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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 étale 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~ Γ 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~  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.

The modularity of the Barth--Nieto quintic and its relatives