Linear relations in families of powers of elliptic curves

Barroero, Fabrizio; Capuano, Laura

doi:10.2140/ant.2016.10.195

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Linear relations in families of powers of elliptic curves

Fabrizio Barroero and Laura Capuano

Vol. 10 (2016), No. 1, 195–214

DOI: 10.2140/ant.2016.10.195

Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_{λ}$ of equation $Y^{2} = X (X - 1) (X - λ)$ , we prove that, given $n$ linearly independent points $P_{1} (λ), \dots, P_{n} (λ)$ on $E_{λ}$ with coordinates in $\bar{ℚ (λ)}$ , there are at most finitely many complex numbers $λ_{0}$ such that the points $P_{1} (λ_{0}), \dots, P_{n} (λ_{0})$ satisfy two independent relations on $E_{λ_{0}}$ . This is a special case of conjectures about unlikely intersections on families of abelian varieties.

Keywords

linear relations, elliptic curves, unlikely intersections

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G50, 11U09, 14K05

Milestones

Received: 29 January 2015

Revised: 1 October 2015

Accepted: 27 November 2015

Published: 14 February 2016

Authors

Fabrizio Barroero
Classe di Scienze Scuola Normale Superiore Piazza dei Cavalieri 7 I-56126 Pisa Italy

Laura Capuano
Classe di Scienze Scuola Normale Superiore Piazza dei Cavalieri 7 I-56126 Pisa Italy

Vol. 10, No. 1, 2016