数理解析研究所講究録958 巻 1996 年 105-110

105

MINIMUM OF POSITIVE DEFINITE QUADRATIC FORMSYOSHIYUKI 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 fromto
. Then there exis aconstant
dependent only on $N$ so that $M$ is represented $b.vN$ if $\min(M)>c(N)$ ,where nuin $(M)$ denotes
positive number represen
leby $M$ .
We know that the assertionis 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 isbest if $rn=1$ , that is
is false. But in the case of
we know at present, isis false if. We do not know
anything the case of $n-m=4$ . Anywav, analyzing the counter-example, we come tothe fullowing two assertionsand.
: There exists a. const ant
dependent only on $N$ so that $llI$ isresen $tedb.\}^{f}N$ if $\min(N)>c’(N)$ a.ndisrepresented byfor 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 latticecontsuch th atis primiti
rep.r.esentedbyprime and $\min(’M’)$ is still large if $\min(M)$ is 1 arge.
forIf the assertionis true, then the assertionis reduced to the apparently
weaker assertion. If the assertionis 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 thatis 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
sertionthe 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 sumnlaryof the proof.
field of rational numbers
the ring of integers,andWe denote by
and their p–adic comple,tions. Terminology and notation on qua.dratic forms are thosescale $s(M)$ deoveron a quadratic spacefrom [K]. For a lattice on
denotes a -module spanned bynotes $\{B(x, y)|x, y\in\Lambda f\}$ , and the norm
denoteit is similarly defined. $dM,$$\{Q(X)|x\in M\}$ . Even for the localization
respectively. A positive lattice means a lattice on a positiveof $M,$
thedefinite quadratic space over Q. We give proofs only for a few assertions.
$\mathrm{i}\dagger‘$$\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}}$$\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{C}\mathrm{r}}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}11\mathrm{t}$$M_{\mathrm{p}}$Definition. For a real number$x$, we define the decimal part$-1/2\leq\lceil x\rceil<1/2$

Note that$\lceil_{\backslash }\prime r1^{2}=\lceil-’.t1^{2}$and$\lceil x\rceil$$x-\lceil x,\rceil\in \mathrm{Z}$by the conditions.

for every real number .
$x$Definition. For positive numbers

$a,$

$b$

, we write$a\ll_{m}b$

if there is a positive number dependent only on $m$ such that $a/b<c$ . If both $a\ll_{m}b$hold, then we write$c$$\mathrm{a}\mathrm{l}\iota \mathrm{d}l)\ll_{m}$$a$$a_{\wedge m}^{\vee}b$

If $m$ is$\mathrm{a}1$)absolute constant, then we omit $m$ .

Definition. For$(s_{\mathrm{s},\mathrm{j}})$is.

$(c_{1}, c_{2})$numbers-diagonal if we have$\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{P}}$$c_{1},$$c_{2}$, we say that a positive definite matrix $S^{(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}]$$(c_{1}, c_{2})$be a positi $ve$ lattice and assume thatin $M$ and for a natural$(B(v_{i}, v_{\mathrm{j}}))$$\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
underrestriction $N|b$ , the lninimunl isyieldsfor some , it is equal to$b,$$\mathrm{f},1\mathrm{l}\mathrm{a}.\mathrm{t}$$\mathrm{t}1_{1}\mathrm{e}$$b_{i}+br_{\mathrm{I}}/N\neq()$$\min(l\downarrow I)$) $\mathrm{t}+br_{i}/N\neq \mathrm{t}\mathrm{J}$

$l$

, and

$\Lambda f$$\Lambda,I’\supset l1I$$\mathrm{J}l’$be positive lattices of rank

be positive numbers] anda 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$r_{1}=1$ , andfinall.

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
andimplies,. 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)

$K \leq(\frac{r_{j}}{2r_{j+1}})^{2}q_{i}$for $j=1,$$\cdot 4\cdot,$$t-\perp$.

is attained a,t $b=1$ . Suppose t,hat an integer give the mininuum
We will show that
$K$ and $|b|\leq N/2$ . The condition $N\{b$ implies $b\neq()$ . First, we claim$b$$I\backslash ^{\Gamma}$(2)$|br_{j}|\leq N/2$

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.

$\square$Corollary 1. Suppose $t=2$ .
$I\zeta\gg\sqrt{q_{1}q_{2}}/N$

Corollary 2. Letif$q_{j},$$Th\rho n$

we have$r_{2}^{2}\wedge\vee\sqrt{q_{1}/q_{\sim^{\supset}}}N$$r_{j},$$t,$.

$N,$ $K$be thor if both $(r_{2}, N)=1$ and $\sqrt{q_{1}/q_{2}}N\ll 1$ .

$\mathrm{o}se$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, be positive numbers, and$N$ a natural
er with $(r_{1}, \cdots, r_{t}, N)=1$ . Put$q_{1},$

nu.m$\cdots$$q_{l}$$\Delta=\prod_{i=1}^{t}q_{i}$$i\mathrm{I}^{\urcorner}he\mathrm{n}$we have$\mathrm{t}f,\mathrm{e}$$\cdots,$ $r_{i}$integers, and finall.

$\gamma$,$I \mathrm{f}:=\min_{b\in \mathrm{Z},N\dagger b}(\sum_{\mathrm{j}=1}^{i}\lceil br_{\mathrm{j}}/N\rceil^{2}qJ’)\backslash$.

follo wing:(1)$I\mathrm{f}\geq \mathrm{m}\vec{\mathrm{l}}\mathrm{n}\{q_{1}, \cdots, q_{i}\}$(2)$r\mathrm{l},$$\mathrm{b}$if$K/\leq_{i}(\Delta/N^{2})^{1}/i$or$I\backslash ^{r}\ll \mathrm{s}(\Delta/N^{2})^{1/9}$$( \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}$$F_{p}\mathrm{i}’l^{rjth}$$\iota^{\gamma}e$,adratic residue symbol with $\chi(0)=0$ .

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 $B(u.v)’\neq 0$ .

(ii)$Supp_{\mathit{0}\mathit{8}}em\geq 4$

and$\dim \mathrm{R}\mathrm{a}\mathrm{d}V\underline{<’\backslash }m-3$

exist, a $su$ bset $T= \{t_{1}, \cdots, t_{4}\}\subset\int_{1}1,2,$ $\cdots,$. Let be a natural nmber. Then there
and a positive number
which
$r$

$n$

$7n_{1}^{1}$$c_{r}$$\ddot{\mathrm{b}}at\mathrm{j}sf_{\mathrm{J}}r$

the following propert. :$\gamma$be bsets ofand assume that $|S_{1}|=3$ andis 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 thatLet$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.

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$$p$$p^{9_{m-}}t+1$,

Let $A^{(m)}=$(that for a,$\mathrm{n}a\mathrm{f}$$r,$$rm\partial,\mathrm{f},ri_{Ce\mathcal{B}}$$v$$A_{1}^{(\cdot,)}||(n-r)m-t$$S=$a.Jld suppose th at) and(resp.

$p^{g_{1}},$

$d$

let

$\cdots,p^{t_{m}}$ -r$t_{1}\leq\cdots\leq t_{m}$$D_{2}$$D_{1}$

an

$t$

$($

.

be an in tegral matrix with $\det A=\pm 1$ . Assume$A_{4}^{(m-r,r)}A^{(}2r))$

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 bothis integral overmatrixvisors over .
$D_{1}$$S_{4}$

integers with $r<m$ . Le.

$\mathrm{v}e$$d$

be regula.
) be elementar. divisors of$p^{g_{m}}$positi$m$

integral matrix an we write$D_{2}\in M_{r}(\mathrm{Z}_{p})$. ,$\cdot$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}}$$\mathit{8}a\mathrm{m}e$, and theelementary$d\mathrm{i}$$\mathrm{Z}_{\mathrm{p}}$Now we can show the following, and by using them we can show the theorem.

. Then
Proposition 2. Let $M$ be a positi lattice suchthere is a positive numb er 6 satisf.ring the following conditi on:contain in $gM$ such th at $[M’ : M]$ is a powethen there is a latticeIf:, 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}$$s(M)\subset p\mathrm{Z}$$\Lambda’I’$$>\delta_{f}$$\mathrm{r}$$s(\Lambda f_{p}’)=\mathrm{Z}_{\mathrm{p}}$$Rem,ark,$. In the Proposition 2, letisrepresented by$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$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 . Thenan suppose that, an $dp$ a prime;
represen tecl byis 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$.

REFERENCES
University Press, 1993.
[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$