Given a family of algebraic varieties over an irreducible scheme, a natural question to ask is what type of properties of the generic fiber, and how do those properties extend to other fibers. For example, the Hilbert irreducibility theorem states that a dominant map from an irreducible variety X defined over a number field to some projective space which is generically of degree d provides a Zariski dense set of degree d points on X. One can also get quantitative estimates for size of the complement which does not carry the generic property.

We will explore this topic from an arithmetic point of view by looking at several scenarios. For instance, suppose we have a 1-dimensional family of pairs of elliptic curves over a number field,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. One may ask how does the property of "being (non-)isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most B, such that the two elliptic curves at the specializations are isogenous?