2005/06/20 by Klaus Hulek, Hulek, Klaus, Helena Verrill +1
Mathematics · #11F03 #11G40 #14F30 #14J32: 14G10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:11F03 #msc:11G40 #msc:14F30 #msc:14G10
paper · pdf · doi:10.48550/arxiv.math/0506388
16 pages
arxiv created 2005/06/20 · openalex publication_date 2005/06/20 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
In their paper Livné 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 Gamma1(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.