Arithmetic and Intermediate Jacobians of Calabi-Yau threefolds

Abstract

This thesis is centered around particular Calabi-Yau threefolds. Borcea \cite{Borcea} and Voisin \cite{Voisin} construct Calabi-Yau threefolds using elliptic curves and K3 surfaces with non-symplectic involutions. This family has an incredible property, that a general member has a mirror pair within this family. We start by investigating if this construction works only for Calabi-Yau threefolds with non-symplectic involutions or with non-symplectic automorphisms of higher order as well. Thereafter, we generalize this construction to Calabi-Yau fourfolds.

After this, we focus on the underlying construction that lead Borcea to the families above, using a product of three elliptic curves with non-symplectic involutions. These threefolds do not come in families, so we cannot ask about mirror symmetry, but if we have models defined over $\Qbb$, we may ask arithmetic questions. Many arithmetic properties of the Calabi-Yau threefolds can be studied via the underlying elliptic curves. In particular, we are able to show (re-establish in the rigid case) that the Calabi-Yau threefolds are all modular by computing their $L$-functions. Then, guided by a conjecture of Yui, we investigate their (Griffiths) intermediate Jacobians and a relationship between their respective $L$-functions.