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

Dragos Ghioca; Liang Chung Hsia; Thomas J. Tucker

doi:10.2140/pjm.2018.295.1

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

Dragos Ghioca, Liang Chung Hsia, Thomas J. Tucker

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 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 σ_Pi, σ_Qi (corresponding to points P_i, Q_i of the generic fibers) of ε_i (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m_1,t,m_2,t, we have [m_i,t](σP_i(t)) = σ_Qi (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. Q_i is a multiple of P_i for some i = 1,2. A special case of our result answers a conjecture made by Silverman.

Original language	English
Pages (from-to)	1-15
Number of pages	15
Journal	Pacific Journal of Mathematics
Volume	295
Issue number	1
DOIs	https://doi.org/10.2140/pjm.2018.295.1
Publication status	Published - 2018

Keywords

Elliptic surfaces
Heights
Unlikely intersections in arithmetic dynamics

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.2140/pjm.2018.295.1

Cite this

@article{dd4f5c3fe7774baf9e16363c8cbf0cc3,

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.}",

note = "Funding Information: We thank Myrto Mavraki and Joe Silverman for several useful conversations. We are grateful to the anonymous referee for numerous comments and suggestions which improved our paper. The research of Ghioca was partially supported by an NSERC Discovery grant. Hsia was supported by MOST grant 104-2115-M-003-004-MY2. Tucker was partially supported by NSF Grant DMS-0101636. Publisher Copyright: {\textcopyright} 2018 Mathematical Sciences Publishers.",

year = "2018",

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.

N1 - Funding Information: We thank Myrto Mavraki and Joe Silverman for several useful conversations. We are grateful to the anonymous referee for numerous comments and suggestions which improved our paper. The research of Ghioca was partially supported by an NSERC Discovery grant. Hsia was supported by MOST grant 104-2115-M-003-004-MY2. Tucker was partially supported by NSF Grant DMS-0101636. Publisher Copyright: © 2018 Mathematical Sciences Publishers.

PY - 2018

Y1 - 2018

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

SN - 0030-8730

VL - 295

SP - 1

EP - 15

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this