The \(L\)-series of certain rigid Calabi-Yau threefolds (Q1976811)

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.