Linear relations in families of powers of elliptic curves

Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_{\lambda }$ of equation $Y^2=X(X-1)(X-\lambda )$, we prove that, given $n$ linearly independent points $P_1(\lambda ),...,P_n(\lambda )$ on $E_{\lambda }$ with coordinates in $\overline{\mathbb{Q}(\lambda )}$, there are at most finitely many complex numbers ${\lambda }_0$ such that the points $P_1({\lambda }_0),...,P_n({\lambda }_0)$ satisfy two independent relations on $E_{{\lambda }_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.

References in corpus (1)

• Rational points on Grassmannians and unlikely intersections in tori

Cited by in corpus (5)

• Unlikely Intersections in families of abelian varieties and the polynomial Pell equation
• CM relations in fibered powers of elliptic families
• Point counting for foliations over number fields
• Isogeny relations in products of families of elliptic curves
• Torsion points with multiplicatively dependent coordinates on elliptic curves