Unlikely intersections in products of families of elliptic curves and the multiplicative group

Abstract

Let E λ be the Legendre elliptic curve of equation Y 2 = X(X − 1)(X − λ). We recently proved that, given n linearly independent points P1(λ), . . . , Pn(λ) on E λ with coordinates in Q(λ), there are at most finitely many complex numbers λ0 such that the points P1(λ0), . . . , Pn(λ0) satisfy two independent relations on E λ 0 . In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two non-isogenous families of elliptic curves and in a family of split semi-abelian varieties.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

“…Now, we present another application of our Theorem . Recently, Ghioca, Hsia and Tucker proved a statement in the spirit of unlikely intersections which is relatively similar to the main result of . Theorem Let be two elliptic surfaces over a curve defined over with generic fibres , and let be sections of (for ) corresponding to points .…”

“…In the Appendix, we will show how the main theorems of imply Theorem while in what follows we deduce an analogous statement for families of abelian varieties without elliptic factors.…”

“…In the previous works , we proved that the intersection of with the union of all flat subgroup schemes of of codimension at least 2 is finite, when is a fibred product of elliptic schemes.…”

“…In the first three cases, the statement of Theorem follows, respectively, from (1)the work of Habegger and Pila ; (2)Theorem 2.1 of ; (3)Theorem 1.3 of . …”

“…Let E λ denote the elliptic curve in the Legendre form defined by Y 2 = X(X − 1)(X − λ). In [34] (see also [33]), Masser and Zannier showed that there are at most finitely many complex numbers λ 0 = 0, 1 such that the two points with coordinates (2, 2(2 − λ 0 )) and (3,6(3 − λ 0 )) both have finite order on the elliptic curve E λ0 . Later, in [35] the same authors proved that one can replace 2 and 3 with any two distinct complex numbers ( = 0, 1) or even choose distinct X-coordinates ( = λ) defined over an algebraic closure of C(λ), provided the points are not identically linearly related.…”