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)

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.