The modularity conjecture for rigid Calabi-Yau threefolds over Q\mathbf{Q}Q (original) (raw)

2001, Kyoto Journal of Mathematics

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