The modularity of the Barth-Nieto quintic and its relatives

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 eÂtale 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 G1 6. By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S1\u856, and Verrill's rigid Calabi-Yau ZA3, which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps