ACTA ARITHMETICA LXXXVIII.3 (1999)

The orders of the reductions of a point in the Mordell–Weil group of an elliptic curve by J. Cheon and S. Hahn (Taejon) In 1886, A. Bang showed that there exists a constant M> 0 so that for each non-zero rational number x, x 6 = ±1, and every integer n> M, there exists a prime number p so that the order of x modulo p is equal to n (see [1]). Since then his result was extended and generalized by other mathematicians. In 1892, K. Zsigmondy found a stronger version (see [8], [3, p. 20]). But most of all, in 1974, A. Schinzel proved that for any number field K there exists a constant M> 0 so that for each x ∈ K × which is not a root of unity, and every integer n> M, there exists a prime ideal ℘ of K so that the order of x modulo ℘ is equal to n (see [4]). Motivated by Y. Ihara’s interpretation of Bang’s theorem (see [2]), in this paper we prove the following elliptic analogue. Theorem. Let E be an elliptic curve over a number field K and let P ∈ E(K) be a point of infinite order. Then for every sufficiently large integer n there exists a prime ℘ of good reduction so that the order of P in the group of points of E modulo ℘ is equal to n. More-over, for all but finitely many P there exists such a prime ℘ for each n> 0. J. Silverman proved the above theorem for elliptic curves defined over Q (see [7]) which we prove first by explicit valuations of division polynomials. To prove the full theorem, we use essentially the same techniques, namely formal groups and heights, as Silverman did. In Schinzel’s result the constant M which depends on the number fieldK was effectively computable. Though we obtain a stronger result for the elliptic analogue, namely n> 0 is enough for all but finitely many P, we could not get an effective estimate for those finitely many exceptions