Simultaneous torsion in the Legendre family

Abstract

We improve a result due to Masser and Zannier, who showed that the set $$ \{\lambda \in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-\lambda)}), (3,\sqrt{6(3-\lambda)}) \in (E_\lambda)_{\text{tors}}\} $$ is finite, where $E_\lambda \colon y^2 = x(x-1)(x-\lambda)$ is the Legendre family of elliptic curves. More generally, denote by $T(\alpha, \beta)$, for $\alpha, \beta \in {\mathbb C} \setminus \{0,1\}$, $\alpha \neq \beta$, the set of $\lambda \in {\mathbb C} \setminus \{0,1\}$ such that all points with $x$-coordinate $\alpha$ or $\beta$ are torsion on $E_\lambda$. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set $T(2,3)$ in question is actually empty. More generally, we obtain an explicit description of the set of parameters $\lambda$ such that the points with $x$-coordinate $\alpha$ and $\beta$ are simultaneously torsion, in the case that $\alpha$ and $\beta$ are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that ${\mathbb Q}(\alpha, \beta)$ has transcendence degree 1. In this case we show that $\#T(\alpha, \beta) \le 1$ and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating $\alpha$ and $\beta$. This leads to a more precise description of $T(\alpha, \beta)$ also in the case when both $\alpha$ and $\beta$ are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs $(\alpha, \beta)$ such that $\#T(\alpha, \beta) \ge 3$.