  We provide two different proofs of an irreducibility criterion for the
preimages of a transverse subvariety of a product of elliptic curves under a
diagonal endomorphism of sufficiently large degree. For curves, we present an
arithmetic proof of the aforementioned irreducibility result, which enlightens
connections to methods used in the context of the Torsion Anomalous Conjecture.
On the other hand, we generalize the result for higher dimensional varieties
using a more geometric approach. Finally, we give some applications of these
results. More precisely, we establish the irreducibility of some explicit
families of polynomials, we provide new estimates for the normalized heights of
certain intersections and images, and we give new lower bounds for the
essential minima of preimages.
