On a conjecture of Buium and Poonen

Abstract

Given a correspondence between a modular curve S and an elliptic curve A, we prove that the intersection of any finite-rank subgroup of A with the set of points on A corresponding to an isogeny class on S is finite. The question was proposed by A. Buium and B. Poonen in 2009. We follow the strategy proposed by the authors, using a result about the equidistribution of Hecke points on Shimura varieties and Serre's open image theorem. The result is an instance of the Zilber-Pink conjecture.Résumé. -Étant donnée une correspondance entre une courbe modulaire S et une courbe elliptique A, nous prouvons que l'intersection de tout sous-groupe de rang fini de A avec l'ensemble de points de A d'une classe isogénie sur S est fini. Le question a été posée par A. Buium et B. Poonen en 2009. Nous suivons la stratégie proposée par les auteurs, utilisant un résultat sur l'équidistribution des points de Hecke sur les variétés de Shimura et le théorème de l'image ouverte de Serre. Le résultat est un cas particulier de la conjecture de Zilber-Pink.

Independence of CM points in elliptic curves

“…One may deduce corollaries analogous to 1.2-1.4 above. The last recovers a result of Baldi [1, Theorem 1.3], (obtained via equidistribution), which is also a special case of results of Dill [9,10], affirming a conjecture of Buium-Poonen [6, Conjecture 1.7]; see the discussion in [1]. Baldi obtains a stronger "Bogomolov"-type result, which we do not.…”

Unlikely intersections between isogeny orbits and curves

“…We can then form the curve S × Ag A g,l , which admits quasi-finite morphisms to S and A g,l , and reduce the conjecture to Corollary 1.4. The conjecture of Buium and Poonen has been proven independently by Baldi in [3] through the use of equidistribution results. He was also able to replace Γ by a fattening Γ for some > 0 (see [3] for the definition of Γ ).…”

“…The conjecture of Buium and Poonen has been proven independently by Baldi in [3] through the use of equidistribution results. He was also able to replace Γ by a fattening Γ for some > 0 (see [3] for the definition of Γ ). Such an extension seems to lie outside the reach of our methods though.…”

Unlikely intersections with isogeny orbits in a product of elliptic schemes

“…So far, results have been obtained only in the cases when V is a curve (Dill [8], Gao [14], Lin-Wang [32]) or Γ contains only torsion points (Gao [14], Habegger [21], Pila [45]). See also [2] and [47] for related results.…”

“…where the lower bound follows from (4.8). We then set a i = 4(b i d i ) 2 . The generalized Vojta-Rémond inequality yields additional constants…”