  In 2007, Bogomolov and Tschinkel proved that given two complex elliptic
curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi_j\colon E_j\to
\mathbb{P}^1$ having different branch loci, the intersection of the image of
the torsion points of $E_1$ and $E_2$ under their respective $\pi_j$ is finite.
They conjectured (also in works with Fu) that the cardinality of this
intersection is uniformly bounded independently of the elliptic curves. As it
has been observed in the literature, the recent proof of the Uniform
Manin-Mumford conjecture implies a full solution of the Bogomolov-Fu-Tschinkel
conjecture. In this work we prove a generalization of the
Bogomolov-Fu-Tschinkel conjecture where instead of even degree-$2$ maps one can
use any rational functions of bounded degree on the elliptic curves as long as
they have different branch loci. Our approach combines Nevanlinna theory with
the Uniform Manin-Mumford conjecture. With similar techniques, we also prove a
result on lower bounds for ranks of elliptic curves over number fields.
