Impossible intersections in a Weierstrass family of elliptic curves

Abstract

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