数理解析研究所講究録998 巻 1997 年 34-43

34

THE ORDERS OF THE REDUCTIONS OF A POINT IN

THE MORDELL-WEIL GROUP OF AN ELLIPTIC CURVE
J. CHEON AND S. HAHN

ABSTRACT. We prove elliptic analogue of Bang’s theorem. Consider an elliptic curve $E$ over a number field$K$. For any non-torsion point $M\in E(K)$ , the order of the reductions $M$

many positive integers as$\mathfrak{p}$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}$runs through all but finitely

runs through all good primes. Moreover, it runs throughbut finitely many points $M\in E(K)$ . At last, we present some examples to deduce$\mathrm{a}\mathrm{U}$positive integers for all$\mathrm{N}\backslash Imf_{M}$for$M\in E(\mathbb{Q})$.

1. Introduction
To begin with, we introduce the famous Bang’s theorem. Let$\{0, \pm 1\}$. Then$a$$a$

be a fixed rational number in$\mathbb{Q}-$can be written as $a=c/d$ for some integers $c\neq 0$ and $d>0$ with $(c, d)=1$ . Let

denote the set of positive prime numbers which do not divide $cd$ anddefine the function$f_{a}$$\mathrm{N}$$D_{a}$

the set of natural numbers. We

as a map$f_{a}$:$D_{a}arrow \mathrm{N}$such that for any $p\in D_{a}$$f_{a}(p)=$Then we can$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\vee$the multiplicative order of (a mod ) in the finite field$p$$\mathrm{F}_{p}$.

Bang’s theorem as follows:

Theorem (Bang). Using the same notations as a bove, the set$\mathrm{N}\backslash Imf_{a}$is linite for any $a\in \mathbb{Q}-\{0, \pm 1\}$ .

This famous theorem was proved by Bang first [2], and by various other mathematicians. In particular,
Zsigmondy proved the stronger version presented here [11] [15]. We say that a prime$p$is a primitive

divisor of $A^{n}-B^{n}$ if $p|A^{n}-B^{n}$ and $p\{A^{m}-B^{m}$ for any $m<n$ , i.e. $f_{a}(p)=n$ .
.
1991 Mathematics Subject Classification. llG05, llG20,
Key words and phrases. Bang’s theorem, Elliptic curves, Reductions, Local height, Division Polynomials.
$14\mathrm{H}52$

35
Theorem (Zigmondy). Let $A,$$B$

be integers with $A>B\geq 1$ and$\mathrm{g}\mathrm{c}\mathrm{d}(A, B)=1$

and let $n\geq 1$ .

(a) $A^{n}-B^{n}$ has a primitive divisor with the following exceptions :(i) $n=1,$ $A-B=1$ .
(ii) $n=2,$ $A+B$ is a power of 2.
(iii) $n=6,$ $A=2,$ $B=1$ .
$(b)A^{n}+B^{n}$has a primitive divisor with the following exceptions :$n=3,$ $A=2,$ $B=1$ .

More general theorem over algebraic number fields by Schinzel is as follows [9]:Theorem (Schinzel). If$\mathrm{g}\mathrm{c}\mathrm{d}(A, B)=1$

divisor for all $n>n_{0}(d)$ , where$d$and $A/B$ is not a root of unity, then $A^{n}-B^{n}$ has a primitiveis the degree of $A/B$ and $n_{0}(d)$ is effict ively computable.

In particular, the first special formulation of Theorem (Bang) was presented by Y. Ihara in [4] [5]together with the sketch of proof using cyclotomic polynomials, which motivated us to study the elliptic

analogue replacing cyclotomic polynomials by division polynomials of elliptic curves. The analogue on
elliptic curves over$\mathbb{Q}$was proved by J. Silverman [12]. In this paper, we extend the result to the cases ofnumber fields and show that$f_{M}$

is surjective for all but finitely many points $M$ of an elliptic curve, whichis the stronger result than Schinzel’s original theorem over number fields since we have only finitely many
exceptional cases. However, we don’t have yet any effective method to compute the number of points $M$

for which$f_{M}$

is not surjective.

2. Some Preliminaries
We introduce some notations to use in this paper.
$\mathrm{N}$: the set of natural numbers.

$\mathbb{Q}$: the set of rational numbers.

$K$: a number field$R$: the ring of integers of $K$$M_{K}$$M_{K}^{\infty}$$M_{K}^{0}$: the complete set of normalized absolute values on: the set of archimedean absolute values in$M_{K}$: the set of non-archimedean absolute values in$v(x)=-\log|x|_{v}$ for $v\in M_{K}$$\mathfrak{p}_{v}=\{x\in R|v(x)>0\}$

for $v\in M_{K}$$M_{K}$

$K$

.

36

$K_{v}$: the completion ofFor$M_{K}$$M_{\mathbb{Q}}$

at

$v$

for $v\in M_{K}$, we use the specific normalization as follows:

To begin with,
(i)

$K$

$M_{\mathbb{Q}}$consists of the followings.

contains one archimedean value, given by$|x|_{\infty}=$(ii) For each prime$p\in \mathbb{Z},$$M_{\mathrm{Q}}$usual absolute value $= \max\{x, -x\}$ .

contains one non-archimedean$|p^{n} \frac{a}{b}|_{\mathrm{p}}=p^{-n}$$M_{K}$

for$b\in \mathbb{Z},$$a,$$\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}\sim$$\mathrm{g}\mathrm{c}\mathrm{d}(p, ab)=1$

consists of all absolute values on $K$ whose restriction to$\mathbb{Q}$value, given by.

is one of the absolute values in

Consider an elliptic curve $E$ over $K$ given by a Weierstrass equation:$E:y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6};a_{i}\in R$

Let

$S$

be a finite subset of$M_{K}^{\infty}\cup${$v\in M_{K}^{0}|E$$M_{K}$

containing

has bad reduction at

Define the local height function

$v$

} {
$\cup$$v\in M_{K}^{0}|v$

as follows :$h_{x,v}$for$h_{x,v}$: $E(K)\backslash \{O\}arrow$$v\in M_{K}\backslash S$

is ramified at$\mathbb{R}$(1)$(x, y)$Note that there is a constant

(2)

$C$

$\mapsto\frac{1}{2}\max\{-v(X), \mathrm{o}\}$, depending only on $E$ , such that$| \hat{h}(M)-\frac{1}{2}hx(M)|<C$

for all $M\in E(K)$ and(3).

$\frac{1}{2}h_{x}(M)=\frac{1}{[K.\mathbb{Q}]}.\sum_{v\in MK}n_{vx,v}h(M)$.

$K/\mathbb{Q}$or divides 2}.

$M_{\mathbb{Q}}$

.

37
where

For

$n_{v}$

denotes the local degree$v\in M_{K}^{0},$$M$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}_{v}$$[K_{v} :\mathbb{Q}_{v}][10][11]$.

denotes the image of $M$ of the reduction map $E(K)arrow E(R/\mathfrak{p}_{v})$ .

For the proof of the main theorem, we need the following famous theorem$[10, \mathrm{p}245]$:

Theorem (Siegel). Let $E/K$ be an elliptic curve with infinite points in $E(K),$ $f\in K(E)$ a non-constanteven ffinction, $v\in M_{K}$ , and$Q\in E(\overline{K})$(4). Then$h_{f}(^{\lim_{P)arrow\infty}} \frac{\log d_{v}(P,Q)}{h_{f}(P)}=0$,

where $h_{f}(P)$ denotes the Weil height of $P$ and $d_{v}(P, Q)$ the -adic distance$v$

If we take$\mathbb{Q}=O$$f_{\mathrm{f}O},mP$

to

$Q$

.

and $f=x$ in the above theorem, we get the following useful corollary.

Corollary. Let $E/K$ be an elliptic curve and $M$ be a non-torsion point of $E(K)$ and $v\in M_{K}$ . Then

(5)$\lim_{narrow\infty}\frac{h_{xv1}(nM)}{h_{x}(nM)}=0$.

3. Main Theorem

Lemma. Let$h_{x,v}(M)>0$$v\in M_{K}\backslash S$

and $M$ be a non-torsion point of $E(K)$ . Suppose that $M$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}_{v}=O$i.e.

. Then(6)$h_{x,v}(nM)=h_{x,v}(M)+v(n)$for any posi tive integer .
$n$

Proof. Let$\mathfrak{M}$$\mathfrak{B}\mathrm{t}=\{x\in K_{v}|v(x)>0\},\hat{E}$the formal group associated to $E$ andwith its usual addition. We have an isomorphism [10,(7)such that $v(z)=v(\log_{\hat{E}}Z)$ for$\log_{\hat{E}}$$z\in \mathfrak{M}$:$\mathrm{I}\mathrm{V}$, Theorem 6.4$(\mathrm{b})$$\hat{G}_{a}(\mathfrak{M})$],$\hat{E}(\mathfrak{M})arrow\hat{G}_{a}(\mathfrak{M})$. Hence we can identify$\hat{E}(\mathfrak{M})$with$\hat{G}_{a}(\mathfrak{M})$.

the additive group

38
Moreover, if we let$E_{1}(K_{v})=\{M\in E(K_{v})|M \mathrm{m}\mathrm{o}\mathrm{d} \mathfrak{p}_{v}=O\}$,

we have an isomorphism [10, VII, Proposition 2.2],

(8)

$z$

:

$E_{1}(Kv)arrow\hat{E}(\mathfrak{M})$such that $v(x[M])=-2v(z(M))$ for any $M\in E_{1}(K_{v})\backslash \{O\}$ , where $x[M]$ denotes the -coordinate of $M$ .
$x$

By (7) and (8), we get $v(x[nM])=v(x[M])-2v(n)$ for any non-torsion point $M\in E_{1}(K_{v})\backslash \{O\}$ ,because$v(z(nM))=v(\log\hat{E}z(nM))=v(n\log_{\hat{E}}z(M))=v(n)+v(z(M))$ .

We complete the proof by (1).
Now we define a function$\square$$f_{M}$

as a map$f_{M}$:$M_{K}\backslash Sarrow \mathrm{N}$such that$f_{M}(v)=$the order of $M$

Theorem. For any non-torsion point $M\in E(K),$all but tinitely$m$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}_{v}$$\mathrm{N}\backslash Imf_{M}$in$E(R/\mathrm{p}_{v})$is finite. In particular,$S(n, M)=$ { $v\in M_{K}\backslash S|nM$

$n$

, put

$\mathrm{m}\mathrm{o}\mathrm{d}$$n= \prod_{i=1}^{S}q_{i}r_{\mathrm{i}}$$\mathfrak{p}_{v}=O$in

for distinct primes$E(R/\mathfrak{p}_{v})$}. Note that$n$$q_{i}$

is surjective for$S(n, M)= \bigcup_{i=1}^{S}S(\frac{n}{q_{\mathfrak{i}}}, M)$. For$v\in M_{K}\backslash S$, there exists$h_{x,v}(nM)=h_{x,v}( \frac{n}{q_{i}}M)+v(q_{i})$and positive integers

belongs to the image of

only if there exists a prime $p \in S(n, M)-\bigcup_{i=1}^{S}S(\frac{n}{q_{1}}, M)$ . (If $n=1,$ let(9)$f_{M}$

any $M\in E(K)$ .

Proof. Given a positive integer

Suppose that.

$i$$\bigcup_{i=1}^{S}S(\frac{n}{q_{\mathfrak{i}}},$$r_{i}$and let$f_{M,\backslash }$if and$M)=\phi$ by notation.)with $1\leq i\leq s$ such that

39
by Lemma. Hence we get(10)$h_{x,v}(nM) \leq\sum_{1i=}\{h_{x,v}S(\frac{n}{q_{i}}M)+v(q_{i})\}$For $v\in S$ , Siegel theorem (5) implies that for any(11)$\epsilon>0$.

,$h_{x,v}(nM)\leq\epsilon h_{x}(nM)$

for all sufficiently large integer .
$n$

If we let

$N= \frac{1}{\lfloor K.\mathbb{Q}\rfloor}.\sum_{v\in S}n_{v}$, by (3), (10) and (11),$\frac{1}{2}h_{x}(nM)\leq\frac{1}{[K.\mathbb{Q}]}.\sum_{sv\in M_{K\backslash }}n_{v}\{\sum_{1i=}hsx.v(\frac{Jl}{q_{i}}M)+v(qi)\}+\frac{1}{[K.\mathbb{Q}]}.\sum_{v\in S}n_{v}\epsilon h_{x}(nM)$(12)$\leq N\epsilon h_{x}(nM)+\sum_{1i=}\frac{1}{2}h_{x}(\frac{n}{q_{i}}SM)+\log n$since$\frac{1}{[K.\mathbb{Q}]}.\sum_{v\in M^{0}}Knvv(q_{i})=\log q_{i}$(13). Hence we get$( \frac{1}{2}-N\epsilon)h_{x}(nM)\leq\frac{1}{2}\sum_{i=1}hSx(\frac{n}{q_{i}}M)+\log n$

On the contrary, if we take$\epsilon<\frac{1}{4N}$, for all sufficiently large integer.

$n$$\frac{1}{2}\sum_{l=1}^{s}h(x\frac{n}{q_{i}}M)+\log n$(by (2) and $s\leq 2\log n$ )$< \sum_{i=1}^{s}\hat{h}(\frac{n}{q_{i}}M)+(1+2C)\log n$(14)(by quadraticity of )$= \sum_{i=1}s\{\frac{n^{2}}{q_{i}^{2}}\hat{h}(M)\}+(1+2C)\log n$$\hat{h}$$< \frac{1}{2}n^{2}\hat{h}(M)+(1+2C)\log n$$( \cdot.\cdot\sum_{p:prim\mathrm{e}}\frac{1}{p^{2}}<\frac{1}{2})$($<(1-2N\epsilon)(n\hat{h}2(M)-^{c)}$$< \frac{1}{2}(1-2N\epsilon)hx(nM)$.

$\cdot.\cdot 1-2N\epsilon>\frac{1}{2}$and$n\gg \mathrm{O}$)(by (2))

It contradicts with (13). Therefore for each sufficiently large integer , there exists
$n$

that

$v \in S(n, M)\backslash \bigcup_{i=1}^{s}S(\frac{n}{q_{\mathrm{i}}}, M)$, i.e. the order of $M$$\mathrm{m}\mathrm{o}\mathrm{d}$$\mathfrak{p}_{v}$in$E(R/\mathfrak{p}_{v})$

is

$n$

.

$v\in M_{K}\backslash S$

such

40
For the second statement, take with$\epsilon$$0< \epsilon<\frac{1}{4N}$. If $h_{x}(M)$ is sufficiently large,$h_{x,v}(nM)<\epsilon h_{\mathrm{r}}(nM)$

for all $n\geq 1$ and all $v\in S$ . Hence for all but finitely many $M\in E(K),$ (13) holds for all $n\geq 1$ . Also, if$\hat{h}(M)$is sufficiently large, then(15)$\frac{1}{2}n^{2}\hat{h}(M)+(1+2C)\log n<(1-2N\epsilon)(n\hat{h}2(M)-^{c)}$for all $n\geq 1$ so that for all but finitely many $M\in E(K),$ (14) holds for all $n\geq 1$ . Therefore for all
but finitely many $M\in E(K),$ $S(n, M)- \bigcup_{i=1}^{S}S(\frac{n}{q}.’ M)\neq\phi$ for all $n\geq 1$ , which implies thatsurjective.

$f_{M}$

is$\square$

4. Some Examples
In this section, we introduce some examples to deduce

Consider an elliptic curve

$E$

over

$\mathbb{Q}$$\mathrm{N}\backslash Imf_{M}$for an elliptic curve defined over Q.

given by a Weierstrass equation:$y^{2}=x^{3}+Ax+B;A,$$B\in \mathbb{Z}$.

For any non-zero $M=(x, y)\in E(\mathbb{Q})$ and an integer , the -coordinate of the point $nM$ is given by$n$.

$x[nM]=$

(16)

where

$\phi_{n}(M)$and$\psi_{n}^{2}(M)$When we write $M=$

are polynomials in$(_{d}^{a}\neg, \overline{d}^{T}b)$

for $a,b$ and$\mathbb{Z}[x]$$x$$\frac{\phi_{n}(M)}{\psi_{n}^{2}(M)}$,, relatively prime in$d\in \mathbb{Z}$with$\mathbb{Q}[x]$$\mathrm{g}\mathrm{c}\mathrm{d}(a, d)=1$

notations; $\hat{\phi}_{m}(M)=d^{2m^{2}}\phi_{m}(M)$ ; $\hat{\psi}_{m}(M)=d^{m^{2}-1}\psi_{m}(M)$ , where weThen (16) becomes(17)$x[nM]= \frac{\phi_{n}(M)}{\psi_{n}^{2}(M)}=\frac{\hat{\phi}_{n}(M)}{d^{2}\hat{\psi}_{n}^{2}(M)}$so that(18)$nM=O$ if and only if $\hat{\psi}_{n}(M)=0$[ $10$ , p107].

and $d>0$ , we set the following$\mathrm{t}\mathrm{a}\tilde{\mathrm{k}}\mathrm{e}\hat{\psi}m(M)$as a positive integer.

41
for a non-zero point(19)$M\in E(\mathbb{Q})$. Hence for a non-zero point$M\in E(\mathbb{Q})$, we have$f_{M}(p)=n$ if and only if $p|\hat{\psi}_{n}(M)$ and $p$ \dag er $\hat{\psi}_{m}(M)$ for any integer $m<n$

Example 1. Let $E$ : $y^{2}=x^{3}+4x+20$ and $M=(1,5)\in E(\mathbb{Q})$ .

following calculation, we may expect that$\mathrm{N}\backslash Imf_{M}=\{1\}$$\Delta_{E}=-2^{7}\cdot 167$.

. From (19) and the.

$\hat{\psi}_{1}(M)=1$$\hat{\psi}_{2}(M)=2\cdot 5$$\hat{\psi}\mathrm{s}(M)=251$$\hat{\psi}_{4}(M)=-2^{2}\cdot 3\cdot 5\cdot 23\cdot 47$$\hat{\psi}_{5}(M)=-97$. 831683$\hat{\psi}_{6}(M)=-2\cdot 5\cdot 251$$\hat{\psi}_{7}(M)=3311717$. 122741447. 438695947$\hat{\psi}_{8}(M)=2\cdot 3\cdot 5\cdot 23\cdot 47\cdot 307\cdot \mathrm{s}019$$\hat{\psi}_{9}(M)=19\cdot 251$

. 4558056703. 94621387$\hat{\psi}_{10}(M)=2\cdot 5\cdot 97\cdot 4261\cdot 831683\cdot 3504419$$\hat{\psi}_{11}(M)=7\cdot 53\cdot 71\cdot 417493\cdot 1343\mathrm{t}\mathrm{K}189$. 2372245029989. 18006074321808164774557$\hat{\psi}_{12}(M)=-2\cdot 3\cdot 5\cdot 23\cdot 47\cdot 251\cdot 5743\cdot 11057\cdot 61224103\cdot 122741447$$\hat{\psi}_{13}(M)=-3083\cdot 3967$. 4895634495812861. 1290001702772707986533036597725024991050866886591$\hat{\psi}_{14}(M)=2\cdot 5\cdot 13\cdot 3311717\cdot 438695947$. 34395428644587863980370679502875987956020839061$\hat{\psi}_{15}(M)=97\cdot 251\cdot 3727\cdot 831683\cdot 680611988239249298432146956724186986216473592705367902544337$

Example 2. Let $E$ : $y^{2}=x^{3}-x+1$ and $M=(1,1)\in E(\mathbb{Q})$ . $\Delta_{E}=-2^{4}\cdot 23$ . From (19) and the following
calculation, we may expect that $\mathrm{N}\backslash Imf_{M}=\{1,2,3,4,5,6,9\}$ .
$\hat{\psi}_{1}(M)=1$$\hat{\psi}_{2}(M)=2$$\hat{\psi}_{3}(M)=2^{3}$$\hat{\psi}_{4}(M)=2^{5}$$\hat{\psi}_{5}(M)=-2^{8}$$\hat{\psi}_{6}(M)---2^{13}$$\hat{\psi}_{7}(M)---2^{16}\cdot 3$$\hat{\psi}_{8}(M)=-2^{21}\cdot 5$42$\hat{\psi}9(M)=-22\tau$$\hat{\psi}_{1}0(M)=233.11$$\hat{\psi}_{11}(M)=2^{4}0.19$$\hat{\psi}_{12}(M)=250.7$$\hat{\psi}_{13}(M)=256.53$$\hat{\psi}14(M)=-2^{6}5.3\cdot 17$$\hat{\psi}_{15}(M)=-275.223$$\hat{\psi}_{16}(M)=-285.5\cdot 103$$\hat{\psi}_{17}(M)=-2^{96}$. 1381$\hat{\psi}_{18}(M)=-2^{10}9.419$$\hat{\psi}_{19}(M)=2^{12}0.13\cdot 509$$\hat{\psi}_{20}(M)=2^{133}\cdot 11$$\hat{\psi}_{21}(M)=2147.32$. 1861. 4903$\hat{\psi}_{22}(M)=2^{161}\cdot 19\cdot 23\cdot 347$$\hat{\psi}_{23}(M)=-21\tau 6.119417$$\hat{\psi}_{24}(M)=-2195.5\cdot 7$.

. 4931$\hat{\psi}_{25}(M)=-2^{208}\cdot 71\cdot 137\cdot 521$$\hat{\psi}_{26}(M)=-2^{225}\cdot 43\cdot 53$. 9281$\hat{\psi}_{27}(M)=-2^{243}\cdot 29485343$$\hat{\psi}_{28}(M)=2^{261}\cdot 3\cdot 17$$\hat{\psi}_{29}(M)=2^{280}$. 4231459. 1243809367$\hat{\psi}_{30}(M)=2^{301}\cdot 11\cdot 59\cdot 223$$\hat{\psi}_{31}(M)=2^{320}$.

. 13597. 23663249429$\hat{\psi}_{32}(M)=-2^{34}1.5\cdot 61\cdot 103\cdot 173$$\hat{\psi}_{33}(M)=-2^{363}\cdot 19$. 1319. 1213 , 20517719$\hat{\psi}_{34}(M)=-2^{385}\cdot 149\cdot 7\mathrm{o}1\cdot 1381$$\hat{\psi}_{35}(M)=-2^{408}\cdot 3\cdot 31\cdot 63649$. 19801. 3046601

REFERENCES

1. M. Ayad, Pcnnts -Entiers des Elliptiques, Manuscripta Math. 76 (1992), 305-324.
$S$

2. A. Bang, Taltheoretiske Untersogelser, Tidskrift f. Math. (5) 4 (1886), 70-80 and 130-137.

43
3. G. D. Birkhoff and H. S. Vandiver, On the Integral Dimsors of $a^{n}-b^{n}$ , Annals of Math. 5 (2) (1904), 173-180.
4. Y. Ihara, On Femat Quotient and $‘ {}^{t}Differentiatim$ of Numbers” (in Japanese), Rims Kokyuroku 810 (Algebraic
Analysis and Number Theory) (1992), 324-341.

5. Y. lhara (Translated and Supplemented by S. Hahn), On Femat Quotient and $‘ {}^{t}Differentiatim$ of Numbers”, Univ. ofGeorgia Math. Preprint Ser. 9 (1994).
6. Dale Husem\"oller, Elliptic Curves, Springer-Verlag, New York, 1987.

7. A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992.
8. P. Ribenboim, Caidan’s Conjecture, Academic Press, 1994.
9. A. Schinzel, Pmmitive Dimsors of the Erpression $A^{n}-B^{n}$ in Algebraic Number Fidds, J. Reine Angew. Math. 268(1974), 27-33.

10. J. Silverman-, The Anthmetic of Elliptic Curves,
11. J. Silverman, The Difference between the$We\iota l$$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}-\dot{\mathrm{v}}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{g}$, 1986.

Heeght and the Canonical Heeght on Elliptic Curves, Math. Comp. 55(1990), 723-743.

12. J. Silverman, Wieferech’s Cmtemon and the $abc$ -Conjecture, Journal of Number Theory 30 (1988), 226-237.
13. J. Tate, The Amthmetic of Elliptic Curves, lnvent. Math. 23$(1974)_{\mathrm{t}}$179-206.

14. L. Washington, Introduction to Cydotomic Fidds, Springer-Verlag, 1979.
15. K. Zsigmondy, Zur Theome der Potenzreste, Monatsh. f. Math. 3 (1892), 265-284.

ELECTRONICS AND TELECOMMUNICATIONS RESEARCH INSTITUTE, 161 KAJONG-DoNc,$\mathrm{Y}_{\mathrm{U}\mathrm{S}\mathrm{o}\mathrm{N}}\mathrm{G}-\mathrm{G}_{\mathrm{U}}$, TAEJON 305-350,

REPU BLIC OF KOREA. E– MAIL ADDRESS : CHEON@CRYPT. KAIST. AC. KR
DEPARTMENT OF MATHEMATICS, KOREA ADVANCED INST[TUTE OF SCIENCE AND TECHNOLOGY, 373-1 GusoNG-DoNc,$\mathrm{Y}\mathrm{U}\mathrm{s}\mathrm{o}\mathrm{N}\mathrm{c}_{-}\mathrm{G}_{\mathrm{U}},$TAEJ ON 305-701, REPUBL$\iota \mathrm{c}$OF KOREA.

-MAIL ADDRESS :$\mathrm{R}$

SGHAHN@MATHX. KAI ST. AC. KR