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

We consider certain Calabi-Yau threefolds of Borcea-Voisin type defined over ℚ.We will discuss the automorphy of the Galois representations associated to these Calabi-Yau threefolds. We construct such Calabi-Yau threefolds as the quotients of products of K3 surfaces S and elliptic curves by a specific involution. We choose K3 surfaces S over ℚ with non-symplectic involution σ acting by -1 on H^2,0(S). We fish out K3 surfaces with the involution σ from the famous 95 families of K3 surfaces in the list of Reid [32], and of Yonemura [43], where Yonemura described hypersurfaces defining these K3 surfaces in weighted projective 3-spaces. Our first result is that for all but few (in fact, nine) of the 95 families of K3 surfaces S over ℚ in Reid-Yonemura's list, there are subsets of equations defining quasi-smooth hypersurfaces which are of Delsarte or Fermat type and endowed with non-symplectic involution σ. One implication of this result is that with this choice of defining equation, (S, σ) becomes of CM type. Let E be an elliptic curve over ℚ with the standard involution ι, and let X be a standard (crepant) resolution, defined over ℚ, of the quotient threefold E × S/ι × σ, where (S, σ) is one of the above K3 surfaces over ℚ of CM type. One of our main results is the automorphy of the L-series of X. 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 the L-series of mirror pairs of Calabi-Yau threefolds of Borcea-Voisin type, and study how L-series behave under mirror symmetry.