SOME MEAN VALUE THEOREMS FOR THE SQUARE OF CLASS NUMBERS TIMES REGULATOR OF QUADRATIC EXTENSIONS(Analytic Number Theory)

SOME MEAN VALUE THEOREMS FOR THE SQUARE OF CLASS

NUMBERS TIMES REGULATOR OF QUADRATIC EXTENSIONS

TAKASHI TANIGUCHI

ABSTRACT. Inthis articlewegiveasurvey of [T1] and [T2] and discuss related topics.

Let$k$beanumberfield,and$\Delta_{k},$ $h_{k}$and$R_{k}$the absolutediscriminant,the classnumber

and the regulator, respectively. In [T2]wefoundthe asymptoticbehavior of themean

valuesof$h_{F}^{2}R_{F}^{2}$ withrespectto $|\Delta_{F}|$ for certain families ofquadratic extensions$F$ of

a fixed number field$k$.

1. INTRODUCTION

This article is

a

survey of [T1] and [T2]. We start with

our

main result. We fix

an

algebraic number field $k$. Let $\mathfrak{M},$ $\mathfrak{M}_{\infty},$ $\mathfrak{M}_{\mathrm{f}},$ $\mathfrak{M}_{\mathrm{R}}$ and $\mathfrak{M}_{\mathbb{C}}$ denote respectively the set of all places of$k$, all infinite places, all finite places, all realplaces and all complexplaces.

For $v\in \mathfrak{M}$ let $k_{v}$ denotes the completion of $k$ at $v$ and if$v\in \mathfrak{M}$ thenlet $q_{v}$ denote the

order of the residue field of$k_{v}$. We let _{$r_{1},$} $r_{2}$, and$e_{k}$ be respectively thenumber ofreal

places, thenumber of complex places, and the number of roots of unity contained in $k$.

We denote by $\zeta_{k}(s)$ the Dedekind zeta function of$k$.

To state

our

result,

we

classify quadratic extensions of $k$ via the splitting type at

places of $\mathfrak{M}_{\infty}$. Note that if $[F : k]=2$, then $F\otimes k_{v}$ is isomorphic to either $\mathbb{R}\cross \mathbb{R}$

or

$\mathbb{C}$ for $v\in \mathfrak{M}_{\mathrm{R}}$ and is $\mathbb{C}\cross \mathbb{C}$ for $v\in \mathfrak{W}$

.

We fixan $\mathfrak{U}t_{\infty}$-tuples $L_{\infty}=(L_{v})_{v\in\varpi\iota_{\infty}}$ where

$L_{v}\in\{\mathbb{R}\cross \mathbb{R}, \mathbb{C}\}$ for $v\in \mathfrak{M}_{\mathrm{R}}$ and $L_{v}=\mathbb{C}\cross \mathbb{C}$ for $v\in \mathfrak{M}_{\mathbb{C}}$. We define

$0(L_{\infty})=$

{

$F|[F$ : $k]=2,$$F\otimes k_{v}\cong L_{v}$ for all $v\in \mathfrak{M}_{\infty}$

}.

Let $r_{1}(L_{\infty})$ and$r_{2}(L_{\infty})$ be the number of real places and complex places of$F\in\Omega(L_{\infty})$,

respectively. (This does not depend

on

the choice of$F.$) For $v\in \mathfrak{M}_{\mathrm{f}}$

we

put

$E_{v}=1-3q_{v}^{-3}+2q_{v}^{-4}+q_{v}^{-5}-q_{v}^{-6}$, $E_{v}’=2^{-1}(1-q_{v}^{-1})^{3}(1+2q_{v}^{-1}+4q_{v}^{-2}+2q_{v}^{-\mathrm{S}})$

.

The following theorem is aspecial

case

of [T2, Theorem 10.12].

Theorem 1.1. We

_fix

an $L_{\infty}$ and _{$v_{1},$}

$\ldots,$

$v_{n}\in \mathfrak{M}_{\mathrm{f}}$ satisfying _{$\mathrm{r}_{2}(L_{\infty})-2r_{2}+n\geq 2$}

.

Then the limit

$\lim_{Xarrow\infty}\frac{1}{X^{2}}$

F.not

$F. \in\Omega(L)|\Delta_{F/\mathrm{k}}|\leq X\mathrm{p}1\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{t}v_{1,\ldots\prime}v_{n}\sum_{\infty}h_{F}^{2}R_{F}^{2}$

enis$ts$, and the value is equal to

$\frac{({\rm Res}_{\epsilon=1}\zeta_{k}(s))^{3}\Delta_{k}^{2}e_{k}^{2}\zeta_{k}(2)^{2}}{2^{r_{1}+r_{2+1}}2^{2r_{1}(L_{\infty})}(2\pi)^{2r_{2}(L_{\infty})}}\cdot\prod_{1\leq i\leq n}E_{v:}’\prod_{v\in \mathfrak{M}_{\mathrm{f}}}.E_{v}v\neq v_{1},..,v_{\hslash}$

.

Density theorems with local conditions at finite places

are

obtained simultaneously.

For details,

see

[T2, Theorem 10.12].

Combined with the result of Kable-Yukie [KY2], we also obtain the limit ofcertain correlation coefficients. For simplicity we state in the

case

$k=\mathbb{Q}$. We state the full

version of this theorem in Section 5.

$T$heorem 1.2. We

fix

a_prime$l$ satisfying_{$l\equiv 1(4)$}. Forany quadratic

field

$F=\mathbb{Q}(\sqrt{m})$

other than$\mathbb{Q}(\sqrt{l})$,

we

put $F^{*}=\mathbb{Q}(\sqrt{ml})$. For apositive number$X$, weput

$A_{l}(X)=\{F|_{F\otimes \mathbb{Q}_{l}isthequadraticunramified}^{[F:\mathbb{Q}]=2,-X<D_{F}<0}$’

extension

_of

$\mathbb{Q}_{l}$.

$\}$.

Then we have

$\lim_{Xarrow\infty}\frac{\sum_{F\in A_{l}(X)}h_{F}h_{F}}{(\sum_{F\in A_{l}(X)}h_{F}^{2})^{1/2}(\sum_{F\in A_{l}(X)}h_{F^{*)^{1/2}}}^{2}}.=\prod_{(\mathrm{f})=-1}(1-\frac{2p^{-2}}{1+p^{-1}+p^{-2}-2p^{-3}+p^{-5}})$

where $( \frac{\mathrm{p}}{l})$ is the Legendre symbol and_$p$ runs through all theprimes satisfying $(_{l}^{\mathrm{g}})=-1$

.

From this theorem, for example, we can observe that if

we

choose $l$

so

that $(_{l}^{\mathrm{g}})=1$

for all smallprimes$p$ then $h_{F}$ and $h_{F}$

.

have strong relation, and ifwe choose $l$ so that

$( \frac{p}{\iota})=-1$ for all small primes$p$then the relations between $h_{F}$ and $h_{F^{*}}$ become weak.

2. TAUBERIAN THEOREM

Our approach to prove the theorems above are the

use

of global zeta functions of

prehomogeneous vector spaces. Beforegiving a sketch oftheproof, we brieflyrecall the

Tauberian theorem to clarify the relation between Dirichlet series

we

consider and the asymptotic formulae in Theorems 1.1 and 1.2. Let $\{a_{n}\}_{n\geq 1}$ be a sequence of positive

numbers. We put

$a(s)= \sum_{n\geq 1}\frac{a_{n}}{n^{s}}$

$(s\in \mathbb{C})$,

$A(X)= \sum_{n\leq X}a_{n}$ $(X>0)$.

Then, roughlyspeaking,theTauberiantheoremsays thatwecanfind

some

informations

ofasymptotic behavior of $A(X)$ as $Xarrow\infty$ from the analytic properties of $a(s)$. The

following is a basic type ofthe Tauberian theorem.

Theorem 2.1. Assume $a(s)$ is holomorphic

for

$\Re(s)>a$ except

for

apole at $s=a$

of

order$b$. Let$\mathrm{c}/(s-a)^{b}$ be the leading

coefficient of

the Laurent $e\varphi ansion$ at $s=a$. Then

$\lim_{Xarrow\infty}\frac{A(X)}{X^{a}(\log X)^{b}}=\frac{\mathrm{c}}{ab!}$.

Hence to prove Theorem 1.1 it is enough to investigate the function

$h_{F}^{2}R_{F}^{2}$

$F: \mathrm{n}\mathrm{o}\mathrm{t}\epsilon \mathrm{p}11\mathrm{t}\cdot \mathrm{t}v_{1},\ldots,v_{n}\sum_{F\in 0(L_{\infty}\rangle}\overline{|\Delta_{F/k}|^{s}}$ .

On the other side this function is more or less the global zeta function of

one

specific

prehomogeneousvector space, andwe canstudy analytic propertiesofthe zetafunctions

3. PREHOMOGENEOUS VECTOR SPACES AND GLOBAL ZETA FUNCTIONS Webrieflyrecallthe definition of prehomogeneous vectorspacesandtheirapplications to numbertheory. For details,

see

[SS] or [Yl, Introduction]. For simplicitywe here give adefinition of

a

certain restricted class instead of the general. Let $k$ be

a

field.

Definition 3.1. An irreducible representationofaconnected reductive algebraic group

$(G, V)$

over

$k$ is called

a

prehomogeneous vector space if

(1) there exists

a

Zariski open$G$-orbit in $V$ and

(2) there exists a non-constant polynomial $P\in k[V]$ and a rational character $\chi$ of $G$

such that $P(gx)=\chi(g)P(x)$ for all$g\in G$ and $x\in V$.

The space of quadratic forms ($\mathrm{G}\mathrm{L}(n)$

,

Sym2

$k^{n}$) is a classical example. Irreducible

prehomogeneous vector spaces over an arbitrary characteristic $0$ algebraically closed

field

were

classified by Sato and Kimura in [SK]. Sato and Shintani [SS] defined global zetafunctions forprehomogeneous vectorspacesif$(G, V)$ isdefined

over

anumber field.

Letus recallthedefinition. Let $k$be

a

number field and A thering of adeles. We denote

by $|\cdot|_{\mathrm{A}}$ the idele

norm.

Let $(G, V)$ be

a

prehomogeneous vector space defined over $k$.

Let $P\in k[V]$ be of minimum degree satisfying (2) in Definition 3.1 (whichis uniqueup

to constant), and$\chi$ be the corresponding character. We put

$T:=\mathrm{k}\mathrm{e}\mathrm{r}(Garrow \mathrm{G}\mathrm{L}(V))$, $\tilde{G}:=G/T$, $V^{\mathrm{S}8}:=\{x\in V|P(x)\neq 0\}$.

Let$\ovalbox{\tt\small REJECT}(V(\mathrm{A}))$ be thespace ofSchwartz-Bruhatfunctionson$V(\mathrm{A})$. We fixaHaar

measure

$dg$

on

$\tilde{G}(\mathrm{A})$.

Definition 3.2. For $\Phi\in\ovalbox{\tt\small REJECT}(V(\mathrm{A}))$ and $s\in \mathbb{C}$ wedefine

$Z( \Phi, s):=\int_{\overline{G}(\mathrm{A})/\tilde{G}(k)}|\chi(g)|_{\mathrm{A}}\sum_{x\in V(k)}..\Phi(gx)dg$

and call it the globalzeta

_function.

Remark 3.3. For$x\in V^{86}(k)$ let$G_{x}=\{g\in\tilde{G}|gx=x\}$and$\tilde{G}_{x}^{\mathrm{o}}$its identity component.

We denoteby $\tau(\tilde{G}_{x}^{\mathrm{o}})$ the unnormalized Tamagawa number of$\tilde{G}_{x}^{\mathrm{o}}$. Roughly speaking the

global zeta function is a counting function of rational orbits $\tilde{G}(k)\backslash V^{\epsilon\epsilon}(k)$

with weight $\tau(\tilde{G}_{x}^{\mathrm{o}})$

.

We do not give the details here, but mention that by

a

standard

modification we have

$Z( \Phi, s)=\sum_{x\in\tilde{G}(k)\backslash V^{\epsilon\epsilon}(k)}\frac{\tau(\tilde{G}_{x}^{\mathrm{o}})}{[\tilde{G}_{x}(k):\tilde{G}_{x}^{\mathrm{o}}(k)]}\int_{\overline{G}(\mathrm{A})/\overline{G}_{[mathring]_{x}}(\mathrm{A}\rangle}|\chi(g_{x})|_{\mathrm{A}}^{s}\Phi(g_{x}x)dg_{x}$ ,

where$dg_{x}$ be

an

appropriate left $\tilde{G}(\mathrm{A})$-invariant

measure.

The interpretation of $\tilde{G}(k)\backslash V^{88}(k)$ and $\tilde{G}_{x}^{\mathrm{o}}$ in terms of field extensions for

preho-mogeneous

vector spaces is first established systematically in the celebrated work of Wright-Yukie [WY].

The remarkabove implies thatifweknow theinformation ofpolestructuresof$Z(\Phi, s)$,

one

can obtain the density theorems of $\tilde{G}(k)\backslash V^{\mathrm{s}\mathrm{s}}(k)$ with weight $\tau(\tilde{G}_{x}^{\mathrm{o}})$. On the other

side it is in general a very difficult problem to describe the principal parts of $Z(\Phi, s)$

explicitly and is

one

of the central problem in the theory of prehomogeneous vector

4. THE SPACE OF PAIRS OF 2 $\cross 2$ MATRICES AND ITS INNER FORMS

Let $\mathfrak{B}$ be aquaternion algebra

over

$k$. We denote by

$\mathfrak{B}^{\mathrm{o}\mathrm{p}}$ the opposite algebra of B.

Let us consider the representation $(G, V)=(G, \rho, V)$ where

(4.1) $G=\mathfrak{B}^{\cross}\mathrm{x}(\mathfrak{B}^{\mathrm{o}\mathrm{p}})^{\mathrm{x}}\mathrm{x}\mathrm{G}\mathrm{L}(2)$, $V=\mathfrak{B}\otimes k^{2}=\mathfrak{B}\oplus \mathfrak{B}$,

and

$\rho(g)(a\otimes v)=(g_{1}ag_{2})\otimes(g_{3}v)$ for $g=(g_{1},g_{2},g_{3})\in G,a\in \mathfrak{B},$ $v\in k^{2}$

.

We regard this representation as arepresentation of the algebraic group $G$

over

$k$

.

If $\mathfrak{B}=\mathrm{M}(2,2)$ then $V$ is the space ofpairs of $2\cross 2$ matrices, and in general $(G, V)$ is an

inner form of this split form. This is

an

example ofprehomogeneous vector space, and

there is

an

interesting interpretationof $\tilde{G}(k)\backslash V^{\mathrm{s}\mathrm{s}}(k)$ and $\tilde{G}_{x}^{\mathrm{o}}$

.

Proposition 4.2. (1) There $e$tists

a non-zero

polynomial $P$

of

$V$ and a rational

char-acter$\chi$

on

$G$ such that $P(gx)=\chi(g)P(x)$

.

(2) There exists the canonical bijection between$\tilde{G}(k)\backslash V\mathrm{a}\mathrm{e}(k)$ and the set

of

isomorphism

classes

_of

separable quadratic algebras

_of

$k$ those are embeddable into $\prime \mathfrak{B}$

.

For

$x\in$

$V^{8\mathrm{S}}(k)$, we denote by $k(x)$ the corresponding algebra.

(3) For$x\in V^{\mathrm{s}\mathrm{s}}(k),\tilde{G}_{x}^{\mathrm{o}}\cong(k(x)^{\mathrm{x}}/k^{\mathrm{x}})^{2}$ as an algebraic group

over

$k$.

with

On the principal parts formula, weproved thefollowing in [T1].

$T$heorem 4.3. Let$\mathfrak{B}$ be a non-split quaternion algebra. Then

$Z(\Phi, s)=Z_{+}(\Phi, s)+Z_{+}(\hat{\Phi}, 2-s)$

$+ \tau(G/T)(\frac{\hat{\Phi}(0)}{s-2}-\frac{\Phi(0)}{s})+\frac{Z_{\mathfrak{B}}(R\hat{\Phi},1/2)}{s-3/2}-\frac{Z_{\mathfrak{B}}(R\Phi,1/2)}{s-1/2}$

.

Here $Z_{+}(\Phi, s)$ is an integral entire as a jfunction

of

$s,$ $\tau(G/T)$ is the Tamagawa number

of

$G/T,$ $R\Phi$ the suitable restriction

of

$\Phi$ to 3(A), $\hat{\Phi}$

a

suitable $Fou$rier transform, and

$Z_{\mathrm{B}}$

.

the zeta

function

of

simple algebra associated to

$\prime \mathfrak{B}$.

Thisis provedby usingthe Fourieranalysis. The globaltheoryincludes rather

case

by

caseexplicit computation. The$k$-rankofthegroup $\tilde{G}$

affectsseriously onthecomplexity

of this computation and if $\mathfrak{B}$ is non-split then the computation becomes quite mild.

This is thereasonwhy weconsidernon-splitforms. For details

on

thistopic

see

Yukie’s treatise [Y1]. We mention that H. Saito’s method [Sa2] is

an

alternative strong tool to establish the global theory in some

cases.

5. CORRELATION COEFFICIENTS Since the split form of(4.1)

(5.1) $G=\mathrm{G}\mathrm{L}(2)\cross \mathrm{G}\mathrm{L}(2)\cross \mathrm{G}\mathrm{L}(2)$, $V=k^{2}\otimes k^{2}\otimes k^{2}$

has a high symmetry, there

are

many $k$-forms of this representation. One interesting $k$-forms is studied by Kable-Yukie in a series ofwork [KY1, Y2, $\mathrm{K}\mathrm{Y}2,$ $\mathrm{K}\mathrm{Y}3,$$\mathrm{K}\mathrm{Y}4$]. We

fixaseparable quadratic algebra $\tilde{k}$

. Wedenote by$H_{2}(\tilde{k})$ betheset ofbinary Hermitian

forms over $\tilde{k}$

.

$H_{2}(\tilde{k})=\{x\in \mathrm{M}(2,2;\tilde{k})|^{t}x^{\sigma}=x\}$. Thegroup $\mathrm{G}\mathrm{L}(2_{7}\tilde{k})$ acts naturally

on

thisspace by $(g,x)\mapsto gx^{t}g^{\sigma}$. Now

$G=\mathrm{G}\mathrm{L}(2,\tilde{k})\cross \mathrm{G}\mathrm{L}(2, k)$, $V=H_{2}(\tilde{k})\otimes k^{2}$

is an outer form of (5.1). For this

case

the following proposition is proved in [KY1]. Proposition 5.2. (1) There exists a non-zero polynomial $P$

of

$V$ and a rational

char-acter$\chi$ on $G$ such that $P(gx)=\chi(g)P(x)$.

(2) There exists the canonical bijection between $\tilde{G}(k)\backslash V^{\epsilon \mathrm{s}}(k)$ and separable quadratic

algebras

_of

$k$

.

For$x\in V^{\mathrm{s}\mathrm{s}}(k)$ we denote by$k(x)$ the coresponding algebra.

(3) For$x\in V\mathrm{a}\mathrm{e}(k),\tilde{G}_{x}^{\mathrm{o}}\cong(\tilde{k}\otimes k(x))^{\mathrm{x}}/\tilde{k}^{\mathrm{x}}$ as an algebraic group

over

$k$

.

Let $\tilde{k}$

and $k(x)$ be different quadratic extensions of $k$. Then the biquadratic field

$\tilde{k}\otimes k(x)$ contains anotherquadratic extensionof$k$

.

We denotethis fieldby$k(x)^{*}$

.

Then

by (3) the Tamagawanumber of$\tilde{G}_{x}^{\mathrm{o}}$is

more or

less the product $h_{k(x)}R_{k(x)}\cross h_{k(x)}\cdot R_{k(x)^{\mathrm{s}}}$

.

We will state the full version of [T2, Section 11]. Let $S$denote a finiteset of places of $k$ containing $\mathfrak{M}_{\infty}$ and $L_{S}=(L_{v})_{v\in S}$

an

$S$-tuple of separable quadratic algebra $L_{v}$ of$k_{v}$.

Let $F$ be a quadratic extension of $k$. We write $F\approx L_{S}$ to

mean

that $F\otimes k_{v}\cong L_{v}$ for

all $v\in S$. Let $X$ be a positivenumber. For convenience, we introduce the abbreviation

$\mathfrak{Q}(L_{S,}.X)=\{F|[F : k]=2, F\approx L_{S},N(\Delta_{F/k})\leq X\}$

.

Here,$N(\Delta_{F/k})$ is the ideal norm ofthe relative discriminant $\Delta_{F/k}$ of$F/k$

.

Definition 5.3. We define

$\mathrm{C}\mathrm{o}\mathrm{r}(L_{S})=\lim_{Xarrow\infty}\frac{\sum_{F\in \mathrm{O}(L_{\mathit{3}},X)}h_{F}R_{F}h_{F}*R_{F}}{(\sum_{F\in \mathfrak{Q}(L_{S},X)}h_{F}^{2}R_{F}^{2})^{1/2}(\sum_{F\in \mathrm{O}(L_{S},X)}h_{F^{*}}^{2}R_{F^{*)^{1/2}}}^{2}}$

.

ifthe limit of the right hand side exists and call it the comlation

_coefficient.

The asymptotic behavior of the numerator in the right hand side as $Xarrow\infty$

was

investigated in [KY2, $\mathrm{K}\mathrm{Y}3,$ $\mathrm{K}\mathrm{Y}4$], while the denominator is considered in [T2]. Hence

wecould find the correlation coefficients for certain types$\mathrm{o}\mathrm{f}k\sim$

and$L_{S}$

.

Let$\mathfrak{M}_{\mathrm{r}\mathrm{m}},$ $\mathfrak{M}_{\mathrm{n}}$ and $\mathfrak{M}_{8}\mathrm{p}$ be the sets of finite places of $k$ which

are

respectively ramified, inert and split

on

extensionto$\overline{k}$

.

For $v\in \mathfrak{M}$andaseparable quadratic algebra$L_{v}$, we definethe separable

quadratic algebra$L_{v}^{*}$ as follows. Let$\sim\sim k_{v}=k\otimes k_{v}$.

$\mathrm{I}\mathrm{f}k_{v}\underline{\simeq}k_{v}\sim\cross k_{v}$thenwe define

$L_{v}^{*}=L_{v}$

for any $L_{v}$. In the

case

$\sim k_{v}$ isa field, if_{$L_{v}=k_{v}\cross k_{v}$} then

we

let $L_{v}^{*}\cong k_{v}\sim$, and if $L_{v}\cong k_{v}\sim$

then we let $L_{v}^{*}=k_{v}\cross k_{v}$

.

Finally in the

case

$\sim k_{v}$ and $L_{v}$

are

distinct fields, we define

$L_{v}^{*}$ the

same

way as

we defined

$F^{*}$ for number fields. Let $\mathfrak{M}_{\mathrm{d}\mathrm{y}}=\{v\in \mathfrak{M}_{i}|v|2\}$. We

proved the following in [T2, Section 11].

Theorem 5.4. Assume$\mathfrak{M}_{\mathrm{r}\mathrm{m}}\cap \mathfrak{M}_{\mathrm{d}\mathrm{y}}=\emptyset$ and$S\supset \mathfrak{M}_{\mathrm{r}\mathrm{m}}$. Let_{$L_{S}=(L_{v})_{v\in S}$} is an S-tuple

of

separable quadratic algebras such that there

are

at leasttwoplaces$v$ with$L_{v}$ are

fields.

$R47ther$

assume

that there are at least twoplaces $v$ with $L_{v}^{*}$ are

fields.

Then we have

6. FURTHER PROBLEMS

Inthe invaluable work [WY], Wright and Yukie found good interpretations of rational orbits for 8

cases

including our

case

(5.1), and discussed the expected density theorems

for those cases. On the other hand, in the process [T1] and [T2] to proveTheorem 1.1,

the technicalheartisto considertheinnerformtohandlethe global theory. The k-forms

ofirreducible reduced regular prehomogeneous vector spacesover local and global fields

are classified byH. Saito [Sal], and we could

see

that

some

other

cases

treated in [WY] have inner forms. In this section, we will discuss the rational orbit decomposition for

some

innerform representations. The proofmaybe appear in aforthcomingpaper. Let

$k$be

an

arbitraryfield. Let $8_{i}$ be theset of isomorphism classes of separable algebras of

$k$ of degree $i$

.

(I) The

case

$(\mathrm{G}\mathrm{L}(3)\cross \mathrm{G}\mathrm{L}(3)\cross \mathrm{G}\mathrm{L}(2), k^{3}\otimes k^{3}\otimes k^{2})$.

Let $\mathcal{D}$ be asimple algebra ofdegree 3 over $k$

.

Then

$G=\mathcal{D}^{\mathrm{x}}\mathrm{x}(\mathcal{D}^{\mathrm{o}\mathrm{p}})^{\mathrm{x}}\cross \mathrm{G}\mathrm{L}(2)$, $V=\mathcal{D}\otimes k^{2}$

isaninner form. Let

₈₃

$(\mathcal{D})$ be theset ofisomorphism classes of separable cubicalgebras

of$k$ those

are

embeddable into D. Then the following propositionholds.

Proposition 6.1. (1) There exists a

non-zero

polynomial $P$

of

$V$ and a rational

char-acter$\chi$ on $G$ such that $P(gx)=\chi(g)P(x)$.

(2) Let $V^{99}=\{x\in V|P(x)\neq 0\}$

.

Then there exists the canonical bijection between

$G(k)\backslash V^{\mathrm{s}\mathrm{s}}(k)$ and $\mathcal{E}_{3}(\mathcal{D})$. $Forx\in V^{\mathrm{S}8}(k)$ we denote by $k(x)\in \mathcal{E}_{3}(\mathcal{D})$ be the

co7Ye-sponding algebra.

(3) For$x\in V^{8S}(k),$ $G_{x}^{\mathrm{o}}\cong k(x)^{\mathrm{x}}\cross k^{\mathrm{x}}$

as

an algebraic group over $k$

.

Romthis proposition, we mayobtain the densityof$h_{F}R_{F}$ ofcubicextensions $F$of$k$.

In the

case

$\mathcal{D}$ is not split, the principal parts of the global zeta function

were

described

in [T1]. It has possible simple pole at $s=0,1/6,4/3,3/2$ and holomorphic elsewhere.

The local theory and the filtering process necessary to obtain the density theorem

are

in progress.

(II) The

case

$(\mathrm{G}\mathrm{L}(4)\cross \mathrm{G}\mathrm{L}(2), \wedge^{2}k^{4}\otimes k^{2})$

.

Let$\mathfrak{B}$beaquaternionalgebraof$k$. Wedenoteby$H_{2}(\mathfrak{B})$be the set of binary Hermitian

forms

over

B. Then

$G=\mathrm{G}\mathrm{L}(2, \mathfrak{B})\cross \mathrm{G}\mathrm{L}(2)$, $V=H_{2}(\mathfrak{B})\otimes k^{2}$

is an inner form. For this casethe following proposition holds.

Proposition 6.2. (1) There exists a

non-zero

polynomial $P$

of

$V$ and a rational

char-acter$\chi$

on

$G$ such that $P(gx)=\chi(g)P(x)$

.

(2) Let $V^{8\mathrm{S}}=\{x\in V|P(x)\neq 0\}$. Then there exists the canonical bijection between

$G(k)\backslash V^{\mathrm{s}\epsilon}(k)$ and $\epsilon_{2}$

.

For $x\in V^{\mathfrak{B}}(k)$ we denote by $k(x)\in\epsilon_{2}$ be the corresponding

algebra.

(3) For $x\in V^{8S}(k)_{f}G_{x}^{\mathrm{o}}\cong(\mathfrak{B}\otimes k(x))^{\mathrm{x}}$ as an algebraic group

over

$k$.

(III) The

case

$(\mathrm{G}\mathrm{L}(6)\cross \mathrm{G}\mathrm{L}(2), \wedge^{2}k^{6}\otimes k^{2})$.

Let $H_{3}(\mathfrak{B})$ be the set of ternary Hermitian forms over B. Then just the same as the

above case,

is

an

inner form. For this

case

$\mathrm{t}$he following proposition holds.

Proposition 6.3. (1) There exists a

non-zero

polynomial $P$

of

$V$ and a rational

char-acter

$\chi$ on$G$ such that $P(gx)=\chi(g)P(x)$

.

(2) Let $V^{\mathrm{s}\mathrm{e}}=\{x\in V|P(x)\neq 0\}$. Then there exists the canonical bijection between

$G(k)\backslash V^{8\mathrm{S}}(k)$ and $\epsilon_{3}$

.

For$x\in V^{\mathrm{s}\mathrm{s}}(k)$ we denote by $k(x)\in\epsilon_{3}$ be the corresponding

algebra.

(3) For $x\in V^{\mathrm{s}\mathrm{s}}(k),$ $G_{x}^{\mathrm{o}}\cong\{g\in(\mathfrak{B}\otimes k(x))^{\mathrm{x}}|\mathrm{N}(g)\in k^{\cross}\}$ as an algebraic group

over

$k$

.

The principalparts ofthe global zeta function for (II) and (III)

are

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