Modularity of the Consani-Scholten quintic (original) (raw)

Modularity of the Consani-Scholten quintic. With an appendix by José Burgos Gil and Ariel Pacetti

Documenta Mathematica

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over Q, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livné method to induced fourdimensional Galois representations over Q. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by José Burgos Gil and the second author.

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.

New realizations of modular forms in Calabi-Yau threefolds arising from ϕ 4 theory

Journal of Number Theory, 2017

Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relation in a general geometric context. We also prove the relation for one example of a modular form of weight 4 and two of weight 3, refine the statement and suggest a method of proving it for four more of weight 4, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).

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

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