Impossible intersections in a Weierstrass family of elliptic curves

Abstract

Consider the Weierstrass family of elliptic curves $E_{\lambda}:y^2=x^3+\lambda$ parametrized by nonzero $\lambda\in\overline{\mathbb{Q}_2}$, and let $P_{\lambda}(x)=(x,\sqrt{x^3+\lambda})\in E_{\lambda}$. In this article, given $\alpha,\beta\in\overline{\mathbb{Q}_2}$ such that $\frac{\alpha}{\beta}\in\mathbb{Q}$, we provide an explicit description for the set of parameters $\lambda$ such that $P_{\lambda}(\alpha)$ and $P_{\lambda}(\beta)$ are simultaneously torsion for $E_{\lambda}$. In particular we prove that the aforementioned set is empty unless $\frac{\alpha}{\beta}\in\{-2,-\frac{1}{2}\}$. Furthermore, we show that this set is empty even when $\frac{\alpha}{\beta}\notin\mathbb{Q}$ provided that $\alpha$ and $\beta$ have distinct $2-$adic absolute values and the ramification index $e(\mathbb{Q}_2(\frac{\alpha}{\beta})~\vert~\mathbb{Q}_2)$ is coprime with $6$. We also improve upon a recent result of Stoll concerning the Legendre family of elliptic curves $E_{\lambda}:y^2=x(x-1)(x-\lambda)$, which itself strengthened earlier work of Masser and Zannier by establishing that provided $a,b$ have distinct reduction modulo $2$, the set $\{\lambda\in\mathbb{C}\setminus\{0,1\}~:~(a,\sqrt{a(a-1)(a-\lambda)}),(b,\sqrt{b(b-1)(b-\lambda)})\in (E_{\lambda})_{tors}\}$ is empty.