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 $e({\mathbb{Q}}_2(\frac{\alpha }{\beta })|{\mathbb{Q}}_2)$ 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.

16 pages

References in corpus (1)

• Preperiodic points for families of rational map