Torsion points and the lattès family

Laura Demarco, Xiaoguang Wang, Hexi Ye

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We give a dynamical proof of a result of Masser and Zannier: for any a ≠b ∈ ℚ\{0,1}, there are only finitely many parameters t ∈ ℂ for which points Pa = (a√ a(a−1)(a−t)) and Pb = (b, √ b(b−1)(b−t)) are both torsion on the Legendre elliptic curve Et = {y2 = x(x−1)(x−t)}. Our method also gives the finiteness of parameters t where both Pa and Pb have small Néron-Tate height. A key ingredient in the proof is an arithmetic equidistribution theorem on ℙ1. For this, we prove two statements about the degree-4 Lattès family ft on ℙ1: (1) for each c ∈ ℂ(t), the bifurcation measure μc for the pair (ft,c) has continuous potential across the singular parameters t = 0,1,∞; and (2) for distinct points a,b ∈ ℂ \ {0,1}, the bifurcation measures μa and μb cannot coincide. Combining our methods with the result of Masser and Zannier, we extend their conclusion to points t of small height also for a,b ∈ ℂ(t).

Original languageEnglish (US)
Pages (from-to)697-732
Number of pages36
JournalAmerican Journal of Mathematics
Volume138
Issue number3
DOIs
StatePublished - Jun 2016

ASJC Scopus subject areas

  • Mathematics(all)

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