The modularity conjecture for rigid Calabi-Yau threefolds over Q (original) (raw)

Rigid Calabi-Yau Threefolds over Q Are Modular

arXiv (Cornell University), 2009

The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a method due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.

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

Rigid Calabi–Yau threefolds over are modular

Expositiones Mathematicae, 2011

ABSTRACT The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a method due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular. Comment: Final version to appear in Expositiones Mathematicae

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.

A modular non-rigid Calabi-Yau threefold

Mirror Symmetry V, 2006

We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem inétale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.

Yui,N.: Quadratic twists of rigid Calabi-Yau threefolds over Q, preprint

2011

We consider rigid Calabi-Yau threefolds defined over Q and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi-Yau threefold over Q is modular so there is attached to it a certain newform of weight 4 on some Γ 0 (N). We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weight 4 on some Γ 0 (N) and integral Fourier coefficients arise from rigid Calabi-Yau threefolds defined over Q (a geometric realization problem).