Automorphy of Calabi-Yau threefolds of Borcea-Voisin type over Q

TL;DR

This paper investigates the automorphy of Galois representations associated with Borcea-Voisin type Calabi-Yau threefolds over Q, linking their moduli spaces to Shimura varieties and exploring mirror symmetry.

Contribution

It demonstrates the automorphy of Galois representations for these Calabi-Yau threefolds and identifies CM points within their moduli spaces, connecting geometry and number theory.

Findings

Existence of CM points in moduli spaces of these threefolds

Automorphy of Galois representations associated to the threefolds

Connection between moduli spaces and Shimura varieties

Abstract

We consider Calabi-Yau threefolds of Borcea-Voisin type over Q. They are constructed from products of K3 surfaces and elliptic curves. We use concrete K3 surfaces and discuss the automorphy of the Galois representations associated to the Calabi-Yau threefolds. The moduli spaces of these Calabi-Yau threefolds are Shimura varieties. Our result shows the existence of a CM point in the moduli space. We also consider mirror symmetry of Calabi-Yau threefolds.