---
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 \[STZ\], 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 \[Y\] 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.'
address:
- 'Institute of Mathematics, Hebrew University of Jerusalem, Givat-Ram, Jerusalem 91904, Israel'
- 'Department of Mathematics, Queen’s University, Kingston. Ontario Canada K7L 3N6'
author:
- Ron Livné
- Noriko Yui
date: 'Version of June 16, 2005'
title: 'The modularity of certain non-rigid Calabi–Yau threefolds'
---

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Introduction
============

Let $X$ be an algebraic threefold and let $f:X\to
{{\mathbb P}}^1$ be a non-isotrivial morphism whose fibers are semi-stable K3 surfaces. Let $S\subset {{\mathbb P}}^1$ be the finite set of points above 