Effective Injectivity of Specialization Maps for Elliptic Surfaces

Tyler R Billingsley, Purdue University

Abstract

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 Q. 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. 1. Given some t0 ∈ Q, how can we effectively determine whether or not the specialization map at t0 is injective? 2. What is the set Σ of t0 ∈ Q such that the specialization map at t0 is injective? The classical specialization theorem of N´eron proves that there is a set S which differs from a Hilbert subset of Q by finitely many elements such that for each t0 ∈ S the specialization map at t0 is injective. We expand this into an effective procedure that determines if some t0 ∈ Q 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 Gusi´c and Tadi´c to include elliptic surfaces with a rational 2-torsion curve.

Degree

Ph.D.

Advisors

Arapura, Purdue University.

Subject Area

Mathematics

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