Impossible intersections in a Weierstrass family of elliptic curves

Abstract

Consider the Weierstrass family of elliptic curves Eλ:y2=x3+λ parametrized by nonzero λ∈Q2‾, and let Pλ(x)=(x,x3+λ)∈Eλ. In this article, given α,β∈Q2‾ such that αβ∈Q, we provide an explicit description for the set of parameters λ such that Pλ(α) and Pλ(β) are simultaneously torsion for Eλ. In particular we prove that the aforementioned set is empty unless αβ∈{−2,−12}. Furthermore, we show that this set is empty even when αβ∉Q provided that α and β have distinct 2-adic absolute values and the ramification index e(Q2(αβ)|Q2) is coprime with 6. We also improve upon a recent result of Stoll concerning the Legendre family of elliptic curves Eλ:y2=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∈C∖{0,1}:(a,a(a−1)(a−λ)),(b,b(b−1)(b−λ))∈(Eλ)tors} is empty.