The \(L\)-series of certain rigid Calabi-Yau threefolds
type: Item
instance of: scholarly article
title: The \(L\)-series of certain rigid Calabi-Yau threefolds
zbMATH Open document ID: 0971.14034
published in: Journal of Number Theory
publication date: 2001-07-04T00:00:00Z
review text: Given a root lattice \(R\), the Weyl chambers form cones of a fan \({\Sigma}_R\), which defines a toric variety \(X({\Sigma}_R)\). The character of the adjoint representation defines a map \(X({\Sigma}_R) \rightarrow {\mathbb P}^1\). Blowing up the base locus of this map gives a fibration \({\phi}: X_R \rightarrow {\mathbb P}^1\). It is shown by \textit{H. A. Verrill} [J. Math. Kyoto Univ. 36, 423--446 (1996; Zbl 0895.14010)] that if \(R\) is a product of root lattices \(A_n\) then the general fibre of \({\phi}\) is a Calabi-Yau variety; and if \(R = A_3\), \(A_1^3 = A_1 \times A_1 \times A_1\), or \(A_1 \times A_2\) then certain double cover \({\pi}:Z_R \rightarrow X_R\) is also a Calabi-Yau variety. In this paper are determined the \(L\)-series of the Calabi-Yau 3-folds \(Z_{A_1^3}\) and \(Z_{A_3}\); the first of these two varieties is also studied by \textit{C. Peters} and \textit{J. Stienstra} [in: ``Arithmetic and complex manifolds'', Proc. Conf., Erlangen 1988, Lect. Notes Math. 1399, 110--127 (1989; Zbl 0701.14037)]. In each of the two cases it is shown that the middle cohomology has dimension 2, and that, up to Euler factors at the primes of bad reduction, the Mellin transform of the \(L\)-series of the middle cohomology is given by a modular form of weight four.
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F23
zbMATH DE Number: 1443344
zbMATH Keywords: Calabi-Yau threefold
zbMATH Keywords: L-series
zbMATH Keywords: root lattice
zbMATH Keywords: Weyl chambers
zbMATH Keywords: toric variety
author: H. A. Verrill
reviewed by: Atanas Iliev
MaRDI profile type: Publication
full work available at URL: jnth.1999.2449
OpenAlex ID: W2061515989
cites work: Modularity of a certain Calabi-Yau threefold
cites work: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
cites work: Birational Calabi--Yau n-folds have equal Betti numbers
cites work: Another congruence for the Apéry numbers
cites work: Monstrous Moonshine
cites work: Q5607047
cites work: Q4858020
cites work: Q4003470
cites work: Q3758032
cites work: Q3871006
cites work: Monodromy and Picard-Fuchs equations for families of $K3$-surfaces and elliptic curves
cites work: Q3478534
cites work: On the Picard-Fuchs equation and the formal Brauer group of certain elliptic \(K3\)-surfaces
cites work: Formal Groups and Congruences for L-Functions
cites work: The modularity conjecture for rigid Calabi-Yau threefolds over \(\mathbb{Q}\)
cites work: On the geometry and arithmetic of some Siegel modular threefolds
cites work: Root lattices and pencils of varieties
cites work: Q4249717
cites work: CONGRUENCES RELATED TO MODULAR FORMS
DOI: 10.1006/JNTH.1999.2449
description: scientific article; zbMATH DE number 1443344