The modularity of the Barth-Nieto quintic and its relatives (Q2736577)

The Barth-Nieto quintic is the variety given by the equations NEWLINE\[NEWLINEN=\left \{\sum^5_{i=0} x_i=\sum^5_{i=0} {1\over x_i}=0 \right\} \subset \mathbb{P}_5.NEWLINE\]NEWLINE The moduli space \({\mathcal A}^2_{1,3}\) of (1,3)-polarized abelian surfaces with full level-2 structure is birationally equivalent to the inverse image \(\widetilde N\) of \(N\) under the double cover of \(\mathbb{P}_5\) branched along the union of the 6 hyperplanes \(\{x_k=0\}\). \textit{W. Barth} and \textit{I. Nieto} [J. Algebr. Geom. 3, 173-222 (1994; Zbl 0809.14027)] have shown that these varieties have Calabi-Yau models, \(Z\) and \(Y\), respectively. In this paper, by applying the Weil conjectures, the authors show that \(Y\) and \(Z\) are rigid. Then they prove that the \(L\)-function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur, and the corresponding modular form is the unique normalized cusp form of weight 4 for the group \(\Gamma_1(6)\).NEWLINENEWLINENEWLINEFinally, by giving explicit maps, the authors show that \(Y\), the fibred square of the universal elliptic curve \(S_1(6) \), and Verrill's rigid Calabi-Yau \({\mathcal Z}_{{\mathcal A}_3}\), which all have the same \(L\)-function, are in correspondence over \(\mathbb{Q}\) (this result should be implied by Tate's conjecture).