## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 45, Number 4 (2005), 667-681.

### On the motive of Kummer varieties associated to $\Gamma_1(7)$ – Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livné and N. Yui)

Klaus Hulek and Helena A. Verrill

#### Abstract

In their paper [LY] Livné and Yui discuss several examples of nonrigid Calabi-Yau varieties which admit semi-stable $K3$-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the modularity of the $L$-function of these examples. The purpose of this note is to point out that the examples which were listed in [LY] but which do not lead to semi-stable fibrations are still modular in the sense that their $L$-function is associated to modular forms. We shall treat the case associated to the group $\Gamma _{1}(7)$ in detail, but our technique also works in the other cases given in [LY]. We shall also make some comments concerning the Kummer construction for fibre products of elliptic surfaces in general.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 45, Number 4 (2005), 667-681.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281651

**Digital Object Identifier**

doi:10.1215/kjm/1250281651

**Mathematical Reviews number (MathSciNet)**

MR2226624

**Zentralblatt MATH identifier**

1106.14023

**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

Hulek, Klaus; Verrill, Helena A. On the motive of Kummer varieties associated to $\Gamma_1(7)$ – Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livné and N. Yui). J. Math. Kyoto Univ. 45 (2005), no. 4, 667--681. doi:10.1215/kjm/1250281651. https://projecteuclid.org/euclid.kjm/1250281651