Communications in Number Theory and Physics

Volume 7 (2013)

Number 4

Automorphy of Calabi-Yau threefolds of Borcea-Voisin type over $\mathbb{Q}$

Pages: 581 – 670



Yasuhiro Goto (Department of Mathematics, Hokkaido University of Education, Hakodate, Japan)

Ron Livné (Institute of Mathematics, Hebrew University of Jerusalem Givat-Ram, Jerusalem, Israel)

Noriko Yui (Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada)


We consider certain Calabi-Yau threefolds of Borcea-Voisin type defined over $\mathbb{Q}$.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 $\mathbb{Q}$ with non-symplectic involution $\sigma$ acting by $-1$ on $H^{2,0}(S)$. We fish out $K3$ surfaces with the involution $\sigma$ from the famous 95 families of $K3$ surfaces in the list of Reid, and of Yonemura, 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 $\mathbb{Q}$ 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 $\sigma$. One implication of this result is that with this choice of defining equation, $(S, \sigma)$ becomes of CM type.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with the standard involution $\iota$, and let $X$ be a standard (crepant) resolution, defined over $\mathbb{Q}$, of the quotient threefold $E \times S / \iota \times \sigma$, where $(S, \sigma)$ is one of the above $K3$ surfaces over $\mathbb{Q}$ 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.

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