118 Proc. Japan Acad., 72, Ser. A (1996) [Vol. 72(A), Elliptic Curves Related with Triangles By Soonhak KWON Department of Mathematics, The Johns Hopkins University, U. S. A. (Communicated by Shokichi IYANAGA, M. J. A., June 11, 1996) In a series of papers [4] [5] [6], T. Ono associated an elliptic curve E to a triangle with sides a, b and c as follows’ 2 3 E’y x + Px2 + Qx, where 1 P=-(a 2+b --c), (a + b + c - 2a-b 2b2c 2ca2). Q We assume abQ 9 e 0 so that this cubic is non- singular. Then one verifies that the elliptic curve ( c c(b a) ) has a point Po= (Xo, Yo) 4 8 Assuming that a, b and c belong to an algebraic number field k, T. Ono obtained a certain condi- tion under which the point Po has an infinite order, and asked whether this condition can be improved (cf. [4,( I )]). In this paper, we assume that a, b and c belong to Q. So the elliptic curve is defined over Q and Po is a rational point. In this case, we will get more precise condition so that Po has an infinite order. Following another setting of T. Ono [4,(II)], we define l, m and n as follows" b+a b-a c l=,m=,n=-. Then, we have 2 E’y x(x + n2) (x + rn n2), andPo= ( n2 lmn) Since rational multiples of l, m, n (etc. a, b, c) give isomorphic elliptic curves, we may assume that l, m, n are integers with (l, m, n) 1. Further we assume lmn :/: O, because in case lmn 0 Po becomes a 2-torsion point. (i.e. we exclude isosceles triangles.) Theorem. Let E be an elliptic curve 2 2 m y =x(x+ -n)(x+ -n), where l, m, n are nonzero integers for which (l, m, n) =1, (l 2 n2) (m 2 n2) (l 2 m2) :/:0. Suppose that E does not satisfy the following two conditions. (i) There exist integers such that 2 2 2 2 22 l 2 ce 2(ce+fl)2 m = (ce+) n (ii) There is a relation among 1, m, n as follows" 1 1 1 1 1 1 --n + --m or 1 rn - -n or 1 1 1 2 12. Then, Po- (n 2, lmn) E(Q) is of infinite order. If E satisfies (i), Po becomes a 3-torsion point, and if E satisfies (ii), Po becomes a 4-torsion point. Proof. In view of the equation of E there exists a point P in E(Q) such that 2P- Po (cf. [2, Th. 4.2]). Suppose that Po is a torsion point. Then by Mazur’s classification of torsion sub- groups of elliptic curves over Q, we have Po 2P 2" (Z/2Z Z/vZ), v 2,4,6,8. From the above relation and since lmn 4= 0, we easily conclude that Po is either a 3-torsion point or a 4-torsion point. Now suppose that Po is a point of order 3, then the torsion subgroup of E is isomorphic to Z/2Z@ Z/6Z and the theorem of K. Ono [3] implies that there exist a positive integer d and relatively prime integers or,/5 such that l 2 n 2 d2a3(a+2fl), m 2 n d2fl(fl + 2a). Since (d2a2fl 2, + d3cr2/5 2(or + fl)2) are points of order 3 (as a simple computation shows) and these are the only 3-torsion points in Q, we have 2 2 n d2c2fl. Thus we get 12 n + d2a(a + 2/) d2a2(a + 2 2 d2cr d2f12 m n + (fl+2a) = (a+ Since we assumed (l, m, n) 1,we get d= 1, and l a (a+) n ,m =fl (ce+/5) =ce2fl 2, where ce and/ are relatively prime integers. Con- versely if l, rn, n satisfy above conditions, then Po must be a 3-torsion point. Next we suppose that Po is a 4-torsion point. Then, since 2Po is a point of order 2, we have 2P o (0,0), or (n 2 l 0), or (n 2- m 2 0) Note that, if (Xo, Yo) is a point of y2= x(x + M)"