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We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over Q. Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.

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Fumiyuki Momose is very much missed. He was a generous warm human being, with immense energy and generosity of spirit, and an extremely gifted mathematician. One of his abiding interests was rational torsion on elliptic curves over number fields, as in [32], [24]. This article is written in his memory.

Torsion points on elliptic curves over all quadratic fields. II Bulletin de la S. M. F., tome 114 (1986), p. 119-122 © Bulletin de la S. M. F., 1986, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/