On the zeta function for the space of binary cubic forms and distributions of discriminants of cubic ring extensions(Automorphic representations, L-functions, and periods)

On

the

zeta

function

for

the

space

of

binary

cubic

forms

and

distributions of

discriminants of cubic ring

extensions

Takashi

Taniguchi (

谷口隆

)

Department

of

Mathematical

Sciences

University of

Tokyo

(

東京大学大学院数理科学研究科

)

1

Introduction

The aim of this note is to give

a

brief introduction

on

applications of Sato-Shintani’s

zeta functions (so called the zeta functions of prehomogeneous vector spaces) to

alge-braic number theory along the line with the author’s preprints $[\mathrm{T}06\mathrm{a}, \mathrm{T}06\mathrm{b}]$, which is a

generalization of Shintani’s

papers

[Sh72, Sh75]. For simplicity

we

mainly consider the

situation of $[\mathrm{T}06\mathrm{a}]$

.

We state

some

of the main results of $[\mathrm{T}06\mathrm{b}]$ in

Section

3.

We start with the main results of this note. Let $k$ be

a

number field and $O$ the ring

ofintegers of$k$

.

Let $\mathfrak{M}_{\mathrm{R}}$ and $\mathfrak{M}_{\mathbb{C}}$ respectively the set of real places and complex places

of $k$. Further let $\mathfrak{M}_{\infty}=\mathfrak{M}_{\mathrm{R}}\mathrm{I}\mathrm{I}\mathfrak{M}_{\mathbb{C}}$

.

We put $r_{1}=\#\mathfrak{M}_{\mathrm{R}},$ $r_{2}=\neq \mathfrak{M}_{\mathrm{C}}$ and $n=[k : \mathbb{Q}]$

.

We denote by $\Delta_{k},$ $h_{k}$ and $\zeta_{k}(s)$ the absolute discriminant, the cl\"ass number and

the

Dedekind

zeta

function

of$k$, respectively.

To classify cubicextensions of$k$ via the splitting typeat places of$\mathfrak{M}_{\infty}$,

we

introduce

the following notation. Let $k_{\infty}=k\otimes_{\mathbb{Q}}$R. We fix

a

separable cubic $k_{\infty}$-algebra $L_{\infty}=$

$\prod_{v\in \mathfrak{M}_{\infty}}L_{v}$, where $L_{v}\in\{\mathbb{R}^{3}, \mathbb{R}\cross \mathbb{C}\}$ if$v\in \mathfrak{M}_{\mathrm{R}}$ and $L_{v}=\mathbb{C}^{3}$ if$v\in \mathfrak{M}_{\mathbb{C}}$

.

Let

$h(L_{\infty}, n):=\#\{(R, F)|_{R\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}F\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}O,\mathrm{a}\mathrm{n}\mathrm{d}N(\Delta_{R/\mathit{0}})=n}^{F\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}k,F\otimes_{\mathrm{Q}}\mathbb{R}\cong L_{\infty}}’$

.

$\}$ .

Here $\triangle_{R/\mathrm{O}}$ is the relative discriminant of $R/O$ (which is an integral ideal of O) and

$N(\Delta_{R/O})$ is its ideal norm. We count pairs $(R, F)$ up to isomorphism. Weput $i(L_{\infty})=$ $\#\{v\in \mathfrak{M}_{\mathrm{R}}|L_{v}=\mathbb{R}^{3}\}$

.

The following is

a

main result of $[\mathrm{T}06\mathrm{a}]$

.

Theorem 1.1 For

any

$\epsilon>0$

,

$\sum_{n<X}h(L_{\infty}, n)=\frac{\mathfrak{U}_{k}}{3^{i(L_{\infty})+r_{2}}}X+\frac{\mathfrak{B}_{k}}{3^{i(L_{\infty})/2}}X^{5/6}+O(X^{\frac{5n-1}{5n+1}+\epsilon})$ $(X-\infty)$,

where

we

put

Remark 1.2 The

case

$k=\mathbb{Q}$ is essentially known by Shintani $[Sh7\mathit{5}]$. In the

formula

above, $X^{5/6}$-term is relevant when _$n=1,2$.

We explain one

more

theorem

we

consider in this note. We call a finite O-algebra

a cubic algebra if it is projective of rank 3 as

an

$\mathcal{O}$-module. We denote by $C(\mathcal{O})$ the

set of isomorphism classes of cubic algebras of $O$

.

For

a

fractional ideal $\alpha$ of $k$,

we

put

$C(\mathcal{O}, a)=\{R\in C(O)|\wedge^{3}R\cong a\}$

.

It is known that $C(\mathcal{O}, a)$ depends only on the ideal

class of $\alpha$ and that $C(O)=\mathrm{I}\mathrm{I}_{\alpha\in \mathrm{C}1(k)}C(\mathcal{O}, \alpha)$ (we

use

the same symbol $a$ to denote its

ideal class.) In general for

a

projective $\mathcal{O}$-module $M$ of rank _$m$, the class of the ideal

isomorphic to $\wedge^{m}M$ is called the Steinitz class of $M$. It is known that finite generated

projective modules over

a

Dedekind domain

are

classified by the rank and the

Steinitz

class. For this fact,

see

Milnor’s

_boo.k

[M71].

We

count the

number of$C(\mathcal{O}, a)$for each $\alpha$

.

More precisely, for each$L_{\infty}$

we

count the

set $C(O, \alpha, L_{\infty})=\{R\in C(O, a)|R\otimes_{\mathrm{Z}}\mathbb{R}\cong L_{\infty}\}$

.

An

interesting phenomenon

we

prove

in the

case

$k$ is

a

quadratic field isthat, the Steinitz class is not uniformly distributedin

the $X^{5/6}$-term if $\mathrm{C}1(k)$ contains

a

non-trivial 3-torsion element. Let $\#(\mathrm{A}\mathrm{u}\mathrm{t}(R))$ be the

cardinality ofthe automorphismsof$R$

as

an $\mathcal{O}$-algebra and$h_{k}^{(3)}$ the numberof3-torsions

of$\mathrm{C}1(k)$ (which is a

power

of3.)

Theorem 1.3 For any $\epsilon>0$,

$R \in C(\mathcal{O},a,L)N(\Delta_{R/\mathcal{O}})\leq X\sum_{\infty}\frac{1}{\#(\mathrm{A}\mathrm{u}\mathrm{t}(R))}=(1+\frac{1}{3^{i(L_{\infty})+r_{2}}})\frac{\mathfrak{U}_{k}}{h_{k}}X+\tau(\alpha)\frac{\mathfrak{B}_{k}h_{k}^{(3)}}{3^{i(L_{\infty})/2}h_{k}}X^{5/6}+O(X^{\frac{4n-1}{4\mathfrak{n}+1}+e})$

as

$Xarrow\infty$. Here

for

$a\in \mathrm{C}1(k)$,

we

put $\tau(\alpha)=1$

if

there exists $\mathrm{b}\in \mathrm{C}1(k)$ such that $a=\mathrm{b}^{3}$ and_{$\tau(\alpha)=0$} otherwise.

Our approach toprove the theorems above

are

the

use

ofthe zeta function theoryof

prehomogeneous vector spaces founded by Sato-Shintani [SS74]. In provingdensity

the-orems, this is an alternative approach to using reduction theory. These two approaches

are

both useful and have different strength. One advantage of zeta function theory

is that

we can

obtain

a

sharp

error

term estimate because

our

zeta function satisfies

the functional equation. For the reduction theory approach,

see

[DH71] or [B05], for example.

2

The

space of

binary

cubic

forms

and

the

zeta

func-tion

of

Sato-Shintani

We first prove Theorem 1.3 and after that Theorem 1.1. We first sketch the proof of Theorem

1.3

and next ofTheorem 1.1. Theorem

1.3

is proved by studying the space

of

binary cubic forms $(\mathrm{G}\mathrm{L}_{2}, \mathrm{S}\mathrm{y}\mathrm{m}^{3}\mathrm{A}\mathrm{f}\mathrm{f}^{2})$both algebraically and analytically. The idealclass

group $\mathrm{C}1(k)$ naturally arises from both parts.

Let $G$ be the general linear group of rank 2 and $V$ the space ofbinary cubic forms;

$G:=\mathrm{G}\mathrm{L}_{2}$,

We

define

the action of $G$

on

$V$ by

$(g \cdot x)(u, v)=\frac{1}{\det(g)}x((u, v)g)$.

The twist by $\det(g)^{-1}$ is to make the representation faithful. For $x=x(u, v)=au^{3}+$

$bu^{2}v+cuv^{2}+dv^{3}\in V$, let $P(x)$ be the discriminant;

$P(x):=b^{2}c^{2}-4ac^{3}-4b^{3}d+18abcd-27a^{2}d^{2}$

.

Then

we

have $P(g\cdot x)=(\det g)^{2}P(x)$

.

2.1

Parameterizations

of

cubic algebras

(algebraic part)

We consider a group

theoretical

parameterization of $C(O, a)$, which is

a

natural

gener-alization of

Delone-Faddeev’s

correspondence [DF64]

over

Z. Deflnition 2.1 We put

$V(k)\supset V_{a}:=a\oplus O\oplus a^{-1}\oplus a^{-2}$

$:=\{au^{3}+bu^{2}v+cuv^{2}+dv^{3}|a\in a, b\in O, c\in a^{-1}, d\in a^{-2}\}$,

$G(k)\supset G_{\mathrm{Q}}:=:=\{|p\in O,$$q\in a,$ $r\in a^{-1},$$s\in O,ps-qr\in O^{\mathrm{x}}\}$

.

Then $G_{\mathrm{Q}}\cdot V_{a}\subset V_{\mathrm{Q}}$

.

Remark 2.2 We

can

regard$V_{\mathrm{Q}}$

as

the space

of

cubic maps

fiom

$O\oplus a$ to

$\alpha\cong\wedge^{2}(O\oplus\emptyset)$

.

Proposition 2.3 (1) There eccists the canonical bijection between $C(O, \alpha)$ and $G_{\mathfrak{g}}\backslash V_{a}$

making the following diagram commutative.

$G_{\mathfrak{g}}\backslash V_{\mathrm{Q}}\downarrow P$ $C(O,a)\downarrow dis\mathrm{c}nminant$

$(O^{\mathrm{x}})^{2}\backslash a^{-2}rightarrow \mathrm{x}\alpha^{2}${integml ideals

of

$\mathit{0}$

}.

Here, the $r\dot{\mathrm{u}}ght$ vertical

arrow

is to take the discriminant, and the low horizontal

arrow

is given by multiplying $a^{2}$

.

Moreover, this diagram is

functo

rial with oespect

to the ring homomorphism

of

Dedekind

domains.

(2) For each $R\in C(O, a)$,

we

denote by $x_{R}$ the corresponding element in $G_{a}\backslash V_{a}$

.

Then

Aut$(R)\cong \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}(G_{a};x_{R}):=\{\gamma\in G_{\alpha}|\gamma\cdot x_{R}=x_{R}\}$ .

Construction

_of

the map For each $R\in C(O, a)$, thebinary cubic form

$x_{R}$: $R/Oarrow\wedge^{2}(R/O)$, $\xi-\xi\wedge\xi^{2}$

can

be regarded

an

element of $G_{\mathfrak{g}}\backslash V_{\mathfrak{g}}$, since $R\in C(O, a)$ implies $R/O\cong O\oplus a$

.

This

map $R\vdasharrow x_{R}$ gives the desiredbijection. For the proof see [

2.2

Zeta

function

(analytic part)

The representation $(G, V)$ is an example of what is called the prehomogeneous vector

space and for such a representation, M. Sato and Shintani [SS74] associated a zeta

function. This zeta function is a Dirichlet series satisfying certain functional equation. Werecallthe adelic version ofthe zetafunction for $(G, V)$. Let$V’=\{x\in V|P(x)\neq 0\}$

.

Let A be the adele ring of $k$

.

We denote by $\mathcal{J}(V(\mathrm{A}))$ the space of Schwartz-Bruhat

functions on $V(\mathrm{A})$

.

Let $\mathrm{C}1(k)^{*}$ be the set of characters of $\mathrm{C}1(k)$. Via the canonical

surjection $\mathrm{A}^{\mathrm{x}}/k^{\mathrm{x}}arrow \mathrm{A}^{\mathrm{x}}/k_{\infty}^{\mathrm{x}}k^{\mathrm{x}}\hat{\mathcal{O}}^{\mathrm{x}}\cong \mathrm{C}1(k)$,

we

regard elements of $\mathrm{C}1(k)^{*}$

as

characters

on

$\mathrm{A}^{\mathrm{x}}/k^{\mathrm{x}}$.

Definition

2.4 For $\Phi\in\triangleleft V_{\mathrm{A}}$), $s\in \mathbb{C}$,

cv

$\in \mathrm{C}1(k)$,

we

define

$Z( \Phi, s,\omega):=\int_{G(\mathrm{A})/G(k)}\omega(\det g)|\det g|_{\mathrm{A}}^{2\epsilon}\sum_{x\in V’(k)}\Phi(g\cdot x)dg$

and call it the global zeta function.

Weconsider the meaning of this function. Asusual, let $\hat{O}=\hat{\mathbb{Z}}\otimes_{\mathrm{Z}}O$where$\hat{\mathbb{Z}}=\prod_{p}\mathbb{Z}_{\mathrm{p}}$

and $\mathrm{A}_{\mathrm{f}}=\hat{O}\otimes_{\mathcal{O}}k$

.

Recall that

we

put _{$k_{\infty}=k\otimes_{\mathbb{Q}}$}R. For

our purpose, we

assume

the

following.

Assumption 2.5 We assume$\Phi\in\triangleleft V(\mathrm{A}))$ to be

of

the

form

$\Phi=\Phi_{\infty}\otimes\Phi_{f}$, where $\Phi_{f}$is

the characte$r\dot{\tau}stic$

function

on

$V(\hat{O})\subset V(\mathrm{A}_{\mathrm{f}})$, and $\Phi_{\infty}$ is

an

arbitrary

Schwartz-Bruhat

function

on$V(k_{\infty})$.

For a fractional ideal $a$,

we use

the same symbol

a

to denote thecorrespondingfinite

idele, which is well defined up to $\hat{O}^{\mathrm{x}}$

-multiple. That is, $\mathfrak{a}\in \mathrm{A}_{\mathrm{f}}^{\mathrm{x}}(\subset \mathrm{A}^{\mathrm{x}})$ is

characterized

by $a=k\cap a\hat{O}$. _{It is known that the double coset} space $G(k_{\infty})G(\hat{O})\backslash G(\mathrm{A})/G(k)$ is

represented by $\mathrm{C}1(k)$. More precisely, we have

$G( \mathrm{A})=\prod_{\alpha\in \mathrm{C}1(k)}G(k_{\infty})G(\hat{O})G(k)$

.

According to this decomposition,

we

define the partial zeta integral by

$Z_{\alpha}( \Phi, s):=\int_{G(k_{\infty})G(\hat{\mathcal{O}})(_{0\alpha}^{10})c(k)/G(k)}|\det g|_{\mathrm{A}}^{2\epsilon}\sum_{x\in V’(k)}\Phi(g\cdot x)dg$

.

Then since$\omega(\det(G(k_{\infty})G(\hat{O})G(k)))=\omega(k_{\infty}^{\mathrm{x}}\hat{O}^{\mathrm{x}}k^{\mathrm{x}})=1$,

we

have

$Z( \Phi, s, \omega)=\sum_{\alpha\in \mathrm{C}1(k)}\omega(a)Z_{a}(\Phi, s)$

.

Deflnition 2.6 (1) Let $\tau_{\infty}$ be the set

of

all possible separable cubic algebras $L_{\infty}$

of

the

form

$\prod_{v\in\varpi\iota}L_{v}$

.

Then set

of

orbits $G(k_{\infty})\backslash V’(k_{\infty})$ corresponds bijectively

to

$\tau_{\infty}$

.

not be

confused to

the

set

_of

$L_{\infty}$ rational points

of

$V.$) We

define

the local zeta

function

at$\mathfrak{M}_{\infty}$ by

$Z_{L_{\infty}}( \Phi_{\infty}, s)=\int_{G(k_{\infty})}|P(g_{\infty}x)|_{\infty}^{s}\Phi_{\infty}(g_{\infty}\cdot x)dg_{\infty}$

where $x$ is an arbitrary element

of

$V_{L_{\infty}}$

.

Here the invariant

measure

$dg_{\infty}$

on

$G(k_{\infty})$

is chosen

so

that $dg=dg_{\mathrm{f}}dg_{\infty}$ where $dg_{\mathrm{f}}$ is the inva$r\cdot iant$ measure on $G(\mathrm{A}_{\mathrm{f}})$ giving

the volume

_of

$G(\hat{O})$

one.

(2) We

_define

$\xi(L_{\infty},a;s)=\sum_{R\in C(\mathcal{O},aL_{\infty})},\frac{(^{\#}\mathrm{A}\mathrm{u}\mathrm{t}(R))^{-1}}{N(\Delta_{R’ \mathcal{O}})^{s}}$

.

Proposition

2.7

We have

$Z_{\mathfrak{g}}( \Phi, s)=\sum_{L_{\infty}\in \mathcal{T}_{\infty}}Z_{L_{\infty}}(\Phi_{\infty}, s)\xi(L_{\infty}, a;s)$

.

Let $G(\hat{O})_{\mathfrak{g}}=(_{\mathrm{A}^{\alpha}}^{10})^{-1}G(\hat{O})(_{0a}^{10})$ and $\Phi_{a}(x)=\Phi((_{0\mathfrak{g}}^{10})x)$. Then since $\Phi_{a}$ is $G(\hat{O})_{\alpha^{-}}$

invariant, $|\det(G(O)_{\alpha}G(k))|_{\mathrm{A}}=1$ and $|\alpha|_{\mathrm{A}}=N(\alpha)^{-1}$,

we

have

$Z_{a}( \Phi, s)=N(a)^{-2s}\int_{G(k_{\infty})G(\overline{\mathcal{O}}_{a})/G(k)\cap G(k)G(\overline{\mathcal{O}})_{a}}\infty|\det g_{\infty}|_{\infty}^{2s}\sum_{x\in V’(k)}\Phi_{a}(g_{\infty}\cdot x)dg_{\infty}dg_{\mathrm{f}}$

.

We

can

easily

see

that, as a subset of$V(k_{\infty})$

or

$G(k_{\infty})$,

$V(k)\cap(_{0a}^{10})^{-1}V(\hat{O})=V_{a}$, $G(k)\cap(_{0a}^{10})^{-1}G(\hat{O})(_{0a}^{10})=G_{\alpha}$

.

Hence

$Z_{a}( \Phi, s)=N(a)^{-2s}\int_{G(k_{\infty})/G_{\mathfrak{g}}}|\det g_{\infty}|_{\infty}^{2\epsilon}\sum_{x\in V_{l}\cap V’(k)}\Phi_{\infty}(g_{\infty}\cdot x)dg_{\infty}\cross\int_{G(\hat{\mathcal{O}})_{a}}dg_{\mathrm{f}}$

.

Since $G(\mathrm{A}_{\mathrm{f}})$ is unimodular, $\int_{G(\hat{\mathcal{O}})_{\alpha}}dg_{\mathrm{f}}=\int_{G(\hat{\mathcal{O}})}dg_{\mathrm{f}}=1$

.

Now by the usual unfolding

method we have

$Z_{a}( \Phi, s)=\sum_{L_{\infty}\in \mathcal{T}_{\infty}}Z_{L_{\infty}}(\Phi_{\infty}, s)(x\in G_{\mathfrak{g}}\backslash (\sum_{\infty}\frac{(\neq \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}(G_{a};x))^{-1}}{N(a)^{2s}|P(x)|_{\infty}^{\epsilon}})V_{\alpha}\cap V_{L})$

.

2.3

Analytic properties

of the

zeta

function and Tauberian

the-orem

Let $\{a_{n}\}$ be a positive sequence. We put

$A(X)= \sum_{n\leq\lambda’}a_{n}$, $a(s)= \sum_{n\geq 1}a_{n}n^{-s}$

.

Then Tauberian theorem states that, from analytic properties of $a(s)$ as a complex

function,

we

can obtain

some

informations

on

the asymptotic behavior of$A(X)$

as

$Xarrow$

$\infty$

.

If$a(s)$ is the Dirichlet series $\xi(L_{\infty}, a;s)$ in Definition

2.6

then $A(X)$ is nothing but

the left hand side of Theorem

1.3.

Hence

we

can

reduce the proof of Theorem 1.3 to

the analysis of $Z(\Phi, s,\omega)$

.

Since

$V$ is

a

vector space,

we can use

the Fourier analysis to

study the zeta functin. The analytic properties of$Z(\Phi, s, \omega)$

was

studied extensively by

Shintani [Sh72] when $k=\mathbb{Q}$ with the trivial character and later generalized by Wright

[Wr85] using adelic language. For $\omega\in \mathrm{C}1(k)^{*}$, let $\delta(\omega)=1$ if

cv

is trivial and $\delta(\omega)=0$

otherwise.

Theorem

2.8

(Shintani [Sh72], Wright [Wr85]) The zeta

_function

$Z(\Phi, s,\omega)$

can

be continued holomorphically to the entire $\mathbb{C}$ except

for

possible simple poles

at

$s=$

$0,1/6,5/6,1$_{. We have}

${\rm Res}_{s=1}Z(\Phi, s, \omega)=\delta(\omega)\Sigma_{a}(\Phi)$, ${\rm Res}_{s=5/6}Z(\Phi, s, \omega)=\delta(\omega^{3})\Sigma_{b}(\Phi)$

for

appropriate invariant distributions $\Sigma_{a},$ $\Sigma_{b}$

.

Also it

satisfies

the

functional

equation

$Z(\Phi, s, \omega)=Z(\hat{\Phi}, 1-s.\omega^{-1})$

where $\hat{\Phi}$

is

an

appropriate Fourier

_{transform of}

$\Phi$

.

For the definitions of $\Sigma_{a},$ $\Sigma_{b}$ and

$\hat{\Phi}$

,

see

[Wr85]. From this theorem, combined with

archimedean local theory,

we

know the functional equation and residue formula of

$\xi(L_{\infty}, a;s)$.

Corollary 2.9 (Datskovsky-Wright [DW86]) The Dirichlet series $\xi(L_{\infty}, a;s)$

can

be continued holomorphically to the whole complex plane except

for

a

simple pole at

$s=1$ and apossible simple pole at$s=5/6$

.

The residues

are

$(1+3^{-r(L_{\infty})-r_{2}})\mathfrak{U}_{k}/h_{k}$ and

$\tau(a)(5/6)\mathfrak{B}_{k}3^{-r(L_{\infty})/2}(h_{k}^{(3)}/h_{k})$, respectively. Also it

satisfies

functional

equation

of

the

form

$\xi(L_{\infty}, a;1-s)=(\Gamma(s)^{2}\Gamma(s+\frac{1}{6})\Gamma(s-\frac{\mathrm{i}}{6}))^{n}\sum_{\lambda\in\Lambda}P_{\lambda}(e^{\pi\sqrt{-1}s}, e^{-\pi\sqrt{-1}s})\hat{\xi}_{\lambda}(s)$,

where

A

is

a

_finite

set, $P_{\lambda}(x, y)$

are

polynomials in $x,$$y$ with degrees do not exceed $2n$

,

and $\hat{\xi}_{\lambda}(s)$

are

certain Dirichlet

series with the absolute

convergence

domains ${\rm Re}(s)>1$

.

Now Theorem 1.3 follows from Sato-Shintani’s Tauberian theorem [SS74] which is

a

2.4

Contributions from

“reducible”

algebras

$V\supset W:=\{v(bu^{2}+cuv+dv^{2})|b, c, d\in \mathrm{A}\mathrm{f}\mathrm{f}\}$.

The step from Theorem 1.3 to Theorem 1.1 is to separate the “reducible” algebras i.e.,

$R\in \mathcal{O}$ with $R\otimes k$ not Pelds. Let us define

$G\supset B:=\{$

(

$0*$

)

$\}$ ,

Then $(B, W)$ is

also

a

prehomogeneous vector space.

Shintani

[Sh75]

_showed

_in

_the

case

$k=\mathbb{Q}$ that the representation $(B, W)$ parameterizes the

reducible

algebras. We

see

in [$\mathrm{T}06\mathrm{a}$,

Section

3] that it is true for

a

general number field. We briefly

recall the

argument. Let $\alpha,$ $\mathrm{c}$ be

non-zero fractional

ideals of$k$

.

Deflnition 2.10 We put

$B(k)\supset B_{a,\mathrm{c}}=\{|t,p\in O^{\mathrm{x}},$ $u\in a^{-1}\mathrm{c}^{-2}\}$ , $W(k)\supset W_{a,\mathrm{c}}=\{y|y_{1}\in \mathrm{c}, y_{2}\in a^{-1}\mathrm{c}^{-1}, y_{3}\in\alpha^{-2}\mathrm{c}^{-3}\}$.

Then $W_{a,\mathrm{c}}$ is $B_{a,\mathrm{c}}$-invariant.

Let $V_{a}^{\mathrm{r}\text{\’{e}}}=$

{

_{$x\in$ $V|R_{x}\otimes k$} is not

a

field}

where

we

denote by _{$R_{x}\in C(O, a)$} the element corresponding to$x\in V_{\mathfrak{g}}$. We fix $a$

.

Proposition 2.11 For each$\mathrm{c}$, there exists the canonicalmap $\psi_{a,t}$: $B_{\alpha,t}\backslash W_{a,\mathrm{c}}arrow G_{a}\backslash V_{\mathrm{Q}}^{\mathrm{r}\mathrm{e}\mathrm{d}}$

which preserve the value

_of

$P$ up to $(O^{\mathrm{x}})^{2}$-multiple. Moreover,

$\prod_{\mathrm{c}\in \mathrm{C}1(k)}B_{a,\mathrm{c}}\backslash W_{\alpha,\mathrm{c}}arrow G_{a}\backslash V_{\alpha}^{\mathrm{r}\text{\’{e}}}$

$is$ “almost bijective”.

For

the

precise meaning

of “almost

bijective”,

see

[$\mathrm{T}06\mathrm{a}$, Proposition 3.12].

We

give

the

construction of $\psi_{a},‘$

.

We fix _$q,$ $s\in k$ such that $q\alpha^{-1}+sO=\mathrm{c}$

.

Then, $q\in a\mathrm{c},$ $s\in \mathrm{c}$,

and also there exist $p\in \mathrm{c}^{-1},$ $r\in a^{-1}\mathrm{c}^{-1}$ such that ps–qr $\in O^{\mathrm{x}}$

.

We can choose such

elementsbecause $O$ is a Dedekind domain. We put $g_{a,\mathrm{c}}=(_{rs}^{pq})\in G(k)$. We define

$\tilde{\psi}_{\mathfrak{g},\mathrm{c}}$:

$W_{\alpha,\mathrm{c}}arrow V_{\alpha,\mathrm{c}}^{\mathrm{r}\mathrm{e}\mathrm{d}}$,

$y\mapsto g_{a,\mathrm{c}}y$

.

We see by computation that $g_{\alpha,\mathrm{c}}^{-1}G_{a}g_{a,\mathrm{c}}\cap B(k)=$ $B_{\alpha,\mathrm{c}}$

.

This shows that the map $\tilde{\psi}_{a,\epsilon}$

induces

a

well defined map $\psi_{a},$

‘: $B_{a},‘\backslash W_{\alpha,\mathrm{c}}arrow G_{a}\backslash V_{a,\mathrm{c}}^{\mathrm{r}\text{\’{e}}}$

.

It turns out that this map does

not depend

on

the choice of$g_{a,\mathrm{c}}$

.

Hence the

same

analytic processyields the asymptotic formula of reducible algebras.

The global theory for $(B, W)$ was done byShintani [Sh75] and the author [$\mathrm{T}06\mathrm{a}$,

Section

4] gave

an

adelic version of Shintani’s treatment. By subtracting this

from

the

formula

3Splitting conditions at

non-archimedean

places

In Theorems 1.1, 1.3 we classify $C(\mathcal{O})$ via the splitting type at infinite places. Recently

the author $[\mathrm{T}06\mathrm{b}]$ consider the

same

problem under imposing finite number of splitting

conditions at non-archimedean places. We state

some

of its main results here.

Let $k$ be

a

general number field. We put $n=[k:\mathbb{Q}]$

.

For

a

place $v$ of$k$ let $k_{v}$ be the

completion of $k$ at $v$

.

Let $T$ be

a

finite set of places. Take

a

separable cubic algebra $L_{v}$

of $k_{v}$ for each $v\in T$ and let $L_{T}=(L_{v})_{v\in T}$ the $T$-tuple. We let

$C(O, L_{T})^{\mathrm{i}\mathrm{r}\mathrm{d}}:= \{R\in C(\mathcal{O})|F\otimes_{k}k_{v}\mathcal{O}F=R\bigotimes_{\cong}kL_{v}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}11v\in T\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}$

.

extension of

$k,$

$\mathrm{a}\mathrm{n}\mathrm{d}\}$ .

We define

$\theta_{L_{T}}^{\mathrm{i}\mathrm{r}\mathrm{d}}(s):=\sum_{R\in C(O,L_{T})^{\mathrm{i}\mathrm{r}\mathrm{d}}}\frac{\#(\mathrm{A}\mathrm{u}\mathrm{t}(R))^{-1}}{N(\triangle_{R/\mathcal{O}})^{\mathrm{s}}}$,

$h_{L_{T}}(X):=\#\{R\in C(O, L_{T})^{\mathrm{i}\mathrm{r}\mathrm{d}}|N(\Delta_{R/\mathit{0}})<X\}$

.

Theorem 3.1 There exist constants$\mathfrak{U}_{L_{T}}$ and $\mathfrak{B}_{L_{T}}$ described explicitly such that;

(1) $\theta_{L_{T}}^{\mathrm{i}\mathrm{r}\mathrm{d}}(s)$ has meromorphic continuation to the whole complex plane which is

holomor-phic

_for

${\rm Re}(s)>1/2$ except

for

simple poles at $s=1$ and5/6 with residues$\mathfrak{U}_{L_{T}}$ and $\mathfrak{B}_{L_{T}}$, respectively, and

(2)

_for

any $\epsilon>0$,

$h_{L_{T}}(X)=\mathfrak{U}_{L_{T}}X+(5/6)^{-1}\mathfrak{B}_{L_{T}}X^{5/6}+O(X^{\frac{5n-1}{6n+1}+\epsilon})$ $(Xarrow\infty)$.

Note that the $X^{5/6}$-term in the formula is relevant only when $n=1,2$

.

We give the

formulae of $\mathfrak{U}_{L_{T}}$ and $\mathfrak{B}_{L_{T}}$

.

We denote by $\mathfrak{M}_{\mathrm{f}}$ the set of all finite places. For $v\in \mathfrak{M}_{\mathrm{f}}$,

let $q_{v}$ be the order of the residue field of $k_{v}$

.

We put

$\theta_{L_{v}}=\#(\mathrm{A}\mathrm{u}\mathrm{t}_{k_{v}- \mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}}(L_{v}))$

.

For a non-archimedean local field $K$ with the order of residue field $q$, we define its local zeta

function by $\zeta_{K}(s)=(1-q^{-\epsilon})^{-1}$

.

The cubic algebra $L_{v}$ is in general a product of local

fields. We define $\zeta_{L_{v}}(s)$

as

the product of the zeta functions of those fields. The relative discriminant $\triangle_{L_{v}/k_{v}}$ is alsodefined

as

the product of relative discriminants of thoselocal

fields. We denote by $\Delta_{L_{v}}$ its norm. We put $i_{\infty}(L_{T})=\#\{v\in \mathfrak{M}_{\mathrm{R}}|L_{v}=\mathbb{R}^{3}\}$. We give

the value in

case

of $T\supset \mathfrak{M}_{\infty}$

.

The general

case

is easily obtained from this by taking a

suitable summation.

Theorem 3.2 When$T\supset \mathfrak{M}_{\infty}$, the constants$\mathfrak{U}_{L_{T}}$ and $\mathfrak{B}_{L_{T}}$

are

given by

$\mathfrak{U}_{L_{T}}=\frac{{\rm Res}_{s=1}\zeta_{k}(s)\cdot(_{k}(2)}{2^{r_{1}+r_{2}+1}3^{i(L_{T})+r_{2}}\infty}\prod_{v\in T\cap \mathfrak{M}_{\mathrm{f}}}\alpha_{v}(L_{v})$,

where

$\alpha_{v}(L_{v})===\frac{(1q_{v})1(1q_{v})2}{(1q_{v^{4}})(1q_{v})5}==\cdot\theta_{L_{v}}^{-1}\Delta_{L_{v}}^{-1}\cdot\frac{\zeta_{L_{v}}(2)}{\zeta_{L_{v}}(4)}$,

$\beta_{v}(L_{v})=\frac{(1-q_{v}^{-1/3})(1-q_{v}^{-1})}{(1-q_{v}^{-10/3})(1-q_{v}^{-4})}\cdot\theta_{L_{v}}^{-1}\triangle_{L_{v}}^{-1}\cdot\frac{\zeta_{L_{v}}(1/3)(_{L_{v}}(5/3)}{\zeta_{L_{v}}(2/3)\zeta_{L_{v}}(10/3)}$

.

Let $v\in \mathfrak{M}_{\mathrm{f}}$. We

see

by computation that _{$\sum_{L_{v}}\alpha_{v}(L_{v})=\sum_{L_{v}}\beta_{v}(L_{v})=1$} where

$L_{v}$

runs

through all the separable cubic algebras of $k_{v}$

.

Hence $\alpha_{v}(L_{v})$ and $\beta_{v}(L_{v})$ give

the proportion ofthe contributions of cubic algebras with local splitting type $L_{v}$

.

The

computation of $\alpha_{v}(L_{v})$ is reduced to the determination of certain orbital volume in

a

$v$-adic vectorspace. The meaning of$\beta_{v}(L_{v})$ is

more

subtleand the computation requires

a

careful local theory.

4

Quartic

case

Let $V=(\mathrm{S}\mathrm{y}\mathrm{m}^{2}\mathrm{A}\mathrm{f}\mathrm{f}^{3})^{*}\otimes \mathrm{A}\mathrm{f}\mathrm{f}^{2}$bethe space ofpairs ofternary quadraticforms. The

group

$G=\mathrm{G}\mathrm{L}_{3}\cross \mathrm{G}\mathrm{L}_{2}$ naturally acts on $V$

as a

linear representation. It is known that there

exist

a non-zero

relative invariant polynomial $P$ in $V$ and _{$V’:=\{x\in V|P(x)\neq 0\}$} is

a

single orbit

over

algebraically closedfields. For any field $k$, Wright and Yukie [WY92]

showedthat the setofnon-degeneraterational orbits$G(k)\backslash V$‘ $(k)$ correspondsbijectively

to the set ofseparable quartic algebras of $k$

.

Hence

we can

regard this representation

as

the quartic analogy of the space ofbinary cubic forms, thus it is naturally to carry

out the similar program tofind the densitytheorems ofdistributions of discriminants of

quartic algebras

or

analytic properties of the Dirichlet series counting quartic algebras.

As for the algebraic part, Bhargava [B04] recently discovered that the set ofintegral

orbits $G(\mathbb{Z})\backslash V(\mathbb{Z})$ corresponds bijectively to the set _{$\{(R, S)\}$} where $R$ is

a

quartic ring

and $S$ is

a

cubic resolvent ring of $R$

over

Z. (For the notion “resolvent ring” due

to Bhargava,

see

[B04].) This correspondence has

a

direct generalization to

over a

Dedekind domain

as we

did for the cubic

case

in Proposition

2.3.

The proof will be appear in elsewhere. Also there

are

three pairs $(P_{i}, W_{i})$ parameterizing the “reducible”

algebras, where $P_{i}$

are

parabolic subgroups and $W_{i}$ their invariant subspaces.

For the analytic part, the global theory of the quartic

case was

achieved by Yukie

[Y93] with large amount of technical computations. The remaining important problem

is the global theory for those $(P_{i}, W_{i})$

.

We hope this to carry out in the future.

Acknowledgment. I would liketo heartily thank for Professor T. Ikedafor giving

me

the opportunity of

a

presentation in this workshop.

References

[B04] M. Bhargava. Higher composition laws III: The parametrization of quartic

rings. Ann. Math., 159:1329-1360,

2004.

[B05] M. Bhargava. On the densityofdiscriminants ofquartic rings andfields. Ann.

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1971.

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[L15] E. Landau.

\"Uber

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[SS74] M.

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[T06a] T. Taniguchi. Distributions of discriminants ofcubic algebras. Preprint 2006,

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Invent. Math., 110:283-314,

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