Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

Text. In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and Q(root-3). Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and Q(root-3), we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and Q(root-3). Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VPhCkJTGB_o. (C) 2010 Elsevier Inc. All rights reserved.