The moduli space of 1Y 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 e Âtale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique

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... rmalized cusp form of weight 4 for the group q 1 6. By Tate's conjecture, this should imply that Y, the ®bred square of the universal elliptic curve S 1 6, and Verrill's rigid Calabi±Yau Z A3 , which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps. c c c y Y 3 N