Torsion points on elliptic curves over fields of low degree (Q919049)

The author has proved before [see Duke Math. J. 53, 157-162 (1986; Zbl 0599.14029) or Bull. Soc. Math. Fr. 114, 119-122 (1986; Zbl 0599.14030)] for certain small primes p the non existence of elliptic curves with rational p-torsion defined over any quadratic field. Here we find an extension of this result, for example theorem 4.1: Let \(p>17\) be a prime \(\equiv 1\) (mod 4) there are only finitely many elliptic curves with rational p-torsion defined over the totality of quadratic extensions of \({\mathbb{Q}}(\sqrt{p})\).