## Journal of Mathematics of Kyoto University

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

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

Ron Livné and Noriko Yui

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/kjm/1250281650

**Mathematical Reviews number (MathSciNet)**

MR2226623

**Zentralblatt MATH identifier**

1106.14025

**Subjects**

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Secondary: 11F23: Relations with algebraic geometry and topology 11F80: Galois representations 14J28: $K3$ surfaces and Enriques surfaces 14J32: Calabi-Yau manifolds

#### 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