The modularity of the Barth-Nieto quintic and its relatives
Abstract
The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third \'etale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group \Gamma_1(6). By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's rigid Calabi-Yau Z_{A_3}, which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.
Keywords
Cite
@article{arxiv.math/0010049,
title = {The modularity of the Barth-Nieto quintic and its relatives},
author = {K. Hulek and J. Spandaw and B. van Geemen and D. van Straten},
journal= {arXiv preprint arXiv:math/0010049},
year = {2007}
}
Comments
30 pages, Latex2e