Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 03/03/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

Inaug.-Diss.--Bonn. / Extra t.p. with thesis statement inserted. Bibliography: p. 54.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1995. / Includes bibliographical references (p. 138-142).

<pre>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.</pre><pre>Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?</pre><pre>What is the set of rational numbers such that the corresponding specialization maps are injective?</pre><pre>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.<br></pre>

This work is concerned with computing the invariant lattices of purely non-symplectic automorphisms of special elliptic K3 surfaces. Brandhorst gave a collection of K3 surfaces admitting purely non-symplectic automorphisms that are uniquely determined up to isomorphism by certain invariants. For many of these surfaces, the automorphism is also unique or the automorphism group of the surface is finite and with a nice isomorphism class. Understanding the invariant lattices of these automorphisms and surfaces is interesting because of these uniqueness properties and because it is possible to give explicit generators for the Picard and invariant lattices. We use the methods given by Comparin, Priddis and Sarti to describe the Picard lattice in terms of certain special curves from the elliptic fibration of the surface. We use symmetries of the Picard lattice and fixed-point theory to compute the invariant lattices explicitly. This is done for all of Brandhorst's elliptic K3 surfaces having trivial Mordell-Weil group.

This thesis discusses some of the invariants of rational elliptic surfaces, namely the Mordell-Weil Group, Mordell-Weil Lattice, and another lattice which will be called the Shioda Lattice. It will begin with a brief overview of rational elliptic surfaces, followed by a discussion of lattices, root systems and Dynkin diagrams. Known results of several authors will then be applied to determine the groups and lattices associated with a given rational elliptic surface, along with a discussion of the uses of these groups and lattices in classifying surfaces.

Etude des propriétés de symétrie des solutions de quelques problèmes aux limites de type elliptique. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished