数理解析研究所講究録 1200 巻 2001 年 82-91

82

On central critical values of the degree fourL-functions for GSp(4) and the matrix
argument Kloosterman sums
(joint work with J. A. Shalika)
Masaaki Furusawa (古澤 昌秋)
Osaka City University (大阪市立大学理学部)
December 19, 2000
1. Motivation
Let Ibe aSiegel eigen cusp form of degree two of weighttoand let$k$with respect$\mathrm{S}\mathrm{p}_{4}(\mathbb{Z})$$\Phi$$(Z)= \sum_{T>0}a(T,\Phi)\exp[2\pi\sqrt{-1}\mathrm{t}\mathrm{r}(TZ)]$

be its Fourier expansion. Here$t_{1},t_{2}$,$t_{3}\in \mathbb{Z}$

and

$T$

$T$

runs over $T=$

is positive definite. For such

$T_{1}\sim T_{2}$A.

$\exists\gamma\in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})\mathrm{s}.\mathrm{t}$$(\begin{array}{ll}t_{1} t_{2}/2t_{2}/2 t_{3}\end{array})$$T_{1}$

and$T_{2}$$T_{2}={}^{t}\gamma T_{1}\gamma$

Let $E$ be an imaginary quadratic field and let
Then we define$D_{E}$

such that

, let

.

be its discriminant.

$B_{E}( \Phi)=\sum_{\{T|\det T=-D_{E}/4\}/\sim}\mathrm{d}\mathrm{e}\mathrm{f}\frac{a(T,\Phi)}{\epsilon(T)}$where. We recall that, by Gauss,there exists abijection between the set $\{T|\det T=-D_{E}/4\}/\sim \mathrm{a}\mathrm{n}\mathrm{d}$the ideal class group of $E$ .
B\"ocherer has proclaimed the following conjecture in 1986 (Preprint
Math. Gottingensis Heft 68)$\epsilon(T)=\#\{\gamma\in \mathrm{S}\mathrm{L}_{2}(\mathbb{Z})|{}^{t}\gamma T\gamma=T\}$

83
Bocherer’s Conjecture . There exists a constanton such that$c_{\Phi}$that depends only$\Phi$$L$(,$\frac{1}{2}$$\Phi$(1)$\otimes\chi_{E})=c_{\Phi}\cdot|D_{E}|^{-k+1}\cdot|B_{E}(\Phi)|^{2}$denotes the spinor (degree four) L-function
for any E. Hereof twisted by the quadratic character corresponding to the quadratic, normalized so that its functional equation is with respectextension$L$ $(s,\Phi \otimes\chi_{E})$$\Phi$$\chi_{E}$$E/\mathbb{Q}$to $s\vdasharrow 1-s$ .

Remarks
1. The center $s=1/2$ is the only critical point in the sense of Deligne.
2. Bocherer, and, later he and Schulze-Pillot (Math. Z. 209 (1992))
verified the assertion for Eisenstein series, SaitO-Kurokawa lifting andYoshida lifting.
3. Kohnen and Kuss have made some numerical experiment on an eigenform of weight 20, which does not belong to the SaitO-Kurokawa
lifting.
4. We may normalize Iso that, hence.
Thus we may regard
as the algebraic part of the special value. It is natural for us to fantasize about the generalized
Birch&Swinnerton-Dyer conjecture, padic interpolation, etc.
5. B\"ocherer did not make any speculation about the constant . It is
important to identify
from the viewpoint of Deligne’s conjecture(Proc. Sympos. Pure Math. 33, 1979) since it is related to theperiodof the special value.
$a(T, \Phi)\in\overline{\mathbb{Q}}$$B_{E}(\Phi)\in\overline{\mathbb{Q}}$$|B_{E}(\Phi)|^{2}$$c_{\Phi}$$c_{\Phi}$$pa\hslash$

Bocherer’s conjecture reminds us of:Waldspurger’s Theorem . (Compositio Math. 54 (1985)) Let $F$ benumber field. Let be an irreducible cuspidal representation of.
For 0, a Hecke character of
There $E$ is a quadratic extension of $F$ ,letdenote the theta series representation of. Assume that
wheredenotes the central character of . Then we have

$a$

$\pi$

$\mathrm{G}\mathrm{L}_{2}(\mathrm{A}_{F})$$\mathrm{A}_{E}^{\mathrm{x}}$$\pi(\Omega)$$\Omega|_{\mathrm{A}_{F}^{\mathrm{X}}}.\omega_{\pi}=1$$\mathrm{G}\mathrm{L}_{2}(\mathrm{A}_{F})$$\pi$$\omega_{\pi}$$L$(,$\frac{1}{2}$$\pi\otimes\pi(\Omega))\neq 0$

84
if and only if there exists a quaternion algebrain the space ofand an automor phic form
Jacquet-Langlands correspondent of of$D$$\pi^{D}$$\varphi^{D}$$\pi$$D^{\mathrm{x}}(\mathrm{A}_{F})$$\int_{\mathrm{A}_{F}^{\mathrm{X}}E^{\mathrm{x}}\backslash \mathrm{A}_{B}^{\mathrm{X}}}\varphi^{D}(t)\Omega(t)d^{\mathrm{x}}t\neq 0$

overcontaining $E$
denotes thewheresuch that$F$$\pi^{D}$.

(2)Remarks
1. Let $BC^{E}(\pi)$ denote the base change lifting of
we have
$L$

$\pi$

to

$\mathrm{G}\mathrm{L}_{2}(\mathrm{A}_{E})$. Then$(s, \pi\otimes\pi (\Omega))=L(s, BC^{E}(\pi)\otimes\Omega)$

and in particular, when$\Omega$

is trivial,$L(s, BC^{E}(\pi))=L(s, \pi)\cdot L(s, \pi\otimes\chi_{E})$wheredenotes the quadratic character corresponding to $E/F$ .
2. When is trivial, there exists the following metaplectic version of
the theorem which might be more familiar (Kohnen-Zagier, Invent,math. 64 (1981) : Let be a normalized eigenform of weight $2k$ withrespect toand let be its Shimura correspondent, $i.e$ .
$\chi_{E}$$\Omega$$f$$)$$\mathrm{S}\mathrm{L}_{2}(\mathbb{Z})$$g$.

$g(z)= \sum_{n\geq 1}b(n)\exp(2\pi\sqrt{-1}nz)\in S_{k+\frac{1}{2}}^{+}(4)$Then for the fundamental discriminant $D$ of $E=\mathbb{Q}(\sqrt{D})$ such that$(-\mathrm{l})^{}$$D>0$ , we have.

(3)$\frac{|b(|D|)|^{2}}{(g,g)}=\frac{(k-1)!|D|^{k-1/2}L(k,f\otimes\chi_{E})}{\pi^{k}(f,f)}$(Here we use the classical normalization for the -function so thatthe functional equation is with respect to $sarrow*2k-s.$ ) We remarkthat (3) implies the non-negativity of the central value $L(k, f\otimes\chi_{E})$which is consistent with the generalized Riemann hypothesis (cf.
Guo, Duke Math. J. 83 (1996) .
3. Similarly, in general, further analysis yields an identity that expresses
the central critical value$)$

L $(1/2,\pi)$ L$(1/2,\pi\otimes\chi_{E})$(resp. L (1/2,$BC^{E}(\pi)\otimes\Omega$))

85
as the square norm of one of these period integrals in (2) multiplied(resp.
) which depends only on(resp.
by aconstant) (Chen and Jacquet, Bull Soc. Math.
(resp.
and ), not on
$E$

$E$

$\pi$

$\pi$

$C_{\pi,E}$$C_{\pi}$$\Omega$

Prance, to appear).
4. The choice of the quaternion algebra which gives the non-zero period
integral (2) is unique and is determined at each place by the local
factor of the -function(see H. Saito, Compositio Math. 85 (1993)).
This is aspecial case of the Gross-Prasad conjecture (Canad. J.
Math. 44 (1992) and ibid 46 (1994) .

$\epsilon-$

$L$

$)$

The original proof by Waldspurger was based on the Weil representation,i.e. theta correspondence. Later Jacquet has given another proof using
formulas,formula. Actually he proved too relative
the relative
one corresponding to the case when is trivial (Ann. scient. Ec. Norm.
isSup. 19 (1986) and the other corresponding to the case whenarbitrary (Compositio Math. 63 (1987)).
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\Omega$$\Omega$$)$

Our Project$\underline{2.}$The ultimate goal of our project is to prove B\"ocherer’s conjecture andformulasits generalization by extending both of Jacquet’s relative.
, A}, whereto{formulas themselves are too techSince our conjectural relative. Acad. Sci.
nical to state here, we refer to the two announcements (Paris 328 (1999), 105-110 and ibid 331 (2000), 593-598) for the details.
formulas.
Instead let us explain the expected consequences of the$F$ be anumber field andFirst we need to introduce some notation. Let$E$ be aquadratic extension of $F$ and let $X(E:F)$ denote the set of
isomorphism classes of the central quaternion algebras over $F$ containing
, let
. For
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{G}\mathrm{S}\mathrm{p}(4)=$$g\in \mathrm{G}\mathrm{L}_{4}|{}^{t}gJg=\lambda J$$J=(_{-1_{2}0^{2}}^{01})$$\in \mathrm{G}\mathrm{L}_{1}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{C}.\mathrm{R}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$E$$\epsilon\in F^{\mathrm{x}}$$D_{\epsilon}=\{$$(\begin{array}{ll}a \epsilon bb^{\sigma} a^{\sigma}\end{array})$$|a$, $b\in E\}$

denotes the unique non-trivial element in the Galois group of $E$
andinduces abijection between
over $F$ . Then
$X$ $(E ; F)$ . Let
denote the involution of . Let
where$\sigma$$F^{\mathrm{x}}/N_{E/F}(E^{\mathrm{x}})$$\epsilon\vdash iD_{\epsilon}$$D_{\epsilon}$$x\vdasharrow\overline{x}$$G_{\epsilon}=\{g\in \mathrm{G}\mathrm{L}_{2}(D_{\epsilon})|g^{*}$$(\begin{array}{ll}0 11 0\end{array})$$g=\lambda(g)$$(\begin{array}{ll}0 11 0\end{array})$,$\lambda(g)\in \mathrm{G}\mathrm{L}_{1}(F)\}$86

where g’F

/3

we have D.

$\ovalbox{\tt\small REJECT}$

6

for g
$\ovalbox{\tt\small REJECT}$

a j3
y 6

Here we note that when e$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\mathrm{M}_{2}$$R_{\epsilon}=\{$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$

subgroup R. of G. by

Let

1 E.

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$and G.

$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$1,

GSp(4, F). Let us define the Bessel$(a a^{\sigma} a a^{\sigma})(\begin{array}{ll}\mathrm{l} X0 1\end{array})$$|a\in E^{\mathrm{x}},\mathrm{t}\mathrm{r}X=0\}$.

be anon-trivial character ofand letbe acharacter of. Then by abuse of notation we denote by acharacter(\^A)
defined by$\psi$$\Omega$$\mathrm{A}_{F}/F$$\mathrm{A}_{E}^{\mathrm{x}}/E^{\mathrm{x}}$$\Omega$$\Omega[(a a^{\sigma} a a^{\sigma})(\begin{array}{ll}1 X0 1\end{array})$$]=\Omega(a)\cdot\psi$

where $\eta\in E$ such that $E=F(\eta)$ and$\mathrm{o}\mathrm{f}R_{\epsilon}$$[$$\mathrm{t}\mathrm{r}\{$ $(\begin{array}{ll}-\eta 00 \eta\end{array})$$\eta^{2}\in F$

$X)]$

.

Conjecture (Furusawa&Shalika) . Let
be an irreducible cuspidal
automorphic representation of. Assume that the central character of is equal to the inverse of.
Then we have$\pi$$\mathrm{G}\mathrm{S}\mathrm{p}_{4}(\mathrm{A}_{F})$$\pi$$\Omega|_{\mathrm{A}_{F}^{\mathrm{X}}}$$L$(,$\frac{1}{2}$$\pi\otimes\pi(\Omega))\neq 0$

if and only if there exists a triple
reducible cuspidal representation offunctorial sense, . having the samethe space of such that

, where,an ir, corresponding toin the-function, anda cusp form in$\epsilon\in F^{\mathrm{x}}$$(\mathrm{e}, \pi_{\epsilon}, \varphi_{\epsilon})$$\pi_{\epsilon}$$G_{\epsilon}(\mathrm{A}_{F})$

$i.e$

$L$

$\pi$

$\varphi_{\epsilon}$$\pi_{\epsilon}$.

(4)$\int_{\mathrm{A}_{F}^{\mathrm{X}}R_{\epsilon}(F)\backslash R_{\epsilon}(\mathrm{A}_{F})}\varphi_{\epsilon}(r)\Omega(r)dr\neq 0$

Moreover, the detailed analysis should yield an identity that expresses
the central critical value of $L(s, \pi\otimes\pi(\Omega))$ as the square norm of one ofthese period integrals (4) multiplied by aconstantwhich depends
only on and $E$ , not on the character of . Also when is trivial, thecentral critical value of $L(s, \pi)L(s, \pi\otimes\chi_{E})$ should be the square normof the period integral (4) multiplied by aconstantwhich depends only$E$
on and not on the quadratic extension .
$C_{\pi,E}’$$\pi$$\Omega$$\Omega$$\mathrm{A}_{E}^{\mathrm{x}}$$C_{\pi}$

$\pi$

87
In particular when is the cuspidal representation ofcorresponding to aholomorphic Siegel eigen cusp form
, by looking
at the Fourier expansion of, it should follow as acorollary of theconjecture that there exists an imaginary quadratic field $E$ and afinite
order Hecke character ofsuch that$\pi$$\mathrm{G}\mathrm{S}\mathrm{p}_{4}(\mathrm{A}_{\mathbb{Q}})$$@=\Phi_{\mathrm{h}\mathrm{o}1}$$\Phi_{\mathrm{h}\mathrm{o}1}$$\mathrm{A}_{E}^{\mathrm{x}}$$L$(,$\frac{1}{2}$$\pi\otimes\pi(\Omega))\neq 0$.

We also speculate that the constantmentioned above is given, in thiscase, essentially as aratio of Petersson norms$C_{\pi}$$\frac{(\Phi_{\mathrm{g}\mathrm{e}\mathrm{n}},\Phi_{\mathrm{g}\mathrm{e}\mathrm{n}})}{(\Phi_{\mathrm{h}\mathrm{o}1},\Phi_{\mathrm{h}\mathrm{o}1})}$

where
denotes ageneric cusp form, i.e. having anon-zero Whittaker Fourier coefficient, corresponding to the same -function as.
It indicates that it is important to study the whole -packet(i.e. all
the automorphic representations giving the same -function)in order tounderstand the nature of the special values of the -function. We remarkthat the constanthere is essentially equal towheredenotes the constant in B\"ocherer’s original conjecture (1).
The first but crucial step to establish aformula is to prove thefundamental lemma, an equality between two local orbital integrals forthe elements in the Hecke algebra. We have proved the fundamental
lemma for the identity element in the Hecke algebra for both of theformulas. We have also proved the Plancherel formula for the Bessel
model which reduces the fundamental lemma for the general element in
the Hecke algebra to an equality between two finite sums of certain local
orbital integrals for the identity element.
The proof of the fundamental lemma for the identity element essentiallyamounts to computing some matrix argument character sums, which weare going to discuss.
$\Phi_{\mathrm{g}\mathrm{e}\mathrm{n}}$$L$$\Phi_{\mathrm{h}\mathrm{o}1}$

$L$

$L$

$L$

$c_{\Phi}\cdot L(1/2, \pi)$$C_{\pi}$$c_{\Phi}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$3. Matrix Argument Kloosterman Sums
From now on we denote byanon-archimedean local field whose
residual characteristic is not equal to two. Let be acharacter of $F$ whose, the ring of integers of $F$ and let be aprime element
conductor isof $F$ . We denote by the cardinality of the residue field
. Let
$F$

$\psi$$O_{F}$$\varpi$$q$$O_{F}/\varpi O_{F}$88be the unique unramified quadratic extension of $F$ andof integers.
First we recall the classical Kloosterman sum defined by$E$$\mathcal{K}\ell(r, s)=\int_{\mathit{0}^{\mathrm{X}}}\psi(r\epsilon +s\epsilon^{-1})$$O_{E}$

be its ring

de

for ,
. Sometimes we call it theKloosterman sum since itis related to the Fourier coefficients of the Poincare series for
. For
, let
$r$

$s\in F^{\mathrm{x}}$$\mathrm{G}\mathrm{L}_{2}$$\mathrm{G}\mathrm{L}_{2}$$a\in O_{F}^{\mathrm{x}}$$\mathcal{H}_{n}(a)=\int_{\mathcal{Z}_{a}}\psi$$[\mathrm{t}\mathrm{r}_{E/F}(\xi)]d\xi$

whereDavenport-Hasse relation:$Z_{a}=\{\xi\in O_{E}^{\mathrm{x}}|N_{E/F}(\xi)\equiv a(\mathrm{m}\mathrm{o}\mathrm{d} \varpi^{n})\}$$\mathcal{H}_{n}(a)=( 1)$”. Here we recall the$q^{-n}\cdot \mathcal{K}\ell(2\varpi^{-n}, 2\varpi^{-n}a)$.

Now let us consider the following matrix argument Kloosterman sums.
”, ,(F) and, let
For$A\in \mathrm{G}\mathrm{L}_{n}(F)$$S$$T\in \mathrm{S}\mathrm{y}\mathrm{m}$$\epsilon$$\in O_{F}^{\mathrm{x}}$(5)$\mathcal{K}(A, S,T,\epsilon)=\int_{\mathcal{X}_{A}}\psi[\mathrm{t}\mathrm{r}(XS+\epsilon TA^{-1}X^{-1}{}^{t}A^{-1})]dX$

where
we may write the right hand side of (5) as$\mathcal{X}_{A}=\{X\in \mathrm{S}\mathrm{y}\mathrm{m}^{n}(F)|XA\in \mathrm{G}\mathrm{L}_{n}(O_{F})\}$$\sum_{V}\psi$

where

$V$

[tr

. Here we remark thatruns over the set of representatives modulo$V\in M_{n}(O_{F})$(6)$(\mathrm{Y}A^{-1}S+\epsilon\cdot A^{-1}VT)]$$A$. Symn$(O_{F})$ofsuch that3 ,$U$ $\mathrm{Y}\in M_{n}(O_{F})$with$(\begin{array}{ll}\mathrm{Y} UA V\end{array})\in \mathrm{S}\mathrm{p}_{2n}$(OF).

As Kitaoka has shown, the Kloosterman sum (6) appears in the Fourier
coefficients of the Poincare series for the Siegel modular group (Nagoya
Math. J. 93 (1984) .
$)$89
Now we restrict ourselves to the case when $n=2$ . Let us define the split
Kloosterman sum by$\mathcal{K}_{\mathrm{s}\mathrm{p}1}(A, \epsilon)=\mathcal{K}(A, S_{1}, S_{1}, \epsilon)=\int_{\mathcal{X}_{A}}\psi[\mathrm{t}\mathrm{r}(S_{1}(X+\epsilon A^{-1}X^{-1}{}^{t}A^{-1}))]dX$

where$S_{1}=(\begin{array}{ll}0 11 0\end{array})$.

Theorem 1. Suppose that$\in\varpi\cdot \mathrm{M}_{2}(O_{F})\cap \mathrm{G}\mathrm{L}_{2}(F)$$A=(\begin{array}{ll}\alpha \beta\gamma \delta\end{array})$.

Then we have$\mathcal{K}_{\mathrm{s}\mathrm{p}1}(A, \epsilon)=\frac{1}{|\Delta|}\{\mathcal{K}\ell(\frac{2\alpha}{\Delta},$$\frac{2\epsilon\delta}{\Delta})+\mathcal{K}\ell(\frac{2\beta}{\Delta}$,$\frac{2\epsilon\gamma}{\Delta})\}$

where $\Delta=\det A$ .
Thus the two by two symmetric matrix argument split Kloosterman
sum reduces to asum of two classical Kloosterman sums.
Let us define the anisotropic Kloosteman sum. For our convenience let, namely,us employ another realization of$\mathrm{M}_{2}(F)$$D_{1}=\{$Let us take y7
Band$\in D_{1}^{\mathrm{x}}$$\in O_{E}^{\mathrm{x}}$$\epsilon\in O_{F}^{\cross}$$\mathcal{Y}_{E}=${$\mathrm{Y}\in D_{1}|$since the matrix

trix$(\begin{array}{ll}1 00 -d\end{array})$$|a$, $b\in E\}$ .

such that E $=F(\eta)$ and $\eta^{2}=d\in F$ . Then for, we define the anisotropic Kloosterman sum by$\mathcal{K}_{\mathrm{a}\mathrm{n}}(B, \epsilon)=\int_{y_{B}}\psi\{\mathrm{t}\mathrm{r}\{$where$(\begin{array}{ll}a bb^{\sigma} a^{\sigma}\end{array})$$(\begin{array}{ll}\eta 00 -\eta\end{array})$

trY $=0$ ,$(\begin{array}{ll}\eta 00 -\eta\end{array})$$( \mathrm{Y}-\frac{\epsilon}{\det B}B^{-1}\mathrm{Y}^{-1}B)]\}d\mathrm{Y}$$\mathrm{Y}B\in \mathrm{G}\mathrm{L}_{2}(O_{E})$}. We call it anisotropic

corresponds to the anisotropic symmetric ma-

in the ordinary$\mathrm{M}_{2}(F)$realization.

90
Theorem 2. Suppose that $n>0$ andWe writewhere$u=\varpi^{m}\epsilon_{u}$$m=\mathrm{o}\mathrm{r}\mathrm{d}(u)$$u\in O_{E}\backslash \{0\}$. Letsuch that $uua\neq 1$ .

$A_{u}=(\begin{array}{ll}1 uu^{\sigma} 1\end{array})$.

1. When $m\geq n$ , we have$\mathcal{K}_{\mathrm{a}\mathrm{n}}(\varpi^{n}A_{u}, \epsilon)=q^{2n}\{(-1)^{n}\mathcal{K}\ell(2\varpi^{-n}, -2\varpi^{-n}\ )+1+q^{-1}\}$.

2. When $0\leq m<n$ , we have$\mathcal{K}_{\mathrm{a}\mathrm{n}}(\varpi^{n}A_{u}, \epsilon)=\frac{(-1)^{n}q^{2n}}{|1-uu^{\sigma}|}\cdot \mathcal{K}\ell(\frac{2\varpi^{-n}}{1-uu^{\sigma}},$$+ \frac{(-1)^{m-n}q^{2n}}{|1-uu^{\sigma}|}\cdot \mathcal{K}\ell$($\frac{-2\varpi^{-n}d\epsilon}{1-uu^{\sigma}})$$\frac{2\varpi^{m-n}}{1-uu^{\sigma}}$,$\frac{-2\varpi^{m-n}\ \epsilon_{u}\epsilon_{u}^{\sigma}}{1-uu^{\sigma}}$).

We also have ageneralization of the Davenport-Hasse relation in ourcase. Letsuch thatand $\det T\neq 0$ . Then for,let${}^{t}T^{\sigma}=T$$T\in \mathrm{M}_{2}(E)$$\epsilon\in O_{E}^{\mathrm{x}}$$\mathcal{H}(T,\epsilon)=\int_{\mathcal{Z}_{T}}\psi$

where&consists of Z
TZ$\in \mathrm{S}\mathrm{y}\mathrm{m}$$\in \mathrm{M}_{2}(O_{E})$

Theorem 3. Let T$\{\mathrm{t}\mathrm{r}_{E/F}[$and$\in \mathrm{M}_{2}(O_{E})$$\epsilon$$\cdot \mathrm{t}\mathrm{r}$($(\begin{array}{ll}0 11 0\end{array})$$]\}dZ$$(T^{-1})^{\sigma}-Z^{\sigma}TZ\in \mathrm{M}_{2}(O_{E})$such that$T=(\begin{array}{ll}a yy^{\sigma} b\end{array})$$y=\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}\epsilon_{y}$)(E) such that${}^{t}T^{\sigma}=T$

write

and

$Z$

,

.

and $\det T\neq 0$ . Let us$\Delta=\det T$

when $y\neq 0$ . Let us define a non-negative integer $m$ by$m= \min\{\mathrm{o}\mathrm{r}\mathrm{d}(a), \mathrm{o}\mathrm{r}\mathrm{d}(b)\}$

1. Suppose that m(a) When $m=0$ and(b) When $m=0$ and$\leq \mathrm{o}\mathrm{r}\mathrm{d}$.

(y).
, we have $H$ $(T,\epsilon)=1$ .
, we have$\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)=0$$0<\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)\leq \mathrm{o}\mathrm{r}\mathrm{d}(y)$$\mathcal{H}(T, \epsilon)=|\Delta|^{-1}(1+q^{-1})$.

91(c) When $m=0$ and$\mathcal{H}$, we have$0\leq \mathrm{o}\mathrm{r}\mathrm{d}(y)<\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)$$(T, \epsilon)=\frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(\Delta y^{-1}})}{|\Delta|}$(d) When $m>0$ and. CZ$( \frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{\Delta},$$\mathcal{H}$$\mathrm{o}\mathrm{r}\mathrm{d}(y)<\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)$$(T, \epsilon)=\frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)}}{|\Delta|}\cdot \mathcal{K}\ell(\frac{2a}{\Delta},$$+ \frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(\Delta y)}}{|\Delta|}\cdot \mathcal{K}\ell$

2. Suppose that m(a) When$>\mathrm{o}\mathrm{r}\mathrm{d}$$\mathrm{o}\mathrm{r}\mathrm{d}(y)=0$(b) When.

, we have$\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)\leq \mathrm{o}\mathrm{r}\mathrm{d}(y)$$\mathcal{H}(T, \epsilon)=\frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(\Delta)}}{|\Delta|}\cdot \mathcal{K}l(\frac{2a}{\Delta},$(e) When $m>0$ and$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}\epsilon\epsilon^{\sigma}\epsilon_{y}\epsilon_{y}^{\sigma}}{\Delta})$$\frac{2\epsilon\epsilon^{\sigma}b}{\Delta})+\frac{1+q^{-1}}{|\Delta|}$.

, we have$\frac{2\epsilon\epsilon^{\sigma}b}{\Delta})$(,$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{\Delta}$$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}\epsilon\epsilon^{\sigma}\epsilon_{y}\epsilon_{y}^{\sigma}}{\Delta}$

).

(y).

, we have, we have$\mathcal{H}(T, \epsilon)=1$.

$m\geq 2\mathrm{o}\mathrm{r}\mathrm{d}(y)$$\mathcal{H}(T,\epsilon)=\frac{1-q^{-1}}{\Delta}$$+ \frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{|\Delta|}\cdot \mathcal{K}\ell\nearrow$(c) When$\mathrm{o}\mathrm{r}\mathrm{d}(y)<m<2\mathrm{o}\mathrm{r}\mathrm{d}(y)$($+ \frac{(-1)^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{|\Delta|}\cdot \mathcal{K}\ell$$;\in O_{F}^{\cross}$,$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}\epsilon\epsilon^{\sigma}\epsilon_{y}\epsilon_{y}^{\sigma}}{\Delta}$).

, we have$\mathcal{H}(T,\epsilon)=\frac{\mathrm{s}\mathrm{g}\mathrm{n}(\epsilon_{y}\epsilon_{y}^{\sigma})^{m}}{|\Delta|}\cdot \mathcal{K}\ell(\frac{2a}{\Delta},$

Here for ($\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{\Delta}$($\frac{2b}{\Delta})$$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}}{\Delta}$,$\frac{2\varpi^{\mathrm{o}\mathrm{r}\mathrm{d}(y)}\epsilon\epsilon^{\sigma}\epsilon_{y}\epsilon_{y}^{\sigma}}{\Delta}$).

, we haveif-1, if1,$\mathrm{s}\mathrm{g}\mathrm{n}(\zeta)=\{$$\zeta\in(\mathcal{O}_{F}^{\cross})^{2}$$\zeta\not\in(O_{F}^{\mathrm{x}})^{2}$

These explicit formulas for the matrix argument Kloosterman sums
might be of some independent interest. Also we mention that it islikely that there exists ageometric interpretation of these formulas, when
the characteristic of $F$ is positive, as in the case of Jacquet-Ye
Kloosterman sums proved by Ngo (Duke Math. J. 96, 1999)$\mathrm{G}\mathrm{L}_{n^{-}}$