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The modularity of the Barth-Nieto quintic and its relatives

K. Hulek·J. Spandaw·B. van Geemen·D. van Straten
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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 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 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.

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K. Hulek, J. Spandaw, B. van Geemen, D. van Straten. 2000-10-05. The modularity of the Barth-Nieto quintic and its relatives. https://arxiv.org/abs/math/0010049

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