Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

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 Trbović. 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 \(\mathbb {Q}(\sqrt{d})\) $\mathbb{Q}(\sqrt{d})$ for all squarefree d with \(|d| < 800\) $|d|<800$ , improving on the previously known range of \(-5< d < 26\) $-5<d<26$ . 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.