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 E1 −→ C and E2 −→ C along with sections Pi, Qi of Ei (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 that [mi,t](Pi)t = (Qi)t on Ei (for i = 1, 2), then at least one of the following conclusions must hold: either (i) there exists a nontrivial isogeny ψ : E1 −→ E2 and also there exist nontrivial endomorphisms ϕi of Ei (for i = 1, 2) such that ϕ2(P2) = ψ(ϕ1(P1)); or (ii) Qi is a multiple of Pi for some i = 1, 2. A special case of our result answers a conjecture made by Silverman. PreliminariesFrom now on, we fix an elliptic surface π : E −→ C, where C is a projective, smooth curve defined over Q. We denote by E the generic fiber of E; this is an elliptic curve defined over Q(C). For all but finitely many t ∈ C(Q), we have that E t := π −1 ({t}) is an elliptic curve defined over Q.

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

“…We will give two applications of the above statement. The first is a Mordell-Lang-type statement (Theorem 8.5) which somehow resembles a recent result of Ghioca, Hsia and Tucker [24]. In the Appendix, we show how their Theorem 1.1 can be deduced from the main results of [2,3].…”

“…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 .…”

Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture

“…It is worth mentioning that before this general theorem, a similar but weaker conclusion (together with other results immaterial here) was proved (with partially independent methods) as Theorem 1.1 of the paper [23] of Ghioca, L. Hsia and Tucker: this assumed A equal to a (fiber) product of two elliptic schemes, and moreover restricting to certain special subgroup schemes, but would be sufficient for our applications here (though probably not for generalisations of them).…”

“…The projection to P 1 is still undefined at four new points, one on each of the four exceptional divisors. Blowing them up again, we obtain a new smooth projective surface X , endowed with a well defined projection to the line, fitting in the diagram (23) X…”

“…L h ) of L p (resp. L h ) on C × C (see diagram 23). Observe that L h intersects the diagonal of C × C at the point (h, h), while L p intersects the diagonal at the complex points of the form (q, q) where q ∈ C is such that the tangent at q passes through p. Now, since p is internal, q cannot be real, so in particular q = h; this implies that (q, q) does not belong to L h .…”