## 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.

Original language | English (US) |
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Pages (from-to) | 21-40 |

Number of pages | 20 |

Journal | Journal of Number Theory |

Volume | 169 |

DOIs | |

State | Published - Dec 1 2016 |

## Keywords

- Attracting fixed point
- Elliptic curve
- Family of Lattès maps
- Impossible intersections
- Preperiodic point
- Torsion

## ASJC Scopus subject areas

- Algebra and Number Theory