1+2\rho\end{smallmatrix}\right)$ then
${}^{t}g_{0}\left(\begin{smallmatrix}2&-1\\\
-1&2\end{smallmatrix}\right)g_{0}=\left(\begin{smallmatrix}&1\\\
1&\end{smallmatrix}\right)$. In this way we realize the adjoint map as a
homomorphism
$\operatorname{Ad}:\operatorname{PGL}(3)\rightarrow\operatorname{SO}_{8}$.
This is the map that is used in the period (2). Restricted to $N(\mathbb{A})$
this leads to the expression

$\text{Ad}\begin{pmatrix}1&x&z\\\ &1&y\\\ &&1\end{pmatrix}=\\\
\begin{pmatrix}1&x&-y&-(xy+\rho z)&-(xy+\rho^{2}z)&x(xy-z)&-zy&z(xy-z)\\\
&1&0&\rho^{2}y&\rho y&xy-z&-y^{2}&y(xy-z)\\\ &&1&x&x&-x^{2}&z&-zx\\\
&&&1&0&-x&-\rho y&\rho xy+\rho^{2}z\\\ &&&&1&-x&-\rho^{2}y&\rho^{2}xy+\rho
z\\\ &&&&&1&0&y\\\ &&&&&&1&-x\\\ &&&&&&&1\end{pmatrix}.$

It was also explicated in [BFG01].

###  3.2. First steps

Since $\Theta$ is automorphic and $\operatorname{Ad}$ is defined over $F$ we
may combine the integral over $[G]$ with the sum over $G(F)$ to obtain

$I(f,\Theta)=\int_{[N]}\int_{G(\mathbb{A})}f(g^{-1}n)\Theta(\operatorname{Ad}(g))\psi(n)\
dg\ dn$

and after the change of variables $g\mapsto ng$ we obtain that

(8)
$I(f,\Theta)=\int_{[N]}\\{\int_{G(\mathbb{A})}f(g^{-1})\Theta(\operatorname{Ad}(n)\operatorname{Ad}(g))\
dg\\}\psi(n)\ dn=\int_{[N]}\Theta_{f}(\operatorname{Ad}(n))\psi(n)\ dn$

where $\Theta_{f}\in\Theta_{\operatorname{SO}_{8}}$ is defined by
$\Theta_{f}(x)=\int_{G(\mathbb{A})}f(g^{-1})\Theta(x\operatorname{Ad}(g))\
dg$.

This means, that in order to obtain a geometric expansion, we are reduced to
showing that

$\int_{[N]}\Theta(\operatorname{Ad}(n))\psi(n)\ dn$

is a sum of factorizable integrals.

As noted in the Introduction, by [GRS97b, Theorem 6.9], the minimal
representation $\Theta_{\operatorname{SO}_{8}}$ can be expressed as a theta
lift using the Weil representation of the metaplectic double cover
$\widetilde{\operatorname{Sp}}_{16}(\mathbb{A})$ of
$\operatorname{Sp}_{16}(\mathbb{A})$. We now make this explicit, and use this
to show the desired expansion. To use the theta lift, we follow Deitmar and
Krieg ([DK91], Section 2) and Kudla-Rallis ([KR94], Section 5) by imposing a
condition on $f$ (see Subsection 3.4 below) at one archimedean place. With
this condition, the theta integral of the Eisenstein series is convergent, and
so the computation applies.

###  3.3. On the Weil representation

We realize the group $\operatorname{Sp}_{16}$ as the group of automorphisms
preserving the alternating matrix $\left(\begin{smallmatrix}{0}&{w_{8}}\\\
{-w_{8}}&{0}\end{smallmatrix}\right)$. For $g\in\operatorname{GL}_{n}$ let
$g^{*}=w_{n}{}^{t}g^{-1}w_{n}$.

For a matrix $Y\in M_{n}(\mathbb{A})$ let ${}_{t}Y=w_{n}{}^{t}Yw_{n}$. The
equality $Y={}_{t}Y$ means that $Y$ is symmetric with respect to the second
diagonal, i.e., $Y=(y_{i,j})$ where $y_{i,j}=y_{n+1-j,n+1-i}$ (or
equivalently, that $w_{n}Y$ is symmetric). It follows from the standard
explicit formulas for the Weil representation $\omega_{\psi}$ of
$\widetilde{\operatorname{Sp}}_{16}(\mathbb{A})$ (see e.g. [GRS11]) that for
$x\in\mathbb{A}^{8}$ and $\phi\in\mathcal{S}(\mathbb{A}^{8})$ we have

$\omega_{\psi}\left(\begin{smallmatrix}{g}&{}\\\
{}&{g^{*}}\end{smallmatrix}\right)\phi(x)=\phi(xg),\ \ \
g\in\operatorname{SL}_{8}(\mathbb{A})$

and

$\omega_{\psi}\left(\begin{smallmatrix}{I_{8}}&{Y}\\\
{}&{I_{8}}\end{smallmatrix}\right)\phi(x)=\psi(xw_{8}{}^{t}Y\
{}^{t}x)\phi(x)=\psi(xYw_{8}\ {}^{t}x)\phi(x),\ \ \ Y={}_{t}Y\in
M_{8}(\mathbb{A}).$

Combined this gives

(9) $\omega_{\psi}(\left(\begin{smallmatrix}{I_{8}}&{Y}\\\
{}&{I_{8}}\end{smallmatrix}\right)\left(\begin{smallmatrix}{g}&{}\\\
{}&{g^{*}}\end{smallmatrix}\right))\phi(x)=\psi(xYw_{8}\,{}^{t}x)\phi(xg).$

Let $P=M\ltimes U$ be the Siegel parabolic subgroup of
$\operatorname{Sp}_{16}$ with unipotent radical
$U=\\{\left(\begin{smallmatrix}{I_{8}}&{Y}\\\
{}&{I_{8}}\end{smallmatrix}\right):{}_{t}Y=Y\in M_{8}\\}$ isomorphic to the
vector space of symmetric matrices in $M_{8}$ and
$M=\\{\operatorname{diag}(g,g^{*}):g\in\operatorname{GL}_{8}\\}$. Let
$P^{\circ}=M^{\circ}\ltimes U$ where
$M^{\circ}=\\{\operatorname{diag}(g,g^{*}):g\in\operatorname{SL}_{8}\\}$ and
let $\operatorname{p}_{M}:P\rightarrow M$ be the projection map to the Levi
subgroup. We consider the right action of $P^{\circ}$ on affine $8$-space
$\mathbb{G}_{a}^{8}$ defined by

$x^{p}=xg,\ \ \ x\in\mathbb{G}_{a}^{8},\ p\in P^{\circ},\
\operatorname{p}_{M}(p)=\operatorname{diag}(g,g^{*})$

and the function $\mathfrak{c}:\mathbb{A}^{8}\times
P^{\circ}(\mathbb{A})\rightarrow\mathbb{C}$ defined by

$\mathfrak{c}(x,p)=\psi(xYw_{8}\,{}^{t}x),\ \ \ x\in\mathbb{A}^{8},\
p=\left(\begin{smallmatrix}{I_{8}}&{Y}\\\
{}&{I_{8}}\end{smallmatrix}\right)\operatorname{p}_{M}(p)\in
P^{\circ}(\mathbb{A}).$

We can rewrite (9) as

(10) $\omega_{\psi}(p)\phi(x)=\mathfrak{c}(x,p)\phi(x^{p}),\ \ \
x\in\mathbb{A}^{8},\ p\in P^{\circ}(\mathbb{A}).$

We denote by $P^{\circ}_{\xi}$ the stabilizer in $P^{\circ}$ of $\xi\in F^{8}$
(considered as an algebraic group over $F$) and by $P^{\circ}(\mathbb{A})_{x}$
the stabilizer in $P^{\circ}(\mathbb{A})$ of $x\in\mathbb{A}^{8}$. Note that

$\mathfrak{c}(\xi,\gamma)=1,\ \ \ \xi\in F^{8},\gamma\in P^{\circ}(F).$

######  Lemma 3.1.

The function $\mathfrak{c}$ satisfies the cocycle condition

$\mathfrak{c}(x,p_{1}p_{2})=\mathfrak{c}(x,p_{1})\mathfrak{c}(x^{p_{1}},p_{2}),\
\ \ x\in\mathbb{A}^{8},\,p_{1},\,p_{2}\in P^{\circ}(\mathbb{A}).$

In particular, it is an automorphic cocycle.

###### Proof.

Write $p_{i}=\left(\begin{smallmatrix}{I_{8}}&{Y_{i}}\\\
{}&{I_{8}}\end{smallmatrix}\right)\left(\begin{smallmatrix}{g_{i}}&{}\\\
{}&{g_{i}^{*}}\end{smallmatrix}\right)$, $i=1,2$. Then
$p_{1}p_{2}=\left(\begin{smallmatrix}{I_{8}}&{Y_{1}+g_{1}Y_{2}(g_{1}^{*})^{-1}}\\\
{}&{I_{8}}\end{smallmatrix}\right)\left(\begin{smallmatrix}{g_{1}g_{2}}&{}\\\
{}&{(g_{1}g_{2})^{*}}\end{smallmatrix}\right)$. Since
$g_{1}Y_{2}(g_{1}^{*})^{-1}w_{8}=g_{1}Y_{2}w_{8}\,{}^{t}g_{1}$ it follows that

$\mathfrak{c}(x,p_{1}p_{2})=\psi(xY_{1}w_{8}\,{}^{t}x+xg_{1}Y_{2}w_{8}\,{}^{t}(xg_{1}))=\mathfrak{c}(x,p_{1})\mathfrak{c}(xg_{1},p_{2})=\mathfrak{c}(x,p_{1})\mathfrak{c}(x^{p_{1}},p_{2})$

as required. ∎

###  3.4. On the theta lift

We fix an embedding of $\operatorname{SL}_{2}\times\operatorname{SO}_{8}$ in
$\operatorname{Sp}_{16}$ such that
$\operatorname{SL}_{2}\times\operatorname{Ad}(N)$ embeds in $P^{\circ}$ (we
later explicate such an embedding) and denote by $(g,h)\mapsto[g,h]$ the
associated embedding of
$\operatorname{SL}_{2}(\mathbb{A})\times\operatorname{SO}_{8}(\mathbb{A})$ in
$\widetilde{\operatorname{Sp}}_{16}(\mathbb{A})$. Note that the action of the
Weil representation directly shows that $P^{\circ}(\mathbb{A})$ splits in
$\widetilde{\operatorname{Sp}}_{16}(\mathbb{A})$. Let

$\tilde{\theta}_{\phi}(g)=\sum_{x\in F^{8}}\omega_{\psi}(g)\phi(x),\ \ \
g\in\widetilde{\operatorname{Sp}}_{16}(\mathbb{A}).$

There is a subspace $\mathcal{S}^{\circ}(\mathbb{A}^{8})$ of
$\mathcal{S}(\mathbb{A}^{8})$ (the image of $\mathcal{S}(\mathbb{A}^{8})$
under the Casimir operator in the center of the universal enveloping algebra
of the Lie algebra of $\operatorname{SL}_{2}$ at one fixed archimedean place)
such that for $\phi\in\mathcal{S}^{\circ}(\mathbb{A}^{8})$ we have that
$g\mapsto\tilde{\theta}_{\phi}([g,h])$ is a rapidly decreasing automorphic
function on $\operatorname{SL}_{2}(\mathbb{A})$. For
$\phi\in\mathcal{S}^{\circ}(\mathbb{A}^{8})$ let

$\Theta_{\phi}(g)=\int_{[\operatorname{SL}_{2}]}\tilde{\theta}_{\phi}([h,g])\
dh,\ \ \ g\in\operatorname{SO}_{8}(\mathbb{A}).$

We then have

$\Theta_{\operatorname{SO}_{8}}=\\{\Theta_{\phi}:\phi\in\mathcal{S}^{\circ}(\mathbb{A}^{8})\\}.$

We consider the distributions on $G(\mathbb{A})$ defined by

$I(f,\phi)=\int_{[N]}\int_{G(\mathbb{A})}f(g^{-1})\int_{[\operatorname{SL}_{2}]}\tilde{\theta}_{\phi}([h,\operatorname{Ad}(ng)])\
dh\ dg\ \psi(n)\ dn,\ \ f\in\mathcal{S}(G(\mathbb{A})),\ \
\phi\in\mathcal{S}^{\circ}(\mathbb{A}^{8}).$

Note that $I(f,\phi)=I(f,\Theta_{\phi})$. Let

$\phi[f](x)=\int_{G(\mathbb{A})}f(g^{-1})\omega_{\psi}([I_{2},\operatorname{Ad}(g)])\phi(x)\
dg.$

Then

$I(f,\phi)=\int_{[N]}\int_{[\operatorname{SL}_{2}]}\tilde{\theta}_{\phi[f]}([h,\operatorname{Ad}(n)])\
dh\ \psi(n)\ dn=\\\ \int_{[N]}\int_{[\operatorname{SL}_{2}]}\sum_{\xi\in
F^{8}}\omega_{\psi}([h,\operatorname{Ad}(n)])\phi[f](\xi)\ dh\ \psi(n)\ dn.$

We therefore need to unfold integrals of the form

(11) $I(\phi)=\int_{[N]}\int_{[\operatorname{SL}_{2}]}\sum_{\xi\in
F^{8}}\omega_{\psi}(h,\operatorname{Ad}(n))\phi(\xi)\ dh\ \psi(n)\ dn.$

###  3.5. The unfolding of $I$

Let $H=\operatorname{SL}_{2}\times N$. For $h=(g,n)\in H(\mathbb{A})$, by
abuse of notation we write $[h]=[g,\operatorname{Ad}(n)]$ and we extend $\psi$
to an automorphic character of $H(\mathbb{A})$ (still denoted by $\psi$) that
is trivial on $\operatorname{SL}_{2}(\mathbb{A})$. With our choice of
embedding of $H(\mathbb{A})$ and by (10) we have

$I(\phi)=\int_{[H]}\sum_{\xi\in F^{8}}\phi(\xi^{[h]})\
\mathfrak{c}_{\psi}(\xi,[h])\ dh$

where
$\mathfrak{c}_{\psi}(\xi,[g,\operatorname{Ad}(n)])=\psi(n)\mathfrak{c}(\xi,[g,\operatorname{Ad}(n)])$.

Let $\Xi_{\operatorname{rel}}$ be a set of representatives for the
$H(F)$-orbits in $F^{8}$ for which
$\mathfrak{c}_{\psi}(\xi,\cdot)|_{H_{\xi}(\mathbb{A})}\equiv 1$. We will later
see that $H_{\xi}(\mathbb{A})$ is unimodular and $[H_{\xi}]$ has finite
volume. Indeed, $H_{\xi}$ is unipotent unless $\xi=0$ in which case
$H_{\xi}=H$. As a consequence of Lemma 3.1 and Remarks 2.1 and 2.2 we can
apply the unfolding principle of Section 2 to obtain

######  Proposition 3.2.

The distribution $I$ given in (11) is a sum of factorizable orbital integrals

$I(\phi)=\sum_{\xi\in\Xi_{\operatorname{rel}}}\mathcal{O}(\xi,\phi)$

where for $\xi\in F^{8}$ relevant, the associated orbital integral is given by

$\mathcal{O}(\xi,\phi)=\int_{H_{\xi}(\mathbb{A})\backslash
H(\mathbb{A})}\phi(\xi^{[h]})\mathfrak{c}_{\psi}(\xi,[h])\ dh.$

Here the measures are normalized so that the volume of each compact quotient
$[H_{\xi}]$ is one. ∎

In order to compare the distributions $I$ and $J$, we must describe the
quantities in this proposition precisely. This is accomplished in Theorem 4.2
below.

##  4\. Determination of the relevant orbits and orbital integrals for $I$

###  4.1. Explication of the set up

Let $V=\operatorname{Span}(e,f)$ be the two dimensional symplectic space with
$\left\langle{e},{f}\right\rangle=1=-\left\langle{f},{e}\right\rangle$ and
$W=\operatorname{Span}(e_{1},\dots,e_{8})$ with
$\left\langle{e_{i}},{e_{j}}\right\rangle=\delta_{i,9-j}$. Fixing the basis
$\\{e,f\\}$ identifies $\operatorname{SL}_{2}$ with $\operatorname{Sp}(V)$ and
the basis $\\{e_{1},\dots,e_{8}\\}$ identifies $\operatorname{SO}_{8}$ with
$\operatorname{SO}(W)$. We also consider $V\otimes W$ as a symplectic space
with the product form. Fixing the basis

$\\{e\otimes e_{1},f\otimes e_{1},\dots,e\otimes e_{4},f\otimes
e_{4},-e\otimes e_{5},f\otimes e_{5},\dots,-e\otimes e_{8},f\otimes e_{8}\\}$

identifies $\operatorname{Sp}_{16}$ with $\operatorname{Sp}(V\otimes W)$. Via
the tensor product embedding
$\operatorname{Sp}(V)\times\operatorname{SO}(W)\hookrightarrow\operatorname{Sp}(V\otimes
W)$ this defines an embedding of

$\operatorname{SL}_{2}\times\operatorname{SO}_{8}\hookrightarrow\operatorname{Sp}_{16}.$

We explicate this embedding.

Let $\lambda:\operatorname{GL}_{2}\rightarrow\operatorname{GL}_{8}$ be the
embedding $\lambda(h)=\operatorname{diag}(h,h,h,h)$. Note that
$\lambda({}^{t}h)={}^{t}\lambda(h)$ and $\lambda(h)^{*}=\lambda(h^{*})$. The
embedding of $\operatorname{SL}_{2}$ in $\operatorname{Sp}_{16}$ is then
$h\mapsto\operatorname{diag}(\lambda(h),\lambda(h^{*}))$.

The embedding of $\operatorname{SO}_{8}$ in $\operatorname{Sp}_{16}$ can be
described as follows. Let $\imath:M_{4}\rightarrow M_{8}$ be the standard
‘tensor with $I_{2}$ map’, that is $\imath(A)=(a_{i,j}I_{2})$, $A=(a_{i,j})\in
M_{4}$ and let $\jmath:M_{4}\rightarrow M_{8}$ be the twisted map
$\jmath(A)=\lambda(\epsilon_{2})\imath(A)=\imath(A)\lambda(\epsilon_{2})$
where $\epsilon_{2}=\operatorname{diag}(1,-1)$. Then, the embedding of
$\operatorname{SO}_{8}$ to $\operatorname{Sp}_{16}$ in terms of the $4\times
4$-blocks is given by

$\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}\imath(a)&\jmath(b)\\\
\jmath(c)&\imath(d)\end{pmatrix}=\operatorname{diag}(\lambda(\epsilon_{2}),I_{8})\begin{pmatrix}\imath(a)&\imath(b)\\\
\imath(c)&\imath(d)\end{pmatrix}\operatorname{diag}(\lambda(\epsilon_{2}),I_{8}).$

Furthermore, we remark that the group homomorphism $\lambda$ and the algebra
homomorphism $\imath$ are such that $\lambda(\operatorname{GL}_{2})$ and
$\imath(M_{4})$ commute with each other.

Recall that $\operatorname{Ad}(N)$ consists of upper-triangular unipotent
matrices in $\operatorname{SO}_{8}$. For

(12) $n=\begin{pmatrix}1&x&z\\\ &1&y\\\ &&1\end{pmatrix}\in N(\mathbb{A})$

we write

$\operatorname{Ad}(n)=\begin{pmatrix}I_{4}&s\\\
&I_{4}\end{pmatrix}\begin{pmatrix}u&\\\ &u^{*}\end{pmatrix}$

with $u=u(n)\in\operatorname{GL}_{4}$ upper-triangular unipotent and
$s=s(n)\in M_{4}$ such that $sw_{4}$ is alternating. We have

$[h,\operatorname{Ad}(n)]=\begin{pmatrix}I_{8}&\jmath(s)\\\
&I_{8}\end{pmatrix}\begin{pmatrix}\imath(u)\lambda(h)&\\\
&\imath(u^{*})\lambda(h^{*})\end{pmatrix},\ \ \
h\in\operatorname{SL}_{2}(\mathbb{A}),\,n\in N(\mathbb{A}).$

Since $w_{8}=\lambda(w_{2})\imath(w_{4})$ we see that
$\jmath(s)w_{8}=\imath(sw_{4})\lambda(\epsilon_{2}w_{2})$. We also note that

$\xi^{[h,\operatorname{Ad}(n)]}=\xi\imath(u)\lambda(h).$

Since $h\epsilon_{2}w_{2}\,{}^{t}h=\epsilon_{2}w_{2}$ it follows that
$x\jmath(s)w_{8}{}^{t}x=x\lambda(h)\jmath(s)w_{8}{}^{t}\lambda(h){}^{t}x$,
$x\in\mathbb{A}^{8}$, $h\in\operatorname{SL}_{2}(\mathbb{A})$ and therefore

(13)
$\mathfrak{c}(\xi,[h,\operatorname{Ad}(n)])=\mathfrak{c}(\xi,[I_{2},\operatorname{Ad}(n)]),\
\ \ n\in N(\mathbb{A})$

is independent of $h\in\operatorname{SL}_{2}(\mathbb{A})$.

We notice from the explicit formula for $\operatorname{Ad}(n)$ in Section 3.1
that

$u=\begin{pmatrix}1&x&-y&-yx-z\rho\\\ &1&&y\rho^{2}\\\ &&1&x\\\
&&&1\end{pmatrix}$

and

$sw_{4}=\begin{pmatrix}&(z-yx)y\rho^{2}&zx\rho&-(yx+z\rho^{2})\\\
(yx-z)y\rho^{2}&&-(z+xy\rho^{2})&y\rho\\\ -zx\rho&z+xy\rho^{2}&&x\\\
yx+z\rho^{2}&-y\rho&-x&\end{pmatrix}.$

Writing $\xi=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})\in F^{8}$ with $\xi_{i}\in
F^{2}$, $i=1,2,3,4$ we therefore have

(14)
$\xi^{[h,\operatorname{Ad}(n)]}=(\xi_{1}h,(\xi_{2}+x\xi_{1})h,(\xi_{3}-y\xi_{1})h,(\xi_{4}+x\xi_{3}+y\rho^{2}\xi_{2}-(yx+z\rho)\xi_{1})h).$

We also conclude that $\jmath(s)w_{8}$ equals

$\displaystyle\begin{pmatrix}0&0&0&(z-xy)y\rho^{2}&0&xz\rho&0&-(xy+z\rho^{2})\\\
0&0&(xy-z)y\rho^{2}&0&-xz\rho&0&xy+z\rho^{2}&0\\\
0&(xy-z)y\rho^{2}&0&0&0&-(xy\rho^{2}+z)&0&y\rho\\\
(z-xy)y\rho^{2}&0&0&0&xy\rho^{2}+z&0&-y\rho&0\\\
0&-xz\rho&0&xy\rho^{2}+z&0&0&0&x\\\ xz\rho&0&-(xy\rho^{2}+z)&0&0&0&-x&0\\\
0&xy+z\rho^{2}&0&-y\rho&0&-x&0&0\\\
-(xy+z\rho^{2})&0&y\rho&0&x&0&0&0\end{pmatrix}$

and this gives us an explicit formula for
$\mathfrak{c}(\xi,[h,\operatorname{Ad}(n)])=\psi(\xi\jmath(s)w_{8}\,{}^{t}\xi)$.

###  4.2. Relevant orbits and their associated orbital integrals

We use the above notation and formulas in order to find an explicit set
$\Xi_{\operatorname{rel}}$ of relevant orbit representatives, compute their
stabilizer and explicate the associated factorizable orbital integrals. Let
$e_{1}=(1,0)$, $e_{2}=(0,1)$ be the standard basis of $F^{2}$ and
$N_{2}=\\{g\in\operatorname{SL}_{2}:e_{2}g=e_{2}\\}$. Whenever we write
$x=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{A}^{8}$ we mean that each
$x_{i}\in\mathbb{A}^{2}$. In particular, we write $x_{i}=0$ if $x_{i}=(0,0)$.

The following elementary observation will be used repeatedly in the orbit
analysis.

######  Lemma 4.1.

Let $\\{v_{1},v_{2}\\}$ be a basis of $F^{2}$. Then there exists
$h\in\operatorname{SL}_{2}(F)$ and $a\in F^{*}$ such that $v_{1}h=e_{2}$ and
$v_{2}h=ae_{1}$.

###### Proof.

Applying $h=\left(\begin{smallmatrix}{}&{1}\\\
{-1}&{}\end{smallmatrix}\right)$ if necessary, we may assume without loss of
generality that $v_{1}=(a,b)$ with $b\neq 0$. Then by applying
$\left(\begin{smallmatrix}{b}&{}\\\ {-a}&{b^{-1}}\end{smallmatrix}\right)$ we
may assume without loss of generality that $v_{1}=e_{2}$. Now write
$v_{2}=(a,b)$. Then $a\neq 0$ and applying
$\left(\begin{smallmatrix}{1}&{-a^{-1}b}\\\ {}&{1}\end{smallmatrix}\right)$
the lemma follows. ∎

For $\phi\in\mathcal{S}(\mathbb{A}^{8})$ we let
$\phi_{K_{2}}\in\mathcal{S}(\mathbb{A}^{8})$ be defined by

$\phi_{K_{2}}(v)=\int_{K_{2}}\phi[v\lambda(k)]\ dk$

where $K_{2}$ is the standard maximal compact subgroup of
$\operatorname{SL}_{2}(\mathbb{A})$.

Let $\xi=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})\in F^{8}$. It follows from (14)
that the following are invariants of $H(F)$-orbits.

  * •

$d_{1}(\xi)=\dim\langle\xi_{1}\rangle$;

  * •

$d_{1,2}(\xi)=\dim\langle\xi_{1},\xi_{2}\rangle$;

  * •

$d_{1,3}(\xi)=\dim\langle\xi_{1},\xi_{3}\rangle$;

  * •

$d_{1,2,3}(\xi)=\dim\langle\xi_{1},\xi_{2},\xi_{3}\rangle$;

  * •

$d_{1,2,3,4}(\xi)=\dim\langle\xi_{1},\xi_{2},\xi_{3},\xi_{4}\rangle$;

In the remainder of this Section, we will analyze the orbits case by case
using these invariants. The following result summarizes this analysis.

######  Theorem 4.2.

A complete set of representatives for the $H(F)$-orbits that are relevant is
given in Lemma 4.3 (orbits with $d_{1}=0$ and $d_{1,2}=d_{1,2,3,4}=1$); Lemma
4.4 (orbits with $d_{1}=0$, $d_{1,2}=d_{1,2,3}=1$ and $d_{1,2,3,4}=2$); Lemma
4.6 (orbits with $d_{1}=1=d_{1,2,3,4}$); Lemma 4.8 (orbits with
$d_{1}=1=d_{1,2}$ and $d_{1,2,3}=2$); Lemma 4.9 (orbits with $d_{1}=1=d_{1,3}$
and $d_{1,2}=2$); and Lemma 4.10 (orbits with $d_{1}=1$ and
$d_{1,2}=2=d_{1,3}$; these are the generic relevant orbits).

When the orbit is relevant, in these Lemmas we also write the associated
orbital integral explicitly. There are other $H(F)$-orbits, and as part of the
proof we will show that they are not relevant. Throughout this analysis we
implicitly use the coordinates (12) for $n\in N(\mathbb{A})$.

####  4.2.1. Orbits with $d_{1,2}=0$ are irrelevant

Suppose $d_{1,2}(\xi)=0$ and write $\xi=(0,0,\xi_{3},\xi_{4})$. Note that

$\mathfrak{c}(\xi,[h,\operatorname{Ad}(n)])=2x\xi_{3}\epsilon_{2}w_{2}\,{}^{t}\xi_{4}$

is independent of $y$ (and $z$) and by (14) we have

$\xi^{[h,\operatorname{Ad}(n)]}=(0,0,\xi_{3}h,(\xi_{4}+x\xi_{3})h).$

In particular,

$\left\\{(I_{2},\begin{pmatrix}1&&\\\ &1&y\\\
&&1\end{pmatrix}):y\in\mathbb{A}\right\\}\subseteq H_{\xi}(\mathbb{A})$

and therefore $\xi$ is not relevant.

####  4.2.2. The $H(F)$-orbits with invariants $d_{1}=0$ and
$d_{1,2}=d_{1,2,3,4}=1$

######  Lemma 4.3.

A complete set of representatives for $H(F)$-orbits with invariants $d_{1}=0$
and $d_{1,2}=d_{1,2,3,4}=1$ is given by

$\\{(0,e_{2},\alpha e_{2},0):\alpha\in F\\}.$

The vector $(0,e_{2},\alpha e_{2},0)$ is relevant if and only if
$\alpha=\rho^{2}$. The associated orbital integral is

$I((0,0,0,1,0,\rho^{2},0,0),\phi)=\int_{\mathbb{A}}\int_{\mathbb{A}^{*}}\phi_{K_{2}}[(0,0,0,t,0,\rho^{2}t,0,xt)]\left|{t}\right|^{2}\
d^{*}t\ \psi(x)\ dx.$

###### Proof.

Let $\xi\in F^{8}$ be such that $d_{1}(\xi)=0$ and
$d_{1,2}(\xi)=d_{1,2,3,4}(\xi)=1$. Then there exists $0\neq v\in F^{2}$ and
$\alpha,\,\alpha^{\prime}\in F$ such that $\xi=(0,v,\alpha
v,\alpha^{\prime}v)$. Since $\operatorname{SL}_{2}(F)$ acts transitively on
non-zero vectors in $F^{2}$ there exists $h_{0}\in\operatorname{SL}_{2}(F)$
such that $vh_{0}=e_{2}$. Let

$n_{0}=\begin{pmatrix}1&&\\\ &1&-\alpha^{\prime}\rho\\\ &&1\end{pmatrix};$

then $\xi^{[h_{0},\operatorname{Ad}(n_{0})]}=(0,e_{2},\alpha e_{2},0)$. Now
for a general $g=(h,n)\in H(F)$ we have

$(0,e_{2},\alpha e_{2},0)^{[g]}=(0,e_{2}h,\alpha
e_{2}h,(x\alpha+y\rho^{2})e_{2}h).$

This is of the form $(0,e_{2},\alpha^{\prime}e_{2},0)$ for some
$\alpha^{\prime}\in F$ if and only if $h\in N_{2}(F)$ and
$x\alpha+y\rho^{2}=0$. We conclude that $(0,e_{2},\alpha e_{2},0)$ and
$(0,e_{2},\alpha^{\prime}e_{2},0)$ are in the same $H(F)$-orbit if and only if
$\alpha=\alpha^{\prime}$ and furthermore

$H_{(0,e_{2},\alpha e_{2},0)}=N_{2}\times\\{n\in N:x\alpha+y\rho^{2}=0\\}.$

We further observe that $\mathfrak{c}((0,e_{2},\alpha e_{2},0),[g])=1$ for all
$g\in H(\mathbb{A})$. It follows that the orbit containing $(0,e_{2},\alpha
e_{2},0)$ is relevant if and only if $\alpha=\rho^{2}$. Thus, for
$\xi=(0,e_{2},\rho^{2}e_{2},0)$ we have

$\mathcal{O}(\xi,\phi)=\int_{\mathbb{A}}\int_{N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}\phi[(0,0,0,1,0,\rho^{2},0,x)\lambda(h)]\
dh\ \psi(x)\ dx.$

The lemma now follows using the Iwasawa decomposition. ∎

####  4.2.3. The $H(F)$-orbits with invariants $d_{1}=0$,
$d_{1,2}=d_{1,2,3}=1$ and $d_{1,2,3,4}=2$

######  Lemma 4.4.

A complete set of representatives for $H(F)$-orbits with invariants $d_{1}=0$,
$d_{1,2}=d_{1,2,3}=1$ and $d_{1,2,3,4}=2$ is given by

$\\{(0,e_{2},\alpha e_{2},\beta e_{1}):\alpha\in F,\beta\in F^{*}\\}.$

The vector $(0,e_{2},\alpha e_{2},\beta e_{1})$ is relevant if and only if
$\alpha=\rho$ and $\beta=\frac{1}{2}\rho^{2}$. The associated orbital integral
is

$\mathcal{O}((0,0,0,1,0,\rho,\frac{1}{2}\rho^{2},0),\phi)=\int_{\mathbb{A}^{*}}\int_{\mathbb{A}}\phi_{K_{2}}[(0,0,0,t,0,\rho
t,\frac{1}{2}\rho^{2}t^{-1},a)]\ da\left|{t}\right|\ d^{*}t.$

###### Proof.

Let $\xi\in F^{8}$ be such that $d_{1}(\xi)=0$,
$d_{1,2}(\xi)=d_{1,2,3}(\xi)=1$ and $d_{1,2,3,4}(\xi)=2$. Then, there exists a
basis $\\{v,w\\}$ of $F^{2}$ and $\alpha\in F$ such that $\xi=(0,v,\alpha
v,w)$. Applying Lemma 4.3 there exist $h_{0}\in\operatorname{SL}_{2}(F)$ and
$\beta\in F^{*}$ such that

$\xi^{[h_{0},I_{8}]}=(0,e_{2},\alpha e_{2},\beta e_{1}).$

Now for a general $g=(h,n)\in H(F)$ we have

$(0,e_{2},\alpha e_{2},\beta e_{1})^{[g]}=(0,e_{2}h,\alpha e_{2}h,[\beta
e_{1}+(x\alpha+y\rho^{2})e_{2}]h).$

This is of the form $(0,e_{2},\alpha^{\prime}e_{2},\beta^{\prime}e_{1})$ for
some $\alpha^{\prime}\in F$ and $\beta^{\prime}\in F^{*}$ if and only if
$h=\left(\begin{smallmatrix}{1}&{a}\\\ {}&{1}\end{smallmatrix}\right)$where
$\beta a+x\alpha+y\rho^{2}=0$ and in this case
$(\alpha^{\prime},\beta^{\prime})=(\alpha,\beta)$. We conclude that
$(0,e_{2},\alpha e_{2},\beta e_{1})$ and
$(0,e_{2},\alpha^{\prime}e_{2},\beta^{\prime}e_{1})$ are in the same
$H(F)$-orbit if and only if $(\alpha^{\prime},\beta^{\prime})=(\alpha,\beta)$
and furthermore

$H_{(0,e_{2},\alpha e_{2},\beta
e_{1})}=\\{(\begin{pmatrix}1&-\beta^{-1}(x\alpha+y\rho^{2})\\\
&1\end{pmatrix},n):n\in N\\}.$

We further observe that

$\mathfrak{c}_{\psi}((0,e_{2},\alpha e_{2},\beta
e_{1}),[h,\operatorname{Ad}(n)])=\psi[(1-2\beta\rho)y+(1-2\beta\alpha)x].$

It follows that $(0,e_{2},\alpha e_{2},\beta e_{1})$ is relevant if and only
if $1-2\beta\rho=1-2\beta\alpha=0$, i.e., if and only if $\alpha=\rho$ and
$\beta=\frac{1}{2}\rho^{2}$. Thus, for $\xi=(0,e_{2},\rho
e_{2},\frac{1}{2}\rho^{2}e_{1})$ we have

$\mathcal{O}(\xi,\phi)=\int_{\operatorname{SL}_{2}(\mathbb{A})}\phi[(0,0,0,1,0,\rho,\frac{1}{2}\rho^{2},0)\lambda(h)]\
dh.$

Using the Iwasawa decomposition we can write this as

$\int_{\mathbb{A}^{*}}\int_{\mathbb{A}}\phi_{K_{2}}[(0,0,0,t,0,\rho
t,\frac{1}{2}\rho^{2}t^{-1},\frac{1}{2}\rho^{2}at)\ da\left|{t}\right|^{2}\
d^{*}t.$

Making the change of variables $a\mapsto 2\rho t^{-1}a$ the lemma follows.

∎

####  4.2.4. The $H(F)$-orbits with invariants $d_{1}=0$, $d_{1,2}=1$ and
$d_{1,2,3}=2$ are not relevant

######  Lemma 4.5.

A complete set of representatives for $H(F)$-orbits with invariants $d_{1}=0$,
$d_{1,2}=1$ and $d_{1,2,3}=2$ is

$\\{(0,e_{2},\beta e_{1},0):\beta\in F^{*}\\}.$

None of them is relevant.

###### Proof.

Let $\xi\in F^{8}$ be such that $d_{1}(\xi)=0$, $d_{1,2}(\xi)=1$ and
$d_{1,2,3}(\xi)=2$. Then, there exists a basis $\\{v,w\\}$ of $F^{2}$ and
$u\in F^{2}$ such that $\xi=(0,v,w,u)$. Applying Lemma 4.3 let
$h_{0}\in\operatorname{SL}_{2}(F)$ and $\beta\in F^{*}$ be such that
$\\{vh_{0},wh_{0}\\}=\\{e_{2},\beta e_{1}\\}$. Let $x_{0},,y_{0}\in F$ be such
that $-u=x_{0}\beta v+y_{0}\rho^{2}w$ and let

$n_{0}=\begin{pmatrix}1&x_{0}&\\\ &1&y_{0}\\\ &&1\end{pmatrix}.$

Then

$\xi^{[h_{0},n_{0}]}=(0,e_{2},\beta e_{1},0).$

Now for a general $g=(h,n)\in H(F)$ we have

$(0,e_{2},\beta e_{1},0)^{[g]}=(0,e_{2}h,\beta e_{1}h,[x\beta
e_{1}+y\rho^{2}e_{2}]h).$

This is of the form $(0,e_{2},\beta^{\prime}e_{1},0)$ for some
$\beta^{\prime}\in F^{*}$ if and only if $h=I_{2}$ and $x=y=0$ and in this
case $\beta^{\prime}=\beta$. We conclude that $(0,e_{2},\beta e_{1},0)$ and
$(0,e_{2},\beta^{\prime}e_{1},0)$ are in the same $H(F)$-orbit if and only if
$\beta^{\prime}=\beta$ and furthermore

$H_{(0,e_{2},\beta e_{1},0)}=I_{2}\times\\{n\in N:x=y=0\\}.$

We further observe that

$\mathfrak{c}((0,e_{2},\beta
e_{1},0),[h,\operatorname{Ad}(n)])=\psi[2\beta(xy\rho^{2}+z)],\ \ \ (h,n)\in
H(\mathbb{A}).$

The inclusion

$\\{I_{2}\\}\times\\{\begin{pmatrix}1&&z\\\ &1&\\\
&&1\end{pmatrix}:z\in\mathbb{A}\\}\subseteq H_{(0,e_{2},\beta
e_{1},0)}(\mathbb{A})$

therefore shows that ${(0,e_{2},\beta e_{1},0)}$ is not relevant for any
$\beta\neq 0$. ∎

This completes the analysis of orbits with $d_{1}=0$.

####  4.2.5. The $H(F)$-orbits with invariants $d_{1}=1=d_{1,2,3,4}$

######  Lemma 4.6.

The set of all $\xi\in F^{8}$ with invariants $d_{1}=1=d_{1,2,3,4}$ consists
of a unique $H(F)$-orbit with representative $(e_{2},0,0,0)$ and it is
relevant. The associated orbital integral is

$\mathcal{O}((0,1,0,0,0,0,0,0),\phi)=\\\
\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}^{*}}\phi_{K_{2}}[(0,t,0,xt,0,-yt,0,zt)]\left|{t}\right|^{2}\
d^{*}t\ dz\ \psi(x+y)\ dx\ dy.$

###### Proof.

Assume that $d_{1}(\xi)=1=d_{1,2,3,4}(\xi)$. Then there exists $0\neq v\in
F^{2}$ and $\alpha_{i}\in F$, $i=1,2,3$ such that
$\xi=(v,\alpha_{1}v,\alpha_{2}v,\alpha_{3}v)$. Let
$h_{0}\in\operatorname{SL}_{2}(F)$ be such that $vh_{0}=e_{2}$ and let

$n_{0}=\begin{pmatrix}1&-\alpha_{1}&\rho^{2}\alpha_{3}+\rho\alpha_{1}\alpha_{2}\\\
&1&\alpha_{2}\\\ &&1\end{pmatrix}.$

Then $\xi^{[h_{0},n_{0}]}=(e_{2},0,0,0)$. For $g=(h,n)\in H(\mathbb{A})$ we
have

$(e_{2},0,0,0)^{[g]}=(e_{2}h,xe_{2}h,-ye_{2}h,-(xy+z\rho)e_{2}h)$

and therefore
$H_{(e_{2},0,0,0)}(\mathbb{A})=N_{2}(\mathbb{A})\times\\{I_{3}\\}$.
Furthermore, $\mathfrak{c}((e_{2},0,0,0),[h])=1$, $h\in H(\mathbb{A})$. It
follows that $\xi=(e_{2},0,0,0)$ is relevant and we have

$\mathcal{O}(\xi,\phi)=\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}\phi[(e_{2},xe_{2},-ye_{2},-(xy+z\rho)e_{2})\lambda(h)]\
dh\ \psi(x+y)\ dx\ dy\ dz.$

The change of variables $z\mapsto-\rho^{2}(z+xy)$ gives

$\mathcal{O}(\xi,\phi)=\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}\phi[(e_{2},xe_{2},-ye_{2},ze_{2})\lambda(h)]\
dh\ dz\ \psi(x+y)\ dx\ dy.$

The lemma follows by applying the Iwasawa decomposition. ∎

####  4.2.6. The $H(F)$-orbits with invariants $d_{1}=1=d_{1,2,3}$ and
$d_{1,2,3,4}=2$ are not relevant

######  Lemma 4.7.

A complete set of representatives of $H(F)$-orbits with invariants
$d_{1}=1=d_{1,2,3}$ and $d_{1,2,3,4}=2$ is

$\\{(e_{2},0,0,\beta e_{1}):\beta\in F^{*}\\}.$

None of them is relevant.

###### Proof.

Assume that $d_{1}(\xi)=1=d_{1,2,3}(\xi)$ and $d_{1,2,3,4}(\xi)=2$. Then there
exists a basis $\\{v,w\\}$ of $F^{2}$ and $\alpha_{i}\in F$, $i=1,2$ such that
$\xi=(v,\alpha_{1}v,\alpha_{2}v,w)$. Applying Lemma 4.3 let
$h_{0}\in\operatorname{SL}_{2}(F)$ and $\beta\in F^{*}$ be such that
$vh_{0}=e_{2}$ and $wh_{0}=\beta e_{1}$. Let

$n_{0}=\begin{pmatrix}1&-\alpha_{1}&\rho\alpha_{1}\alpha_{2}\\\
&1&\alpha_{2}\\\ &&1\end{pmatrix}.$

Then $\xi^{[h_{0},n_{0}]}=(e_{2},0,0,\beta e_{1})$. For $g=(h,n)\in
H(\mathbb{A})$ we have

$(e_{2},0,0,\beta e_{1})^{[g]}=(e_{2}h,xe_{2}h,-ye_{2}h,\beta
e_{1}h-(xy+z\rho)e_{2}h).$

This is of the form $(e_{2},0,0,\beta^{\prime}e_{1})$ for some
$\beta^{\prime}\in F^{*}$ if and only if $x=y=0$ and
$h=\left(\begin{smallmatrix}{1}&{a}\\\ {}&{1}\end{smallmatrix}\right)$ with
$\beta a-z\rho=0$. It follows that $(e_{2},0,0,\beta e_{1})$ and
$(e_{2},0,0,\beta^{\prime}e_{1})$ are in the same $H(F)$-orbit if and only if
$\beta=\beta^{\prime}$ and

$H_{(e_{2},0,0,\beta
e_{1})}(\mathbb{A})=\\{(\begin{pmatrix}1&\beta^{-1}z\rho\\\
&1\end{pmatrix},\begin{pmatrix}1&&z\\\ &1&\\\
&&1\end{pmatrix}):z\in\mathbb{A}\\}.$

We further note that

$\mathfrak{c}((e_{2},0,0,\beta e_{1}),[g])=\psi[2\beta(xy+z\rho^{2})]$

and therefore we can deduce that $(e_{2},0,0,\beta e_{1})$ is not relevant.

∎

####  4.2.7. The $H(F)$-orbits with invariants $d_{1}=1=d_{1,2}$ and
$d_{1,2,3}=2$

######  Lemma 4.8.

A complete set of representatives of $H(F)$-orbits with invariants
$d_{1}=1=d_{1,2}$ and $d_{1,2,3}=2$ is

$\\{(e_{2},\alpha e_{2},\beta e_{1},0):\alpha\in F,\,\beta\in F^{*}\\}.$

The vector $(e_{2},\alpha e_{2},\beta e_{1},0)$ is relevant if and only if
$1+2\beta\rho\alpha^{2}=0$. For $\alpha\in F^{*}$ and
$\xi[\alpha]=(e_{2},\alpha e_{2},-\frac{1}{2}\rho^{2}\alpha^{-2}e_{1},0)$ the
associated orbital integral is

$\mathcal{O}(\xi[\alpha],\phi)=\\\
\int_{\mathbb{A}^{*}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\phi_{K_{2}}[(0,t,0,t(\alpha+x),-\frac{1}{2}\rho^{2}\alpha^{-2}t^{-1},-ty,-\frac{1}{2}\rho^{2}\alpha^{-2}t^{-1}x,t(y\rho^{2}\alpha-
xy-z\rho))]\\\
\psi[-\rho\alpha^{-1}xy-\rho^{2}\alpha^{-1}z+\alpha^{-2}xz+x+y]\ dx\ dy\
dz\left|{t}\right|^{2}\ d^{*}t.$

###### Proof.

Assume that $d_{1}(\xi)=1=d_{1,2}(\xi)$ and $d_{1,2,3}(\xi)=2$. Then there
exists a basis $\\{v,w\\}$ of $F^{2}$, $\alpha\in F$ and $u\in F^{2}$ such
that $\xi=(v,\alpha v,w,u)$. Let $x_{0},\,z_{0}\in F$ be such that
$u=z_{0}\rho v-x_{0}w$ and let

$n_{0}=\begin{pmatrix}1&x_{0}&z_{0}\\\ &1&\\\ &&1\end{pmatrix}.$

Applying Lemma 4.3, let $h_{0}\in\operatorname{SL}_{2}(F)$ and $\beta\in
F^{*}$ be such that $vh_{0}=e_{2}$ and $wh_{0}=\beta e_{1}$. Then
$\xi^{[h_{0},n_{0}]}=(e_{2},(\alpha+x_{0})e_{2},\beta e_{1},0)$. For
$g=(h,n)\in H(\mathbb{A})$ we have

$(e_{2},\alpha e_{2},\beta e_{1},0)^{[g]}=(e_{2}h,(\alpha+x)e_{2}h,(\beta
e_{1}-ye_{2})h,(\beta xe_{1}+(y\rho^{2}\alpha-xy-z\rho)e_{2})h).$

This is of the form $(e_{2},\alpha^{\prime}e_{2},\beta^{\prime}e_{1},0)$ for
some $\alpha^{\prime}\in F$ and $\beta^{\prime}\in F^{*}$ if and only if
$x=0$, $z\rho=y\rho^{2}\alpha$ and $h=\left(\begin{smallmatrix}{1}&{a}\\\
{}&{1}\end{smallmatrix}\right)$ with $\beta a=y$. It follows that
$(e_{2},\alpha e_{2},\beta e_{1},0)$ and
$(e_{2},\alpha^{\prime}e_{2},\beta^{\prime}e_{1},0)$ are in the same
$H(F)$-orbit if and only if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$
and

$H_{(e_{2},\alpha e_{2},\beta
e_{1},0)}(\mathbb{A})=\\{(\begin{pmatrix}1&\beta^{-1}y\\\
&1\end{pmatrix},\begin{pmatrix}1&&y\rho\alpha\\\ &1&y\\\
&&1\end{pmatrix}):y\in\mathbb{A}\\}.$

We further note that

$\mathfrak{c}((e_{2},\alpha e_{2},\beta
e_{1},0),[g])=\psi[2\beta\alpha(xy\rho^{2}+z)-2\beta xz\rho].$

We therefore have

$\mathfrak{c}_{\psi}((e_{2},\alpha e_{2},\beta
e_{1},0),[g])=\psi[(1+2\beta\alpha^{2}\rho)y],\\\
g=(\begin{pmatrix}1&\beta^{-1}y\\\
&1\end{pmatrix},\begin{pmatrix}1&&y\rho\alpha\\\ &1&y\\\ &&1\end{pmatrix})\in
H_{(e_{2},\alpha e_{2},\beta e_{1},0)}(\mathbb{A}).$

We conclude that $(e_{2},\alpha e_{2},\beta e_{1},0)$ is relevant if and only
if $1+2\beta\alpha^{2}\rho=0$ and for $\beta=-\frac{1}{2}\rho^{2}\alpha^{-2}$
we have

$\mathcal{O}(0,1,0,\alpha,\beta,0,0,0),\phi)=\\\
\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\operatorname{SL}_{2}(\mathbb{A})}\phi[(0,1,0,\alpha+x,\beta,0,\beta
x,-z\rho)\lambda(h)]\ dh\ \psi(x-\rho^{2}z-2\beta\rho xz)\ dx\ dz.$

Alternatively, we can also express the orbital integral as the iterated
integral

$\int_{N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\phi[(0,1,0,\alpha+x,\beta,-y,\beta
x,y\rho^{2}\alpha-xy-z\rho)\lambda(h)]\\\
\psi[2\beta\alpha(xy\rho^{2}+z)-2\beta xz\rho+x+y]\ dx\ dy\ dz\ dh$

where the integral over
${N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}$ makes sense
only after integration over $y$ and $z$. Applying the Iwasawa decomposition,
this equals

$\int_{\mathbb{A}^{*}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\phi_{K_{2}}[(0,t,0,t(\alpha+x),t^{-1}\beta,-ty,t^{-1}\beta
x,t(y\rho^{2}\alpha-xy-z\rho))]\\\ \psi[2\beta\alpha(xy\rho^{2}+z)-2\beta
xz\rho+x+y]\ dx\ dy\ dz\left|{t}\right|^{2}\ d^{*}t.$

The lemma follows.

∎

####  4.2.8. The $H(F)$-orbits with invariants $d_{1}=1=d_{1,3}$ and
$d_{1,2}=2$

######  Lemma 4.9.

A complete set of representatives of $H(F)$-orbits with invariants
$d_{1}=1=d_{1,3}$ and $d_{1,2}=2$ is

$\\{(e_{2},\beta e_{1},\alpha e_{2},0):\alpha\in F,\,\beta\in F^{*}\\}.$

The vector $(e_{2},\beta e_{1},\alpha e_{2},0)$ is relevant if and only if
$\beta=\frac{1}{2}\rho\alpha^{-2}$. For $\alpha\in F^{*}$ and
$\xi[\alpha]=(e_{2},\frac{1}{2}\rho\alpha^{-2}e_{1},\alpha e_{2},0)$ the
associated orbital integral is

$\mathcal{O}(\xi[\alpha],\phi)=\int_{\mathbb{A}^{*}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\phi_{K_{2}}[(0,t,\frac{1}{2}\rho\alpha^{-2}t^{-1},tx,0,t(\alpha-y),\frac{1}{2}\alpha^{-2}t^{-1}y,t(x\alpha-
xy-z\rho))]\\\ \psi[\alpha^{-2}(xy-z)y-\rho(xy\rho^{2}+z)\alpha^{-1}+x+y]\ dx\
dy\ dz\left|{t}\right|^{2}\ d^{*}t.$

###### Proof.

Assume that $d_{1}(\xi)=1=d_{1,3}(\xi)$ and $d_{1,2}(\xi)=2$. Then there
exists a basis $\\{v,w\\}$ of $F^{2}$, $\alpha\in F$ and $u\in F^{2}$ such
that $\xi=(v,w,\alpha v,u)$. Let $y_{0},\,z_{0}\in F$ be such that
$u=z_{0}\rho v-y_{0}\rho^{2}w$ and let

$n_{0}=\begin{pmatrix}1&&z_{0}\\\ &1&y_{0}\\\ &&1\end{pmatrix}.$

Applying Lemma 4.3 let $h_{0}\in\operatorname{SL}_{2}(F)$ and $\beta\in F^{*}$
be such that $vh_{0}=e_{2}$ and $wh_{0}=\beta e_{1}$. Then
$\xi^{[h_{0},n_{0}]}=(e_{2},\beta e_{1},(\alpha-y_{0})e_{2},0)$. For
$g=(h,n)\in H(\mathbb{A})$ we have

$(e_{2},\beta e_{1},\alpha e_{2},0)^{[g]}=(e_{2}h,(\beta
e_{1}+xe_{2})h,(\alpha-y)e_{2}h,(x\alpha e_{2}+y\rho^{2}\beta
e_{1}-(xy+z\rho)e_{2})h).$

This is of the form $(e_{2},\beta^{\prime}e_{1},\alpha^{\prime}e_{2},0)$ for
some $\alpha^{\prime}\in F$ and $\beta^{\prime}\in F^{*}$ if and only if
$h=\left(\begin{smallmatrix}{1}&{a}\\\ {}&{1}\end{smallmatrix}\right)$ with
$\beta a+x=0$, $y=0$ and $z\rho=x\alpha$ and in this case
$(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$. It follows that
$(e_{2},\beta e_{1},\alpha e_{2},0)$ and
$(e_{2},\beta^{\prime}e_{1},\alpha^{\prime}e_{2},0)$ are in the same
$H(F)$-orbit if and only if $(\alpha,\beta)=(\alpha^{\prime},\beta^{\prime})$
and

$H_{(e_{2},\beta e_{1},\alpha
e_{2},0)}(\mathbb{A})=\\{(\begin{pmatrix}1&-\beta^{-1}x\\\
&1\end{pmatrix},\begin{pmatrix}1&x&\rho^{2}x\alpha\\\ &1&\\\
&&1\end{pmatrix}):x\in\mathbb{A}\\}.$

We further note that

$\mathfrak{c}((e_{2},\beta e_{1},\alpha
e_{2},0),[g])=\psi[2\beta(xy-z)\rho^{2}y-2(xy\rho^{2}+z)\alpha\beta].$

We therefore have

$\mathfrak{c}_{\psi}((e_{2},\beta e_{1},\alpha
e_{2},0),[g])=\psi[(1-2\rho^{2}\alpha^{2}\beta)x],\ \ \
g=(\begin{pmatrix}1&-\beta^{-1}x\\\
&1\end{pmatrix},\begin{pmatrix}1&x&\rho^{2}x\alpha\\\ &1&\\\
&&1\end{pmatrix}).$

We conclude that $(e_{2},\beta e_{1},\alpha e_{2},0)$ is relevant if and only
if $2\rho^{2}\alpha^{2}\beta=1$ and in this case, the associated (iterated, as
in the previous lemma) orbital integral is

$\int_{N_{2}(\mathbb{A})\backslash\operatorname{SL}_{2}(\mathbb{A})}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\phi[(0,1,\beta,x,0,\alpha-y,y\rho^{2}\beta,x\alpha-
xy-z\rho)\lambda(h)]\\\
\psi[2\rho^{2}\beta(xy-z)y-2(xy\rho^{2}+z)\alpha\beta+x+y]\ dx\ dy\ dz\ dh.$

Again applying the Iwasawa decomposition, the lemma follows. ∎

####  4.2.9. The generic $H(F)$-orbits

######  Lemma 4.10.

A complete set of representatives of $H(F)$-orbits with invariants $d_{1}=1$
and $d_{1,2}=2=d_{1,3}$ is

$\\{(e_{2},\beta e_{1},\beta^{\prime}e_{1},0):\beta,\,\beta^{\prime}\in
F^{*}\\}.$

They are all relevant. For $a,\,b\in F^{*}$ and
$\xi[a,b]=(e_{2},ae_{1},be_{1},0)$ the associated orbital integral is

$\mathcal{O}(\xi[a,b],\phi)=\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}^{*}}\\\
\phi_{K_{2}}[(0,t,t^{-1}a,t(x+as),t^{-1}b,t(bs-y),t^{-1}(xb+y\rho^{2}a),t[(xb+y\rho^{2}a)s-(xy+z\rho)])]\left|{t}\right|^{2}\
d^{*}t\ ds\\\ \psi[x+y+2a(xy-z)y\rho^{2}-2bxz\rho]\ dx\ dy\ dz.$

###### Proof.

Let $\xi\in F^{8}$ be such that $d_{1}(\xi)=1$ and
$d_{1,2}(\xi)=2=d_{1,3}(\xi)$. Then there exist $v,\,w,\,w^{\prime},\,u\in
F^{2}$ such that $\\{v,w\\}$ and $\\{v,w^{\prime}\\}$ are bases of $F^{2}$ and
$\xi=(v,w,w^{\prime},u)$. Let $a,\,b\in F$ be such that $w^{\prime}=av+bw$ and
note that $b\neq 0$. Let $x_{0},\,y_{0},\,z_{0}\in F$ be the unique common
solutions to the equations

$\begin{cases}w^{\prime}-y_{0}v=b(w+x_{0}v)\\\
u+x_{0}w^{\prime}+y_{0}\rho^{2}w-(x_{0}y_{0}+z_{0}\rho)v=0.\end{cases}$

Applying Lemma 4.3 let $h_{0}\in\operatorname{SL}_{2}(F)$ and $\beta\in F^{*}$
be such that $(vh_{0},(w+x_{0}v)h_{0})=(e_{2},\beta e_{1})$ and let

$n_{0}=\begin{pmatrix}1&x_{0}&z_{0}\\\ &1&y_{0}\\\ &&1\end{pmatrix}.$

Then

$\xi^{[h_{0},\operatorname{Ad}(n_{0})]}=(e_{2},\beta e_{1},b\beta e_{1},0).$

For $g=(h,n)\in H(F)$ we have

$(e_{2},\beta e_{1},\beta^{\prime}e_{1},0)^{[g]}=(e_{2}h,(\beta
e_{1}+xe_{2})h,(\beta^{\prime}e_{1}-ye_{2})h,[(x\beta^{\prime}+y\rho^{2}\beta)e_{1}-(xy+z\rho)e_{2}]h)$

and we can deduce that this is of the form
$(e_{2},\beta_{1}e_{1},\beta_{1}^{\prime}e_{1},0)$ for some
$\beta_{1},\,\beta_{1}^{\prime}\in F^{*}$ if and only if $g$ is the identity.
Thus, $H_{(e_{2},\beta e_{1},\beta^{\prime}e_{1},0)}$ is trivial and therefore
$(e_{2},\beta e_{1},\beta^{\prime}e_{1},0)$ is relevant. We also note that

$\mathfrak{c}((e_{2},\beta
e_{1},\beta^{\prime}e_{1},0),[g])=\psi[2\beta(xy-z)y\rho^{2}-2\beta^{\prime}xz\rho].$

We therefore have

$\mathcal{O}((e_{2},\beta e_{1},\beta^{\prime}e_{1},0),\phi)=\\\
\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\mathbb{A}}\int_{\operatorname{SL}_{2}(\mathbb{A})}\phi[(0,1,\beta,x,\beta^{\prime},-y,x\beta^{\prime}+y\rho^{2}\beta,-(xy+z\rho))\lambda(h)]\
dh\\\ \psi[x+y+2\beta(xy-z)y\rho^{2}-2\beta^{\prime}xz\rho]\ dx\ dy\ dz.$

The lemma follows by the Iwasawa decomposition. ∎

This completes the proof of Theorem 4.2. ∎

##  5\. Unfolding the distribution $J$

Recall that $G^{\prime}(\mathbb{A})$ denotes the adelic $3$-fold cover of
$\operatorname{SL}_{3}(\mathbb{A})$ and $G^{\prime}(F)$ denotes the embedded
image of $\operatorname{SL}_{3}(F)$ in $G^{\prime}(\mathbb{A})$. For
notational convenience we suppress this embedding and write elements of
$G^{\prime}(F)$ by matrices. We consider the action
$g\cdot(n_{1},n_{2})=n_{1}^{-1}gn_{2}$ of $N(F)\times N(F)$ on
$G^{\prime}(F)$. The element $g$ or its orbit is called relevant if
$(n_{1},n_{2})\mapsto\psi(n_{1}^{-1}n_{2})$ is trivial on the stabilizer
$(N\times N)_{g}(\mathbb{A})$ of $g$. For $a,b\in F^{*}$ let
$t(a,b)=\operatorname{diag}(a,a^{-1}b,b^{-1})$, and let

$w_{1}=\begin{pmatrix}&-1&\\\ 1&&\\\ &&1\end{pmatrix},\ \ \
w_{2}=\begin{pmatrix}1&&\\\ &&-1\\\ &1&\end{pmatrix}.$

Then using the Bruhat decomposition, one sees that a complete set
${\Xi}^{\prime}_{\operatorname{rel}}$ of representatives for the relevant
orbits is given as follows:

  1. (1)

$t(a,a^{2})=aI_{3}$, $a\in\mu_{3}\subseteq F^{*}$;

  2. (2)

$t(a^{-2},a^{-1})w_{1}w_{2}=\begin{pmatrix}&&a^{-2}\\\ a&&\\\
&a&\end{pmatrix}$, $a\in F^{*}$;

  3. (3)

$t(-a,a^{2})w_{2}w_{1}=\begin{pmatrix}&a&\\\ &&a\\\ a^{-2}&&\end{pmatrix}$,
$a\in F^{*}$;

  4. (4)

$t(b^{-1},a^{-1})w_{1}w_{2}w_{1}=\begin{pmatrix}&&b^{-1}\\\ &-a^{-1}b&\\\
a&&\end{pmatrix}$, $a,\,b\in F^{*}$.

(See for example [BFG88], where the determination of relevant orbits is
carried out in the language of Poincaré series; the adelic ingredients are the
same.) Note that the notion of relevance does not depend on the cover, though
the orbital integral that we consider does.

To give the needed orbital integrals, we fix once and for all an embedding of
$\mu_{3}$ in $\mathbb{C}^{*}$ and consider the genuine Hecke algebra
$\mathcal{H}_{\operatorname{gen}}(G^{\prime}(\mathbb{A}))=\\{f^{\prime}\in
C_{c}^{\infty}(G^{\prime}(\mathbb{A})):f^{\prime}(zg)=zf^{\prime}(g)\\}$. For
$f^{\prime}\in\mathcal{H}_{\operatorname{gen}}(G^{\prime}(\mathbb{A}))$ and
$g\in G^{\prime}(F)$ relevant we consider the orbital integral

$\mathcal{O}^{\prime}(g,f^{\prime})=\int_{N(\mathbb{A})\times
N(\mathbb{A})/(N\times
N)_{g}(\mathbb{A})}f^{\prime}(n_{1}^{-1}gn_{2})\psi(n_{1}^{-1}n_{2})\ dn_{1}\
dn_{2}.$

Here $N(\mathbb{A})$ is embedded in $G^{\prime}(\mathbb{A})$ by the trivial
section $n\mapsto(n,1)$. Explicitly, according to the relevant orbits we have

  1. (1)

$\mathcal{O}^{\prime}(t(a,a^{2}),f^{\prime})=\int_{N(\mathbb{A})}f^{\prime}(an)\psi(n)\
dn$, $a\in\mu_{3}$;

  2. (2)

$\mathcal{O}^{\prime}(t(a^{-2},a^{-1})w_{1}w_{2},f^{\prime})=\int_{N(\mathbb{A})\times
U_{2,1}(\mathbb{A})}f^{\prime}(n^{-1}t(a^{-2},a^{-1})w_{1}w_{2}u)\psi(n^{-1}u)\
dn\ du$;

  3. (3)

$\mathcal{O}^{\prime}(t(-a,a^{2})w_{2}w_{1},f^{\prime})=\int_{U_{2,1}(\mathbb{A})\times
N(\mathbb{A})}f^{\prime}(u^{-1}t(-a,a^{2})w_{2}w_{1}n)\psi(u^{-1}n)\ du\ dn$;

  4. (4)

$\mathcal{O}^{\prime}(t(b^{-1},a^{-1})w_{1}w_{2}w_{1},f^{\prime})=\\\
\int_{N(\mathbb{A})\times
N(\mathbb{A})}f^{\prime}(n_{1}^{-1}t(b^{-1},a^{-1})w_{1}w_{2}w_{1}n_{2}))\psi(n_{1}^{-1}n_{2})\
dn_{1}\ dn_{2}$

where

$U_{2,1}(\mathbb{A})=\\{\begin{pmatrix}I_{2}&v\\\
&1\end{pmatrix}:v\in\mathbb{A}^{2}\\}.$

######  Proposition 5.1.

The distribution $J$ given in (4) is a sum of factorizable orbital integrals

$J(f^{\prime})=\sum_{\xi^{\prime}\in{\Xi}^{\prime}_{\operatorname{rel}}}\mathcal{O}(\xi^{\prime},f^{\prime})$

where the relevant orbits and the associated orbital integrals are given
above.

## Part II. Local Theory: The Fundamental Lemma

##  6\. Notation and main result

Throughout Part II we set the following notation. Let $F$ denote a non-
archimedean local field, $\mathcal{O}$ the ring of integers of $F$,
$\mathfrak{p}$ its maximal ideal, $\varpi$ a uniformizer in $F$ (a generator
of $\mathfrak{p}$), $k_{F}=\mathcal{O}/\mathfrak{p}$ the residual field of
$F$, $p$ the characteristic of $k_{F}$, $q$ the number of elements in $k_{F}$,
${\operatorname{val}}:F^{*}\rightarrow\mathbb{Z}$ the standard valuation so
that ${\operatorname{val}}(u\varpi^{n})=n$ for $u\in\mathcal{O}^{*}$,
$n\in\mathbb{Z}$, and $S^{n}=\\{s^{n}:s\in S\\}$ for any subgroup $S$ of
$F^{*}$ and $n\in\mathbb{N}$. Assume that $F$ contains a primitive cube root
of unity $\rho$ and set $\mu_{3}=\langle\rho\rangle$.

Fix a character $\psi$ of $F$ with conductor $\mathcal{O}$ (that is, $\psi$
restricted to $\mathcal{O}$ is trivial and restricted to $\mathfrak{p}^{-1}$
is non-trivial). We denote by ${\bf{1}}_{A}$ the characteristic function of a
set $A$. We normalize integration so that $\int_{\mathcal{O}}\,dx=1$,
$\int_{\mathcal{O}^{*}}\,d^{*}x=1$.

The main work is to compare the two local orbital integrals corresponding to
the big cells. These are families of integrals parameterized by $F^{*}\times
F^{*}$. The first is defined by the formula

(15) $I(a,b)=\int_{F}\int_{F}\int_{F}\int_{F}\int_{F^{*}}\\\
{\bf{1}}_{\mathcal{O}^{8}}[(0,t,t^{-1}a,t(x+as),t^{-1}b,t(bs-y),t^{-1}(xb+y\rho^{2}a),t[(xb+y\rho^{2}a)s-(xy+z\rho)])]\left|{t}\right|^{2}\
d^{*}t\ ds\\\ \psi[x+y+2a(xy-z)y\rho^{2}-2bxz\rho]\ dx\ dy\ dz.$

This is the local integral obtained from Lemma 4.10.

In order to define the second we need to set some further notation. Consider
the local metaplectic three fold cover $\widetilde{\operatorname{SL}}_{3}(F)$
of $\operatorname{SL}_{3}(F)$. It is a central extension

$1\rightarrow\mu_{3}\rightarrow\widetilde{\operatorname{SL}}_{3}(F)\rightarrow\operatorname{SL}_{3}(F)\rightarrow
1.$

As a set, we identify $\widetilde{\operatorname{SL}}_{3}(F)$ with
$\operatorname{SL}_{3}(F)\times\mu_{3}$ and the group operation is given by
$(g_{1},z_{1})(g_{2},z_{2})=(g_{1}g_{2},z_{1}z_{2}\sigma(g_{1},g_{2}))$ where
$\sigma$ is a certain $2$-cocycle of $\operatorname{SL}_{3}(F)$. This cocycle
is described, following Matsumoto [Mat69], in Bump and Hoffstein [BH87, §2].

Let $N$ be the group of upper triangular unipotent matrices in
$\operatorname{SL}_{3}(F)$ and $K=\operatorname{SL}_{3}(\mathcal{O})$. The map
$n\mapsto(n,1):N\rightarrow\widetilde{\operatorname{SL}}_{3}(F)$ is an
imbedding of $N$ in $\widetilde{\operatorname{SL}}_{3}(F)$ and this way we
view $N$ as a subgroup of $\widetilde{\operatorname{SL}}_{3}(F)$. The group
$K$ also admits a splitting in $\widetilde{\operatorname{SL}}_{3}(F)$. There
is a map $\kappa:K\rightarrow\mu_{3}$ such that the map
$g\mapsto(g,\kappa(g))$ imbeds $K$ into
$\widetilde{\operatorname{SL}}_{3}(F)$. By abuse of notation we also denote by
$\psi$ the character of $N$ that satisfies $\psi(u)=\psi(u_{1,2}+u_{2,3})$,
$u\in N$.

Fix an embedding of $\mu_{3}$ in $\mathbb{C}^{*}$ and let
$f_{0}:\widetilde{\operatorname{SL}}_{3}(F)\rightarrow\mathbb{C}$ be defined
by

$f_{0}(g,z)=\begin{cases}z\kappa(g)^{-1}&g\in K\\\
0&\text{otherwise}\end{cases}\ \ \ g\in\operatorname{SL}_{3}(F),\
z\in\mu_{3}.$

The second family of integrals is defined by

(16) $J(a,b)=\int_{N}\int_{N}f_{0}((u_{1}g_{a,b}u_{2},1))\psi(u_{1}u_{2})\
du_{1}\ du_{2}$

where

$g_{a,b}=\begin{pmatrix}&&b^{-1}\\\ &-a^{-1}b&\\\
a&&\end{pmatrix}\in\operatorname{SL}_{3}(F).$

Let $(\cdot,\cdot)_{3}:F^{*}\times F^{*}\rightarrow\mu_{3}$ be the cubic
Hilbert symbol. Our main result is the following comparison.

######  Theorem 6.1 (The fundamental lemma for the big cell orbital integrals-
unit element).

Assume $p>3$. For any $a,\,b\in F^{*}$ we have

$I(a,b)=(c,d)_{3}\ J(c,d),\ \ \ \text{where}\ \ \ c=-54a,\ d=54b.$

In the final section, we also compare the orbital integrals for the other
relevant orbits (Theorem 11.7).

##  7\. Ingredients needed for the comparison

We develop the ingredients that are needed to establish Theorem 6.1. From now
on we assume $p>3$.

###  7.1. The cubic Hilbert symbol

We begin by recalling the basic properties of the cubic Hilbert symbol. They
will be used throughout our computation without further mention. For
$x,\,y,\,z\in F^{*}$ we have

  * •

$(y,x)_{3}=\overline{(x,y)_{3}}$

  * •

$(xy,z)_{3}=(x,z)_{3}(y,z)_{3}$

  * •

$(x,1-x)_{3}=1$, $x\neq 1$

  * •

$(x,y)_{3}=1$ for all $y\in F^{*}$ if and only if $x\in F^{*3}$

  * •

$(x,u)_{3}=1$ for all $u\in\mathcal{O}^{*}$ if and only if
$3|{\operatorname{val}}(x)$.

The last property is a consequence of our assumption that the residual
characteristic $p>3$. We further observe that by this assumption and Hensel’s
Lemma $1+\mathfrak{p}\subseteq F^{*3}$ and consequently

$(x+y,z)_{3}=(x,z)_{3}\ \ \ \text{whenever}\ \ \
\left|{y}\right|<\left|{x}\right|.$

###  7.2. Cubic Gauss sums

For future reference we recall here a formula that is essentially the
calculation of the absolute value squared of a cubic Gauss sum. This result is
standard and the proof is omitted.

######  Lemma 7.1.

Let $a\in F$ be such that $\left|{a}\right|=q$. Then

$\int_{\mathcal{O}^{*}}\int_{\mathcal{O}^{*}}(a,uv^{-1})_{3}\,\psi(a(u+v))\
du\ dv=q^{-1}.$

∎

###  7.3. Kloosterman integrals

For $y\in F^{*}$ and $a,b\in F$ consider the integral

$\mathcal{K}(y;a,b)=\int_{\mathcal{O}^{*}}(y,u)_{3}\,\psi(au+bu^{-1})\ du.$

We provide a formula for $\mathcal{K}(y;a,b)$ in terms of Kloosterman sums
over the residual field, the function

$\delta_{3\mid{\operatorname{val}}(x)}=\begin{cases}1&3\mid{\operatorname{val}}(x)\\\
0&\text{otherwise}\end{cases},\ \ \ x\in F^{*}$

and the function $\mathfrak{s}$ on $F^{*}$ defined by

$\mathfrak{s}(x)=\sum_{z\in\mathfrak{p}^{\ell-e}/\mathfrak{p}^{\ell}}\psi(xz^{2})$

where $\ell\in\mathbb{Z}$ and $e\in\\{0,1\\}$ are such that
$\left|{x}\right|=q^{2\ell-e}$. Note that
$\ell=\lfloor\frac{1-{\operatorname{val}}(x)}{2}\rfloor$ and the summand
$\psi(xz^{2})$ is well defined (independent of the class of $z$ mod
$\mathfrak{p}^{\ell}$). In particular, $\mathfrak{s}(x)=1$ if
${\operatorname{val}}(x)$ is even. It is easy to verify the basic property of
$\mathfrak{s}$:

(17) $\mathfrak{s}(xy^{2})=\mathfrak{s}(x),\ \ \ x,y\in F^{*}.$

In the evaluation below and subsequent formulas, if $x\in F^{*}$ is a square
then we write $\sqrt{x}$ for a square root of $x$ in $F$; all formulas will be
independent of the choice of square root.

######  Lemma 7.2.

We have

$\mathcal{K}(y;a,b)=\begin{cases}(1-q^{-1})\delta_{3\mid{\operatorname{val}}(y)}&\max(\left|{a}\right|,\left|{b}\right|)\leq
1\\\ q^{-1}\sum_{u\in
k_{F}^{*}}(y,u)_{3}\,\psi(au+bu^{-1})&\max(\left|{a}\right|,\left|{b}\right|)=q\\\
q^{-\lfloor\frac{1-{\operatorname{val}}(a)}{2}\rfloor}(y,ab^{-1})_{3}\,\sum_{\epsilon=\pm
1}\psi(2\epsilon\sqrt{ab})\mathfrak{s}(\epsilon\sqrt{ab})&\max(\left|{a}\right|,\left|{b}\right|)>q\text{
and }a^{-1}b\in\mathcal{O}^{*2}\\\
0&\max(\left|{a}\right|,\left|{b}\right|)>q\text{ and
}a^{-1}b\not\in\mathcal{O}^{*2}.\end{cases}$

###### Proof.

If $\max(\left|{a}\right|,\left|{b}\right|)\leq q$ then the integrand
$(y,u)_{3}\psi(au+bu^{-1})$ depends only on $u+\mathfrak{p}$ and therefore
$\mathcal{K}(y;a,b)=q^{-1}\sum_{u\in k_{F}^{*}}(y,u)_{3}\psi(au+bu^{-1})$. If
in addition $a,b\in\mathcal{O}$ then $\psi(au+bu^{-1})=1$ for all $u\in
k_{F}^{*}$. This explains the formula when
$\max(\left|{a}\right|,\left|{b}\right|)\leq q$.

Assume now that $\max(\left|{a}\right|,\left|{b}\right|)\geq q^{2}$ and let
$\ell\geq 1$ and $e\in\\{0,1\\}$ be such that
$\max(\left|{a}\right|,\left|{b}\right|)=q^{2\ell-e}$. We decompose the
integral by writing $u=v(1+z)$ with
$v\in\mathcal{O}^{*}/(1+\mathfrak{p}^{\ell})$ and $z\in\mathfrak{p}^{\ell}$.
Since $(1+z)^{-1}=\sum_{i=0}^{\infty}(-z)^{i}\in 1-z+\mathfrak{p}^{2\ell}$ and
$bv^{-1}\mathfrak{p}^{2\ell}\subseteq\mathcal{O}$ we conclude that

$\mathcal{K}(y;a,b)=\sum_{v\in\mathcal{O}^{*}/(1+\mathfrak{p}^{\ell})}(y,v)_{3}\psi(av+bv^{-1})\int_{\mathfrak{p}^{\ell}}\psi((av-
bv^{-1})z)\ dz.$

We have

$\int_{\mathfrak{p}^{\ell}}\psi((av-bv^{-1})z)\ dz=\begin{cases}q^{-\ell}&av-
bv^{-1}\in\mathfrak{p}^{-\ell}\\\ 0&\text{otherwise.}\end{cases}$

Note that $av-bv^{-1}\in\mathfrak{p}^{-\ell}$ if and only if
$\varpi^{2\ell-e}(av^{2}-b)\in\mathfrak{p}^{\ell-e}$ and by assumption
$\ell-e\geq 1$. By Hensel’s lemma, there exists $v\in\mathcal{O}^{*}$ such
that $av-bv^{-1}\in\mathfrak{p}^{-\ell}$ if and only if
$a^{-1}b\in\mathcal{O}^{*2}$ and in this case $v$ is of this form if and only
if $v\in\pm v_{0}+\mathfrak{p}^{\ell-e}$ where $v_{0}^{2}=a^{-1}b$. Note that
for $v=\pm v_{0}(1+z)$ with $z\in\mathfrak{p}^{\ell-e}$ we have

$(y,v)_{3}=(y,\pm v_{0})_{3}=(y,v_{0}^{-2})_{3}=(y,ab^{-1})_{3},$
$av+bv^{-1}\in\pm(av_{0}+bv_{0}^{-1}+(av_{0}-bv_{0}^{-1})z+bv_{0}^{-1}z^{2})+b\mathfrak{p}^{3(\ell-e)}$

and by assumption $av_{0}-bv_{0}^{-1}=0$ and
$b\mathfrak{p}^{3(\ell-e)}\subseteq\mathcal{O}$ so that

$\psi(av+bv^{-1})=\psi(\pm(av_{0}+bv_{0}^{-1})+bv_{0}^{-1}z^{2}).$

Note further that $av_{0}+bv_{0}^{-1}=2av_{0}=2bv_{0}^{-1}$ and that
$bv_{0}^{-1}$ is a square root of $ab$. The lemma readily follows. ∎

We remark that the analysis above of the case
$\max(\left|{a}\right|,\left|{b}\right|)\geq q^{2}$ is tantamount to an
evaluation of the sum using the method of stationary phase.

###  7.4. Cubic exponential integrals

For $a,b\in F$ consider the integrals

$\mathcal{C}(a,b)=\int_{\mathcal{O}}\psi(ax+bx^{3})\ dx\ \ \ \text{and}\ \ \
\mathcal{C}_{0}(a,b)=\int_{\mathcal{O}^{*}}\psi(au+bu^{3})\ du.$

######  Lemma 7.3.

We have

$\mathcal{C}(a,b)=\begin{cases}1&\max(\left|{a}\right|,\left|{b}\right|)\leq
1\\\ 0&\left|{b}\right|<\left|{a}\right|=q\\\ q^{-1}\sum_{x\in
k_{F}}\psi(ax+bx^{3})&\left|{a}\right|\leq\left|{b}\right|=q\\\
0&\max(\left|{b}\right|,q)<\left|{a}\right|\\\
q^{-\lfloor\frac{1-{\operatorname{val}}(a)}{2}\rfloor}\sum_{\epsilon=\pm
1}\psi(\epsilon\frac{2}{9}a\sqrt{-3ab^{-1}})\mathfrak{s}(\epsilon\frac{1}{9}a\sqrt{-3ab^{-1}})&\left|{a}\right|=\left|{b}\right|>q\text{
and }ab^{-1}\in\mathcal{O}^{*2}\\\ 0&\left|{a}\right|=\left|{b}\right|>q\text{
and }ab^{-1}\not\in\mathcal{O}^{*2}\end{cases}$

and

$\mathcal{C}_{0}(a,b)=\begin{cases}1-q^{-1}&\max(\left|{a}\right|,\left|{b}\right|)\leq
1\\\ -q^{-1}&\left|{b}\right|<\left|{a}\right|=q\\\ q^{-1}\sum_{u\in
k_{F}^{*}}\psi(au+bu^{3})&\left|{a}\right|\leq\left|{b}\right|=q\\\
q^{-\lfloor\frac{1-{\operatorname{val}}(a)}{2}\rfloor}\sum_{\epsilon=\pm
1}\psi(\epsilon\frac{2}{9}a\sqrt{-3ab^{-1}})\mathfrak{s}(\epsilon\frac{1}{9}a\sqrt{-3ab^{-1}})&\max(\left|{a}\right|,\left|{b}\right|)>q\text{
and }ab^{-1}\in\mathcal{O}^{*2}\\\ 0&\text{otherwise.}\end{cases}$

In particular,

$\mathcal{C}(a,b)=\begin{cases}q^{-1}+\mathcal{C}_{0}(a,b)&\max(\left|{a}\right|,\left|{b}\right|)\leq
q\\\
\mathcal{C}_{0}(a,b)&\max(\left|{b}\right|,q^{2})\leq\left|{a}\right|.\end{cases}$

######  Remark 7.4.

The lemma excludes the computation of $\mathcal{C}(a,b)$ when
$\max(\left|{a}\right|,q)<\left|{b}\right|$. In fact, in this case we have

$\mathcal{C}(a,b)=\begin{cases}q^{-\ell}\sum_{x\in\mathfrak{p}^{\lfloor\frac{\ell+1-e}{2}\rfloor}/\mathfrak{p}^{\ell}}\psi(ax+bx^{3})&\left|{a}\right|\leq
q^{\ell}\\\ 0&\text{otherwise}\end{cases}$

where $\ell\geq 1$ and $e\in\\{0,1\\}$ are such that
$\left|{b}\right|=q^{2\ell-e}$. However, we omit the proof for this case since
it is never used in our work.

###### Proof.

The two formulas are straightforward when
$\max(\left|{a}\right|,\left|{b}\right|)\leq q$. Assume now that
$\max(\left|{b}\right|,q^{2})\leq\left|{a}\right|$ and let $\ell\geq 1$ and
$e\in\\{0,1\\}$ be such that $\left|{a}\right|=q^{2\ell-e}$. Since
$b\mathfrak{p}^{2\ell}\subseteq\mathcal{O}$, for $y\in\mathcal{O}$ we have

$\int_{y+\mathfrak{p}^{\ell}}\psi(ax+bx^{3})\
dx=\psi(ay+by^{3})\int_{\mathfrak{p}^{\ell}}\psi((a+3by^{2})z)\
dz=\begin{cases}q^{-\ell}\psi(ay+by^{3})&a+3by^{2}\in\mathfrak{p}^{-\ell}\\\
0&\text{otherwise.}\end{cases}$

It follows that

$\mathcal{C}(a,b)=q^{-\ell}\sum_{\begin{subarray}{c}x\in\mathcal{O}/\mathfrak{p}^{\ell}\\\
a+3bx^{2}\in\mathfrak{p}^{-\ell}\end{subarray}}\psi(ax+bx^{3})\ \ \
\text{and}\ \ \
\mathcal{C}_{0}(a,b)=q^{-\ell}\sum_{\begin{subarray}{c}u\in\mathcal{O}^{*}/(1+\mathfrak{p}^{\ell})\\\
a+3bu^{2}\in\mathfrak{p}^{-\ell}\end{subarray}}\psi(au+bu^{3}).$

Note that for $y\in\mathcal{O}$ we have $a+3by^{2}\in\mathfrak{p}^{-\ell}$ if
and only if $\varpi^{2\ell-e}(a+3by^{2})\in\mathfrak{p}^{\ell-e}$ and that by
assumption $\mathfrak{p}^{\ell-e}\subseteq\mathfrak{p}$ while
$\varpi^{2\ell-e}a\in\mathcal{O}^{*}$. Consequently, if $y\in\mathcal{O}$ is
such that $a+3by^{2}\in\mathfrak{p}^{-\ell}$ then $|a|=|b|$ and
$y\in\mathcal{O}^{*}$. It follows that indeed in this case
$\mathcal{C}(a,b)=\mathcal{C}_{0}(a,b)$. Note that
$-3=(1+2\rho)^{2}\in\mathcal{O}^{*}$. It further follows from Hensel’s lemma
that there exists $y\in\mathcal{O}^{*}$ such that
$a+3by^{2}\in\mathfrak{p}^{-\ell}$ if and only if
$ab^{-1}\in\mathcal{O}^{*2}$. The vanishing of $\mathcal{C}(a,b)$ when
$\max(\left|{b}\right|,q)<\left|{a}\right|$ and when
$\left|{a}\right|=\left|{b}\right|>q$ and $ab^{-1}\not\in\mathcal{O}^{*2}$
follows.

Assume now that $\left|{a}\right|=\left|{b}\right|>q$ and
$ab^{-1}\in\mathcal{O}^{*2}$ and let $v_{0}\in\mathcal{O}^{*}$ be such that
$v_{0}^{2}=-3ab^{-1}$. Then for $y\in\mathcal{O}$ we have
$a+3by^{2}\in\mathfrak{p}^{-\ell}$ if and only if
$y\in\pm\frac{1}{3}v_{0}+\mathfrak{p}^{\ell-e}$.

Since $b\mathfrak{p}^{3(\ell-e)}\subseteq\mathcal{O}$, for
$z\in\mathfrak{p}^{\ell-e}$ and $x=\pm(\frac{1}{3}v_{0}+z)$ we have

$ax+bx^{3}\in\pm(\frac{2}{9}av_{0}+bv_{0}z^{2})+\mathcal{O}.$

Consequently,

$\mathcal{C}(a,b)=q^{-\ell}\sum_{\epsilon=\pm
1}\psi(\epsilon\frac{2}{9}av_{0})\mathfrak{s}(\epsilon bv_{0}).$

Since by assumption $\frac{1}{9}ab^{-1}\in F^{*2}$ it follows from (17) that
$\mathfrak{s}(\epsilon bv_{0})=\mathfrak{s}(\epsilon\frac{1}{9}av_{0})$ and
the lemma follows. ∎

Once again, the proof above may be regarded as a use of the method of
stationary phase.

For future reference, for $\ell\in\mathbb{Z}$ let

$\mathcal{C}_{\ell}(a,b)=\int_{{\operatorname{val}}(x)=\ell}\psi(ax+bx^{3})\
dx.$

The change of variables $x\mapsto tx$ shows that

(18)
$\mathcal{C}_{\ell}(a,b)=\left|{t}\right|\mathcal{C}_{\ell-{\operatorname{val}}(t)}(ta,t^{3}b),\
\ \ t\in F^{*}.$

As an immediate consequence of Lemma 7.3 we record here that

(19) $\mathcal{C}_{\ell}(a,b)=0\ \ \ \text{whenever}\ \ \
\max(q^{-\ell}\left|{a}\right|,q^{-3\ell}\left|{b}\right|)>q\ \ \ \text{and}\
\ \ \left|{a^{-1}b}\right|\neq q^{2\ell}.$

###  7.5. The comparison of cubic exponential integrals and Kloosterman
integrals

Duke and Iwaniec [DI93] established the following identity between cubic
exponential and Kloosterman sums. Let $\chi$ be an order three character of
$k_{F}^{*}$ ($\chi$ and $\chi^{-1}$ are the only such characters) and $\xi$ a
non-trivial character of $k_{F}$. Then

(20) $\sum_{x\in k_{F}}\xi(x+ax^{3})=\chi(a)^{-1}\sum_{u\in
k_{F}^{*}}\chi(u)\xi(u-3^{-3}a^{-1}u^{-1}).$

This identity, together with the work above which may be regarded as an
application of the method of stationary phase, gives the following key
comparison of cubic exponential and Kloosterman sums.

######  Corollary 7.5.

We have

$\mathcal{C}(a,-3^{-3}c^{-1}d^{-1}a^{3})=(t,c^{-1}d)_{3}\mathcal{K}(t;c,d)$

whenever

  * •

either $\left|{a}\right|=\left|{c}\right|=\left|{d}\right|=q$ and
$3\nmid{\operatorname{val}}(t)$

  * •

or $\left|{a}\right|=\left|{c}\right|=\left|{d}\right|>q$.

###### Proof.

If $\left|{a}\right|=\left|{c}\right|=\left|{d}\right|=q$ then

$\mathcal{C}(a,-3^{-3}c^{-1}d^{-1}a^{3})=q^{-1}\sum_{x\in
k_{F}}\psi(ax-3^{-3}c^{-1}d^{-1}a^{3}x^{3})$

while

$\mathcal{K}(t;c,d)=q^{-1}\sum_{u\in k_{F}^{*}}(t,u)_{3}\psi(cu+du^{-1}).$

If in addition $3\nmid{\operatorname{val}}(t)$ then the character
$(t,\cdot)_{3}$ is not trivial on $\mathcal{O}^{*}$ and we apply (20) with

$\xi(x+\mathfrak{p})=\psi(ax)\ \ \ x\in\mathcal{O}\ \ \ \text{and}\ \ \
\chi(u+\mathfrak{p})=(t,u)_{3},\ \ \ u\in\mathcal{O}^{*}$

to obtain

$\mathcal{C}(a,-3^{-3}c^{-1}d^{-1}a^{3})=q^{-1}(t,-3^{3}cda^{-2})_{3}\sum_{u\in
k_{F}^{*}}(t,u)_{3}\xi(u+cda^{-2}u^{-1}).$

Note that

$(t,-3^{3}cda^{-2})_{3}(t,a^{-1}c)_{3}=(t,c^{-1}d)_{3}$

and therefore the change of variables $u\mapsto a^{-1}cu$ proves the desired
identity.

If $\left|{a}\right|=\left|{c}\right|=\left|{d}\right|>q$ the lemma follows by
comparing the formulas in Lemmas 7.2 and 7.3. ∎

###  7.6. An additional computation involving cubic exponential integrals

We present one additional computation involving a cubic exponential integral.
We will use this evaluation when an orbital integral involves Kloosterman
integrals without a cubic character – that is, integrals $\mathcal{K}(t;c,d)$
with $\left|{c}\right|=\left|{d}\right|=q$ but with
$3\mid{\operatorname{val}}(t)$ – and so the result of Duke-Iwaniec is not
applicable.

######  Lemma 7.6.

For $\left|{a}\right|=\left|{b}\right|\leq q^{-3}$ we have

$3+\left|{a}\right|^{-1}\sum_{k=0}^{2}\mathcal{C}_{{\operatorname{val}}(a)-1}(b^{-1}+a^{-1}\rho^{k},-3^{-3}a^{-1}b^{-1})=\begin{cases}q&-(ab^{-1})^{3}\in
1+\mathfrak{p}\\\ 0&-(ab^{-1})^{3}\not\in 1+\mathfrak{p}.\end{cases}$

###### Proof.

Note that by (18) with $t=\varpi^{-1}a$ we have

$\left|{a}\right|^{-1}\mathcal{C}_{{\operatorname{val}}(a)-1}(b^{-1}+a^{-1}\rho^{k},-3^{-3}a^{-1}b^{-1})=q\,\mathcal{C}_{0}(\varpi^{-1}(ab^{-1}+\rho^{k}),-3^{-3}\varpi^{-3}a^{2}b^{-1})$

and by assumption $-3^{-3}\varpi^{-3}a^{2}b^{-1}\in\mathcal{O}$ so that by
Lemma 7.3 we have

$q\,\mathcal{C}_{0}(\varpi^{-1}(ab^{-1}+\rho^{k}),-3^{-3}\varpi^{-3}a^{2}b^{-1})=\begin{cases}q-1&-ab^{-1}\in\rho^{k}+\mathfrak{p}\\\
-1&-ab^{-1}\not\in\rho^{k}+\mathfrak{p}.\end{cases}$

Since $-ab^{-1}\in\rho^{k}+\mathfrak{p}$ for at most one $k\in\\{0,1,2\\}$ and
this is the case if and only if $(-ab^{-1})^{3}\in 1+\mathfrak{p}$ the lemma
follows. ∎

##  8\. The integral $J(a,b)$

In this section we establish the following formula for $J(a,b)$.

######  Proposition 8.1.

For $\left|{b}\right|\leq\left|{a}\right|$ we have

$(a,b)_{3}\,J(a,b)=$ $\begin{cases}1&\left|{a}\right|=\left|{b}\right|=1\\\
2q&\left|{a}\right|=\left|{b}\right|=q^{-1}\\\
3\left|{a}\right|^{-1}+\left|{ab}\right|^{-1}\sum_{\ell=\lfloor\frac{{\operatorname{val}}(a)+1}{2}\rfloor}^{{\operatorname{val}}(a)-1}\sum_{k=0}^{2}\mathcal{C}_{\ell}(b^{-1}+\rho^{k}a^{-1},-3^{-3}a^{-1}b^{-1})&\left|{a}\right|=\left|{b}\right|\leq
q^{-2}\\\
\left|{b}\right|^{-1}\mathcal{C}(b^{-1},-3^{-3}a^{-1}b^{-1})&\left|{b}\right|<\left|{a}\right|=1\\\
\left|{ab}\right|^{-1}\sum_{k=0}^{2}\mathcal{C}_{\frac{{\operatorname{val}}(a)}{2}}(b^{-1}+\rho^{k}a^{-1},-3^{-3}a^{-1}b^{-1})&\left|{b}\right|<\left|{a}\right|<1\text{
and }2|{\operatorname{val}}(a)\\\ 0&\left|{b}\right|<\left|{a}\right|<1\text{
and }2\nmid{\operatorname{val}}(a)\\\ 0&\left|{a}\right|>1.\end{cases}$

Furthermore,

(21) $J(b,a)=\overline{J(a,b)},\ \ \ a,\,b\in F^{*}.$

###  8.1. Fixing coordinates

By definition

$J(a,b)=\int_{A[a,b]}\kappa(u_{1}g_{a,b}u_{2})^{-1}\psi(u_{1}u_{2})\ du_{1}\
du_{2}$

where

$A[a,b]=\\{(u_{1},u_{2})\in N\times N:u_{1}g_{a,b}u_{2}\in K\\}.$

We fix coordinates on $N\times N$ as follows. In our computation we set

$u_{i}=\begin{pmatrix}1&x_{i}&z_{i}\\\ &1&y_{i}\\\ &&1\end{pmatrix}\in N,\ \ \
i=1,2$

so that

(22)
$u_{1}g_{a,b}u_{2}=\begin{pmatrix}az_{1}&az_{1}x_{2}-a^{-1}bx_{1}&az_{1}z_{2}-a^{-1}bx_{1}y_{2}+b^{-1}\\\
ay_{1}&ay_{1}x_{2}-a^{-1}b&ay_{1}z_{2}-a^{-1}by_{2}\\\
a&ax_{2}&az_{2}\end{pmatrix}.$

We also recall that an element $g=(g_{i,j})\in\operatorname{SL}_{3}(F)$
satisfies $g\in K$ if and only if $g_{i,j}\in\mathcal{O}$ for all $1\leq
i,\,j\leq 3$.

###  8.2. A functional equation

We observe the following properties on the big Bruhat cell.

######  Lemma 8.2.

Let $u_{1},u_{2}\in N$, $a,b\in F^{*}$ and $g=u_{1}g_{a,b}u_{2}$. Then

  * •

$\sigma(g,g^{-1})=1$.

  * •

Consequently, if in addition $g\in K$ then
$\kappa(g^{-1})=\kappa(g)^{-1}=\overline{\kappa(g)}$.

###### Proof.

The second part is immediate from the first since

$1=\kappa(I_{3})=\kappa(gg^{-1})=\kappa(g)\kappa(g^{-1})\sigma(g,g^{-1}).$

For the first part note that $g_{a,b}^{-1}=g_{b,a}$ and

$\sigma(g,g^{-1})=\sigma(g_{a,b},g_{b,a})=1$

by the explicit formulas for $\sigma$ on monomial matrices in [BH87, §2]. ∎

The following consequence of the lemma reduces the computation of $J$ to the
case $\left|{b}\right|\leq\left|{a}\right|$.

######  Corollary 8.3.

We have

$J(b,a)=\overline{J(a,b)},\ \ \ a,\,b\in F^{*}$

###### Proof.

Since $g_{a,b}^{-1}=g_{b,a}$ the map
$(u_{1},u_{2})\mapsto(u_{2}^{-1},u_{1}^{-1})$ maps $A[a,b]$ bijectively to
$A[b,a]$. Since $\psi(u_{2}^{-1}u_{1}^{-1})=\overline{\psi(u_{1}u_{2})}$ the
Corollary is now immediate from Lemma 8.2. ∎

###  8.3. Computation of $\kappa$

We provide a formula for $\kappa(g)$ for almost all $g$ in the intersection of
$K$ with the big Bruhat cell (outside a measure zero set). Throughout the
computation we freely use coordinates of $g$ as in (22).

The fact that $g\mapsto(g,\kappa(g))$ is a splitting of $K$ in
$\widetilde{\operatorname{SL}}_{3}(F)$ is expressed by the equality

$\kappa(g_{1}g_{2})=\kappa(g_{1})\kappa(g_{2})\sigma(g_{1},g_{2}),\ \ \
g_{1},\,g_{2}\in K.$

Our computation uses properties of $\kappa$ which we recall below and an
algorithm of Bump and Hoffstein for computing $\sigma(g_{1},g_{2})$. The
algorithm is based on an explicit Bruhat decomposition of $g_{1}$ and it will
be more convenient to write such a decomposition with factors in
$\operatorname{GL}_{3}(F)$ rather than in $\operatorname{SL}_{3}(F)$. For this
reason we consider $\widetilde{\operatorname{SL}}_{3}(F)$ as a subgroup of the
cubic cover $\widetilde{\operatorname{GL}}_{3}(F)$ of
$\operatorname{GL}_{3}(F)$ constructed in Bump and Hoffstein [BH87] (following
Matsumoto). We then freely use the algorithm for the computation of the two-
cocycle $\sigma$ in [BH87, §2]. In particular, in terms of the Bruhat
decomposition, for $h\in\operatorname{GL}_{3}(F)$ we denote by $R(h)$ the
unique monomial (i.e., scaled permutation) matrix such that $h\in NR(h)N$.
Furthermore, we continue to denote by $\kappa$ an extension to a splitting of
$\operatorname{GL}_{3}(\mathcal{O})$ in $\widetilde{\operatorname{GL}}_{3}(F)$
and freely use the following properties of $\kappa$ (see [KP84, §1]):

  * •

$\kappa$ is trivial on signed permutation matrices and on upper-triangular
matrices in $\operatorname{GL}_{3}(\mathcal{O})$,

  * •

$\kappa(\operatorname{diag}(g,1))=\kappa(\operatorname{diag}(1,g))=\kappa_{2}(g)$,
$g\in\operatorname{GL}_{2}(\mathcal{O})$ where

$\kappa_{2}(g)=\begin{cases}(c,d\det g^{-1})_{3}&0<\left|{c}\right|<1\\\
1&\text{otherwise}\end{cases},\ \ \ g=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{GL}_{2}(\mathcal{O}).$

######  Lemma 8.4.

Let

$g=\begin{pmatrix}0&a&b\\\ u&0&0\\\ x&c&d\end{pmatrix}\in K$

with $x,\,c\neq 0$. Then

$\kappa(g)=(u,x)_{3}\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=\begin{cases}(u,x)_{3}&\left|{c}\right|=1\\\
(c,d)_{3}(u,xc^{-1})_{3}&\text{otherwise.}\end{cases}$

###### Proof.

Note that $g=ksh$ with

$k=\begin{pmatrix}1&&\\\ &1&\\\ &u^{-1}x&1\end{pmatrix},\ \ \
s=\begin{pmatrix}&1&\\\ 1&&\\\ &&1\end{pmatrix},\ \ \
h=\begin{pmatrix}u&0&0\\\ 0&a&b\\\ 0&c&d\end{pmatrix}\in K$

and therefore $\kappa(g)=\kappa(ks)\kappa(h)\sigma(ks,h)$. Clearly,
$\kappa(h)=\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ and $\kappa(ks)=\sigma(k,s)$. We compute that
$\sigma(k,s)=1$ and $\sigma(ks,h)=(u,x)_{3}$ using the algorithm in [BH87,
§2]. ∎

######  Lemma 8.5.

Let

$g=\begin{pmatrix}a&0&b\\\ c&0&d\\\ x&u&0\end{pmatrix}\in K$

with $x,\,c\neq 0$. Then

$\kappa(g)=\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)(c,u^{-1}x)_{3}=\begin{cases}(c,dx)_{3}&\left|{c}\right|<1\\\
(c,x)_{3}&\left|{c}\right|=1.\end{cases}$

###### Proof.

Note that $g=hsk$ with

$h=\begin{pmatrix}a&b&0\\\ c&d&0\\\ 0&0&u\end{pmatrix},\ \ \
s=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix},\ \ \
k=\begin{pmatrix}1&0&0\\\ u^{-1}x&1&0\\\ 0&0&1\end{pmatrix}\in K$

and therefore $\kappa(g)=\kappa(h)\kappa(sk)\sigma(h,sk)$. Clearly,
$\kappa(h)=\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ and $\kappa(sk)=\sigma(s,k)$. We compute that
$\sigma(s,k)=1$ and $\sigma(h,sk)=(c,u^{-1}x)_{3}$ using the algorithm in
[BH87, §2]. ∎

######  Lemma 8.6.

Let

$g=\begin{pmatrix}a&0&b\\\ x&u&0\\\ c&y&d\end{pmatrix}\in K$

with $u\in\mathcal{O}^{*}$ and $x,\,c,\,xy-cu\neq 0$. Then

$\kappa(g)=(x,c)_{3}(c-u^{-1}xy,u^{-1}x)_{3}\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c-u^{-1}xy}&{d}\end{smallmatrix}\right)=\begin{cases}(x,c)_{3}(c-u^{-1}xy,xd)_{3}&\left|{cu-
xy}\right|<1\\\ (x,c)_{3}(c-u^{-1}xy,x)_{3}&\left|{cu-
xy}\right|=1.\end{cases}$

###### Proof.

Note first that $g=sk$ where

$s=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix},\ \ \
k=\begin{pmatrix}a&0&b\\\ c&y&d\\\ x&u&0\end{pmatrix}\in K$

so that $\kappa(g)=\kappa(s)\kappa(k)\sigma(s,k)$. Note that $\kappa(s)=1$ and
we compute that

$\sigma(s,k)=(x,c)_{3}$

using the algorithm in [BH87, §2]. In order to compute $\kappa(k)$ we note
that $k=nhk^{\prime}$ where

$n=\begin{pmatrix}1&0&0\\\ 0&1&u^{-1}y\\\ 0&0&1\end{pmatrix}\in N\cap K,\ \ \
h=\begin{pmatrix}a&b&0\\\ c-u^{-1}xy&d&0\\\ 0&0&u\end{pmatrix},\ \ \
k^{\prime}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ u^{-1}x&1&0\end{pmatrix}\in K$

and therefore
$\kappa(k)=\kappa(hk^{\prime})=\kappa(h)\kappa(k^{\prime})\sigma(h,k^{\prime})$.
Clearly, $\kappa(h)=\kappa_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c-u^{-1}xy}&{d}\end{smallmatrix}\right)$ (note that $\det h=\det g=1$).
Furthermore, it is easy to see that $\kappa(sk^{\prime})=1$ and therefore also
that $\kappa(k^{\prime})=\sigma(s,sk^{\prime})=1$. We further compute that

$\sigma(h,k^{\prime})=(c-u^{-1}xy,u^{-1}x)_{3}$

using the algorithm in [BH87, §2]. The lemma follows. ∎

######  Lemma 8.7.

Let $g=u_{1}g_{a,b}u_{2}\in K$ with $a,\,b\in F^{*}$ and $u_{1},\,u_{2}\in N$.

  1. (1)

if $\left|{a}\right|=1=\left|{b}\right|$ then $\kappa(g)=1$.

  2. (2)

if $\left|{a}\right|=1>\left|{b}\right|$ then
$\kappa(g)=(b,a)_{3}(a^{-1}b,y_{2})_{3}$.

  3. (3)

if $\left|{a}\right|,\,\left|{b}\right|<1$ then at most one of $ay_{1}$ and
$ax_{2}$ is in $\mathcal{O}^{*}$ and

    1. (a)

if $\left|{ay_{1}}\right|=1$ then
$\kappa(g)=(b,a)_{3}(y_{1}y_{2},ab^{-1})_{3}(y_{2},y_{1})_{3}$

    2. (b)

if $\left|{ax_{2}}\right|=1$ then
$\kappa(g)=(b,ax_{2})_{3}(z_{2}x_{2}^{-1}-y_{2},b^{-1}ax_{2})_{3}$

    3. (c)

if $\left|{ay_{1}}\right|,\,\left|{ax_{2}}\right|<1$ then
$ay_{1}x_{2}-a^{-1}b\in\mathcal{O}^{*}$ and if in addition $y_{1}\neq 0$ then

$\kappa(g)=(b,a)_{3}(y_{1},ab^{-1})_{3}(ay_{1}(x_{2}y_{2}-z_{2}),ay_{1}x_{2}-a^{-1}b)_{3}(b,x_{2}y_{2}-z_{2})_{3}.$

###### Proof.

If $\left|{a}\right|=1=\left|{b}\right|$ then $u_{1},\,u_{2}\in K$ and
therefore $\kappa(g)=\kappa(g_{a,b})=1$. If
$\left|{a}\right|=1>\left|{b}\right|$ then
$x_{2},\,y_{1},\,z_{1},\,z_{2}\in\mathcal{O}$ and therefore
$\kappa(g)=\kappa\left(\begin{smallmatrix}{}&{g_{0}}\\\
{a}&{}\end{smallmatrix}\right)$ where
$g_{0}=\left(\begin{smallmatrix}{-a^{-1}bx_{1}}&{b^{-1}-x_{1}y_{2}a^{-1}b}\\\
{-a^{-1}b}&{-a^{-1}by_{2}}\end{smallmatrix}\right)$. Note that
$\left(\begin{smallmatrix}{}&{g_{0}}\\\
{a}&{}\end{smallmatrix}\right)=\left(\begin{smallmatrix}{g_{0}}&{}\\\
{}&{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}{}&{I_{2}}\\\
{a}&{}\end{smallmatrix}\right)$ and that
$\sigma(\left(\begin{smallmatrix}{g_{0}}&{}\\\
{}&{1}\end{smallmatrix}\right),\left(\begin{smallmatrix}{}&{I_{2}}\\\
{a}&{}\end{smallmatrix}\right))=1$. Therefore

$\kappa\left(\begin{smallmatrix}{}&{g_{0}}\\\
{a}&{}\end{smallmatrix}\right)=\kappa_{2}(g_{0})=(-a^{-1}b,-by_{2})_{3}=(b,a)_{3}(a^{-1}b,y_{2})_{3}.$

For the rest of the proof assume that $\left|{a}\right|,\,\left|{b}\right|<1$.
Since the (2,2)-entry of $g$ is $ay_{1}x_{2}-a^{-1}b\in\mathcal{O}$ we cannot
have both $ay_{1}$ and $ax_{2}$ in $\mathcal{O}^{*}$. If
$\left|{ay_{1}}\right|=1$ then

$k=\begin{pmatrix}1&\frac{a^{-1}b-y_{1}ax_{2}}{y_{1}a}&\frac{a^{-1}by_{2}-y_{1}z_{2}a}{y_{1}a}\\\
&1&\\\ &&1\end{pmatrix},\ \ \
k^{\prime}=\begin{pmatrix}1&-\frac{z_{1}a}{y_{1}a}&\\\ &1&\\\
&&1\end{pmatrix}\in N\cap K,$

so that $\kappa(g)=\kappa(k^{\prime}gk)$. Note that

$k^{\prime}gk=\begin{pmatrix}0&\frac{(z_{1}-x_{1}y_{1})b}{y_{1}a}&\frac{(z_{1}-x_{1}y_{1})y_{2}b}{y_{1}a}+b^{-1}\\\
y_{1}a&0&0\\\ a&\frac{b}{y_{1}a}&\frac{y_{2}b}{y_{1}a}\end{pmatrix}$

and from Lemma 8.4 it follows that

$\kappa(g)=(b(ay_{1})^{-1},b(ay_{1})^{-1}y_{2})_{3}(ay_{1},ab^{-1}ay_{1})_{3}=(b,a)_{3}(y_{1}y_{2},ab^{-1})_{3}(y_{2},y_{1})_{3}.$

If $\left|{ax_{2}}\right|=1$ then

$k=\begin{pmatrix}1&&\\\ &1&\frac{-z_{2}a}{x_{2}a}\\\ &&1\end{pmatrix},\ \ \
k^{\prime}=\begin{pmatrix}1&&\frac{x_{1}a^{-1}b-z_{1}x_{2}a}{x_{2}a}\\\
&1&\frac{a^{-1}b-y_{1}x_{2}a}{x_{2}a}\\\ &&1\end{pmatrix}\in N\cap K,$

so that $\kappa(g)=\kappa(k^{\prime}gk)$. Note that

$k^{\prime}gk=\begin{pmatrix}x_{1}x_{2}^{-1}a^{-1}b&0&(z_{2}x_{2}^{-1}-y_{2})x_{1}a^{-1}b+b^{-1}\\\
x_{2}^{-1}a^{-1}b&0&(z_{2}x_{2}^{-1}-y_{2})a^{-1}b\\\ a&x_{2}a&0\end{pmatrix}$

and from Lemma 8.5 it follows that

$\kappa(g)=(x_{2}^{-1}a^{-1}b,(z_{2}x_{2}^{-1}-y_{2})b)_{3}=(b,ax_{2})_{3}(z_{2}x_{2}^{-1}-y_{2},b^{-1}ax_{2})_{3}.$

Finally, assume that $\left|{ax_{2}}\right|,\,\left|{ay_{1}}\right|<1$. Any
two by two minor of an element of $K$ must have an entry in $\mathcal{O}^{*}$.
Since the (1,3)-minor of $g$ is
$\left(\begin{smallmatrix}{ay_{1}}&{ay_{1}x_{2}-a^{-1}b}\\\
{a}&{ax_{2}}\end{smallmatrix}\right)$, the condition $g\in K$ implies that
$ay_{1}x_{2}-a^{-1}b\in\mathcal{O}^{*}$. Therefore

$k=\begin{pmatrix}1&&\\\
&1&\frac{a^{-1}by_{2}-y_{1}z_{2}a}{y_{1}ax_{2}-a^{-1}b}\\\ &&1\end{pmatrix},\
\ \
k^{\prime}=\begin{pmatrix}1&\frac{x_{1}a^{-1}b-z_{1}x_{2}a}{y_{1}ax_{2}-a^{-1}b}&\\\
&1&\\\ &&1\end{pmatrix}\in N\cap K$

and $\kappa(g)=\kappa(k^{\prime}gk)$. Note that $k^{\prime}gk$ has the form

$k^{\prime}gk=\begin{pmatrix}*&0&*\\\ y_{1}a&y_{1}ax_{2}-a^{-1}b&0\\\
a&x_{2}a&\frac{(x_{2}y_{2}-z_{2})b}{y_{1}ax_{2}-a^{-1}b}\end{pmatrix}.$

Assuming furthermore that $y_{1}\neq 0$ the rest of the computation follows
from Lemma 8.6. ∎

###  8.4. Evaluation of $J(a,b)$: some simple cases

####  8.4.1. The case $\max(\left|{a}\right|,\left|{b}\right|)>1$

######  Lemma 8.8.

If $\max(\left|{a}\right|,\left|{b}\right|)>1$ then $J(a,b)=0$.

###### Proof.

For $g=(g_{i,j})\in\operatorname{SL}_{3}(F)$ let $\Delta_{1}(g)=g_{3,1}$ and
$\Delta_{2}(g)=\det\left(\begin{smallmatrix}{g_{2,1}}&{g_{2,2}}\\\
{g_{3,1}}&{g_{3,2}}\end{smallmatrix}\right)$. Note that $\Delta_{1}(g)=a$ and
$\Delta_{2}(g)=b$ for every $g\in Ng_{a,b}N$. Consequently, $A[a,b]$ is empty
and therefore $J(a,b)=0$ unless $a,b\in\mathcal{O}$. ∎

####  8.4.2. The case $\left|{a}\right|=\left|{b}\right|=1$

######  Lemma 8.9.

If $a,b\in\mathcal{O}^{*}$ then

$(a,b)_{3}\,J(a,b)=1.$

###### Proof.

If $a,b\in\mathcal{O}^{*}$ then $(a,b)_{3}=1$ and it easily follows from (22)
that $A[a,b]=(N\cap K)\times(N\cap K)$. Since $\kappa$ is bi-$(N\cap
K)$-invariant, $\kappa(g_{a,b})=1$ and $\psi$ is trivial on $N\cap K$ the
lemma follows. ∎

####  8.4.3. The case $1=\left|{a}\right|>\left|{b}\right|$

######  Lemma 8.10.

If $1=\left|{a}\right|>\left|{b}\right|$ then

$(a,b)_{3}\,J(a,b)=\left|{b}\right|^{-1}\mathcal{C}(b^{-1},-3^{-3}a^{-1}b^{-1}).$

###### Proof.

It follows from (22) and Lemma 8.7 part (2) that

$J(a,b)=(a,b)_{3}\int(ab^{-1},y_{2})_{3}\psi(x_{1}+y_{2})\ dx_{1}\ dy_{2}$

where the integral is over $x_{1}$ and $y_{2}$ such that

$x_{1}b,y_{2}b,b^{-1}-x_{1}y_{2}a^{-1}b\in\mathcal{O}$

or equivalently such that

$y_{2}b\in\mathcal{O}^{*},\ x_{1}\in(y_{2}b)^{-1}ab^{-1}+\mathcal{O}.$

Integrating over $x_{1}$ we obtain that

$J(a,b)=(a,b)_{3}\int_{\left|{y_{2}}\right|=\left|{b}\right|^{-1}}(ab^{-1},y_{2})_{3}\psi(y_{2}+ab^{-2}y_{2}^{-1})\
dy_{2}$

and after the change of variables $y_{2}\mapsto b^{-1}y_{2}$ this becomes

$\left|{b}\right|^{-1}\int_{\mathcal{O}^{*}}(ab^{-1},y_{2})_{3}\psi(b^{-1}y_{2}+ab^{-1}y_{2}^{-1})\
dy_{2}=\left|{b}\right|^{-1}\mathcal{K}(ab^{-1};b^{-1},ab^{-1}).$

The lemma now follows from Corollary 7.5. ∎

###  8.5. Calculation of $J(a,b)$ when
$1>\left|{a}\right|\geq\left|{b}\right|$: subdivision and evaluations of
$J_{1}$, $J_{2}$

Assume now that $1>\left|{a}\right|\geq\left|{b}\right|$ and consider the
conditions

  1. (1)

$\left|{ay_{1}}\right|=1$

  2. (2)

$\left|{ax_{2}}\right|=1$

  3. (3)

$\left|{ay_{1}}\right|,\,\left|{ax_{2}}\right|<1$.

Let

$J_{i}(a,b)=\int_{A_{i}[a,b]}\kappa(u_{1}g_{a,b}u_{2})^{-1}\psi(u_{1}u_{2})\
du_{1}\ du_{2}$

where $A_{i}[a,b]$ is the intersection of $A[a,b]$ with the set defined by the
condition $(i)$ for $i=1,2,3$. By Lemma 8.7 we have

(23) $J(a,b)=J_{1}(a,b)+J_{2}(a,b)+J_{3}(a,b).$

####  8.5.1. Computation of $J_{1}$

######  Lemma 8.11.

Let $1>\left|{a}\right|\geq\left|{b}\right|$ then

$(a,b)_{3}\,J_{1}(a,b)=\begin{cases}q&\left|{a}\right|=\left|{b}\right|=q^{-1}\\\
0&\text{otherwise.}\end{cases}$

###### Proof.

Note that according to Lemma 8.7 we have

$J_{1}(a,b)=(a,b)_{3}\int(y_{1}y_{2},a^{-1}b)_{3}(y_{1},y_{2})_{3}\psi(x_{1}+x_{2}+y_{1}+y_{2})\
dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\ dz_{1}\ dz_{2}$

where the integral is over

$ay_{1}\in\mathcal{O}^{*},\,az_{1},az_{2},ax_{2},y_{1}x_{2}a-a^{-1}b,y_{1}z_{2}a-y_{2}a^{-1}b,\,z_{1}x_{2}a-x_{1}a^{-1}b,z_{1}z_{2}a-x_{1}y_{2}a^{-1}b+b^{-1}\in\mathcal{O}$

or equivalently

(24) $ay_{1}\in\mathcal{O}^{*},\
az_{1},\,az_{2},\,x_{2},\,y_{1}z_{2}a-y_{2}a^{-1}b,\,x_{1}a^{-1}b,\,z_{1}z_{2}a-x_{1}y_{2}a^{-1}b+b^{-1}\in\mathcal{O}.$

We first show that $J_{1}(a,b)=0$ unless
$\left|{a}\right|=\left|{b}\right|=q^{-1}$. Indeed, for
$u,\,v\in\mathcal{O}^{*}$ the variable change

$(x_{1},y_{1},z_{1},y_{2},z_{2})\mapsto(ux_{1},vy_{1},uvz_{1},u^{-1}y_{2},u^{-1}v^{-1}z_{2})$

preserves the domain of integration. Averaging over $u,\,v$ we see that the
integral $J_{1}(a,b)$ factors through

$\int_{\mathcal{O}^{*}}(v,a^{-1}by_{2})_{3}\psi(vy_{1})\
dv\int_{\mathcal{O}^{*}}(u,ab^{-1}y_{1})_{3}\psi(ux_{1}+u^{-1}y_{2})\ du.$

Note that in the domain of integration
$\left|{y_{1}}\right|=\left|{a}\right|^{-1}>1$ and for such $y_{1}$ it follows
from Lemma 7.2 that $\int_{\mathcal{O}^{*}}(v,a^{-1}by_{2})_{3}\psi(vy_{1})\
dv=0$ unless $\left|{a}\right|=q^{-1}$. Next, assume that
$\left|{b}\right|<\left|{a}\right|=q^{-1}$. Note that in the domain (24) we
have

(25) $\left|{y_{2}}\right|=\left|{b}\right|^{-1}>q\ \ \ \text{and}\ \ \
\left|{x_{1}}\right|=\left|{ab^{-1}}\right|<\left|{y_{2}}\right|.$

Indeed, in the domain of integration

$\left|{ay_{1}z_{2}}\right|,\,\left|{az_{1}z_{2}}\right|\leq\left|{a}\right|^{-1}<\left|{b}\right|^{-1}$

and consequently,

$\left|{a^{-1}by_{2}}\right|\leq\left|{a}\right|^{-1}\ \ \ \text{and}\ \ \
\left|{a^{-1}bx_{1}y_{2}}\right|=\left|{b}\right|^{-1}.$

Combined with the condition $a^{-1}bx_{1}\in\mathcal{O}$ we obtain (25).
Consequently, it follows from Lemma 7.2 that
$\int_{\mathcal{O}^{*}}(u,ab^{-1}y_{1})_{3}\psi(ux_{1}+u^{-1}y_{2})\ du=0$.

Assume now that $\left|{a}\right|=\left|{b}\right|=q^{-1}$. The domain of
integration (24) is equivalently characterized by

$\left|{y_{1}}\right|=\left|{y_{2}}\right|=q,\,x_{2},\,az_{1}\in\mathcal{O},\,z_{2}\in
a^{-2}by_{1}^{-1}y_{2}+\mathcal{O},\,x_{1}\in
ab^{-1}y_{2}^{-1}(az_{1}z_{2}+b^{-1})+\mathfrak{p}$

and in particular, the integrand is independent of $x_{1},\,x_{2},\,z_{1}$ and
$z_{2}$ in this domain. After integrating over $x_{1}$ and $x_{2}$ we further
integrate over $z_{1}$ and $z_{2}$. After the variable change

$(y_{1},y_{2})\mapsto(a^{-1}y_{1},a^{-1}y_{2})$

we have

$J_{1}(a,b)=(b,a)_{3}q^{2}\int_{\mathcal{O}^{*}}\int_{\mathcal{O}^{*}}(a^{-1},y_{2}y_{1}^{-1})_{3}\psi(a^{-1}(y_{1}+y_{2}))\
dy_{1}\ dy_{2}.$

The lemma now follows from Lemma 7.1. ∎

####  8.5.2. Computation of $J_{2}$

######  Lemma 8.12.

Let $1>\left|{a}\right|\geq\left|{b}\right|$ then

$(a,b)_{3}\,J_{2}(a,b)=\begin{cases}q&\left|{a}\right|=\left|{b}\right|=q^{-1}\\\
0&\text{otherwise.}\end{cases}$

###### Proof.

Note that according to Lemma 8.7 we have

$J_{2}(a,b)=(a,b)_{3}\int(x_{2},b)_{3}(b^{-1}ax_{2},z_{2}x_{2}^{-1}-y_{2})_{3}\psi(x_{1}+x_{2}+y_{1}+y_{2})\
dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\ dz_{1}\ dz_{2}$

where the integral is over

$ax_{2}\in\mathcal{O}^{*},\
az_{1},\,az_{2},\,ay_{1},\,y_{1}x_{2}a-a^{-1}b,\,y_{1}z_{2}a-y_{2}a^{-1}b,\,z_{1}x_{2}a-x_{1}a^{-1}b,\,z_{1}z_{2}a-x_{1}y_{2}a^{-1}b+b^{-1}\in\mathcal{O}.$

Note further that

$z_{1}z_{2}a-x_{1}y_{2}a^{-1}b+b^{-1}=x_{2}^{-1}z_{2}(z_{1}x_{2}a-x_{1}a^{-1}b)+a^{-1}bx_{1}(x_{2}^{-1}z_{2}-y_{2})+b^{-1}$

and the domain of integration is equivalently characterized by

$ax_{2}\in\mathcal{O}^{*},\
az_{1},\,az_{2},\,y_{1},\,y_{2}a^{-1}b,\,z_{1}x_{2}a-x_{1}a^{-1}b,\,a^{-1}bx_{1}(x_{2}^{-1}z_{2}-y_{2})+b^{-1}\in\mathcal{O}.$

Since the variable change

$(x_{1},\,x_{2},\,y_{2},\,z_{1},\,z_{2})\mapsto(u^{-1}x_{1},\,vx_{2},\,uy_{2},\,u^{-1}v^{-1}z_{1},\,uvz_{2})$

preserves the domain of integration for every $u,\,v\in\mathcal{O}^{*}$ an
argument analogous to the proof of Lemma 8.11 shows that $J_{2}(a,b)=0$ unless
$\left|{a}\right|=\left|{b}\right|=q^{-1}$.

Assume now that $\left|{a}\right|=\left|{b}\right|=q^{-1}$. After the variable
change $y_{2}\mapsto y_{2}+x_{2}^{-1}z_{2}$ we have

$J_{2}(a,b)=(a,b)_{3}\int(x_{2},b)_{3}(b^{-1}ax_{2},y_{2})_{3}\psi(x_{1}+x_{2})\
dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\ dz_{1}\ dz_{2}$

where the integral is over

$\left|{x_{2}}\right|=q,\
az_{1},\,az_{2},\,y_{1},\,y_{2},\,z_{1}x_{2}a-x_{1}a^{-1}b,\,a^{-1}bx_{1}y_{2}-b^{-1}\in\mathcal{O}$

or equivalently

$\left|{x_{1}}\right|=\left|{x_{2}}\right|=q,\,az_{2},\,y_{1}\in\mathcal{O},\,z_{1}\in
a^{-2}bx_{2}^{-1}x_{1}+\mathcal{O},\,y_{2}\in ab^{-2}x_{1}^{-1}+\mathfrak{p}.$

Note that in this domain $ax_{2},\,y_{2}\in\mathcal{O}^{*}$ and therefore

$(x_{2},b)_{3}(b^{-1}ax_{2},y_{2})_{3}=(x_{2}y_{2},b)_{3}=(ax_{1}^{-1}x_{2},b)_{3}.$

That is, the integrand is independent of $y_{1},\,z_{1},\,y_{2},\,z_{2}$ in
the domain of integration. Integrating over these variables and applying the
variable change

$(x_{1},x_{2})\mapsto(a^{-1}x_{1},a^{-1}x_{2})$

we obtain that

$J_{2}(a,b)=(b,a)_{3}q^{2}\int_{\mathcal{O}^{*}}\int_{\mathcal{O}^{*}}(b,x_{1}x_{2}^{-1})_{3}\psi(a^{-1}(x_{1}+x_{2}))\
dx_{1}\ dx_{2}.$

The lemma now follows from Lemma 7.1. ∎

###  8.6. The calculation of $J(a,b)$ when
$1>\left|{a}\right|\geq\left|{b}\right|$: evaluation of $J_{3}$

We turn to the evaluation of the third summand that contributes to $J(a,b)$.
Recall that by definition

$J_{3}(a,b)=\int_{A_{3}[a,b]}\kappa(u_{1}g_{a,b}u_{2})^{-1}\psi(u_{1}u_{2})\
du_{1}\ du_{2}$

where $A_{3}[a,b]$ is defined by the conditions (coordinates are as in (22))

  1. a)

$ay_{1}\in\mathfrak{p}$

  2. b)

$ax_{2}\in\mathfrak{p}$

  3. c)

$az_{1}\in\mathcal{O}$

  4. d)

$az_{2}\in\mathcal{O}$

  5. e)

$y_{1}x_{2}a-a^{-1}b\in\mathcal{O}$

  6. f)

$y_{1}z_{2}a-y_{2}a^{-1}b\in\mathcal{O}$

  7. g)

$z_{1}x_{2}a-x_{1}a^{-1}b\in\mathcal{O}$

  8. h)

$z_{1}z_{2}a-x_{1}y_{2}a^{-1}b+b^{-1}\in\mathcal{O}$.

In the proof of the following simple lemma we refer to this list of defining
conditions.

######  Lemma 8.13.

Let $a,b\in F^{*}$ be such that $\left|{a}\right|<1$.

  1. (1)

Every element of $A_{3}[a,b]$ satisfies $bx_{1}\in\mathfrak{p}$.

  2. (2)

Every element of $A_{3}[a,b]$ satisfies
$ay_{1}x_{2}-a^{-1}b,az_{1},az_{2}\in\mathcal{O}^{*}$.

###### Proof.

It follows from b and c that
$\left|{az_{1}x_{2}}\right|<\left|{z_{1}}\right|\leq\left|{a}\right|^{-1}$.
Part 1 of the lemma therefore follows from g. For an element of $A_{3}[a,b]$
the matrix (22) reduces $\bmod\mathfrak{p}$ to upper triangular form. Part 2
follows. ∎

Note that the subset of $A_{3}[a,b]$ with $y_{1}=0$ is of measure zero (in
fact, it is empty if $\left|{b}\right|<\left|{a}\right|$). It therefore
follows from Lemma 8.7 that

(26)
$J_{3}(a,b)=(a,b)_{3}\int_{A_{3}^{\prime}[a,b]}(y_{1},a^{-1}b)_{3}(ay_{1}x_{2}-a^{-1}b,ay_{1}(x_{2}y_{2}-z_{2}))_{3}(x_{2}y_{2}-z_{2},b)_{3}\\\
\psi(x_{1}+x_{2}+y_{1}+y_{2})\ dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\ dz_{1}\ dz_{2}$

where $A_{3}^{\prime}[a,b]$ is the subset of elements of $A_{3}[a,b]$
satisfying $y_{1}\neq 0$.

####  8.6.1. Integration over $z_{1}$

We begin the computation of $J_{3}$ by integrating over the single variable
$z_{1}$. We use the coordinates $(x_{1},x_{2},y_{1},y_{2},z_{2})$ in $F^{5}$.
For a set $D$ in $F^{5}$ let $D^{\prime}$ be the set of elements in $D$ such
that $y_{1}\neq 0$. Let $D[a,b]$ be the domain defined by the conditions

  1. a)

$ay_{1}\in\mathfrak{p}$

  2. b)

$ax_{2}\in\mathfrak{p}$

  3. c)

$az_{2}\in\mathcal{O}^{*}$

  4. d)

$y_{1}-a^{-1}by_{2}\in\mathcal{O}$

  5. e)

$x_{2}-a^{-1}bx_{1}\in\mathcal{O}$

  6. f)

$\left|{1-bx_{1}y_{2}}\right|=\left|{ab^{-1}}\right|$.

In the rest of the section we refer to this list of conditions.

######  Lemma 8.14.

Let $\left|{b}\right|\leq\left|{a}\right|<1$. We have

$J_{3}(a,b)=(a,b)_{3}\left|{a^{-1}b}\right|\int_{D^{\prime}[a,b]}(y_{1},ab^{-1}(ay_{1}x_{2}-a^{-1}b))_{3}(ax_{2}y_{2}-1,ay_{1}x_{2}-a^{-1}b)_{3}\\\
(b,az_{2}(ax_{2}y_{2}-1))_{3}\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})dx_{1}\
dx_{2}\ dy_{1}\ dy_{2}\ dz_{2}.$

###### Proof.

Applying the variable change

$x_{1}\mapsto az_{1}x_{1},\ \ \ y_{2}\mapsto az_{2}y_{2}$

to (26) and Lemma 8.13 we have

$J_{3}(a,b)=(a,b)_{3}\int(y_{1},a^{-1}b)_{3}(ay_{1}x_{2}-a^{-1}b,y_{1}(ax_{2}y_{2}-1))_{3}(z_{2}(ax_{2}y_{2}-1),b)_{3}\\\
\psi(az_{1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})\ dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\
dz_{1}\ dz_{2}$

where integration is now over the domain defined by $y_{1}\neq 0$ and

$ay_{1},\,ax_{2}\in\mathfrak{p},\,az_{1},\,az_{2}\in\mathcal{O}^{*},\,ay_{1}x_{2},\,y_{1}-a^{-1}by_{2},\,x_{2}-a^{-1}bx_{1},az_{1}z_{2}(1-bx_{1}y_{2})+b^{-1}\in\mathcal{O}.$

Note that the conditions

$az_{1},\,az_{2}\in\mathcal{O}^{*},\,az_{1}z_{2}(1-bx_{1}y_{2})+b^{-1}\in\mathcal{O}$

are equivalent to the conditions

$az_{2}\in\mathcal{O}^{*},\left|{1-bx_{1}y_{2}}\right|=\left|{ab^{-1}}\right|,\,z_{1}\in-(az_{2})^{-1}(1-bx_{1}y_{2})^{-1}b^{-1}+a^{-1}b\mathcal{O}.$

Furthermore, it follows from Lemma 8.13 that
$bx_{1}\in\mathfrak{p}\subseteq\mathcal{O}$ and therefore in the domain of
integration

$\psi(az_{1}x_{1})=\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1})$

so that the integrand is independent of $z_{1}$ in its $a^{-1}b\mathcal{O}$
coset. Integrating over $z_{1}$ we obtain

$J_{3}(a,b)=(a,b)_{3}\left|{a^{-1}b}\right|\int(y_{1},a^{-1}b)_{3}(ay_{1}x_{2}-a^{-1}b,y_{1}(ax_{2}y_{2}-1))_{3}(z_{2}(ax_{2}y_{2}-1),b)_{3}\\\
\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})\ dx_{1}\
dx_{2}\ dy_{1}\ dy_{2}\ dz_{2}$

where the integral is over the subdomain of elements of $D^{\prime}[a,b]$ that
satisfy $ay_{1}x_{2}\in\mathcal{O}$. However, this condition is satisfied by
any element of $D[a,b]$. Indeed, if for an element of $D^{\prime}[a,b]$ either
$x_{2}$ or $y_{1}$ is in $\mathcal{O}$ this follows from the conditions a and
b, otherwise it follows from d and e that
$\left|{x_{2}}\right|=\left|{a^{-1}bx_{1}}\right|$ and
$\left|{y_{1}}\right|=\left|{a^{-1}by_{2}}\right|$ and therefore
$\left|{ay_{1}x_{2}}\right|=\left|{a^{-1}b^{2}x_{1}y_{2}}\right|$. The
equality f implies that
$\left|{bx_{1}y_{2}}\right|\leq\left|{ab^{-1}}\right|$. The statement and the
lemma follow. ∎

####  8.6.2. On the domain $D[a,b]$

Let $D_{0}[a,b]$ be the subset of elements in $D[a,b]$ that satisfy either
$a^{-1}bx_{1}\in\mathcal{O}$ or $a^{-1}by_{2}\in\mathcal{O}$, or equivalently
(by d and e) either $x_{2}\in\mathcal{O}$ or $y_{1}\in\mathcal{O}$ and let
$D_{>0}[a,b]=D[a,b]\setminus D_{0}[a,b]$ be its complement. Further let
$D_{0x}[a,b]$(resp. $D_{0y}[a,b]$) be the subset of elements in $D[a,b]$ that
satisfy $x_{2}\in\mathcal{O}$ (resp. $y_{1}\in\mathcal{O}$).

######  Lemma 8.15.

Let $\left|{b}\right|\leq\left|{a}\right|<1$.

  1. (1)

The set $D[a,b]$ is preserved by the variable interchange

(27) $(x_{1},y_{1})\leftrightarrow(y_{2},x_{2}).$

  2. (2)

Every element of $D[a,b]$ satisfies $by_{2},\,bx_{1}\in\mathfrak{p}$.

  3. (3)

The set $D_{0}[a,b]$ is empty unless $\left|{a}\right|=\left|{b}\right|$.

  4. (4)

If $\left|{a}\right|=\left|{b}\right|$ then the set $D_{0x}[a,b]$ is defined
by the conditions

(28) $x_{1},\,x_{2}\in\mathcal{O},ay_{1}\in\mathfrak{p},\,y_{2}\in
ab^{-1}y_{1}+\mathcal{O},\,az_{2}\in\mathcal{O}^{*}.$

  5. (5)

If $\left|{a}\right|=\left|{b}\right|$ then the set $D_{0y}[a,b]$ is defined
by the conditions

(29) $y_{1},\,y_{2}\in\mathcal{O},ax_{2}\in\mathfrak{p},\,x_{1}\in
ab^{-1}x_{2}+\mathcal{O},\,az_{2}\in\mathcal{O}^{*}.$

  6. (6)

If $\left|{a}\right|=\left|{b}\right|$ then every element of
$D^{\prime}_{0}[a,b]$ satisfies

$(y_{1},a^{-1}b)_{3}(ay_{1}x_{2}-a^{-1}b,y_{1}(ax_{2}y_{2}-1))_{3}(z_{2}(ax_{2}y_{2}-1),b)_{3}=(z_{2},b)_{3}.$

  7. (7)

For every element of $D^{\prime}_{>0}[a,b]$ we have

$(y_{1},a^{-1}b)_{3}(ay_{1}x_{2}-a^{-1}b,y_{1}(ax_{2}y_{2}-1))_{3}(z_{2}(ax_{2}y_{2}-1),b)_{3}=(x_{1},bx_{1}y_{2}-1)_{3}(z_{2},b)_{3}.$

###### Proof.

Part 1 is straightforward from the defining properties a–f of $D[a,b]$. It
follows from a and d that an element of $D[a,b]$ satisfies $a^{-1}by_{2}\in
a^{-1}\mathfrak{p}$ or equivalently $by_{2}\in\mathfrak{p}$. Part 2 now
follows further applying the symmetry (27).

By part 2 an element of $D_{0x}[a,b]$ satisfies $bx_{1}y_{2}\in
ay_{2}\mathcal{O}\subseteq ab^{-1}\mathfrak{p}$. If
$\left|{b}\right|<\left|{a}\right|$ this contradicts f. By the symmetry (27)
if $\left|{b}\right|<\left|{a}\right|$ then $D_{0y}[a,b]$ is also empty. Part
3 follows.

Suppose that $\left|{a}\right|=\left|{b}\right|$. It is easy to see that an
element of $D_{0x}[a,b]$ satisfies the conditions (28). For the reverse
inclusion note that an element of $F^{5}$ satisfying (28) satisfies $by_{2}\in
ay_{1}+b\mathcal{O}\subseteq\mathfrak{p}$ and therefore also
$bx_{1}y_{2}\in\mathfrak{p}$. Consequently,
$\left|{1-bx_{1}y_{2}}\right|=1=\left|{ab^{-1}}\right|$. Part 4 now easily
follows. Part 5 follows from part 4 and the symmetry (27). Every element of
$D_{0}[a,b]$ satisfies $ay_{1}x_{2},ax_{2}y_{2}\in\mathfrak{p}$ while, by
assumption, $a^{-1}b\in\mathcal{O}^{*}$ and therefore

$(t,ay_{1}x_{2}-a^{-1}b)_{3}=(t,a^{-1}b)_{3}\ \ \ \text{and}\ \ \
(t,ax_{2}y_{2}-1)_{3}=1,\ \ \ t\in F^{*}.$

Part 6 now follows by properties of the Hilbert symbol.

For part 7 we return to the more general setting
$\left|{b}\right|\leq\left|{a}\right|<1$. For an element of $D_{>0}[a,b]$ the
conditions d and e imply that

$\left|{x_{2}}\right|=\left|{a^{-1}bx_{1}}\right|>1\ \ \ \text{and}\ \ \
\left|{y_{1}}\right|=\left|{a^{-1}by_{2}}\right|>1.$

In particular, from conditions d and e it follows that

$(ay_{1},t)_{3}=(by_{2},t)_{3},\ \ \ t\in F^{*}$

and it easily follows that

$ay_{1}x_{2}-a^{-1}b\in a^{-1}b(bx_{1}y_{2}-1)+\mathfrak{p}\ \ \ \text{and}\ \
\ ax_{2}y_{2}-1\in bx_{1}y_{2}-1+ab^{-1}\mathfrak{p}.$

Combined with (f) it follows that

$(t,ay_{1}x_{2}-a^{-1}b)_{3}=(t,a^{-1}b(bx_{1}y_{2}-1))_{3}\ \ \ \text{and}\ \
\ (t,ax_{2}y_{2}-1)_{3}=(t,bx_{1}y_{2}-1)_{3},\ \ \ t\in F^{*}.$

Part 7 now follows from the properties of Hilbert symbols. ∎

Based on Lemma 8.15, if $\left|{b}\right|\leq\left|{a}\right|<1$ we have

(30) $J_{3}(a,b)=\mathcal{J}_{0}(a,b)+\mathcal{J}_{>0}(a,b)$

where

(31)
$\mathcal{J}_{>0}(a,b)=\left|{a^{-1}b}\right|\int_{D^{\prime}_{>0}[a,b]}(x_{1},bx_{1}y_{2}-1)_{3}(az_{2},b)_{3}\\\
\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})\ dx_{1}\
dx_{2}\ dy_{1}\ dy_{2}\ dz_{2}$

and $\mathcal{J}_{0}(a,b)=0$ unless $\left|{a}\right|=\left|{b}\right|$ in
which case

$\mathcal{J}_{0}(a,b)=\int_{D^{\prime}_{0}[a,b]}(az_{2},b)_{3}\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})\
dx_{1}\ dx_{2}\ dy_{1}\ dy_{2}\ dz_{2}.$

####  8.6.3. The computation of $\mathcal{J}_{0}$

######  Lemma 8.16.

Let $\left|{a}\right|=\left|{b}\right|<1$. We have

$(a,b)_{3}\mathcal{J}_{0}(a,b)=\begin{cases}2\left|{a}\right|^{-1}&3\nmid{\operatorname{val}}(a)>1\\\
(1+q^{-1})\left|{a}\right|^{-1}&3|{\operatorname{val}}(a)\\\
0&{\operatorname{val}}(a)=1.\end{cases}$

###### Proof.

Suppose that $\left|{a}\right|=\left|{b}\right|<1$. Write $\text{\j}_{1}$,
resp. $\text{\j}_{2}$, resp. $\text{\j}_{3}$ for the integral of

$(az_{2},b)_{3}\psi(-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+y_{1}+x_{2}+az_{2}y_{2})$

over the domain $D^{\prime}_{0x}[a,b]$, resp. $D^{\prime}_{0y}[a,b]$, resp.
$D^{\prime}_{0x}[a,b]\cap D^{\prime}_{0y}[a,b]$ so that

$\mathcal{J}_{0}(a,b)=\text{\j}_{1}+\text{\j}_{2}-\text{\j}_{3}.$

By Lemma 8.15, part 4, an element of $D_{0x}[a,b]$ satisfies
$-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1},\,x_{2}\in\mathcal{O}$ and
$az_{2}y_{2}\in a^{2}b^{-1}z_{2}y_{1}+\mathcal{O}$. Therefore

$\text{\j}_{1}=\int_{a^{-1}\mathcal{O}^{*}}\int_{a^{-1}\mathfrak{p}}\int_{ab^{-1}y_{1}+\mathcal{O}}\int_{\mathcal{O}}\int_{\mathcal{O}}(az_{2},b)_{3}\psi((a^{2}b^{-1}z_{2}+1)y_{1})\
dx_{1}\ dx_{2}\ dy_{2}\ dy_{1}\ dz_{2}.$

Integrating over $x_{1},x_{2}$ and $y_{2}$ and applying the variable change
$z_{2}\mapsto a^{-2}bz_{2}$ we have

(32)
$(a,b)_{3}\,\text{\j}_{1}=\left|{a}\right|^{-1}\int_{\mathcal{O}^{*}}(z_{2},b)_{3}\int_{a^{-1}\mathfrak{p}}\psi((z_{2}+1)y_{1})\
dy_{1}\ dz_{2}.$

If $\left|{a}\right|=q^{-1}$ then $a^{-1}\mathfrak{p}=\mathcal{O}$ and

$\int_{\mathcal{O}}\psi((z_{2}+1)y_{1})\ dy_{1}=1$

is independent of $z_{2}\in\mathcal{O}^{*}$. Since
${\operatorname{val}}(b)=1$, we conclude in this case that

$(a,b)_{3}\,\text{\j}_{1}=\left|{a}\right|^{-1}\int_{\mathcal{O}^{*}}(z_{2},b)_{3}\
dz_{2}=0.$

If $\left|{a}\right|<q^{-1}$ then $a\mathfrak{p}^{-1}\subseteq\mathfrak{p}$
and

$\int_{a^{-1}\mathfrak{p}}\psi((z_{2}+1)y_{1})\
dy_{1}=\begin{cases}q^{-1}\left|{a}\right|^{-1}&z_{2}\in-1+a\mathfrak{p}^{-1}\\\
0&z_{2}\in\mathcal{O}^{*}\setminus(-1+a\mathfrak{p}^{-1}).\end{cases}$

Consequently,

$(a,b)_{3}\,\text{\j}_{1}=\left|{a}\right|^{-2}q^{-1}\int_{-1+a\mathfrak{p}^{-1}}(z_{2},b)\
dz_{2}=\left|{a}\right|^{-1}.$

By part 5 of Lemma 8.15 an element of $D_{0y}[a,b]$ satisfies
$y_{1},\,az_{2}y_{2}\in\mathcal{O}$ and
$-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1}x_{1}\in-
ab^{-1}(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1})x_{2}+\mathcal{O}$. Therefore

$\text{\j}_{2}=\int_{a^{-1}\mathcal{O}^{*}}\int_{a^{-1}\mathfrak{p}}\int_{ab^{-1}x_{2}+\mathcal{O}}\int_{\mathcal{O}}\int_{\mathcal{O}}(az_{2},b)_{3}\psi((1-ab^{-1}(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1})x_{2})\\\
dy_{1}\ dy_{2}\ dx_{1}\ dx_{2}\ dz_{2}.$

Note that in the domain of integration we have $bx_{1}y_{2}\in\mathfrak{p}$.
Applying the variable change $z_{2}\mapsto ab^{-2}(1-bx_{1}y_{2})^{-1}z_{2}$
the integrand becomes independent of $x_{1}\in ab^{-1}x_{2}+\mathcal{O}$ and
$y_{1},y_{2}\in\mathcal{O}$ and after integrating over these three variables
we obtain

$(a,b)_{3}\,\text{\j}_{2}=\left|{a}\right|^{-1}\int_{\mathcal{O}^{*}}\int_{a^{-1}\mathfrak{p}}(z_{2},b)_{3}\psi((1-z_{2}^{-1})x_{2})\
dx_{2}\ dz_{2}.$

If $\left|{a}\right|=q^{-1}$ we again see that the integral factors through
$\int_{\mathcal{O}^{*}}(z_{2},b)\ dz_{2}$ and therefore $\text{\j}_{2}=0$. If
$\left|{a}\right|<q^{-1}$ then the computation is similar to that of
$\text{\j}_{1}$. We have

$(a,b)_{3}\,\text{\j}_{2}=\left|{a}\right|^{-2}q^{-1}\int_{1+a\mathfrak{p}^{-1}}(z_{2},b)\
dz_{2}=\left|{a}\right|^{-1}.$

Finally, it easily follows from parts 4 and 5 of Lemma 8.15 that
$D_{0x}[a,b]\cap D_{0y}[a,b]$ is characterized by the conditions

$x_{1},\,x_{2},\,y_{1},\,y_{2}\in\mathcal{O},\,az_{2}\in\mathcal{O}^{*}.$

Consequently, after integrating over $x_{1},x_{2},y_{1}$ and $y_{2}$ and
applying the variable change $z_{2}\mapsto a^{-2}bz_{2}$ we have

$(a,b)_{3}\,\text{\j}_{3}=\left|{a}\right|^{-1}\int_{\mathcal{O}^{*}}(z_{2},b)\
dz_{2}=\left|{a}\right|^{-1}(1-q^{-1})\delta_{3|{\operatorname{val}}(b)}.$

The lemma follows. ∎

####  8.6.4. A simplification of $\mathcal{J}_{>0}(a,b)$

######  Lemma 8.17.

Let $\left|{b}\right|\leq\left|{a}\right|<1$. Then

$\mathcal{J}_{>0}(a,b)=\sum_{\ell=\lfloor\frac{{\operatorname{val}}(a)+1}{2}\rfloor}^{{\operatorname{val}}(a)-1}\mathcal{J}_{\ell}(a,b)$

where

$\mathcal{J}_{\ell}(a,b)=\left|{a}\right|^{-2}\int_{\left|{y_{2}}\right|=q^{-2\ell}\left|{b}\right|^{-1},\,\left|{1-y_{2}}\right|=\left|{ab^{-1}}\right|}(b^{-1}\varpi^{\ell},y_{2}-1)_{3}\\\
\mathcal{K}(b;-ab^{-2}\varpi^{\ell}(1-y_{2})^{-1},\varpi^{-\ell}y_{2})\mathcal{K}(b^{-1}(y_{2}-1)^{-1};a^{-1}\varpi^{\ell},a^{-1}by_{2}\varpi^{-\ell})\
dy_{2}.$

In particular, $\mathcal{J}_{>0}(a,b)=0$ if $\left|{a}\right|=q^{-1}$.

###### Proof.

Note that the domain $D_{>0}[a,b]$ is characterized by the conditions

$\left|{ab^{-1}}\right|<\left|{x_{1}}\right|,\left|{y_{2}}\right|<\left|{b}\right|^{-1};\,az_{2}\in\mathcal{O}^{*};\,y_{1}-a^{-1}by_{2},\,x_{2}-a^{-1}bx_{1}\in\mathcal{O};\,\left|{1-bx_{1}y_{2}}\right|=\left|{ab^{-1}}\right|.$

In particular, if $\left|{a}\right|=q^{-1}$ then $D_{>0}[a,b]$ is empty and
$\mathcal{J}_{>0}(a,b)=0$. For the rest of this section assume that
$\left|{a}\right|<q^{-1}$. Applying the variable changes $y_{1}\mapsto
y_{1}+a^{-1}by_{2}$ and $x_{2}\mapsto x_{2}+a^{-1}bx_{1}$ to (31), the
integrand becomes independent of $y_{1},\,x_{2}\in\mathcal{O}$. After
integrating over these two variables we obtain the expression

$\mathcal{J}_{>0}(a,b)=\left|{a^{-1}b}\right|\int(x_{1},bx_{1}y_{2}-1)_{3}(az_{2},b)_{3}\\\
\psi((a^{-1}b-(bz_{2})^{-1}(1-bx_{1}y_{2})^{-1})x_{1}+(a^{-1}b+az_{2})y_{2})\
dx_{1}\ dy_{2}\ dz_{2}$

where integration is over the domain defined by the conditions

$az_{2}\in\mathcal{O}^{*},\ \ \
\left|{ab^{-1}}\right|<\left|{x_{1}}\right|,\left|{y_{2}}\right|<\left|{b}\right|^{-1}\
\ \ \text{and}\ \ \ \left|{1-bx_{1}y_{2}}\right|=\left|{ab^{-1}}\right|.$

Next, applying the variable change

$x_{1}\mapsto az_{2}x_{1},\ y_{2}\mapsto(az_{2})^{-1}y_{2}$

we have

$\mathcal{J}_{>0}(a,b)=\left|{a^{-1}b}\right|\int(x_{1},bx_{1}y_{2}-1)_{3}(az_{2},b(bx_{1}y_{2}-1))_{3}\\\
\psi(bz_{2}x_{1}-ab^{-1}(1-bx_{1}y_{2})^{-1}x_{1}+a^{-2}by_{2}z_{2}^{-1}+y_{2})\
dx_{1}\ dy_{2}\ dz_{2}$

where integration is over the same domain. Now applying the variable change

$z_{2}\mapsto z_{2}x_{1}^{-1},\,y_{2}\mapsto y_{2}(bx_{1})^{-1}$

we have

$\mathcal{J}_{>0}(a,b)=\left|{a}\right|^{-1}\int\left|{x_{1}}\right|^{-2}(b,x_{1})_{3}(az_{2},b(y_{2}-1))_{3}\\\
\psi(bz_{2}-ab^{-1}(1-y_{2})^{-1}x_{1}+a^{-2}y_{2}z_{2}^{-1}+b^{-1}y_{2}x_{1}^{-1})\
dx_{1}\ dy_{2}\ dz_{2}$

where integration is over the domain defined by

$\left|{ab^{-1}}\right|<\left|{x_{1}}\right|=\left|{az_{2}}\right|<\left|{b}\right|^{-1},\
\ \ \left|{ax_{1}}\right|<\left|{y_{2}}\right|<\left|{x_{1}}\right|\ \ \
\text{and}\ \ \ \left|{1-y_{2}}\right|=\left|{ab^{-1}}\right|.$

Note further that the condition
$\left|{1-y_{2}}\right|=\left|{ab^{-1}}\right|$ implies that
$\left|{y_{2}}\right|\leq\left|{ab^{-1}}\right|$ and combined with the
condition $\left|{ab^{-1}}\right|<\left|{x_{1}}\right|$ it implies that
$\left|{y_{2}}\right|<\left|{x_{1}}\right|$. Thus the condition
$\left|{y_{2}}\right|<\left|{x_{1}}\right|$ may be omitted.

We express the integral $\mathcal{J}_{>0}(a,b)$ as a sum over
$\ell=1,\dots,{\operatorname{val}}(a)-1$ of integrals over the subdomain where
$\left|{x_{1}}\right|=\left|{b}\right|^{-1}q^{-\ell}$, that is, over the
domain defined by the conditions

$\left|{x_{1}}\right|=\left|{b}\right|^{-1}q^{-\ell},\
\left|{z_{2}}\right|=\left|{ab}\right|^{-1}q^{-\ell},\
q^{-\ell}\left|{b^{-1}a}\right|<\left|{y_{2}}\right|\ \ \ \text{and}\ \ \
\left|{1-y_{2}}\right|=\left|{ab^{-1}}\right|.$

Applying the variable change

$x_{1}\mapsto b^{-1}\varpi^{\ell}x_{1},\ z_{2}\mapsto
a^{-1}b^{-1}\varpi^{\ell}z_{2}$

to the $\ell$-th summand we have

$\mathcal{J}_{>0}(a,b)=\left|{a}\right|^{-2}\sum_{\ell=1}^{{\operatorname{val}}(a)-1}\int(b^{-1}\varpi^{\ell},y_{2}-1)_{3}\int_{\mathcal{O}^{*}}\int_{\mathcal{O}^{*}}(b,x_{1})_{3}(z_{2},b(y_{2}-1))_{3}\\\
\psi(a^{-1}\varpi^{\ell}z_{2}-ab^{-2}\varpi^{\ell}(1-y_{2})^{-1}x_{1}+a^{-1}b\varpi^{-\ell}y_{2}z_{2}^{-1}+\varpi^{-\ell}y_{2}x_{1}^{-1})\
dx_{1}\ dz_{2}\ dy_{2}\\\
=\left|{a}\right|^{-2}\sum_{\ell=1}^{{\operatorname{val}}(a)-1}\int(b^{-1}\varpi^{\ell},y_{2}-1)_{3}\\\
\mathcal{K}(b;-ab^{-2}\varpi^{\ell}(1-y_{2})^{-1},\varpi^{-\ell}y_{2})\mathcal{K}(b^{-1}(y_{2}-1)^{-1};a^{-1}\varpi^{\ell},a^{-1}by_{2}\varpi^{-\ell})\
dy_{2}$

where in the $\ell$-th integral $y_{2}$ is integrated over the domain defined
by the conditions

$q^{-\ell}\left|{b^{-1}a}\right|<\left|{y_{2}}\right|\ \ \ \text{and}\ \ \
\left|{1-y_{2}}\right|=\left|{ab^{-1}}\right|.$

It follows from Lemma 7.2 that in the domain of integration above we have

$\mathcal{K}(b^{-1}(y_{2}-1)^{-1};a^{-1}\varpi^{\ell},a^{-1}by_{2}\varpi^{-\ell})=0$

unless

$\left|{y_{2}}\right|=q^{-2\ell}\left|{b}\right|^{-1}.$

Indeed, if $\ell<{\operatorname{val}}(a)-1$ then
$\left|{a^{-1}\varpi^{\ell}}\right|>q$ and for
$\ell={\operatorname{val}}(a)-1$ we have
$\left|{a^{-1}\varpi^{\ell}}\right|=q$ while in the domain of integration
$\left|{a^{-1}by_{2}\varpi^{-\ell}}\right|\geq q$. Since in the domain of the
$\ell$-th integral we have $\left|{y_{2}}\right|\leq\left|{ab^{-1}}\right|$
and since $q^{-2\ell}\leq\left|{a}\right|$ if and only if
$\ell\geq\lfloor\frac{{\operatorname{val}}(a)+1}{2}\rfloor$ the lemma follows.
∎

In order to proceed with the computation of $\mathcal{J}_{\ell}(a,b)$ we
separate into the two cases:

  * •

Case 1: $\ell<{\operatorname{val}}(b)-1$ or $\ell={\operatorname{val}}(b)-1$
and $3\nmid{\operatorname{val}}(b)$.

  * •

Case 2: $\ell={\operatorname{val}}(b)-1$ and $3|{\operatorname{val}}(b)$.

The computation in Case 1 requires some preparation that we carry out first.

#### 8.6.5.

Let $a,\,b\in F^{*}$ and let $\ell$ be an integer such that

  * •

either $\left|{a}\right|=\left|{b}\right|\leq q^{-2}$ and
$\frac{{\operatorname{val}}(a)}{2}\leq\ell\leq{\operatorname{val}}(a)-1$

  * •

or $\left|{b}\right|<\left|{a}\right|<1$ and
$\ell=\frac{{\operatorname{val}}(a)}{2}$.

For $j\geq 1$ let

$\Gamma_{j}=\\{(x,y)\in\mathcal{O}\times\mathcal{O}:x^{3}-y^{3}\in
a\mathfrak{p}^{-j}\\}$

and let

$G_{j}^{\ell}(a,b)=\int_{\Gamma_{j}}\psi(\varpi^{\ell}b^{-1}x-3^{-3}\varpi^{3\ell}a^{-1}b^{-1}x^{3}+\varpi^{\ell}a^{-1}y)\
dx\ dy.$

Note that the dependence of $\Gamma_{j}$ on $a$ is only via
${\operatorname{val}}(a)$. Our goal in this subsection is to compute
$G_{\ell}^{\ell}(a,b)$ and $G_{\ell+1}^{\ell}(a,b)$. Let
$j\in\\{\ell,\ell+1\\}$ and set

$m={\operatorname{val}}(a)-j.$

Note that with the above assumptions $m\geq 0$. We begin with the explication
of $\Gamma_{j}$.

######  Lemma 8.18.

We have the disjoint union

$\Gamma_{j}=\left(\mathfrak{p}^{-\lfloor-\frac{m}{3}\rfloor}\times\mathfrak{p}^{-\lfloor-\frac{m}{3}\rfloor}\right)\sqcup\mathop{\sqcup}\limits_{n=0}^{\lfloor\frac{m-1}{3}\rfloor}\mathop{\sqcup}\limits_{k=0}^{2}\\{(x,y):\left|{x}\right|=q^{-n},\
y\in\rho^{k}x+\mathfrak{p}^{m-2n}\\}.$

(In particular, if $m=0$ then $\Gamma_{j}=\mathcal{O}\times\mathcal{O}$.)

###### Proof.

The case $m=0$ is straightforward. Assume that $m>0$. By definition we have

$\Gamma_{j}=\\{(x,y)\in\mathcal{O}\times\mathcal{O}:x^{3}-y^{3}\in\mathfrak{p}^{m}\\}.$

Note that for $(x,y)\in\Gamma_{j}$ we have $x^{3}\in\mathfrak{p}^{m}$ if and
only if $y^{3}\in\mathfrak{p}^{m}$ and that for $x\in F$ we have
$x^{3}\in\mathfrak{p}^{m}$ if and only if
$x\in\mathfrak{p}^{-\lfloor-\frac{m}{3}\rfloor}$. Suppose that
$(x,y)\in\Gamma_{j}$ with $x^{3}\not\in\mathfrak{p}^{m}$ and set
$\left|{x}\right|=q^{-n}$. By assumption $0\leq
n\leq\lfloor\frac{m-1}{3}\rfloor$ and thus
$\left|{x}\right|=\left|{y}\right|$. Note that in the decomposition

$y^{3}-x^{3}=(y-x)(y-\rho x)(y-\rho^{2}x)\in\mathfrak{p}^{m}$

two of the three factors must also be of absolute value $q^{-n}$ and the third
must therefore be in $\mathfrak{p}^{m-2n}$. This gives the inclusion of
$\Gamma_{j}$ in the disjoint union. The above decomposition also explains the
other inclusion. ∎

######  Lemma 8.19.

With the above assumptions we have

$G_{\ell+1}^{\ell}(a,b)=0$

and