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MINIMUM OF POSITIVE DEFINITE QUADRATIC
FORMS(Analytic Number Theory)
KITAOKA, YOSHIYUKI
数理解析研究所講究録 (1996), 958: 105-110
1996-08
http://hdl.handle.net/2433/60458
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Type
Textversion
Departmental Bulletin Paper
publisher
Kyoto University
数理解析研究所講究録
958 巻 1996 年 105-110
105
MINIMUM OF POSITIVE DEFINITE QUADRATIC FORMS
YOSHIYUKI KITAOKA(北岡良之)
Department of Mathematics, School of Science
Nagoya University
$(k\delta\prime z\mathrm{x}\tau \mathrm{E})$
We are concerned with reresentation of positive definite quadratic forms by a positive
definite quadra,tic form. Let us consider the following assertion
,
: Let $M,$ $N$ be positi $ve$ definite adra $tic$ la ttices over with rank
a $n\mathrm{d}rank(N)=n$ respectively. We
that the localiz a
is repre,
for every prime , th at is the is an isom etry from
to
. Then there exis a
constant
dependent only on $N$ so that $M$ is represented $b.vN$ if $\min(M)>c(N)$ ,
where nuin $(M)$ denotes
positive number represen
le
by $M$ .
We know that the assertion
is true if $n\geq 2m+3$ a.nd there are several results.
But, the present problem is whether the condition $7l\geq 2m+3$ is the best possible or
. It is known that this is
best if $rn=1$ , that is
is false. But in the case of
we know at present, is
is false if
. We do not know
anything the case of $n-m=4$ . Anywav, analyzing the counter-example, we come to
the fullowing two assertions
and
.
: There exists a. const ant
dependent only on $N$ so that $llI$ is
resen $tedb.\}^{f}N$ if $\min(N)>c’(N)$ a.nd
is
represented by
for every
$(M)=?\uparrow\iota$
$\mathrm{Z}$
$\mathrm{A}_{1n,n}$
$q\mathrm{u}$
$\mathrm{a}ss\mathrm{u}me$
$N_{\mathrm{p}}$
$p$
$\mathrm{s}e\mathrm{n}tedb.\gamma$
$t\mathrm{i}o\mathrm{J}’ M_{\mathrm{p}}$
$\mathrm{r}e$
$N_{p}$
$ll/I_{\mathrm{p}}$
$t\epsilon$
$\mathrm{c}(N)$
$tl_{l}e$
$t_{\sim}\rho d$
$a,\sim \mathrm{s}t$
$\mathrm{A}_{m,n}$
$\mathrm{t}1^{-}1\mathrm{e}$
$\mathrm{n}\mathrm{o}1_{}$
$\mathrm{A}_{1,4}$
$\mathrm{w}\mathrm{h}\mathrm{a}\mathrm{t}_{\iota}$
$m\geq\tau_{)}\sim,$
$\mathrm{t}\mathrm{h}\mathrm{a}\dagger,$
$n-m\underline{<_{\backslash }}3$
$\mathrm{A}_{t)}\iota,n$
$\mathrm{i}\mathrm{I}1$
$\mathrm{A}\mathrm{P}\mathrm{W}_{m,n}$
$\mathrm{A}\mathrm{P}\mathrm{W}_{m,n}$
$\mathrm{R}_{m,n}$
$c’(l\backslash ^{\tau}’)$
$T^{\rho}arrow p-$
$M_{p}$
$p\mathrm{r}i_{I}\mathrm{n}e_{I)}$
$pril\mathit{1}\mathit{1}iti\mathrm{v}\rho.\iota_{\mathrm{J}^{\gamma}}$
$N_{\mathrm{p}}$
.
: There is a lattice
cont
such th at
is primiti
rep.r.esented
by
prime and $\min(’M’)$ is still large if $\min(M)$ is 1 arge.
for
If the assertion
is true, then the assertion
is reduced to the apparently
weaker assertion
. If the assertion
is false,
it becomes possible to
make a counter-example to the assertion
.
matter of fact,
is true but
is false, and it yields examples of $N$ such that
is false.
Anywa it is important to study the behaviour of the minimum of quardatic la.ttices
when we vary thenl. Our aim is to show
$M^{l}$
$\mathrm{R}_{m,n}$
$N_{p}$
$e\mathrm{v}er\mathrm{J}^{f}$
$\mathrm{a}i\mathrm{n}ingM$
$\mathit{1}VI_{\mathrm{P}}^{;}$
$p$
$\mathrm{R}_{m,n}$
$\mathrm{A}\mathrm{P}\mathrm{W}_{m,n}$
$\mathrm{A}_{m,n}$
$\mathrm{t}_{}1_{1}\mathrm{e}\mathrm{n}$
$\mathrm{R}_{m,n}$
$\mathrm{A}_{?7,n},$
$\mathrm{R}_{1,4}$
$\mathrm{A}\circ,$
,
$\mathrm{a}$
$\Lambda_{1,4}$
$‘ \mathrm{y}$
$\mathrm{V}^{\rho}ly$
$\mathrm{A}\mathrm{P}\mathrm{W}_{1,4}$
106
Theorem. The assertion
$\mathrm{R}_{m,n}$
is true if $n-m>3,$
$?7,$
or $n=2\uparrow n\geq 12$ .
$\geq 2m+1$
is false, we can construct a counter-example to the asRemark. If the assertion ,
as above. When the case of $n<2\prime n$ seeIns to have a different nature fronu
sertion
the case of $n\geq 2m$ .
$\mathrm{R}_{n,n}$
$\mathrm{A}_{m,n}$
To prove , we are involved in analytic nulnber theory. The rest is a brief sumnlary
of the proof.
field of rational numbers
the ring of integers,
and
We denote by
and their p–adic comple,tions. Terminology and notation on qua.dratic forms are those
scale $s(M)$ deover
on a quadratic space
from [K]. For a lattice on
denotes a -module spanned by
notes $\{B(x, y)|x, y\in\Lambda f\}$ , and the norm
denote
it is similarly defined. $dM,$
$\{Q(X)|x\in M\}$ . Even for the localization
respectively. A positive lattice means a lattice on a positive
of $M,$
the
definite quadratic space over Q. We give proofs only for a few assertions.
$\mathrm{i}\dagger‘$
$\mathfrak{t}_{\mathrm{L}}\mathrm{h}\mathrm{e}$
$\mathrm{Z},$
$\mathrm{Q},$
$\mathrm{Q}_{p}$
$\mathrm{Z}_{\mathrm{p}}$
$V$
$\mathrm{A}l$
$\mathrm{Q},$
$n(\Lambda I)$
$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}$
$\mathrm{Z}$
$d\Lambda I_{\mathrm{p}}$
$ll,f_{\mathrm{p}}$
$\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{C}\mathrm{r}}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}11\mathrm{t}$
Definition.
$M_{\mathrm{p}}$
For a real number , we define the decimal part
$x$
$-1/2\leq\lceil x\rceil<1/2$
Note that
Definition.
$\lceil_{\backslash }\prime r1^{2}=\lceil-’.t1^{2}$
and
$\lceil x\rceil$
$x-\lceil x,\rceil\in \mathrm{Z}$
by the conditions
.
for every real number .
$x$
For positive numbers
$a,$
$b$
, we write
$a\ll_{m}b$
such that $a/b<c$ . If both
$a\ll_{m}b$
, we say that a positive definite matrix
$S^{(m)}=$
if there is a positive number dependent only on
hold, then we write
$c$
$\mathrm{a}\mathrm{l}\iota \mathrm{d}l)\ll_{m}$
$a$
$a_{\wedge m}^{\vee}b$
If
$m$
is
$\mathrm{a}1$
)
Definition.
$(s_{\mathrm{s},\mathrm{j}})$
is
$m$
absolute constant, then we omit
numbers
-diagonal if we have
For
$(c_{1}, c_{2})$
$\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{P}}$
$c_{1},$
$c_{2}$
$m$
.
.
$\mathrm{c}_{1}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(s1,1, \cdots, s_{m,m})<S<c_{2}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(s1,1, \cdots, s_{m,m})$
If is in the Siegel donlain , then there exist positive nunibers
-diagonal (see Ch.2 in [K]).
so that is
$S$
$\mathfrak{S}$
$S$
$(C_{1}{}_{)}C_{2})$
$c_{1},$
.
$c_{2}$
dependent on
$\mathfrak{S}$
107
Lerrlma 1.
-diagonal. For a
have
$L\cdot et,$
$M=\mathrm{Z}[v_{1}, \cdots, v_{m}]$
$(c_{1}, c_{2})$
be a positi $ve$ lattice and assume that
in $M$ and for a natural
$(B(v_{i}, v_{\mathrm{j}}))$
$\mathit{1}\mathrm{J}\mathrm{r}imiti\mathrm{V}\mathrm{e}el\mathrm{e}\mathrm{m}e\mathrm{n}tw=\sum_{i=1}^{1n}r_{i}v_{i}$
is
$nu\mathrm{m}be\mathrm{r}$
$N_{\mathit{1}}\mathfrak{s}\mathrm{v}e$
$1 \mathrm{n}\mathrm{i}\mathfrak{n}(M+\mathrm{Z}[(\{)/N1)_{\wedge c}^{\vee}1,c_{2}\mathrm{n}?\mathrm{i}\mathrm{n}(\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}(M),\min\sum_{=}^{m}b\in \mathrm{Z},N\{bi1\lceil br_{i}/N\rceil^{2}q(v_{i})l\cdot$
Proof.
Since there are positive constants
$c_{1}.
$C_{1}{}_{)}C_{2}$
so
$\mathrm{t}\mathrm{h}\mathrm{a},\mathrm{t}$
\sum_{1=1}^{m}x_{i}Q2(vi)<Q(\sum^{m}x_{i}v,))<C_{2}i=1\dot{\mathrm{z}}\sum_{=1}^{m}X^{2}iQ(vi)$
,
putting
$:= \sum_{=i1}^{m}xiQ2(\prime vi)$
$Q^{\iota}( \sum_{i=1}^{m}xivi)$
,
we have
$\min_{Q}(M+\mathrm{Z}[w/N])_{\wedge Q}^{\vee}1,\mathrm{c}2(\min_{Q’}M+\mathrm{Z}[u’/N])$
$= \min(’\sum_{i=1}^{n}(bi+br_{i}/N)^{2}Q(vt))$
where integers $b_{i}(^{}i=1, \cdots, m)$ should satisfy
under
restriction $N|b$ , the lninimunl is
yields
for some , it is equal to
$b,$
$\mathrm{f},1\mathrm{l}\mathrm{a}.\mathrm{t}$
$\mathrm{t}1_{1}\mathrm{e}$
) $\mathrm{t}+br_{i}/N\neq
$l$
$b_{i}+br_{\mathrm{I}}/N\neq()$
$\min(l\downarrow I)$
\mathrm{t}\mathrm{J}$
, and
$\Lambda f$
$\Lambda,I’\supset l1I$
$\mathrm{J}l’$
be positive lattices of rank
be positive numbers] and
a natu number.
we have
$tq_{1_{\rangle}}\cdots,$
$\gamma N$
If
$i$
$b$
$\mathrm{t}1\overline{1}\mathrm{a}\mathfrak{t}_{}\mathrm{t}1^{-}1\mathrm{e}$
$q_{i}$
$r_{1},$
$r\mathrm{a}\mathit{1}$
.
$\square$
$M=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}l|p’$
$\min(M’)\leq 1\mathrm{n}\mathrm{i}\mathrm{n}(M)\leq[M’ : \mathrm{J}/f]^{2}\min(M’)$
Theorem 1. Le
$r_{1}=1$ , and
finall.
for some . By noting
condition $N$ \dag er
$\prime i$
$\min(\min(M),\min\sum_{=}^{m}b\in \mathrm{Z}|N\{bi1\lceil br_{t}/N1^{2}Q(vi))$
Remark. Let
and
implies
,
. Then the condition
.
$\cdots,$
$r_{\mathit{5}}$
non-ze $ro$ integers
$Th\mathrm{P}l1$
)
$:=b \in \mathrm{Z}N;1\mathrm{J}1\mathrm{i}|\mathrm{n}b(_{j}\sum_{=1}^{i}\lceil br_{j}/\mathit{1}\mathrm{V}1\wedge)’$
$qj$
$\geq\min((\frac{r_{1}}{2r_{2}})^{2}q_{1},$
$\cdots,$
$( \frac{r_{i-1}}{2r_{l}})^{2}q_{i-1},$
$N-2 \sum^{t}rj=1\mathrm{i}^{q}j)2$
.
$\dagger 1^{\gamma}i\mathrm{t}\cdot h$
108
Proof.
Suppose that
(1)
$K \leq(\frac{r_{j}}{2r_{j+1}})^{2}q_{i}$
is attained
We will show that
$K$ and $|b|\leq N/2$ . The condition
$I\backslash ^{\Gamma}$
(2)
for $j=1,$
$\cdot 4\cdot,$
$t-\perp$
.
Suppose t,hat an integer give the mininuum
implies $b\neq()$ . First, we claim
a,t $b=1$ .
$N\{b$
$|br_{j}|\leq N/2$
$b$
for
$j=\perp,$
$\cdots,$
$t$
.
it is true because of $r_{1}=1$ . Suppose that (2) is true for $j=i$ ; then we llave
$|br_{i}|\leq N/2$ and hence
.
, which yields
Now using (1), we
$N/2$ . Thus (2) has been shown inductively.
and then
The condition (2) implies
When
$j=1_{\tau}$
$|b|\leq\sqrt{I\mathrm{f}/q_{i}}N/|r,$
$I\iota’\geq\lceil br_{\dot{\mathrm{t}}}/N\rceil^{2}q_{t}=(br_{\mathrm{i}}/N)^{2}qi$
$|$
$1\mathrm{l}\mathrm{a}\mathrm{v}\mathrm{e}|br_{i+1}|\underline{<’.}\sqrt{J\mathrm{t}’/q;}\cdot N/|r_{i}|\cdot|r_{\mathrm{a}+1}|\leq|r_{i}|/(2|r_{t+}1|)\cdot N/|r_{i}|\cdot|?_{t}+1|=$
$\lceil br_{\mathrm{j}}/N\rceil^{2}=(br_{j}/N)^{2}$
$I \mathrm{i}’=\sum_{j=1}^{i}(brj/N)^{2}qj=b^{2}/N^{2}\sum^{1}rjqjj=12\geq N^{-2}\sum_{=j1}^{9}rj2q_{j}$
This completes the proof.
Corollary 1. Suppose
$I\zeta\gg\sqrt{q_{1}q_{2}}/N$
Corollary 2. Let
if
$q_{j},$
$\square$
$t=2$ .
$Th\rho n$
we have
$r_{2}^{2}\wedge\vee\sqrt{q_{1}/q_{\sim^{\supset}}}N$
$r_{j},$
$t,$
.
$N,$ $K$
be th
or if both
$\mathrm{o}se$
$(r_{2}, N)=1$
and
$\sqrt{q_{1}/q_{2}}N\ll 1$
in Theorem 1, and put
$\Delta:=\mathrm{I}^{t}\mathrm{I}q_{k},$
$\eta_{j}:=\frac{|r_{j}|}{N^{(\mathrm{j}1})/t\Delta_{j^{\prime 2}}1}k=1kj$
$\Delta_{j}:=\Delta^{-(}i-1)/i\mathrm{I}_{<}\mathrm{I}^{q}k,$
for $j=1,$
$\cdots,$
$t$
.
$\mathrm{T}l_{l}eD\mathrm{v}veha\mathrm{b}’\underline{\theta}$
(i)
4
(
$\frac{\Delta}{N^{2}})^{-1/i}I_{\mathrm{L}}’$
$\geq \mathrm{l}\mathrm{n}\mathrm{i}\mathrm{n}((\eta_{1}/\eta_{2})2,$
$\cdots,$ $( \eta_{t-1}/\eta‘)2,\sum_{j=1}^{i}\eta_{\mathrm{j}}^{2}(\Delta/N2)^{1}-j/t(\mathrm{I}_{k}\mathrm{I}j<\leq iq_{k}1^{-},1)$
$\geq\min((\eta_{1}/\eta_{2})^{2}, \cdots, (\eta_{i1}-/\eta_{t})2\eta_{t}:)2$
(ii) $\eta_{1}=1$ ,
(iii) if
$q_{1}\geq q_{2}\geq\cdots\geq q_{i}$
To undestand
$I\backslash ^{r}$
, then we
$h\mathrm{a}\mathrm{V}^{\rho}\Delta_{j}\geq 1$
for $j=1,$
$\cdots$
, it is better to give an estimate from above.
, .
$t$
.
109
Proposition 1. Let
$N$ a natural
er with
$q_{1},$
nu.m
be positive numbers, and
$(r_{1}, \cdots, r_{t}, N)=1$ . Put
,
$\cdots$
$\mathrm{b}$
$q_{l}$
$\Delta=\prod_{i=1}^{t}q_{i}$
$i\mathrm{I}^{\urcorner}he\mathrm{n}$
we have
$\mathrm{t}f,\mathrm{e}$
,
$r_{i}$
$\cdots,$
$I \mathrm{f}:=\min_{b\in \mathrm{Z},N\dagger b}(\sum_{\mathrm{j}=1}^{i}\lceil br_{\mathrm{j}}/N\rceil^{2}qJ’)\backslash$
integers, and finall.
$\gamma$
.
follo wing:
(1)
$I\mathrm{f}\geq \mathrm{m}\vec{\mathrm{l}}\mathrm{n}\{q_{1}, \cdots, q_{i}\}$
(2)
$r\mathrm{l},$
if
$K/\leq_{i}(\Delta/N^{2})^{1}/i$
or
$I\backslash ^{r}\ll \mathrm{s}(\Delta/N^{2})^{1/9}$
$( \Delta/N^{2})^{1/9}\ll_{t}\min\{q_{1}, \cdots, q_{i}\}$
.
We must study the distribution of isotropic vectors in a quadratic space over a finite
prime field to take account of the condition at a finite prime in the assertion
. For
an odd prinle
denotes the prime field with elements.
$\mathrm{R}_{n.’
$\mathrm{P},$
$p$
$F_{\mathrm{p}}$
Lemlna 2. Let $V=F_{p}[e_{1}, e_{2}]$ be a regular quadratic space over the field
.
quadra $tic$. form Q.
en for every positive int eger $If<p$ , we a
$Tl_{1}$
$h$
$| \sum_{\leq 1\leq xH}’\chi(Q(xe_{1}+c2))|\leq 2\sqrt{p}\log p+1$
$w\iota_{ler}e\chi$
1},$
stands for the
$q\mathrm{u}$
adratic residue symbol with
$\chi(0)=0$
$F_{p}\mathrm{i}’l^{rjth}$
$\iota^{\gamma}e$
,
.
The proof is routine.
Theorem 2. Let $V=F_{\mathrm{p}}[e_{1}, \cdots)e_{m}](m\geq 3)$ be a nadratic space
. Then we
.
have the following assertions:
{ for some $i,j(i\neq j)$ . Then for any
that
(i)
$(k\neq i, j)$ , there are element.
$y_{j}=\pm 1$ and $u\in V$ so that
$q$
$Q(e_{i})=0_{\backslash }.B(e_{i}, e_{j})\neq$
$s_{\mathrm{u}pP^{O}}.\mathrm{q}e$
$o\mathrm{V}_{-}^{\rho\Gamma F}p$
$)$
$x_{k}\in F_{l)}$
$\mathrm{s}y_{i}\in F_{p},$
.
$v:=yiei+yje_{j}+ \sum_{k\neq i,j}xke_{k}$
$i_{\mathit{8}}$
isotropic a.nd
(ii)
$B(u.v)’\neq 0$
$Supp_{\mathit{0}\mathit{8}}em\geq 4$
and
.
$\dim \mathrm{R}\mathrm{a}\mathrm{d}V\underline{<’\backslash }m-3$
. Let be a natural nmber. Then there
and a positive number
which
$r$
$n$
a $su$ bset
the following propert. :
Let
be bsets of
and assume that $|S_{1}|=3$ and
is a union of at
sets of consecutive integers. If $p>c_{r}$ and
, then
$x_{2}=\pm
1,$
ments
for
and $u\in V$
such that
exist,
$T= \{t_{1}, \cdots, t_{4}\}\subset\int_{1}1,2,$ $\cdots,$
$7n_{1}^{1}$
$c_{r}$
$\ddot{\mathrm{b}}at\mathrm{j}sf_{\mathrm{J}}r$
$\gamma$
$S_{1},$
$S_{2}$
$\mathrm{s}u$
$S_{2}$
$F_{\mathrm{p}}$
$\mathrm{m}o,\mathrm{s}tr$
$a\mathit{1}^{\backslash }eele$
$|S_{2}|>5\uparrow\cdot\sqrt{p}\log p$
$x_{1}\in F_{\mathrm{p}},$
$x_{3}\in\llcorner\backslash _{1}’,$
$x_{4}\in S_{2;}*?/j\in F_{\mathrm{p}}$
$v= \sum_{j=1}x_{j\mathrm{t}_{\dot{j}}}4e+\sum_{\tau i\not\in}y\dot{i}ei$
is isotropic and
$B(u, v)\neq 0$
.
$i\not\in T$
$ther\theta$
110
To colnbine stories at the infinite prime and at a finite prinue, we need the following.
Theorenl 3. Let
$S^{(m)}$
be a
Let
.
$p^{9_{m-}}t+1$
$A^{(m)}=$
that for a
,
(
,
$\mathrm{n}a\mathrm{f}$
.
$\cdot$
integers with $r<m$ . Le.
an let
integral matrix an we write
be a prim
$r\rho,gular\epsilon y\cdot \mathrm{m}metriC$
$D_{1}\in M_{m-\Gamma}(\mathrm{Z}_{\mathrm{P}}),$
$re\mathrm{s}p$
$p$
$r,$
be regula.
) be elementar. divisors of
$p^{g_{m}}$
$rm\partial,\mathrm{f},ri_{Ce\mathcal{B}}$
$v$
$A_{1}^{(\cdot,)}||(n-r)m-t$
positi
$m$
$S=$
a.Jld suppose th at
) and
(resp.
$p^{g_{1}},$
be an in tegral matrix with
$A_{4}^{(m-r,r)}A^{(}2r))$
$\cdots,p^{t_{m}}$
-r
$($
$t_{1}\leq\cdots\leq t_{m}$
$D_{2}$
$D_{1}$
$d$
$\det A=\pm 1$
.
. Assume
ural um ber ,
$n$
$e$
$A_{4}\equiv 0$
mod
$p^{e},$
$t_{m-,\backslash } \sim<e+t_{1}\leq\min(t_{m}+1, t_{m-r+1})$
$S[A]\equiv$
mod
$p^{t_{m}+1}$
have the same elementary divisors and
and
Then
and
and both
is integral over
matrix
visors over .
$D_{1}$
$S_{4}$
$t$
$\mathrm{v}e$
$d$
$D_{2}\in M_{r}(\mathrm{Z}_{p})$
,
and
$e\mathrm{n}\iota \mathrm{t}m\mathrm{b}e\mathrm{r}$
$S_{43}^{-1}S$
$S_{1}-S_{4^{-1}}[s_{3}]$
$\mathrm{Z}_{p}$
.
$S_{3}\equiv 0$
$D_{2}h\mathrm{a}\backslash \prime e$
mod
the
$p^{e+i_{1}}$
$\mathit{8}a\mathrm{m}e$
, and the
elementary
$d\mathrm{i}$
$\mathrm{Z}_{\mathrm{p}}$
Now we can show the following, and by using them we can show the theorem.
Proposition 2. Let $M$ be a positi lattice such
there is a positive numb er 6 satisf.ring the following conditi on:
contain in $gM$ such th at
then there is a lattice
If
, and $\min(M’)\geq p^{1/4}$ .
of prim $ep,$
$th_{\partial,t\mathrm{a}}\mathrm{r}\mathrm{n}\mathrm{k}(M)\geq 4,$
$\mathrm{v}e$
)
$\gamma_{\vee}$
$\Lambda’I’$
$>\delta_{f}$
$s(M)\subset p\mathrm{Z}$
$[M’ : M]$
. In the Proposition 2, let
is
represented by
$M_{p}$
by
$\mathrm{r}\mathrm{e}_{\mathrm{P}^{\mathrm{r}\mathrm{e}\mathrm{S}}\mathrm{J}}\mathrm{e}\mathrm{n}\dagger \mathrm{e}\mathrm{d}$
$N_{p}$
$N_{\mathrm{p}}$
and that
$N$
is a powe
$\mathrm{r}$
:
$s(\Lambda f_{p}’)=\mathrm{Z}_{\mathrm{p}}$
$Rem,ark,$
. Then
be a positive lattice of rank $2m$ and assume tluat
is primitivelv
is unimodular if $p>\delta$ . Then
$N_{p}$
$\Lambda f_{\mathrm{p}}’$
.
and $N$ be positi lattices of rank $(\Lambda r)=m\geq 6$ and rank $(N)=$
Proposition 3.
is represented by . Then
an suppose that
, an $dp$ a prime
;
represen tecl by
is prim
if $q\neq p,$
there is a lattice $M’(\supset M)$ such that
depends
on
and
where
and 1nin
$ve$
$I_{J}et\Lambda^{\text{ノ}}f$
$2mreRpeCtiVel_{\mathrm{J}\prime}r$
$num\mathrm{b}e\mathrm{r}_{;}$
$d$
$M_{p}$
$WI_{q}’=lll_{q}$
$( \Lambda I^{f})>c(N_{p})\min(M)^{c}’ \mathrm{p}1$
$N_{p}$
only
$on\uparrow n$
$N_{\mathrm{p}}$
’
$l\mathcal{V}I_{\mathrm{P}}$
$it\mathrm{i}v\theta.ly$
$N_{p}$
$c_{p}$
$c(N_{\mathrm{p}})d..epe\mathrm{r}\mathit{1}dso\mathrm{J}2\mathrm{J}y$
.
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[K] Y.Kitaoka, Arithmetic of $quadrai|c$ forms,
quations over $Fin\dagger iep;_{eld}$ , An elementary App roach, Springer Lecture Notes in
[S] W.M.S
Math, vol. 536, Springer-Verlag, 1976.
[V] I.M.Vinogradov, Elemen of Number Theory, Dover Publications, 1954.
$\mathrm{c}_{\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{b}_{\Gamma \mathrm{i}\mathrm{d}\mathrm{g}}}\mathrm{e}$
$(_{-}.\mathrm{h}_{1}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{t},$
$E^{l}$
$i\epsilon$