Automorphy of Calabi-Yau threefolds of Borcea-Voisin type over Q (original) (raw)

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

Communications in Number Theory and Physics, 2013

We consider certain Calabi-Yau threefolds of Borcea-Voisin 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 , and of Yonemura [38], where Yonemura described hypersurfaces defining these K3 surfaces in weighted projective 3-spaces.

The automorphy of certain K3 fibered Calabi-Yau threefolds with involution over Q

arXiv (Cornell University), 2012

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 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.

The modularity of certain non-rigid Calabi–Yau threefolds

Journal of Mathematics of Kyoto University - J MATH KYOTO UNIV, 2005

Let XXX be a Calabi-Yau threefold fibred over mathbbP1\mathbb{P}^{1}mathbbP1 by non-constant semi-stable K3 surfaces and reaching the Arakelov-Yau bound. In [25], X. Sun, Sh.-L. Tan, and K. Zuo proved that XXX is modular in a certain sense. In particular, the base curve is a modular curve. In their result they distinguish the rigid and the non-rigid cases. In [17] and [28] rigid examples were constructed. In this paper we construct explicit examples in non-rigid cases. Moreover, we prove for our threefolds that the “interesting” part of their LLL-series is attached to an automorphic form, and hence that they are modular in yet another sense.

The modularity conjecture for rigid Calabi-Yau threefolds over mathbfQ\mathbf{Q}mathbfQ

Kyoto Journal of Mathematics, 2001

We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice A 3. Our proof is based on geometric analysis. 1. The L-series of Calabi-Yau threefolds Let Q be the field of rational numbers, and letQ be its algebraic closure with Galois group G := Gal(Q/Q). Let X be a smooth projective threefold defined over Q or more generally over a number field. Definition 1.1. X is a Calabi-Yau threefold if it satisfies the following two conditions: (a) H 1 (X, O X) = H 2 (X, O X) = 0, and (b) The canonical bundle is trivial, i.e., K X O X. The numerical invariants of Calabi-Yau threefolds Let X be a Calabi-Yau threefold defined over Q, and letX = X × QQ. The (i, j)-th Hodge number h i,j (X) of X is defined by h i,j (X) = dimQH j (X, Ω iX). The condition (a) implies that h 1,0 (X) = h 2,0 (X) = 0, and the condition (b) that h 3,0 (X) = h 0,3 (X) = 1. The number h 2,1 (X) represents the number of deformations of complex structures on X, and h 1,1 (X) is the number of Hodge (1, 1)-cycles on X. By using Hodge symmetry and Serre duality, we obtain

On the global moduli of Calabi-Yau threefolds

arXiv (Cornell University), 2017

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial-even though in most of these cases the Picard group is infinite.

3 New Examples of Calabi–Yau Threefolds and Genus Zero Surfaces

2016

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K 2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

Modular Calabi–Yau Threefolds of Level Eight

International Journal of Mathematics, 2007

In the studies on the modularity conjecture for rigid Calabi–Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate conjecture, correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper, we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.