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. Livn\'e and N. Yui)

Abstract

In their paper Livn\'e and Yui (math.AG/0304497) discuss several examples of non-rigid 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 their paper, 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 treat the case associated to the group Gamma_1(7) in detail, but our technique also applies to many other cases. We further make some comments concerning the Kummer construction for fibre products of elliptic surfaces in general.