A dictionary of modular threefolds
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Abstract
The thesis deals with the modularity conjecture for three-dimensional
Calabi-Yau varieties. This is a generalization of the work of A. Wiles
and others on modularity of elliptic curves. Modularity connects the
number of points on varieties with coefficients of certain modular forms.
In chapter 1 we collect the basics on arithmetic on Calabi-Yau manifolds,
including general modularity results and strategies for modularity proofs.
In chapters 2, 3, 4 and 5 we investigate examples of modular Calabi-Yau
threefolds, including all examples occurring in the literature and many
new ones. Double octics, i.e. Double coverings of projective 3-space
branched along an octic surface, are studied in detail.
In chapter 6 we deal with examples connected with the
same modular forms.
According to the Tate conjecture there should be correspondences between
them. Many correspondences are constructed explicitly. We finish by
formulating conjectures on the occurring newforms, especially their levels.
In the appendices we compile tables of coefficients of weight 2 and weight 4
newforms and many examples of double octics.