RATIONAL POINTS ON FREY ELLIPTIC CURVES y 2 = x3 n2x Ilker INAM, Osman BIZIM, Ismail Naci CANGUL
Abstract In this work, we study the rational points on elliptic curves of the form y 2 = x3 n2 x over …nite …elds in a di¤erent way, i.e. using only elementary number theory. We calculate the number of rational points over Fp modulo 8. We show that there are two possible cases where p 1 or 3 (mod 4). In the former case we …nd a classi…cation of the number of points, while in the latter case, we know that there are p + 1 points on the curve by the supersingular curve theory.
1
Introduction
1
Let F be a …eld of characteristic not equal to 2 or 3. An elliptic curve E de…ned over F is given by an equation y 2 = x3 + Ax + B 2 F[x]
(1)
where A; B 2 F so that 4A3 + 27B 2 6= 0 in F. The set of all solutions (x; y) 2 F F to this equation together with a point , called the point at in…nity, is denoted by E(F), and called the set of F-rational points on E: The value (E) = 16(4A3 + 27B 2 ) is called the discriminant of the elliptic curve E. For a more detailed information about elliptic curves in general, see [1]. The E(F) forms an additive abelian group having identity . Here by de…nition, P = (x; y) for a point P = (x; y) on E: It has always been interesting to look for the number of points over a given …eld F: In [2], three algorithms to …nd the number of points on an elliptic curve over a …nite …eld are given. Also in [3], [4] the number of rational points on Frey elliptic curves is found. In this paper we only use elementary number theory in a di¤erent way. A similar work has been done for the elliptic curves y 2 = x3 + cx in [5]. The third author is supported by the Uludag University Research Fund, Project No: F2004/40. 1 AMS 2000 Subject Classi…cation Number : 11G20, 14G05 Keywords: Elliptic curves over …nite …elds, rational points
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In [14], starting with a conjecture from 1952 of Dénes which is a variant of Fermat-Wiles theorem, Merel illustrates the way in which Frey elliptic curves have been used by Taylor, Ribet, Wiles and the others in the proof of FermatWiles theorem. Serre, in [15], gave a lower bound for the Galois representations on elliptic curves over the …eld Q of rational points. In the case of a Frey curve, the conductor N of the curve is given by the help of the constants in the abc conjecture. In [12], Ono recalls a result of Euler, known as Euler’s concordant forms problem, about the classi…cation of those pairs of distinct non-zero integers M and N for which there are integer solutions (x; y; t; z) with xy 6= 0 to x2 + M y 2 = t2 and x2 + N y 2 = z 2 :When M = N , this becomes the congruent number problem, and when M = 2N , by replacing x by x N in E(2N; N ); a special form of the Frey elliptic curves is obtained as y 2 = x3 N 2 x: Using Tunnell’s conditional solution to the congruent number problem using elliptic curves and modular forms, Ono studied the elliptic curve y 2 = x3 + (M +N )x2 +M N x denoted by EQ (M; N ) over Q. He classi…ed all the cases and hence reduced Euler’s problem to a question of ranks. In [9], Parshin obtaines an inequality to give an e¤ective bound for the height of rational points on a curve. In [11], the problem of boundedness of torsion for elliptic curves over quadratic …elds is settled. Here we shall deal with these problems in the …nite …eld case. Let us denote Frey elliptic curves by En , the set of Fp -rational points on En by En (Fp ) and the set of quadratic residues in mod p by Qp , and let Np;n be the cardinality of the set En (Fp ): We prove the following: Theorem 1 If p
1 (mod 4) is a rational prime then 0 (mod 8) 4 (mod 8)
Np;n =
if n 2 Qp if n 2 = Qp .
In [1], by supersingular curve theory, we know that if p Np;n = p + 1: To prove the theorem, we need some preliminaries:
2
Preliminaries
In the following,
a p
denotes the Legendre symbol.
Proposition 2 ([6]) Let p be an odd prime. Then p 1 X y2 + c p y=0
=
p
1 1
2
if c 6 0 (mod p) if c 0 (mod p).
3 (mod 4) then
By this proposition we can adopt this result to elliptic curves y 2 = x3 as follows:
n2 x,
Corollary 3 Let p be an odd prime. Then p 1 X i2
n2 p
i=0
=
1 1
p
if n 6 0 (mod p) if n 0 (mod p).
Proof. By Proposition 2, if one takes c = n2 then the result follows since "n 0 (mod p) if and only if n2 0 (mod p)". One can calculate the number of rational points on the elliptic curves y 2 = x3 n2 x by Legendre symbol and Wilson theorem which are important in elementary number theory. Now, we need the corollary below. In the following theorem we assume that n is not zero modulo p. Theorem 4 If p p
1 (mod 4) is prime, then 1
2 Y
i2
n2
i=1 i 6= n
+1 1
=
p
if p if p
1 (mod 8) 5 (mod 8).
Proof. By Corollary 3, we get p 1 X i2 i=0
Since p
p
n2 p
n2 p
=
i=1
n2 p
= n2 p
n2 2 Qp . Hence
1,5 (mod 8), so that p
+2
1
2 X i2
1:
= +1. Therefore
1
2 X i2
n2 p
i=1
=
1.
i2
n2
(1)
Let us denote M=
1
i
p
1
:
2
p
= +1
and N=
1
i
p
1 2
:
i2
n2 p
=
1 .
By (1), we conclude that the cardinality of the set N is greater than the one of M . Because for i = n and i = p n, we know that i2 n2 0(mod p). Further, either n or p n lies between 1 and p 2 1 . Therefore, for only one value 3
of n or p n, i2 n2 could be divisible by p. That is, for only this value, i2 n2 = 0. For all other values of i, this symbol is +1 or 1. Also by (1), p as there is an even number of summands, the number of 1’s should be one p
more than the number of +1’s. The product
jN j
to ( 1)
3
p
= ( 1)
i2
n2 p
i=1 i 6= n
1 4
1
2 Y
thus evaluates
.
Proof of Theorem 1.
First, we need the following: Proposition 5 ([5]) If p p
1 (mod 4) is a prime then
1
2 Y i p i=1
=
+1 1
if p if p
1 (mod 8) 5 (mod 8).
Hence the number of rational points on the Frey elliptic curves, …rstly depends on the number of points which are of order two. Because, by [8], we know that points which satisfy 2P = where being the in…nity point, are of the form (x; 0). Let us denote E[m] = fP 2 En (Fp ) : mP = g : Lemma 6 Let En be a Frey elliptic curve. Then the cardinality of the set E[2] is 4. Proof. By [8], we know that points which satisfy 2P = are of form (x; 0). Also (p x; 0); (0; 0) and are on the elliptic curve, so that the result is clear. Now we shall compute p
1
2 Y
i=1 i 6= n
p
i p
i2
n2 p
=
1
2 Y
i=1 i 6= n
p
i p
i=1 i 6= n
First of all, we need the following Proposition 7 i. Let p
1 (mod 8) be prime, then
4
1
2 Y
i2
n2 p
.
p
1
2 Y
i p
i=1 i 6= n ii: Let p p
=
+1 1
if n 2 Qp if n 2 = Qp .
5 (mod 8) be prime, then 1
2 Y
i=1 i 6= n
i p
=
1 +1
if n 2 Qp if n 2 = Qp.
Proof. i: Let p
1 (mod 8) be prime. In this case there are an odd number of n = 1 so elements in the product, so if n 2 Qp ; then p p
1
2 Y
i=1 i 6= n and if n 2 = Qp , then
n p
=
i p
= +1,
i p
=
1 so p
1
2 Y
i=1 i 6= n
1.
ii: Let p
5 (mod 8) be prime. In this case there are an odd number of n elements in the product, so if n 2 Qp ; then = +1 so p p
1
2 Y
i=1 i 6= n and if n 2 = Qp , then
n p
=
i p
=
i p
= +1.
1,
1 so p
1
2 Y
i=1 i 6= n
5
p
Let p
1 (mod 8): Thus if n 2 Qp ; then by Proposition 7
and by Theorem 4 p
1
2 Y
i=1 i 6= n
p
Likewise if n 2 = Qp ; then
p
1
2 Y
i=1 i 6= n
n2 p
i2 n2 i p p 0 (mod 8):
even integer, so by Lemma 6, Np;n p
i2
i=1 i 6= n
that is the number of i’s satisfying
i=1 i 6= n
i=1 i 6= n
i p
p
=
= +1,
= +1 with 1
i2 p
i=1 i 6= n i p
n2
i2
n2
p 4 (mod 8):
=
=
1.
1 with 1
5 (mod 8): Thus if n 2 Qp ; then by Lemma 8,
and by Theorem 4 1
i=1 i 6= n
p
even integer, so by Lemma 6, Np;n Likewise if n 2 = Qp ; then
i2
i=1 i 6= n
that is the number of i’s satisfying
i
i=1 i 6= n
is
=
1,
1
2 Y
i=1 i 6= n
n2 p
i p
i i2 n2 p p 0 (mod 8):
= +1;
= +1 with 1
i
1
2 Y
p 1 2
1
2 Y
i p
p
is an
1, and by Theorem 4
p
p
p 1 2
i
1
2 Y
i p
an odd integer, so by Lemma 6, Np;n
2 Y
= +1,
1
2 Y
That is the number of i’s satisfying
Let p
i p
1
2 Y
i p
1
2 Y
i p
= +1, and by Theorem 4
6
p 1 2
is an
p
1
2 Y
i=1 i 6= n
p
i p
1
2 Y
i2 p
i=1 i 6= n i p
n2
i2
=
1.
n2
p 1 = 1 with 1 i 2 is p an odd integer, so by Lemma 6, Np;n 4 (mod 8). Note that we count the point (0; 0) in Lemma 6, in which case i = n. Therefore the proof of Theorem 1 is complete.
That is the number of i’s satisfying
The authors would like to thank the referee for his valuable and helpful remarks which made the paper more readable.
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[10] Frey, G., L-series of elliptic curves: results, conjectures and consequences, Proc. of the Ramanujan Centennial Int. Conference (Annamalainagar, 1987), 31-43, RMS Publ., 1, Ramanujan Math. Soc., Annamalainagar, 1988. [11] Kamienny, S., Some remarks on torsion in elliptic curves, Comm. Alg. 23 (1995), no. 6, 2167-2169. [12] Ono, K., Euler’s concordant forms, Acta Arith. 78 (1996), no. 2, 101-123. [13] Darmon, H., Diamond, F., Taylor, R. L., Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem, (Hong Kong, 1993), 2-140, Internat. Press, Cambridge, MA, 1997. [14] Merel, L., Arithmetic of elliptic curves and Diophantine equations, Les XXemes Journees Arithmetiques (Limoges, 1997), J. Theor. Nombres Bordeaux 11 (1999), no. 1, 173-200. [15] Serre, J.-P., Propriétés galoisiennes des points d’ordre …ni des courbes elliptiques, Invent. Math. 15 (1972), 259-331. Uluda¼ g University Faculty of Arts and Science Department of Mathematics 16059 Gorukle/Bursa TURKEY
[email protected],
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