The modularity of the Barth-Nieto quintic and its relatives

By K. Hulek, J. Spandaw, Bert van Geemen, Duco van Straten. 2000

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The moduli space of O1; 3U-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 G1O6U. By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S1O6U, 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73a76785850793ca370ea87113357895c343184d