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)
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.
Publié le : 2005-05-15
Classification:
11G40,
11F23,
11F80,
14J28,
14J32
@article{1250281651,
author = {Hulek, Klaus and Verrill, Helena A.},
title = {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)},
journal = {J. Math. Kyoto Univ.},
volume = {45},
number = {4},
year = {2005},
pages = { 667-681},
language = {en},
url = {http://dml.mathdoc.fr/item/1250281651}
}
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., Tome 45 (2005) no. 4, pp. 667-681. http://gdmltest.u-ga.fr/item/1250281651/