Effective Injectivity of Specialization Maps for Elliptic Surfaces

This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.

Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?

What is the set of rational numbers such that the corresponding specialization maps are injective?

The classical specialization theorem of Neron proves that there is a set S which  differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.