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

Dragos Ghioca, Liang Chung Hsia, Thomas J. Tucker

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(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.

Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalPacific Journal of Mathematics
Volume295
Issue number1
DOIs
Publication statusPublished - 2018 Jan 1

Fingerprint

Pi
Elliptic Curves
Theorem
Isogenies
Elliptic Surfaces
Fiber
Curve
Integer

Keywords

  • Elliptic surfaces
  • Heights
  • Unlikely intersections in arithmetic dynamics

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A variant of a theorem by Ailon-Rudnick for elliptic curves. / Ghioca, Dragos; Hsia, Liang Chung; Tucker, Thomas J.

In: Pacific Journal of Mathematics, Vol. 295, No. 1, 01.01.2018, p. 1-15.

Research output: Contribution to journalArticle

Ghioca, Dragos ; Hsia, Liang Chung ; Tucker, Thomas J. / A variant of a theorem by Ailon-Rudnick for elliptic curves. In: Pacific Journal of Mathematics. 2018 ; Vol. 295, No. 1. pp. 1-15.
@article{495104c80f6d47a1a26362f5a99f4d43,
title = "A variant of a theorem by Ailon-Rudnick for elliptic curves",
abstract = "Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(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 = "Elliptic surfaces, Heights, Unlikely intersections in arithmetic dynamics",
author = "Dragos Ghioca and Hsia, {Liang Chung} and Tucker, {Thomas J.}",
year = "2018",
month = "1",
day = "1",
doi = "10.2140/pjm.2018.295.1",
language = "English",
volume = "295",
pages = "1--15",
journal = "Pacific Journal of Mathematics",
issn = "0030-8730",
publisher = "University of California, Berkeley",
number = "1",

}

TY - JOUR

T1 - A variant of a theorem by Ailon-Rudnick for elliptic curves

AU - Ghioca, Dragos

AU - Hsia, Liang Chung

AU - Tucker, Thomas J.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(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.

AB - Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(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.

KW - Elliptic surfaces

KW - Heights

KW - Unlikely intersections in arithmetic dynamics

UR - http://www.scopus.com/inward/record.url?scp=85044200457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044200457&partnerID=8YFLogxK

U2 - 10.2140/pjm.2018.295.1

DO - 10.2140/pjm.2018.295.1

M3 - Article

AN - SCOPUS:85044200457

VL - 295

SP - 1

EP - 15

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -