The modularity of certain non-rigid Calabi-Yau threefolds

Details The modularity of certain non-rigid Calabi-Yau threefolds The modularity of certain non-rigid Calabi-Yau threefolds

2003-04-30

arxiv.org/abs/math/0304497v2

Mathematics

Number Theory

19 pages; some corrections made; see also related submission by Hulek-Verrill

Scientific paper

Let $X$ be a Calabi--Yau threefold fibred over ${\mathbb P}^1$ by non-constant semi-stable K3 surfaces and reaching the Arakelov--Yau bound. In [STZ], 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 [SY] and [V] 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.

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