TY - JOUR

T1 - Torsion points and the lattès family

AU - Demarco, Laura

AU - Wang, Xiaoguang

AU - Ye, Hexi

N1 - Funding Information:
Research supported by the National Science Foundation and the University of Illinois at Chicago
Publisher Copyright:
© 2016 by Johns Hopkins University Press.

PY - 2016/6

Y1 - 2016/6

N2 - 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).

AB - 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).

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U2 - 10.1353/ajm.2016.0026

DO - 10.1353/ajm.2016.0026

M3 - Article

AN - SCOPUS:84969963741

SN - 0002-9327

VL - 138

SP - 697

EP - 732

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 3

ER -