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

作者：Goto Yasuhiro^*; Livne Ron; Yui Noriko

来源：Communications in Number Theory and Physics, 2013, 7(4): 581-670.

DOI：10.4310/CNTP.2013.v7.n4.a2

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 sigma acting by -1 on H-2,H-0 (S). We fish out K3 surfaces with the involution sigma 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. %26lt;br%26gt;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%26apos;s list, there are subsets of equations defining quasi-smooth hypersurfaces which are of Delsarte or Fermat type and endowed with non-symplectic involution sigma. One implication of this result is that with this choice of defining equation, (S, a) becomes of CM type. %26lt;br%26gt;Let E be an elliptic curve over Q with the standard involution iota, and let X be a standard (crepant) resolution, defined over Q, of the quotient threefold E x S/iota x sigma, where (5, a) 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. %26lt;br%26gt;The moduli spaces of these Calabi-Yau threefolds are Shimura varieties. Our result shows the existence of a CM point in the moduli space. %26lt;br%26gt;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.

• 出版日期2013-12