Irreducibility criteria for the preimages of a transverse variety under endomorphisms of products of elliptic curves

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 images.