### arXiv:1803.04946 [math.NT]AbstractReferencesReviewsResources

#### On a conjecture of Buium and Poonen

Published 2018-03-13Version 1

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 problem was conjectured 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.

**Comments:**Comments are welcome

**Categories:**math.NT

Bounds for Serre's open image theorem for elliptic curves over number fields

arXiv:0709.0132 [math.NT] (Published 2007-09-02)

On the index of the Heegner subgroup of elliptic curves

Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves