The number of twists with large torsion of an elliptic curve

Abstract

For an elliptic curve $E/\mathbb {Q}$, we determine the maximum number of twists $E^d/\mathbb {Q}$ it can have such that $E^d(\mathbb {Q})_{tors}\supsetneq E(\mathbb {Q})[2]$. We use these results to determine the number of distinct quadratic fields $K$ such that $E(K)_{tors}\supsetneq E(\mathbb {Q})_{tors}$. The answer depends on $E(\mathbb {Q})_{tors}$ and we give the best possible bound for all the possible cases.

Appendix

Table 1 gives examples for every case possible from Theorem 1, excluding those when the choice of $E(\mathbb {Q})_{tors}$ uniquely determines the number of twists with large torsion. The second column gives an example of such an elliptic curve, the third column gives the number of quadratic twists of $E$, and in the fourth column all the $d$-s such that $E^d(\mathbb {Q})$ has large torsion are listed, with $E^d(\mathbb {Q})_{tors}$ given in brackets for every listed $d$.

Table 2 gives examples for every case possible from Theorem 2, excluding those when the choice of $E(\mathbb {Q})_{tors}$ uniquely determines the number of quadratic fields in which the torsion of $E$ grows. The values in the columns are listed in a similar manner as in Table 1.