Vol. 295, No. 1, 2018

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A variant of a theorem by Ailon–Rudnick for elliptic curves

Dragos Ghioca, Liang-Chung Hsia and Thomas J. Tucker

Vol. 295 (2018), No. 1, 1–15
Abstract

Given a smooth projective curve C defined over ¯ and given two elliptic surfaces 1 C and 2 C along with sections σPi,σQi (corresponding to points Pi,Qi of the generic fibers) of i (for i = 1,2), we prove that if there exist infinitely many t C( ¯) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi(t) on i (for i = 1,2), then at least one of the following conclusions must hold:

i. There exist isogenies φ : E1 E2 and ψ : E2 E2 such that φ(P1) = ψ(P2). ii. Qi is a multiple of Pi for some i = 1,2.

A special case of our result answers a conjecture made by Silverman.

Keywords
heights, elliptic surfaces, unlikely intersections in arithmetic dynamics
Mathematical Subject Classification 2010
Primary: 11G50
Secondary: 11G35, 14G25
Milestones
Received: 8 March 2017
Revised: 10 November 2017
Accepted: 3 January 2018
Published: 13 March 2018
Authors
Dragos Ghioca
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Liang-Chung Hsia
Department of Mathematics
National Taiwan Normal University
Taipei
Taiwan
Thomas J. Tucker
Department of Mathematics
University of Rochester
Rochester, NY
United States