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Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13]  discussed the k-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 that
for all elliptic curves E defined over a quadratic field k ().
Let . denote the modular curve associated with the normalizer of a non-split Cartan group of level N., where N. is an arbitrary integer. The curve is denned over Q and the corresponding scheme over ℤ[1/N] is smooth . If N. is a prime, the genus formula for . is given in [5,6]. The curve . has genus 0 if N < 11 and has genus 1. Ligozat  has shown that the group of Q-rational points on has rank 1. If the genus g(N). is greater than 1, very little is known about the Q-rational points of . Since under simple conditions imaginary quadratic fields with class number 1 give an integral point on these curves, Serre and others have asked whether all integral points are obtained in this way .
Let N be an integer ≥ 1. The affine modular curve Y0(N) parameterizes isomorphism classes of pairs (E; F), where E is an elliptic curve defined over ℂ, the field of complex numbers, and F is a cyclic subgroup of order N. The compacti-fication X0(N) is an algebraic curve defined over ℚ.