UC Santa Barbara

The goal of this dissertation is to develop tools for studying genus one fibered Calabi-Yau 3-folds without section, with an eye towards applications in F-theory.

In particular:

Locally trivial fibrations are well-understood via the work of Mark Gross and others - we know that there are only finitely many families parametrizing all of them, we know they are classified by the Tate-Shafarevich group and we can use Ogg-Shafarevich theory to classify all such fibrations with a common Jacobian.

Since we have an genus one fibration structure, we can always find an equation describing the generic fiber as a genus one curve in some projective vector bundle. Furthermore, we can obtain an equation for the Jacobian from an equation for the torsor.

The work of Mark Gross is especially relevant and helpful, but it has two shortcomings:

Ogg-Shafarevich theory can only be used to classify fibrations whose geometry is reasonably nice - it requires a smooth base, with a simple normal crossings divisor, and only classifies fibrations without multiple fibers. We will see that there are fibrations which simultaneously fail all of these conditions.

It is important to be able to relate arithmetic phenomena to degenerations on elliptic fibrations in every possible codimension. Purity for the Brauer group means we shouldn't expect to gain precise information about degenerations in codimension 2. The Jacobian formulae can be used to obtain information about codimension 2 singularities by brute force. However, the problem there is that the formulae become too messy to actually be used to analyze codimension 2 singularities.

In Part I and Part II, we go over the necessary background. In Part III, we begin our analysis of torsors. We start by reviewing the theory of torsors of index 2 and 3. We will see that the data of the torsor can be recovered from the data a trace zero point on the Jacobian. In the final chapter, we propose using \emph{trace zero varieties} as a tool for studying torsors in general, and explain why this approach is well suited for answering some outstanding questions in F-theory.

Trace zero points are part of the data used to study torsors in either of the other two settings. If we have a class in $\Sh$, it can be represented by a cocycle, and the cocycle is determined by a set of trace zero points.

Trace zero points are easy to parametrize. In the final chapter, we construct a variety that parametrizes pairs consisting of an elliptic curve and a point of trace zero on that elliptic curve.

Trace zero points can be thought of as a generalizing of torsion points. We have a good understanding of moduli space for torsion pairs, and in fact we will use those moduli spaces to bound torsion on elliptically fibered Calabi-Yau 3-folds.

We will also explain how we hope to use these ideas in future work in the last chapter.