Vol. 10, No. 1, 2016

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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
Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve Eλ of equation Y 2 = X(X 1)(X λ), we prove that, given n linearly independent points P1(λ),,Pn(λ) on Eλ with coordinates in (λ) ¯, there are at most finitely many complex numbers λ0 such that the points P1(λ0),,Pn(λ0) satisfy two independent relations on Eλ0. 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