## Journal of Mathematics of Kyoto University

### The modularity of certain non-rigid Calabi–Yau threefolds

#### Abstract

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 [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.

#### Article information

Source
J. Math. Kyoto Univ., Volume 45, Number 4 (2005), 645-665.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281650

Digital Object Identifier
doi:10.1215/kjm/1250281650

Mathematical Reviews number (MathSciNet)
MR2226623

Zentralblatt MATH identifier
1106.14025

#### Citation

Livné, Ron; Yui, Noriko. The modularity of certain non-rigid Calabi–Yau threefolds. J. Math. Kyoto Univ. 45 (2005), no. 4, 645--665. doi:10.1215/kjm/1250281650. https://projecteuclid.org/euclid.kjm/1250281650