Rigid Calabi-Yau Threefolds Over Q Are Modular: A Footnote to Serre (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.

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 conjecture for rigid Calabi-Yau threefolds over Q

2000

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.

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

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).

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.

Quadratic twists of rigid Calabi-Yau threefolds over QQ\QQQQ

arXiv (Cornell University), 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).