The modularity of certain non-rigid Calabi-Yau threefolds

Let X be a Calabi-Yau threefold fibred over ℙ¹ 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 X 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 L-series is attached to an automorphic form, and hence that they are modular in yet another sense.