Papers › Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
Barinder S. Banwait, Maarten Derickx
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We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbovi\'c. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields ℚ(√(d)) for all squarefree d with |d| < 800, improving on the previously known range of -5 < d < 26. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.
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