Given a smooth projective curve
$C$
defined over
$\overline{\mathbb{Q}}$ and given
two elliptic surfaces
${\mathcal{\mathcal{E}}}_{1}\to C$
and
${\mathcal{\mathcal{E}}}_{2}\to C$ along with sections
${\sigma}_{{P}_{i}},{\sigma}_{{Q}_{i}}$ (corresponding to
points
${P}_{i},{Q}_{i}$ of the generic
fibers) of
${\mathcal{\mathcal{E}}}_{i}$ (for
$i=1,2$), we prove that if there
exist infinitely many
$t\in C\left(\overline{\mathbb{Q}}\right)$ such
that for some integers
${m}_{1,t},{m}_{2,t}$,
we have
$\left[{m}_{i,t}\right]\left({\sigma}_{{P}_{i}}\left(t\right)\right)={\sigma}_{{Q}_{i}}\left(t\right)$
on
${\mathcal{\mathcal{E}}}_{i}$ (for
$i=1,2$), then
at least one of the following conclusions must hold:
i. There exist isogenies
$\phi :{E}_{1}\to {E}_{2}$
and
$\psi :{E}_{2}\to {E}_{2}$ such
that
$\phi \left({P}_{1}\right)=\psi \left({P}_{2}\right)$.
ii.
${Q}_{i}$ is a
multiple of
${P}_{i}$
for some
$i=1,2$.
A special case of our result answers a conjecture made by Silverman.
