Let He a quadratic field and E an elliptic curve defined over k. The authors [8,12,13] [23] discussed the ^-rational points on E of prime power order. For a prime number p, let n = n{k,p) be the least non negative integer such thatfor all elliptic curves E defined over a quadratic field k ([15]). For prime numbers p < 300, p ψ 151, 199, 227 nor 277, we know that n(k, 2) = 3 or 4, n(k 9 3) = 2, n(k, 5) = n(k, 7) = 1, n(k, 11) = 0 or 1, n(k, 13) = 0 or 1, and n(k, p) = 0 for all the prime numbers p ^> 17 as above (see loc. cit.). It seems that n(k, p) = 0 for all prime numbers p ^ 17 and for all quadratic fields k. In this paper, we discuss the iV-torsion points on E for integers N of products of powers of 2, 3, 5, 7,11 and 13. Let

No abstract

No abstract

Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.

No abstract